COLLECTED PAPERS O F CARL WIEMAN
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OF CARL WIEMAN Carl E. Wieman...
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COLLECTED PAPERS O F CARL WIEMAN
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COLLECTED PAPERS
OF CARL WIEMAN Carl E. Wieman Director of the Carl Wieman Science Education Initiative University of British Columbia Distinguished Professor of Physics Director of the University of Colorado Science Education Initiative University of Colorado
world scientific NEW JERSEY
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LONDON
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SINGAPORE
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BElJlNG
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S H A N G ~ A I* H O N G K O N G
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TAIPEI
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CHENNAI
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COLLECTED PAPERS OF CARL WIEMAN Copyright 0 2008 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, orparts thereoj may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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ISBN- 13 978-98 1-270-4 15-3 ISBN- 10 98 1-270-4 15-9 ISBN-13 978-981-270-416-0 (pbk) ISBN- 10 98 1-270-4 16-7 (pbk)
Printed in Singapore by World Scientific Printers (S)Pte U d
CONTENTS Introduction
1
Precision Measurement and Parity Nonconservation
5
T. W. Hansch, S. A. Lee, R. Wallenstein and C. Wieman, “Doppler-free two-photon spectroscopy of hydrogen ls-2~:’Phys. Rev. Lett. 3 , 3 0 7 (1975).
9
C. E. Wieman and T. W. Hansch, “Doppler-free laser polarization spectroscopy,” Phys. Rev. Lett. 36,1170 (1976).
12
R. Feinberg, T. Hansch, A. Schawlow, R. Teets and C. Wieman, “Laser polarization spectroscopy of atoms and molecules,” Opt. Comm. l8,227 (1976).
16
C. E. Wieman and T. W. Hansch, “Precision measurement on the 1s Lamb shift and of the ls-2s isotope shift of H and D,” Phys. Rev. A 22, 192 (1980).
17
C. E. Wieman and S. L. Gilbert, “Laser frequency stabilization using mode interference from a reflecting reference interferometer,” Opt. Lett. 1,480 (1982).
31
S. L. Gilbert, R. Watts and C. E. Wieman, “Hyperfine structure measurement of the 7s state of cesium,” Phys. Rev. A 21,581 (1983).
34
R. N. Watts, S. L. Gilbert and C. E. Wieman, “Precision measurement of the Stark shift of the 6s-7s transition in atomic cesium,” Phys. Rev. A 21, 2769 (1983).
36
S. L. Gilbert, R. N. Watts and C. E. Wieman, “Measurement of the 6s --+7s M1 transition in cesium with the use of crossed electric and magnetic fields,” Phys. Rev. A 29, 137 (1984).
40
S. L. Gilbert, M. C. Noecker and C. E. Wieman, “Absolute measurement of the photoionization cross section of the excited 7s state of cesium,” Phys. Rev. A 29, 3150 (1984).
47
S. L. Gilbert, M. C. Noecker, R. N. Watts and C. E. Wieman, “Measurement of parity nonconservation in atomic cesium,” Phys. Rev. Lett. 2680 (1985).
51
R. N. Watts and C. E. Wieman, “The production of a highly polarized atomic cesium beam,” Opt. Comm. s , 4 5 (1986).
55
S. L. Gilbert and C. E. Wieman, “Atomic-beam measurement of parity nonconservation in cesium,” Phys. Rev. A 34,792 (1986).
59
S. L. Gilbert, B. P. Masterson, M. C. Noecker and C. E. Wieman, “Precision measurement of the off-diagonal hyperfine interaction,” Phys. Rev. A 3 , 3 5 0 9 (1986).
71
C. E. Wieman, M. C. Noecker, B. P. Masterson and J. Cooper, “Asymmetric line shapes for weak transitions in strong standing wave fields,” Phys. Rev. Lett. 54, 1738 (1987).
75
C. E. Tanner and C. E. Wieman, “Precision measurement of the Stark shift in the 6 s 1/2 + 6P3/, cesium transition using a frequency-stabilized laser diode,” Phys. Rev. A 3 4 162 (1988).
79
C. E. Tanner and C. E. Wieman, “Precision measurement of the hyperfine structure of the 133Cs 6P3,, state,” Phys. Rev. A 38, 1616 (1988).
83
z,
V
vi
M. C. Noecker, B. P. Masterson and C. E. Wieman, “Precision measurement of parity nonconservation in atomic cesium: A low energy test of the electroweak theory,” Phys. Rev. Lett. 61,310 (1988).
85
B. P. Masterson, C. Tanner, H. Patrick and C. E. Wieman, “High brightness, high purity spin polarized cesium beam,” Phys. Rev. A 47,2139 (1993).
89
C. E. Wieman, “Parity nonconservation in atoms; past work and trapped atom future,” in Proceedings of the Workshop on Traps for Antimatter and Radioactive Nuclei, J. Hyperfine Int.
8l,27 (1993).
96
C. E. Wieman, S. Gilbert, C. Noecker, P. Masterson, C. Tanner, C. Wood, C. Cho and M. Stephens, “Measurement of parity nonconservation in atoms,” in Proceedings of the 1992 ‘Enrico Fermi’ Summer School, Varenna, Italy, Course CXX Frontiers of Laser Spectroscopy, (eds.) T. W. Hansch and M. Inguscio (North Holland, 1994), 240.
104
L. Young, W. T. Hill 111, S. Sibener, S. D. Price, C. E. Tanner, C. E. Wieman and S. R. Leone, “Precision lifetime measurements of Cs 6p2P and 6p2P,/, levels by single-photon counting,” 112 Phys. Rev. A 50,2174 (1994).
184
D. Cho, C. S. Wood, S. C. Bennett, J. L. Roberts and C. E. Wieman, “Precision Measurement of the Ratio of Scalar to Tensor Transition Polarizabilities for the Cesium 6s-7s Transition,” Phys. Rev. A 55, 1007 (1997).
192
C. S. Wood, S. C. Bennett, D. Cho, B. P. Masterson, J. L. Roberts, C. E. Tanner and C. E. Wieman, “Measurement of parity nonconservation and an anapole moment in cesium.” Science 275, 1759 (1997).
197
S. C. Bennett, J. L. Roberts and C. E. Wieman, “Measurement of the dc Stark shift of the 6s -+ 7 s transition in atomic cesium,” Phys. Rev. A 59, R16 (1999).
202
S. C. Bennett and C. E. Wieman, “Measurement of the 6 s --t 7 s transition polarizability in atomic cesium and an improved test of the standard model,” Phys. Rev. Lett. a 2 4 8 4 (1999).
205
W. C. Haxton and C. E. Wieman, “Atomic parity nonconservation and nuclear anapole moments,” Ann. Rev. Nucl. Part. Sci. 2, 261 (2001).
209
C. E. Wieman, “Pursuing Fundamental Physics with Novel Laser Technology,” in Laser Physics at the Limits (Springer Verlag, 2002), H. Figger, D. Meschede and C. Zimmermann (eds.)
242
Laser Cooling and Trapping
251
R. N. Watts and C. E. Wieman, “Stopping atoms with diode lasers,” Laser Spectroscopy VII, Proceedings of the Seventh International Conference, Hawaii, June 24-28, 1985, eds. T. W. Hansch and Y. R. Shen (Springer-Verlag, 1985), 20.
255
R. N. Watts and C. E. Wieman, “Manipulating atomic velocities using diode lasers,” Opt. Lett. 11,291 (1986).
257
D. E. Pritchard, E. L. Raab, V. Bagnato, R. N. Watts and C. E. Wieman, “Light traps using spontaneous forces,” Phys. Rev. Lett. 57, 310 (1986).
260
C. E. Tanner, B. P. Masterson and C. E. Wieman, “Atomic beam collimation using a laser diode with a self-locking power-buildup cavity,” Opt. Lett. g, 357 (1988).
264
vii
D. Sesko, C. G. Fan and C. E. Wieman, “Production of a cold atomic vapor using diode-laser cooling,” J. Opt. SOC.Am. B 5, 1225 (1988).
267
D. W. Sesko and C. E. Wieman, “Observation of the cesium clock transition in laser cooled atoms,” Opt. Lett. 14,269 (1989).
270
D. Sesko, T. Walker, C. Monroe, A. Gallagher and C. Wieman, “Collisional losses from a light force atom trap,” Phys. Rev. Lett. 63, 961 (1989).
273
T. Walker, D. Sesko and C. Wieman, “Collective behavior of optically trapped neutral atoms,” Phys. Rev. Lett. 64,408 (1990).
277
D. Sesko, T. G. Walker and C. Wieman, “Behavior of neutral atoms in a spontaneous force trap,” J. Opt. SOC.Am. B 8,946 (1991).
28 1
C. Monroe, W. Swann, H. Robinson and C. Wieman, “Very cold trapped atoms in a vapor cell,” Phys. Rev. Lett. 65, 1571 (1990).
294
C. Monroe, H. Robinson and C. Wieman, “Observation of the cesium clock transition using laser-cooled atoms in a vapor cell,” Opt. Lett. 16,50 (1991).
298
E. A. Cornell, C. Monroe and C. E. Wieman, “Multiply-loaded, ac magnetic trap for neutral atoms,” Phys. Rev. Lett. 67, 2439 (1991).
301
C. E. Wieman, C. Monroe and E. Cornell, “Fundamental physics with optically trapped atoms,” in Laser Spectroscopy X, (ed.) M. Ducloy (World Scientific, 1992), 77.
305
K. Lindquist, M. Stephens and C. Wieman, “Experimental and theoretical study of the vapor-cell Zeeman optical trap,” Phys. Rev. A 46,4082 (1992).
311
C. Monroe, E. Cornell and C. Wieman, “The low (temperature) road toward Bose-Einstein condensation in optically and magnetically trapped cesium atoms,” in Proceedings of the International School of Physics “Enrico Fermi”, Course CXVIII, Laser Manipulation of Atoms and Zons, (eds.) E. Arimondo, W. D. Phillips and F. Strumia (North Holland, 1992), 361.
320
C. R. Monroe, E. A. Cornell, C. A. Sackett, C. J. Myatt and C. E. Wieman, “Measurement of Cs-Cs elastic scattering at T = 30 pK,” Phys. Rev. Lett. 70,414 (1993).
343
C. J. Myatt, N. R. Newbury and C. E. Wieman, “Simplified atom trap using direct microwave modulation of a diode laser,” Opt. Lett. 47,649 (1993).
347
S. L. Gilbert and C. E. Wieman, “Laser cooling and trapping for the masses,” Opt. Photonics News 4, 8 (1993).
350
M. Stephens, K. Lindquist and C. Wieman, “Optimizing the capture process in optical traps,” J. Hyperfine Int. 203 (1993).
356
M. Stephens and C. E. Wieman, “High collection efficiency in a laser trap,” Phys. Rev. Lett. 72,3787 (1994).
369
M. Stephens, R. Rhodes and C. Wieman, “Study of wall coatings for vapor-cell laser traps,” J. Appl. Phys. 76,3479 (1994).
373
C. Wieman, G. Flowers and S. Gilbert, “Inexpensive laser cooling and trapping experiment for undergraduate laboratories,” Am. J. Phys. 317 (1995).
383
a,
a,
viii
N. R. Newbury, C. J. Myatt, E. A. Cornell and C. E. Wieman, “Gravitational sisyphus cooling of 87Rbin a magnetic trap,” Phys. Rev. Lett. 74,2196 (1995).
397
N. R. Newbury, C. J. Myatt and C. E. Wieman, “S-wave elastic collisions between cold ground state 87Rbatoms,” Phys. Rev. A 51, R2680 (1995).
40 1
M. J. Renn, D. Montgomery, 0. Vdovin, D. Z. Anderson, C. E. Wieman and E. A. Cornell, “Laser-guided atoms in hollow-core optical fibers,” Phys. Rev. Lett. 75, 3253 (1995).
406
C. J. Myatt, N. R. Newbury, R. W. Ghrist, S. Loutzenhiser and C. E. Wieman, “Multiply loaded magneto-optical trap,” Opt. Lett. 21, 290 (1996).
410
N. R. Newbury and C. E. Wieman, “Resource Letter TNA-1: Trapping of neutral atoms,” Am. J. Phys. 64, 18 (1996).
413
M. J. Renn, E. A. Donley, E. A. Cornell, C. E. Wieman and D. Z. Anderson, “Evanescent-wave guiding of atoms in hollow optical fibers,” Phys. Rev. A R648 (1996).
416
Z. T. Lu, K. L. Corwin, M. J. Renn, M. H. Anderson, E. A. Cornell and C. E. Wieman, “Low-velocity intense source of atoms from a magneto-optical trap,” Phys. Rev. Lett. 77, 3331 (1996).
420
Z.-T. Lu, K. L. Convin, K. R. Vogel and C. E. Wieman, “Efficient collection of ’”Fr into a vapor cell magneto-optical trap,” Phys. Rev. Lett. 79,994 (1997).
424
K. L. Corwin, S. J. M. Kuppens, D. Cho and C. E. Wieman, “Spin-polarized atoms in a circularly polarized optical dipole trap,” Phys. Rev. Lett. 1311 (1999).
428
S. J. Kuppens, K. L. Corwin, K. W. Miller, T. E. Chupp and C. E. Wieman, “Loading an optical dipole trap,” Phys. Rev. A 62,013406-1 (1999).
432
S. Duerr, K. W. Miller and C. E. Wieman, “Improved loading of an optical dipole trap by 01 1401-1 (2001). suppression of radiative escape,” Phys. Rev. A
445
Bose-Einstein Condensation
449
M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman and E. A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor,” Science 269, 198 (1995).
453
D. S. Jin, J. R. Ensher, M. R. Matthews, C. E. Wieman and E. A. Cornell, “Collective excitations of a Bose-Einstein condensate in a dilute gas,” Phys. Rev. Lett. -7 420 (1996).
457
J. R. Ensher, D. S. Jin, M. R. Matthews, C. E. Wieman and E. A. Cornell, “Bose-Einstein condensation in a dilute gas: Measurement of energy and ground-state occupation,” Phys. Rev. Lett. 77,4984 (1996).
46 1
C. E. Wieman, “The Richtmyer memorial lecture: Bose-Einstein condensation in an ultracold gas,” Am. J. Phys. 64, 847 (1996).
465
C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell and C. E. Wieman, “Production of two overlapping Bose-Einstein condensates by sympathetic cooling,” Phys. Rev. Lett. 78, 587 (1997).
489
D. S. Jin, M. R. Matthews, J. R. Ensher, C. E. Wieman and E. A. Cornell, “Temperature-dependent damping and frequency shifts in collective excitations of a dilute Bose-Einstein condensate,” Phys. Rev. Lett. 73,764 (1997).
493
s,
a,
a,
ix
E. A. Burt, R. W. Ghrist, C. J. Myatt, M. J. Holland, E. A. Cornell and C. E. Wieman, “Coherence, correlations, and collisions: What one learns about Bose-Einstein condensates from their decay,” Phys. Rev. Lett. 79,337 (1997).
497
J. Williams, R. Walser, C. Wieman, J. Cooper and M. Holland, “Achieving steady state Bose-Einstein condensation,” Phys. Rev. A 57, 2030 (1998).
501
C. E. Wieman and E. A. Cornell, ‘The Bose-Einstein condensate,” Sci. Am. 2 x 4 0 (1998).
508
D. S . Hall, M. R. Mattews, J. R. Ensher, C. E. Wieman and E. A. Cornell, “Dynamics of component separation in a binary mixture of Bose-Einstein condensates,” Phys. Rev. Lett. 8 1 1539 (1998).
515
D. S. Hall, M. R. Matthews, C. E. Wieman and E. A. Cornell, “Measurements of relative phase in two-component Bose-Einstein condensates,” Phys. Rev. Lett. sl, 1543 (1998).
519
M. R. Matthews, D. S . Hall, D. S. Jin, J. R. Ensher, C. E. Wieman and E. A. Cornell, “Dynamical response of a Bose-Einstein condensate to a discontinuous change in internal state,” Phys. Rev. Lett. 243 (1998).
523
J. L. Roberts, N. R. Claussen, J. P. Burke, Jr., C, H. Greene, E. A. Cornell and C. E. Wieman, “Resonant magnetic field control of elastic scattering in cold 85Rb,”Phys. Rev. Lett. 81,5109 (1998).
528
E. A. Cornell, J. R. Ensher and C. E. Wieman, “Experiments in dilute atomic Bose-Einstein condensation,” in Bose-Einstein Condensation in Atomic Gases, Proceedings of the International School of Physics “Enrico Fermi” Course CXL, Italian Physical Society, M. Inguscio, S. Stringari and C. E. Wieman, (eds.) (October 1999).
533
M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S . Hall, M. J. Holland, J. E. Williams, C. E. Wieman and E. A. Cornell, “Watching a superfluid untwist itself Recurrence of Rabi oscillations in a Bose-Einstein condensate,” Phys. Rev. Lett. 83, 3358 (1999).
585
M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S . Hall, C. E. Wieman and E. A. Cornell, “Vortices in a Bose-Einstein condensate,” Phys. Rev. Lett. g,2498 (1999).
589
S. L. Cornish, N. R. Claussen, J. L. Roberts, E. A. Cornell and C. E. Wieman, “Stable 85Rb Bose-Einstein condensates with widely tunable interactions,” Phys. Rev. Lett. 851795 (2000).
593
B. P. Anderson, P. C. Halijan, C. E. Wieman and E. A. Cornell, “Vortex precession in Bose-Einstein condensates: observations with filled and empty cores,” Phys. Rev. Lett. 85,2857 (2000).
597
J. L. Roberts, N. R. Claussen, S. L. Cornish and C. E. Wieman, “Magnetic field dependence of ultracold inelastic collisions near a Feshbach resonance,” Phys. Rev. Lett. -58 728 (2000).
601
J. L. Roberts, N. R. Claussen, S. L. Cornish, E. A. Donley, E. A. Cornell and C. E. Wieman, “Controlled collapse of a Bose-Einstein condensate,” Phys. Rev. Lett. -68 4211 (2001).
605
J. L. Roberts, J. P. Burke, Jr., N. R. Claussen, S. L. Cornish, E. A. Donley and C. E. Wieman, “Improved characterization of elastic scattering near a Feshbach resonance in ”Rb,” Phys. Rev. A 64,024702-1 (2001).
609
a,
X
E. A. Donley, N. R. Claussen, S. L. Cornish, J. L. Roberts, E. A. Cornell and C. E. Wieman, “Dynamics of collapsing and exploding Bose-Einstein condensates,” Nature 412, 295 (2001).
612
N. R. Claussen, E. A. Donley, S. T. Thompson and C. E. Wieman, “Microscopic dynamics in a strongly interacting Bose-Einstein condensate,” Phys. Rev. Lett. -98 010401 (2002).
617
E. A. Donley, N. R. Claussen, S. T. Thompson and C. E. Wieman, “Atom-molecule coherence in a Bose-Einstein condensate,” Nature 417,529 (2002).
62 1
E. A. Cornell and C. E. Wieman, “Nobel Lectures: Bose-Einstein condersation in a dilute gas: The first 70 years and some recent experiments,” Rev. Mod. Phys. 74(3), 875 (2002).
626
N. R. Claussen, S. J. J. M. F. Kokkelmans, S. T. Thompson, E. A. Donley, E. Hodby and C. E. Wieman, “Very-high-precision bound state-spectroscopy near a 85RbFeshbach resonance,” Phys. Rev. A 67,060701 (2003).
645
S. T. Thompson, E. Hodby and C. E. Wieman, “Spontaneous dissociation of 85RbFeshbach molecules,” Phys. Rev. Lett. 3,020401-1 (2005).
649
E. Hodby, S. T. Thompson, C. A. Regal, M. Greiner, A. C. Wilson, D. S. Jin, E. A. Cornell and C. E. Wieman, “Production efficiency of ultracold Feshbach molecules in Bosonic and Fennionic systems,” Phys. Rev. Lett. 94, 120402-1 (2005).
653
S. T. Thompson, E. Hodby and C. Wieman, Ultracold molecule production via a resonant oscillating magnetic field, Phys. Rev. Lett. 95, 190401-1 (2005).
657
Science Education
661
K. B. MacAdam, A. Steinbach and C. Wieman, “A narrow band tunable diode laser system with grating feedback, and a saturated absorption spectrometer for Cs and Rb,” Am. J. Phys. 6 3 1098 (1992).
665
C. Wieman, G. Flowers and S. Gilbert, “Inexpensive laser cooling and trapping experiment for undergraduate laboratories,” Am. J. Phys. 63,3 17 (1995).
679
N. R. Newbury and C. E. Wieman, “Resource Letter TNA-1: Trapping of neutral atoms,” Am. J. Phys. 64, 18 (1996).
693
C. Wieman, “Good science and business practices also yield positive educational results,” Laser Focus World, Comment (April 2004).
696
C. Wieman, “Firming up physics,” AAPT Announcer 34,6 (Summer 2004).
698
K. Perkins and C. Wieman, “Webwatch free on-line resource conncects real-life phenomena with science,” Phys. Edu. 93(January 2005).
699
K. Perkins, W. Adams, M. Dubson, N. Finkelstein, S. Reid, C. Wieman and R. LeMaster, “PhET Interactive simulations for teaching and learning physics,” Phys. Teach. (in press, 2005).
702
W. K. Adams, K. K. Perkins, M. Dubson, N. D. Finkelstein and C. E. Wieman, “The design and validation of the Colorado learning attitudes about science survey,” PERC Proc. (in press, 2004).
710
K. K. Perkins and C. E. Wieman, “The surprising impact of seat location on student performance,” Phys. Teach. 43,30 (2005).
714
xi
K. K. Perkins, W. K. Adams, N. D. Finkelstein, S. J. Pollock and C. E. Wieman, “Correlating student beliefs with student learning using the Colorado learning attitudes about science survey,” PERC Proc. (2004).
718
C. Wieman, “Minimize your mistakes by learning from those of others,” Phys. Teach. 43,252 (2005).
722
C. E. Wieman, “Engaging students with active thinking,” Peer Rev. (Winter 2005).
724
C. Wieman and K. Perkins, “Transforming physics education,” Phys. Today (November 2005), 36.
725
S. B. McKagan and C. E. Wieman, “Exploring Student Understanding of Energy through the Quantum Mechanics Conceptual Survey,” PERC Proc. (accepted).
739
W. K. Adams, K. K. Perkins, N. Podolefsky, M. Dubson, N. D. Finkelstein and C. E. Wieman, “New instrument for measuring student beliefs about physics and learning physics: The Colorado learning attitudes about science survey, Phys. Rev. Spec. Top. PER (submitted), 2005.
743
Development of Research Technology
757
B. R. Brown, G. R. Henry, R. W. Keopcke and C. E. Wieman, “High-resolution measurement of the response of an isolated bubble domain to pulsed magnetic fields,” IEEE Trans. Magnetics 11, 1391 (1975).
761
S. L. Gilbert and C. E. Wieman, “Easily constructed high vacuum valve,” Rev. Sci. Instr. 53, 1627 (1982).
764
D. W. Sesko and C. E. Wieman, “High frequency Fabry-Perot phase modulator,” Appl. Opt. 2 h 1663 (1987).
766
G. J. Dixon, C. E. Tanner and C. E. Wieman, “432-nm source based on efficient second-harmonic generation of GaA 1 As diode-laser radation in self-locking external resonant cavity,” Opt. Lett. Irl, 73 1 (1989).
769
C. Wieman and L. Hollberg, ‘‘Using diode lasers for atomic physics,” (invited review) Rev. Sci. Instrum. 62,l(1991).
772
H. Patrick and C. E. Wieman, “Frequency stabilization of a diode laser using simultaneous optical feedback from a diffraction grating and a narrowband Fabry-Perot cavity,” Rev. Sci. Instrum. 62,2593 (1991).
792
C. Sackett, E. Cornell, C. Monroe and C. Wieman, “A magnetic suspension system for atoms and bar magnets,” Am. J. Phys. 6l, 304 (1993).
795
P. A. Roos, M. Stephens and C. E. Wieman, “Laser vibrometer optical based on feedback-induced frequency modulation for a single-mode laser diode,” Appl. Opt. 6754 (1996).
801
K. L. Corwin, Z.-T. Lu, C. F. Hand, R. J. Epstein and C. E. Wieman, “Frequency-stabilized diode laser with the Zeeman shift in an atomic vapor,” Appl. Opt. 37( 15), 3295 (1998).
809
z,
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Introduction
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The simplest way to characterize my physics career is that I blast atoms with laser light. I actually have been doing this from my early undergraduate days in Dan Kleppner’s lab at MIT. However I like to think that my work has been guided by somewhat more elevated intellectual theme and as a result my research career has followed a somewhat meandering but iairly continuous path that is reflected in my papers. The general theme is that I looked for how I could use the technical capabilities of laser light to study atoms in new ways and in new regimes, with a preference for tackling relatively fundamental physics questions. This meant that I was not only carrying out new physics experiments, but I was also working to advance the technology itself and experimental techniques. As a graduate student with Ted Hansch, I was interested in utilizing the new (at that time) capabilities of narrowband tunable lasers to do high resolution laser spectroscopy in hydrogen. This simplest of all atoms could be quantitatively understood in terms of basic quantum electrodynamics, and thus lasers allowed more precise measures of the energy splittings and correspondingly new tests of QED. By the end of my graduate school days, however, I had become convinced that QED had been so well established that testing it was unlikely to lead to any interesting surprises. The theory of neutral currents and electroweak unification was just coming into vogue at that time, and as such it was the natural extension of my interest. Here was an area where one could do high precision spectroscopy in atoms to measure parity nonconservation (PNC), and thereby explore territory that was far less trod than QED. Of course the downside was that the PNC effects were incredibly tiny and so the experiments were very difficult. This challenge however appealed to my inclination to see just how far one could push experimental capabilities. By pushing the state of the art in several aspects of laser spectroscopy, my group was able to see, and then ultimately measure parity violation in atomic cesium to quite high precision. These were very difficult experiments requiring large amounts of technology development followed by many years of tedious careful measurements and vast amounts of checking and double-checking for possible experimental errors. This was the work that first established my reputation as a scientist, and in retrospect, there were a lot of easier paths that I might have followed. In the midst of doing the PNC experiments, we used portions of the apparatus to carry out numerous other precision measurements on the cesium atom to test the atomic theory that was needed to interpret our parity violation results in terms of fundamental physics. This work also honed our experimental skills and provided publications in the long intervals between each new PNC result. Ultimately, we achieved a much more precise measurement of PNC in atoms, and a correspondingly more precise low energy test of the Standard model of elementary particle physics that, as of this date, is still significantly more precise than any other work. Rather to our surprise, in our second generation measurement we observed that there was the strong suggestion of a small part of the atom PNC that depended on the nuclear spin. After long agonizing over this unexpected and undesired effect in the experiment, we discovered that in fact there were theoretical predictions that such an effect might arise from a strange and never before seen entity called an “anapole moment”. Our third and last PNC experiment, which took more than ten years to complete was sufficiently precise that it provided a reasonably good measurement of the anapole moment. Although this result attracted significant interest and raised a number of significant questions about parity nonconservation in the nuclear forces, it is a bit remarkable that as of this writing, no experimental group other than ours has been able to achieve sufficient sensitivity to observe an anapole moment in any system, now some
3
twenty years after our original observation. To carry out the third generation of atomic parity nonconservation experiments required four tunable very narrowband lasers. At that time the only laser technology available were dye lasers and although, or perhaps because, we were experts in that technology, we knew that the cost and complexity of four suitable dye lasers would be unreasonable. So my graduate student Rich Watts and I began to explore the capabilities of inexpensive, but rather badly behaved, diode lasers. This led to many years of development of diode laser technology and the application of diode lasers in atomic physics research. One of our first and most recognized applications was to show that diode lasers could be used to slow atoms, thereby reducing the cost of the laser system required for atom slowing by nearly a factor of one thousand compared to what had been used. Although the original work was intended merely to show off the capabilities of the inexpensive new laser technology, this work led me to a long and profitable involvement of exploring new capabilities to use light to cool and trap atoms and the study of the light-atom and atom-atom interactions in this new ultracold atom regime. After some years of that work, we had progressed to the point that I thought my group understood enough about the atomic physics and we had developed enough technological capabilities for cooling and trapping atoms in various ways that it would be worth taking the gamble of pursuing the “holy grail” of Bose-Einstein condensation. Eric Cornell joined me in that quest and after five years of work we were successful. After making BEC, there were so many new and exciting experiments that could be done with BEC, and those experiments were so easy compared to the PNC work, that BEC soon became the dominant focus of my research. Throughout most of my career I have been interested in physics education. I always had many undergraduates working in my research labs, and I regularly worked on innovations for teaching undergraduates. I was always struck by the way that students seem to learn little or nothing in classes towards becoming an actual functioning physicist. I could see this in the courses that I taught and in seeing them starting to work in my lab as undergraduate and graduate students. This was in dramatic contrast to the way that a few years in the research lab routinely transformed them into highly competent physicists; something that 16+ years of schooling seemed incapable of doing. From this, it was clear to me that there was some sort of intellectual process present in the research lab that was sorely missing from the traditional education process. As my physics research career was reaching the point where it seemed like there were few if any new heights to reach, I became increasingly interested in science education as a research activity and an area where I might be able to make a substantial impact. I began to see how doing careful research on how people learn physics, and how guiding one’s teaching by the results of that research could work as well in education as it did in the physics lab. This convinced me that science education could be dramatically improved if teachers could be persuaded to break with tradition and follow a new more scientific approach. After receiving the Nobel Prize and realizing the potential for using the stature that comes with the Prize to advance this idea, I have devoted an increasingly large amount of my time and effort to education. This involves both carrying out research in physics education and serving as a public advocate for improved science education and how to achieve it.
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Precision Measurement and Parity Nonconservation
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This work begins with spectroscopy in hydrogen. It involved lots of development of laser to achieve new capabilities. Hidden in the results of the spectroscopy is the development of the method of high power pulsed amplification of a CW laser, as well as the development of the first single frequency tunable blue laser. I also still have fond memories of the excitement at coming up with the idea for polarization spectroscopy as a young graduate student. It was the first time I invented a new experimental technique that was better than what was in use, and so, in my own mind at least, made me feel like a real physicist. As an assistant professor, I staked my career on measuring parity violation in atoms. Fortunately, it did not occur to me for a long time how dangerous a path this was, because I was so wildly and naively optimistic about the difficulty of the experiments. This totally unrealistic optimism in the face of difficulties has stood me in good stead throughout my career. It was getting severely strained though by the time we successfully completed our first PNC measurement. The final PNC measurement was also enormously difficult as a host of unanticipated and unanticipatable problems arose as we pushed far beyond the limits that we or anyone else had previously reached. I had an enormous sense of relief and satisfaction with the completion of that work. Although it did not receive nearly the attention of BEC, I think that it some sense PNC was my best work. It certainly advanced the state of the art and surpassed the work of the competition by a much larger factor.
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VOLUME 34, N U M B E R 6
PHYSICAL REVIEW LETTERS
10 FEBRUARY 1975
Doppler-Free Two-Photon Spectroscopy of Hydrogen I S-2S:g T. W. Hansch,? S. A . Lee, R. Wallenstein,$ and C. WiemanD Department of P h y s i c s , Stanford University, Stanford, Califonzia 94,705 (Received 23 December 1974) We have observed t h e 1.5-2stransition in atomic hydrogen and deuterium by Dopplerf r e e two-photon spectroscopy, using a frequency-doubled pulsed dye laser at 2430 .&. Simultaneous recording of the absorption spectrum of t h e Balmer-p line at 4860 A, using t h e fundamental dye-laser output, allowed u s to precisely compare the energy int e r v a l s 1s-2s and 2 S , P - 4 S , P , D and to determine the Lamb shift of the 1s ground s t a t e to be 8 . 3 i 0 . 3 GHz (D) and 8 . 6 1 0 . 8 GI-Iz (11).
We have observed transitions from the 1s ground state of atomic hydrogen and deuterium to the metastable 2s state, using Doppler-free two-photon s p e c t r o ~ c o p y . " ~ The atoms a r e excited by absorption of two photons of wavelength 2430 A, provided by a frequency-doubled pulsed dye laser, and the excitation is monitored by observing the subsequent collision-induced 2P-lS fluorescence a t the L, wavelength 1215 A. Linewidths smaller than 2% of the Doppler width were achieved with two counter-propagating light beams, whose Doppler shifts cancel. The fundamental dye-laser wavelength a t resonance 4860 coincides with the visible Balmer-P line, and simultaneous recording of the absorption profile of this line permits a precise comparison of the energy intervals 1s-2s and 2 S , P - 4 S , P , D. From our first preliminary measurements we have determined the Lamb shift of the 1s ground state to be 8.3-tO.3 GHz (D) and 8.6*0.8 GHz (H), in good agreement with theory. The only previous measurement of the Lamb shift of the 1s state of deuterium, 7.9*1.1 GHz, has been reported by Herzberg,4 who used a difficult absolute-wavelength measurement of the La line. The hydrogen-1S Lamb shift has never been measured before. Numerous have pointed out that it would be very desirable to observe the 1s-2s transition in hydrogen by Doppler-free two-photon spectroscopy. The +-see lifetime of the 25 state promises ultimately an extremely narrow resonance width. The resolution obtained in the present experiment is already better than that achieved in our recent study of the Balmer-a line by saturation spectroscopy,6 and the implications for a future even more precise measurement of the Rydberg constant a r e obvious. We utilized a dye-laser system, consisting of a pressure-tuned dye-laser oscillator with optional confocal-filter interferometer7 and two subsequent dye-laser amplifier stages, pumped
by the same 1-MW nitrogen l a s e r (Molectron UV 1000) a t 1 5 pulses/sec. This l a s e r generates 102sec-long pulses of 30-50-kW peak power at 4860 A with a bandwidth of about 120 MHz (1-2 GHz without confocal filter). A 1-cm-long crystal of lithium formate monohydrate (Lasermetrics) generates the second harmonic with a peak power of about 600 W. A detailed description of this l a s e r system will be published elsewhere. The ground-state hydrogen atoms are produced by dissociation of H, o r D, gas in a Wood-type discharge tube (1 m long, 8 mm diam, typically 0.1-0.5 T o r r , 1 5 mA). The atoms a r e carried by gas flow and diffusion through a folded transf e r tube about 25 cm in length into the Pyrex observation chamber (Fig. 1). This chamber has two side a r m s with quartz Brewster windows to transmit the uv l a s e r light and a MgF, (originally LiF) window for the observation of the emitted L, photons. A thin coating of syrupy phosphoric acid is applied to all Pyrex walls to reduce the catalytic recombination of the atoms. The uv l a s e r light is focused into the chamber
WOOD DISCHARGE
PRESSURE-TUNED DYE LASER
?f
LITHIUM FORMATE FREQUENCY DOUBLER S O L A R BLIND PHOTOMULTIPLIER
FIG. 1. Experimental setup.
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-1 'F
I S L A M B SHIFT
2430
(b)
.k,
--c
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1 1 1 1 1 1 1 l I I I I I I l I I I I I I I I l l l l l i l l l l
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FIG. 2. (a) Absorption profile of the deuterium Balmer-p line with theoretical fine structure; ( b ) sirnultaneously recorded two-photon resonance of deuterium
1s-2.5.
to a spot s i z e of about 0.5 mm diam. The t r a n s mitted beam is refocused into the cell by a spherical m i r r o r to provide a standing-wave field. The separation between illuminated region and MgF, window is kept s m a l l (1-2 mm) to reduce the l o s s of L, photons due to resonance trapping. The transmitted L, photons a r e detected by a solar-blind photomultiplier (EMR 541 J). An int e r f e r e n c e filter with 6% t r a n s m i s s i o n at 1215 reduces the off-resonance background signal to l e s s than one r e g i s t e r e d photon p e r s e v e r a l hundred laser pulses. The multiplier output i s processed by a gated integrator (Molectron LSDS) with a n effective gate opening t i m e of 200 y s e c and is electronically divided by a signal proportional to the s q u a r e of the uv l a s e r intensity. Figure 2(b) shows a two-photon spectrum of deuterium 1S-2S recorded with moderate resolution (no confocal-filter interferometer). The low Doppler-broadened pedestal is caused by twophoton excitation by each of the linearly polarized uv b e a m s individually and could be eliminated by the use of c i r c u l a r l y polarized light.3 The signal at resonance corresponds to about 10-20 r e g i s tered La photons p e r pulse, and r e m a i n s within the s a m e o r d e r of magnitude when the H, o r D, gas is diluted by He up to a ratio of 1OOO:l. The expected d e c r e a s e in the number of excited meta-
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stable a t o m s i s apparently largely compensated over a wide r a n g e by a concomitant reduction of the loss of L, photons due to resonance trapping and quenching. Despite the lack of a n e a r - r e s o nant intermediate state, the two-photon fluorescence is comfortably strong, and it should easily be possible to o b s e r v e the signal a t a considerably lower total gas p r e s s u r e (loD4T o r r ) , where p r e s s u r e broadening and shifts would become unimportant if the 2s-2P transitions w e r e induced by an applied rf field. F i g u r e 2(a) shows the absorption spectrum of the deuterium Balmer-P line which w a s simultaneously recorded by sending a s m a l l fraction of the blue dye-laser light through a 15-cm-long c e n t e r section of the positive column of a WoodThe posidischarge tube (0.2 T o r r D,, 25 &). tions of the indicated theoretical fine-structure components w e r e located by a computer fit of the line profile. We have m e a s u r e d the separation of the 1s-2s resonance f r o m the strongest component (2P,,,-4D5,,) in the B a l m e r - 8 spectrum to be 3.38 f 0.08 GHz f o r deuterium and 3.3 0.2 GHz f o r hydrogen (in t e r m s of the blue-light f r e quency). The corresponding theoretical s e p a r a tion$ a r e 3.420 and 3.422 GHz, respectively. These separations would be l a r g e r if the 1s s t a t e were not r a i s e d above its D i r a c value by the Lamb shift (theoretically 8.172 GHz for D and 8.149 GHz for H) and our measurement can be interpreted as a determination of the groundstate L a m b shift. A considerable improvement in a c c u r a c y c a n be expected when a high-resolution saturation spectrum6 of the Balmer-P line is used f o r the comparison. A two-photon s p e c t r u m of hydrogen 1s-2s with the laser operating in its high-resolution mode is shown in Fig. 3 (scan t i m e about 2 min). The linewidth is limited by the laser bandwidth of about 120 MHz (in the blue). The s p e c t r u m reveals.two hyperfine components, separated by the difference of the hyperfine splittings of lower and upper s t a t e s , as expected f r o m the selection r u l e AF =O.' It is not difficult to c o m p a r e t h e observed signal strength with theoretical estimates, using Eq. (7) of Ref. 2, derived f o r steady-state conditions. In the p r e s e n t experiment the a t o m s are excited by light pulses whose t i m e duration 7 is s h o r t compared to the inverse linewidth re of the twophoton transition, and which have a n e a r - F o u r i e r transform-limited bandwidth A o = T-'. One can show with the help of time-dependent perturbation theoryg that Eq. (7) of Ref. 2 in this c a s e still
*
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2430.7
tl 0
F:
1*1
0-0
I
4
I
I
I
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I
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D Y E LASER F R E Q U E N C Y D E T U N I N G ( G H z I
FIG. 3. High-resolution two-photon spectrum of hydrogen IS-2s with resolved hyperfine splitting.
correctly predicts the effective two-photon transition r a t e if one replaces the linewidth reby the laser bandwidth A o . W e have numerically evaluated this expression in a calculation similar to that of Ref. 5, but with inclusion of the continuum in the summation over intermediate states, and obtain a two-photon transition r a t e per atom of I?= 7 x 1 0 - 4 1 2 / A ~where , the light intensity I is measured in W/cm2, the laser bandwidth Am in MHz, and r in sec". F o r a comparison with the experiment we consider a H, partial p r e s s u r e of 2 X10'* Torr and 10%dissociation, i,e,, a density of 1.4X10l3 1s atoms/cm3, s o that the loss by resonance trapping is not important. Our estimate then predicts about 3 X l o 5 excited metastable atoms per pulse over a 1-cm path length, With a detection solid angle of 0.02~4.rrsr, a filter transmission of 6%, and a multiplier quantum efficiency of lo%, we expect about thirty registered photons per l a s e r pulse, in reasonable agreement with the present observations. We have also calculated the shift of the 1S-2.S two-photon resonance frequency caused by the intense uv radiation (ac Stark effect). By numerically evaluating Eq. (1) of Liao and Bjorkholm," we estimate a n intensity shift of about 5.5 Hz/ (W/cmz) o r l e s s than 2 MHz under the present operating conditions. The light intensity and
10 FEBRUARY 1975
hence the shift can, in principle, be reduced without loss of resonance signal, by decreasing the l a s e r bandwidth and increasing the interaction time with the atoms. A hundredfold improvement in resolution should be obtainable with the present pulsed dye-laser system, if the hydrogen cell is placed inside a narrow-band confocalfilter interferometer. W e are grateful to Professor T.. Fairchild for lending u s a superb La interference filter of his design. And we thank Professor N. Fortson for stimulating discussions and Professor A. Schawlow for his continuous stimulating interest i n this research.
*Work supported by the National Science Foundation under Grant No. MPS74-14786A01, by the U . S. Office of Naval Research under Contract No. ONR-0071, and by a Grant from the National Bureau of Standards. tAIfred P. Sloan Fellow 1973-1975. $NATO Postdoctoral Fellow. 8 Hertz Foundation Predoctoral Fellow. 'L. S . Vasilenko, V . P. Chebotaev, and A . V . Shishaev, P i s ' m a Zh. Eksp. T e o r . Fiz. 12,161 (1970) [JETP Lett. 12,113 (1970)). *B. Cagnac, G. Grynberg, and F. Biraben, J. Phys. (Paris) 3, 845 (1973). 3 F . Biraben, B. Cagnac, and G. Grynberg, Phys. Rev. Lett. 3, 643 (1974); M. D . Levenson and N. Bloembergen, Phys. Rev. Lett. 32, 645 (1974); T . W. HLnsch, K. C . Harvey, G. Meisel, and A . L. Schawlow, Opt. Commun. 50 (1974). 4G. Herzberg, P r o c . Roy. SOC., S e r . A 34, 516 (1956). k.V . Baklanov and V . P. Chebotaev, Opt. Coinmun. 1 2 , 312 (1974). G T .W. HXnsch, M. H. Nayfeh, S. A . L e e , S. M. C u r r y , and I. S. Shahin, Phys. Rev. Lett. 32, 1336 (1974). 'R. Wallenstein and T. W. Hhhsch, Appl. Opt. 13, 1625 (1974). *J. D. Garcia and J . E . Mack, J . Opt. SOC.Amer. E, 654 (1965). 'A. Gold, in Quantum Optics, Proceedings of the International School of Physics '%nrico Fermi," Course 4 2 , edited by R. J. Glauber (Academic, New York, 1969). lop.F. Liao and J . E . Bjorkholm, Phys. Rev. Lett. 34, 1 (1975).
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VOLUME 36, NUMBER 20
P H Y S I C A L REVIEW LETTERS
17 MAY 1976
Doppler-Free Laser Polarization Spectroscopy* C. Wiemant and T. W. Hsnsch Department of Physics, Stanford U n i v e r s i t y , Stanford, California 94305 (Received 25 March 1976)
We have demonstrated a sensitive new method of Doppler-free spectroscopy, monitoring the nonlinear interaction of two monochromatic laser beams in an absorbing gas via changes in light polarization. The signal-to-background ratio can greatly surpass that of saturated absorption. Polarization spectra of the hydrogen Balmer-P line, recorded with a cw dye laser, reveal the Stark splitting of single fine-structure components in a Wood discharge. We report on a sensitive new technique of highresolution laser spectroscopy based on light-induced birefringence and dichroism of an absorbing gas. Polarization spectroscopy i s related to saturated-absorption1 o r saturated-dispersion' spectroscopy, but offers a considerably better signal-to-background ratio. It is of particular interest for studies of optically thin samples or weak lines and permits measurements even with weak o r fluctuating l a s e r sources. We have studied the Falmer-@line of atomic H and D near 4860 A by the new method, using a single-frequency cw dye laser. The spectra reveal for the f i r s t time the Stark splitting in the weak axial electric field of a Wood-type gas discharge. For saturation spectroscopy it is well known that the signal magnitude depends on the relative polarization of saturating beam and The possibility and advantage of using the resulting optical anisotropy in a sensitive polarization detection scheme seem to have gone unexplored, however. On the other hand, the phenomena of light-induced birefringence and dichroism a r e quite common in optical-pumping experiments with incoherent light source^.^ Recent related experiment^^'^ encourage us to expect that highresolution polarization spectroscopy will also prove useful f o r studies of two-photon absorption and stimulated Raman scattering in gases. The scheme of a polarization spectrometer is shown in Fig. 1. A linearly polarized probe beam from a monochromatic tunable l a s e r is sent through a gas sample, which is shielded from external magnetic fields to avoid Faraday rotation. Only a small fraction of this beam reaches a photodetector after passing through a nearly crossed linear polarizer. Any optical anisotropy which changes the probe polarization will alter the light flux through the polarizer and can be detected with high sensitivity. Such an anisotropy can be induced by sending a second. circularly polarized, l a s e r beam in nearly the oppo-
site direction through the sample. In the simplest case both beams have the same frequency w and a r e generated by the same laser. A s in conventional saturation spectroscopy, a resonant probe signal is expected only near the center of a Doppler-broadened absorption line where both beams a r e interacting with the same atoms, those with essentially zero axial velocity. For a quantitative description we can decompose the probe into two circularly polarized beams, rotating in the s a m e (+) and in the opposite (-) sense as the polarizing beam. As long a s the probe is weak these two components can be considered separately. The polarizing beam in general induces different saturation, i.e., changes in absorption coefficient, Aa' and Aa-, and in refractive index, An' and An-, for these components. A difference Aa'-Acu- describes a circular dichroism which will make the probe light elliptically polarized, and a difference An' -Art- describes a gyrotropic birefringence which will rotate the axis of polarization. A s long as these polarization changes a r e small, the complex field amplitude behind the blocking polarizer is given by
where E , is the probe amplitude, 6 i s some small angle by which the polarizer is rotated from the
r X/4 P L A T E
FIG. 1. Scheme of laser polarization spectrometer,
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perfectly perpendicular position, and 1 is the absorption path length. For low intensities, within third-order perturbation theory* and in the limit of large Doppler widths, the interaction of two counterpropagating circularly polarized beams with two levels of angular momentum J and J' can be described simply in t e r m s of a velocity-selective hole burning' for the different degenerate sublevels of orientational quantum number m , where the axis of quantization is chosen along the direction of propagation. As in conventional saturation spectroscopy' the absorption change as a function of the l a s e r frequency is a Lorentzian function with the natural linewidth yab:
ha+= Acr-/d = - + c Y , , I / I ~ ~1 ~+ X( '). (1 - 5 / ( 4 P d5A~~ACY+=
(2)
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Here, a, is the unsaturated background absorption, I is the intensity of the polarizing beam, Isatis the saturation parameter, and x =(w-wOb)/ yabdescribes t h e l a s e r detuning f r o m resonance. The absorption change corresponds t o the imaginary p a r t of a complex third-order susceptibility, whose r e a l part results in a concomitant change in refractive index, An*= -$Aaixc/w, in agreement with the Kramers-Kronig relation. The magnitude of the anisotropy is described by the parameter d and depends on the angular momenta and the decay rates yJ and y J # of the involved states. If spontaneous re-emission into the lower state is ignored, the steady-state signal contributions of the different orientational sublevels can be added with the help of simple sum rules4 to yield
+ 4 ~ 2)+ for J = J',
j ( 2 3 + 5 r J + 3)/(122- 2) for J = J'+ 1. where Y = ( y j - y j t ) / ( y j + yp). By inserting these results into Eq. (l), we obtain the light flux a t the probe detector
1=I,,[8z + O$sx/( 1 +x') +(is)'/(
1+xZ)],
( 4)
where I, is the unattenuated probe power and s = -i(1 -d)aoZI/Isat gives the maximum relative intensity difference between the two counter-rotating probe components. In practice we have to add an amount I,[ to account f o r the finite extinction ratio t of the polarizer. For a perfectly crossed polarizer, i.e., 6 = 0 , the combined effects of dichroism and birefringence lead to a Lorentzian resonance signal. The signal magnitude is proportional to s2, i.e., it drops rapidly for small s. Small signals can be detected with higher sensitivity a t some small bias rotation 6>>s. The last t e r m in Eq. (4) can then be neglected and the birefringent polarization rotation produces a dispersion-shaped signal on a constant background. If, a s in many practical situations, laser intensity fluctuations a r e the primary source of noise, a figure of merit for the sensitivity is the signal-to-background ratio. Compared t o saturated-absorption spectroscopy, this ratio is improved by a factor (1 -d)O/4(Q2 + 5) which reaches its maximum (1- d ) / 845 for 8 = dt. In addition to the improved sensitivity, such a dispersion-shaped signal is of obvious interest for the locking of the l a s e r frequency to some resonance line. It is also noteworthy that its
(3)
f i r s t derivative h a s a linewidth s m a l l e r than half the natural width, which can greatly facilitate the spectroscopic resolution of closely spaced line components. For the alternative scheme of a linearly polarized saturating beam, rotated 45" with respect to the probe polarization, it can be shown in analogous fashion that the signal always remains Lorentzian, independent of the polarizer angle 6 . As in saturated-absorption spectroscopy, crossover signals a r e expected halfway in between two resonance lines which s h a r e a common upper o r lower level. A third-order nonlinear susceptibility tensor, applicable t o this situation, has actually been calculated p r e v i ~ u s l y .It~ predicts that the ratio A~i-/ha'= d for certain angular momentum states can exceed 1, unlike the expression (3), and hence give r i s e to signals with inverted polarization rotation. For the experimental study of the hydrogen Balmer-P line we used a cw jet-stream dye laser (Spectra-Physics Model No. 375) with 7-diethylamino-4-methyl-coumarin in ethylene glycol, pumped by an uv argon l a s e r (Spectra-Physics Model No. 171). Single-frequency operation was achieved with an air-spaced intracavity etalon (free spectral range 30 GHz; m i r r o r reflectivity 30%) and two additional fixed uncoated quartz etalons (thicknesses 0.1 and 0.5 mm). At 1 W pump power, the l a s e r provid$s single-mode output of about 10 mW near 4860 A with a linewidth of
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~
about 20 MHz. The l a s e r can be scanned continuously over - 4 GHz by applying linear ramp voltages to the piezotransducers of the cavity end m i r r o r and the air-spaced etalon. As in the previous saturated-absorption experiment,g the hydrogen atoms were excited to the absorbing n = 2 state in a Wood-type discharge tube (1 m long, 8 mm in diameter, 0.2 T o r r , 3 mA dc current). The laser light i s sent through a 40-cm-long center section of the positive column. Probe and polarizing beam a r e each about 1 mm in diameter and have powers of 0.1 and 1 mW, respectively, A mica sheet s e r v e s as a h / 4 retarder for the polarizing beam. To avoid residual Doppler broadening due to a finite crossing angle we operated with collinear beams, replacing m i r r o r M , in Fig. 1 with an 80% beam splitter which transmits part of the probe. Standard Glan - Thomp s on prism polarize r s (Karl Lambrecht) were employed for the probe beam. Birefringence in the quartz windows of the gas cell due to internal strain was reduced by squeezing the windows gently with adjustable clamps. We achieve extinction ratios of or better in this way, and the possible improvement over saturated-absorption spectroscopy in signalto-background ratio is on the order of 100-1000. The probe light which passes the blocking polari z e r is sent through a spatial filter to eliminate incoherent light emitted by the gas discharge and scattered light from the polarizing beam, Its intensity is monitored with a photomultiplier. Figure 2 shows a portion of the Balmer-8 spectrum plotted versus time during a l a s e r scan of about 5 min duration. To record the derivative of dispersion-shaped resonances, the dye l a s e r was frequency modulated by adding a s m a l l audiofrequency voltage to the cavity-mirror tuning ramp, and the resulting signal modulation was detected with a phase-sensitive amplifier. The three strongest theoretical fine-structure transitions in this region a r e shown on top for comparison. Hyperfine splitting is ignored. The positions of possible crossover lines due to a common upper o r lower level are indicated by arrows. Obviously, the polarization spectrum reveals many more components. These have to be a s cribed to the Stark splitting in the axial electric field of the positive discharge column, We have calculated the theoretical Stark pattern for an axial field of 10 V/cm by diagonalizing the Hamiltonian. lo The positions of the strongest Stark components and their respective crossover lines are indicated in Fig. 2 and agree well with
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10 vcm-'
I
0
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L A S E R FREOUENCY DETUNING
GHZ ---c
FIG. 2. Polarization spectrum of a portion of the deuterium Balmer-P line. The three strongest fine-structure components and the positions of the strongest Stark components for an axial electric field of 10 V/cm are shown on top for comparison. Crossover lines due to a common upper ( t ) or lower ( + ) level are indicated by arrows.
the observed spectrum. The splitting of the upp e r 4P,D,/, level is essentially a linear function of the electric field, and i t s magnitude can easily be determined to within a few percent from the observed spectrum. The theoretically expected line strengths and the signs of the crossover lines are also in satisfactory agreement with the experiment. The polarization spectrum of light hydrogen looks almost identical to that of deuterium and shows no indication for the 170-MHz hyperfine splitting of the 2.9 state; the components originating in the F = 0 state are missing, because atoms in this level cannot be oriented. The observed Stark pattern changes quite drastically if the l a s e r beams a r e displaced from the tube axis, indicating the presence of additional radial electric fields due to space and surface charges of cylindrical symmetry. Polarization spectroscopy of the hydrogen Ralmer lines thus opens new possibilities for sensitive plasma diagnostics. The spectrum in Fig. 2 also clearly reveals the different natural linewidths of components originating in the short-living 2P state and the
15 VOLUME36, NUMBER 20
PHYSICAL REVIEW LETTERS
l o n g e r - l i v i n g 2s state. T h e narrowest o b s e r v e d components have a width of about 40 MHz, c o r r e sponding t o a r e s o l u t i o n of about 6 p a r t s i n lo*, i.e., they exhibit m o r e than an o r d e r of m a g n i tude i m p r o v e m e n t o v e r o u r earlier p u l s e d - l a s e r saturation ~ p e c t r a .A~ substantial improvement in t h e a c c u r a c y of t h e 1s L a m b s h i f t is e x p e c t e d , i f s u c h a polarization s p e c t r u m is u s e d as a refe r e n c e f o r t h e 1s-2s two-photon s p e ~ t r u m .A~ still f u r t h e r i m p r o v e m e n t i n r e s o l u t i o n should b e p o s s i b l e i f t h e l a s e r linewidth is r e d u c e d by frequency stabilization. A t low e l e c t r i c f i e l d s t h e n a t u r a l linewidth of t h e q u a s i - f o r b i d d e n 2s-4s t r a n s i t i o n is only about 1 MHz. A m e a s u r e m e n t of t h e H-D isotope shift t o better t h a n 0.1 MHz would c o n f i r m o r i m p r o v e t h e i m p o r t a n t r a t i o of e l e c t r o n mass to proton mass, a n d a n a b s o l u t e wavelength o r f r e q u e n c y m e a s u r e m e n t t o better than 6 MHz would yield a new i m p r o v e d value for t h e Rydberg constant. We are p r e s e n t l y e x p l o r i n g t h e s e a n d o t h e r p o s s i b i l i t i e s for new p r e c i s i o n
measurements. We are indebted to P r o f e s s o r A. L. Schawlow for h i s s t i m u l a t i n g interest in this work, and we
17 MAYI976
thank J, E c k s t e i n f o r h i s help i n c a l c u l a t i n g Eq.
(3). *Work supported by the National Science Foundation under Grant No. NSF 14786, and by the U. S. Office of Naval Research under Contract No. N00014-75-C-0841. tHertz Foundation Predoctoral Fellow. 'P. W. Smith and T. W. H b s c h , Phys. Rev. Lett. -26 9 740 (1971). 'C, Borde, G , Carny, B. Decomps, and L. Pottier, C. R. Acad. Sci., Ser. B E , 381 (1973). 'T. W. H k s c h and P. Toschek, 2 . Phys. 266, 213 (1970). '111. Sargent, ID, M. 0. Scully, and W. E. Lamb, Jr., Laser Physics (Addison-Wesley, London, 1974). %V. Happer, Prog, Quantum Electron. 1, 53 (1970). 'P. F. Liao and G. C. Bjorklund, Phys. Rev. Lett. 36, 584 (1976). 'D. Heiman, R. W. Hellwarth, M. D. Levenson, and G. Martin, Phys. Rev. Lett. 36, 189 (1976). 'M. Dumont, thesis, University of Paris, 1971 (unpublished). 'S. A, Lee, R. Wallenstein, and T. W. Hgnsch, Phys. Rev. Lett. 35, 1262 (1975). 10 J. A. Blackman and G . W. Series. J. P h v a R 6, 1090 (1973).
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U7
traveling in opposite directions, a resonant signal is expected only near the center of a Doppler-broadened absorption line, where both beams are interacting with the same atoms, those with essentially zero axial velocity. With perfectly crossed polarizers, the combined effects of dichroism and buefringence lead to a Lorentzian signal of essentially the natural line width. If the analyzing polarizer is slightly rotated from the perpendicular position, the birefringent polarization rotation produces a dispersion-shaped signal on a small background. With an extinction ratio of lo-', as can be readily achieved with commercial prism polarizers, the ratio of signal to background can surpass that of saturated absorption spectroscopy by 2 or 3 orders of magnitude. To study the hydrogen Balmer-P line, we employed a singlefrequency cw dye laser near 4860 A. The achieved resolution exceeds that of earlier saturation spectra by more than an order of magnitude. The resolved Stark splitting permits the quantitative mapping of axial and radial electric fields in the discharge plasma. To study molecular sodium in an oven at 300°C by lower level labeling, we used two nitrogen-laser-pumped pulsed dye lasers. One provided a broadband probe beam (Ax p. 300 A, centered at 4880 A), the other a monochromatic (Av = 1 GHz) circularly polarized saturating beam, tuned to a molecular X-B transition. The probe light passing through the crossed polarizers is analyzed by a grating spectrograph. Light is transmitted at all wavelengths corresponding to A J = f 1 transitions with the same lower state as the pumped transition. It will be demonstrated how this method can greatly simplify the analysis of unknown atomic or molecular spectra.
LASER POLARIZATION SPECTROSCOPY OF ATOMS AND MOLECULES3 R. FEINBERG, T.W. HANSCH, A.L. SCHAWLOW, R.E. TEETS and C. WlEMAN Department of Physics, Stonford University, Stonford, Gzlifornw 94305, USA
We report on a sensitive new technique of nonlinear laser spectroscopy, based on light induced birefringence and dichroism of an absorbing gas. The technique is related to saturated absorption spectroscopy, but can offer a substantially better ratio of signal to background. Doppler-free polarization spectra of the hydrogen Balmer-line reveal, for the fust time, the Stark splitting of single fine-structure components in the 10 V/cm axial electric field of a Wood-type gaspischarge, and open numerous interesting possibilities for new precision measurements and for sensitive plasma diagnostics [ 11. We have also applied polarization spectroscopy to lower-level labeling [2] of Naz, to provide a useful technique for unraveling the complexities of molecular spectra. The resuiting spectra resemble those of laserexcited fluorescence, but they provide direct information about the spectroscopic constants and quantum numbers of the upper state rather than the lower state. Experimentally, the new technique is rather simple: a probe laser beam is sent t h r o u a a gas sample between crossed polarizers. The light flux reaching a photodetector is a sensitive indicator for any optical anisotropy of the sample. Such an anisotropy can be induced by sending a second, circularly polarized, laser beam nearly collinearly through the probed gas region. By differentially changing the population of various angular momentum sublevels via saturation and optical pumping, this second produces a circular dichrohm and gyrotropic birefringence. For two monochromaticlaser beams of the same frequency,
References (1 ] C. Wieman and T.W. Hansch, Phys. Rev. Letters (1976), accepted for publication. [2] M.E. Kaminsky, R.T. Hawkins, F.V. Kowalski and A.L. Schawlow, Phys. Rev. Letters 36 (1976) 671.
3 Work supported by the National Science Foundation under Grant NSF-14786,and by the U.S. Office of Naval Research under Contract N00014-75C-0841.
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P H Y S I C A L REVIEW A
VOLUME 2 2 , NUMBER 1
JULY 1980
Precision measurement of the 1S Lamb shift and of the 1s-2sisotope shift of hydrogen and deuterium C. Wieman* and T. W. Hansch Department of Physics, Stanford Uniuersity, Stanford, California 94305 (Received 14 September 1979) A precision measurement of the IS Lamb shift of atomic hydrogen and deuterium, using high-resolution laser spectroscopy is reported. The 1S -2stransition was observed by Doppler-free two-photon spectroscopy, using a single-frequency cw dye laser near 4860 A with a nitrogen-pumped pulsed dye amplifier and a lithium formate frequency doubler. T h e n = 2-1 Balmer-P line was simultaneously recorded with the fundamental cw dye-laser output in a lowpressure glow discharge, using sensitive laser polarization spectroscopy. From a comparison of the two energy intervals a ground-state Lamb shift of 8151 k 3 0 MHz has been determined for hydrogen and 8177 30 M H z for deuterium, in agreement with theory. The same experiments yield a tenfold improved value of the IS-2S isotope shift 670992.3 f6.3 MHz and provide the first experimental confirmation of the rclativistic nuclear recoil contribution to hydrogenic energy levels.
-+
result gives the f i r s t experimental confirmation of the relativistic nuclear recoil correction to hydrogenic energy levels. Traditionally, Lamb shifts of S states have been measured by exciting radiofrequency trinsitions to a nearby P state. This technique cannot be used for the ground state, because t h e r e is no P state in the n = 1 level, and a measurement of the 1s shift is consequently considerably more difficult. Herzberg5 attempted the direct approach of measuring the absolute wavelength of the Lyman-a, line to sufficient precision, but such an experiment i s beset by many problems. First, the Lyman-a line (1215 A ) is in the vacuum-ultraviolet region of the spectrum where precision wavelength measurements a r e difficult. Secondly, traditional emission spectroscopy is complicated because emission sources a r e strongly self-reversed, while absorption spectroscopy is plagued by spurious background lines in continuum sources. Thirdly, the Doppler width of the Lyman-a line is about 40 GHz at room temperature, four times l a r g e r than the expected Lamb shift. And finally, if all these difficulties could be surmounted t o obtain a precise value for the Lyman-a! energy, the present uncertainty of the Rydberg constant‘ prevents one from determining the 1s Lamb shift t o better than one part in 1000. Recent advances in high-resolution laser spectroscopy together with improvements in dye-laser technology have made it possible to determine the 1s Lamb shift in a different manner, however, which avoids all these difficulties. This approach, as first described in Ref. 2, u s e s l a s e r spectroscopy t o precisely compare the Balmer-P (n = 2 to 4) transition energy with 2 of the Lyman-a energy. If the Bohr formula were c o r r e c t these two intervals would be exactly the same, & of the
I. INTRODUCTION
The measurement of Lamb shifts in hydrogenic atoms has played a vital role in the development of quantum electrodynamics (QED). Since the f i r s t measurement of the splitting between the 2S,,, and PI,, levels by Lamb and Retherford,’ Lamb shifts have been measured for many hydrogenic states. Some of these measurements a r e among the most precise tests of quantum electrodynamics. One measurement which was notably missing, however, was that of the shift of the 1s ground state. This paper reports on the last of a s e r i e s of three increasingly precise measurements of t h i s quantity performed at Stanford University. Unlike previous Lamb-shift measurements, these experiments a r e based on high-resolution l a s e r spectroscopy. A first and rather preliminary experimental value of the 1s Lamb shift was obtained by m n s c h et al.‘ at the time of the first observation of 1S-2S two-photon excitation in hydrogen. This was followed by the m o r e careful measurement of Lee et aZ.,3combining the techniques of Dopplerf r e e two-photon spectroscopy and saturated absorption spectroscopy. In the present measurements we have made a number of important technical improvements which have enabled us to further reduce the uncertainty to -+30MHz. Among these improvements have been the development of a new, highly sensitive technique of Doppler-free laser spectroscopy, “polarization s p e c t r ~ s c o p y , ”and ~ the construction of a cw dye-laser oscillator with pulsed dye-laser amplifier which offers substantially better power and bandwidth than the previously used pulsed dye-laser system. The new l a s e r has also enabled US to measure the Lyman-a! isotope shift for hydrogen and deuterium to within k6.3 MHz. This 22 -
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@ 1980 The American Physical Society
18 P R E C I S I O N M E A S U R E M E N T O F T H E 1s L A M B S H I F T A N D O F . . .
22
Rydberg energy. Actually they differ by t h e ground-state Lamb shift plus small, well-measured QED and fine-structure corrections t o the excited-state energies. Thus a n accurate comparison of the two intervals allows one to determine the 1s Lamb shift. The comparison of the two energies is made using a powerful and highly monochromatic dye l a s e r with a wavelength near 2860 as illustrated in Fig. 1. The frequency-doubled output of this l a s e r is used to excite two-photon transitions from the IS ground state to the metastable 2s state. Simultaneously, the fine-structure spectrum of the Balmer-P line i s observed using the fundamental l a s e r output, and the position of some component of this line is measured relative to the two-photon reso-
EOHR
.......
4
DlRAC
193
nance. A s can be seen in Fig. 1, a major contribution t o this separation originates from the 1s Lamb shift. Both wavelengths a r e n e a r 4860 A, and one avoids the problems of vacuum-ultraviolet spectroscopy. Furthermore, Doppler broadening of the 15-2s transition can easily be avoided by excitation with counterpropagating beams.’ And because the measurement only involves the comparison of two hydrogen transitions, the uncertainty of the Rydberg constant is unimportant. 11. BACKGROUND
A. Hydrogen energy levels and 1.9 Iamb shift
The energy levels of hydrogen can be written in the form7
QED
-
3
2 2p3/2
/2s1/2 ‘2PI/Z
i
4
_ -
~
-4-
IS-2s TWO-PHOTON EXCITATION WITH
H g FINE STRUCTURE
(b)
where E D is the energy predicted by the Dirac equation using the reduced m a s s Rydberg constant, E R is the additional nuclear recoil correction predicted by the Breit equation for a relativistic two-body system, E N is the correction due t o nuclear s i z e and nuclear s t r u c t u r e effects, and E L , a sum of many t e r m s , gives the QED corrections. In this paper we will be somewhat loose with the t e r m “ground-state Lamb shift,” and use it to r e f e r to the total deviation of the 1s energy from the Dirac energy, i.e., the s u m E , (IS)+ E,(lS) + E,(lS). A detailed list of all the known contributions t o this shift with their most recently computed numerical values is given in Ref. 7. The relativistic nuclear recoil t e r m is given t o lowest o r d e r by
k-n-4
IS-2s TWO-PHOTON RESONANCE (C)
i
I
P I S L A M E SHIFT ! x 114
w
-5 o ~(GHZ) L A S E R FREOUENCY DETUNING-
FIG. 1. Top: Simplified diagram of hydrogen energy levels and transitions. The Dirac fine structure and QED corrections for n =1, 2, and 4 are shown on an enlarged scale. Hyperfine structure and Stark effect are ignored, except for showing the weakly Stark-allowed transition 2S112- 4S1/2. Bottom: Fine-structure spectrum of the Balmer-P line and relative position of the 1s-2s two-photon resonance as recorded with the second harmonic frequency. The dashed line and the dashed arrows above give the hypothetical position of the 1s-2s resonance if there were no 1s Lamb shift. An experimental value of the 1s Lamb shift has been determined from the frequency interval Av between the two-photon resonance and a crossover resonance observed in a polarization spectrum of Balmer-,9 halfway between the fine-structure components 2Si/z4Pi/2 and 2Si/2-*1/2.
where m is the electron m a s s and M the nuclear mass. This recoil shift, which has never been experimentally verified, amounts to -23.81 MHz for hydrogen 1s and -11.92 MHz f o r deuterium 1s. The respective 1.9 shifts due to nuclear structure a r e 1.00*0.05 and 6.78k0.09 MHz. Including these corrections, the 1s Lamb shift has a theoretical value of 8149.43 *0.08 MHz for hydrogen and 8172.23*0.2 MHz for deuterium. The previous measurements of the 1s Lamb shift and the nonlinear spectroscopic techniques involved in this work have been discussed elsewhere in various degrees of completeness.’* 3 1 8 i 9 However, for the sake of clarity we shall give a brief review. B. Doppler-free two-photon spectroscopy of IS-2.9
The technique of Doppler-free two-photon spec-
19
c.
194
W I E M A N A N D 1’.
troscopy, reviewed i n Ref. 10, h a s been cen t r al t o all m e a s u r e m e n t s of t h e 1s L amb shift. F i r s t suggested by Vasilenko et nl. ,11 two-photon excitation of g a s a t o m s with two counterpropagating l a s e r b e a m s can provide n a r r o w resonance s i g n al s f r e e of f i r s t - o r d e r Doppler broadening, b ecau s e t h e a t o m s s e e the two b e a m s with opposite, and thus canceling, Doppler shifts. Two-photon excitation of hydrogen 1.7-2sr e q u i r e s ultraviolet radiation n e a r 2430 of relati.vely high intensity, because t h e r e is no n ear - r es o n an t i n te rmediate level. The t r an s i t i o n r a t e f o r cw excitation h a s been computed numerically by s e v e r a l authors.”’ l 3 According t o Gontier and Trahin,l2 the two-photon absorption c r o s s s ect i o n i s of the order
A
u(Aj=2,75X10-171gcm2,
(3)
w h e r e 1 is t h e light intensity in W/cm2 and g t h e in v e r se atomic transition linewidth i n sec. The expected n at u r al linewidth of t h e transition i n t h e a b s e n ce of collisions i s only about 1 Hz, limited by t h e 4 - s e c l i f et i me of t h e 2 s s t a t e . While c u r r e n t l y available lasers a r e much too broadband to r e a ch t h i s limit, t h ey a r e m o r e than sufficient t o o b s e r v e t h e transition. For a pulsed laser we can easily es t i mat e t h e excitation probability with the help of time-dependent secondo r d e r perturbation theory.14 Assuming, f o r s i m plicity, a s q u a r e excitation pulse of t i m e duration T, intensity I , frequency o, and t r a n s f o r m l i m ite d bandwidth A w =a/T we obtain a n excitation probability p e r a t o m
w h e r e wIs-Dsdenotes t h e at o mi c resonance frequency. At exact resonance, t h e r i g h t mo s t f r a c tion simplifies t o T2; i.e., t h e effective absorption c r o s s section, compared t o t h e s t ead y - s t at e result (Eq. 3), is reduced by a f a c t o r T / 2 g . With a pulse length of 7 n s e c an d a n intensity I = 2 X 10G W/cm2 (approximately t h e experimental conditions), we find a n excitation probability p e r at o m of about 2 X 1 0 - 3 . Since t h e density of 1s a t o m s c a n e a si l y b e as l a r g e as 1014/cm3, t h e signal c a n be substantial even if t h e detection efficiency is poor. C. Previous measurements of the 1.5 lamb shift and limitations
In both previous meas u r emen t s of t h e 1s Lamb shift,2s3t h e frequency-doubled output of a pulsed d y e - l a se r oscillator-amplifier system15 was used t o excite t h e 1s-2s transition, and t h e excitation was detected by observing t h e Lyman-o! radiation emitted i n t h e collision-induced 2.5-1s decay.
w.
22 -
HANSCH
In the f i r s t experiment,‘ the 1s-2s interval w a s compared with t h e n = 2-4 inte rva l by sim ply recording a Doppler-broadened absorption of the Ba.lmer-/3 line in a glow disc ha rge plasma, using the fundamental dye -la se r output. The second measurement3 achieved ;Isubstantial improvement in a c c u r a c y by using the technique of sa tura te d a bsorption spe c trosc opy to obtain better resolution of the Ealnier-P line, again with a portion of the funda m e nta l dye -la se r beam. But despite sub-Doppler linewidths, the fine s t r u c t u r e of t h i s line re m a ine d pa rtly unresolved, and t h e quoted a c c u r a c y of the 1s La m b shift was s t i l l a lm ost e ntire ly limited by the inadequate resolution of this re fe re nc e line. I>. Laser polarization spectroscopy
Searching f o r ways to im prove the resolution of the Ba lm e r-P line, we developed t h e technique of laser polarization ~ p e c t r o s c o p y a, ~method of Doppler-free spe c trosc opy which offers considerably higher sensitivity than conventional sa tura te d absorption spectroscopy. This techniqlie enabled us to obse rve the Ralmer-P line with a low-power cw dye l a s e r in a mild glow discharge. Single Stark components of fine-structure line s could b e readily resolved in t h e s m a l l axial c l e c t r i c field of the disc ha rge plasma. This s u c c e s s opened the way f o r the pre se nt improved m e a sure m e nt of the 1.5 La m b shift. The b a s i c concept of polarization spectroscopy is r a t h e r simple: A line a rly polarized probe laser be a m is s e n t through a g a s s a m p l e and p a s s e s through a ne a rly c r o s s e d l i n e a r pola riz e r, be fore reaching a detector. Any optical anisotropy in the s a m p l e which changes t h e probe polarization can thus b e detected with high sensitivity. Such a n anisotropy is introduced by a second laser be a m of the s a m e frequency, which is counterpropagating and c irc ula rly polarized. In t h i s case, a s s u m i n g low intensities, one de te c ts a Doppler-free signa l as given by Eq. (4) of Ref. 4:
[
1 11, (I>” llxz]’
I=I, 62f6 --+
-
(5)
where I is t h e detected intensity, I , t h e incident probe intensity, and 8 is t h e rotation angle of the analyzing pola riz e r f r o m t h e perfectly c r o s s e d orientation. The frequency detuning f r o m the center of the Doppler-broadened line is de sc ribe d b y the normalized p a r a m e t e r x = (w w a b ) / y a b ,w h e r e w is the l a s e r frequency, wab is t h e tra nsition f r e quency, and yab is the na tura l linewidth of the transition. The p a r a m e t e r s gives t h e maximum relative intensity difference between t h e rightand left-circularly-polarized components of the probe and is defined by s =-$(1- d ) a. I I/Isatr w he re 2 i s t h e s a m p l e length, a,, is t h e unsaturated
-
20 22
P R E C I S I O N M E A S U R E M E N T O F T H E 1s L A M B S H I F T A N D O F ...
absorption coefficient at line center, Isatis the transition saturation parameter, as defined in Ref. 15, and d is the ratio of the light-induced changes in the absorption coefficient for right- and leftcircularly-polarized light. In the chosen mode of operation the analyzing polarizer is rotated s o that 4 8 s i s about 10 times larger than (s/4)', and a derivative signal is obtained by giving the lassr a small frequency modulation and recording the corresponding signal modulation with a phase sensitive amplifier. The resulting line shape, the derivative of x/(l + x ) , appears bell shaped with inverted wings, and its width is less than $ the natural linewidth.
LASER
INTERFEROMETER SHIFTER
/
,-
POLARIZER
I
'0w 7%
PMT
PULSED
nI. APPARATUS
A schematic overview of the apparatus is given by Fig. 2. The output of a single-frequency cw
dye laser (upper left) is split into four beams. One of these is sent into a frequency-marker interferometer (top) for the purpose of calibration. Two of the beams become the polarizing and probe beam of the polarization spectrometer (center). Finally, the fourth beam provides the input for a pulsed dye-laser amplifier system. The amplified beam is sent through a frequency-doubling crystal which generates the ultraviolet radiation for the two-photon spectrometer (bottom right). Because of its complexity, this entire setup is spread over three separate optical tables. A. cw dye-laser oscillator
The cw laser is a Spectra Physics model 375 folded cavity jet s t r e a m dye l a s e r which has been modified to provide tunable single-frequency operation at blue wavelengths, using a solution of 7-diethylamino-4-methylcoumarinin ethylene glycol. The standard tuning elements, a dielect r i c tuning wedge filter and a 0.1-mm uncoated quartz etalon a r e augmented by two additional intracavity etalons. The first is an angle-tuned solid uncoated quartz etalon 0.5 mm thick. The second is an a i r spaced etalon with a gap of 5 mm (free spectral range 30 GHz) and 30% m i r r o r reflectivity. The spacing can be varied with a piezotransducer, and the etalon is contained in a temperature stabilized oven. For cavity fine tuning the laser output m i r r o r is mounted on a piezotransducer. To decrease the linewidth the entire l a s e r is completely enclosed in a large, nearly airtight, wooden box. The dye laser is pumped by a Spectra Physics model 171 argon-ion laser. The pump threshold for single-frequency operation at 4860 is about 600 mW. An output of 20 mW i s obtained with 2-W pump power. During the experiment the dye l a s e r
195
AMPLIFIERS FREOUENCY DOUBLER
1- ABSORPTION TWO-PHOTON CELL
FIG. 2. Schematic of experiment. The Balmer-P line is recorded by polarization spectroscopy (center), while the 1s-2s transition is observed by two-photon excitation with the second harmonic of the dye-laser frequency (bottom).
operated typically at 7.5-mW output power. The dye-laser frequency can be continuously scanned over 4 GHz by applying tuning ramp voltages to the piezotransducers of the cavity end mirr o r and the air spaced etalon. During the experiment the linewidth was on the order of 5 to 10 MHz [full width at half maximum (FWHM)] , although the width decreased to about 2 MHz when the pneumatic vibration isolation of the optical table was activated. The kequency modulation (about 10-MHz sweep amplitude) needed to record a derivative signal is obtained by adding an audio frequency a c voltage (4 V, 2 kHz) to the ramp voltage of the cavity mirror. The dye-laser output beam is sent into an acousto-optic modulator. Diffraction from a traveling acoustic wave in a block of fused quartz generates three beams, one upshifted in frequency (+37.60 MHz), one downshifted (-37.60 MHz), and one with its frequency unchanged. The primary function of this device in our experiment is that of a beam splitter, but the frequency shift improves the signal-to-noise ratio of the polarization spectrometer, as will be discussed later. The upshifted beam, containing about 0.75 mW of power, is sent into the frequency-marker interferometer. The downshifted beam (also 0.75 mW) becomes the probe beam of the polarization spectrometer, while the unshifted beam is used as the polarizing beam. About 30% of this unshifted beam is split off with a partly reflecting m i r r o r and sent into the pulsed amplifier system.
21 196
C . WIEM.43' A N D T. W . HANSCII
22
B. Freqilencp-marker interferometer
D. Polarization spectrometer
The frequency-marker interferometer i s a s e m i confocal interferometer, consisting of two dielect r i c m i r r o r s cemented onto the ends of a quartz tube. It has a finesse of 1 5 , and i t s transmission peaks a r e separated by 113.142 33(15) MHz. This separation was used for frequency calibration in all reported measurements. The thermal drift of this frequency marker was 0.2 to 1.5 MHz/min and typically remained constant to better than 10% o v e r 5 h. The m a r k e r separation was determined to within 2 p a r t s in lo4, by mechanically measuring the length of the spacer. To find a more accurate value, the dye l a s e r was tuned to the center of a transmission peak, and the l a s e r frequency was measured with a precise fringe-counting digital wave meter.lG Then the l a s e r was tuned to another peak, about lo3 fringes away, and the frequency was measured again. From the frequency difference and the known marker spacing the number of fringes between the two peaks can be determined exactly, which in turn yields an improved m a r k e r separation. The procedure can then be repeated with a new, l a r g e r frequency interval. Taking several iterative steps in this manner, the separation between adjacent o r d e r s could be quickly determined to the quoted accuracy.
A simpiified scheme of the polarization spectrometer is included in Fig. 2. The polarizing beam is sent through a quarter wave plate to change its polarization from linear to circular. Two lenses of focal length f =7.8 cm form a 1 : 1 telescope (not shown in Fig. 2) which focuses the beam (original diameter 3 mm) to a waist diame t e r of about 0.2 mm at the center of the hydrogen discharge tube a t 75-cm distance. The probe beam, coming from the opposite direction, passes through an identical telescope to nearly match the confocal parameters of the polarizing beam. To ensure nearly perfect linear polarization the probe is sent through a prism polarizer before entering the discharge tube. It c r o s s e s the polarizing beam near the center of the discharge tube at an angle of 2 mrad. After emerging from the Wood tube, the probe is sent into a second, "analyzing" polarizer whose axis is rotated nearly 90 degrees relative to the first one. Both polarizers a r e cemented Glan Thomson prism polarizers (Karl Lambrecht) with a rated extinction ratio of By using small (about 2mm diameter) and fairly well collimated beams, however, we typically achieve extinction ratios better than During the experiment, the angle of the analyzing polarizer was adjusted s o that the 0s contribution to the signal'was about 5 to 10 times l a r g e r than the s2 contribution. This angle depended on the experimental conditions and varied between 4X and 3 X rad. Probe light which passes through the analyzing polarizer is sent through a spatial filter at 1.5-m distance from the discharge tube (lens of f=7.8 cm, pinhole of 50-pm diameter), and its intensity is monitored by a photomultiplier (RCA 1P28) followed by a lock-in amplifier (PAR model JB5). The internal sine wave reference of this amplifier provides the signal for the 2-kHz dye-laser f r e quency modulation. Although this apparatus can be assembled quite easily, certain pains must be taken t o attain the l a r g e signal-to-noise ratio offered by polarization spectroscopy. The first requirement i s a good extinction ratio for the probe beam. Considering the small rotation angle of the analyzing polarizer it i s obvious that a n extinction ratio much worse than would have been a serious limitation in the reported experiments. Several different meas u r e s a r e taken to a s s u r e this ratio: First, fairly good polarizers a r e used and the beams through them a r e kept small and collimated; second, t h e r e a r e a s few optical components as possible in between the polarizers, and for those which a r e unavoidable (the discharge windows) the birefringence
C. Hydrogen discharge tube
In o r d e r to observe the Balmer-P line in absorption, hydrogen atoms a r e excited in a Wood-type glow discharge tube, similar to those described in Refs. 6 and 9. The tube i s 138-cm long with an inner diameter of 1.5 cni, and the walls a r e coated with orthophosphoric acid to prevent catalytic r e combination of the atoms. Wet molecular hydrogen from a n electrolytic generator provides a continuous gas flow through the tube. A l a r g e a r e a cold aluminum cathode ensures a stable discharge. The l a s e r beams pass through a 60-em-long center section of the positive discharge column. The windows a r e formed by pieces of quartz microscope slides, cemented onto tube extensions with T o r r Seal adhesive. By gently squeezing these windows with small transverse clamps their birefringence can be reduced until extinction ratios better than a r e observed between crossed polarizers. During the experiment the tube was operated at p r e s s u r e s between 0.1 and 1.0 t o r r and at currents between 5 and 20 mA, and the absorption at resonance ranged from as high as 10% to l e s s than l%, depending on p r e s s u r e and current.
22 22
-
P R E C I S I O N M E A S U R E M E N T O F T H E 1s L A M B S H I F T A N D O F
is minimized; finally, all surfaces a r e kept clean to avoid depolarizing scattering. It is also important to minimize any light other than the probe light which reaches the detector. The two primary sources of such background in the present experiment a r e the gas discharge and backscattering of the polarizing beam. The spatial filter reduces the light from both these sources to below the residual probe intensity (10-710). Originally we were facing the problem, however, that the small amount of scattered polarizing light which still reached the detector was coherent with the probe light, causing the detector to respond to an interference between the two electric fields. Slow phase fluctuations resulted in considerably increased noise. The use of the acousto-optic frequency shifter eliminates this problem by causing the interference term to oscillate at a frequency too high for the detector to respond to it. We found that vibrating one of the m i r r o r s reflecting the polarizing beam, for instance, by mounting it on a radio speaker, also eliminates this noise source. The frequency shifter i s preferred, however, because it also reduces amplitude fluctuations of the dye laser due to feedback into the cavity, a problem to which this dye l a s e r is particularly sensitive. E. Pulsed dye-laser amplifier
The pulsed laser amplifier is shown in Fig. 3. The system worked at first t r y more than adequately for this experiment, and little effort was made to optimize the geometry. The pump source is a Molectron W 1000 nitrogen laser operating at 15 pulses per s e c and producing 10-nsec-long pulses of about 650-kW peak power. Approximately 10% of t h i s light is used to pump the first dye amplifier stage, 30% is sent to the second, and 60% to the third stage. The dye cells and the transverse focusing of the pump light into them a r e identical to those used in the pulsed oscillatoramplifier forerunner of this With 1-mW cw input power a gain of about lo5 in peak power is readily achieved in the first stage, but the pulse width is only about 3-4 nsec (FWHM). The power of the amplified beam after the first stage is about 5 to 10 times larger than the amplified spontaneous fluorescence into the same solid angle. The second stage gives a (saturated) gain of about 45 and some pulse stretching, while the third stage gives a gain of 22 and stretches the pulse to about 7 nsec (FWHM). The third stage is saturated to such an extent that it produces the same peak power under almost any alignment condition. Optimizing the alignment broadens the roughly Gaussian-shaped pulse in time, however, and reduces the bandwidth accordingly, as we
TO
...
197
DOUBLING CRYSTAL
f
IMW NITROGEN LASER
25cm
\DIAPHRAGM AMPLIFIER
I
FIG. 3. Pulsed dye-laser amplifier.
found empirically. Pulse energies of about 5 mJ with less than 5% fluctuations are easily obtained, and the spatial beam profile can show a much closer resemblance to a Gaussian TEM,, mode than was normally attained with the earlier pulsed 0s cillato r -amplifi e r system. In hindsight the most serious problem in the present experiment is the fact that the pulsed amplifiers not only broaden but also shift and distort the spectrum. To monitor this shift, a portion of the amplified output beam is sent through a confocal interferometer (2-GHz f r e e spectral range, finesse 150) as indicated in Fig. 4. The transmitted light is measured by two photodiodes; one of these records the pulsed light while the other registers only the simultaneously present collinear cw light. Thus, as the l a s e r is scanned over the transmission peak of the interferometer, the pulsed and cw spectra are obtained simultaneously and can be compared. F. Two-photon spectrometer
Also shown in Fig. 4 are details of the twophoton spectrometer which is almost identical to that used in the earlier experiments.2s3 The output of the pulsed amplifier is focused into a lithium formate frequency-doubling crystal (Lasermetrics) using a lens of f = 1 8 cm. The crystal is 1 cm long and phase matching near 4860 is achieved through angle tuning. With this geometry
23 22
C . W I E M A N A N D 1'. W . H A N S C H
198
'
I 55cm
I
1
\ L I T~. HIUM
FORM ATE FREQUENCY DOUBLER
-
expect a 2s lifetime of about 10 nsec a t a p r e s s u r e of 0.1 t o m , which s e e m s t o a g r e e with our (imprecise) observations. However, t h e r e i s substantial radiation trapping and l o s s of photons through nonradiative decay mechanisms which a r e not well understood. To minimize this loss of signal the ultraviolet beams a r e kept as close to the side window a s possible (distance l e s s than 1 mm). The emerging Lyman-or photons a r e detected with a s o l a r blind photomultiplier (EMR 5415). Scattered l a s e r light is reduced to a negligible level by a Lyman-a interference filter in front of the detector (Matra Seavom Co., 15% peak transmission, 90 I% bandwidth). The output current from the photomultiplier is sent into a gated integrator. G. Data processing
BEAM FROM PULSED A M P L I F I E R SYSTEM
FIG. 4. Details of two-photon spectrometer.
A l l data from the experiment, a r e converted into digital form (12 bits accuracy) and stored on magnetic disk for processing with a minicomputer (Hewlett - Packard 2 1OOA). IV. EXPERIMENTAL PROCEDURE AND ANALYSIS
the doubling efficiency approached 2%, and most data were taken at pulse energies of about 7 pJ. The crystal showed signs of burning after about 15 000 pulses at such a power level. (With tighter focusing it was possible t o obtain over 7% conversion efficiency, but the crystal burned much m o r e rapidly.) The generated ultraviolet beam i s separated f r o m the fundamental beam by a Brewster-angle quartz p r i s m and then collimated with a Suprasil lens (f=1000 mm). The ultraviolet intensity is monitored by observing the light reflected off this lens with a photomultiplier with attenuating diffuser. A s it p a s s e s through the hydrogen absorption cell the beam has a nearly rectangular c r o s s section of 0.2 by 0.4 mm. A flat m i r r o r immediately a f t e r the cell is used t o reflect the beam back onto itself t o provide the required standing-wave field. The two-photon absorption cell is the s a m e as described in Refs. 2 and 3. The ground-state hydrogen atoms a r e produced in a Wood-type discharge and flow and diffuse into the absorption cell through a 25-cm-long folded transfer tube, coated with phosphoric acid. The discharge was run with a current of 20 mA and at p r e s s u r e s between 1.0 and 0.05 t o r r of wet hydrogen. The absorption cell is about 10 c m long with quartz windows f o r the ultraviolet beams on the ends s e t at the Biewster angle. The 1s-2s excitation is monitored by observing the Lyman-a! radiation emitted through a magnesium fluoride side window. The vacuum-ultraviolet radiation i s due t o collisional mixing of the 2.5 and 2P states. F r o m measurements of the collision cross sections" we
A major portion of our efforts were devoted to the investigation of systematic line shifts, as summarized in Table I. The experimental procedure was divided into five distinct portions; the investigation of systematic shifts of the hydrogen two-photon line, the measurement of the s e p a r ation of hydrogen and deuterium two-photon lines, the study of shifts of the Balmer-P reference line, the measurement of the separation between the hydrogen two-photon line and the Balmer-/3 reference line, and finally, the measurement of the hydrogen-deuterium separation of the Balmer-/3 line. The result of the first two p a r t s gives the H-D 1s-2s isotope shift while the first, third, and fourth part yield the hydrogen 1s Lamb shift. The deuterium 1s Lamb shift is obtained by combining results of all five portions. A. Systematic shifts of the 1s-2sline
The only systematic shift of the two-photon signal studied experimentally was that due t o the p r e s s u r e in the two-photon absorption cell. First the pressure was set at 0.05 t o r r and the l a s e r was scanned -5 times over a n -I-GHz range containing the resonance line. During these scans the computer sampled the inputs to six channels on the analog t o digital converter 100 t i m e s p e r sec. Each sweep lasted -25 s e c so the sampling points were about 0.25 MHz apart. The six data inputs were the two-photon signals, the intensity of frequencydoubled (2430-A) light, the frequency-marker signal, the pulsed signal f r o m the spectrum analyzer monitoring the amplified beam, the cw signal from
24 P R E C I S I O N M E A S U R E M E N T O F T H E 1s L A M B S H I F T A N D O F
22 -
...
199
TABLE I. Systematic line shifts considered. a
-
1s-2s two-photon resonance
pressure a c Stark effect laser line shape Balmer-j3 line (4S112-2S1 /,-4P1 polar iz er angle polarizer angle pressure, electric field zero-pressure Stark shift and unresolved hyperfine structure ac Stark effect discharge current frequency-marker thermal drift
meas calc meas/calc
MHz/torr MHZ/MW cm-' of linewidth
0 +5 +0.6
-13
*9%
15 12 -28
*4
crossover) meas caIc meas
MHz/sB" MHz/sB-' MHz/torr
+6
-4.5 * 2 -0.28
calc calc meas meas
MHz MHZ/W cm-2 MHz/niA AfHz/h
-0.1 i: 0.2 60 t l
aShift v (observed) - v (true) in terms of fundamental dye-laser frequency. bGood approximation for 1 sO-'l< 0.2, as in all measurements. See Eq. (5). 'These results a r e not independent. See text for discussion. dTypical value for a given run.
that analyzer, and last, the scanning voltage applied to the cw laser end m i r r o r piezodrive. All pulsed signals were processed by gated integrators. The output proportional to the pulse a r e a was held constant in between laser pulses. After recording these scans the pressure was changed to 1.0 t o r r and 5 more scans w e r e made with the same recording procedure. This was followed by 5 scans again at 0.05 t o r r and then 5 at 1.0 torr. The entire procedure took about 30 min. The first step in the analysis of these data was t o set the frequency scale for each scan by finding the centers of the frequency-marker peaks. For each peak t h e computer generated a smoothed curve where each point of the smoothed curve was the average of 10 adjacent raw data points. For example, the 95th point on the smoothed curve was the average of the 91st through 100th raw data points. The line center for the smoothed curve was found at the Q, and $ peak-height points and the average of these points was taken t o be the frequency-marker center. Although there was essentially no uncertainty in finding the center of a given peak, imperfections in the laser piezodrive caused a 2-3% random fluctuation in the peak-to-peak spacing causing a corresponding uncertainty of the frequency scale. All subsequent analysis in this and the remaining part of the experiment was done in terms of frequency scales determined using this procedure. The two-photon spectra were first normalized by dividing the signal by the square of the 2430-A intensity. This caused no observable shift of the lines, but did reduce the pulse to pulse amplitude fluctuations slightly. Next the spectra were
a,
smoothed in the same manner as used with the frequency marker except that a 40-point average was used. While such a smoothing routine could conceivably cause the peak frequency to shift, for our lineshapes the shift would be less than 0.4 MHz and hence negligible compared to the statistical uncertainty. Examples of such spectra are shown in Figs. 5 and 6(b). After smoothing, the centroid of the line including both hyperfine components was calculated. The values for each set of 5 runs were averaged and the resulting four points (at 0.05, 1, 0.05, and 1 torr) were plotted as a function of the time of acquisition t o determine the thermal drift of the frequency marker. The drift rates shown by the high- and low-press u r e points were the same, and an appropriate correction was made. B. Hydrogendeuterium IS-2s isotope shift
The next stage of the experiment was the measurement of the 1s-2s isotope shift. F o r this, the
tI
1
0
1
1
0.2
1
1
1
1
'
!
-
0.4 0.6 168.0 DYE-LASER FREQUENCY TUNING
,
I
168.2
*
(GHz)
FIG. 5. Two-photon spectrum of 1s-2s transition in hydrogen (left) and deuterium (right).
C . WIEMAN AN D T. W. HANSCH
200
I
4800
I
I
4600
I
I
4400
I
-
4M)
t
I
iS - 2 5 POLARIZATION
SPECTRUM
I
I
200 0 FREQUENCY I M H z )
TWO-PHOTON SPECTRUM
FIG. 6. (a) Portion of the polarization spectrum of the hydrogen Balmer-fi line. @) Two-photon spectrum of hydrogen 1.9-2s transition. The frequency-marker signal at the top has been recorded simultaneously.
two-photon absorption cell was filled with a mixture of H and D at 0.05 t o r r . A crude preliminary measurement using a digital wave meter w a s made to determine the separation of the hydrogen and deuterium lines to within one free spectral range of the frequency marker. The following procedure was then used to determine the additional fraction. Several (4 to 5) scans of the hydrogen two-photon line were made and the spectra stored as previousl y described and then the l a s e r frequency was shifted t o the deuterium line and the procedure repeated. The laser wavelength was returned to the hydrogen line and the sequence repeated until a total of 52 spectra were obtained. The data were analyzed by finding the frequency scale and normalizing and smoothing the twophoton spectra just as before. At t h i s stage in the analysis however, it was necessary to confirm that the deuterium and hjjdrogen line shapes were not systematically different (except for the difference in hyperfine splitting, of course). Because the two-photon line shape was not simple and could change noticeably over periods on the order of a day such a confirmation that it did not change with wavelength was necessary. This was verified by two tests. First, spectra of the fundamental pulsed l a s e r taken at the wavelength of the hydrogen line, the deuterium line, and -100 A on either side were compared and found to be identical. While it is impossible to make a direct correspondence between this spectrum and the twophoton line shape, as will be discussed later, it was observed that changes in one were always correlated with changes in the other. The second test was to compare the two two-photon line shapes directly, a procedure which was complicated by the deuterium hyperfine splitting being
22 -
only partially resolved. This problem was s u r mounted by taking the line shape observed for a single hydrogen hyperfine component and using it to generate a predicted deuterium spectrum using the known hyperfine splitting and intensity ratio for deuterium. The generated line shape was then compared with the observed deuterium spectra obtained at similar times using only the overall amplitude as a free parameter. These were found to match to within the statistical fluctuations, confirming that the hydrogenand deuterium line shapes were the same. Now the centroids of the lines were calculated and these values plotted as a function of time. The drift of the frequency marker was then calculated and appropriate corrections made. C. Systematic shifts of Balmer-0 line
The next and longest portion of t h e experiment was the investigation of systematic shifts of the Balmer-6 signals. The basic procedure was to repeatedly scan the cw dye laser to obtain a polarization spectrum containing the 2S,,4P,,, line, the 2S,, 4S,, line (Stark allowed), and their crossover, l9 while changing various parame t e r s [see Fig 6(a)]. We restricted ourselves to these lines because they a r e narrow and much simpler to interpret than the other components. During each scan the computer sampled and stored the inputs to three channels on the analog-todigital converter (ADC) at 100 times/sec. The three inputs were the l a s e r scanning voltage, t h e frequency-marker signal, and the output of the lock-in amplifier monitoring t h e polarization spectrometer phototube. The first systematic effect studied was the shift of the polarization spectra line centers due to the addition of the small derivative-shaped S0 t e r m to the larger bell-shaped S2term. To measure t h i s shift several scans were made with the polarizer in the + 0 orientation then it was rotated to - 8 , which causes a shift in the opposite direction, and several more scans were recorded. This was r e peated at several times to determine the frequency-marker drift, and at a variety of angles. The following analysis procedure of the polarization spectra was used for this and all subsequent portions of the experiment. First t h e spect r a were smoothed in the standard way. Between one and ten points were used in the smoothing average, depending on experimental conditions, with tests made to ensure this introduced no line shifts, The line center was then determined at regular height intervals (every 5% of the 2S,, - 4 P y , line and every 10% of the crossover) and the average of these centers found. The result of the tests of the 0 dependence
-
26 22 -
PRECISION MEASUREMENT OF THE
showed that f o r all values of 0 used in the subsequent measurements the shift was l e s s than 3 MHz, i n agreement with the theoretical estimates. A s part of all subsequent measurements the data were taken with the polarizer in both + and - orientations and the results averaged, thereby canceling the shift to within a negligible uncertainty. The effects of the discharge environment (cur rent, pressure, electric field) on the Balmer-P reference line were also investigated. First the effect of the current was studied by recording a s e r i e s of spectra at discharge currents of 20 and 11 mA, alternating repeatedly between the two settings. Next the effects of the pressure and the electric field in the discharge were measured. Because only the pressure could be independently varied and this in turn affected the electric field it was necessary to obtain a composite correction. In these measurements the same data-acquisition procedure was used as for the current shift measurements, except that the current was set at 11 mA and the pressure varied. Two data runs were used; one which took data at 0.16, 0.41, and 0.70 t o r r and the second at 0.39, 0.27, and 0.72 torr. The pressure readings were made using a thermocouple gauge. A total of 80 spectra were recorded. D. Comparison of 1s-2s transition with Balmer-0 line
The fourth stage of the experiment was the measurement of the separation between the hydrogen two-photon peak and the crossover line between the 2SII2- 4pl12and the 2SlI,- 4S,,, transitions. This crossover w a s chosen as the reference line for the Lamb-shift measurement because it was l e s s sensitive to the electric field perturbation than the other lines. In this stage seven data channels were read into the computer; the combination of the three cw and four pulsed data lines previously described. The two-photon absorption cell was operated at 0.05 t o r r and the polarization spectrometer discharge ran at 0.27 t o r r and 11 mA. I-GHz scans with the same sampling rate as before were used. First, 5 scans were made over the twophoton line; then as rapidly a s possible (-1 min) the l a s e r wavelength was reset and the Balmer-0 crossover line was scanned several times. Examples of such spectra a r e shown in Fig. 6. The l a s e r was then reset to the two-photon line and the procedure repeated until a total of 45 twophoton spectra and 51 Balmer-P spectra were obtained. The Balmer-P spectra were analyzed using the same procedure a s before, but it was necessary to use a different procedure in the analysis of the two-photon lines because of their asymmetric line shape. The difficulties caused by t h i s line shape
1s
L A M B S H I F T A N D O F ...
20 1
will be discussed in Sec. V. After normalizing and smoothing the two-photon spectra as before, the line centers were found at equal intervals in the region between 30 and 90% of the peak height, This region was used because at lower than 30% the peak is dramatically skewed towards lower frequency, while above 90% the amplitude fluctuations make a center meaningless. The 10 cent e r points could usually be fit with a straight line to within statistical fluctuations. This line w a s extrapolated to the pe& maximum and that point taken to be the “peak” frequency. This value w a s typically about 5 MHz higher in frequency than the line center at the 50% point. Once these centers were found the data were averaged and graphed a s before. Since this data run lasted several hours the frequency-marker drift was fit with a secondorder polynomial (the second-order correction was about the same size a s the statistical spread), and the data corrected. E. Hydrogen-deuterium Balmer-0 isotope shift
The last quantity measured was the isotope shift between the hydrogen and deuterium Balmer -p lines. For this the polarization spectrometer discharge was operated with a 50-50 mixture of hydrogen and deuterium. Polarization spectra of the 2S,,4SS1/,, 2SlI,- 4P,,, , and crossover region were recorded in the usual way alternating between the hydrogen and deuterium lines. This was done at several discharge pressures to ensure that the two had the same systematic shifts. The spectra were analyzed in the standard manner. V. RESULTS A. Systematic shifts of the 1s-2s line
In this and all subsequent sections, numerical results are given in t e r m s of the fundamental dyel a s e r frequency unless otherwise stated. All systematic line shifts a r e given as v (measured)-u (true). The 1s-2s pressure shift was determined to be less than 5 MHz/torr, and since the isotope and Lamb-shift measurements were made at 0.5 t o r r this caused a negligible correction and uncertainty to the result. The ac Stark effect“ was estimated by measuring the 2430-A beam size (0.02X 0.04 cm’) and peak power (0.9-1.1 kw) and using t h e result in Ref. 21 of 0.6 Hz/W/cm shift. This gave a shift of 0.8 f 0.2 MHz. The largest uncertainty in the results is caused by t h e difficulty to relate the line shape of the twophoton signal to the t r u e atomic transition frequency. The atomic linewidth of t h e 1s-2s transi-
27 20 2
c.
WIEMAN A N D T.
tion can certainly be neglected compared to the l a s e r bandwidth. In linear spectroscopy the observed line shape would then simply be given by the laser intensity spectrum, which can be readily meascred with a spectrum analyzer. For twophoton excitation, however, the calculation of the line shape, P (d), involves the convolution of the field spectrum,
where E ( o ) is the component of the electric field of the l a s e r at frequency w. A serious problem arises now, because this integral depends on the relative phases as well a s the amplitudes of the field components, whereas an optical spectrum analyzer can determine only the intensity o r amplitude. In the present experiment the situation is aggravated because there a r e two nonlinear processes, frequency doubling and two-photon excitation, which can each introduce uncertain line shifts and distortions between the measured l a s e r intensity spectrum and the observed two-photon signal. If the relative phases in the amplitude spectrum were truly arbitrary one could create almost any signal line shape. But fortunately the possible phase variations in the present experiment are quite limited by the origin of the l a s e r pulse. We have used various models for these phase v a r i ations, together with the measured laser intensity spectrum, to numerically calculate possible twophoton line shapes. By comparing the calculated and observed line shapes we have determined which model gives t h e best agreement and have used this model to estimate a correction to the observed transition frequency. We have also considered which models correspond to realistic extremes for the possible phase variations, and have used the corresponding corrections to set our uncertainty. The interferometer in Fig. 4 measures the pulsed l a s e r intensity spectrum and relates it to the cw laser frequency. For an ideal laser amplifier t h i s spectrum would be Gaussian, with a Fouriertransform-limited linewidth determined by the pulse length. In this case, there would be no phase variation and t h e peak frequency would be the same as that of the cw laser. The observed spectrum, however, appeared downshifted in its peak frequency by 17 MHz, the linewidth was almost twice as large as the transform limit, and the line shape appeared slightly asymmetric with a small but long tail on the low frequency side. This would imply a shift of the measured 1s-2s frequency of -68 MHz. If the phases varied rapidly and randomly from one frequency interval to the next, the square of
w.
HANSCH
22 -
the convolution integral6 could be replaced by a convolution integral of the intensity spectrum. In this limit there would be no shift between the atomic frequency and the peak of the observed signal. The line profile calculated for this case, however, was 20% narrower and more symmetric than our observed line. Next we investigated the (also quite unrealistic) limit of zero phase variations across the amplitude spectrum, given by the square root of the intensity spectrum. In this case the calculated line profile was drastically skewed, due to the long tail in the spectrum, and had little resemblance to the observed line. Since the long tail is, in all likelihood, due to some brief, rapid frequency chirping, it seemed more realistic to remove t h i s tail and to assume constant phase only for the remainder of the amplitude spectrum before compting the convolution integral. This approximation reproduced the observed line shape f a i r l y well, although the calculated Iinewidth w a s about 10% too large. The predicted resonance peak had a lower frequency. A s a case between these two limits we continued to assume a constant phase in the integrand of the convolution integral, but restricted the integration limits. The rationalization for this is that f r e quency components separated by much more than the Fourier-transform-limited width should be uncorrelated. The calculated peak shape and position is fairly insensitive to what is used as the central frequency of the integration. It matches t h e observed shape quite well when our limits a r e about 2.5 Fourier-transform-limited linewidths on either side of the frequency of maximum amplitude. In t h i s case the predicted shift of the 1s-2s f r e quency is -40 MHz. Because our approach cannot be entirely rigorous we have cautiously chosen e r r o r bars which more than cover the discussed limiting cases, assuming a shift of -40 f 28 MHz. The uncertainty of this shift is much larger than the &3-MHz statistical limit. Of course, the corrections of the observed fundamental frequency a r e one quarter as large. B. Systematic shifts of the Balmer-5 line
The largest correction to the position of the Balmer-P reference line is that due to pressure and electric-field effects. The positions of the %-, 4 P I I ,line and the 4S,I,-2S,,,-4P,, crossover both changed linearly with the pressure to within the statistical accuracy (-1 MHz), although the shifts were due to both pressure and electric field. Taking advantage of that behavior we have extrapolated their positions to zero pressure. The shift from 0 t o 0.27 torr is -17.0i 3.4 MHz for t h e 2S,,-4P,, peak and -7.6-+1.6 MHz for the
28 22 -
P R E C I S I O N M E A S U R E M E N T O F T H E 1s L A M B S H I F T A N D O F ...
crossover, with the uncertainty primarily due to the inaccuracy of the pressure readings. The residual “zero -pressure” electric field can be found by comparing the extrapolated separation between the two peaks with theoretical calculations. Knowing this field w e can determine the residual Stark shift of the crossover line. Because the separation between the two peaks i s approximately 4 times more sensitive to the electric field than the crossover position, the correction of the latter should have very little uncertainty. One difficulty arises, though, because both the 4Sl1, state and the 4Pl,, state possess unresolved hyperfine structure, and we do not know with certainty to what extent the hyperfine interaction is decoupled by collisions in the discharge plasma. Weber and Goldsmith” have found in later investigations of the hydrogen Balmer-a line that the hyperfine splitting of the 3Pl1, state is almost completely destroyed a t helium pressures above 0.1 torr. Unresolved hyperfine splitting is of little consequence in linear spectroscopy where it does not shift the center of gravity of a spectral line. In polarization spectroscopy, however, we expect, in general, some line displacement due to unresolved hyperfine structure, and collisional decoupling can lead to measurable line shifts. In order to consider two limits we have calculated the Stark shift n = 4 energy levels as a function of electric field by numerically diagonalizing the HamiltonianZ3without and with inclusion of hyperfine interaction. A comparison of calculated polarization spectra with the extrapolated observed separation between the two peaks yields a zero-pressure electric field of 4.3 *O.l V/cm if hyperfine interactions can be ignored. The corresponding Stark shift of the crossover position
TABLE 11. Separation A v = v
amounts to -2.6k0.2 MHz. If the hyperfine splitting were fully present the calculated zero-press u r e field would equal 4.6 0.2 V/cm and the combined Stark and hyperfine s h i f t of the crossover would be -6.5+0.2 MHz. The e r r o r bars in the final analysis a r e chosen large enough to span both limits. Including the pressure shifts discussed earlier we then a r r i v e at a total shift of - 1 2 . 2 i 2 . 6 MHz for the position of the crossover line. The a c Stark effect at a beam intensity of 5 + 2 W/cm’ causes a calculated additional shift of -1.4 0.5 MHz of the crossover position. No shift of the line with discharge current between 11 and 20 mA was observed within the statistical e r r o r limits.
*
*
C. Hydrogen 1s lamb shift
The hydrogen ground-state Lamb shift is found from the separation between the Balmer-0 c r o s s ) and t h e over line (4S1,,F=,,1-2S11ZF=1-~11zF=o,1 ISF = = two-photon line, recorded at the fundamental dye -laser frequency. Under the experimental conditions described in Sec. IV D we measure a separation A v = u (HB crossover) - v(lS-2S)/4 of 4760.4* 1.2 MHz. After applying the systematic corrections summarized in Table 11, we a r r i v e a t a true separation of 4764.8 i 7.6 MHz. The theoretical separation, including all t e r m s (7)except for the 1s Lamb shift, is 2’727.0 MHz. From a comparison, we find a 1s Lamb sh i f t of 8151 *30 MHz, which is in excellent agreement with the theoretical value‘ of 8149.43*0.08 MHz. The experimental uncertainty is dominated by the frequency shift introduced by the pulsed dye-laser amplifiers
.
(H,crossover) - v (1S-2S)/4.
Raw measurement (statistical average from 45 two-photon spectra at 0.05 torr, 51 Balmer-P spectra at 0.27 torr, 11 mA)
4760.4 f 1.2 MHz
Systematic corrections (a) 1.9-2s line laser line shape ac Stark effect (b) Balmer-P line pressure, electric field, hyperfine structure ac Stark effect
total correction corrected separation AV
203
-10.0 +0.8
* 7.0 MHz * 0.1
12.2 f 2.6 MHz 1.4 * 0.5 4.4 * 7.5 MHz 4764.8 i 7.6 MHz
29 22 -
C . W I E M A N A N D T . W . HANSCH
204
D . Hydrogen-deuterium IS-2sshift
The IS-2s isotope shift could be measured with much l e s s systematic uncertainty because the twophoton line shape was the same for hydrogen and deuterium. We found an experimental separation of 670992.3 6.3 MHz between the two line centroids. This result is in good agreement with the theoretical prediction7 of 670 994.96 i 0.81 MHz and is about ten times more precise than that of the best previous experimental value.3 This r e sult is of interest in, its own right because it gives the first confirmation of the predicted relativistic nuclear recoil corrections [Eq. ( 2 ) ] .
*
E. Balmer-fl isotope shift and deuterium IS lamb shift
The isotope shif t between the 2 S U Z F= 4F’1,2F - 41 component of the Balmer-0 line was measured to be 167 783.9 i 1.2 MHz. No systematic dependence on pressure o r voltage could be detected, since both isotopes a r e affected in nearly the same way. A correction of +0.7*0.7 MHz has been included, though, to allow for a difference in unresolved 4 P hyperfine splitting, as discussed in Sec. IV B. The result agrees with the theoretical value? of 167 783.7+0.2 MHz. The Balmer-P isotope shift, together with the 1s-2s isotope shift and the hydrogen 1s Lamb shift has been used to derive a n experimental deuterium 1s Lamb shift of 8177 30 MHz in agreement with the theoretical result? of 8172.23i0.12 MHz
*
before been tested. The techniques used offer the possibility of considerable improvements in both these measurements. In traditional excited-state Lamb-shift measurements, the broad P state linewidth has always limited the possible precision. Our approach does not have t h i s limitation. With purely technical improvements, such as better l a s e r s and hydrogen atomic beams it could far surpass the precision of present measurements of any Lamb shift.
It should, for instance, be possible t o achieve a Balmer-j3 reference linewidth as narrow as 1
MHz by observing the 2s-4s dipole-forbidden line component in a two-step process (one optical photon +one radiofrequency photon),24o r simply by single-photon transitions in the presence of a weak electric field. The two-photon linewidth could be reduced by using a l a s e r amplifier of longer pulse duration, o r by using the technique of multiple pulse excitation.Z5*2G It would be particularly interesting to improve the 1s-2s isotope s h i f t measurement because the present theoretical uncertainty is limited by the electron-to-proton-mass ratio, and thus an improved experiment could be used to obtain a better value for this ratio. ACKNOWLEDGMENTS
Using laser spectroscopic techniques we have measured t h e 1s Lamb shift in hydrogen and deuterium with improved precision and we find good agreement with theory. We have also improved the measurement of the 1s-2shydrogen-deuterium isotope shift and find it too agrees with theory; it agrees, in particular, with t h e predicted relativistic nuclear recoil correction which h a s never
W e are indebted to Dr. John E. M. Goldsmith for a computation of the hydrogen Stark effect, taking into account hyperfine interactions. We are also grateful t o Professor Arthur L. Schawlow for his stimulating interest in this work, and we thank Frans Alkemade and Kenneth Sherwin for skilled technical assistance. Finally we would like to thank Dr. Dirk J. Kuizenga for lending us a superb acousto-optic frequency shifter of his design. This work was supported by the National Science Foundation under Grant No. NSF 9687, and by the Office of Naval Research under Cont r a c t No. ONR N00014-78-C-0403.
* P r e s e n t a d d r e s s : Randall Laboratory of Ph y sics, Univ e r s i t y of Michigan, Ann Ar b o r , Michigan 48105. lW. E. Lamb, Jr. and R . C . R eth er f o r d , Phys. Rev. 79, 549 (1950). *T. W. Hznsch, S. A . L e e , R. Wallenstein, and C. Wieman, P hys . Rev. Lett. 34, 307 (1975). 3S. A . Lee, R. Wallenstein, and T. W. Hlnsch, Phys. Rev. Lett. 35, 1262 (1975). 4C. W i e m a n a n d T. W. Hansch, Phys. Rev. Lett. 36, 1170 (1976). 5G. H e rz be rg. Proc. R. Soc. London, Ser. A S , 516 (1956).
6T. W. Hznsch, M. H. Nayfeh, S . A. Lee, S. M. C u r r y , and I. S . Shahin, Phys. Rev. Lett.32, 1336 (1974). ‘G.W. Erickson, J. Phys. Chem. Ref. Data j, 831 (1977). ‘T. W. Hansch, in Tunable L a s e r s and Applications, edited b y A . Mooradian et a1 Sp r i n g er Series in Optical Sciences (Springer, New York. 1976), Vol. 3, p. 326. ’T. W. H h s c h , Phys. T o d a y z , 34 (1977). ‘ON.Bloembergen and M. D. Levenson, in High Resolution L a s e r Spectroscopy, Topics in Applied Ph y si cs (Springer, New York, 1976), Vol. 13, p. 315. “L. S. Vasilenko, V. P. Chebotaev, and A. V. Shtshaev,
.
VI. CONCLUSIONS AND FUTURE IMPROVEMENT
.,
30 22 -
PRECISION MEASUREMENT OF THE
P i s ' m a Zh. Eksp. Teor. Fiz. 12,161 (1970) [ J E T P Lett, 1 1 3 (1970)). "Y. Gontier and M . T r a h i n , Ph y s. Lett. H,463 (1971). 13F. Ba s s a ni, J. J. Fo r n ey , and A . Qu attr o p an i, Phys. Rev. L e tt. 1070 (1977). 14T. W. H&sch, in Proceedings of the Intrmational SchooE of Physics "Enrico Fermi,'' Course L X I V on Nonlinear Spectroscopy, Varenna. Italy, 1975 (NorthHolland, New York, 1977), p. 17. "T. W. Hgnsch, M. D. Levenson, and A . L. Schawlow, Phys. Rev. L e t t . 2 , 946 (1971). 16F. V. Kowalski, R. T. Hawkins, and A . L. Schawlow, J. Opt. SOC.Am. E, 965 (1976). IrR. Wallenstein and T . W. H&sch, Opt. Commun. 14, 353 (1975). '*W. L. F i t e s , R . T. Braclunan, D. G. H u m m e r , and 363 (1959); R . F. Stebbings, Phys. Rev. 9,
12,
2,
124,
1s
LAMB SHIFT A N D OF
...
205
2051 (1961). ''T. W. Hgnsch, I. S. Shahin, and A . L . Schawlow, Phys. Rev. Lett. 7 ,707 (1971). ''A. M. Bonch-Bruevich and V. A . Khodovoi, Usp. Fiz. N a u k E , 71 (1967) [Sov. Phys: Usp. 1(1, 637 (1968)l. 'IS. A . L e e , Ph.D. t h e s i s , Stanford U n i v er si t y , M. L. R e p o r t No. 2460 (1975). "E. W. Weber and J. E. M. Goldsmith, Ph y s. Lett. 95 (1979). 23G. U r d e r s , Ann. Phys. (Leipzig) 2,308 (1951). 24E. W. Weber and J. E. M. Goldsmith, Phys. Rev. Lett. 41, 940 (1978). Teets, J. N. Eck st ei n , and T. W. H h s c h , Ph ys . Rev. Lett. 3, 760 (1977). 26J. N. Eck st ei n , A . I. Fer g u so n , and T. W. H s s c h , Ph y s. Rev. Lett. 40, 847 (1978).
B,
's
480
OPTICS LETTERS / Vol. 7, No. 10 / October 1982
Laser-frequency stabilization using mode interference from a reflecting reference interferometer C. E. Wieman and S. L. Gilbert Department of Physics, University of Michigan, Ann Arbor, Michigan 48109 Received May 17,1982 We present a new method for locking the frequency of a laser to a reference-interferometer cavity. For a nonmode-matched input beam, the light reflected off a cavity contains an interference between the wave fronts corresponding to the various cavity modes. A detector placed a t the proper position on the interference pattern provides a signal proportional to the imaginary component of the reflected field. As a function of laser frequency, this signal is dispersion shaped and can be used as the error signal for electronic frequency stabilization.
The frequency stability of a single-mode laser can generally be improved by using an electronic servo system to lock it to the resonance of a passive reference The most important component of this procedure is the character of the error signal that is fed back to the laser. In this Letter we present a simple new method for obtaining an error signal that is proportional to the 90" -phase-shifted or relatively imaginary component of the field reflected off the cavity. This signal is obtained by detecting the interference between the wave fronts in the reflected beam corresponding to resonant and off-resonance modes of the reference cavity. Traditional locking techniques have used the amplitude of the signal transmitted through the interferometer. One popular approach' is to lock on the side of the transmission peak, but this method is severely limited by the problem that a frequency jump of only half of the peak width can throw the system out of lock. A second traditional approach2is to modulate the frequency of the cavity or the laser and do phase-sensitive detection. One obtains the derivative of the Airyfunction transmission peak and uses it to lock to the center of the peak. This method, as well as requiring modulation and phase-sensitive detection, inherently limits the response time of the feedback. It is also poor a t recovering from large frequency jumps because the signal decreases rapidly as the frequency moves more than a linewidth away from the resonance. Many years ago Pound,3 in working with stabilization of microwave oscillators, realized that the amplitude of the imaginary component of the field reflected off a cavity has nearly ideal characteristics for a feedback signal. As the input frequency varies, this function passes through zero rapidly and has long wings, which aid in recovering from sudden large changes. Recently, this fact was rediscovered in the context of laser stabilization. Two methods for observing this imaginary component and using it for laser stabilization have been demonstrated. Hall and Drever et d 4used the method of putting modulation sidebands on the laser and looking at the reflected beam with phase-sensitive de-
tection at the sideband frequency. This technique, although quite successful, has the practical disadvantage that the need for high-frequency sidebands on the laser and high-frequency detection and signal processing add considerable technical complexity. Hansch and Couillaud5have presented an alternative technique in which a polarizing element is inserted in the reference cavity and the ellipticity of the polarization of the reflected beam is measured. This approach works well but also has certain practical complications because of the need for a polarization-sensitive reference cavity. In general, the most desirable features of a reference cavity are stability and high finesse, but it is normally quite difficult to insert a polarizing element into a high-finesse cavity without appreciably degrading both of these characteristics. Our approach has the same advantages as these methods but is technically simple. Consider the field ERCreflected by a mode-matched interferometric cavity. The standard result6 is
E; is the incident field, 6 = 4rL/X,and R1and TI are the reflection and transmission coefficients, respectively, of the input mirror. R is the amplitude ratio for successive round trips, so the finesse is rdR/( 1 - R ) . 6, the phase difference between the fields on successive round trips in the cavity, is equivalent to frequency. In Fig. 1A we plot Im(ERc)/E; versus 6 to illustrate the desirable dispersion-shaped resonance mentioned earlier. The underlying concept of the approach that we use to isolate this term is shown schematically in Fig. 2. Detector 2 looks at a portion of the pattern formed by the interference of a reference beam Eo and the beam reflected off the cavity, ERC. The detected intensity
Reprinted from Optics Letters, Vol. 7, page 480, October 1982 Copyright 0 1982 by the Optical Society of America and reprinted by permission of the copyright owner.
31
32 October 1982 / Vol. 7, No. 10 / OPTICS LETTERS Mirror
is 12
481
+
+
= ~ E R CE012 = I0 I R c 2 Re(ERc)Re(Eo) 2 Im(ERc)Im(Eo).
+
Writing EOas
Eo =
+
aces 8 + iJ&sin
(2)
= 10
8,
(3) Detector 1
+ IRC+ 2Jz COS 0 Re(ERc)
+ 2Jz sin 0 Im(ERc).
1- 1 7 (4)
The phase angle 8 will be just the difference in the path lengths for the two beams divided by the wavelength. If detector 2 is now positioned so that it sees only the portion of the interference pattern near 0 = go",
+ I R C + 2J&
(5) Detector 1sees the entire interference pattern, so, to a good approximation, 12
= 10
I1 = I0
h(ERC).
+ IRC,
(6)
and the difference, I Z - I1 = 2& Im(ERc). One point that cannot be neglected, however, is the effect of changes in 8. In general, [Im(ERc) Sin 8 + Re(ERc) COS 01. (7) This function has the important property that, for all values of 8 except close to 0" and 180", the frequency dependence of Re(ERc) can be neglected and I2 - I1 can be approximated by
Iz - 11= 2&
The accuracy of this approximation is shown in Fig. 2B. The solid curve is the exact value of ( 1 2 - I 1 ) / 2 a as given by Eq. (7) for 8 = lo", and the dashed curve is the value given by formula (8). The similarity between the two curves rapidly improves as 8 increases toward 90". For comparison purposes we note that the horizontal scale is the same as in Fig. l A , whereas the ver-
Fig. 1. A, Im(ERc)/E; versus 6. B, solid curve, 12 - I1/(24&
Ei);dashed curve, approximation from formula (8). Vertical scale is expanded by a factor of 5.76 from A.
Reference Interferometer Attenuatpr Variable
we have 12
I
Differential Amulifier
12-11
h Diaphragm
Detector 2
Fig. 2, Schematic showing laser-stabilizer concept.
tical scale is expanded by 5.76 (= l/sin 10") and its origin offset by 5.67 (=l/tan 10"). The implications of this dependence on 8 are that, over a wide range of 8, the feedback signal will effectively have the same shape but will have a 0-dependent zero offset. Such an offset, if it can vary with time, is undesirable in a feedback signal. With a setup such as that shown in Fig. 2, extreme care would be necessary to achieve the adequate path-length (8) stability of a fraction of a wavelength. An alternative approach that avoids this problem is the use of the beam reflected off a non-mode-matched cavity. This beam can be described as an appropriate superposition of the modes of the cavity, and the detection of the interference between these modes can provide the desired signal. Relative to a plane wave, the phase of the wave front corresponding to each mode is given in Ref. 7 as +'m,n
= (m
+ n + l)arctan(Xz/awi),
(9)
where m and n are the transverse mode numbers, z is the propagation distance from the waist, and 00 is the diameter of the beam waist. For simplicity let us consider an input beam composed of two modes, one having a resonant frequency near the laser frequency and a second with a resonant frequency far away. From Eq. (1)it is evident that the amplitude and the phase of this latter mode will have very little frequency dependence and hence can serve as a reference signal. The interference of the two modes is then described by Eq. (4). The phase, 8 = +'m,n - +p,,,,,, can be varied by changing the distance between the interferometer and the detector. The relevant length scale for this variation is wi.rr/X,which, for a confocal resonator, is one half of the mirror separation. The transverse variations in the amplitudes of the two modes ( z fixed) are described by the usual products of Hermite and Gaussian function^.^ By evaluating these one can calculate the x and y coordinates where the two mode amplitudes have the same ratio as their spatially integrated amplitudes measured by detector 1. The optimum detector position is then the x , y coordinate where the amplitudes are largest and in this ratio and where the z coordinate is such that 8 = 90". In general, the reflected beam will be a sum of many
33 482
OPTICS LETTERS / Vol. 7. No. 10 / October 1982
modes with different amplitudes and phases a t any given point in space. Calculating the ideal detector position in that case is a rather messy problem. However, finding an acceptable, though not ideal, detector position is a much simpler task because of the b' dependence discussed previously. As is shown by formula (81, the required frequency dependence is obtained for nearly any value of 19and therefore of z . Thus the sole requirement is finding an x , y position where only the Im(ERC) term of formula (8) is left after the two detector signals are subtracted. Such a position is easily found by using the simple empirical procedure given below. We investigated this approach experimentally using a Spectra-Physics Model 380A ring dye laser and a Trope1 Model 240 spectrum analyzer as a reference cavity. The dye laser provided a single-mode output at -540 nm; rhodamine 110 dye was used. The Model 240 spectrum analyzer is a confocal spherical FabryPerot interferometer with a 1.5-GHz free spectral range and a finesse of -150. The detectors were silicon photodiodes. In these experiments we scanned the reference cavity with the laser free running and examined the signals obtained. With the scan turned off, we then used the signal to lock the laser to the reference cavity. We first constructed a rather crude version of the setup shown in Fig. 2 to verify the basic concept of formulas (1)-(8). We obtained signals much like that shown in Fig. 1. Within a few minutes, however, the zero level drifted as much as the height of the peaks. This is hardly surprising since the temperature of our laboratory fluctuates by several degrees centigrade. As was mentioned previously, this is rather unsuitable for laser stabilization. Then to observe mode interference we used the same setup but removed the top mirror so only a single beam went to the detectors. To obtain a dispersion-shaped signal the following procedure was used: Detector 2 was moved across the beam profile while the attenuator in front of detector 1was adjusted to provide a zero error signal when the laser frequency was far off resonance. This approach provided good error signals for a considerable range of input beam diameters and curvatures and detector-interferometer separation. Both the wave-front diameter and curvature were varied by roughly 2 orders of magnitude with the values for the TEMoo mode of the interferometer being at approximately the geometric center of these ranges. The detector-interferometer separation has been as much as 20 times the interferometer length. In nearly all these cases, the signal-to-noise ratio was limited to the same value by imperfectly subtracted laser-amplitude fluctuations. In principle, of course, the optimum signal would be obtained by using an input beam composed of only two modes and positioning the detector as calculated in the manner discussed above. We wish to
emphasize, however, that in practice no such effort is necessary to obtain high-quality signals. After setting the detector position, we turned off the scan of the reference cavity and locked the laser frequency to it. This was done by amplifying the detector-difference signal and feeding it to the galvanometer-driven plates and piezoelectrically movable mirror, which are the standard tuning elements of the dye laser. The frequency stability was measured by using a second Model 240 spectrum analyzer. This feedback reduced the peak-to-peak frequency jitter from -50 MHz in a 10-sec period to under 0.5 MHz. We have also used this approach to lock the laser to a homemade semiconfocal interferometer, which has a 1.5-MHz linewidth. With -95% of the input beam coupled into the TEMoo mode, good error signals and locking were still obtained. We have presented a new method of stabilizing the frequency of a laser by locking it to the resonance of a passive reference cavity. By using mode interference, an error signal is obtained that is proportional to the imaginary component of the reflected field. This signal has nearly ideal characteristics for feedback stabilization. The method works with any reference cavity and requires the simplest of optics and electronics. This technique will apply equally well to the problem of locking the center of a cavity resonance to a laser frequency. This is particularly useful for obtaining intracavity power buildup for second-harmonic generation. This work is supported in part by a Precision Measurements Grant from the National Bureau of Standards, in part by National Science Foundation grant no. PHY-8111118, and in part by a grant from Research Corporation. S. L. Gilbert would like to acknowledge support from the following sources: Laporte Fellowship, AWIS Meyer-Schutzmeister Fellowship, and University of Michigan Rackham Predoctoral Fellowship. We thank R. Watts and J. Ward for useful discussions. References 1. R. L. Barger, M. S. Sorem, and J. L. Hall, Appl. Phys. Lett.
22,573 (1973). 2. A. D. White, IEEE J. Quantum Electron. QE-1, 349 (1965). 3. R. V. Pound, Rev. Sci. Instrum. 17,490 (1946). 4. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, and A. J. Munley, Joint Institute for Laboratory Astrophysics, Boulder, Colorado 80309 (personal communication). 5 . T. W. Hansch and B. Couillaud, Opt. Commun. 35, 441 (1980). 6. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), Chap. 7.6; Ref. 5. 7. H. Kogelnik and T. Li, Appl. Opt. 5,1550 (1966).
PHYSICAL REVIEW A
VOLUME 27, NUMBER 1
JANUARY 1983
Hyperfine-structure measurement of the 7s state of cesium S. L. Gilbert, R. N. Watts, and C. E. Wieman Physics Department, Universiw of Michigan. Ann Arbor, Michigan 48109 (Received 23 September 1982) The hyperfine structure of the 7Sl/2 state of 133Cshas been measured with the use of laser spectroscopy on a cesium atomic beam. We find the magnetic dipole coupling constant A =545.90(09) MHz. From the amplitude of the hyperfine components we find the ratio of the scalar to tensor polarizabilities ( I a / p I ) for the 6s -7s transition to be 9.80(12).
The measurement of hyperfine structure has long been an important probe of atomic structure. For this reason, the hyperfine structure of the ground and excited states of alkali atoms has been studied extensively.' Information on the 7s1/2 state of cesium is particularly important at the present time because of its role in the study of parity violation in atoms.2 In this paper we present a measurement of the hyperfine structure of the 7s,/2 state of 133Csobtained by studying the laser excited 6S1/2-7S1/2 transition in the presence of an electric field. By measuring the separation of the hyperfine components we determine the hyperfine splitting of the 7S,12state to considerably higher precision than all other measurements of alkali excited-state hyperfine splittings. In addition, by comparing the amplitudes of the components we obtain the ratio between the scalar and tensor polarizabilities of the 6S1/2-7S1/2 Stark dipole matrix element. A measurement of this ratio is of interest because the previous two measurements of this quantity are in serious d i ~ a g r e e m e n t . ~ . ~ The SIR levels of cesium are split into hyperfine doublets with total angular momentum F = 3 and 4. Thus, in a dc electric field ijf, a linearly polarized laser will excite four transitions having the intensities and frequency separations shown below: 6 s ~ '4I S F ~ , 143=21f12E:, IAvl = A 7 S h f r
6S~,3'7s,r,3,
133'7fl'E:
6s~4'7sF-4,
1.,.,'15f12E:
+28a2Ei, 1Avz = A6S h i - A7s h i
+36a2Ei,
neglected. Since A6S hf is the primary standard of frequency, measuring the ratio of either Avl or Av3 to AVZ Av, yields hl. Although the individual transition energies are electric field dependent, the separations, to a very good approximation, are not.s The ratio alp is obtained from the ratios 1 4 4 1 1 3 4 and I 3 3 / 1 4 3 . The apparatus used is shown schematically in Fig. 1. A collimated beam of atomic cesium is produced in a manner similar to that given in Ref. 6. A secondary collimator further reduces the divergence to about 0.025 rad. This beam intersects a standing wave laser field at right angles in a region of static electric field. The static field is obtained by applying 5000 V to optically transparent-electrically conducting coated glass plates (not shown in figure) which sit 0.5 cm above and below the laser beam. The standing wave field is provided by a spherical mirror FabryPerot interferometer (power enhancement interferometer). The output from a single frequency ring dye laser (Spectra Physics model No. 380) with homemade frequency stabilization7 is circularly polarized and modematched into the interferometer. One of the interferometer mirrors is mounted on a piezoelectric transducer, thereby allowing the interferometer resonance to be electronically locked to the laser frequency. The dye laser output is typically 0.1 to 0.2 W at 540 nm and the interferometer increases the electromagnetic field strength by a factor of 12.5. The short term laser frequency jitter is about 0.5 MHz peak to peak. Five percent of the dye laser beam is sent into an
+
IAv.3 = A7s hf
CESIUM
6s~-4+7&-3, 134=21f12E:.
(1) The subscripts I and II are with respect to the direction of laser polarization, a and /3 are the scalar and tensor p~larizabilities,~ respectively, and AbS hf and h[ are the hyperfine splittings of the 6 S and 7 s states. In addition to the electric dipole intensities shown, each transition has a magnetic dipole amplitude. For the electric fields we used ( 5000 V/cm), these contribute less than 1 part in lo4 and can be
i1.V
'
FREQUENCY MARKER INTERFEROMETER !
-
I
POWER ENHANCEMENT lNTERFERoMETER DETECTOR
DETECTOR
VACUUM CHAMBER
0
FIG. 1. Schematic of apparatus.
27
58 1
34
01983 The American Physical Society
35 582
S. L. GILBERT, R. N . WATTS, AND C. E. WIEMAN
additional fixed length Fabry-Perot interferometer which is thermally insulated and hermetically sealed (frequency marker interferometer). The resonances of this cavity provide a frequency scale. The 6s -7s transition is detected by a silicon photodiode which sits under the lower electric field plate. This diode monitors the 850- and 890-nm light which is emitted in the 6P1/2,3/2- 6 s step of the cascade decay of the 7s state. A colored glass cutoff filter prevents the scattered 540-r1m light from reaching the detector. The photodiode output goes into a preamp followed by a precision voltage divider. The linearity of the photodiode preamp combination is better than 1 part in lo4. The voltage divider is used to reduce the 3 -3 and 4 -4 transitions signals by a factor of 155.44, making them approximately the same size as the undivided 4 -3 and 3 -4 transitions. Other photodiodes measure the light transmitted through the frequency marker and power enhancement interferometers. These three signals and the voltage ramp which scans the laser frequency are digitized and stored by a PDP-ll computer for off-line analysis. Spectra were obtained by scanning the laser frequency over each of the hyperfine transitions. For each transition the scan lasted about 1 min and covered about 1 GHz. For each of the three separations a set of about 15 scans were made by alternating between the two transitions of interest. Typical scans are shown in Fig. 2. The 20-MHz linewidth is primarily due to the cesium beam divergence. The marker interferometer is intentionally adjusted to obtain the rather complex signal shown. This structure corresponds to exciting a number of nondegenerate modes and is useful for checking for and correcting scan nonlinearities. Analysis and results. To find the hyperfine splitting hf, the spectra were first normalized to the instantaneous laser intensity. The line centers were then found in terms of the frequency marker scale. This was complicated by the thermal drift of the marker interferometer. To correct for the drift, the position of each transition relative to the frequency marker was found as a function of time. Because the thermal time constant of the interferometer was long, the drift was quite linear with time during each set and varied between 0 and 0.3 MHz per minute for the different sets. Having thus determined the drift, the results were corrected appropriately. Once the line separations were measured in terms of the frequency 'E. Arimondo, M. Inguscio, and P. Violino, Rev. Mod. Phys. 49, 31 (1977). 2M. A. Bouchiat and C. Bouchiat, J. Phys. (Paris) 35, 899 (1974). )M. A. Bouchiat and L. Pottier, J. Phys. (Paris) Lett. 36, L189 (1975). 4J. Hoffnagle et al., Phys. Lett. 143 (1981).
m,
27 -
FIG. 2. Typical spectra.
marker, the separation between the F = 4 -4 and F = 3 -4 lines was used to determine the exact marker spacing. It is amusing to note that the interferometer length is thus known in terms of the fundamental frequency standard. Using this scale, Avl and Av3 were found to be in excellent agreement and their combined average was 2,183.59(36) MHz = 7 s , / 2 hyperfine splitting. This gives the magnetic dipole coupling constant A =545.90(09) MHz, which is in good agreement with the previous result of 546.3(30) M H Z . ~The uncertainty is primarily due to uncertainties in scan linearity and the correction for the drift of the frequency marker. To find 1 a/PI, the area under the four peaks were measured and the ratios Z33/143 and Z44/134 determined. The ratio la/PI was then found using Eq. (1). The light transmitted through the enhancement interferometer was found to be imperfectly circularly polarized, resulting in a correction of 0.6%. From 133/Z43we obtained la/pI =9.91(08), and from Z 4 / Z 3 4 , la//3I =9.68(08). Because these results did not seem consistent with a simple statistical variation we took the combined value to be 9.80(12), where the uncertainty covers both values. This is in good agreement with the value 9.91(1 1) of Ref. 4, but disagrees with the value 8.8(4) given in Ref. 3. This work was supported by the National Science Foundation and the Research Corporation. S. L. Gilbert and R. N. Watts would like to acknowledge support from the University of Michigan Rackham fellowship program. S. L. Gilbert is also pleased to acknowledge receipt of an Association for Women in Science Meyer-Schutzmeister fellowship. SM. Mizushima, Quantum Mechanics of Atomic Spectra and Atomic Structure (Benjamin, New York, 1970), Sec. 10-4. 6P. F. Wainwright, M. J. Algaurd, G . Baum, and M. S. Lubell, Rev. Sci. Instrum. 9, 571 (1978). 'C. E. Wieman and S. L. Gilbert, Opt. Lett. 1,480 (1982). ER.Gupta, W. Happer, L. K. Lam, and S. Svanberg, Phys. Rev. A &, 2792 (1973).
MAY 1983
VOLUME 27, NUMBER 5
PHYSICAL REVIEW A
Rapid Communications ~~
The Ropid Communications section is intended for the accelerated publication of important new results. Manuscripts submitted to this section are given priority in handling in the editorial office and in produclion. A Rapid Cornrnunicalion may be no longer than 3% printed pages and must be accompanied by an absiroci. Page proofs are sent 10 authors. but, because of rhe rapid publication schedule, publication is not delayed for receipt of corrections unless requested by the author.
Precision m e a s u r e m e n t of t h e S t a r k shift of the i n atomic cesium
6s-7stransition
R. N. Watts, S. L. Gilbert, and C. E. Wieman Randall Physics Laborarory, University of Michigan. Ann Arbor, Michigan 48109 (Received 29 November 1982)
We have measured the Stark shift of the 6 s - 7 s transition in a beam of atomic cesium with the use of laser spectroscopy. We find this shift to be 0.7103(24) Hz (V/cm)-*. From this value we determine the static polarizability of the 7 s state, a 7 S = [ 6 1 1 1 ( 2 1 ) l a i ,and the 7P-7S oscillator strength, f 7 P , 7 S = 1.540(7). Finally, we derive a new empirical value for the 6 s - 7 s Starkinduced electric dipole transition probability
INTRODUCTION
cesium and find the frequency shift as a function of electric field. From the literature value for (Y6S and Eq. (21, we determine a l s .
While alkali atom ground- and first excited-state polarizabilities have been studied extensively, little information is available on the polarizabilities of the excited S states.' In this paper, we present a highresolution measurement of the Stark shift of the 6S7s transition of cesium in a dc electric field. From this, we derive the static electric dipole polarizability of the 7s state and obtain a precise value for the oscillator strength f 7 p . l ~ . W e then use f 7 p . 7 ~to make an improved empirical determination of the 6s-7S Stark-induced electric dipole transition amplitude. This amplitude plays an important role in determining the size of the parity-violating interaction recently observed in cesium.2 In a static electric field E, the energy of an atomic S state shifts by an amount I
hens=- i a n s E 2
I
APPARATUS
The experimental apparatus shown in Figs. 1 and 2 is the same as that described in Ref. 4 with the addition of an iodine saturated-absorption spectrometer. A dense highly collimated beam of atomic cesium intersects a standing wave-laser field at right angles in a region of static electric field. The standing wave is produced by a frequency-stabilized cw ring dye laser mode matched into an electronically tunable, spherical-mirror Fabry-Perot interferometer. The laser has a typical output of 150 m W and a shortterm frequency jitter of 0.5 MHz peak to peak. With
(1)
where a d is the static polarizability for that state. For shifts much less than the fine-structure splitting, ad does not depend upon the total angular momen~ electric field tum quantum numbers F o r M F . The makes it possible to drive electric dipole transitions between parity-mixed S states of different n. This transition frequency, expressed in atomic units, is displaced by
1 DYE LASER
', n
5-
FREQUENCY' FREQUENCY
-
CESIUM OVEN
x 14
n
I
1
iy \
POWEk POWER ENHANCEMENT MARKER INTERFEROMETER INTERFEROMETER :, ODETECTOR DETECTOR
In our experiment, we drive the
6s-7s transition in
CHAMBER
FIG. 1. Schematic of apparatus.
21
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36
01983 The American Physical Society
37
2110
R . N . WATTS, S. L. GILBERT, AND C. E. WIEMAN
FIG. 2. Detail of interaction region.
I
I
I
I
150
100
50
0
1
50
100
AV (MHZ)
the resonant frequency of the interferometer locked to the laser frequency, the electromagnetic field strength inside the interferometer is 12.5 times that of the incident laser beam. The 6 s - 7 s transition driven by the laser is monitored by observing the 850- and 890-nm light that is emitted in the 6P-6s leg of the 7S-6P-6S cascade decay. This light is detected by a cooled silicon photodiode that sits below the electric field plates. Glass color filters not shown in Fig. 2 prevent scattered 540-nm laser light from reaching the detector. The static electric field is produced by applying 1 to 13 kV to a pair of 5 x 7.5-cm2-coated glass plates 0.9208(5) cm apart. To assure field uniformity, the laser and cesium beams intersect in a 3-cm-long line in the middle of the electric field region. The top field plate is coated with evaporated gold. The lower field plate is coated with a thin layer of chromium having an optical transmission of 75% and a surface resistivity of 7 kCl/o. The voltage applied to the field plates is monitored with a voltage divider and digital voltmeter. Five percent of the laser beam is sent into a thermally isolated, hermetically sealed, fixed-length Fabry-Perot interferometer (frequency marker interferometer). The resonances of this cavity set the frequency scale but thermal drifts make it unsuitable for use as a frequency reference. For this, 30% of the laser beam is sent into a Doppler-free saturatedabsorption iodine spectrometer identical to that described in Ref. 5. Iodine is used because the X ' Z : ( v " - O ) -B311&(v'= 29) J = 81 R-branch transition overlaps the 6&-4-7s~-4 cesium hyperfine transition.6 The iodine hyperfine lines that result as the laser is scanned over the cesium transition serve as stable frequency references. Signals from the frequency marker interferometer, the silicon photodiode, and the iodine spectrometer are digitized and stored by a Digital Equipment Corporation PDP11/23 computer for off-line analysis. Spectra were obtained by scanning the laser over the cesium transition for electric fields of 1.1, 5.7,
FIG. 3. Combination of frequency scans at 1.1 and 11.8 kV/cm. (a) Representative frequency marker interferometer spectrum. (b) Iodine saturated-absorption spectrum. (c) Cesium peaks (amplitude of 11.8-kV/cm peak reduced by factor of 100).
-
9.0, 11.8, and 14.0 kV/cm. Each scan took approximately 30 sec and several scans were made for each value of the electric field. A combination of two such scans appears in Fig. 3. As discussed in Ref. 4 the frequency marker interferometer had been adjusted to give the rather complex structure displayed. The iodine peaks of interest have been arbitrarily labeled (I through e. We believe that the small dip in the center of the cesium peaks at higher (> 6 kV/cm) electric fields is associated with saturation of the transition and has no effect o n the line center. ANALYSIS A N D RESULTS
To find the frequency shifts as a function of electric field, we first chose iodine peak d as an arbitrary zero. Using the frequency marker resonances, which were calibrated to 5 parts in los in Ref. 4, we determined the separations of peaks u through e. Then, using the iodine peaks, we found the displacement from the arbitrary zero of each cesium peak. Uncertainty in determining a peak center was typically 0.1 5 MHz. The data along with the straight-line fit to Eq. (2) is shown in Fig. 4. In this drawing, the error bars are smaller than the data points. The slope d(Av)/ d ( E 2 ) is 0.7103(24) Hz (V/cm)-' with a correlation coefficient of 0.999 98. The statistical uncertainty in the slope is 0.0010 Hz (V/cm)-*. The systematic error is dominated by the uncertainty in the electric field. The separation between field plates was measured to l part in 1800. The voltage divider was checked with the assistance of M. Misakian at NBS to the 0.1% level and the digital voltmeter is good to 0.1%. This combines to give a 0.15% uncertainty in
38
21
PRECISION MEASUREMENT OF THE STARK SHIFT OF T H E . . .
2771
quency tables of Moore,l3 we find 140
f77=1.540(7) .
/
/
/
-2 40
1
E~ ( lo6 V2/crnz)
FIG. 4. Stark-frequency shift vs square of electric field Error bars are smaller than data points.
the electric field and an uncertainty of 0.0021 Hz (V/cm)-2 in the slope. Using Eq. (2) and Q6S= [402(8)Iad from Ref. 7 , we find
This value includes a 0.15% correction due to the inclusion of fine structure in Eq. ( 3 ) . The quoted uncertainty consists of 0.006 from the combined experimental uncertainties and 0.003 from a 25% theoretical uncertianty in A a l s . A 25% uncertainty is reasonable: The calculations of Hofsaess for cesium are usually within 10% and, in the worst case, within 25% of the measured principal series oscillator strengths f n 6 , and the photoionization cross sections. To our knowledge, this value for f 7 7 is one of the most accurately determined oscillator strengths. It is in fairly good agreement with a theoretical value of f 7 7 = 1.574 based on the work of Hofsaess. In a manner analogous to the definition of a7s, the Bouchiats have defined a quantity a that is a measure of the transition probability between the 6 s and 7 s states of cesium in the presence of a electric field.14 This quantity may also be written in terms of experimentally determined oscillator strengths
a7~=[6111(21)I~O3 . This value for (27s can be compared with [6673(386)1ad as determined by Hoffnagle, Telegdi, and W e k 8 The difference in the uncertainties is explained by noting that the second measurement was made in a heated vapor cell. This gave linewidths 40 times larger than those of our experiment and limited the size of the electric field that could be applied. This, in turn, forced Hoffnagle et a/. to make precise measurements of shifts that were at most only 1% of the linewidth.
-
DISCUSSION
The 7s state polarizability can also be expressed in a straightforward manner in terms of oscillator strengths. Following Refs. 8 and 9, we ignore fine structure, expand Eq. (11, and find f67
f77
a 7 s = 1 + 7 - + A a 7 . y w76 W77
,
(3)
where w n n , wnr.n,pand f n t n =f n l p , m . The term A a 7 s contains the contributions from discrete states n’ 2 8, the continuum, and autoionizing states. This quantity has not been measured experimentally but, from the theoretical work of Hofsaess”, ” and Safinya,12 has been estimated by Hoffnagle to be 590d. Using Hoffnagle’s value of f 67 = - 0.573 ( 13 ) and the fre-
IT. M. Miller and B. Bederson, Adv. At. Mol. Phys. 13, I (1977); R . Marrus, D . McColm, and J. Yellin, Phys. Rev. 147, 55 (1966). 2M-A. Bouchiat, J . Guena, L. Hunter, and L. Pottier, Phys.
As in Eq. (31, A a contains the experimentally unknown contributions from the discrete states n’ 3 8, the continuum, and the autoionizing states. Using the theoretical inputs of Hofsaess and Froese-Fischer,” Hoffnagle estimates P a to be - 4.400). Taking Hoffnagle’s value for f 6 7 , f 6 6 = 1.068(21) from Ref. 16, and our value for f 7 7 , Eq. (4) gives a = - [263.7(27)1ad. A 1% correction due to finestructure splitting has been included. T h e experimental uncertainties contribute 2.500) and, as before, we have assumed a 25% (1.10;) uncertainty in ACZ. This value for a is in excellent agreement with a value of - 262.6 calculated by Hoffnagle using the theoretical input of Hofsaess. This work was supported by the National Science Foundation and the Research Corporation. S.L.G. and R.N.W. would like to acknowledge support from the University of Michigan Rackham Fellowship Program. S.L.G. is also pleased to acknowledge receipt of an Association of Women in Science MeyerSchutzmeister Fellowship. We thank M . Misakian for his help in checking our voltage divider and J. C. Zorn for many helpful discussions.
Lett.
w,358 (1982).
)A. Khadjavi and A. Lurio, Phys. Rev.
167,128 (1968).
4S. L. Gilbert, R. N . Watts, and C . E. Wieman, Phys. Rev. A 27, 581 (1983).
39
2112
R. N. WATTS, S. L. GILBERT,A N D C. E. WIEMAN
SW. Derntroder, Laser Spectroscopy (Springer, New York, 19811, p. 489. 6J. D . Simmons and J . T. Hougen, J . Res. Nat. Bur. Stand. Sect. A 7 l , 25 (1977). 'R. W.Molof ef a/., Phys. Rev. A lo,1131 (1974). 8J. Hoffnagle, V. L. Telegdi, and A. Weis, Phys. Lett. 457 (1981). 9J. Hoffnagle, dissertation (Swiss Federal Institute of Technology, Zurich, 1982) (unpublished). 1°D. Hofsaess, Z. Phys. A 281, 1 (1977).
m,
21
"D. Hofsaess, At. Data Nucl. Data Tables 24, 285 (1979). I2K. A. Safinya, T. F. Gallagher, and W. Sander, Phys. Rev. A 22, 2672 (1980). 13C.E. Moore, Atomic Energy Levels, NBS Circular No. 467 ( U S . GPO, Washington, D.C., 1949). 14M. A. Bouchiat and C. Bouchiat, J . Phys. (Paris) 3.493 (1975). ISC. Froese-Fischer, At. Data 4, 301 (1972). 16L. N. Shabanova ef a/.,Opt. Spectrosc. (USSR) 47, 1 (1979).
VOLUME 29, NUMBER 1
PHYSICAL REVIEW A
JANUARY 1984
Measurement of the 6s +7S M1 transition in cesium with the use of crossed electric and magnetic fields S . L. Gilbert, R. N. Watts, and C. E. Wieman Department of Physics, University of Michigan, A n n Arbor, Michigan 48109 (Received 21 June 1983; revised manuscript received 4 August 1983) Th e forbidden 6S+7S magnetic dipole transition amplitude in cesium has been measured by laser spectroscopy of an atomic beam in crossed electric and weak magnetic fields. Th e M I amplitude was determined by observing the change in the transition rate caused by intcrfcrence with a Stark-induced El amplitude. Th e result for the nuclear-spin-independent amplitude is -42.10(80) X 1O-'pB; the result for the nuclear-spin-dependent amplitude is 7.59(55)X 1 0 - 6 p ~ . These values disagree with earlier measurements but they ar e in good agreement with theory. Th e experimental approach is well suited to measuring parity-violating neutral-current interactions.
INTRODUCTION
The 6s-to-7S magnetic dipole ( M 1) transition in cesium has received considerable attention recently because of its role in the study of parity violation in atoms. However, the mechanism responsible for this very small transition amplitude ( -lo5 times smaller than an allowed M I transition) has remained unclear. For some time the dominant mechanism was thoughtIs2 to be the fourth-order product of the interconfiguration and spin-orbit interactions. Recent more accurate calculations334of this product, however, have shown that it is at least an order of magnitude smaller than the experimental value measured by Bouchiat and Pottier' and later by Hoffnagle et a1.6 Stimulated by this, Flambaum et u I . ~proposed that a third-order contribution to the M 1 amplitude would exist and calculated its size. However, this value was nearly a factor of 2 larger than the experimental value, as discussed in Appendix A. Several author^'^^^^ have pointed out that, in addition to this nuclear-spin-independent component of the amplitude, the off-diagonal hyperfine interaction would give rise to a smaller nuclear-spindependent component. There was even poorer agreement between theoretical and experimental values for this component! This was particularly puzzling because the calculation of this quantity is straightforward and can be directly related to well-known hyperfine splittings. We report here the measurement of both components of the M1 amplitude using a new technique which yields higher sensitivity than previously possible and avoids a number of sources of systematic error which may have affected earlier work. Our results resolve the previous disagreements between theoretical predictions and experimental values. The experimental technique used has considerable promise for studying panty-violating neutralcurrent effects in atoms. We conclude with a brief discussion of this future application. The experimental approach employs the interference between the M 1 amplitude and a larger electric dipole ( E 1 ) amplitude. The measurement of a small amplitude by observing its interference with a larger known amplitude has been applied to a variety of problems but was first dis-
cussed in this context by the Bouchiats.' Neglecting parity-violating effects, only M 1 or higher multipole transitions can take place between states of the same parity in an unperturbed atom. The application of a weak dc electric field, however, creates a small admixture of states of opposite parity which gives rise to a "Stark-induced'' E 1 transition amplitude. They pointed out that this amplitude can interfere with both the M 1 amplitude and a panty-violating E 1 amplitude arising from neutralcurrent interactions. Thus the measurement of these two amplitudes via this interference involves very similar experimental considerations. In the absence of a magnetic field, both of these interference terms can create a polarization of the excited state but cannot affect the transition rate.' A number of impressive measurements of small M1 (see Refs. 5,7,8) and parity-violating E 1 amplitudes'*" have been made by determining this polarization. However, such measurements suffer from a loss in sensitivity because the state polarization cannot be measured directly. It must be inferred from the degree of polarization of light absorbed or emitted in a transition from the excited state. In general this will be less than the atomic polarization and can be a source of systematic error in determining the amplitude of interest. Our method avoids these difficulties because the interference is manifested as a direct contribution to the transition rate. This requires that the Zeeman sublevels be resolved. We achieve this by using narrowband laser light to excite transitions in an atomic beam in the presence of a weak magnetic field. The concept of such interference terms affecting the transition rate when a magnetic field is present has been discussed a number of times, though never experimentally demonstrated. The idea was implicit in the experiment to search for panty violation in hydrogen proposed by Lewis and Williams." Following that, it was discussed explicitly by several authors in the context of possible techniques for the measurement of parity violation in heavy atom^.^'"^ Recently Bouchiat and PoirierI4 have extended that discussion to the closely related problem of measuring weak M 1 amplitudes. All these proposed methods require magnetic fields on the order of 1 kG resulting in a complex relationship between the data and the amplitudes 137
29 -
40
@ 1984 The
American Physical Society
41 138
S. L. GILBERT, R. N. WATTS, AND C. E. WIEMAN
-
29 -
LASER BEAM
FIG. 2. Schematic of excitation region. Magnetic field coils and the wire mesh E field plates below and above the plane of intersection of the two beams are not shown.
&, are the same. Thus only a single line is observed and FIG. 1. Cesium energy-level diagram showing hyperfine and weak field Zeeman structure of 6s and 7s states.
of interest. A significant difference between these proposals and our experiment is that we require only tens of Gauss. This small field makes the interpretation of the data quite simple, as we will discuss, and substantially reduces a number of possible systematic errors.
THEORY
The use of crossed atomic and laser beams provides inherently narrow transition linewidths with little background. A n additional experimental consideration however is the configuration of the magnetic (g),the static (g), and oscillating ( E ' ) electric fields. The magnetic field, if weak, causes the Zeeman sublevels to split according to A E / h =mgFpBB,where m is the quantum number for the z component of the total angular momentum F. For transitions between states with the same values of and 5 the resulting spectrum is dramatically different depending on whether is parallel or perpendicular to T. For the parallel case the selection rules for the Stark-induced transitions are A F = O and Am =O. In the weak-magnetic-field limit the energies of all the Am = O transitions remain the same because the initial and final values of F, and hence
the interference terms do not contribute to the transition rate. Only in intermediate and high magnetic fields do the ground- and excited-state Zeeman levels shift differently allowing transitions to different Zeeman sublevels to be resolved. For the case 2 perpendicular to F t h e selection rules are A F = O , f 1, and Am =0, f1. This gives rise to a multiplet structure even in the weak-field regime. Because the different lines in the multiplet involve transitions between different Zeeman sublevels, the transition rate for any given line will contain a contribution from the interference term. Thus the necessary resolution is achieved in quite a weak (tens of Gauss) magnetic field which has the advantages previously mentioned as well as simplifying the experimental apparatus. We shall illustrate the other features of the experimental technique by considering in more detail the 6S-7S hyperfine transitions in cesium which were studied. These can be seen on the cesium energy level diagram in Fig. 1. Although other possible field configurations could be + used, in particular, E along the laser propagation direction, in the interest of brevity we shall limit our discussion to the configuration shown in Fig. 2. The static electric and magnetic fields are perpendicular and the transition is excited by laser light propagating perpendicularly to "E and g, with Pparallel to 5. The Stark-induced E 1 amplitude for a transition from an initial state 6SFmto a final state 7SF.m.is given by
where C,$Am'is proportional to the usual Clebsch-Gordan coefficient and is tabulated in Appendix B. The vector transition polarizability p, introduced in Ref. 2, is given by
42 MEASUREMENT OF THE 6S+7S M1 TRANSITION IN . . .
29
139
I
The M 1 amplitude for this transition is given by f
d ~ ~ ' ( M 1 ) = ( 7 S F , , , I ~i6S,cm) ..~ .
(3)
The selection rules are AF=O,&l and Am = +1. This amplitude can be written as the sum of two terms proportional to M and (F-F'IMhf, respectively, where M is the nuclear-spin-independent component and Mhf is the nuclear-spin-dependent component. This gives
The notable feature of this equation is that the transition rate contains a pure Stark-induced E 1 term, fi2E2,plus an E I-M 1 interference term-the sign of which depends on E and Am. Using Eq. (6) and the fact g ~ = 4 =- g F Z 3 , we can now obtain the spectrum for the three Zeeman multiplets of interest. 6SF=4+7SF=3. The spectrum contains eight lines where each line strength R (i) is given by
d ; ~ ' ( M l ) = [ M + ( F - - F ' M l , f ] ( C ~ , ~ - , )(Am = + l ) and
(4)
.d;Z'(M 1 ) = [ M + ( F - F ) ) M h f ] ( -cj?,,mil) (Am = - 1 ) . For each transition between particular Zeeman sublevels, 6SFm+7sFm., the transition rate I , is the square of the sum of the E 1 and M 1 amplitudes,
I d ( E l ) + d ( M l )1 ' =
I d ( E 1 ) l 2 + 2 ~ ( E l ) ~ ( M 1/ ). d + ( M 1 ) I 2 .( 5 )
For the case we are interested in 1 . d ( M 1 ) 1 ~-'=~&P2E2-2PEM).
rn=-3-+4
FIG. 3. (a) Theoretical spectrum. The solid lines are the
I &'(El) I * contributions while
the dashed lines are the
2 d ( E 1) d ( M1) contributions on an expanded vertical scale.
The dashed lines have been shifted slightly to the right for ease of viewing. Actually both lines occur at the same frequency and the observed intensity will be the sum of the two contributions. (b) Scan of 6 s ~ = 4 + 7 s ~ =transition 3 with B = 100 G.
This analysis used the weak-field limit for the Zeeman effect which assumes no mixing of different hyperfine states by the magnetic field. In the field regime of interest this mixing can be accurately calculated using secondorder perturbation theory. We find its effect is to change the relative transition strengths slightly from the weakfield limit. For a typical field of 40 G the changes range between 0% and 3.4% for the various lines of the multiplets discussed. However, for the spectral lines measured in this paper, the size of the change is unimportant because the corresponding E 1 and M 1 amplitudes are affected by exactly the same amount. Thus the ratio of the two amplitudes, which is the quantity of interest, is independent of the mixing of the hyperfine states. For all three hyperfine transitions the interference t e r n s are odd under reversal of 2, 5,and the changing of the excitation frequency to the opposite multiplet component, providing a simple experimental way to isolate and mea-
43 29 -
S. L. GILBERT, R. N. WATTS, AND C. E. WIEMAN
140
sure them. The experiment consists of carrying out these reversals and detecting the resulting fractional modulation in the transition rate. In principle any single reversal would be sufficient but in practice the extra reversals are valuable for eliminating systematic errors. EXPERIMENT
We have used this technique to measure the ratios of the magnetic dipole amplitude to the Stark-induced amplitude 8,for the 6 s ~ = 3 + 7 s ~ = 4the , ~ S F = , + ~ S F = and ~, 6SF=4-f7SF=4 transitions in cesium. The F = 3 + 4 and F = 4 4 3 transitions were measured, then the apparatus was modified slightly and all three transitions measured. In a previous work15 we determined the absolute value of
8. The experiment is shown schematically in Fig. 2. A narrow-band dye-laser beam intersects a collimated beam of atomic cesium in a region of perpendicular electric and magnetic fields. The 6S+7S transition rate is monitored by observing the 850- and 890-nm light emitted in the 6P3/2,1/2*6S1/2 step of the 7 s decay. The dye laser is a Spectra-Physics model No. 380 dye laser with homemade frequency and amplitude stabilizers. The output power is typically 500 m W with a linewidth of -100 KHz. The cesium beam, which is produced by a two-stage oven to reduce the dirner fraction, is collimated to 0.015 rad. The 2.5-cm-long region of intersection of the two beams is imaged onto a silicon photodiode by a 5-cm-long gold-coated cylindrical mirror with flat ends. The scattered 540-nm light is blocked by a colored glass cutoff filter in front of the detector. This filter is coated with an optically transparent conductive coating. Fine wire mesh was placed above and below the interaction region and the electric field was produced by applying voltage (typically 1.6 KV) to the upper mesh and grounding the lower. For the first run the mesh spacing was 0.5 cm while for the second a new collector was used with a 0.6-cm spacing. A 40-G magnetic field is provided by a 25-cm-diam Helmholtz pair. Data was obtained by locking the laser frequency to the peak of a particular line of the multiplet and reversing the electric field every 0.25 sec. During each half cycle of electric field the detector output was integrated, digitized, and stored by a Digital Equipment Corporation model No. PDP-11 computer. For each of the three hyperfine transitions data was taken on the extreme high-and lowfrequency lines of the Zeeman multiplets as these provide the largest signals. This was done for both directions of magnetic field providing a total of four data sets for each hyperfine line. A t 1-2-min intervals during these measurements the electric field was set t o zero t o determine the baseline for the transition. To test for any asymmetry in the reversal of 3, data was also taken at zero magnetic field for laser frequencies both on and well off the transitions. To test for frequency-dependent background signals the laser frequency was scanned over the transitions both with and without magnetic field and the spectra recorded. One such scan with a rather large magnetic field is shown in Fig. 3(b). The 14 M H z linewidth is due to the residual
-
-
I
I
I
0
0.5
1.0 V (GHI)
1
1.5
I
2.0
FIG. 4. Scan of transition with B = O
Doppler shift from the cesium beam divergence. In order to obtain an absolute value for the M 1 amplitude it is necessary to know the dc electric field. We determined this by measuring the Stark shift of the 6SF=4+7S,7=4 transition as described in Ref. 15. Once the Stark shift was known the field was found using the polarizability given in that reference. DATA ANALYSIS A N D RESULTS
The data analysis primarily consisted of taking the difference between the positive electric field and negative electric field transition rates and dividing it by the average rate. From Eqs. (6)-(8), this is simply (9)
where Meff is M + M , f , M - M h f , and M , for the three transitions studied. However, considerable additional effort was devoted to determining and correcting for possible systematic errors. First we considered systematic errors which could have been introduced by various background signals. The cesium oven and, to a lesser extent, the scattered laser light gave appreciable background signals which were independent of laser frequency. The only effect of these signals on our measurement was to cause a slow uniform drift in the detector zero level, typically corresponding to 1% of the transition rate per minute. This drift was determined from the E = O data and did not introduce a significant uncertainty. The scans of laser frequency over the transition showed a broad frequency-dependent background signal centered on the atomic transition frequency. This can be seen in Fig. 4. We ascribe this signal to a n isotropic background gas of cesium in the interaction region. The line shape was accurately determined from the spectra taken with B=O. The background pedestal could be fitted well by a 575-MHz-wide Gaussian curve. The height varied between 0.040(2) and 0.067(2) times the height of the 14-
44 29 -
MEASUREMENT OF T H E 6S--t7S M1 TRANSITION I N
MHz-wide peak for the different runs. This pedestal contributes to both the average ( B 2 E 2 )transition rate and the E 1-M 1 modulation. The contribution to the modulation is considerably diluted over that given in Eq. (9) because the linewidth is much broader than the Zeeman splitting. Using the measured line shape we calculated the correction needed because of the presence of the pedestal (11-26%). We do not observe any other frequencydependent background-in particular, that due to molecular cesium which was significant in the work discussed in Refs. 5 and 6. A number of possible systematic errors associated with the field reversals were also checked. The use of three independent mechanisms to reverse the sign of the interference term greatly reduces such errors but a number of independent experimental tests were also made. These included tests of the follo_wing:_(1) detector sensitivity depending on direction of E or B, (2) imperfect B reversal, (3) imperfect reversal, (4) error due to ??not perfectly linearly polarized perpendicular to E, ( 5 ) misalignment of B with respect to E and Z,and (6) pickup of '*E fieldswitching transients. None of these sources were found to be significant at the level of the experimental uncertainty except the "E field switching transients. These were determined from the modulation signal measured when no laser light was present. It was found to be -9% of the interference term. Since this spurious signal is always the same sign, its effect averages to zero when the data for different directions of magnetic field and different lines of the multiplet are averaged. In order to obtain an absolute value for the M 1 amplitude from the ratio given by Eq. (9) it is necessary to know E. From the Stark shift of the 6SFS4+7SFE4 transition we obtained 2934U5) V/cm for the first data run. For the second run, values of E between 2661(10) and 2997(10) V/cm were used. The values obtained from the Stark shift agreed well with less accurate ( 5 % ) values calculated by dividing the applied voltage by the mesh separation. The dependence of the transition line shape on electric field also provides a test of the field uniformity. We find that the 14-MHz linewidth is changed by less than 1% in going from 625 to 5000 V/cm while the line center shifts 17.5 M H z [Av=O.71 Hz(V/cm)-2 from Ref. 151. This implies that any spatial variation of the field strength is substantially less than 1%. In Fig. 5 we present the fractional changes in the transition rate observed for the different lines. We have put in corrections for the pedestal and the electric field transients. For display purposes we have normalized all the data to a field of 2997 V/cm. Comparing across each row one can see the expected sign changes. Also the agreement in the magnitudes provides further confirmation that all the reversals are working as planned. From this data and Eq. (9) we obtain from the first run in V/cm
-
-
141
1 1 1 1
''1
1
T
FIG. 5. Fractional changes in transition rates. Direction of and - while the H and the magnetic field is indicated by the L indicate the multiplet line. The highest fi-equency line is labeled H and the lowest L . The AR's in the first and third columns were negative, as indicated. Each dotted line is the average of the four points in that row. All the data was normalized to 2997 V/cm.
+
M-Mhf
= - 34.59( 80)
B -=_
, M f M h f =-24.10(70) , I3
29.43(62),
P where this value for M / B is based on only the F = 4 + 4 data. In Table I we combine these results and compare with earIier measurements. With the exception of the value for ( M -kfhf)/fl our results disagree with the earlier measurements. These earlier measurements of M / P were made using A F = O transitions while for our first data run we derived this quantity from A F = + l and -1 transitions. The disagreement between these led us to speculate that perhaps, contrary to Eq. (4), there was some unsuspected M 1 component which contributed equally to the AF#O lines and did not contribute to the A F = O transitions. However, the agreement between the value of M / f l we obtained from the A F = 1 and - 1 transitions [-29.83(40) V/cm] and the value we subsequently obtained from the F = 4 + F = 4 transition rules out such an idea. Using /3=26.6(4)ai as discussed in Appendix A we obtain
+
M
= -42.10(
80) X 10-6p8 ,
Mhf=7.59(55)X 10-6p,
.
Considering the uncertainty in the calculation, the value for M is in agreement with the recent value of -63 X TABLE I. Measurements of M 1 Amplitudes
Ref. 5 M (V/cm) R
Mhf -
and for the second run in V/cm
. ..
M
- 23.2U.3)
Ref. 6
This work
-26.2(1.7)
-29.73(341
- 34.3(2.1)
- 35.21(56)
-0.3 l(3)
-0.180(13)
45 142
S . L. GILBERT, R. N . WATTS, A N D C. E. WIEMAN
calculated by Flarnbaum, Khriplovich, and Sushkov4 for the contribution due to the product of first-order interconfiguration interaction and second-order off-diagonal spinorbit interaction. M h f arises from the mixing of states of different n by the hyperfine interaction. Hoffnagle6 has shown it can simply be expressed as (theory)
where A w and ~ A ~ W , ~are the accurately known hyperfine splittings of the 6s and 7s states.I6 This value is expected to be quite accurate, and is in good agreement with our measured value. EXTENSIONS OF THE TECHNIQUE
The experimental approach of crossed beams in a weak magnetic field can also be used to measure the parityviolating E 1 amplitude [ d ( ElPv)] arising from neutralcurrent mixing of the § . and P states. As mentioned earlier this interferes with the Stark-induced amplitude in a manner similar but not identical to the M1 amplitude. For the field configuration we have discussed, one significant difference, which was mentioned in the Bouchiats early work,2 is that the parity-violating mixing matrix element is imaginary relative to the Stark mixing matrix element. Thus no interference will be present for linearly polarized light. This can be remedied by using elliptically polarized light, Z=C’l+iC’l,, where the I and / I are with respect to E. NOW iFI1 creates a parity-violating amplitude which has the same phase and hence interferes with the Stark-induced amplitude. Carrying through a similar analysis as before (including the mixing of hyperfine states) for the 6SF=4-+7SF=3 transition one obtains the same Clebsch-Gordan coefficients and the same spectrum as for the M 1 case shown in Fig. 3(a). Th e only difference is that now the vertical scale for the interference terms is proportional to ( e l E f i ) [ E @ ( E lPv)]. Like the M 1 case the interference terms change sign with a reversal of E, 5,and multiplet component. However, unlike the M 1 case they do not change sign if the laser light propagation direction is reversed” and they have the additional signature that they change sign when the “handedness” of the light is reversed (ze,l+-iell). It might also be noted that the ratio between interference and pure Stark-induced contributions t o the transition rate can be enhanced by making elI /el > 1. The apparatus previously described is thus well suited to measure this parity-violating term. Besides the use of elliptically polarized light, the only significant change needed is the addition of a power buildup interferometer cavity of the type we have previously This provides 150 times more laser power i n the interaction region. Because the signal to noise ratio is limited entirely by detector noise this should give a corresponding improvement in the signal-to-noise ratio while suppressing the E 1-M 1 interference. For a panty-violating amplitude of the size measured by Bouchiat et al. lo ( l o p 4 times the M l ) , such a signal-to-noise ratio would allow a one standard-deviation measurement of E I,, with an integrat-
-
-
29 -
ing time on the order of 10 min. The apparatus has potential for considerable future improvement. For the measurement of M 1 amplitudes the addition of a traveling-wave ring buildup cavity would improve the signal-to-noise ratio by 100. Also the use of an optically pumped atomic beam would provide a 16fold increase in signal for measurements of both the d ( M 1 ) and .d(E lpv)amplitudes and eliminate the need for a magnetic field. Note added in prooj We have learned that Bouchiat, Cuena, and Pottier have recently remeasured M / / 3 and measured Mhf/P, and they now obtain results in excellent agreement with ours.
-
ACKNOWLEDGMENTS
We are pleased to acknowledge the assistance of M. C. Noecker on this experiment and D. Kleppner for a critical reading of the manuscript. This work was supported by the National Science Foundation and in part by Research Corporation. One of us (R.N.W.) was supported by a University of Michigan Rackham Fellowship. APPENDIX A: THE VALUE OF /S
A certain amount of confusion has been caused by the value of the vector transition polarizability /3 used in Ref. 5 . The quantity actually measured in that work was the value of M / f i given in Eq. (13) of that reference and listed in our Table I. By using a value of 8.8(4) for the ratio of scalar to vector transition polarizability, 1 a//3 1 , and a theoretical value of ( - 30%~:) for a , they quoted an absolute value for M of 4.24(34)X 1 p B 1 . Subsequent measurements by Hoffnagle et d 6and ourselves,16 and a remeasurement by Bouchiat et al. have obtained values for I a / P [ of 9.91(11), 9.80(12),and 9.90(10), respectively. We have experimentally determined15 a to be -263.7(27)ai. From the average of these results we take l a / P I =9.9(1) and arrive at a value of /3=26.6(4)ai. Using this value for P, the M / P given in Ref. 5 gives M = 3 . 3 ( 2 ) X 10-’pB which is farther from the calculated value of 6.3 x 10-’pB than the number originally quoted. In the text we have taken care to only compare the measured ratios M , f f / P . To derive an absolute value for M and M h f from our data we used 8 = 2 6 . 6 ( 4 ) a i which was obtained as described.
’*
APPENDIX B: CFAm’COEFFICIENTS
We find the C;km’ coefficients to be as follows: c4 4;mr-1 m‘ =++[(5-m ’)(4+m ’)]l” ,
c $ $ +=-+[(5+rn’)(4-m’)1’/~ ~ ~;;l-~
,
, l+$[(4+m = ’)(5+m ’)]1/2 ,
= ++[(4-m’)(5-m‘)11/’
~ 3,m’ 4 , ~ ’ +
c ~ Z : -=-+[(3+m ~ ’)(4+m ’)]1/2
,
= - +[(3-m’)(4-m’)]1/2
,
~ 43 ,.mm, +’ l
c:;Z~.-~=-+[(4-rn’)(3+m’)l1/’
,
.
c 3 3 :m m’ , + l = + + [ ( 4 + r n ’ ) ( 3 - ~ ~ ’ ) ] ~ / ~
46 MEASUREMENT O F T H E 6S+7S M1 TRANSITION I N . . .
29
1M. Phillips, Phys. Rev. 88,202 (1952). *M. A. Bouchiat and C. Bouchiat, J. Phys. (Paris) 35, 899 (1974); 36, 493 (1975). 3D7V. Neuffer and E. D. Commins, Phys. Rev. A 16,1760 (1977). 4V. V. Flambaum, I. B. Khriplovich, and 0. P. Sushkov, Phys. Lett. 177 (1978). 5M. A. Bouchiat and L. Pottier, J. Phys. (Paris) Lett. 37, L79 (1976). 65. Hoffnagle, L. Ph. Roesch, V. Telegdi, A. Weis, and A. Zehnder, Phys. Lett. @A,143 (1981); J. Hoffnagle, Ph.D. thesis, Swiss Federal Institute of Technology, Zurich, 1982 (unpublished). 'S. Chu, E. D. Commins, and R . Conti, Phys. Lett. 96 (1977). *W. Itano, Phys. Rev. A 22, 1558 (1980). 9P. Bucksbaum, E. Commins, and L. Hunter, Phys. Rev. Lett. 48, 607 (1982). loM. A. Bouchiat, J. Guena, L. Hunter, and L. Pottier, Phys. Lett. 358 (1982). "R. R. Lewis and W. L. Williams, Phys. Lett. B 59, 70 (1975).
m,
a,
m,
143
12M. A. Bouchiat and L. Pottier, in Proceedings of the International Workshop on Neutral Current Interactions in Atoms, (Cargese, 19791, edited by W. L. Williams (unpublished); M. A. Bouchiat, M. Poirer, and C. Bouchiat, J. Phys. (Paris) 40, 1127 (1979). 13P. Bucksbaum, in Proceedings of the International Workshop on Neutral Current Interactions in Atoms (Cargese, 19791, edited by W. L. Williams (unpublished). We are aware that an experiment along the lines discussed in this reference is being worked on by P. Drell and E. D. Commins (private communication). I4M. A. Bouchiat and M. Pokier, J. Phys. (Paris) 43, 729 (1982). lSR. N. Watts, S. L. Gilbert, and C. E. Wieman, Phys. Rev. A 27, 2769 (1983). 1%. L. Gilbert, R. N. Watts, and C. E. Wieman, Phys. Rev. A 27, 581 (1983). I'M. A. Bouchiat and L. Pottier, Lnser Spectroscopy III, edited by J. L. Hall and J. L. Carlsten (Springer, Berlin, 1977). 18M. A. Bouchiat, J. Guena, L. Hunter, and L. Pottier, Opt. Commun. e , 3 5 (1983).
PHYSICAL REVIEW A
VOLUME 29, NUMBER 6
JUNE 1984
Absolute measurement of the photoionization cross section of the excited 7s state of cesium S. L. Gilbert, M. C. Noecker, C. E. Wieman Department of Physics, University of Michigan, Ann Arbor, Michigan 48109 (Received 5 December 1983) We report the first measurement of the absolute cross section for photoionization of the 7 s state of cesium. The measurement employed a new technique in which the density of the excited-state atoms was determined by the amount of fluorescence. The cross section for photoionization by 540-nm light is 1.14( 1O)X cm2. We also propose a second new technique for the absolute measurement of photoionization cross sections which is based on modulated fluorescence.
I. INTRODUCTION
The photoionization behavior of alkali-metal atoms has been a subject of considerable theoretical and experimental study. Cesium has been of particular interest lately because the effects of spin-orbit interaction and core polarization are more pronounced than in the lighter alkali metals. This has been quite significant because the spin-orbit interaction gives rise to the Fano effect which is used for producing polarized electrons. However, these effects make the theoretical calculation of the photoionization cross section more difficult, particularly for S states. Simple quantum-defect theories which are adequate for light alkali metals give rather poor agreement with experimental results for cesium. Weisheit' and Norcross' have carried out semiempirical calculations in which they included both core polarization and the spin-orbit interaction, while ab initio calculations by Chang and Kelly,3 Johnson and Soff; and Huang and Starace' treated the spin-orbit interaction but neglected core polarization. Lahiri and Manson6 have done a simple Hartree-Slater calculation for a number of low-lying states. Attempts to check the accuracy of these theoretical approaches have been limited by the lack of accurate and consistent experimental data on absolute cross sections. In modem times there have been only three absolute measurements of photoionization cross sections for S states of cesium, all of which were for the 6 S ground state. Marr and Creek' obtained results for the ultraviolet region of the spectrum which were about a factor of 2 larger than the values calculated by either Weisheit or Norcross. However, Cook et a1.' repeated this measurement and obtained values in agreement with those calculations but with uncertainties of ?30%. Grattan er aL9 have also measured this cross section at a wavelength in the vacuum ultraviolet with similar results. We report here the absolute measurement of the cross section for the 7 s state accurate to +9%. This measurement used the new technique of "fluorescence normaiization" in which the excited-state atomic density was determined from the amount of fluorescence. This technique avoids the uncertainties in determining the molecular background and the ground-state atomic density which have limited these previous absolute measurements. 29 -
We shall conclude with the discussion of another technique for the absolute measurement of photoionization cross sections. The technique proposed is an extension of fluorescence normalization but under some conditions has substantial advantages over fluorescence normalization and other techniques which have previously been used.
11. EXPERIMENTAL METHOD A N D APPARATUS
For this measurement a beam of cesium atoms in an electric field was excited to the 7s state by a cw dye laser. This normally forbidden transition is allowed due to the small (5 parts in lo') mixing of S and P states by the static electric field. The number density of 7s atoms was determined from the amount of fluorescence emitted as the atoms spontaneously decayed. The laser radiation also caused a small fraction of the 7 s atoms to be photoionized, and the resulting ion current was measured. Strictly speaking, the process observed is two-photon photoionization with a resonant intermediate state. It is an extremely good approximation to treat this as two single-photon processes, however, because both the 6S-7.S transition and the 7S-+continuumtransition were far from saturation. The density of excited atoms was lo-* times the density of ground-state atoms, and only 1 part in lo4 of the excited atoms was photoionized. This allows the 7 s photoionization cross section to be determined from the fluorescence signal, the laser power, and the photoionization current. The apparatus is shown schematically in Fig. 1. A highly collimated beam of atomic cesium intersected a standing-wave laser field at right angles in a region of static-electric field. The cesium beam was produced by a two-stage oven to reduce the dimer fraction. The output nozzle was a microchannel plate which produced a beam with a cross section 0.5X2.5 cm2 and a half-angle divergence of -0.05 rad. This passed through a multislit collimator which reduced the divergence in the direction of the laser beam to 0.013 rad. A final 2-cm-wide aperture just before the intersection with the laser beam provided a precisely defined intersection geometry 2.0 cm long with a diameter equal to that of the laser beam (0.05 cm). The cesium beam density (-5X 109/cm3) was uniform to 3150
47
@ 1984 The
American Physical Society
48
29 -
ABSOLUTE MEASUREMENT OF THE PHOTOIONIZATION CROSS INTERFEROMETER MIRROR
...
3151
fl
il
FIG. 1 . Schematic of apparatus.
within a few percent over the intersection region because the total distance from nozzle to intersection (6 cm) was much shorter than the distance scale for beam-density redistribution along the direction of the laser (100 cm). This uniformity was confirmed using a hot wire detector. The standing-wave laser field was produced by an amplitude- and frequency-stabilized cw ring dye laser mode-matched into an electronically tunable, semiconfocal Fabry-Perot interferometer buildup cavity. The laser had a typical output of 250 mW and a short-term frequency jitter of 0.5 MHz peak to peak. With the resonant frequency of the interferometer locked to the laser frequency, the electromagnetic field strength inside the interferometer corresponded to two linearly polarized traveling waves, each with 116 times the power incident on the buildup cavity. A calibrated photodiode measured the light transmitted through the cavity. The intensity profile for the laser beam in the cavity is that of the lowest-order eigenmode of a semiconfocal cavity with a 25-cm focal-length curved mirror." This is a Gaussian profile with a spot-size radius of 0.021 cm at the input mirror and 0.029 cm at the output mirror. Less than 1% of the power was contained in higher-order modes. The 7s-state population was monitored by observing the 850- and 890-nm light that was emitted in the 6P-6S step in the 7S-6P-6S cascade decay. This light was detected by a cooled silicon photodiode 0.5X 5.5 cm2 that sat 9 mm below the interaction region. Glass color filters not shown in Fig. 1 prevented scattered 540-nm laser light from reaching the detector. The static-electric field was produced by applying voltage to the top field plate and grounding the lower. The laser and cesium beams intersected in the middle of the 5 x 7.5 cm2 electric field region. The top field plate was coated with evaporated gold and the lower field plate was coated with an electrically conducting optically transparent (84%) coating. The photoionization current was measured using an ammeter in the line providing voltage to the top field plate. Data were obtained by scanning the laser over the 6SF=4+7SF=4 Stark-induced transition and simultaneously recording the fluorescence signal and the photoionization current. Typical data are shown in Fig. 2. Scans
100 200 300 LASER FREQUENCY ( M H t )
FIG. 2.
(a)
Photoionization current vs laser frequency. (b)
Fluorescence detector current vs laser frequency. Zero of the laser frequency scale was chosen arbitrarily.
were made at several voltages between 1500 and 3000 V/cm and with both positive and negative voltage. The fluorescence line shape is composed of a narrow (14 MHz) resonance peak superimposed on a low broad (575 MHz) pedestal. The pedestal arises from a diffuse background vapor of cesium in the interaction region and is discussed in more detail in Ref. 1 1 . The photoionization current has an identical line shape but it has an additional background component which is independent of laser frequenCY.
The 7 s number density was calculated from the 7sstate lifetime and the total emitted fluorescence. The determination of the total fluorescence is inherently the least accurate and aesthetically the most unpleasant part of the experiment, as it involves finding the detector quantum efficiency and the detection solid angle. The detector size, uniformity, and angular dependence of the response were measured. The quantum efficiency was given as 0.80(4) by the manufacturer.12 The detection solid angle was calculated by numerical integration using the geometry of the apparatus and the measured transmission of the lower field plate and filters at various angles of incidence. This included the light which was reflected off the upper field plate, the reflectivity of which was measured. There was a negligible contribution due to reflection off other surfaces in the apparatus since these were all far from the interaction region and painted flat black. The laser power in the buildup cavity was determined by dividing the transmitted power by the transmission coefficient for the output mirror. The power measurement was made using a photodiode calibrated against a Coherent Inc. power meter recently calibrated to a NBS standard.
49 3152
S. L. GILBERT, M. C. NOECKER, A N D C. E. WIEMAN 111. RESULTS
their value for the Cooper minimum would explain the discrepancy. They believe the uncertainty in the calculation of this minimum is considerably larger than 0.001 eV.
The total photoionization current is given by I=e J-s(F)~(F)~~F,
29 -
(1)
r
where S(i)is the laser photon flux, n (F) is the density of 7 s atoms, and u is the photoionization cross section of interest. The fluorescence signal is
IV. EXTENSIONS OF PRESENT WORK
The present work has provided the first measurement of a photoionization cross section for an excited S state of cesium. This provides a good test of the different theoretical treatments of photoionization of cesium, a better test than has been possible using the less accurate and disparate values measured for the ground state. However, it is obviously desirable to measure the dependence of this cross section on wavelength. This could be done using the 21gaw2 fluorescence normalization technique if a second laser was U= (3) used to do the photoionization. The obvious choice for a FSor ’ second laser would be a relatively high-power pulsed laser. where So is the total number of laser photons per second However, such a laser would allow the use of a new techand w is the Gaussian beam radius at the interaction renique for measuring the cross section which is something gion. Our measurements give the cross section in cm2 as of a hybrid between fluorescence normalization and the popular saturation te~hnique,’~ but can have significant (T= 1.14( 1O)xl o p i 9. advantages over both approaches. The significant contributions to the 9% uncertainty are This technique, henceforth called modulated fluores5% for detector quantum efficiency, 6% for laser power, cence, could be used quite generally for determining 3% in measurement of photoionization current, 3% for excited-state photoionization cross sections. In the modusolid angle, and 2% for the 7s lifetime. lated fluorescence technique one would excite the atoms to There are several possible sources of additional systhe state of interest with one laser and monitor the change tematic error which we have considered. First, there was in the fluorescence when the photoionizing laser (or lamp) a background current which we attribute primarily to is pulsed. The photoionization rate would then be simply photoemission from surfaces; however, this background the fractional change in the fluorescence signal divided by was eliminated in the analysis by using only the 14 MHzthe lifetime of the state. The only quantities which must wide components of the measured currents shown in Fig. be determined absolutely are the lifetime of the state and 2. A second possible source of error was multiplication of the photoionizing laser intensity. In this respect it is simithe photoionization products through collisions with neular to the saturation technique and superior to fluorestral atoms or secondary surface emission. This can be cence normalization. However, it has a significant advanruled out by the observation that the measured photoionitage over the saturation approach in that it requires conzation cross section was independent of the strength of the siderably lower photoionization rates and hence lower applied field. At the highest field there was a substantial laser power. In the saturation method it is necessary to increase in the noise on the photoionization current, howachieve ionization rates which are at least comparable and ever, suggesting some multiplication or arcing phenomepreferably several times larger than the spontaneous decay na. Therefore only the values obtained at the lowest elecrate. However, the modulated fluorescence technique will tric field (1500 V/cm) were used in obtaining u. Another work with photoionization rates which are a small fracpossible source of systematic error is anisotropic radiation tion of this. The necessary fraction is ultimately limited trapping. Calculations indicate this should be quite small. by the signal-to-noise ratio of the fluorescence measureHowever, we also checked this empirically by measuring ment, but this ratio is characteristically quite high. There the ratio of fluorescence signal to beam density as a funcare many cases where the necessary photoionization rate, tion of beam density. When the density was increased RpI could be 10-2-10-3 of the decay rate. from 5 to 40 times lo9 atoms/cm3 this ratio decreased by The three techniques mentioned are thus complementaless than 20%. This indicates that radiation trapping is ry to each other since each has a range of experimental not a significant effect for the density ( 5 x 109/cm3) at conditions where it is generally superior. If unlimited which the photoionization measurement was made. laser power is available ( RpI > 1/71 the saturation techThe cross section we obtain can be compared with the nique would usually be best because it is usually easier to theoretical value obtained by Lahiri and Mansom6 Their measure photoionization current than fluorescence. For calculation predicts a value about one half of what we an intermediate rate [ loL3(1/ r )< RPI < 1/ r ] the modulatmeasure. However, 540 nm is in a region where their caled fluorescence technique would be superior. Finally, for culated cross section has a very strong dependence on eneven lower photoionization rates, such as in the experiergy (wavelength) so that an error of only 0.001 eV in ment reported, the problem of obtaining a sufficiently where 4 is the detection efficiency and T the 7s lifetime.” As mentioned previously, the 6s -7s excitation rate and 7 s photoionization rate are much smaller than 1/ r . This allows us to assume that n (it) has the same Gaussian form as S(F). With this substitution, Eqs. ( 1 ) and (2) can easily be solved to yield
2
ABSOLUTE MEASUREMENT OF T H E PHOTOIONIZATION CROSS . . .
high signal-to-noise ratio to observe the modulated fluorescence becomes worse than the additional difficulty of determining the absolute fluorescence detection efficiency. In this case the fluorescence normalization technique would be the best choice.
'J. Weisheit, Phys. Rev. A 5, 1621 (1972). Norcross, Phys. Rev. A 7, 606 (1973). 3J. Chang and H. Kelly, Phys. Rev. A 5, 1713 (1972). 4W. R.Johnson and G. Soff, Phys. Rev. Lett. So, 1361 (1983). *K. N. Huang and A. F. Starace, Phys. Rev. A 19,2335 (1979); 22, 318 (1980). 65. Lahiri and S. Manson (private communication). 7G. V. Marr and D. M. Creek, Proc. R. Soc. London, Ser. A 304, 233 (1968). *T. B. Cook, F. B. Dunning, G. W. Foltz, and R. F. Stebbings, Phys. Rev. A 15,1526 (1977). 9K. Grattan, M. Hutchinson, and E. Theocharous, J. Phys. B 13, 2931 (1980). 2D.
3153
ACKNOWLEDGMENTS
We are pleased to acknowledge the assistance in this work of R. N. Watts and T. Miller and the encouragement of D. Norcross. This work was supported by the National Science Foundation.
IOA. E. Seigman, Introduction to Lasers and Masers (McGrawHill, New York, 1971), Chap. 8. "S. L. Gilbert, R. N. Watts, and C. E. Wieman, Phys. Rev. A 29, 137 (1984). I2This is a custom-made photodiode supplied and calibrated by Hughes Aircraft Co., Industrial Products Division. I3J. Hoffnagle, V. L. Telegdi, and A. Weis, Phys. Lett. 457 (1981). I4R. V. Arnbartzumian, N. P. Furzikov, V. S. Letokhov, and A. A. Puretsky, Appl. Phys. 2, 335 (1976); U. Heinzmann, D. Schinkowski, and H. D. Zeman, ibid. l2, 113 (1977); A. V. Smith, D. E. Nitz, J. E. M. Goldsmith, and S. J. Smith, Phys. Rev. A 22, 577 (1980).
m,
VOLUME55, NUMBER 24
P H Y S I C A L R E V I E W LETTERS
9 DECEMBER 1985
Measurement of Parity Nonconservation in Atomic Cesium S. L. Gilbert, M. C. Noecker, R. N. Watts, and C. E. Wieman(’) Joint Institute for Laboratory Astrophysics, Universi@of Colorado and National Bureau af Standards, Boulder. Colorado 80309 (Received 1 1 September 1989 A new measurement of parity nonconservation in cesium is reported. The experimental technique involves measurement of the 6s- 7s transition rate by use of crossed atomic and laser beams in a region of perpendicular electric and magnetic fields. Our results are I m l PNC//3= - 1.65 f 0.13 mV/crn and C t - 2 zk 2. These results are in agreement with previous measurements in cesium and the predictions of the electroweak standard model. This experimental technique will allow future measurements of significantly higher precision.
-
PACS numbers: 35.10.Wb. 11.30.Er.12.30.C~
The standard model for the electroweak interaction predicts a parity-nonconserving (PNC) neutral-current interaction between electrons and nucleons. In 1974 the Bouchiats’ proposed that this effect might be observable in large-Z atoms, thus generating a decade of experimental effort. Measurements of PNC have now been made in bismuth and lead by observation of the optical rotation of light,’ and in thallium3s4 and cesiums by use of the technique of Stark interference. Here we report a new measurement of parity nonconservation in cesium which is more precise than the previous measurements in atoms and is approaching the precision of the best high-energy test of the electroweak theory. This result is in good agreement with previous measurements in cesium. By comparing the PNC observed on two different hyperfine transitions we also set a much lower limit for the proton-axialvector PNC contribution. Parity-nonconservation measurements in atoms are valuable because they test the electroweak theory in a different regime from that probed by high-energy experiments. As well as being sensitive to a very different energy scale for the exchange of virtual particles, atomic experiments also involve a nearly orthogonal set of electron-quark couplings. Because of this, atomic PNC measurements, when combined with high-energy results, can provide useful tests of the electroweak radiative corrections and alternatives to the standard model. Deriving information about the basic neutral-current interaction from atomic PNC measurements requires knowledge of the atomic wave functions. Cesium is a particularly good atom in this respect because of its single-electron character. This enables a more direct and accurate calculation of the wave function than is possible for other heavy atoms. The basic experimental concept has been discussed previously6 but we will review the essential points. The PNC interaction in an atom mixes the S and P eigenstates, allowing a small electric dipole ( E 1) transition amplitude between states of the same parity. In all atomic PNC experiments, this parity-nonconserving 2680
amplitude (APNC) is measured by observation of its interference with a much larger parity-conserving amplitude. In our experiment, the parity-conserving amplitude is a “Stark-induced” E 1 amplitude (A st) created by the application of a dc electric field to mix S and P states. The early Stark interference experim e n t ~ ~measured ,’ the spin polarization of the excited state due to this PNC interference. In our experiment, we are able to observe the interference directly in the transition rate by applying a magnetic field to break the degeneracy of the Zeeman levels. The idea of using electric and magnetic fields was proposed by a number of authors’ and used in the recent thallium PNC measurement of Drell and C o m m i n ~ .A~ similar technique was also demonstrated in our measurement6 of the Cs 6S 7s magnetic dipole amplitude ( A M l ) . A unique feature of our approach is the use of crossed laser and atomic beams. This yields narrow transition linewidths which have the important experimental advantage of allowing the use of a small ( < 100 G) magnetic field. Other desirable features of an atomic-beam experiment include the reduction of collisions, radiation trapping, and molecular backgrounds. We measure the PNC interference term on both the 6 & = 4 - + 7SF,_, and 6 S F E 3 - +7SF,-4 hyperfine lines shown in Fig. 1. The basic field configuration for the experiment is shown in Fig. 2. A standing-wave laser beam along the 9 axis excites transitions in a region with an electric field (E)in the k direction and a magnetic field (B) along the 2 axis. The laser field has polarization d = eZ2 i e x % , where ex and ez are real. For any transition between particular Zeeman sublevels ( m and m“)the transition probability is +
+
I-I~S~+AMMI+A~NCI~,
(1)
where each A is a function of F, F , m, and m’. Using the results of Gilbert* we substitute for the amplitudes in Eq. (1) and obtain for F Z F ,
r;y=
[pZE2€,2 T 2pE€, I m 8 PNCEx1 F’m’ 2 X ( C p m ) 6m,mtkl
@ 1985 The American Physical Society 51
(2)
52
PHYSICAL R E V I E W LETTERS
VOLUME55, NUMBER 24
9 DECEMBER 1985 INTERFEROMETER MIRROR
1.36~ 1.47~
‘LASER
BEAM
0
FIG. 2. Interaction region and field configuration.
FIG. 1. Cesium energy-level diagram showing hyperfine and weak-field Zeeman structure of 6s and 7s states.
plus negligibly small terms involving only 2? PNC and A M 1 . The first term in the brackets is the pure Starkinduced transition rate where p is the vector transition polarizability defined in Ref. 1. The second term is the interference between Ast and the much smaller amplitude A p N c . The quantity i g p N C is the PNC electric-dipole reduced matrix element. The coefficients C are related to Clebsch-Gordan coefficients and are tabulated in Ref. 6. In the low-magnetic-field limit, the spectrum of the F = 4- 3 transition is composed of eight lines with strengths, R ( i ) , given by where i = - 3 to +4. The two outermost lines of the multiplet involve only a single transition ( m = 4- 3 and m = - 4 to - 3, respectively) while the other lines are each the sum of a A m = 1 and a A m = - 1 transition. This spectrum (identical to the F = 3 4 spectrum) is shown in Fig. 3, where the transition rate for each line is the sum of spectra (a) and (b). As we discussed in Ref. 6, the magnetic-field-induced mixing of hyperfine states is small and has no effect on the experiment, although it does cause the slight asymmetry seen in Fig. 3 (c). From Fig. 3 and Eqs. (2) and (3) it is now easy to understand the essence of the experiment. The laser is set to one of the end lines of the multiplet and, by reversing various fields, we change the sign of the PNC interference term without affecting the larger pure Stark-induced rate. Thus we isolate the PNC
term by observing the modulation in the transition rate with reversals of the E field, the B field, and the sign of cX (handedness of polarization). An additional reversal (“m” reversal) of the interference term is achieved by a change of the laser frequency to the other end of the multiplet. The use of four independent reversals is extremely helpful in the suppression of possible systematic errors. The experimental setup is similar to that used in
+
+
FIG. 3. 6&-47SF-3 transition. (a) Theoretical pure Stark-induced spectrum. (b) Theoretical parity-nonconserving interference spectrum on expanded scale. (c) Experimental scan of the transition with B = 70 G .
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53 VOLUME55, NUMBER 24
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Ref. 6. About 500 mW of laser light is produced by a ring dye laser at the 540-nm transition frequency. Servocontrol systems provide a high degree of frequency and intensity stabilization. A Pockels cell with k X/4 voltage applied selects the handedness of the laser polarization. Following the Pockels cell the laser beam enters a Fabry-Perot power-buildup cavity, the length of which is controlled to keep it in resonance with the dye-laser frequency. This produces a standing wave in the cavity with a field which is nearly 20 times larger than the incident field. An intense, well-collimated cesium beam intersects this standing wave at right angles in a line 2.7 cm long. The atomic beam is produced in a two-temperature oven with a capillary-array nozzle followed by a multislit collimator. At the intersection the cesium flux is 1x 10" atoms cm-2 s-'. The interaction region is shown in Fig. 2. Situated 2 mm above and below the line of intersection are flat glass plates with transparent electrically conductive coatings. The dc electric field is produced by application of positive or negative voltage to the top plate and grounding of the lower. Data were taken with use of values of the electric field ranging from 2.0 to 3.2 kV/cm. A 70-G magnetic field parallel to the atomic beam is produced by Helmholtz coils. 7s transition rate is monitored by obserThe 6 s vation of the light which is produced by the ~ P , J ~ , ~6sJ branch ~ - + of the 7s decay. This light is detected by a cooled silicon photodiode below the lower field plate. A gold cylindrical mirror above the top field plate images the interaction region onto the detector. Colored-glass filters in front of the detector block the scattered green laser light. The detector is carefully shielded to reduce electrical pickup, and is quite insensitive to the magnetic field. The output of the photodiode is amplified and sent into a gated integrator controlled by a PDP-11/23 computer. This computer also controls the B, E, and P (polarization) reversals. The reversal rates are 0.02, 0.2, and 2 Hz, respectively, with regular 180" phase shifts introduced in the switching cycles. After a brief dead time to avoid transient effects, the detector current is integrated, digitized, and stored for each half cycle of P. The m reversal is done manually every 30 min. A data run consisted of about 8 h of data accumula3 and tion divided equally between the F -4F = 3 4 transitions. Systematic error checks (discussed below) were made at the beginning and end of each run. Typical experimental conditions were E =2500(15) V/cm and ex/rz==0.94(l), giving a detector current of 3 x lo-'' A and a parityFor the data nonconserving fraction of 1.3 x used here the noise was 2 to 3 times worse than the statistical shot-noise limit, primarily because of noise from laser-light-induced fluorescence in the optics. +
This resulted in an integration time of 20 to 30 min for a ? 100% measurement of the PNC contribution. The data were analyzed by finding the fraction of the , modulated with P,E, B, and transition rate, A p ~ c that m. From Eq. (2) it can be seen that ApNC = 2 ( e x / e , ) (Im %'&E,6). A small calibration correction, 5.0(5)%, was made to account for the incomplete resolution of the lines in the multiplet. Systematic errors, namely, contributions to the signal which mimic the parity nonconservation under all reversals, were a fundamental concern in the design and execution of the experiment. Our approach to the identification and measurement of these contributions was similar to that used in earlier Stark-interference The transition rate was derived for the general case, allowing for all possible componenfs of E, B, Q , and the oscillating magnetic field, r x k . Each of these components was given a reversing and a nonreversing (stray) part. With use of empirically determined limits, all terms which could contribute false signals amounting to greater than 1% of the true PNC were then identified and a set of auxiliary expefiments was designed to measure them. These terms and most of the auxiliary experiments have direct counterparts in the work discussed in Refs. 3-5. One test, however, which is unique to this experiment is the in situ measurement of the birefringence of the buildup-cavity mirror coatings. Table I shows the results from a typical data run along with all the significant corrections due to false PNC signals. The average of all the data runs has a systematic correction of 14(1)%. As a result of space limitations a detailed discussion of these correction terms will be given in a subsequent paper. We made a number of other tests to confirm that there are no additional sources of systematic error. Among these were the introduction of known nonreversing fields, misalignments, and birefringences. All of these produced false PNC signals which agreed with the sizes predicted by the calculation discussed TABLE I. Raw data and corrections to a typical run. AEz and AE, are stray electric fields. 6 represents the birefringence of the buildup-cavity mirror coatings and M is the M1 matrix element.
-
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9 DECEMBER 1985
Fractional modulation ( X lo6) A.PN= (raw data) 4-3 3-4
Corrections A E, EJ ( E , )~ A EyB,/ (E X 5 ,) Mt'/ (PEx)
- 1.82(40)
- 1.49(45) -0.02(1) +0.23(4) O.Ol(2)
-
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VOLUME55, NUMBER 24
P H Y S I C A L R E V I E W LETTERS
above. Also, analysis of the data showed that on all time scales, from minutes to days, the distribution of values for I m I PNC/Pwas completely statistical. This included data taken with two different sets of buildupcavity mirrors, different electric-field plates, several complete realignments of the experiment, and different dc electric fields. Our results are
I
- 1.51 k0.18 mV/cm I m O PNC/P= - 1.80 -t 0.19 mV/cm - 1.65 k 0.13 mV/cm
-
(F=4(F=3
31, 41,
(average),
where the uncertainty includes all sources of error. This is in good agreement with the value of - 1.56 rt 0.17 k 0.12 mV/cm reported by Bouchiat el a/.’ for the average of measurements made o n the 4 - 4 and 3 - 4 hyperfine lines. With p - 2 7 . 3 ( 4 ) a d (see Ref. 91, we have
9 DECEMBER 1985
Because the uncertainty is almost entirely statistical we believe that we can achieve significantly higher precision with future refinements to this experiment. The most immediate gain will come from improved optics which will increase the signal size while decreasing the noise. A more substantial change is the use of a spin-polarized atomic beam. This would provide an increase in signal and allow a number of interesting options on the experimental design. W e have developed a polarized cesium beam of the necessary purity and intensity’‘ and will be exploring these options in the future. W e are pleased to acknowledge the donation of equipment by Jens Zorn and valuable discussions with Dr. J. Ward, Dr. A. Gallagher, Dr. J. Sapirstein, and Dr. W.Johnson. This work was supported by the National Science Foundation. One of us (C.E.W) is an A. P. Sloan Foundation fellow.
x 10”eao. I m g P P N C -0.88(7) =
Using a calculated value for the atomic matrix element we can compare our measurement with the predictions of the standard model for the weak charge, Q,. Since a discussion of the atomic-physics calculation is beyond the scope of this paper, we will simply take the range of reasonable values to be g P ~ ~ (0.85 = = to 0.97) x 10-”ieao(Qw/N) and refer the reader elsewhere” for a review of this subject. When combined with our measured I PNC we obtain - Qw in the range (71 to 81) f 6, where 6 is the experimental uncertainty. This is in agreement with the value of 70 k 4 predicted by the standard model. This agreement provides an improved test of the electroweak theory at low energies and has several specific implications as discussed by Robinett and Rosner” and Bouchiat and Piketty.12 These include improved limits on the radiative corrections to the electroweak theory and on the masses and couplings for additional neutral bosons. From the comparison of the PNC measurements made on the two hyperfine lines13 we find that the proton-axial-vector-electron-vector coupling constant is C,= - 2 ? 2. This is in agreement with the predicted value of 0.1 and is a substantial improvement over the previous experimental limit of IC2,,l < 100 (Ref. 5 ) . To obtain a more precise value for Qw from future experiments the uncertainty in the atomic theory must be reduced. There are presently a number of groups working on this problem and improvements may well be forthcoming in the near future. It is worth noting that this challenge is leading to new ideas and insights into atomic-structure calculations.
(a)Also at Department of Physics, University of Colorado, Boulder, Colo. 80309. ‘M. A. Bouchiat and C. Bouchiat, J. Phys. (Paris) 35, 899 (1974), and 36,493 (1975). 2A review of experiments in this field is provided by E. N. Fortson and L. L. Lewis, Phys. Rep. 113. 289 (1984). 3P. H. Bucksbaum, E. D. Commins, and L. R. Hunter, Phys. Rev. D 24, 1134 (1981). 4P. S. Drell and E. D. Commins, Phys. Rev. Lett. 53, 968 (1984). 5M. A. Bouchiat, J. Guena, L. Hunter, and L. Pottier, Phys. Lett. 117B, 358 (19821, and 134B, 463 (1984). %. L. Gilbert, R. N. Watts, and C. E. Wieman, Phys. Rev. A 29, 137 (1984). ’R. R. Lewis and W. L. Williams, Phys. Lett. B 59, 70 (1975); M. A. Bouchiat, M. Poirer, and C. Bouchiat, J . Phys. (Paris) 40, 1127 (1979); P. H . Bucksbaum, in Proceedings of the Workshop on Parity Violation in Atoms, Cargbse, Corsica, 1979 (unpublished). 83. L. Gilbert, Ph.D. thesis, University of Michigan, 1984 (unpublished); also see Ref. 7 for a similar derivation. 9This is a semiempirical value of p obtained with use of the approach discussed in Ref. 6 and references therein. The value given here utilizes updated experimental and theoretical inputs which will be discussed in a subsequent paper. low. R. Johnson, D. S. Guo, M. Idrees, and J . Sapirstein, Phys. Rev. A 32, 2093 ( 1 9 8 9 , and a subsequent paper including first-order corrections (to be published). IIR. Robinett and J. Rosner, Phys. Rev. D 25, 3036
(1982).
12C.Bouchiat and C. Piketty, Phys. Lett. 128B, 73 (1983). 13V.N. Novikov et u / . , Zh. Eksp. Teor. Fiz. 73, 802 (1977) [Sov. Phys. JETP 46, 420 (1977)l. 14R.N. Watts and C. E. Wieman, to be published.
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Volume 57, number 1
OPTICS COMMUNICATIONS
1 February 1986
THE PRODUCTION OF A HIGHLY POLARIZED ATOMIC CESIUM BEAM R.N. WATTS and C.E. WIEMAN
'
Joint Institute for Laboratory Astrophysics, University of Colorado and National Bureau of Standards, Boulder, CO 80309, USA Received 23 October 1985
We have produced a cesium beam of l o i 4 atoms s - ' with more than 98% of the atoms in a single spin state. The beam was polarized by optically pumping it with a single laser diode whose frequency was switched between the 6s( F = 4)-6p,,,( F = 5) and 6 s ( F = 3)-6p3,,(F= 4) transitions at 100 kHz. To d o this, it was necessary to determine the frequency modulation characteristics of the laser. The same laser was used to simultaneously probe the atomic population distribution. We find this to be a remarkably simple and inexpensive way to produce a highly spin polarized beam.
There has long been an interest in producing spinpolarized beams of alkali atoms. Because such beams are in a single well-defined quantum state, they are invaluable tools for studying atomic collisions with other atoms, electrons, or surfaces [I]. In addition, they are useful in a variety of spectroscopic measurements. For example, we plan to use a spin polarized beam in an improved measurement of atomic parity violation [ 2 ] ,Finally, by transferring the electron polarization of one species to the nucleus of another, spin polarized atoms are used to produce polarized nuclei for targets and ion sources [3]. In this paper, we present a new method to spin polarize a beam using optical pumping. We cause the output frequency of a single diode laser to jump rapidly between two different 6 s - 6 ~ ~ 1hyperfine 2 transitions in cesium. These transitions are chosen to deplete one ground state hyperfine level and to polarize the other. We find that, using this technique, we can put essentially 100% of the beam into the 6s(F= 4,Mf = 4) spin state. Polarized beams were originally produced by state selection with an inhomogeneous magnetic field. However, this technique severely limited both the beam intensity and the degree of polarization. The invention
'
of tunable dye lasers dramatically improved this situa. tion by making it possible to optically pump atomic beams. In optical pumping, an atom repeatedly absorbs circularly polarized light and decays spontaneously. After several such cycles the atom is spin polarized along the direction of the laser. The difficulty in this technique lies in the large ground state hyperfine splitting present in all alkalis compared to the narrow linewidth of the pumping laser. Generally, only one of the two hyperfine levels can be in resonance with the laser and so only those atoms will be polarized. Several clever schemes have been devised for overcoming this problem. In one method, an atomic beam is state selected with a hexapole magnet to remove the atoms in one hyperfine level. The atoms in the remaining level are then polarized [4]. For small hyperfine splittings, acoustooptic frequency modulators [5,6] or a broad band multimode laser [6] have been used to pump both hyperfine levels. A fourth approach has been to mix the two hyperfine levels by applying large microwave fields [7]. All of these methods are relatively complicated, In addition, the best polarizations achieved have ranged from 70 to 90%. Diode lasers have some obvious advantages for spin polarization. First, they are far less expensive than the dye lasers which have been used previously. In addition, they are much easier to use, being simple to operate and virtually maintenance free. Finally, they pos-
Also Department of Physics, University of Colorado, Boulder, CO 80309, USA. Sloan Foundation Fellow.
0 030-4018/86/$03.50 0Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) 55
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1 February 1986
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sess remarkable frequency modulation characteristics. We have found that it is possible to sweep the frequency of a diode laser by as much as 15 GHz within a microsecond. This makes them uniquely suited to polarizing an atomic beam: by jumping quickly between two transitions, one laser can pump both hyperfme levels. As an added bonus, these modulation characteristics make it possible to use the pumping laser to simultaneously determine the degree of beam polarization, In our experiment, we polarize cesium into the 6s(F = 4,Mf = 4) spin state shown in fig. la. This is done in two steps. First, the circularly polarized laser is tuned to the 6s(F = 3)-6p3p(F = 4) transition. This excited state decays to both the F = 3 and F = 4 ground state levels, so after a few transitions per atom the F = 3 population is completely pumped into the F = 4 state. The laser frequency is then tuned to the 6s(F = 4)-6p3p(F = 5) transition. This excited state can only decay to the F = 4 ground state and so after approximately ten absorptions and emissions the atom is completely polarized. For this process to work the transit time of the atom through the pumping region must be longer than the period of the frequency switching cycle, and must be much longer than the inverse of the transition rate, We find that diode lasers can easily meet both these criteria. To determine the resulting polarization of the cesium beam we measure both the depletion of the F = 3 state and the polarization of the F = 4 state. We make both of these measurements using a small fraction of the beam from the pumping laser. The F = 3 state de-
b’
pletion is obtained by measuring the absorption (or lack thereof) by the optically pumped atomic beam during the time the laser is tuned to the F = 3-4 transition. The F = 4 state polarization is derived from the measurement of the atomic beam circular dichroism (difference in the absorption of right versus left circularly polarized light) while the laser is in resonance with the F = 4-5 transition. Having this information and knowing the matrix elements that connect the various Zeeman levels, we can determine the degree of beam polarization. The success of our technique depends critically on the fast frequency response of a diode laser. Because diode lasers are routinely used in the communications industry, their intensity modulation characteristics have been well studied into the Gbit s-1 regime, In contrast, relatively little work has been done on their frequency modulation characteristics, especially in the 10 kHz to 1 MHz region. It is known that the lasing wavelength of a diode laser depends on the index of refraction of the active region which, in turn, depends on the diode temperature. Changes in injection current affect the temperature of the lasing junction [8] and thus change the wavelength. We have investigated the speed at which this happens using a Hitachi HLP1400 laser and the setup shown in fig. 2. The injection current to the laser consisted of two parts: a dc current of about 100 mA and a 5 kHz square wave. We measured the time delay between the edge of the square wave and when the laser frequency matched the resonance of the Fabry-Perot interferometer. By changing the interferometer resonant frequency by known amounts and determining the time delay for each change, we thus produced a graph of laser fre-
--
- 5 -4
6201 MHz
3 2
LASER
V V POLARIZER
PD
85218
I
_I -2- 3
4
MF
Fig. 1. Cesium energy level diagram showing the transitions used to spin polarize the beam. The u’s give the relative tranxition probabilities out of the F = 4, Mf= 4 level.
46
Fig. 2. Schematic of apparatus used to determine diode laser frequency modulation response.
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OPTICS COM?vIUNICATIONS
I
3 k
'0
10
20
30
40
50 60 t (psec)
70
80
90
100
Fig. 3. Frequency shift versus time for a 5.6 mA injection current step.
1 February 1986
changes could be made even faster. It is somewhat remarkable that the laser frequency settles to 1 part in lo3 of such a large jump after only 1.5 ps. We also note that this frequency change is accompanied by about a 15% change in intensity. This has no effect on the pumping process, however, since for the laser power used, both transitions are well saturated. A schematic for the optical pumping experiment is shown in fig. 4. The cesium bean! is produced in a twotemperature oven heated to about 100°C. The atomic beam is collimated to 25 mrad divergence by a glass capillary array nozzle on the oven followed by a multislit collimator [ 9 ] .The beam typically has an intensity of 1014 s-l and a density of about 1010/cm3 as determined by a tungsten hot wire detector. The laser is mounted on a copper heat sink and is temperature stabilized to a few mK using a thermoelectric cooler The dc current is stabilized to 1 part in lo6 and added to it is the 100 kHz frequency switching component discussed above. The laser has a bandwidth of 30 MHz
*.
quency against time as shown in fig. 3. For times between 0.1 and 100 ps, we found that the curve of frequency versus time could be fitted as the sum of two exponential response curves with time constants of about 2 ps and 17 ps. We were subsequently able t o determine the shorter time constant somewhat more accurately by looking at the response of the laser t o a 0.2 ps current spike. The time constants were found to vary only slightly when the amplitude of the square wave was changed from 1 to 6 mA. It is interesting to note that these results are consistent with those given in ref. [8] for a different laser and larger modulations. This reference explains the shorter time constant as the time required for the chip to reach internal thermal equilibrium. The larger time constant is the time for the chip to come into equilibrium with the heat sink on which it is mounted. Knowing the response function of the laser, we could then make the laser jump rapidly between the two transitions used to polarize our beam. For slow (less than 10 kHz) frequency switching one simply adds a square wave component to the injection current. We need to switch at 100 kHz or faster, however, and hence must compensate for the exponential character of the frequency change. This was done by sending the square wave through two RC highpass filters. We found that by using time constants of 1.5ps and 32 ps, and by adjusting the relative amplitudes of the two filters appropriately, we could force the laser to jump the necessary 8.2 GHz in 1.5 ps and then change its frequency by less than 10 MHz over the remaining 3.5 /JS of a 100 kHz half cycle. Smaller frequency
* These lasers were wavelength selected to lase at 852 * 5 nm at 25°C and 10 mW output power. This is necessary to insure that they can be tuned to the cesium transition but does not affect the& modulation characteristics.
BEAM SPLITTER
PROBE BEAM
-1T b
P O L A R I ZERS
X/4
PLATES
REMOVABLE
ABSORPTION DETECTOR
t
X I 2 PLATE
FLUORESCENCE D E T E CTO R
Fig. 4. Schematic of apparatus used to spin polarize and probe the cesium beam.
47
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Volume 57, number 1
OPTICS COMMUNICATIONS
and a long-term drift of no more than 200 MHz in 5 hours. The output of the laser is collimated to a 0.5 cm diameter beam, then split into a saturating pump beam (the transition saturates at 1 mW/cm2) and a weak probe beam. The light is circularly polarized using a linear polarizer and a quarter wave plate. In addition, a half wave plate can be inserted into either the pump or the probe beams to reverse their respective polarizations. The power in the pump beam is about 5 mW at the interaction region while the probe beam has an intensity of about 0.1 mW. The laser beams intersect the cesium beam at right angles. Detectors monitor the absorption of the probe beam and the fluorescence of the pump beam. From the fluorescence signal we obtain an error signal which we feed back to the injection current to lock the center laser frequency to the 10 MHz wide cesium transition. The quantization axis is defined by a 1 G magnetic field directed along the laser beams, An atom with the most probable velocity spends about 20 ps in the pumping beam which is only two full periods of the laser switching cycle. However this is time for nearly 350 excitations and spontaneous decays. Thus an atom must only spend slightly more than one half cycle (5 ps) in the pump region to be polarized. This explains why even the atoms in the high velocity tail of the Doppler distribution are sufficiently pumped. To confirm this we modulated the laser at 200 kHz and obtained the same polarization. As mentioned previously the atomic beam polarization is obtained by monitoring the absorption of the probe laser beam. While the laser is tuned to the F = 3-4 trmsition we measure an absorption of 2.6% when the pump beam is blocked and an absorption of 0 0.1%when the pump is unblocked. This implies a 100 f 5% depletion of the F = 3 hyperfine level. From the calculated transition rates and interaction times we estimate that this level is at least 99% depleted, which agrees with the measured value. Next we measure the difference in the absorption of right and left circularly polarized light while the laser is tuned to the F = 4-5 transition. We find the ratio of the different absorptions is 33 f 5. From the ClebschGordan coefficients shown in fig. l b , this ratio would be 45 for an atom in the P = 4,Mf = 4 ground state. Thus our result can be interpreted in one of two ways (or some linear combination of the two). First, all the atoms are in the Mf = 4 state and the polarization of the probe light is imperfect. An addition of 0.8% of
*
48
1 February 1986
the wrong circular polarization would give the ratio observed. Given the quality of the optics used, we think that this is a very likely possibility. Alternatively, in the unlikely event that the laser polarization is perfect, some fraction of the population must be in the other Mf levels. Although the fraction implied by the results is quite dependent on the distribution among the Mf levels, if we assume that it is distributed uniformly, the measured ratio implies that theMf = 4 level contains 98% of all the atoms. Because the laser linewidth is twice the residual Doppler width, the entire atomic beam is being sampled by the probe laser beam. Thus we believe that essentially all the atoms are in the F = 4,Mf = 4 state. Simply reversing the laser polarization, of course, puts the atoms in the M f = -4 state. We have produced an intense, very highly polarized beam of atomic cesium in a simple, inexpensive, and compact manner. This technique will make possible an improved measurement of parity violation in cesium and has many other applications. With presently available diode lasers, this technique can easily be applied to rubidium and potassium as well, We would like to thank Dana Anderson, Alan Gallagher and Linden Lewis for helpful discussions. This work is supported by the National Science Foundation.
References [l] H. Kleinpoppen, Adv. At. Mol. Phys. 15 (1979) 423, and references therein. [2] S.L. Gilbert, M.C. Noecker, R.N. Watts and C.E. Wieman, to be published. [3] F.P. Calaprice, W. Happer, D.F. Schreiber, M.M. Lowry, E. Miron and X. Zeng, Phys. Rev. Lett. 54 (1985) 174. [4] D. Hih, W. Jitschin and H. Kleinpoppen, Appl. Phys. 25 (1981) 39; W. Dreves, W. Kamke, W. Broermann and D. Fick, Z. Phys. A303 (1981) 203. [ 5 ] G . Baum, C.D. Caldwell and W. Schroder, Appl. Phys. 21 (1980) 121. [6] J.T.Gusma and L.W. Anderson, Phys. Rev. A28 (1983) 1195. [7] W. Dreves, H. Jansch, E. Koch and D. Fick, Phys. Rev. Lett. 50 (1983) 1759. [8] P. Melman and W.J. Carlsen, Appl. Optics 20 (1981) 2694. [9] S.L. Gilbert,M.C. Noecker andC.E. Wieman, Phys. Rev. A29 (1984) 3150.
PHYSICAL REVIEW A
VOLUME 34, NUMBER 2
AUGUST 1986
Atomic-beam measurement of parity nonconservation in cesium S. L.Gilbert* and C. E. Wieman Joint Institute f o r Laboratory Astrophysics, University of Colorado and National Bureau of Standards, Campus Box 440, Boulder, Colorado 80309-0440 and Physics Department, Uniuersity of Colorado, Boulder, Colorado 80309-0440 (Received 4 April 1986) We present a new measurement of parity nonconservation in cesium. In this experiment, a laser excited the 6s-7s transition in an atomic beam in a region of static electric and magnetic fields. The quantity measured was the component of the transition rate arising from the interference between the parity nonconserving amplitude, LfP,,,, and the Stark amplitude, P E . Our results where C2p is the proton-axialare Im%",,,/~= - 1.65+0.13 mV/cm and C2,= - 2 i 2 , vector-electron-vector neutral-current coupling constant. These results are in agreement with previous less precise measurements in cesium and with the predictions of the electroweak standard model. We give a detailed discussion of the experiment with particular emphasis on the treatment and elimination of systematic errors. This experimental technique will allow future measurements of significantly higher precision.
a wealth of precise experimental data on the various properties of cesium ground and excited states which can be used for testing and refining calculations of its wave functions. In a previous paper6 we briefly presented the results of our first measurement using this new technique. Although we expect considerable future improvement, this measurement is already more precise than previous measurements of atomic PNC and is approaching the pre-
I. INTRODUCTION
The standard model of electroweak unification has stimulated considerable interest in atomic parity nonconservation (PNC) over the last decade. This theory predicted a PNC neutral-current interaction between electrons and nucleons which would mix the parity eigenstates of an atom. Although the standard model has now been tested with moderate precision in a variety of experiments using high-energy accelerators, atomic P N C data can provide unique and complementary information about this interaction. This is because the atomic case probes a very different energy scale and is sensitive to a different set of electron-quark coupling constants. Thus precise atomic data would allow one to measure the radiative corrections' to the electroweak theory and to explore the possible alternatives to the standard model over a larger parameter space. In pursuit of this goal, measurements of panty nonconservation have now been carried out on bismuth,* lead,3 thallium> and cesium.536 The approximately Z3 dependence of the PNC mixing is the reason for the emphasis on high-Z atoms. Aside from some ambiguity in the early bismuth results, all of the data are now in agreement with the predictions of the standard model. While this work has provided significant new information, its importance has been limited by two factors. The first has been the level of precision of the experimental results, and the second is the difficulty in relating the observations to the fundamental electron-nucleon interaction because of the complexities of the atomic structure. To overcome these limitations we have developed a new experimental technique which will allow precise measurements on cesium. Cesium has the virtue that it is the simplest heavy atom, having one S-state electron outside a filled inner core. Thus it is highly single-electron in character and calculations of its structure are more direct and accurate than for other heavy atoms. In addition, there is
1.36~ 1.47U
FIG. 1. Cesium-energy-level diagram showing hyperfine and weak-field Zeeman structure of the 6 s and 7s states.
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59
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@ 1986 The American Physical Society
60 34 -
ATOMIC-BEAM MEASUREMENT OF PAPHTY . . .
cision of the best high-energy tests of the electroweak theory. Here we give a detailed discussion of this experiment with particular emphasis on the critical issue of the treatment and elimination of systematic errors. The PNC interaction in an atom mixes S and P eigenstates, allowing a small electric dipole ( E 1 ) transition amplitude between states of the same parity. Similar to all atomic PNC experiments, we measure this small parity nonconserving amplitude ( A,,, by observing its interference with a larger parity conserving amplitude. In our experiment, the parity conserving amplitude is a "Stark induced" E 1 amplitude ( AST) created by applying a dc electric field to mix S and P states. The use of this amplitude was first suggested by the Bouchiats' and it was used in the cell experiments discussed in Refs. 4 and 5 . In our experiment we use a laser to excite the transition of interest in an atomic beam. The use of an atomic beam nearly eliminates the Doppler broadening and hence we obtain very narrow transition lindwidths. This enables us to observe the PNC interference directly in the transition rate by applying a small magnetic field (70 G) to break the degeneracy of the Zeeman levek8 Other advantages of an atomic-beam experiment include the reduction of collisions, radiation trapping, and molecular backgrounds. A final important feature of our approach is that the transitions can be detected with high efficiency. In this paper we will first discuss the theory of the experiment in Sec. 11. In Sec. I11 we will discuss the apparatus and experimental procedure and in Sec. IV we cover the treatment of systematic errors. In Sec. V we
793
present the results and in Sec. VI discuss the future improvements to the experiment. 11. THEORY
In this section we present the basic theory needed to understand and interpret the experiment. We are interested in the excitation of the 6s state of cesium to the 7 S state by a resonant electromagnetic field in the presence of static electric and magnetic fields. This problem has been previously discussed for the case of large magnetic fields and broad transition linewidths.' Here we consider the case of narrow linewidths and a weak magnetic field which has a relatively simple analytic solution. There are three relevant transition amplitudes which can give rise to this excitation; a Stark induced electric dipole, a magnetic dipole, and the PNC electric dipole. We will first consider the total transition amplitude between a particular ground state (6S,F,m)and excited state (7S,F',m')level, where m ( m ' ) is the z component of the total angular momentum F (F').This derivation applies to the general case of arbitrary dc electric and laser field orientations. Combining this result for the amplitude with knowledge of how the Zeeman levels shift in a weak magnetic field, we will then derive the transition spectrum for the particular field configuration we have chosen. Figure 1 is the cesium-energy-level diagram for the transitions of interest in this experiment. With a static electric field, E, and an oscillating (laser) electric field, E, the Stark-induced transition amplitude between the perturbed 6S(F,m)and 7 S ( F ' , m ' )states is
AST(F,m;F',m')=(7S,F',m'1 -d.e 16S,F,m )
where d is the electric dipole operator. The coefficients CFA"' are proportional to the usual Clebsch-Gordan coefficients and are tabulated in Appendix A. The quantities a and P are the scalar and vector transition polarizabilities respectively and are given by7
Similarly, the magnetic dipole transition amplitude between these states is h
AMI ( F ,m ;F',m ') = f ( k x E ),S,,,,
+ [ L ( k X +i ( k X h
h
E )x
E ) y lSm,rn'? I
] MC;Am'
(3)
where k is the laser propagation vector. M is the highly forbidden magnetic dipole ( M 1 ) matrix element defined as
M = ( 7 S Ip,/c 1 6 s ) ,
(4)
where p, is the z component of the magnetic dipole operator. The panty nonconserving potential, VpN,-, mixes S and P states and gives rise to a transition amplitude given by
61 S. L. GILBERT AND C. E. WIEMAN
794
Combining all three transition amplitudes, the transition probability between particular sublevels (rn and m') is then
I = I AST+AMl+APNC I
2
(7)
where each A is a function of F, F ' , m , and m ' . We chose the experimental design to maximize the interference between AS* and A p N C while minimizing the AsT-A,, interference and other effects which may mimic the P N C signal. It should be noted that the PNC interference term is dependent on m. If the rn levels are degenerate and equally populated this term will sum to zero in the total rate for the F+F' transition. A magnetic field is introduced to break the degeneracy of the m levels and hence avoid this cancellation. The field configuration used" is shown in Fig. 2. A standing wave laser beam along the ^y axis excites transitions in a region of dc electric field in the ^x direction and dc magnetic field ( B ) along the 2 axis. The laser field is elliptically polarized with E = E Z h Z + i E x 2 where E~ and E, are real. Using this field configuration and substituting the amplitudes from Eqs. (l), (31, and ( 5 ) into Eq. (71, we obtain for F#F',
plus negligibly small terms involving only g p and ~ M. The quantities ~ , k + and E:- represent the hz components of the laser field for the ^k=+? and k = - 9 laser beam propagation directions resuectivelv, and E,= 2 I E,k++E,k- I . The first term in Eq. (8)is the pure %ark-induced transition rate and the second term is the interference between AST and the much smaller amplitude A p N C . This corresponds to the pseudoscalar 6G.EX B _
1:
where 6 represents the handedness of the laser polarization. The third term is the interference between As, and A, For our standing-wave laser field, ~ , k = E: - , which leads to a cancellation of the As, - A , I interference. The problem of imperfect cancellation of this term and other effects due to imperfect E, E, and B fields will be addressed in Sec. IV. In the weak magnetic field limit, each Zeeman sublevel is shifted in frequency by an amount Av=mgFpBB,where g F x 4 =- & = 3 = for s states. Applying this to the field configuration of our experiment, we find the spectrum of the 6 S ( F =4)-t7S(F = 3 ) transition to be composed of eight lines with strengths, R (i), given by
,.
+
The two outermost lines of the rnultiplet involve only a single Zeeman transition ( m=4+3 and rn = - 4 - + - 3 , respectively) while the other lines are each the sum of a Am = + 1 and a Am = - 1 transition. The F =4+3 spectrum is shown in Fig. 3 where the transition rate for each line is the sum of contributions (a) and (b). In the weakfield limit, the spectrum for the F = 3-4 transition is identical. These spectra are modified slightly due to magnetic
~
I
A = 0.35MHzIG
17L
FIG.2. Field configuration for experiment.
FIG. 3. 6SF=4+7SF=, transition. (a) Theoretical pure Stark-induced spectrum. (b)Theoretical panty nonconserving interference spectrum on expanded scale. (c) Experimental scan of the transition with B=70 G.
62 34 -
ATOMIC-BEAM MEASUREMENT OF PARITY . . .
field induced mixing of hyperfine states. This causes the right-left asymmetry of the peak heights in Fig. 3(c), for example. However, the mixing is quite small and can be accurately calculated using first-order perturbation theory. We calculate that a 7 0 G magnetic field causes the extreme left- and right-hand peak heights of the F =4+3 transition to differ by about 5%. The asymmetry is smaller on the F = 3 + 4 transition by the ratio of the 7 s to 6 s hyperfine splittings (-0.25). The magnetic field induced mixing, however, is not important in our experiment because we only make measurements on the two outermost lines of the multiplet where the A s , and A p ~ c contributions are affected equally. Thus the ratio ApNC/As*, which is the quantity of interest, is independent of this mixing. 111. EXPERIMENT
A laser, tuned to one of the end lines of the multiplet shown in Fig. 3, excited the 6 S - 7 S transition in cesium. We monitored the transition rate by measuring the amount of 850- and 890-nm light emitted in the 6P,/2,3/2-+6S step of the 7 S - + 6 P + 6 S decay sequence. The essence of the experiment can be understood from Eq. (8) and Fig. 3. The parity nonconserving interference term in Eq. (8) changes sign under the reversal of the E field, the B field, and the sign of E, (handedness of laser polarization). This causes a slight change in the overall transition rate, and hence provides a means of isolating the PNC interference term from the much larger pure Stark-induced term. A n additional reversal ("m" reversal) was achieved by changing the laser frequency to the other end of the multiplet. We have used this technique to measure the ratio of the P N C amplitude to the Stark-induced 6S(F=4)-+7S(F=3) and amplitude for the 6 S ( F = 3 ) - 7 S ( F = 4 ) transitions. A. Apparatus
The basic experimental setup is shown in Fig. 4. Laser light at 540 nm was produced by a dye laser and the beam passed through several optical elements before entering a vacuum chamber. Inside the vacuum chamber the laser beam was coupled into a Fabry-Perot interferometer, referred to as the power-buildup cavity (PBC) in Fig. 4 . The PBC was maintained in resonance with the laser which resulted in a large standing wave field inside the cavity. This field induced transitions in a cesium beam in
FIG. 4. Schematic of apparatus. PC no. 1 and PC no. 2 are the intensity stabilization and polarization control Pockels cells, respectively. D1 is the transition detector (actually situated below the cesium beam) and 0 2 is the PBC transmission detector.
195
a region of static electric and magnetic fields. A silicon photodiode detected the fluorescence emitted by the decay of the excited state. In the following paragraphs, we will discuss each of the key elements of the apparatus. A Spectra-Physics Model No. 380 ring dye laser produced approximately 500 m W of light at 540 nrn. We found it necessary to reduce the frequency fluctuations of the laser. To accomplish this, a few percent of the laser output power was sent to a stable reference interferometer cavity and an error signal was derived from the cavity reflection using the Hansch-Couillaud method." The error signal was used to control the galvanometer driven plates and a piezoelectric transducer (PZT) mounted mirror in the dye-laser cavity. This reduced the laser linewidth to about 100 kHz. A Brewster angle galvanometer driven plate within the reference interferometer allowed cavity optical length adjustment. This in turn produced laser frequency tuning when the laser was locked to the reference cavity. As will be discussed below, the long-term stability of the reference cavity was insured by locking it to the cesium transition frequency. The main laser beam passed through many optical elements. The first element was an electro-optic modulator (EOM) which produced small 4-MH-z frequencymodulation (FM) sidebands on the laser. As discussed below, this was necessary for the scheme which we used to hold the power-buildup cavity in resonance. Following the electro-optic modulator, two lenses modematched the laser into the lowest-order spatial mode of the PBC. The next element was a Pockels cell which, in combination with a linear polarizer, enabled stabilization of the laser intensity with active feedback. Following the intensity stabilization Pockels cell, the laser passed through a Faraday rotation optical isolator which isolated the laser from reflected beams. At the output of the optical isolator, the laser light was linearly polarized at 45"with respect to the ^x axis defined in Fig. 2. The next component in the laser beam path was the polarization control element which set the ellipticity and handedness of the laser polarization. The polarization control element was made up of a h / 2 retardation plate followed by a longitudinal single-crystal [potassium dihydrogen phosphate (KDP)] Pockels cell. When 1.85 kV was applied, the Pockels cell produced a 1 / 4 (90") phase retardation on the x component of the laser field. We adjusted the ellipticity of the light by rotating the (linear) laser polarization at the input of the Pockels cell with the h / 2 plate. The handedness of the laser polarization was then changed by reversing the voltage applied to the Pockels cell (+h/4-+ -h/4 retardation). We reversed this voltage using a double-pole double-throw vacuum relay. The Pockels cell had a 1-cm aperture and provided quite uniform birefringence across the 0.05-cm-diameter laser beam. It was necessary to temperature stabilize the Pockels cell and isolate it from air currents to minimize variations in the resultant laser polarization. With these precautions, the birefringence of the Pockels cell was stable to a few parts in lo5 over the course of an 8-h data run. Following the polarization control element, the laser beam entered the vacuum chamber (pressure 3 X lo-' Torr) and was coupled into the power-buildup cavity.
63 S. L. GILBERT A N D C. E. WIEMAN
196
The PBC was a spherical mirror Fabry-Perot interferometer with a mirror separation of 23 cm. The flat, partially transmitting ( R=98.5% and T=1.3%) input mirror was mounted on a piezo-electric transducer. The second mirror had a reflectivity of 99.8% and a 50-cm radius of curvature. The power-buildup cavity was maintained in resonance with the laser using the FM sideband stabilization technique.12 We implemented this by using a fast photodiode to detect the laser beam which was reflected off the PBC. The output of the photodiode went to a phase-sensitive demodulator operating at the EOM 4MHz drive frequency. This produced an error signal which was then fed to the PZT mounted input mirror of the power-buildup cavity. When locked on resonance, the laser field within the power-buildup cavity was 20 times that of the incident laser beam. We monitored the power in the PBC by detecting the light which was transmitted through the second PBC mirror. The intensity at this detector was held constant to better than one part in lo5 per sec”’ by a feedback loop which controlled the intensity stabilization Pockels cell. We found that our signal-to-noise ratio was improved by about a factor of 2 by inserting a linear polarizer with its axis along the z direction in front of this detector. This means that we were only stabilizing the intensity of the field component ( E, ) which drove the pure Stark-induced term in Eq. (8). The standing-wave laser field in the PBC was crossed by an intense, collimated beam of cesium atoms. The cesium beam was produced in a two-stage oven where the nozzle region was maintained hotter than the main body to reduce the Cs2 dimer fraction. The exit of the oven was a Galileo Electro-optics glass capillary array of 10 p m by 0.05-cm channels with an area of 0.5 cmX2.5 cm. The cesium beam was further collimated in the y direction by passing through a stainless-steel multislit collimator. Several liquid-nitrogen cooled copper plates were placed upstream and downstream of the collimator to pump away cesium and thereby reduce the background cesium pressure. A small amount of background gas still remained, however, showing up as a broad pedestal amounting to a few percent of the transition signal size. At the intersection with the laser beam, the cesium intensity was lOI5 atomscm-2 s-’ with a full angle divergence of 0.04 rad in the y direction. The laser cesium beam interaction region is shown in more detail in Fig. 5. Not shown in the figure is a 30cm-diameter Helmholz pair which produced the 70-G magnetic field. The intersection of the beams was a cylinder 0.05 cm in diameter by 2.7-cm long. Two millimeters above and below this line were optically transparent, electrically conductive coated (InSn02) flat glass plates which had dimensions 2.5 cm x 5.0 cm. A dc electric field of k2.5 kV/cm was produced by applying a positive or negative voltage to the top plate and grounding the lower plate. As with the polarization (P)reversal, this voltage was reversed using a high-voltage double-pole double-throw vacuum relay. As will be discussed in Sec. IV, the plates were heated to avoid stray electric fields. We supplied f W to each plate by running ac (17 kHz) 100-R conductive coatings. This current through the
-
34 I N T E R FE R O M E T E R
MIRROR
‘LASER
\
,
BEAM
FIG. 5 . Detail of the interaction region.
current was sent through isolation transformers to reject any dc component and to isolate the heater supply from the high voltage. The 850- and 890-nm light from the decay of the 7 5 state was detected by a liquid-nitrogen cooled rectangular silicon photodiode (active area 0.5 X 5.6 cm) situated below the lower field plate. A gold-coated cylindrical mirror above the top plate imaged the interaction region onto this detector. Colored glass filters in front of the detector blocked the scattered green laser light while passing the infrared. The output of the photodiode went to a low-noise preamplifier. The detector-preamp combination had a frequency response of -400 Hz (fjde) with a noise equivalent power of 8x lo-” W/Hz”’. The signal from the preamp was sent to a gated integrator controlled by a Digital Equipment Corporation PDPll/23 computer which also stored the integrated data. A more detailed discussion of the data acquisition scheme will be given in the following section. An additional frequency stabilization loop was necessary to remove the effects of thermal drift of the reference cavity. To accomplish this we dithered the laser frequency at 330 Hz by feeding a sine wave to the galvonometer driven Brewster angle plate in the reference cavity. The -2-MHz amplitude of this dither gave rise to a slight modulation on the cesium transition signal. The transition signal, along with the 330-Hz reference, was sent to a lock-in amplifier which provided a very low-frequency correction signal for the reference cavity. We were careful to make sure that this modulation did not produce any signals at the parity reversal frequencies. B. Data acquisition and analysis A typical data run consisted of 8 h of data accumulation divided equally between the F=4-+3 and F = 3 - + 4 transitions. For each hyperfine transition, the laser frequency was locked to the extreme high- or low-frequency line of the multiplet shown in Fig. 3(c). A PDP11/23
64 34
ATOMIC-BEAM MEASUREMENT OF PARITY. . .
computer produced the 'ITL (transistor-transistor logic) signals which controlled the P, E, and B reversals. The reversal rates were 2, 0.2, and 0.02 Hz,respectively, with regular 180" phase shifts introduced in the switching cycles. The transition detector signal was integrated, digitized, and stored for each half cycle of P . Brief deadtimes after each field reversal were necessary to avoid transient effects. The deadtimes used were 25 ms for the P reversal and an entire P cycle (450 ms) for the E and B reversals. After 30 min of data acquisition, the laser was tuned to the other end of the hyperfine multiplet and data acquisition was continued. For normalization purposes, the average signal size was measured using a digital voltmeter and recorded for each data file. The integrator output was also calibrated using this voltmeter. Tests for systematic errors, discussed in the following section, were made before and after each 8 h run. We analyzed each data file to determine the fraction of the total signal which modulated with the P, E and B reversals. From Eq. (8) this fraction, disregarding systematic effects, is (101 The m reversal was implemented by subtracting the APNC for the low-frequency line from that obtained for the high-frequency line of the Zeeman multiplet. In our measurements E, /E, was close to l and E fields between 1750 and 3000 V/cm were used. With a typical voltage of 2500 V/cm we obtained a detector current of 3X A and a parity nonconserving fraction of 1.3 X
-
-
IV. CONTRIBUTIONS DUE TO SYSTEMATIC EFFECTS Systematic errors were of fundamental concern in the design and execution of this experiment. This led to the use of four independent reversals to identify the P N C signal. The quality of each of the reversals was better than one part in lo4. Thus, in principle, we would only require two reversals to cleanly resolve the PNC signal. The two extra reversals provide redundancy which greatly reduces the potential systematic error, since nearly all of the factors which affect the transition rate are at most correlated with only one reversal. The primary concern then becomes the small imperfections in the various field orientations and field reversals which can give signals that mimic the parity nonconserving signal under every reversal. As discussed below, we have identified and measured all such possible errors. Our approach to the identification of these contributions was similar to that used in earlier Stark interference experiments. Using Eqs. (1) and (3) we derived the transition rate for the general case, allowing for all possible components of E,B,E, and the oscillating magnetic field k X E. Each of these components was given a reversing and nonreversing part to characterize its behavior under the P, E, B, and m parity reversals. In this analysis, the 9 axis was taken to be along the laser beam. This means that E~ is absent by definition. The ^x axis of the system was defined to be along the component of the applied E field (i.e., the reversing E field) which was perpendicular to the 3 axis. With this definition, the
797
2 component of the reversing part of E ( E, is absent but there can be a nonreversing stray field part, AE,. These definitions give the following behavior for the E field upon reversal:
The P reversal ( E,
+
- E,
)
can be characterized as
where E, and E, are real. We considered all the combinations of these field components which contribute to the 6S-7S transition rate. Using rough empirical limits for these components, all terms which could be greater than 0.1% of the true PNC were identified. Effectively, this limit means that we needed only to consider terms which changed sign with all four reversals and involved no more than two components which were either stray or misaligned fields. The three terms which satisfy these criteria, along with their typical values, are listed in Table I. Counterparts of all of these terms were considered in earlier P N C experiments, as discussed in Ref. 13. The first term in the table arises from an electric field misalignment (E,) and a stray E field (AE,).The second term is due to the stray E field in the y direction (My ) and a misalignment ( B , ) of the magnetic field. The component B, causes a mixing of states within a particular hyperfine level, and, to a smaller extent mixing of different hyperfine levels with the same principal quantum number. We calculated the size of this mixing using first-order perturbation theory. The third term in Table I is due to the A s T - A w l interference shown in Eq. (8). As mentioned previously, this interference is suppressed by its change in sign under reversal of the laser propagation direction, ^k. For our standing-wave field, this suppression factor, relative to the P N C interference, is about lo3 (PBC output mirror R-99.8%). A second suppression comes from the fact that, though the M1 interference mimics the P N C interference under the E, B, and m reversals, it does not change sign under the P reversal. However, imperfections in the P reversal, such as those indicated in Eq. (121, can cause a significant amount of the M1 interference to leak
TABLE I. Terms which mimic the PNC signal, given as a ratio to the pure Stark induced transition rate of Eq.(8). Term
Average size relative to PNC term 0.01
0.04 0.17
65
34 -
S. L. GILBERT AND C. E. WIEMAN
798
-
through since M / ( I m P p N c ) lo4. Of particular concern is the birefringence in the power-buildup-cavity outputmirror coating as discussed in Ref. 14. This can give rise to an asymmetry between the fields of the counterpropagating laser beams which changes sign with the P reversal. There is an additional contribution to this term which involves the birefringence of the input mirror and other optics. However, these birefringences also cause a modulation in the transition signal size when the polarization is reversed which is about lo5 times larger than the PNC systematic. We observed this modulation and added birefringence to cancel it. With this cancellation the PNC systematic involving these birefringences is negligible leaving only that due to the output mirror. For a more detailed discussion of this point see Ref. 14 or Ref. 15. We have designed a set of auxiliary experiments to measure the fields which contribute to terms 1-3 in Table I. Our basic philosophy was that we should be able to monitor all the possible systematics while we were taking P N C data. T o do this we used the atoms themselves to measure the fields which give rise to the three terms in Table I. This was done without changing the basic experimental configuration. As can be seen below, the procedures used were somewhat excessive for the level of precision of the present measurement. However, much of this effort was preparation for the more precise measurements we plan to make using this technique. The following is a description of each of these auxiliary experiments. (a) AE, / E x measurement. Conditions: linearly polarized laser light E = E , R + E ~ ~ ; IB I =O. The F = 4 - + 4 Stark-induced transition rate from Eq. (1) is given by
I : = 9 a ’ ( E e ~ ) ~%p2 + I E X EI
(17)
This measurement was made simultaneously with measurement (b) and the stored data were analyzed to obtain E,, / E x and AE,, / E x . (d) B x / B z measurement. Conditions: same as in (b) but with B; =O. These were the same conditions as used in the P N C measurement. The fractional modulation of the transition rate with the P,B, and m reversals is (&E’)P,B,m
I:;’
= 2 - -E,, - - - -Bx . Ex 4
Ex
(18)
Ez
This measurement takes no additional time as it was derived from the raw parity nonconservation data. The desired quantity, B x / B z , was obtained from Eq. (18) since E y / E , was known from (b) and E , / E , was measured for each data run. (e) (6~,k+- & - ) measurement. Conditions: circular laser polarization; I B I =O; cesium-beam collimator tilted such that the Cs beam is no longer perpendicular to the laser beam. In this situation, it is simplest to think of the power-buildup-cavity laser field as being made up of two superimposed traveling waves with opposite propagation vectors, ^k = +9 and k - = -9. Due to the tilted collimator, a particular cesium transition is now split into two peaks; one corresponding to the Doppler-shifted resonance with ^k component of the laser field, and the second corresponding to the Doppler-shifted resonance with the ^k component. Using this method we were able to clearly resolve these peaks, as shown in Fig. 6. For the F =4-+3 transition, the transition rates for these two peaks are h
+
= 9 d E ; ~ t+; ~ E , E , E , E) ,
(13)
since a / p - 10 and Ex >>Ey,Ez. The modulation (amplitude) of this transition rate with the E reversal is [ h l : ] =9a2( ~ ExAE,E; + E x AE,E,E,)
The fields B, and B; were measured using a gaussmeter and again E, / E , = 1. With this information and the measurement represented by Eq. (16), we solved for E,, / E x . (c) A E y / E x measurement. Conditions: same as in (b). The measurement is identical to that outlined in (b) with the addition of the E reversal
.
(14)
The above measurement was made and the laser polanzation was then rotated to E’=E,S-E~~. This causes the second term in Q. (14) to change sign. The difference between these two measurements, divided by the overall transition rate yields
+
(19) ~ i
i l
ll
ll
l
ll
ll
l
I--
Knowing E, / E , = 1.O, we obtained AEz / E x . (b) Ey / E x measurement. Conditions: circularly polarized light ( E = E z 2 + i E x B ) ; B,=70 G an additional magnetic field in the 9 direction, B;=O.15B2. In this measurement we monitored the transition rate on the two outermost lines for both the F =4-3 and F =3+4 Zeeman rnultiplets. The effect of the additional magnetic field, B;, is the replacement of Bx with ( B , +Bx,) in term 2 of Table I. The fractional modulation of the transition rate with the P,B;, and m reversals is then (16)
LASER FREQUENCY C H A N G E ( M H z )
FIG. 6 . Scan of the 6SF=4-7S~=4 transition with the cesium-beam collimator tilted so that the ^k and ^k - peaks are resolved ( B = O G ) . +
66 -
~~
ATOMIC-BEAM MEASUREMENT OF PARITY . . .
34
The difference in fractional modulation with the P reversal for the two peaks is then
(20)
wherewehaveused I ~ , k + l = l ~ , k - l = ( l / f i ) I E , l . The expression in Eq. (20) is identical to the coefficient of M / ( P E , ) in term 3 of Table I. The ratio M / ( O E ) has been measured previous~y.’~ Measurements (a), (b), and (c) were made at the beginning and end of each data run to guard against the possibility of AE, and AE, changing with time. These measurements took about l h to complete and, when combined with measurement (d), resulted in an uncertainty for terms 1 and 2 in Table I which was typically an order of magnitude smaller than the statistical uncertainty in the PNC measurement. As discussed below, measurement (el was only made if the PBC output mirror had been moved or rotated. A number of additional tests were made to verify that these field imperfections gave the false signals predicted by Table I, both in magnitude and sign. In each test, a particular term was enhanced and the systematic tests (a)-(e) were made. Parity nonconservation data were then taken and the predictions of the systematic tests were compared with the measured false P N C signals. Nonreversing E fields AEy and AE, of 3 volts/cm (AEy,,/Ex were produced by applying dc voltages across each field plate. The large reversing Bx field ( B X / B , = 3 x lo-’) was produced with an external coil that was reversed with the B, coil. A mirror with a large coating birefringence was put into the PBC to enhance term 3 in Table I. For each of these tests, the prediction of the systematic tests agreed with the measured false P N C to within the uncertainties of the measurements ( 10-20 %). We have carried out extensive studies of the effects which give rise to the terms of Table I. These studies achieved two goals: they led to modifications in the apparatus which reduced the size of the necessary corrections, and reduced the time variation of the systematic errors. This latter point is by far the most important. The measurements described above allow us to measure all the corrections to a high degree of accuracy relative to the P N C rate in a short time. This means it is not particularly important how big these corrections are but it is crucial to know if they vary during the time we then spend taking P N C data. We found that the stray electric fields in particular could be highly time dependent and quite large if preventative measures were not taken. Especially troublesome was the fact that every material we tested tended to acquire stray electric fields after it was exposed to the cesium beam for some time. We tried many different kinds of field plates before arriving at the heated conductive coatings we presently use. Purely empirically we have determined that if these field plates are kept somewhat above room temperature (but not too far above) they have
-
799
very desirable characteristics. When first put into the apparatus they had some modest initial 9 and 2 stray fields, perhaps 0.25 V/cm. After brief exposure to the cesium beam the stray fields would drop to less than 0.05 V/cm; one to two parts in lo5 of the total applied E field. We saw only very slow subsequent drifts in these stray fields. The misaligned fields ( Ey and B, ) were quite stable as expected since all the components of the apparatus were rigidly mounted. The Ey field depends on the alignment of the electric field plates with the laser beam. We found that we could set this alignment to make Ey / E , = lop4. However, a nominal alignment of Ey / E x ~r lo-’ was used as a compromise between enhancing the signal for measurement (d) and minimizing term l in Table I. Magnetic field coils in the f , 9, and 2 directions were used to shim out stray (nonreversing) and misaligned (reversing) B fields. Although most of these fields do not produce false PNC signals, they can give rise to signal modulations with the P reversal which complicated the systematic tests. Thus we found i t worthwhile to eliminate them. The appropriate nonreversing shim field settings were determined by measuring these fields with a gaussmeter. These fields were reduced to about 10 mG. The reversing field shims were set by monitoring the signal modulation with the P, B, and m reversals. After this adjustment, B , / B , was typically 3 x lop3. The possibility of E and B field inhomogeneity along the 1-in line of intersection of the cesium and laser beams was also investigated. The measurements (a)-(d) are only sensitive to the average value of the quantities A E y / E x , AE,/ E x , E, / E x , and B, /B,. These average values were then combined to calculate terms 1 and 2 in Table I. This approach is not strictly correct, however, if both components which make up a single term have spatial inhomogeneity. We tested for this possibility by taking measurements (a)-(d) under the normal conditions and then repeated them with about half the length of the cesium beam blocked. As we expected, these data showed that there were indeed spatial variations in the stray E fields (AE, and AEz) of roughly 50%. However, Ey and B, were found to be homogeneous to better than 10%. This confirms that there is only one inhomogeneous field in each of the terms, and therefore our measurements are valid. The corrections we measure before and after each data run support the conclusion that the stray electric fields vary little with time. The second term in Table I was always found to be the same before and after to within the statistical uncertainty. The average size of this correction for a data run was 3.5% of the P N C with an uncertainty of about the same size. The first term varied by more than the statistical uncertainty on about half the runs. This was hardly grounds for concern, however, since the average value for this correction was 0.4% of the P N C and the typical statistical uncertainty was half of that. When there was variation, we used the average of the two corrections. The error bars were then taken to cover both values, which at worst differed by 0.4% of the PNC. Since the statistical uncertainty in the P N C measurement was about 20% for each data run it is clear that it was not really necessary to check the stray fields before and after
67 ~~~~
S. L. GILBERT AND C. E.WIEMAN
800
each run. Our previous experience, however, made us wary of relying on their constancy until a considerable amount of supporting evidence had been obtained. The only apparatus dependence to term 3 of Table I is the coating birefringence of the PBC output mirror, or to be more precise, the product of the coating birefringence times the angle between the birefringence axis and the x axis. Measurement (e) listed above is a very sensitive way to determine the actual correction due to this term. However, we found the following procedure was a simpler way to study the general characteristics of the birefringence. With the vacuum chamber up to air, the transmission through a linear polarizer following the PBC was monitored while the circular polarization of the laser was reversed. The modulation of this signal showed a periodic dependence with the rotation of the output mirror (PBC on resonance), due to the combination of the output mirror coating and substrate birefringences. The generally smaller contribution due to the substrate birefringence was determined by the same procedure but with the input mirror removed. Using this approach the axis and amount of the output-mirror coating birefringence could be determined. We investigated a number of mirrors from different manufacturers in this manner. In agreement with Ref. 14 we found that the coating birefringence was a general property which varied from mirror to mirror, but for a particular mirror it was largely the same across the entire surface. We did find that there were occasional, usually very local, regions where the birefringence could be significantly different, however. This is in contrast to the results reported in Ref. 14 but we believe this is because the method used in that reference was insensitive to local variations. Based on a limited number of test samples we now believe the birefringence is predominantly determined by the geometry of the coating facilities when the mirror is made. We saw no temporal variations in the coating birefringence. Before taking P N C data we rotated the output mirror to reduce term 3 in Table I. If the angle between the birefringence axis and the x axis is zero, this term vanishes. Because of the spatial variation in the coating and the substrate we could only set this angle to within a few degrees, however. The residual birefringence correction to the P N C data was then determined as described in measurement (e). Because earlier tests had shown there was no time variation to this correction term, measurement (e) was repeated only if the mirror was moved or the laser alignment was changed. We found that laser alignment had very little effect, but rotation of the mirror made a substantial difference, as shown below. For about one third of the data the correction due to 49(4)% of the PNC, for the second third it term 3 was was -58(4)%, and for the remainder it was -0.4(1.0)%. Although these corrections are relatively large, they can be accurately determined and hence are not a serious problem here. However, we have now obtained mirrors with far lower birefringence which will be used in future work. Another conceivable systematic effect we considered was one due to a dependence of the detector sensitivity on the direction of E and/or B. We tested for this effect by
+
34 -
monitoring the detector signal while reversing both E and B. A stable light level was provided by a light-emitting diode. We found that there was no change in detector sensitivity at the part in 10’ level. The existence of the other reversals makes any residual effect from this source negligible. We have also considered the effects of motional E and B fields which arise because the atoms are moving through magnetic and electric fields and we also find these to be negligible. We believe that there are no significant contributions that mimic the P N C signal which have not been taken into account. It should be noted that the uncertainty in determining these corrections is predominantly statistical. Thus improved signal to noise will not only reduce the statistical uncertainty on the P N C results but will also reduce the uncertainty in the corrections. The only remaining source of systematic error is in the calibration of the experiment. This calibration involves measuring the dc electric field, the ratio E, /E=, and determining the contributions to the observed detector current which are not represented in Eq. (9). Such contributions have often been called “dilutions” in earlier papers on this subject. The electric field was determined from the applied voltage and the measurement of the separation of the field plates. The 0.5% uncertainty to this calibration came entirely from the separation measurement. We have previously shown that more accurate field measurements can be made by observing the Stark shift of the cesium atoms, but that was unnecessary for this experiment. We by measuring the polarization of the determined L,/E, light which was transmitted by the power-buildup cavity using a linear polarizer and a photodiode. To measure the background signals we periodically set the static electric field to zero and observed the detector current. This was then subtracted off the signal observed with the E field on. The principal source of background was laser induced fluorescence of the optics which was typically 0.15 times the cesium signal. Though this background did not introduce any systematic uncertainty, it did increase the overall noise by about a factor of 2 and thus increased the statistical uncertainty in our results. The only additional background we observed came from cesium molecules and was about a factor of 5 smaller. We tested for any E field dependent background by tuning the laser frequency well off the transition and measuring the signal for E on and off. This set an upper limit of times the atomic transition signal for such background. A small calibration correction was needed because of the incomplete resolution of the lines in the &man multiplet, as seen in the experimental spectrum of Fig. 3. This correction was obtained in the following way. First, we scanned the laser to obtain the transition spectra for both I B I = O G and 1 B 1 =70 G. The 70 G spectrum, such as that shown in Fig. 3(c), was then fitted as the sum of eight individual lines where each line was assumed to have the 0 G line shape. From this fit we found the contribution of the overlapping lines and, using Eq. (9) we calculated the appropriate correction. This was done for each data run and the correction was typically 4% with negligible uncertainty.
68
ATOMIC-BEAM MEASUREMENT OF PARITY . . .
34
V. RESULTS AND CONCLUSIONS Ten data runs were made in the manner described in Sec. III. The signal-to-noise ratio was typically two or three times worse than that expected in the shot noise limited case. This extra noise was due primarily to the scattered-light-induced fluorescence background mentioned previously. This resulted in an integration time of 20—30 min for a 100% measurement of the PNC term. Our combined results for the ten data runs are -1.5110.18 mV/cm ( J F=4->3), -1.8010.19 mV/cm (F = 3~»4) , — 1.6510.13 mV/cm (average), where the quoted uncertainty includes all sources of error. As discussed earlier, the uncertainty is dominated by the purely statistical contribution. Our value is in good agreement with the value of —1.56±0.17±0.12 mV/cm reported by Bouchiat et al. for the average of measurements made on the F=4—»4 and 3—»4 hyperfine transitions.5 Using /3=27.3(4)do as discussed in Appendix B, we obtain Im^ P N C =-0.88(7)XlO-"ea 0 . To relate this to the weak charge Qw, or equivalently sin2^, it is necessary to know the value of the matrix element in Eq. (6). As we mentioned in Ref. 6 there is some uncertainty in the theoretical evaluation of this quantity. The most extensive calculation has been carried out by Druba et al.11 and their result of § > PNC =0.88(3) X \Q~lliea0(Qw/N) is probably the best value to take for this quantity and its uncertainty. However, a very conservative view would allow a range from 0.85 to 0.97 as can be seen in Ref. 18. It is likely that new results will be forthcoming in the near future which will clarify and hopefully improve this situation. Using the value of Ref. 17 our experimental results give &, =-78+6+3 for the cesium experiment. Where the first uncertainty is due to our experimental uncertainty and the second is due to the theoretical uncertainty. This is in good agreement with the standard model value'7 using sin2^ obtained from the mass of the W boson, e io =-71.0±l. 713.0 for the standard model prediction. The experimental value of Qw can also be used to obtain the weak mixing angle. Using the renormalized weak charges' for the proton and neutron this gives sin2^ =0.257+0.02810.014 for the cesium experiment. A comparison of the PNC measurements for the two hyperfine lines provides information on the nucleon spindependent coupling constants. Novikov et a/.19 have calculated the difference in the PNC between the F=4—>3 and 3—»4 lines using a shell model for the nucleus. They
801
find that the difference is the flip of one proton spin with an estimated uncertainty of 30%. Using this result and our measurements of the two hyperfine lines we find
C2p=~2±2 where C2p is the proton-axial-vector—electron-vector neutral-current coupling constant. This is in agreement with the predicted value of 0.1 and is a substantial improvement over the previous experimental limit of C 2 p > 1 where E is the error signal with no feedback. Thus one wants to make
this gain large in order to push the error signal as close to zero as possible, which is equivalent to forcing the laser frequency to be the Same as the reference fiequency. The key p i n t in designkg any servo loop is that there are always time delays in the system, and much of the design is based on dealing with these time delays. This can be best understood by considering the laser mirror. The mirror is attached to a piezo electric transducer (PZT)which will stretch it when voltage is applied. Although the PZT is a hard
ceramic and the rest of the mount is a solid metal piece, on the scale of frequency errors and
136
31 corresponding distances relevant here (atomic diameters or less), there is no such thing as a
stiff mount. In fact, at these atomic scales the stiffest PZT is incredibly spongy and can compress many wavelengths. Therefore, a PZT can be best visualized as a spring with the
mass of a mirror mounted on it, as shown in Fig. 16. We then move the back end of the piezo to try to correct the position of the mirror and compensate for random vibrations. If
one considers how a mass that is attached to a wall by a spring responds to sinusoidal motions of the wall as a function of frequency, this is just a driven harmonic oscillator. For frequencies below the resonant frequency, the response is in phase with the drive. However, above the resonant frequency of the mass-spring system, there is a 1800 phase lag and the mirror moves in the opposite direction of the driving force. This shows the primary problem encountered in designing any semo system; if the amount of feedback is the Same amplitude and phase relative to the error signal for all frequencies, the servo works fine for correcting for errors that fluctuate at frequencies lower than the resonant frequency. However, above
the resonance frequency, this 1800 phase shift causes positive feedback and the system becomes an oscillator if the feedback gain is greater than 1.
This is obviously unacceptable. A straightfoward solution is to make the gain smaller than 1 for frequencies above the resonant frequency and make it larger than 1 for lower frequencies. This gain is produced simply by having the compensation electronics include a simple low-pass filter. The normal way to operate a servo system containing such a filter is
to turn up the gain until it just starts to oscillate at the 1800 phase shift point, and then reduce the gain slightly so that it stops oscillating. This provides a stable sew0 system.
1.17
32 Notice that the gain at low frequencies is set by the response @base lag) of the system at high frequencies. This is characteristic of any servo system. Here we have presented the simplest possible compensation. In a more advanced servo design, one would put in more elaborate compensation involving electronic circuits that change the phase shifts and gain with frequency. With such systems one can optimize the gain at particular frequencies where the noise might be especially large or where one wants
to have the system be particularIy stable, such as at the frequency where the data are being
acquired. Other reasons for more elaborate compensation are to improve the transient
response so the system can recover more rapidly from a sudden shock. In the remainder of
this section we provide a few more examples of relatively straightforward compensation and how one can deal with various kinds of phase lags in systems. Several methods have been developed for designing servo systems - frequency response methods, the root locus method, and state space methods. The method that one
chooses depends on the requirements of the sew0 design, such as transient response and steady state enor. We will describe frequency response methods from an experimentalist’s point of view. We have chosen this method because it is very easy to measure a system’s frequency response with modem signal analyzers. The basis of the fiequency response method is the Bode plot. The Bode plot shows a system’s gain and phase as a function of frequency. Both the system to be controlled and the compensation have characteristic Bode plots. When designing a servo, one first mmures the frequency response of the system to be controlled and then designs a compensation circuit
that tailors the open loop frequency response to provide the desired control. It is important
138
33 to keep in mind thbr high gain at frequencies where the phase is less than 1800 provides good
control, but low gain is required at frequencies where the phase is greater than or equal to 180".
As an example of how to tailor the frequency response of a system with compensation, consider a simple harmonic oscillator. Many physical systems can be modelled as damped harmonic oscillators. The Bode plot of a harmonic oscillator is shown is Fig. 17. There are
two different goals one can have in trying to control this system: (i) to minimize the steadystate error, i.e., to have a large dc gain; or (ii) to maximize the bandwidth of the servo, i.e., to provide damping of the resonant peak with feedback.
As we mentioned previously, the simplest way to prevent oscillation is to make the gain smaller than 1 at frequencies at which the phase shift is greater than 180"; a Bode plot
of such compensation (an integrator, or a low-pass filter) is shown in Fig. 18a. Figure 18b shows the resultant frequency response for the oscillatorcompensation system. Note that the gain at low frequencies has increased, but the phase shift has reached 180" at a lower frequency. We now have a larger dc gain but a smaller bandwidth. The controlled oscillator will lock to the reference signal well at dc but will have a slow transient response and more
noise at higher frequencies. Suppose instead one compensates by adding a "phase lead" (e.g., a differentiator) to the compensation.so that the phase shift of the oscillatorampensation system has not yet reached 180" at the resonance. This allows the resonance to be artificially damped. Figure 19 shows the resultant frequency response for a harmonic oscillator compensated with a phase lead. Note that compared to Fig. 18, the gain near the resonant frequency is large,
34 and the dc gain has decreased. ' f i s system will have a faster transient response, but more
error in locking to the reference signal at dc. There is in general a trade-off between bandwidth (fast transient response) of a servo and dc gain (small steady-state error) of a servo. One way to think of it is in terms of
integrators and differentiators. An integrator will generally provide less steady-state error because it has "memory" to make accurate adjustments at low frequencies. On the other hand, fluctuations that are fast compared to the integration time will be "washed out," and,
as a result, the bandwidth of the system will be reduced. Differentiators predict the future performance of the system by looking at the slope of the error signal, and therefore increase the bandwidth. However, because a differentiator is compensating for future fluctuations it can slightly over or under compensate, leading to less steady-state accuracy.
Systems with more complicated frequency responses than that of a simple harmonic oscillator can be controlled by extending these ideas. A compensation circuit's phase and gain characteristics are tailored to provide the best compromise of bandwidth versus dc
response. We now return to discuss how this stabilization is applied to the actual PNC experiment. The power buildup cavity is stabilized by moving the mirrors with piezoelectric transducers. Then laser frequency is locked to the power buildup cavity by a combination of
elements, most of which are standard in cw dye lasers. First, we have a rotating plate on a galvanometer that changes the optical length of the laser cavity. This has a rather slow
response, but a large dynamic range. Second, one of the mirrors is mounted on a PZT. This
can change the cavity length with a frequency response extending to about 50 kHz. Third,
140
35 the fastest feedback is provided by an electrwptic modulator. This has a unity gain frequency of about 2 MHz, but can only correct for rather small errors in the frequency. In addition to the frequency stabilization, to stabilize the optical power inside the power buildup cavity, we sense the light transmitted by the output mirror and hold it constant using acoustooptic or electmoptic modulators to control the incident laser power. One significant difficulty in this experiment is the fact that the power transmitted by this mirror does not
seem to be exactly proportional to the power inside the cavity at the parts in 106 level. This discrepancy has been an ongoing problem which we do not yet fully understand.
5.7. Field Reversals and Signal Processing
With all the necessary frequencies, intensities, and lengths stabilized, one then has to
be concerned about reversing the various fields as precisely as possible, without upsetting the semwontrol systems. The electric field flip is accomplished by reversing the voltage applied to the electric field plates. Initially we tried a sinusoidal reversal, but this gave unacceptably large electrical pickup on the detector. We then switched to a square wave modulation with a few milliseconds of dead time after each reversal before taking data, to
allow the transients to die away. The primary problem in obtaining a perfect electric field reversal is the stray fields. There is considerable black magic we have learned for the preparation and handling of the plates which keeps the stray fields to a minimum, typically, a few tens of mV/cm. For the actual voltage reversal, we have experimented with various solid state and mechanical switches and have obtained the cleanest reversals when we use high voltage
141
36 relays. These have the minor annoyance that they are somewhat slow (less than -40 Hz), but the reversal is much more exact than with any solid state devices we have found. Mercury relays are faster but are limited in the voltage they can handle. To reverse the polarization, we use the same high voltage relays to flip the voltage applied to the Pockels
cell that provides a quarter wave retardation. In this reversal, the major problem is the birefringence of the Pockels cell which drifts with temperature. However, with careful temperature stabilization this can be reduced to a reasonable level. The magnetic field reversal is the easiest; we simply reverse the currents flowing through mils using solid state switches. One of the major concerns when doing any of these reversals is to avoid upsetting any of the servo loops. This takes considerable care and involves the use of various sample-and-holds circuits and gates with precise timing to isolate the servos from the transients. We have succeeded in keeping everything stable enough that the noise while flipping the various fields is as low as when there are no reversals. The signal processing is the final part of the apparatus, and it is hirly simple. The current from the photodiode is sent into a very low noise current-@voltage converter, as mentioned above. The output voltage of this amplifier is monitored with two different systems. The first is relatively crude (1 part in l@),and simply monitors the overall dc signal level for normalization. The second system detects the small changes in the signal due to the PNC modulation. First, the signal passes through a low noise amplifier which
subtracts off a constant voltage so that the dc output is close to zero. This near-zero signal is sent into a gated integrator and the signal is integrated during the time between each reversal. At the end of each interval the output from the integrator is digitized and stored in a
142
37 computer. Each of these numbers is stored with its appropriate label as to the state of E, B,
m, and polarization. Then the computer carries out the next reversal, resets the gated integrator, and the sequence is repeated. Using the offset and gated integrator in this
manner, we avoid the dynamic range problem encountered in trying to measure a very small modulation on top of a large signal.
6.
Systematic Errors Most of the time spent taking data in these experiments is devoted to the study and
reduction of potential systematic errors. Our approach to dealing with systematic errors
follows the same general analysis used in the earlier cesium and thallium experiments. This procedure starts by considering the most general possible case of both dc and ac electric and magnetic fields which have components in the X, Y,and Z directions. Thus we have 12 possible field components. Next we look at all the combinations of these fields that can give
rise to a 6s + 7s transition, either electric dipole or magnetic dipole. We allow each of these 12 field components to have both flipping and non-flipping (henceforth known as
"stray" parts).
We then go through the exhaustive list of combinations that produce terms
that mimic the parity nonconservation by reversing with all of the possible various reversals.
Then we measure the size of these 24 different field components and, in the process, try to make the stray and misaligned fields as small as possible. We have been able to reduce them to typically 10'' to
of the main applied field. Using the measured sizes of the different
components, we look at all the vast number of combinations that mimic PNC,and see which
38
ones are simcant. The 10" to 10-~values effectively mean that any terms involving
more than two stray or misaligned components are negligibly small compared to the true (10'
- 10-6) PNC. At the end of this exercise we found there are three terms that contain
two small components, and these are listed in Table 1. The first of these terms involves a stray electric field in the Y direction times the (misaligned) magnetic field in the X direction, The second term involves a stray electric field in the 2 direction times a misaligned
component of electric field in the Y direction. And the third term is a product of El and M1 transition amplitudes times a mirror birefringence factor. We measure each of the fields and the birefringence involved in these terms while the experiment is running and subtract off their contributions. To do this we run a set of
-
auxiliary experiments simultaneously, or interleaved with the PNC data acquisition. These auxiliary experiments involve observing the effects on the 6s
7s atomic transition rate of
different hyperfine transitions, different laser polarizations, and application of additional E or
B fields. Two points should be emphasized about dealing with systematic errors in this manner. First, it is important to use the atoms themselves so that the same region of space
is sampled at nearly the same time as the PNC experiment. Second, the auxiliary experiments must be designed to allow systematic corrections to be measured with an
uncertainty that is much less than the statistical uncertainty in the parity nonconservation experiment. It is highly desirable to have a measurement time much shorter than that
required to take the parity violation data. If one fails to achieve this, then the uncertainty of an experiment increases because much of the running time is spent in taking data on systematic errors and little on the measurement itself.
144
39
In the experiments we have designed, achieving the necessary unceainty requires a small fraction of the PNC integration time. In Table 1, we show the different sizes of the
systematic uncertainties for our 1988 experiment, how much they vary from one run to another, and the average correction and uncertainty. It can be seen that the typical corrections are a few percent or less, and most importantly, the uncertainties in all of these cOrrections in a given day are less than 196, and thus much smaUer than the statistical uncertainty. An obvious question is, "Is this analysis foolproof, or did we miss something?" In fact
we did miss something; there is no analysis that is absolutely foolproof. We overlooked a
small correction the first time through, although we caught it well before we were ready to publish a result. However, it is educational to discuss the statistical analysis procedure we
used to discover this systematic error. This Same type of analysis can (and probably should) be used in any precision measurement. It involves using a 2 test in a particular way to track down systematic errors. The Allan variance used to characterize frequency standards is
related to this approach. Our data consist of a large set of numbers, each number corresponding to a current which was integrated for 0.1 s. In the entire data set there are roughly lo' such numbers
stored in the computer for analysis. The fist step is to find the scatter in the numbers which is due purely to the statistics and has no contribution from any systematic source. This is
accomplished by looking at the fluctuations on the shortest possible time scale where the statistical fluctuations are large. This gives us a standard deviation, Q, which is most likely
145
40
to be purely statistical. In our case,we are doubly sure that it is truly statistical because it
corresponds to the shot noise limit for the signal. Having found u, we collect the data into various bin sizes, for example, the frrst million data points would be one bin, the second million would be the second bin, and so on for all the data. This produces 10 bins of data, and we can now predict how the average
values in each bin should distribute based on u, and the number of points in the bin
(cave=c
m).This hypothetical distribution is then compared with the actual distribution. Specifically we find the value of 2 for the distribution Using
Q
to obtain an uncertainty for
the value in each bin, then we look up the probability for having that value of
2. If the
resulting probability is 0.5 or larger, we are confident there was no systematic error that
varied on a time scale of the length of a bin that would be significant relative to the statistical uncertainty. Now, by choosing different bin Sizes we probe for variations on different time scales,
This is quite important because if the bin size is much larger or much smaller than the time scale for the variations in some systematic error, the 2 will probably look reasonable. However, when one chooses a bin size that corresponds to the time scale of the variations, suddenly, the probability will be very low, indicating the presence of some unknown systematic error. This approach does not have to be limited to binning the data by time. It
is equally useful to bin it according to any other factor that may lead to some systematic errors. For example, one might also bin the data according to room temperature to search for temperaturedependent systematic errors.
146
41 Of course this approach only works if the systematic errors vary
- if they are always
constant you will never see them. However, one can make them vary by changing everything about the experiment that might be important, such as realigning all the optics or
replacing critical components. Again, we binned the data corresponding to the different configurations and performed the 9 test. This is a remarkably sensitive test for potential systematic errors, and, although it is not generally taught, it is important to keep in mind in
any precision experiment. In our case it revealed that we had neglected to consider the EIMl interference correction associated with the off-resonance excitation of a m level. This excitation is forbidden at zero magnetic field but could occur because of the second-order
Zeeman effect. Having carried out all the detailed studies of systematic errors and
2 tests we finally
achieved the result
ImbNc
B
-1.639(47)(08) mV/cm
F=4-F'=3
-1.513(49)(08) mV/cm
F=3-F=4
-1.576(34)(08) mV/cm
(average)
.
The size of the parity nonconservhg mixing is given in terms of the equivalent amount of dc
electric field that would be neceSSary to give the Same miXing of S and P states. As shown, we have measured this mixing for two different hyperfine transitions, the 6S, F = 4 -+ 7S, F' = 3 and the 6$, F = 3 + 7S, F' = 4. In both cases, the amount of mixing corresponds to about 1.5 mV/cm. The average of these two is the most important quantity, as we will discuss below. We have measured this to an uncertainty of 2 %, which is dominated by the
0.034 mV/cm statistical uncertainty. The systematic uncertainty is about 1/4 of the size of
147 ~~~
~
42
the statistical uncertainty. It should be noted that this systematic uncertainty is different from many systematic uncertainties, in that it is actually a true statistical uncertainty in the evaluation of the systematic correction. Therefore if the statistical signal-to-noise in the experiment is improved, this uncertainty will be reduced.
In Fig. 20, we show a comparison of the different experimental measurements of parity nonconsewation in cesium, the most thoroughly measured atom. On top are the two experimental results of the Paris group in '82 and '84, below is our 1985 result, and our 1988 result, with its 2% uncertainty. There is good agreement among all of these numbers.
This gives one a Certain amount of confidence that no tremendous systematic errors are being overlooked.
In Table 2, we show a summary of the results from all atomic parity nonconsewation experiments. In the first section are the optical rotation experiments which looked at the 648
nm line of bismuth. These results are somewhat controversial in that the results from the
three groups showed substantial discrepancy, as did the theoretical calculations. In retrospect, the former was probably due to systematic mors that were not sufficiently controlled. More recent optical rotation experiments have shown better consistency, and the uncertainties are mostly in the 15 - 30% range. The one exception is the recent Oxford measurement in bismuth which has an uncertainty of only 2%. The Stark-induced interference experiments are given at the bottom of this table. Most of these are the cesium measurements we have already mentioned, plus there is the one thallium result from Berkeley with an uncertainty of 28 %.
148
43
Before we can consider what these measurements tell us about elementary particle physics, we much return to the atomic structure issue. The quantity that is experimentally measured is the GpNc mixing, which is equal to Q, times the T~ matrix element. This matrix element is found by calculating the atomic structure. Thus, in order to obtain Q, to 1%, both the experiment and the matrix element calculation must be accurate to better than 1%.
As mentioned earlier, the calculation of -y5 varies considerably in accuracy from one atom to another. In Table 2 we have given the accuracy quoted for the best calculations for each atom. Here we will limit our discussion to the cesium atom for which there have been the most abundant and most accurate calculations. Two basic approaches have been employed for these calculations. The first is the semi-empirical method which has been used in Paris, Oxford, Colorado and elsewhere. This approach uses experimental data to determine wave functions which are then used to find the matrix element. This technique is relatively easy. However, it is difficult to make a rigorous evaluation of the accuracy of the calculation, since all the relevant experimental data have already been incorporated into the calculation. The estimates for the uncertainty in these calculations are as small as 2%. The second approach is to use ub initio relativistic many-body perturbation theory. "he need for accurate cesium PNC calculations has spurred major advances in this field, although the calculations are very long and difficult. The most recent and most accurate results have
come from the Novosobirsk group of Flambaum, Suskov, et al., who have achieved a 2 % uncertainty, and the Notre Dame group of Blundel, Saperstein and Johnson who have now reached 1%uncertainty. The advantage of this calculational approach is that there is a fairly
149 ~
~
44 clear prescription for evaluating the accuracy of the calculations. The most direct way is to simply use the same calculational technique to determine many properties of the atoms and compare these with experimental data. In this case, this means calculating hyperfine splittings, oscillators strengths between many transitions, energy levels, and fine structure splittings for cesium and other alkali atoms. Fortunately, a tremendous mount of experimental data is available for comparison. In all cases, the agreement between the calculations and the experiments has been within 1%. Another technique for estimating the
uncertainty in these calculations is to estimate the size of the uncalculated higher order terms in the perturbation series expansion. This approach also gives an uncertainty of about 1%.
7.
Implications In this section, we will consider the implications of the Colorado measurement of PNC
in cesium. We will first discuss the significance of the comparison of the two different hyperfine transitions. This difference between the two numbers, A = 0.126 (68) mV/cm, is probably not zero. More specifically, this value indicates a 97% probability that A is greater
than zero. When we made this measurement, we did not anticipate a non-zero result at this level and therefore spent a considerable amount of time trying to determine what was wrong
with the data. The result, however, stubbornly persisted. Only later did we discover that an effect of nearly this size had been predicted. The primary difference beiween these two transitions is that the nuclear spin is reversed relative to the electron spin. Thus, A is a measure of the nuclear spindependent contribution to the PNC signal. Two processes have been discussed which would cause a
150
45
Rxlear spin dependent parity nonconservation. The first is simply the electron-quark portion
of the weak neutral current which depends on the spin of the quarks. This interaction is characterid by the C;, and C,coefficients. As we mentioned earlier, because of the size
of these coefficients and the fact that the effect is proportional to the total nuclear spin (and not proportional to the number of quarks), this contribution is much smaller than the weakcharge contribution. However, it has also been pointed out3 that there is a substantially larger contribution, called the nuclear anaple moment, which arises from weak interactions
within the nucleus. The effect of these weak interactions (both charged and neutral) are to mix the parity eigenstates of the nucleus, leading to a parity nonconseming electromagnetic
current in the nucleus. This current takes the form of a toroidal helix, and therefore has no long-range electric or magnetic fields. Thus, it gained the name "anapole moment." This phenomenon was first proposed by Zel'dovich in 1957 in the general context of parity violation in charged systems."
It is not well hown because people shortly thereafter
decided such an effect could never be measured. However, because the cesium electrons penetrate the nucleus, they spend some time inside the toroidal helix and thereby detect its existence. The coupling to the electrons is purely electromagnetic, but because the underlying nuclear currents are parity violating, it leads to parity violation in the electronic transition. There has been a significant amount of interest in this nuclear anapole moment by the nuclear physics community and several authors have calculated the expected size. The first calculations were by Khriplovich and Flambaum3 and their estimates are consistent with our observations. &ton
et aI.26have also made similar calculations, but have treated the
151
46
nuclear physiLs rather differently. Finally, Bouchiat and Piketty have done a calculation which is not consistent with our result.27 We have been told by V. Flambaum that the differences in these calculations are not due to any fundamental difference in the theory, but are a problem of the basic interpretation of nuclear PNC from other experiments. Depending
on how one chooses to interpret the other experiments, it is possible to obtain very different constants which characterize PNC interactions in the nucleus. This emphasizes the need for more accurate data in this field. There is hope that these nuclear anapole moment measurements can provide these data. The nuclear anapole moment is unique in that it is a PNC distortion of the nuclear ground state. The previous measurements on nuclear PNC
have observed parity mixing of excited, and often rather distorted, nuclear states where there
is considerable uncertainty about the nuclear wave functions. Thus, it is clear that future improvements in atomic PNC precision should substantially improve the understanding of nuclear parity nonconsewation. Obviously, the uncertainty due to the nuclear physics is a serious issue in the interpretation of atomic parity nonconservation. If we had measured only a single transition, it would seriously compromise our ability to test the Standard Model. Fortunately, if we take the average of the measurements on the two hyperfine transitions, as opposed to the difference, the nuclear spin dependent part cancels out. In this way, we also cancel out any questions involving the nuclear structure, which is critical in allowing a precision test of the Standard Model.
152
47
From this average a d the Notre Dame matrix element calculation, we obtain a weak charge, Q, = 71.0 f 2% f 1%. If one assumes the Standard Model is correct, one can then from this extract a value of sin2& which is equal to sin20, = 0.223 i 0.007 (expeshental) t 0.003 (theoretical).
(14)
This value of sin2eWcan now be compared with values obtained from other experiments such
as the measurement of the 2, mass or the neutrino scattering results, as shown in Table 3. In addition to these two measurements, there are many other measurements from high energy experiments which can be used, but which have lower precision or involve other properties
of the &. We have omitted the latter group because, while these are reputed to be independent measurements, the variations in the values of sin%, obtained are much smaller than the quoted uncertainties. This leads to unrealistic 2 probabilities and suggests that
these measurements are not truly independent. The comparison of the values of sin2b provides a precise test of the Standard Model. It is worth noting that in other tests of the Standard Model, particularly those involving the comparison of neutrino scattering and Z,data, the uncertainty in the mass of the top quark introduces an uncertainty of 0.003 in the relative values of
However, the
comparison of the atomic and the Z,mass values are unique in that dependence on the top quark mass is essentially identical in the two cases. Because the atom is sensitive to a different set of electron quark couplings and a different energy scale, this comparison, however, is very sensitive to new physics which is not contained in the Standard Model.
Proposed examples of such new physics include technicolor and extra 2 bosons, which occur in many models.
153
48
Figure 21 shows the values of
Gdversus C,,”as determined by a model-independent
analysis of the experiments, along with the Standard Model prediction. The crossed-hatch
area is the constraint from the SLAC deep inelastic electron scattering experiments. In contrast, the constraint set by our cesium experiment and the Notre Dame theory is the narrow solid line which is nearly orthogonal and, in particular, constrains the value of C1,
much more severely. The crossed line shows the values allowed by the SU,
X
U,
Wineberg/Salam/Glashow theory. The point on that line is determined by the value chosen for sin2Bw. This figure shows quite clearly that any new physics that would cause a change
in the value of C,, but not affect the other coupling constants, would only be revealed by the atomic parity nonmnservation measurements. The arrows on this figure show how a few popular proposed models would shift the values of these two coupling constants to a different place in the plane. Because the current atomic physics line passes through the Standard Model point, there is no indication of the existence of new physics. However, this does put
constraints, in some cases quite severe, on the parameters of models that propose such new physics. One example which has drawn considerable attention in the last few years, is how atomic PNC results constrain the proposed mechanism known as technicolor, or more generally, dynamical symmetry-breaking involving heavy particles. This type of new physics has been characterized in terms of S and T parameters which enter directly into QW.**
Generic technimlor models predict the S parameters should be around +2 or somewhat larger.% From the comparison of results just mentioned, one finds that the atomic PNC yields a value of S which is -2.7 f 2 2 1.1 as given in Ref. 29. Thus, one finds that
154
49
technicolor is on somewhat shaky ground, although the atomic PNC experimental results are
not good enough to completely rule it out. Finally, atomic PNC provides the best constraints on many models that involve additional neutral Z bosons. While there are many papers on this subject (see references in They consider 11 Ref. 2), we note particularly the paper by Mahanthappa and M0ha~at.m.~~ different models with additional Z’s which have been proposed, and they find that in 8 of
these 11 cases, atomic PNC provides the most severe constraints. Thus, it is clear that the cesium PNC results are providing information on elementary particle interactions that is not available from any other source at the present time.
8.
Future Improvements
8.1. Near Term While atomic PNC experiments are providing useful information, it is clear that more precise results would be desirable and useful. The mass of the 2, is now known to around 1 part in
Id. If atomic PNC results could be improved to that level, we would have a 10-fold
improvement in the test of the Standard Model and correspondingly improved sensitivity to possible new physics. With this in mind, we would like to discuss our efforts to improve the cesium PNC results. Work is also under way to improve atomic parity nonconservation measurements by several other groups: In Paris,the Bouchiat group is building a new experiment that involves stimulated emission probing of the excited state in cesium. At Oxford and Washington, experiments are under way to obtain more precise optical rotation measurements in thallium. At Berkeley, efforts continue to obtain a more precise Stark
155
50
inference measurement in thallium. All of these experiments h v e been under development for a number of years, and we hope to have results in the not too distant future.
In our efforts to improve the Colorado 1988 experiment, our primary focus has been
on improving the signal-to-noise ratio. This is clearly the major limitation of our experiment since the statistical uncertainty was much larger than the systematic uncertainties. We also
took into account the fact that the scattered laser light was a major nuisance requiring frequent realignment of the optics, and the transparent conducting coatings were a substantial problem. The coatings deteriorated under exposure to the cesium vapor, and lead to frequent interruptions in the experiment while they were replaced. With these issues in mind we built a new apparatus that uses an optically pumped atomic beam which, in principle, should provide 16 times more atoms since there are 16 possible m levels of the 6s state. We now have better mirrors for our power buildup cavity; these increase the buildup by about a factor
of 10, resulting in a total buildup of 15,000. A third improvement is using downstream detection of the 6S+7S excitation. This concept is illustrated in Fig. 22, which shows the schematic of the new apparatus. After leaving the oven, the cesium atomic beam is optically pumped into a single F and
mF level by light from two diode lasers which drive two hyperfine transitions of the 6S+6P3, transition. The atoms in the single m-level then propagate down the atomic beam and intersect the power buildup cavity beam where they are excited to the 7s state. They then have a 70% probability of decaying back down into the 6s hyperfine level which was previously depleted. The atoms continue down the optical beam in this state until they reach the probe region. In this region, light from another diode laser again excites the 6S+6P3,2
156
51 transition. However, here we excite a cycling transition (F=4
+
F'=5 or F=3 -. F'=2).
On a cycling transition the atom returns only to the Same initial state and hence can be excited many times. Typically lo00 IR photons are scattered for each 6S-.7S excitation. We detect this fluorescence to determine the 6-7s excitation rate. This detection scheme provides a substantial amount of amplification, yielding a detection of about 200 photons per S+7S transition, instead of the 0.3 detected in the previous apparatus.
In addition, since the detection takes place at a different region from the excitation region, we can now construct our electric field plates out of any material. This greatly simplifies their construction and increases their longevity. Also, scattered light from the green laser light is now negligible, as is detector noise, because the signal size is much larger. All of these improvements would suggest that the experiment should be much easier.
In fact, there have been major headaches and delays with this approach, and it is educational to consider what has gone wrong and what lessons a n be learned about doing experiments at
the frontier of laser spectroscopy. We will now discuss the unexpected problems we encountered in making this "improved" experiment work. The fist problem was noise in the optical pumping and resonance fluorescence detection regions due to the fact that we were using diode lasers. Diode lasers have very rapid (ns) fluctuations in the optical phase. Through a somewhat obscure process, this leads to very low frequency fluctuations in the atomic transition rate.
This was quite puzzling when we first observed it, and has now been explained in a series of papers by Zoller and collaborators.3o While this has become interesting atomic/optical physics to a number of people, to us it is a major experimental problem. To avoid the
157
52
problem one must have a feedback system capable of providing gigahertz bandwidth correction signals in order to eliminate the noise at 10-20 Hz,where we detect our signals.
In the first attempt we used optical feedback from narrow-band resonators, as demonstrated by Hollberg and coworkers.31 This approach gave low noise signals, but the locking was not reliable enough to allow three diode lasers to operate for reasonable periods of time. After considerable additional work, we settled on using optical feedback from diffraction gratings.32 These gave much more reliable performance, but the noise levels were still unacceptably high. We solved this problem by finding a laser manufacturer whose
instruments gave superior performance when operated with grating feedback. Thus we have finally succeeded in producing a very reliable source of diode laser light which provides very good signal-to-noise in excitation of narrowband atomic transitions. Specifically, we can
now achieve a noise-to-signal ratio of 3 x
/ Hz'" when exciting a 10 MHz wide
atomic transition. The necessafy laser has a combination of optical, mechanical, and current feedback as shown in Fig. 23. The grating which provides the optical feedback is mounted
on a piezoelectric transducer which allows mechanical adjustment of the grating position.
This holds constant the length, and hence the frequency, of the optical cavity. For faster corrections to the cavity resonant frequency we servo the laser current to achieve the highest level of stabilization. The second major problem in the new experiment was background atoms in the supposedly empty F state. It is relatively easy to deplete one F level of a low intensity atomic beam very well (< 10-4) . However, with a more intense beam there are a number of mechanisms that can repopulate the empty level. For example, collisions between atoms in
158
53 the beam and surfaces or other atoms (parhcularly oxygen) is the first mechanism. We have eliminated this s o u ~ c eby improving the vacuum and carefully positioning the collimating surfaces. A second contribution to the background, which appears to be from atoms in the
wrong state, actually comes from the excitation of the other hyperfine line by the tail of the spectral distribution of light in the probe beam. We have eliminated this source by sending
the probe light through an interferometer filter cavity which blocks out the tails of the spectral distribution. The third and most serious source of atoms in the wrong F state has
been the multiple scattering of the optical pumping light. The optical pumping process produces fluorescence that can travel down the atomic beam and re-excite the atoms out of the desired state. The number of atoms pumped back into the empty state scales as the square of the atomic beam intensity. We have found several ways to reduce this background: multiple pumping beams (the "clean-up" beam in Fig. 22), picking the optical pumping transition which minimizes the scattered fluorescence, and using the photon blocking collimator. This is a collimator with very thin black vanes which allows only the highly collimated photons to pass through. Finally, even with all these steps, the background was still too large, and we ultimately had to reduce our atomic beam intensity. In spite of all
these setbacks and delays, this new, improved experiment is now operational and we are taking data with a signal-@-noise ratio several times better than that of the 1988 experiment. A very painful lesson has been brought home to us in carrying out this improved
experiment. When one is probing a region of technology and physics which is unexplored, it
is important to step warily, and to keep all your options open. In terms of an experiment,
this means you should keep the apparatus flexible and be ready to adapt, as we mentioned
54
earlier. In this experiment we were somewhat seduced by the fact that this approach seemed to solve all our old problems, and we committed ourselves to a design that turned out to be
filled with major unexpected difficulties.
8.2. Long Term As the experimental accuracy improves beyond 196, the principal limitation on the
usefulness of atomic PNC will become the atomic theory. There have been credible speculations that it will be possible to calculate the theory in wium to a part in
Id.
However it is not clear when these calculations will be completed, and the question of how to check their accuracy becomes a major issue.
We have begun a longer term experimental project that avoids the atomic theory question. The basic idea is to compare precise measurements of atomic PNC for different
isotopes of cesium. The weak charge is sensitive to the number of neutrons, and hence will change for different isotopes. The atomic matrix element however, depends on the electronic structure and is almost independent of the number of neutrons. If one then looks at appropriate combinations and ratios of experimental results, for example
the atomic matrix element will drop out, leaving a ratio of weak charge which can be directly compared with Standard Model predictions. In this manner we hope to achieve measurements that can be compared with the Standard Model predictions at the part in I d
160
55
level. There are two major obstacles to carrying out these experiments. First is the need for even better signal-@noise ratios. Second, and most critical, is the need to carry out PNC measurements with small atomic samples, rather than the many grams used in the atomic
beam measurements. This requirement is neceSSary because all the other isotopes of cesium are radioactive and can only be obtained and used in small quantities. We propose to overcome both of these obstacles by using the new technology of laser trapping. This will improve the signal-to-noise ratio because it is possible, even easy, to obtain optical thicknesses in trapped atom samples 10 or 100 times larger than can be achieved in our atomic beam. It is more difficult to show that optical trapping will allow the experiments to be done with very small atomic samples (10'' atoms). We are currently working on this problem. The approach we are using starts with a very small sample of a given isotope (short-lived
isotopes will be produced at an accelerator, while longer lived isotopes can be brought to our
lab), whichis injected into a special cell where the atoms will be efficiently captured by a laser trap (Fig. 24). We have Carried out detailed studies on capturing atoms from a vapor
and we are currently developing wall Coatings which will allow the cesium atoms to bounce
around inside the cell without sticking until they are captured. Preliminary work with silane
coatings has been quite encouraging. Once the atoms are captured, PNC measurements can then be carried out in the cold
dense samples. If all goes according to plan, the next decade will see high precision measurements of PNC in a number of cesium isotopes. This will provide detailed information on the nuclear anapole moment and a very precise test of the Standard Model.
161
56
Acknowledgments This work has been supported by the National Science Foundation.
162
57 References
1. M. A. Bouchiat and C. Bouchiat, J. Phys. (Paris) 3 (1974)899. 2. P. Langacker, M.-X.Luo, and A. Mann, Rev. Mod.Phys. @ (1992) 87.
3. V. V. Flambaum and I. B. Khriplovich, JE"P
(1980) 835;V. V. Flambaum, I. B.
Khriplovich, and 0. P. Sushkov, Phys. Lett. B
(1984)367.
4. F. Curtis-Michel, Phys. Rev. 138B (1965)408. 5. M.A. Bouchiat and C. C. Bouchiat, Phys. Lett.
m,(1974) 111.
6. D. Z.Anderson, University of Colorado, private communication. 7. M.A. Bouchiat, J. Guena, L. Hunter, and L. Pottier, Phys. Lett. 1178 (1982)358.
8. R. Conti, P. Bucksbaum, S. Chu, E. Commins, and L. Hunter, Phys. Rev. Lett. 42 (1979)343;E. Commins, P. Bucksbaum, and L. Hunter, Phys. Rev. Lett
(1981)
640; P. H. Bucksbaum, E. D. Commins, and L. R. Hunter, Phys. Rev. D 24 (1981)
1134. 9.
S. L. Gilbert, M.C. Noecker, R. N. Watts, and C. E. Wieman, Phys. Rev. Lett. 3 (1985)2680.
10. M.C. Noecker et al., Phys. Rev. Lett. 61 (1988)310. 11. P. S. Drell and E. D. Commins, Phys. Rev. Lett. 3 (1984)968.
12. C. Wieman et al., Phys. Rev. Lett. 58 (1987)1738.
13. P. E. G. Baird et al., Phys. Rev. Lett. 3 (1977)798. 14. L. L. Lewis et al., Phys. Rev. Lett. 3 (1977)795. 15. L.M.Barkov and M.S. Zolorotev, JETP 22 (1978)357. 16. T.P. Emmons et al., Phys. Rev. Lett.
a (1983)2089.
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58 17.
V. A. Dzuba, V. A. Flambaum, P. G. Silvestrov, and 0. P. Sushkov, Europhys. Lett.
2 (1988) 413. 18. J. H.Hollister et al., Phys. Rev.
Lett 46 (1981) 642. (1989) 147.
19.
V. A. Dzbua, V. A. Flambaum, and 0. P. Sushkov, Phys. Lett A
20.
M. J. D. MacPherson et al., Phys. Rev. Lett.61 (1991) 2784.
21.
T. Wolfenden, B. Baird, and P. Sanders, Europhys. J.
22.
V. A. Dzuba, V. A. Flambaum, P. G. Silvestrov, and 0. P. Sushkov, J. Phys. B 2 p
15 (1991) 731.
(1987) 3297. 23.
M. A. Bouchiat et al., J. Phys. (Paris) 47 (1986) 1709.
24.
S. A. Blundell, W. R. Johnson, and J. SapirSteh, Phys. Rev.
25.
Ya.B. Zel'dovich, Zh.Eksp. Teor. Fiz. 3 (1958) 1531 [Sov. Phys. JETP 2 (1957)
Lett. 65 (1990) 141.
11841.
a (1989) 949.
26.
W. C. Haxton, E. M.Henley, and M. J. Musolf,Phys. Rev. Lett.
27.
C. Bouchiat and C. Piketty, 2. Phys. C
28.
W. Marciano and D. Rosner, Phys. Rev. Lett. 65 (1990) 2963.
29.
K. T. Mahanthappa and P. K. Mohapatra,Phys. Rev. D
30.
T. Haslwanter, H.Ritsch, J. Cooper, and P. Zoller, Phys. Rev. A 3 (1988) 5652.
31.
B. Dahmani, L. Hollberg, and R. Drullinger, Opt. Lett.
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32.
K. MacAdam, A. Steinbach, and C. Wieman, Am. J. Phys.
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a (1991) 3093.
164
59
Table 1. Potential Systematic Errors Systematic Contributiona
(El Mlf)(Am= f1)
(Am=O)
Rangeb
Average
Daily
All Data
Uncertainty
-0.3 964+ 1.1 96
+0.3 96
0.4 %
-1.3%++0.4%
-0.1%
0.4%
-0.8 %++4.8 A
+1.7%
0.6% (AF=-l)
-1.1 %++6.8%
+2.4%
0.9% (AF=+l)
-0.3 %++0.6%
+0.04%
0.04% (AF=-1)
-1.6%++0.1%
-0.23%
0.06% (AF=
+ 1)
L\Er and aE,are nonreversing electric held components, Bxand E,, are misaligned magnetic
t
and electric field components, and f represents the birefringence of the coating on the output
mirror. 'The range shows largest and smallest daily corrections.
165
60 Table 2. Summary of PNC resuLs Ancient (controversial) history Bismuth
Experiment 0xfordl3 Univ. of Wash14 Novosobirskls
variations
Pb
(Wash. '83)16
f28 %
Bi
(Wash. '8 1)'* (Oxford '9 l)')
*18% *2%
(Oxford '9 l)*'
f15%
*3
293 nm (F3erkeley '85)"
*28%
f6%?=
Modern civilized
Wide
wide variations
(?) e a :
Tl 1 . 3 ~
*10%'0(?)17 f 15 % (?)19 II
% (?)22
Stark induced interference: mall
cs (Paris '84-'86)23
(Col.'85)9 (Col. '88)"
*12% *12% *2%
all agree with Standard Model
*1
ll
w
%24
166
61 Table 3. Sin2 0, values.
W mass
-
Neutrino scattering
0.2320f0.0007
4
0.233 f 0.003 f 0.005
167
62 FIGURE CAPTIONS FIG. 1. Parity violating neutral current interaction between electrons and nucleons predicted by the Weinberg-Salaam-Glashow electroweak
theory contrasted with the normal electromagnetic interaction. FIG. 2. Basic experimental set-up for optical rotation PNC experiments. FIG. 3. Basic experimental set-up for the first generation Stark interference PNC experiments.
FIG. 4. Cesium energy level diagram. FIG. 5. Theoretical spectrum of the 6S,F=3 + 7S, F'=4 transition of cesium in a weak magnetic field. (a) The pure Stark induced rate (A;). @) The AE APNCinterference terms multiplied by 106.
FIG. 6. General layout of the Boulder cesium PNC experiment. The coordinate system in the interaction region is defined by a dc electric field, E, a dc magnetic field, g, and the angular momentum, u, of the laser photons.
FIG. 7. Experimental spectrum observed when the laser is scanned over the 6S,
F=3 4 7S, F'=4 transition of cesium in a 70 G magnetic field. FIG. 8. Dependence of the signal-to-noise ratio on the electric field. FIG. 9. Cesium beam wllimation. FIG. 10. Multipass system for reusing laser light. FIG. 11. Laser power buildup cavity. TIis the transmission of the input mirror, T2is the transmission of the second mirror.
FIG. 12. Fluorescence collection and detection.
168
63 FIG. 13. Overall schematic of the apparatus for the 1985 and 1988 Boulder cesium PNC experiment.
FIG. 14. Detail of the interaction region. FIG. 15. Basic sew0 loop. FIG. 16. Spring analogy for PZT. FIG. 17. Bode plot for damped harmonic oscillator. FIG. 18. (a) Bode plot for low pass filter. (b) Bode plot for harmonic oscillator plus low
pass filter compensation. FIG. 19. Bode plot for harmonic oscillator plus phase lead compensation. FIG. 20. Comparison of the experimental measurements of PNC in cesium. FIG. 21. Constraints on the Cld and C,, coupling constants by experimental measurements. The cross-hatched region is from SLAC deep inelastic scattering data, while the solid line is from atomic PNC. The SU, x U,line is the Standard Model value
as a function of sin2 8,. FIG. 22. Schematic of the new Boulder cesium PNC apparatus. FIG. 23. Schematic of the diode laser control system. FIG. 24. Laser trap cell.
169
e
N
e
N PNC Weak Neutral
Figure 1
Figure 2
e
Coulomb
P
170
polarized state
circular polarizer detector
Figure 3
171
540nm
v
m +4
-4
-3
+3 Figure 4
172
-
m=3 tom=4
m = -3 tom=-4
I
I
Figure 5
cs beam
Figure 6
t
173
100 MHr
H
Figure 7
Nshot
G
c 4
en
0 U
E fieId Figure 8
174
capiIlary array
multislit colI imator
I
0 absorb more strongly from the left-moving laser beam ( L ) than from the right-moving beam ( R ) even though the left-moving beam is less intense. We first show how this can be done by use of a static external field to shift the resonant frequency of the atoms. Consider a two-level atom and imagine that R is tuned below its resonant frequency while L is tuned above it. Assume also that the intensities are adjusted so that there is no force at z=O and that there is a magnetic field gradient in the ?i direction, dB,/dz > 0. As the atom moves toward positive z, its transition frequency will be Zeeman tuned toward the blue, bringing it closer into resonance with L and farther from resonance with R. Obviously if dB,/dz is sufficiently large, the increased absorption of photons from L will more than offset its decreased intensity, resulting in a net spontaneous force back toward z = 0 and consequent stability along 2. Since this configuration is already stable along f and i, it is a spontaneous lightforce trap in violation of the OET. To achieve damping of the velocity (a useful trap must have cooling in addition to stability), R should be detuned slightly below resonance ( r / 2 for maximum damping where r is the natural linewidth) and L should be
-
21 J U L Y1986
tuned either close to resonance or several I’ above it. The velocity dependence of the force from R, which provides damping, is much greater than that from L and hence dominates and cools. For either tuning, to make the center of the trap an equilibrium point, the powers in the two beams must be different. Although real atoms have more than two levels, a spin-polarized alkali atom, such as sodium in the F,M= 2 , 2 level excited with circularly polarized (m+ light, is a sufficiently good approximation to this two-level case. A plot of acceleration versus longitudinal position and velocity for sodium in such a trap is shown in Fig. 2. Stability results from the fact that the spatial derivative of the force is negative near z = O (and for zero velocity the sign of the force changes). Damping occurs because the higher the velocity of the atom, the larger is the force which is opposing it. In the second example we demonstrate how a static field which changes the orientation of the atoms can be used to produce a stable trap. In this case the laser beams have different linear polarizations, say R polarized along i and L along 9. Application of a magnetic field of constant amplitude but changing direction can cause atoms to interact differently with the two beams provided that the atoms follow the field adiabatically. In particular, for Na atoms in the F , M = 2 , 2 state, transitions to F,M=3,3 are not excited by light polarized parallel to the axis of quantization (the local magnetic field). Consequently, if the B field is a helix that twists toward k for z > 0 and toward i for z < 0, the configuration can be made stable. If the atom moves
I
-5
I
I
-4
-3
I -2
I 0
-I
I
I
2
I
I
3
4
5
(cm)
FIG. 2. Accelerations felt by sodium atoms with various velocities in a light trap having frequency-discriminated counter-propagating beams and a magnetic field gradient of 4 G/cm. At z = 0, beam L, tuned r/2 to the red side of resonance, has an intensity IL =0.8/,,; beam R, tuned r/10 to the blue side of resonance, IR -0.151,,. Both L and R are
right-circularly polarized and use f/2 optics.
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sufficiently far in the + i direction, B will be parallel to f and the atom will absorb photons only from the (weaker) L beam. Motion in the - 2 direction will produce a similar restoring force from the R beam. Moreover, both beams can be tuned on the red side of the resonance to provide viscous damping of the velocity. Since the light is not circularly polarized, transitions to F = 3, M = 2 or 1, which destroy the polarization, may also occur unless the magnetic field causes sufficient Zeeman splitting. Approximate modeling shows that a high fraction of population can be maintained in the F,M = 2 , 2 state for appropriate B fields and laser detunings. Our final example demonstrates how optical pumping of the atoms can be used to obtain a stable trap. Assume now that the laser beams have opposite angular momentum (e.g., u + for L and C T - for R ) . Consider a transition where the F quantum number of the excited state is less than that of the ground state but not equal to zero (e.g., Cs, F = 3 I;‘ 2 , which must decay back to F - 3 ) . If atoms are exposed to two beams of different intensity, the atoms become optically pumped so that they absorb more photons from the weaker beam than from the stronger. This can be understood from the transition probabilities shown in Fig. 3. An atom in the M=O state, for z > 0 will be quickly pumped into the M = 2 or 3 sublevel by the stronger u + beam. At that point, the atom can only absorb u- photons from the other beam which is directed toward the center of the trap. In addition, since the matrix elements strongly favor M = 1 decays, the atom will continue to absorb mostly CT- photons, thus generating an average longitudinal restoring force. This peculiar intensity dependence makes it necessary to tune the laser above resonance to provide longitudinal velocity damping. This weakly heats the atoms in the transverse direction. Thus in a real trap it would be necessary to provide additional transverse damping, for example, by use of an asymmetric trap geometry or adding weak “optical molasses’16 beams along and f . Phase-space trajectories for an atom in this trap with such additional cooling beams are shown
21 JULY 1986
in Fig. 4. These show that the atoms quickly are compressed into a region which is a fraction of a millimeter in size and have residual random velocities on the order of several centimeters per second corresponding to T. A weaker version of this trap results from use of orthogonal linear polarizations. We have calculated the performance of the above traps and variations on them. While it is not the purpose of this paper to give detailed results, it seems worthwhile to give the general scale to stimulate consideration of optical traps based on the concepts of this paper. With available dye-laser powers, spontaneous-
5 0 P (cm’secl
- -
+
-2
6p3/2
-I
0
I
(a) vx ( c r n l s t c ) 100
1-50 t
u
2
F=2
-150
6s F.3
-
-2
-3
_I
0
I
2
3
FIG. 3. Relative transition probabilities for the 6 S ( F 3) -6P3,1(F 2 ) transition in cesium.
312
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(b) FIG. 4. Phase space plots in z and x for a cesium atom of particular initial position and velocity in a light trap having polarization discriminated ( u +and IT- counterpropagating beams along the z axis and weak optical “molasses” in the x-y plane. At z 0 the trap beam characteristics are area 1
-
cm2, intensity 0.5 mW/cm2, f12, and detuning r/2. The “optical-molasses” beams are plane waves with an intensity of 0.025 mW/cm2and detuning of - r/2.
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force traps such as those we have described can be constructed with 10-cm dimensions, but 1 cm is more practical with inexpensive optics. The 1-cm size has confinement forces that are a fraction ( < f ) of the unidirectional spontaneous force. Integration of this force over the size of the trap gives a depth of several kelvins. However, a particle of this energy will not necessarily be confined in the trap since Doppler shifts can cause the forces to be significantly smaller for atoms with this much energy. For example, a I - K Cs atom has a Doppler shift of 6r and will interact much more weakly with the radiation than the above calculation of the depth assumes. This trap depth is nearly 100 times deeper and 1OIs times larger than that obtained for gradient-force traps of the type recently demonstrated by Chu et The spontaneous-force optical traps we have proposed are relatively simple ones designed to illustrate general ways in which the optical Earnshaw theorem can be circumvented. More complicated geometries can probably exploit these approaches more fully, particularly ones which provide more direct cooling and confining in the x-y plane. As a simple example, the addition of weak optical molasses along 2 and $, mentioned above, will make any of these traps perform better. Also, additional beams may be needed to ensure that trapped atoms do not escape to untrapped hyperfine ground states such as exist in all alkali atoms. The particular examples we have chosen illustrate three ways the internal degrees of freedom of the atom can be exploited to trap atoms, but there are many additional possibilities. Other types of optical pumping exist, such as pumping between two different hyperfine levels. Also, external static or oscillating fields can be used in a variety of ways to affect how an atom absorbs radiation." Probably all of these can be used to produce light-force traps. Finally, it may be possible to avoid the restrictions of the optical Earnshaw theorem by use of saturation' or absorption. The latter is particularly attractive: An optically dense cloud of Na will experience a maximum spontaneous N/m2 (corresponding to radiative pressure of 5 mW/cm2), enough to contain an atom density of 5 x lOI3/cm3 at T-0.25 mK according to the perfectgas law. Indeed, gas pressure limits the confinement density of a spontaneous-force trap, but this limit may be increased by using a weaker transition with a correspondingly lower ultimate temperature. We have pointed out ways in which spontaneous
-
-
-
21 JULY 1986
light forces may be used to make traps for atoms or ions. The advantages of this spontaneous-force approach as compared to other proposed neutral-particle light traps include large physical extent, reasonably deep wells, and experimental simplicity. Time-varying laser intensity or frequency is not required and the simple designs proposed are quite forgiving to optical misalignment or intensity mismatch. We hope our suggestions will remove the optical Earnshaw theorem as a practical barrier to the design of spontaneous-force light traps and will open the way to the realization of successful traps of different types. We acknowledge useful conversations with S . L. Gilbert, C. E. Tanner, and D. Z. Anderson. This work was supported by the National Science Foundation and the Office of Naval Research. One of us (C.E.W.) is an A.P. Sloan Foundation fellow.
(a)Permanent address: Instituto de Fisica e Q u h i c a de Sio Carlos da Universidad de SBo Carlos, SHo Carlos, Brazil. IA. Ashkin and J. P. Gordon, Opt. Lett. 8, 511 (1983). *See, for example, the feature issue on the mechanical effects of light, J. Opt. Soc. Am. B 2, 1707-1860 (1985). 3J. Prodan, A. Migdall, W. D. Phillips, I. So, H. Metcalf, and J. Dalibard, Phys. Rev. Lett. 54, 992 (1985). 4W. Ertmer, R. Blatt, J. L. Hall, and M. Zhu, Phys. Rev. Lett. 54, 996 (1985). 5R. N. Watts and C. E. Wieman, Opt. Lett. (to be published). %. Chu, L. Hollberg, J. E. Bjorkholm, A. Cable, and A. Ashkin, Phys. Rev. Lett. 5 5 , 48 (1985). 'V. G. Minogin, Kvant. Elektron. (Moscow) 9, 505 (1982) [Sov. J. Quantum Electron. 12, 299 (1982)l. *V. G. Minogin and J. Javainen, Opt. Commun. 43, 119 (1982). 9A. Ashkin, Opt. Lett. 9, 454 (1984). I0S. Chu, J. Bjorkholm, and A. Ashkin, and A. Cable, following Letter [Phys. Rev. Lett. 57, 314 (19861. lwhile these fields do exert forces themselves on the atoms, here we are limiting our discussions to cases, such as the magnetic fields mentioned above, where such forces are many orders of magnitude weaker than the spontaneous force. Hence it is most useful to think of the fields as merely providing a control for the spontaneous force. It is possible to have hybrid traps which are produced by a combination of forces from large static fields and radiation pressure. An interesting possibility for such a trap which employs a large magnetic field for confinement and detuning has been pointed out to us by P. Could, private communication.
313
Reprinted from Optics Letters, Vol. 13, page 357, May 1988. Copyright 0 1988 by the Optical Society of America and reprinted by permission of the copyright owner.
Atomic beam collimation using a laser diode with a self-locking power-buildup cavity Carol E. Tanner, Bernard P. Masterson, and Carl E. Wieman Toint Institute for Laboratory Astrophysics, Department of Physics, University of Colorado, Boulder, Colorado 80309-0440 Received January 11,1988;accepted February 18,1988 We have demonstrated a self-locking power-buildup cavity for laser diodes. This device requires only a few simple optical elements and can provide a standing wave containing as much as 1000 times the power emitted by the laser diode. With this device we have obtained an intense standing wave of tunable light that was used to collimate a cesium atomic beam. We have studied the power and frequency dependence of the beam collimation.
Experimental complications are encountered when one considers extending the work of Ref. 5 to a SBC. The first is that the laser will not lock stably to a cavity if there is more than a few percent feedback to the laser. However, one wants to couple all the laser power into the cavity, so the cavity-laser coupling must have a strong directional dependence. A second, slightly more difficult, problem is that for most uses of a power-buildup cavity one would like all the power in the lowest-order (TEMoo) mode of the cavity, but the light that is matched into this mode will have a minimum of reflection at the cavity resonance rather than the maximum needed for locking. If one is interested only in frequency control this is not a problem. As discussed in Ref. 5 , one simply tilts the interferometer relative to the incident beam to excite a higher-order mode; but this solution is unacceptable for a powerbuildup cavity. We solve both of these problems by sending the laser light through a circular polarizer into a slightly birefringent cavity. In this way, one can couple all the light into the lowest-order mode and still obtain the appropriate amount of feedback, since the only light that can return through the circular polarizer is the small amount that has been inside the interferometer and has been transformed into the opposite circular polarization. The simplest implementation of this design is a two-spherical-mirror Fabry-Perot interferometer in which one relies on the nearly universal birefringence of dielectric mirror coatings.6 For the research described below, we use a three-mirror folded cavity because it is more flexible, allowing the birefringence to be adjusted. One can thus vary the intracavity power over a larger range by attenuating the incident light and still maintain sufficient feedback for locking. Also, it is easier to change the beam-waist size in this configuration. The locking of the laser to the cavity depends on the phase as well as on the amplitude of the optical feedback, so the distance between the cavity and the laser must be controlled. If the feedback is optimized, the laser will stay locked for any phase that is not extremely close to 180 deg. Therefore, if one can tolerate very small (megahertz) gaps in the frequency coverage, it is
In the past few years a number of new techniques have been demonstrated for manipulating the position and velocity of atoms by using laser light. These techniques have provided an entirely new level of control in studying the properties of atoms and molecules. We have been working to make these exciting new capabilities widely accessible by showing how they can be implemented with simple inexpensive laser diOne particular technique that was recently demonstrated is the use of the strong dipole forces exerted by an intense standing wave to cool atoms3 and channel them into the field nodes.4 This approach relies on intense fields and thus would not have seemed well suited to the use of low-power laser diodes. However, in this Letter we introduce a new experimental tool, the self-locking power-buildup cavity (SBC), which allows one to use a laser diode to obtain relatively high-intensity standing waves in a simple manner. The SBC has many obvious applications, ranging from particle detection by light scattering to second-harmonic generation and nonlinear spectroscopy. In this Letter we demonstrate the use of a SBC to collimate a beam of atomic cesium, employing the stimulated emission technique in Ref. 3. The SBC exploits the unique sensitivity of laser diodes to external optical feedback. Dahmani et aL5 recently demonstrated that this property can be used to lock the output frequency of a laser diode to a Fabry-Perot interferometer. This dramatically reduces the laser linewidth with no electronic feedback. It is straightforward to extend that work to obtain a high-power SBC. The idea of a power-buildup cavity for a laser is not new. It is well known that the circulating power inside a resonant interferometer cavity can be much larger than the incident power, but in conventional buildup cavities it is necessary to have an electronic servo system to keep the cavity resonance frequency the same as the laser frequency. The higher the buildup desired, the narrower the resonance width, and therefore the more elaborate the electronics. The SBC relies on optical feedback to lock the laser frequency to the cavity resonance and thus requires no electronics. Furthermore, the locking generally improves as the buildup increases. 0146-9592/88/050357-03$2.00/0
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OPTICS LETTERS / Vol. 13, No. 5 / May 1988 PZT
U
Fig. 1. Schematic of the apparatus. The SBC has not been drawn to scale. It actually has a total length of 24 cm, and the fold angle is approximately 6 deg. LD1 is the prepumping laser that drives the 6S112 F = 3 to 6P3/2 F’ = 4 transition. LD2 is the laser that is locked to the cavity. X/4, quarterwave plate; PZT; piezoelectric transducer.
sufficient that the distance does not rapidly change by more than one-half wavelength. In the research discussed below, this distance could change substantially because the laser was on an optical table and the SBC was in an atomic beam machine. These changes would cause the laser frequency to jump. T o prevent this, we stabilized the laser-SBC distance by using a movable mirror. A simple electronic servo system adjusted the mirror position to maximize the power inside the SBC by holding the separation (or phase, when scanning the frequency) constant. With SBC’s we have achieved a buildup factor (ratio of intracavity traveling wave power to incident power) of approximately 1000. This factor is limited only by the quality of the mirrors and thus could be many thousands with very good mirrors. For a typical lowpower laser diode, even a buildup of 100 means that one has 1 W of tunable power in each traveling wave. The frequency of the light can be tuned over the full range of the laser by changing the resonant frequency of the cavity and the laser current and temperature. We emphasize that this is a simple device to construct and use-it uses only a few optical elements and requires little or no electronics. We have used the standing-wave field in such a cavity to collimate a beam of atomic cesium. A standing-wave field will exert a force on an atom owing to the induced dipole moment, and for intense fields this force can be much larger than the spontaneous force due to the photon momentum. As discussed in Ref. 7 , this force has a velocity-dependent component that is positive (heating) when the light is at a frequency lower than a resonant atomic transition and negative (dampingor cooling) when the light is tuned above the transition. Aspect et a1.3 have demonstrated this effect by using light from a tunable dye laser to collimate a beam of cesium atoms. A schematic diagram of our apparatus is shown in Fig. 1. A thermal beam of atomic cesium is collimated to a full-angle divergence of 15 mrad using small apertures. Initially, all the atoms are put into the 6 S ~ /F2 = 4 hyperfine ground state by optical pumping. Then they pass through the standing-wave field in the SBC and are detected 30 cm beyond the SBC by a movable tungsten hot-wire detector. We measure the flux as a
function of position, which reflects the horizontal velocity distribution of the beam. To produce the standing-wave field, the 852-nm output of a laser diode is sent through a circular polarizer and is mode matched into a folded cavity as previously described. With a SBC, it is easy to reach an intensity high enough that the dipole heating is much larger than the expected cooling. T o avoid this problem, we use a cavity with a flat input mirror of only 85% reflectivity. The other two mirrors, with 5- and 2.5-cm radii, have reflectivities greater than 99%. The separation of the latter mirrors, slightly less than 5 cm, is adjusted so the light beam has a waist diameter of 0.12 cm and is highly collimated where it intersects the cesium beam a t right angles. The maximum power in the cavity is 0.18 W, with a linewidth much less than 1.0 MHz. The standing-wavefield primarily excites the 6S1/2 F= 4 to 6P3/2 F‘ = 5 transition, which is a cycling transition in that the only decay channel is back to the initial state. The F’ = 4 excited state is 250 MHz lower in energy and thus farther off resonance than the Fr = 5 state, but its effect is significant, particularly when the laser is tuned to the red of the F’ = 5 line. We measured the resulting cesium beam velocity profile as a function of laser power and frequency. In Fig. 2 we show the largest collimation (laser tuned to blue) and decollimation (tuned to red) that we achieved. This figure illustrates the dramatic improvement in beam collimation achievable with this method and is similar to the collimation reported in Ref. 3 in which a tunable dye laser and much lower intensity were used. The narrow peak in the center of the curve obtained with red detuning is a reproducible signal t h a t comes from very low-velocity atoms trapped a t the antinodes of the standing wave.3
POSITION RESOLUTION -VELOCITY
RESOLUTION
-l 3 m l s e c
v
= I m/rec-1
rnm
0 4
3
2
1
0
1
2
3
4
HOT WIRE POSITION (mm)
Fig. 2. Signal as a function of hot-wire position. The squares show the beam profile with no laser present, while the crosses show the profile with red detuning and the circles show it for blue detuning. The transverse velocities corresponding to the hot-wire positions are also shown. The intracavity intensity was 21 W/cm2.
266
May 1988 / Vol. 13, No. 5 / OPTICS LETTERS
00 2 - 3 0
-20
-10
4 10
0
DETUNING
20
cavities, we have achieved intensities high enough that we actually observed decollimation for all frequencies because the heating was so large. From the data in Fig. 3 we also extract the detuning for optimum collimation as a function of intensity. This is plotted in Fig. 4,and one can see that it scales linearly with the Rabi frequency. The simplicity and low cost of a diode-laser-SBC system makes it attractive for collimating atomic beams in many applications. With minor modifications to the cavity geometry, one could produce a standing wave that intersects itself a t right angles, providing two-dimensional collimation. The velocity-dependent dipole force has two significant advantages for beam collimation over the spontaneous or radiation pressure force that has also been used for this purpose.8 First, the dipole force is much greater so that the collimation can be accomplished in a much shorter distance; second, each spontaneously emitted photon accomplishes far more cooling. This means that for a fixed laser power, a far more intense atomic beam can be collimated. In addition to its use as an atomic lens, a SBC also offers a convenient way to study the behavior of the atoms in intense standingwave fields. For example, one could use two intracavity phase modulators to shift the position of the standing wave back and forth a t a few megahertz and thereby excite the vibrational resonance of the atoms in the potential wells produced by the field. This would provide detailed information on the atom-field interaction.
5 0r
40
30
(r)
Fig. 3. Dependence of the hot-wire signal on laser detuning when the hot wire is at the center of the cesium beam. The four curves represent different values of intracavity power and are characterized by their respective Rabi frequencies at the center of the Gaussian profile of the beam. The maximum intensity is 21 W/cm2 (corresponding to WRabi = 195 r).
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This research was supported by the National Science Foundation. C. E. Wieman is an Alfred P. Sloan Foundation Fellow.
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In Fig. 3 we show the frequency dependence of the collimation for various values of intracavity power. The curves in this figure were obtained by placing the hot wire a t the center of the cesium beam and observing the signal while ramping the SBC (and hence the laser) frequency from 160MHz to the red of the transition (30 times the 5.3 MHz natural linewidth = I') to 40 r to the blue. These data show that the maximum collimation achievable at a given intensity increases with intensity up to a peak of about 17 W/cm2, which corresponds to a resonant Rabi frequency of 176 I'. As can be seen from curve 4, if the intensity is increased further the collimation decreases, presumably because of the increased dipole heating. With other
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References '
'
1. R. N. Watts and C. E. Wieman, Opt. Lett. 11,291 (1986). 2. D. Sesko, C. G. Fan, and C. E. Wieman, "Production of a cold atomic vapor using diode-laser cooling," J. Opt. SOC. Am. B (to be published). 3. A. Aspect, J. Dalibard, A. Heidmann, C. Salomon, and C. Cohen-Tannoudji, Phys. Rev. Lett. 57,1688 (1986). 4. C. Salamon, J. Dalibard, A. Aspect, H. Metcalf, and C. Cohen-Tannoudji, Phys. Rev. Lett. 59,1659 (1987). 5. B. Dahmani, L. Hollberg, and R. Drullinger, Opt. Lett. 12, 876 (1987). 6. For a discussion of birefringence in mirror coatings see S. L. Gilbert and C. E. Wieman, Phys. Rev. A 34,792 (1986), and reference therein. 7. J. Dalibard and C. Cohen-Tannoudji, J. Opt. SOC.Am. B 2, 1707 (1985); A. P. Kazantsev, Zh. Eksp. Teor. Fiz. 66, 1599 (1974); V. G. Minogin and 0. T. Serimaa, Opt. Commun. 30, 373 (1979); J. P. Gordon and A. Ashkin, Phys. Rev. A 21,1606 (1980), A. P. Kazantsev, V. S. Smirnov, G. I. Surdutovich, D. 0. Chudesnikov, and V. P. Yakovlev, J. Opt. SOC.Am. B 2,11731 (1985). 8. V. I. Balykin and A. I. Sidorov, Appl. Phys. B 42, 51 (1987).
Reprinted from Journal of the Optical Society of America B, Val. 5, page 1225, June 1988 Copyright 0 1988 by the Optical Society of America and reprinted by permission of the copyright owner.
Production of a cold atomic vapor using diode-laser cooling D. Sesko, C. G.Fan, and C. E. Wieman Joint Institute for Laboratory Astrophysics, University of Colorado and National Bureau of Standards, University of Colorado, Boulder, Colorado 80309-0440 Received November 30,1987; accepted january 25,1988 = 852 nm) to damp the motion of atoms in a cesium vapor. We have been able to contain more than lo7 atoms for 0.2 sec and cool them to a temperature of 100?i:o p K in this viscous photon medium (the so-called optical molasses).
We have used the light from diode lasers (X
In the past few years there has been a flurry of work demonstrating new ways to use laser light to control the positions and velocities of However, nearly all of this work involves rather elaborate and expensive technology. We have been working to achieve these capabilities by using technology that is simple enough to be practical for use in a wide variety of atomic- and molecular-physics experiments. In a previous paper3 we presented a simple and inexpensive way to stop a beam of cesium atoms by using frequencychirped diode lasers. In this paper we present the implementation of optical molasses by using diode lasers and discuss improved results on atom stopping. The cooled125 p K ) and vapor produced in this work was colder (2' significantly denser (-lo8 atoms/cm3) than was previously obtained with sodium in optical molasses2 produced by a dye laser. The idea of cooling an atomic sample by using counterpropagating laser beams was originally proposed by Hansch and Schawlow4 in 1975. They pointed out that if an atom sees counterpropagating laser beams that are tuned slightly below a transition frequency, the Doppler shift will cause the atom to absorb preferentially those photons moving opposite to its velocity. The momentum imparted by these photons will cause the atom to slow down and thus be cooled. In 1985 Chu et aL2demonstrated this idea in a vapor of sodium atoms and coined the name optical molasses. They sent a beam of relatively slowly moving sodium atoms into a region where six laser beams intersected in a cross. These beams cooled the atoms to a temperature, T = hy/2k, which is the theoretically predicted2 minimum temperature one can achieve when exciting a resonance line of width y. Also, they showed that the photons produced a highly viscous medium that confined the atoms for a fraction of a second before they could diffuse out of it. This production of free atoms that are far colder than anything previously obtainable opens up a rich new area for experiments, e.g., using these atoms for ultra-high-resolution spectroscopy or studying the interactions of these cold atoms with surfaces or other atoms. Such experiments are far more practical if diode lasers can be used to cool the atoms. Tunable diode lasers have a number of important advantages over the dye lasers that were used exclusively in the early work on laser cooling.
The most striking feature, of course, is their much lower cost. Other advantages, which are well known, are that they are. simpler to use; because there are no optical elements to get misaligned and dirty, i t is easy to change the output frequency rapidly, and they have good intensity stability. An advantage that is not well known is that it is quite easy to obtain diode-laser linewidths of much less than 1 MHz. A free-running diode laser has a typical linewidth of 30 MHz, but Dahmani and co-workers5 have recently shown that a small amount of optical feedback from a Fabry-Perot interferometer will lock the laser frequency to that of the interferometer and can reduce the linewidth by more than a factor of 1000. In this experiment we utilized this technique; thus the laser linewidth is much smaller than the 5-MHz natural linewidth of the transition. To produce ultracold cesium atoms we first started with a thermal atomic beam with an average velocity of 250 m/sec. These atoms were slowed to a few hundred centimeters per second by a beam of counterpropagating resonant laser light. The slowly moving atoms then drifted into a region where intersecting laser beams, which were tuned slightly below the center of the atomic resonance, formed the optical molasses. This light cooled the atoms and held them for an extended period of time. We studied the atoms by observing their fluorescence while in this region. A schematic of the apparatus for this experiment is shown in Fig. 1. A beam of cesium effused from an oven into one end of a vacuum chamber, and the frequency-chirped laser beam entered from the opposite end. This initial slowing portion of the experiment was identical to that discussed in Ref. 3, except that the stopping laser was locked to an interferometer cavity and the stopping distance was extended to 90 cm. T o lock the laser, a small portion of the output beam was sent into a 5-cm confocal cavity, which was a few centimeters from the laser. The cavity was tilted slightly so that the beam entered a t an angle relative to the cavity axis. Of the order of 1%of the laser power returned to it and caused its frequency to lock to that of the external cavity. If the free-running laser frequency was within approximately 500 MHz of the cavity resonance, the optical feedback would pull the laser frequency to the cavity resonance. This allowed us to tune the laser frequency coarsely by using temperature and current in the usual manner and then to do fine
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J. Opt. Soc. Am. B/Vol. 5, No. 6/June 1988 PUMPING LASER 2 , MOLASSES LASER
V
LASER
C OVEN PHOTODIOD
Fig. 1. Schematic of the apparatus. The third molasses beam, which was perpendicular t o the other two, is not shown. FP1 and FP2 are the Fabry-Perot locking cavities. The saturated absorption spectrometers are labeled SASl and SASS.
tuning by changing the resonant frequency of the cavity by using a piezoelectric transducer to translate one of the mirrors. The stopping laser drove the 6 S ~ = 4 6P3/2,~=5 resonance transition. A second laser, which was not locked to a cavity, was tuned to the 65'~=3 6 P 3 / 2 , ~ = transition 4 to ensure that atoms were not lost to the F = 3 ground state. The frequencies of these two lasers were swept from approximately 500 MHz below the respective transition frequencies to within a few megahertz of the transition. The first half of the chirp does little slowing since the average initial velocity of the atoms is -250 m/sec, but the additional time is needed to allow slower atoms to populate the downstream end of the beam between successive chirps. After a series of chirps, the stopping laser frequency was quickly (=30 gsec) shifted away from the resonant frequency, The frequency of the stopping laser, just before shifting, determined the final velocity of the atoms as they entered the molasses. The F = 3 state depletion laser remained a t the same frequency that it had a t the end of the ramp and thus ensured that the F = 3 state also remained depleted in the molasses. The frequencies of both lasers were monitored using small cesium-saturated absorption spectrometers. The optical molasses was produced by light from a third laser that was also locked to a cavity. The output of this laser passed through an attenuator and an optical isolator and was then split into three linearly polarized beams, each containing approximately 0.5 mW of power, that intersected each other a t right angles. The intersection region was slightly less than 1 cm in diameter and overlapped the cesium beam. The three beams were reflected back onto themselves by dielectric mirrors. A small part of the output from the molasses laser was used to obtain a saturated absorption spectrum in a cesium cell. This spectrum provided an error signal that was fed back to the cavity to hold the laser F = 5 transition. frequency on the red side of the F = 4 These three lasers were sufficient to cool the atoms. However, to obtain @her densities in the molasses by multiple loading, we used a fourth (unstabilized) laser, as described below. A silicon photodiode monitored the fluorescence from the molasses region. The cooling chirp slowed the atoms in the cesium beam to several hundred centimeters per second or less. We found that using a cavity-locked stabilized laser significantly im-
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proved the efficiency of this process. The number of stopped atoms was approximately 2.5 times larger with a stabilized laser than it was with an unstabilized laser, as was used in our previous work. The number of fast background atoms was correspondingly decreased relative to the unstabilized case. Apparently some of the laser-frequency fluctuations were large enough to cause the frequency to change more abruptly than the maximum allowable ramp rate. If this happens, some of the atoms are not decelerated enough to stay in resonance with the laser, and they are no longer slowed. In the present work we obtained approximately 5 X lo6 stopped atoms/cm3, and there were few fast-moving background atoms until a few milliseconds after the stopping chirp was finished. The narrower laser linewidth also permitted much better control over the final velocity of the atoms. Once the atoms were slowed to low velocity they could be caught by the molasses. The fluorescence from the molasses region showed a rapid rise during the stopping laserfrequency chirp, followed by a long decay indicating slow departure of the atoms. The behavior of the atoms in the optical molasses depended quite critically on the velocity they had as they entered the molasses region. To obtain reproducible results, the end of the frequency ramp had to remain constant to within approximately 1 MHz (I MHz corresponds to u = 100 cm/sec). If the atoms were moving too slowly they did not penetrate into the molasses, but instead they piled up on the surface and quickly diffused away. Best results were obtained when they had a final velocity of several hundred centimeters per second so that they penetrated the entire molasses region. This also produced a more dense sample since a larger volume of stopped atoms went into the molasses. In this situation the fluorescence signal from the molasses region would continue to rise for several milliseconds after the stopping laser was switched off. If the atomic velocity was increased further the atoms would fly right through the molasses. Once the atoms were stuck in the molasses they diffused out very slowly. This is illustrated in the plot of the fluorescence as a function of time in Fig. 2. As one would expect, the diffusion time was a sensitive function of loading velocity, alignment, and molasses laser power and frequency. The longest l/e decay time that we observed was 0.2 sec. We found that the conditions for maximum decay time were intensity per beam of approximately one half the 1mW/cm2
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TIME ( s e c 1
Fig. 2. Real-time trace of the fluorescence. The black dots on the left-hand side show the level after each new bunch of atoms was loaded. After eight bunches the loading was stopped, and the righthand side shows the subsequent decay of the fluorescence as the atoms diffuse away.
269 Sesko et al.
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Fig. 3. The dots show the fraction of the initial fluorescence that remained after the molasses laser was blocked for the time intervals shown. The solid line is the theoretical fit. Note that the data point a t 41 msec is much lower than the theoretical curve because of gravitational effects.
saturation intensity, frequency detuning between 0.5 and 1 natural linewidth, and loading velocity of a few hundred centimeters per second. These values are in reasonable agreement with what is predicted for these parameters when using the results in Ref. 6, although the intensities are a bit higher than expected. Because the atoms remained in the molasses 20 times longer than the time for a stopping chirp (10 msec), one could obtain much higher densities by repeated loading of the molasses. However, this required a fourth laser to provide the F = 3 state depletion in the molasses while another batch of atoms was being stopped. In the left-hand side of Fig. 2 one can see the effects of loading eight bunches. The dots show the fluorescence level after each additional bunch was loaded. After eight bunches the loading was stopped, and there was a slow decay of the fluorescence. By using multiple loading we were able to increase the number of atoms in the molasses by more than a factor of 10. From the amount of fluorescence, we estimate the maximum number of atoms we contained in the molasses to be approximately 5 X lo7. This was bright enough, when observed with an infrared viewer, that it could be seen in a well-lit room. We determined the temperature of this vapor by measuring the spread of the atoms in the dark, as was done in Ref. 2. After the atoms had been in the molasses for approximately 25 msec, the molasses laser was quickly switched off. A brief time later the light was turned back on, and the decrease in
1227
the fluorescence was measured. In Fig. 3 we show the fluorescence as a function of the time the light was off. After more than 35 msec with the light off, the subsequent fluorescence was dramatically less (falling off rapidly with longer times without light). We believe that this was due to the gravitational acceleration of the atoms. We fitted the points in Fig. 3 with the calculated curve for the expansion of a uniform sphere of atoms with a Maxwell-Boltzmann velocity distribution. This curve will be changed somewhat by the gravitational acceleration, but this effect appears to be small for times less than 35 m/sec. From this fit we find the temperature to be 100?itop K . There is considerable uncertainty in this measurement because the decay was a strong function of the distribution of the atoms, and we could see that the distribution was only vaguely spherical and could change somewhat from one batch to the next. Nevertheless, this value is in good agreement with the predicted2limit, hr/ 2k = 125 pK for this transition, and is colder than any cooled atoms or ions previously reported. At this temperature the rms velocity for these atoms is only approximately 15 cml sec. We have achieved an extremely cold and moderately dense atomic vapor by cooling atoms with light from inexpensive diode lasers. With the light on, this vapor will remain for a fraction of a second, and with the light off, the atomic motion is largely determined by gravitational acceleration. The technology required to achieve this remarkable behavior is simple enough that such atoms could be produced routinely for use in precision spectroscopy or collision studies.
ACKNOWLEDGMENTS This work was supported by the National Science Foundation and the Office of Naval Research. We are thankful for the assistance of C. Tanner and L. Hollberg in the diodelaser development. The authors are also with the Department of Physics, University of Colorado, Boulder, Colorado 80309-0440.
REFERENCES 1. For a review of this field, see the feature issue on the mechanical effects of light, J. Opt. Soc. Am. B 2,1706 (1985). 2. S. Chu, L. Hollberg, J. E. Bjorkholm, A. Cable, and A. Ashkin, Phys. Rev. Lett, 55,48 (1985). 3. R. N. Watts and C. E. Wieman, Opt. Lett. 11,291 (1986). 4. T. W. Hansch and A. L. Schawlow, Opt. Commun. 13,68 (1975). 5. B. Dahmani, L. Hollberg, and R. Drullinger, Opt. Lett. 12, 876 (1987). 6. J. P. Gordon and A. Ashkin, Phys. Rev. A 21, 1606 (1980); R. J. Cook, Phys. Rev. A20,224 (1979);Phys. Rev. Lett. 44,976 (1980).
Reprinted from Optics Letters
Observation of the cesium clock transition in laser-cooled atoms D. W. Sesko and C. E. Wieman Joint Institute for Laboratory Astrophysics, University of Colorado, Boulder, Colorado 80309 Received October 3,1988; accepted December 12, 1988
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We have used the light from diode lasers to produce a nearly stationary (u 15 cm/sec) sample of atomic cesium in optical molasses that is entirely in the F = 3 hyperfine state. In this sample we excite the 9.2-GHz 6s F = 3, m = 0 to F = 4, rn = 0 clock transition. Most of the atoms remain for -20 msec in the 0.4-cm3 observation region. We observe that transitions take place by monitoring the fluorescence when the atoms are illuminated with light tuned to the 6.9 F = 4 to 6P3/2F = 5 transition. Rabi resonance linewidths of less than 50 Hz are obtained.
There has been a great deal of interest recently in the use of laser light to stop and cool atomic samples.*-3 It has been suggested frequently that such atoms would be useful for precision spectroscopy and frequency standards.l However, this potential has yet to be realized. In this Letter we present a step toward this goal by using laser-cooled atoms t o obtain narrow linewidths on the 6 s F = 3, m = 0 to 6 s F = 4, m = 0 transition in atomic cesium. This is the so-called clock transition since it provides the present definition of the second. Although we are interested only in the linewidth that can be obtained and not in the many details that must be addressed in making an actual clock or frequency standard, this research does have two notable features with respect to such devices. First, the interaction volume, which must be extremely well shielded against stray magnetic fields in a real clock, is less than 1 cm3. Second, the cooling of the atoms and the detection of the transition are done using low-power diode lasers. These features allow one to contemplate a relatively compact and portable frequency standard based on this technology. The experimental technique is simple and uses the following time sequence. First, atoms in a cesium atomic beam are slowed and cooled by using the pressure exerted by the laser light. Then the laser light is turned off in such a way that all the cold atoms are left in the F = 3 ground state. During the subsequent dark period microwaves excite the transition to the F = 4 state. The resulting population of the F = 4 state is then determined by switching on a laser that excites this hyperfine state and observing the resulting fluorescence. The relevant transitions are shown in Fig. 1. The primary effort for this experiment is in the laser cooling of the atoms, but we have discussed this in detail previously3 and thus only briefly review it here. Light from a cavity-stablized diode laser is sent counterpropagating to a cesium atomic beam. The output frequency of this laser is swept from 500 MHz below the 6 s F = 4 to 6P3/2 F = 5 transition to 4 MHz below the resonance using a 10-msec ramp. This provides atoms with a velocity of a few meters per second, and the atoms are caught in optical molasses that is created by using the light from a second laser to produce three intersecting orthogonal standing waves. This
light is tuned approximately 10 MHz below the 6 s F = 4 to 6P312 F = 5 transition. In addition there is a third laser, which weakly illuminates the atomic beam and the molasses region and is used t o depopulate the F = 3 ground state. It does this by exciting the 6 s F = 3 to 6P312 F = 4 transition. Its frequency is ramped along with the stopping laser and then held on the transition during the time the atoms are in the molasses. In this manner we produce an atomic sample in the middle of the vacuum chamber with a residual velocity of approximately 15 cm/sec. To carry out microwave spectroscopy on this sample we irradiate it with approximately 7 nW/cm2 of radiation a t 9.2 GHz using a microwave horn driven by a microwave frequency synthesizer. Although the microwave radiation is always present, its effect is unob-
n 4-4 5
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Fig. 1. Energy-level diagram for cesium. 1989 Optical Society of America
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Table 1. Chronology of the Experiment Time
Function
Laser 1 (Stopping and Probe)
Laser 2 (Hyperfine Pump)
Laser 3 (Molasses)
I. 0-10 msec
Stop atoms u, = 400 mlsec uf = 4 mlsec
6s F = 4 6P3iz F = 5 chirp t o compensate for Doppler shift
6s F = 3 76P3iz F = 4 chirp
11. 10-30 msec
Molasses cooling ui = 4 mlsec uf = 15 cm/sec
Off resonance (500 MHz)
6 s F = 3 6P312 F = 4 on resonance
6SF=4+6?3/2F= 5 detuned -10 MHz
111. 30-33 msec
Off-resonant Blocked 6 s F = 4 6P312F = 4 optically pumps cooled atoms into 6s F = 3
Blocked
6s F = 4 6P312F = 5 detuned -10 MHz
Excite clock transition
Blocked
Blocked
Blocked
Blocked
+
6SF=4+6P3/2F= 5 detuned -10 MHz
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+
+
IV. 33-53 msec
V. 53-53.5 msec Probe F = 4 state population
Blocked 6s F = 4 6P3i2 F = 5 on resonance +
VI. Change microwave frequency and repeat sequence
servable when the la& light is present. T o observe the clock transition we follow the time sequence shown in Table 1. After the atoms have been cooled in the molasses for 20 msec (any time from 2 to 80 msec works as well) the F = 3 depopulating laser is switched off using a mechanical shutter, and 3 msec later the main F = 4 to F = 5 molasses beams are also blocked by a shutter. During this 3 msec all the atoms are pumped into the F = 3 ground state owing to the offresonant 6 s F = 4 to 6P312 F = 4 transition. There is then an interval of time during which all lasers are blocked and only microwave radiation is present. All transitions other than the 6 s F = 3, m = 0 to F = 4, m = 0 are shifted far out of resonance by a 0.5 G magnetic field. The microwave power is adjusted to give a ir pulse during this time. At the end of this interval the stopping laser (laser l),which is now tuned to the peak of the 6 s F = 4 to 6P312 F = 5 transition, is unblocked. The fluorescence produced by this laser beam is a direct measure of how many atoms have made transitions to the F = 4 ground state. The fluorescence comes in a brief pulse because the atoms are repumped into the F = 3 state in approximately 0.5 msec. However, during this pulse there are still a few thousand photons scattered for each atom that makes the transition to the F = 4 state, and we detect a few percent of these using a photodiode. The entire time sequence takes 54 msec when the dark period is approximately 20 msec. Because the crude mechanical shutters that we use tend to overheat, we then wait an additional 150 msec before repeating the sequence. If one used acousto-optic shutters to obtain the optimum data rate, it would be possible to obtain a data pulse every 35 msec with a 20-msec microwave transition time. After each cycle the microwave frequency is incremented, and this sequence is repeated. We find that good signals can be obtained for any period of darkness between 1 and 20 msec. For times longer than that the signals drop rapidly owing to the atoms’ leaving the region illuminated by the probe
beam. We believe that the 20-msec interval is primdrily determined by gravitational acceleration and the geometry of the laser beams. The atoms form a somewhat irregular clump in the center of the molasses that is roughly 0.5 cm on a side, and the probe beam is a cylinder with a diameter of 0.5 cm. Thus the atoms simply fall out of the observation region in 20 msec. With a dark period of slightly more than 20 msec we obtain the spectrum shown in Fig. 2. This spectrum represents 5280 55-msec data sequences (each point is the average of 24). It has a linewidth of 44 Hz. If the microwave radiation were pulsed to give a Ramsey separated oscillatory field, i t would give a 23-Hz linewidth. This can be compared with the 26-He linewidths that are currently obtained in the NBS-6 atomic clock that uses a 4.3-m-long transition region.* 1.0
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FREQUENCY CHANGE ( Hz ) Fig. 2. Experimental spectrum and theoretical fit of the 9.2-GHz clock transition in cesium ( 6 S P = 3, m = 0 to 6 s F = 4, m = 0 ) .
272 March 1,1989 / Vol. 14, No. 5 / OPTICS LETTERS
We believe that the signal-to-noise ratio can be improved greatly, since the actual number of photons produced is large. However, instabilities in the frequency of the stopping laser, which is not actively stabilized, cause substantial pulse-to-pulse fluctuations in the number and spatial distribution of the atoms in the molasses. We believe that if this laser were stabilized the signal-to-noise ratio would be larger. It would also be possible to obtain longer interaction times and hence a narrower resonance by changing the geometry or by making an atomic “fountain” as is being built by Ertmer et aL5 The amount of improvement that can be obtained in this fashion is somewhat limited, however, because the gain in time only goes as the square root of the distance. We have demonstrated that it is practical to use laser-cooled atoms t o obtain narrow transition linewidths in small interaction volumes. We hope that this will stimulate the consideration of this technology for high-precision spectroscopy and actual frequency standards.
271
We are pleased to acknowledge the assistance of Chris Monroe and Steve Lundeen and the loan of the microwave equipment from D. Wineland, J. Hall, J. Levine, and D. Anderson. This research was supported by the U.S. Office of Naval Research and the National Science Foundation. The authors are also with the Physics Department, University of Colorado, Boulder, Colorado.
References 1. For a review of this field see the feature issue on the mechanical effects of light, J. Opt. SOC.Am. B 2, 1706 (1985). 2. S. Chu, L. Hollberg, J. E. Bjorkholm, A. Cable, and A. Ashkin, Phys. Rev. Lett. 55,48 (1985). 3. D. Sesko, C. G. Fan, and C. E. Wieman, J. Opt. SOC.Am. B 5, 1225 (1988). 4. D. J. Glaze, H. Hellwig, D. W. Allan, and S. Jarvis, Jr., Metrologia 13,17 (1977). 5. W. Ertmer, R. Blatt, J. L. Hall, and M. Zhu, Phys. Rev. Lett. 54,996 (1985).
PHYSICAL R E V I E W LETTERS
V O L U M E63, N U M B E R9
28 AUGUST1989
Collisional Losses from a Light-Force Atom Trap D. Sesko, T. Walker, C. Monroe, A. Gallagher,(a)and C. Wieman Joint Institute for Laboratory Astrophysics, University of Colorado and National Institute of Standards and Technology, and Department of Physics, University of Colorado, Boulder, Colorado 80309-0440 (Received 5 J u n e 1989)
W e have studied the collisional loss rates for very cold cesium atoms held in a spontaneous-force optical trap. In contrast with previous work, we find that collisions involving excitation by the trapping light fields are the dominant loss mechanism. We also find that hyperfine-changing collisions between atoms i n the ground state can be significant under some circumstances. PACS numbers: 32.80.Pj. 34.50.Rk
Spontaneous-force light traps ’,* have provided a way to obtain relatively deep static traps for neutral atoms. These allow one to produce samples containing large numbers of very cold atoms. In this paper we present an experimental study of the collisions which eject atoms from such a trap. These collisions are of considerable interest because the temperatures of the trapped atoms ( l o p 4 K) are far lower than in usual atomic collision experiments. The theory of such low-energy collisions and their novel features have been discussed by several aut h o r ~ . ~Perhaps -~ the most notable feature is that the collision times are very long, and the collision dynamics are dominated by long-range interactions and spontaneous emission. These collisions also have important implications with regard to potential uses of optically trapped atoms. For many applications the maximum density that can be obtained is a critical parameter, and these collisions limit the attainable density. There have been two experimental studies of collisions in optical traps. Gould et al. measured the cross section for associative ionization of sodium. However, there is no evidence that this process is significant in limiting trapped-atom densities, and for some atoms, including cesium, it is energetically forbidden. Prentiss et al. studied the collisional losses which limited the density of sodium atoms which were held in a spontaneous-force trap. Their surprising and unexplained results were a direct stimulus for our work. In particular, they observed no dependence of the loss rate on the intensity of the trapping light. This was quite surprising, because a ground and an excited atom interact at long range via the strong l l r resonant dipole interaction, and a portion of the excited-state energy can be converted into sufficient kinetic energy to allow the atoms to escape from the trap. By comparison, two atoms in their ground states interact only through a much weaker short-range l / r 6 Van der Waals attraction, and even when such collisions occur, they may not produce significant kinetic energy to cause trap loss. This implies that the dominant collisional loss mechanism would involve the excited atomic states, and thus depend on the intensity of the light which causes such excitations.
In this paper we present measurements which show that, in contrast with Ref. 9, the collisional loss rate has a marked dependence on the trap laser intensity. We will present strong circumstantial evidence that the dependence at very low intensities is due to hyperfinechanging collisions between ground-state atoms. We believe that the loss rates at higher intensities are associated with collisions involving excited states, and are the type discussed by Gallagher and P r i t ~ h a r d . ~ As discussed in Ref. 7, these collisions are very different from normal ground-excited-state atomic collisions in which two initially distant atoms, A and A * , approach, collide, and separate in a time much less than the radiative lifetime of the excited state. In contrast, for these very-low-temperature collisions the absorption and emission of radiation in the midst of the collision drastically alter the motion. In particular, if the excitation takes place when the two atoms are far apart ( R > 1000 A) they will reradiate before being pulled into the small-R region where energy transfer occurs. However, if they are sufficiently close when excited, they can be pulled close enough together for substantial potential energy to be transferred into kinetic energy before decaying. The two dominant transfer processes are excited-state fine-structure changes and radiative redistribution. In the first, A* changes its fine-structure state in the collision and the pair acquire a fine-structureinterval worth of kinetic energy. The second process, radiative redistribution, refers to A-A * reemitting a photon which, because of the A-A * attractive potential, has substantially less energy than that of the photon which was initially absorbed. This energy difference is transferred to the subsequent kinetic energy of the ground-state atoms. The trap loss rate depends on the probability of exciting such “close” A-A * pairs, and this probability is determined by the frequency and intensity of the exciting radiation. Light which is tuned to the red of the atomic resonance frequency, VO, excites pairs which are closer together (and shifted in energy) and thus is more effective at causing trap loss than light which is at VO. We tested this hypothesis by examining how the loss
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@ 1989 The American Physical Society 273
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rate changes when the trapped atoms are illuminated by an additional laser tuned far from resonance. It is particularly straightforward to interpret this second measurement because such a red-detuned “catalysis” laser is very ineffective at exciting isolated atoms and, hence, has a negligible effect on the trap depth. Also, as discussed below, it is considerably simpler to theoretically predict the collisional loss rate for the case of large detuning. Much of the apparatus for this experiment was the same as in our earlier work with cesium in optical molasses, and the arrangement of laser and atomic beams is very similar to that discussed in Ref. 10. A beam of cesium atoms effused from an oven and passed down a I-m tube into a UHV chamber (lo-’’ Torr). In this chamber the atoms were trapped using a Zeeman-shift spontaneous-force trap2 formed by having three perpendicular laser beams which were reflected back on themselves. The beams were all circularly polarized, with the polarizations of the reflected and incident beams being opposite. They had Gaussian profiles with diameters of about 0.5 cm, and were tuned to the 6 S p 4 to 6Py2f-s transition of cesium. Magnetic field coils which were 3 cm in diameter and arranged in an anti-Helmholtz configuration provided a field which was zero a t the middle of the intersection of the six beams, and had a longitudinal gradient of 5.1 G/cm. In addition to the main trap laser beams, there were beams to ensure the depletion of the F - 3 hyperfine ground state in the trap. Thus all the trapped atoms were in the F = 4 ground state. Two other lasers, the “catalysis” and “stopping and probing” lasers, were used on occasion as discussed below. A11 the lasers were diode lasers whose output frequencies were controlled by optical and electronic feedback and had short-term linewidths and long-term stabilities of well under 1 MHz. The fluorescence from the trapped atoms was observed with both a photodiode, which monitored the total fluorescence, and a charged-coupled-device television camera which showed the size and shape of the atomic cloud. The spatial resolution was about 10 pm. Data acquisition involved the following sequence: First, atoms were loaded into the trap by opening a flag which blocked the cesium beam. When the trap laser intensity was very low the stopping laser was used to slow the atoms for loading as in Ref. 10. After a few seconds the cesium and stopping laser beams were again blocked, the video image of the trap was digitized and stored, and the total fluorescence signal was measured as a function of time for the next 205 s. In a separate measurement, we determined the density and the excited-state fraction by loading the trap under the same conditions and observing the absorption of a weak probe beam by the cloud with the trapping light on, and 1 ms after it had been switched off. To measure the temperature of the trapped atoms the probe was moved a few mm to the side of the cloud and the time-of-flight spectrum was observed after the trap light was turned off, as in Ref. 1 1 . The temper962
28 AUGUST1989
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ature was found to be between 2.5 and 4 x 10 K. In the initial stages of this work we loaded the trap with as many atoms as possible. This produced a large low-density cloud of trapped atoms which had a loss rate which was proportional to the total number of trapped atoms. The behavior of the cloud in this large-number regime showed a variety of complex cooperative behaviors which will be discussed in a later publication. We subsequently discovered that, when the number of atoms in the trap decreased, the diameter of the cloud diminished, while the density increased. When the number was much lower, the cloud became a small Gaussian sphere with a diameter of about 1.7x l o - * cm. Further reduction in the number of atoms then only reduced the density but not the diameter. In Fig. 1 we show the fluorescence signal from the cloud in this constant-diameter regime, along with a fit which assumes a loss rate of the form d n / d t = -an - p n 2 , where n is the density of atoms in the trap. We also assume that the fluorescence is proportional to the density. The figure shows that this is the appropriate dependence, and the n dependence indicates that collisions between the trapped atoms are causing loss from the trap. The a term is responsible for a simple exponential decay. For all the data presented here, we were careful to ensure that the initial number of atoms in the trap ( 3 l o~4 ) was low enough that the cloud diameter remained constant. This was verified both by directly measuring the cloud diameter and by checking that the fluorescence decay had the expected time dependence. As mentioned earlier, we measured collisional losses under two different conditions. The first approach, similar to that used in Ref. 9, was to simply examine the rate at which atoms were lost from the trap as a function of
10”
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TlME (SEC) FIG. 1. Fluorescence from the cloud of trapped atoms as function of time after the loading is terminated. The solid line is a fit to the data taking the loss rate t o be an + p n 2 , where the initial density was 1.2xlO9/cm3, giving aP1/(132 s), and ~-7.6xI0-”cmJ/s.
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the intensity of the trap laser light. The quantity of interest was the coefficient p, since a depends only on the background pressure in the vacuum chamber. T o find p, we fitted the fluorescence decay curves as in Fig. 1 to get pn0, and from the absorption measurements and the video image we obtained the initial density, no. The digitized image of the cloud showed that the fluorescence was a Gaussian distribution as one would expect. The initial density was between 1 and 4 x l o 9 atoms/cm3, and the probe-beam absorption was 2.5% to 10%. In Fig. 2 we show the dependence of p on the total intensity of all trap laser beams for a trap laser detuning of 5 M H z (I natural linewidth). The scatter in the points provides a reasonable estimate of the uncertainty in the measurements. We should mention that there was far more scatter in the data before considerable effort was expended to improve the quality of the laser beam wave fronts, the alignment, and the measurement of the trap size. W e interpret the data as follows. The large p values seen in Fig. 2 for intensities less than 4 mW/cm2 are due to hyperfine-changing collisions between ground-state atoms. Two atoms acquire velocities of 5 m/s if they collide and one changes from the 6Sj7-4 hyperfine state to the lower F - 3 state. For high trap laser intensities, atoms with this velocity cannot escape, but at lower intensities the trap is too weak to hold them. We also find that if the beams are misaligned slightly, a similarly large+ loss rate is observed for all intensities. W e estimate that hyperfine-changing collisions between ground-state atoms due to the Van de Waals interaction will give a p in the range lo-'' to l o - ' ' cm3/s, which is consistent with the data shown in Fig. 2. Above 4 mW/cm2, p appears to increase linearly with intensity. W e believe that this is due to the energy-transfer collisions of the type discussed above.
28 AUGUST1989
We studied these collisions in more detail by using an additional laser to illuminate the atoms in the trap. As mentioned earlier, measuring the p caused by this laser when it is detuned to the red provides a particularly good way to test the Gallagher-Pritchard model for these collisions. The data acquisition in this case was exactly the same as before. In Fig. 3 we show the results. These data were taken with a trap laser intensity of 13 mW/cm2 and a detuning of 5 MHz. The solid line is the prediction obtained using the Gallagher and Pritchard model.I2 For detuning greater than 100 M H z the curve turns over simply because of the decrease in the number of pairs with such small separations. (A frequency shift of 100 M H z corresponds to a separation of about 500 A.) The agreement between theory and experiment is excellent in view of the estimates used in the calculation, which are discussed below, and the experimental uncertainties. It is much less straightforward to compare the results shown in Fig. 2 with theory because the laser detuning is much smaller. The calculation in Ref. 7 neglects the excited-state hyperfine splitting and the initial atomic velocity which are substantially more important effects in the small-detuning case. When the laser is tuned close to the atomic resonance, the initial interatomic force is weak because the colliding atoms are relatively far apart. As a result, the probability of the atom getting in to small R before radiating has some dependence on its initial velocity. This probability also depends very sensi-
I
lo-'"
'.U
lo-" 0.0 -1200.0
-900.0
-600.0
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0.0
DETUNING OF CATALYST LASER (MHz)
lo-" 0.0
3.0
6.0
9.0
12.0
15.0
Intensity (mW/m2)
FIG. 2. Dependence of p on the total intensity in all the trap laser beams. The solid line indicates the prediction of the Gallagher-Pritchard model, and would be a straight line if the scale were linear.
FIG. 3. Dependence of p on detuning of the catalysis laser. The zero of detuning corresponds to the center of gravity of the 6P3/2 hyperfine states. The absence of data for a detuning of less than 300 M H z is caused by the need to be well away from all 6S-to-6P3/2 hyperfine transitions to avoid perturbing the trap. These data are for a catalysis laser intensity of 24 mW/cm2. The dotted line shows the value of p which is obtained when the catalysis laser is not present. The solid line indicates the predictions obtained using the Gallagher-Pritchard model.
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tively on the shape of the potential, which, a t these distances, will be strongly affected by the excited-state hyperfine interaction and clearly will not be t h e simple C3/R dependence which was assumed. This could easily change the potential enough to shift the theoretical result by a factor of 5 , and hence would explain the difference between the magnitudes of the theoretical and experimental results. T h e theory does predict that p should increase linearly with intensity which is what we observe. For the larger detunings used with the catalysis laser, the theory is less sensitive to the shape of the potential curves and the atomic velocity, because the probability of reaching small R is approaching 1. However, a factor of 2 error is still not surprising. It is not clear why our results differ from those of Prentiss et a/.,but we do not think it is likely that sodium behaves fundamentally differently from cesium. O n e possibility is that for the range of intensities they used the sum of hyperfine-changing and excited-state collisional losses remained roughly constant. W e have shown that collisions which depend on the trap laser intensity will b e an important factor in determining the densities attainable in optical traps. This will be true of all types of optical traps, but the exact loss rate will depend on the frequency and intensity of the laser light present. Also, ground-state hyperfinechanging collisions can be a n important loss mechanism if the trapped atoms are not in the lowest hyperfine state and the trap is not sufficiently deep. W e are pleased to acknowledge the assistance of W. S w a m with the construction and operation of the stabilized diode lasers. This work was supported by the Office of Naval Research and the National Science
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Foundation (")Staff member, Quantum Physics Division, National Institute of Standards and Technology, Boulder, CO 80309-0440. ID. Pritchard, E. Raab, V. Bagnato, C. Wieman, and R. Watts, Phys. Rev. Lett. 57, 310 (1986). 2E. L. Raab, M. G. Prentiss, A. E. Cable, S. Chu, and D. E. Pritchard, Phys. Rev. Lett. 59, 2631 (1987). 3J. Vigue, Phys. Rev. A 34,4476 (1986). 4P. Julienne, Phys. Rev. Lett. 61, 698 (1988). sP. Julienne, in Aduonces in Loser Science III, edited by A. C. Tam, J. L. Gole, and W. C. Stwalley, AIP Conference Proceedings No. 172 (American Institute of Physics, New York, 1988), p. 308. This presents a rough quantummechanical estimate for the radiative redistribution loss, as contrasted with the semiclassical treatment of A. Gallagher and D. Pritchard, preceding Letter, Phys. Rev. Lett. 63, 957
(1989). 6D. Pritchard, in Electron and Atom Collisions, edited by P. Lorents (North-Holland, Amsterdam, 1986). p. 593. 'Gallagher and Pritchard, Ref. 5. 8P. L. Gould, P. D. Lett. P. S. Julienne, W. D. Phillips, H. R. Thorsheim, and J. Weiner, Phys. Rev. Lett. 60, 788 (1988). 9M. Prentiss, A. Cable, J. E. Bjorkholm, S. Chu, E. Raab, and D. Pritchard, Opt. Lett. 13, 452 (1988). 'OD. Sesko, C. G. Fan, and C . Wieman, J. Opt. SOC.Am. B 5 , 1225 (1988). "P. D. Lett et al., Phys. Rev. Lett. 61, 169 (1988). 121n the terminology of Ref. 7, we have used the values C3-49 eV A 3 and qj-0.033 in evaluating the GallagherPritchard model. qj is estimated from a plot of measured fine-
structure-changing collision rates versus the Landau-Zener adiabaticity parameter. We also estimated the probability of sufficient radiative redistribution of energy to cause trap loss to be 0.027, using a trap depth of 1 K.
VOLUME 64, NUMBER 4
PHYSICAL REVIEW LETTERS
22 J A N U A R Y 1990
Collective Behavior of Optically Trapped Neutral Atoms Thad Walker, David Sesko. and Carl Wieman Joint Institute f o r Laboratory Astrophysics, University of Colorado and National Institute of Standards and Technology. and Departrnenr of Physics, University of Colorado, Boulder, Colorado 80309-0440 (Rcccivcd 5 October 1989)
We describe experiments that show collective behavior in clouds of optically trapped neutral atoms. This collective behavior is demonstrated in a variety of observed spatial distributions with abrupt bistable transitions between them. These distributions include stable rings of atoms around a small core and clumps of atoms rotating about the core. The size of the cloud grows rapidly as more atoms are loaded into it, implying a strong long-range repulsive force between the atoms. We show that a force arising from radiation trapping can explain much of this behavior. P A C S numbers: 32.80.Pj.42.50.Vk
ing rate and the collisional loss’ rate. Two other lasers were used to deplete the F - 3 hyperfine ground state in both the trapped atoms and the atomic beam. All the lasers were diode lasers with long-term stabilities and linewidths well under 1 MHz. The trapped atoms were observed by a photodiode, which monitored the total fluorescence, and a chargecoupled-device (CCD) television camera, which showed the size and shape of the cloud of atoms. The optical thickness of the cloud was determined by measuring the absorption of a weak probe beam 1 ms after the trap light was turned off. To measure the temperature of the atoms, the probe beam was moved a few mm to the side of the trap and the time-of-flight spectrum was observed after the trap light was rapidly turned off, as described in Ref. 5. By pushing the atoms with an additional laser, as was done in Ref. 2, we determined that the trapping potential was harmonic out to a radius of 1 mm with a spring constant of 6 K/cm2. In this experiment, care was taken to obtain a symmetric trapping potential. The Earth’s magnetic field was zeroed to 0.01 G, and the trapping laser beams were spatially filtered, collimated, and carefully aligned with respect to the magnetic field zero. Before these precautions were taken we observed a variety of random shapes and large density variations within the trap. Three separate well-defined modes of the trapped atoms were observed, depending on the number of atoms in the trap and on how we aligned the trapping beams. The “ideal-gas” mode occurred when the number of trapped atoms was less than 40000. The atoms formed a small sphere with a constant diameter of approximately 0.2 mm. In this regime, the density of the sphere had a Gaussian distribution, as expected for a damped harmonic potential, and increased linearly in proportion to the number of atoms. The atoms in this case behaved just as one would expect for an ideal gas with the measured temperature and trap potential. When the number of atoms was increased past 40000, the behavior deviated dramatically from an ideal gas, and strong long-range repulsions between the atoms became apparent. In this “static” mode the diameter of
In the past few years there have been major advances in the trapping and cooling of neutral atoms using laser light. It has recently become possible to hold relatively large samples of atoms for minutes at a time and cool them to a fraction of a mK using a spontaneous-force optical trap.’.’ One would expect these atoms to behave like any other sample of neutral atoms, namely they would move independently as in an ideal gas, except when they undergo short-range collisions with other atoms. In this paper we report on experiments which show that contrary to these expectations, optically trapped atoms behave in a highly collective fashion under most conditions. The spatial distribution of the cloud of atoms is profoundly different from that of an ideal gas, and we observe dramatic dynamic behavior in the clouds. We believe that this unusual behavior arises from a longrange repulsive force between the atoms which is the result of multiple scattering of photons by the trapped atoms. This force is sensitive to the modification of the emission and absorption profiles of the atom in the laser field. In the past, this force has been neglected in any system smaller and colder than stars. However, because of the extremely low temperature and relatively high densities of optically trapped atoms, it becomes very important. The apparatus for this experiment has been described e l ~ e w h e r e . ~ ,The ~ trap was a Zeeman-shift spontaneous-force trap,* and was formed by the intersection of three orthogonal retro-reflected laser beams. The beams came from a diode laser tuned 5-10 M H z below the 6Si12 F-4 to 6 P 3 j ~F-5 transition frequency of cesium. The beams were circularly polarized with the retro-reflected beams possessing the opposite circular polarization. A magnetic field was applied that was zero at the center of trap and had a vertical field gradient (5-20 G/cm) 2 times larger than the horizontal field gradient. Bunches of slowed cesium atoms from an atomic beam were loaded into the trap by chirping a counterpropagat~ were ing “stopping” laser at 20 times a ~ e c o n d . We able to load about 2~ lo6 atoms per cooling chirp. The number of atoms in the trap was determined by the load408
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the cloud smoothly increased with increasing numbers of atoms. Instead of a Gaussian distribution, the atoms in the trap were distributed fairly uniformly, as shown in Fig. l(a). To study the growth of the cloud we measured the number of atoms versus the diameter of the cloud using a calibrated CCD camera. The results are shown in Fig. 2. T o fully understand the growth mechanism the temperature as a function of diameter was also measured. For a detuning of 7.5 MHz, the temperature increased from the asymptotic value of 0.3 up to 1 .O mK as the cloud grew to a 3 mm diam. If the cloud were an ideal gas, the temperature increase would need to be orders of magnitude higher to explain this expansion. One distinctive feature of this regime is that the density did not increase as we added more atoms to the trap.
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When all the trapping beams were reflected exactly back on themselves we obtained a maximum of 3 . 0 l~o * atoms in this distribution. In contrast, if the beams were slightly misaligned in the horizontal plane, we observed unexpected and dramatic changes in the distribution of the atoms. The atoms would collectively and abruptly jump to “orbital” modes when the cloud contained approximately lo8 atoms. The number of atoms needed for a transition depended on the degree of misalignment. The shapes of these orbital modes are illustrated in Fig. 1 11 (b), l k ) , and l ( d ) show the top view, and I(e) shows the view from the side]. We first observed the ring around a center shown in Fig. l ( b ) and then, by strobing the camera, discovered it was actually a clump of atoms orbiting counterclockwise about a central ball
FIG. 1 . Spatial distributions of trapped atoms. (a) Below 10’ atoms the cloud forms a uniform density sphere. (b) Top view of rotating clump of atoms without strobing. (c) Top view of ( b ) with the camera strobed at I10 Hz. (d) Top view of a continuous ring. ( c ) Side view of (d). Horizontal full scale for (a), (d), and (e) is 1 .O cm; for (b) and ( c ) it is 0.8 cm. 409
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1.2
0.8
0.4
0.0 1 00
I
1.0
1
2.0
I
3.0
Number of atoms
I
4.0
I 5.0
(XIO')
FIG. 2. Plot of the diameter (FWHM) of the cloud of atoms as a function of the number of atoms contained in the cloud. For the full figure the magnetic field gradient is 9 G/cm and 16.5 G/cm for the inset. The laser detuning is -7.5 MHz, and the total laser intensity is 12 mW/cm2. The solid lines show the predictions of the model described in the text.
as shown in Fig. 1 (c). Note that there is a tenuous connection between the clump and the center, and that the core has an asymmetry which rotates around with the clump. As more atoms were slowly added to the trap there were several abrupt increases in the radius of these orbits. The clumps orbited at well-defined frequencies between 80 and 130 Hz. W e have also observed a continuous ring encircling a ball of atoms as shown in Fig. 1 (d). In this case we assume that the atoms are still orbiting, but no clumping of the atoms was seen and there was no connection between the ring and the central ball. We observed only one stable radius (2.5 mm) for this ring. The ring is shaped by the intersection geometry of the laser beams with the corners of the ring corresponding to the centers of the beams. The atoms lie in a horizontal plane about 0.5 mm thick [Fig. l(e)l. As mentioned above, the formation of these stable orbital modes depended on the alignment of the trapping beams in the horizontal plane. The beams had a Gaussian width of 6 mm and the return beams were misaligned horizontally by 1-2 mm (4-8 mrad) at the trap region. It was observed that the direction of the rotation corresponded to the torque produced by the misalignment. The degree of the misalignment affected which of the orbital modes would occur. Typically, the atoms would switch into the rotating clump for misalignments of 1-1.5 mm and into the continuous ring for 1.5-2 mm. If the light beams were misaligned even further, the cloud would be in the ring mode for any number of atoms. The rotational frequencies of the atoms were measured by strobing the image viewed by the camera or by observing a sinusoidal modulation of the fluorescence of the atoms on a photodiode. The frequency depended on the detuning of the trapping laser, the magnetic field gradient, and the average intensity of the trapping light. 410
22 JANUARY1990
For each of these, the frequency of the rotation increased as the spring constant of the trap was increased. We also discovered we could induce the atoms to jump into rotating clumps by modulating the magnetic field gradient at a frequency within 8 H z of the rotational frequency. The transition between the static and orbital modes showed pronounced hysteresis. When the number of atoms was slowly increased until it reached lo8, the cloud would jump in < 20 msec from the static mode to the rotating clump and the trap would then lose approximately half its atoms in the next 50-100 msec. This orbital mode would remain stable until there was a small fluctuation in the loading rate. Then the cloud would suddenly switch back to the static mode without losing any atoms. The number of atoms would have to build up to lo8 again before it would jump back into the orbital mode. We have also observed hysteresis between the static and orbital modes when the trap would jump into the continuous ring. The trap started with 10' atoms in the static mode and then 80% of these atoms would jump into a continuous ring with the remaining 20% in the central ball. For certain loading rates the cloud would repeatedly switch back and forth between the two modes. The expansion of the cloud with increasing numbers of atoms and the various collective rotational behaviors clearly show that strong, long-range forces dominate the behavior of the atomic cloud. Since normal interatomic forces are negligible for the atom densities in the trap (lO'o-lO'' ~ m - ~ )other , forces must be at work. In particular, since the optical depth for the trap lasers is on the order of 0.1 for our atom clouds ( 3 for the probe absorption at the resonance peak), forces resulting from attenuation of the lasers or radiation trapping can be important. The attenuation force is due to the intensity gradients produced by absorption of the trapping lasers, and has been discussed by Dalibard in the context of optical molasses.6 This force compresses the atomic cloud and so cannot explain our observations. In the following, we demonstrate that radiation trapping produces a repulsive force between atoms which is larger than this attenuation force and thus causes the atomic cloud to expand. W e will also show how radiation trapping leads to the rotational orbits we observe. The attenuation force, FA,for small absorption, obeys the relation
The atom density is R. the cross section for absorption of the laser light is uL,and the incident intensity of a single laser beam is I,. The absorbed photons must subsequently be reemitted and can then collide with other atoms. Although this was neglected in Ref. 6, this reabsorption of the light (radiation trapping) provides a repulsive force, FR, which is larger than FA. Two atoms separated a dis-
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tance d in a laser field of intcnsity I repel each other with a force
1 FRI
~ a ~ a ~ ~ / ~ r C d ~ ,
(2)
where for simplicity we have assumed isotropic radiation and unpolarized atoms. (The observed fluorescence is unpolarized.) The cross section UR for absorption of the scattered light is in general different than U L due to the differing polarization and frequency properties of the scattered light.7 For a collection of atoms in an optical trap, the radiation trapping force obeys
V.F l p = 6 u ~ c r ~ l , n l c .
(3)
Comparison of (1) and (3) shows that the net force between atoms due to laser attenuation and radiation trapping is repulsive when UR > U L . In the limit that the temperature can be neglected, FR+FA must balance the trapping force - k r . This implies a maximum tichievable density in the trap of n m a x - c k / 2 a ~ ( a R -c~)l-. This allows us to explain the behavior shown in Fig. 1. Once the density reaches nmax.increasing the number of atoms only produces an increase in the size of the atomic cloud. To compare this model to experiment we numerically solve the above equations with the finite temperature taken into account. The only free parameter in the calculation is UR/OL - 1. A value of C R / U L - 1 -0.3 gives the solid curve shown in Fig. 2, which agrees well with experiment. We have estimated U R / U L by convoluting the emission and absorption profiles for a two-level atom' in a ID standing wave field. The intense laser light causes acStark shifts, giving rise to the well-known Mollow triplet emission spectrum. The absorption profile is also modified by the laser light, being strongly peaked near the blue-shifted component of the Mollow triplet, causing the average absorption cross section for the emitted light to be greater than that for the laser light. We calculate alp la^ - I -0.2 for our detuning and intensity ( 1 - - 2 mW/cm2). While this is less than the value of 0.3 needed to match the data in Fig. 2, it is very reasonable that our twenty-level atom in a 3D field should differ from our simple estimate by this amount. The inset of Fig. 2 shows that the expansion of the cloud deviates from this model for sizes greater than 1.5 mm. However, the precise behavior in this region is very sensitive to alignment and is not very reproducible. Other effects not included in the above model, such as magnetic field broadening, optical pumping, multiple levels of the atom, the spatial dependence of the laser fields, and multiple ( > 2) scattering of the light may also be important in this region. We can explain the rings of Fig. 1 by considering the motion of atoms in the x-y plane when the laser beams are misaligned. A first approximation to the force produced by this misalignment is F I = k ' i x r . The atoms are also subject to a harmonic restoring force - k r and a
22 JANUARY 1990
damping force --ydr/dr. If we ncgicct radiation trapping, we find circular orbits can exist with angular frequency o - k ' / y , if k ' / y - ( k / r n ) Since the orbits may have any radius, this clearly does not explain the formation of rings. If, however, we add a force ( a N / r * ) i due to the radiation pressure from a cloud of N atoms within the orbit radius R,we find circular orbits exist only for (4)
if o < ( k / r n ) Thus the effect of the radiation from the inner cloud of atoms is to cause circular orbits at a particular radius 6.e.. rings are formed), and to allow circular orbits for a large range of misalignments. This is in accord with our observations that the orbits always encompass a small ball of atoms (Fig. 1). We have done a more detailed calculation including the Gaussian profiles of the lasers which gives a rotational frequency of 130 Hz and an orbit diameter of 3.5 mm for the conditions of Fig. I (d). While the simple model of the radiation trapping force we have presented explains the expansion of the cloud and the existence of the circular rings, there are several interesting aspects of the observed phenomena which are not explained. These include the formation of rotating clumps of atoms and the dynamics of the transitions between different distributions. We have produced dense cold samples of optically trapped atoms. We find that this unique new physical system shows a fascinating array of unexpected collective behavior at densities orders of magnitude below where such behavior was expected. Much of this collective behavior can be explained by the interaction between the atoms in the trap due to their radiation fields. However, the detailed distributions and the transition dynamics deserve considerable future study. This work was supported by the Office of Naval Research and the National Science Foundation. We are pleased to acknowledge helpful discussions with T. Mossberg and the contributions made by C. Monroe and W. Swann to the experiment.
ID. Pritchard, E. Raab, V. Bagnato, C. Wieman, and R. Watts, Phys. Rev. Lett. 57, 310 (1986). 2E. L. Raab, M. G . Prentiss, A. E. Cable, S. Chu, and D. E. Pritchard, Phys. Rev. Lett. 59, 2631 (1987). 3D. Sesko, T. Walker, C. Monroe, A. Gallagher, and C. Wieman, Phys. Rev. Lett. 63,961 (1989). 4D.Sesko, C. G. Fan, and C. Wieman, J. Opt. SOC.Am. B 5, 1225 (1988).
5P. D. Lett ef 01.. Phys. Rev. Lett. 61, 169 (1988).
6J. Dalibard, Optics Commun. 68, 203
(1988).
7B. R. Mollow, Phys. Rev. 188, 1969 (1969); Phys. Rev. A 5, 2217 (1972); D. A. Holm, M. Sargent, 111, and S. Stenholm, J. Opt. Soc. Am. B 2, 1456 (1985). 41 1
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a reprint from Journal of the Optical Society of America B
Behavior of neutral atoms in a spontaneous force trap D. W. Sesko, T. G. Walker, and C . E. Wieman Joint Institute for Laboratory Astrophysics, University of Colorado, and National Institute of Standards and Technology, Boulder, Colorado 80309-0440 Received March 27, 1990;accepted October 10, 1990 A classical collective behavior is observed in the spatial distributions of a cloud of optically trapped neutral atoms. They include extended uniform-density ellipsoids, rings of atoms around a small central b d , and clumps of atoms orbiting a central core. The distributions depend sensitively on the number of atoms and the alignment of the laser beams. Abrupt bistable transitions between different distributions are seen. This system is studied in detail, and much of this behavior can be explained by the incorporation of long-range interactions between the atoms in the equation of equilibrium. It is shown how attenuation and multiple scattering of the incident photons lead to these interactions.
1. INTRODUCTION
by a balance between the above forces and the trapping force. This paper reports on a detailed study of the behavior of optically trapped atoms and the comparison with a theoretical model that includes effects of radiation trapping and the attenuation of the trapping light. In Section 2 we present the theory, in Section 3 we describe the experimental apparatus, and in Section 4 we compare the results of the theoretical calculations with the experimental data.
Over the past few years rapid progress has been made in the ability to produce large optically trapped samples of cold atoms.'.' There have been orders-of-magnitude improvements in the number of atoms trapped and the densities achieved. The deepest optical traps are the Zeeman-shift spontaneous force traps. We have used these to trap 4 x lo8 atoms with densities of more than 10" ~ m - ~At . these densities and numbers the gas of cesium atoms becomes optically thick, and forces arising from this optically dense vapor can have a profound effect on the behavior of the cloud. In a recent paper3 we described a variety of collective effects that arise from this force between the atoms, and we presented a model to explain this behavior. Here we provide more detailed experimental studies and analysis. The neutral atoms in these traps might naYvely be expected to behave as an ideal gas except when they undergo short-range ~ o l l i s i o n s . Although ~~~ there have been speculations concerning deviations from classical ideal-gas behavior, these centered on Bose-Einstein condensation. This quantum-mechanical phenomenon occurs at temperatures much lower and densities much higher than in the present experiments and is quite different from the behavior discussed here. In this paper we consider collective behavior in which the atomic motion is classical in mature and the behavior is analogous to that observed in plasmas and charged-particle beams. At the densities that we studied, one would not expect such behavior if only conventional interatomic forces between the neutral atoms were involved because the interactions would be dominated by the few nearest neighbors and not involve the cloud as a whole. In this situation the behavior would be much like that of an ideal gas. In contrast, we find that atoms in a spontaneous force trap actually show ideal-gas behavior only for small clouds of atoms. Larger clouds show dramatic dynamic and collective behavior, as Fig. 1 illustrates. We propose that this behavior arises from a strong long-range coupling between the atoms due to the multiple scattering of photons and from the attenuation of the trapping beams as they pass through the cloud. The density and the size of the cloud of atoms are determined 0740-32241911050946-13$05.00
2. THEORY A. Growth of Cloud We consider the force on a trapped atom to be made up of three contributions. The first is the trapping force produced by the laser beams and the Zeeman shifts of the atomic energy levels. The two others are interatomic forces between the trapped atoms. These are the attenuation force, caused by atomic absorption of the laser photons, and the radiation trapping force, arising from the atoms' reradiating the absorbed photons, which are subsequently scattered a second time by other atoms. The first two forces compress the cloud of atoms, while the latter causes it to expand. The three forces are illustrated in Fig. 2 and will be discussed in turn. The trapping force is a damped-harmonic force and was discussed previously.',' The trap is of the Zeeman-shift spontaneous force type, but the following analysis is suitable for any type of spontaneous force trap forming a damped-harmonic potential. The trapping force can be derived if we consider the force on an atom exerted by three pairs of intersecting circularly polarized beams. Each of these beams is reflected back on itself but with the opposite polarization. We define the polarization of the light with respect to the atoms such that r + (r-) polarization drives the Am = +1 (Am = -1) transitions. These beams intersect at the minimum of a magnetic field of magnitude B, = BfZ. By favoring absorption of one polarization over the other, the magnetic field gradient produces a harmonic potential to first order' in each direction such that F = - k r . The Doppler shift provides a damping that makes the atomic motion overdamped by typically a factor of 10. 0 1991 Optical Society of America
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Fig. 3 . Spatial distributions of trapped atoms. (a)With fewer than 10' atoms, the cloud forms a uniform-density sphere. (b) Top view rotating around the nucleus in a counof ratating clump without strobing. (c) View of (b) with camera strobed at 110 Hz. terclockwise direction. (d) Top view of rotating clump without strobing but with a than in (b) (the clump is one quarter of the area of the total fluorescence). (e) Top view of continuous ring. (f) Side view of (e). The horizontal full scale for (a), (d), (e), and (f) is 1.0 cm. For (b) and (c) it is 0.8 cm.
are in the trap, the attenubecomes important. The atThe first is a slight reduction of the spring constant of the trap. Much more important, however, is the force resulting from the local intensity imbalance produced by the absorption, as Figs. Z(a) and Z(c) show. This force was discussed in the context of optical molasses by Dalibard' and Kazantsev et a1." Finally, the intensity of the retroreflected beams is reduced by the first pass through the cloud. This causes an uninteresting shift in the pos
The strength of the force associated with the local intensity imbalance is found in the following way. The lasers are initially propagating in opposite directions along the x axis with intensity I , and are attenuated by the trapped atoms. We define an absorption function
AxW = ( 4 I:%ntr)dx,
(1)
where (VL) is the absorption cross section for the incident laser beams and n(r) is the atomic density o f the cloud.
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where AT is the total absorption of light across the cloud and Z+(I-) is the intensity of the beam traveling in the positive (negative) coordinate direction. The two unattenuated beams are assumed to have the same intensity I,. The force due t o the intensity differential between the positive and negative x beams is then
FA,@')= -(+~)IrnAx(F)/c,
(3)
where c is the speed of light. By following the same arguments for they and z directions, we find that the attenuation force F A obeys the relation 0.7
-
V .F A I
0.6
-0.15
-0.09
I
-0.03
x
(4
2.5
-2.5
1
I -0.15
I
I
-0.09
I
I
0.03
0.09
I
I
-0.03
I
I
I
x
0.15
(4
I
I
0.03
0.09
I 0.15
(4
+
Since our maximum absorption of the trapping beams was approximately 20%, we can use the small-absorption approximation (accurate to 2%). Then the spatial dependence of the intensity as a function of x is Ix,(r) = I41 - [AT(xz)
* Ax(r)l/2),
(2)
-6(~~)~I,n/c.
(4)
The negative sign indicates that this attenuation force compresses the cloud or, equivalently, is an effective attractive force between the atoms. This is the force on the atoms due only to the absorption of the trap laser photons. However, for any real atom these photons must be reemitted, and the re-emitted photons also exert a force. This force is illustrated in Figs. 2(b) and 2(c). It is clear that this must be a repulsive force, because in the process of one atom emitting a photon and a second atom absorbing it, the relative momentum of the atoms is increased by 2hk. In a laser field of intensity I, an atom absorbs and subsequently reradiates energy at rate ( U L ) ~ . Thus the intensity of light radiated by one atom at the position of a second atom located a distance d away is
The force on the second atom due to the light emitted by the first is (Irad/c)(uF), where (UF) is the absorption cross section for the scattered photons. A critical point is that the reradiated light is different from the incident light because of such effects as frequency redistribution and depolarization of the scattered light. Thus the average absorption cross section for the scattered fluorescence light, (Q), is different, in general, from the cross section for the trapping photons, (uL). Using Eq. (5), we obtain a repulsive force between the atoms of magnitude
The preceding argument may be extended from two atoms to an arbitrary distribution of atoms, in analogy to Gauss's law of electrostatics, if we assume that an incident photon is unlikely to scatter more than twice. The radiation trapping force thus takes the form
(c) Fig. 2. Forces that arise within an optically thick cloud of atoms. The diameter of the cloud is 0.2 cm. (a) The change in the intensity of the trapping light due to absorption across the cloud of atoms produces the attenuation force Fax a (uL)(I+ - I-). (h) The spontaneous emission of two atoms separated by a distance d produces the repulsive radiation trapping force between the atoms, F Rcc~( n ~ ) d - ' . (c) The three forces in units of kelvins per centimeter. Note that the total force is FT= - k x FA^ + Fnx = 0 within the cloud.
=
FR(r)=
(uF)(uL)z n ( r ' r) m r' d3r', ~
4TC
(7)
and
v . FR = ~ ( U F(UL)I-TZ(r)/C. )
(8)
Here we have replaced the total average intensity I with the average intensity of each of the six trapping beams, I, = I/6,so that Eq. (8)is in the same form as the attenuation force in Eq. (4). Note that the force falls off as l / r 2 for an atom outside the cloud distribution just as it does for a point charge outside a charge distribution. To have the same repulsive force as that due to radiation trapping under our conditions, an atom would need a charge of aptimes the charge of the electron. proximately 5 x
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We may formulate another derivation of the radiation trapping force for a spherical distribution of atoms of uniform density by using the principle of conservation of photons. Here we have that the total outward force on a spherical shell of radius R enclosing N radiating atoms is given by
where the total reradiated intensity in a laser field of intensity I = 61, is
Ir&= (uL)IN/4rR2.
(10)
This result agrees with the solution of Eq. (7) for a spherical cloud of uniform density. It is a good approximation to what we observe for many experimental conditions. The net force between the atoms due to laser attenuation and radiation trapping can be obtained by adding these two forces. Comparing Eqs. (4) and (8), we see that the net force is repulsive if (uF) > (uL). This condition states that if the cross section for absorption of the reradiated light is greater than that for incident laser light, then the repulsive force due t o the re-emission of photons is greater than the attractive force due to the attenuation of the trapping beams. The resulting net repulsive force leads to the expansion of the cloud and thus limits the density of the atoms. In the limit of zero temperature, if the atoms are in equilibrium the attenuation and radiation trapping forces must balance the trapping force - k r . Empirically, we know that for large clouds the temperature is not greatly important, since the observed value of thermal energy is much less than the trap potential energy (ksT ( u L ) . If (UF) IQ ) , then the net force is always compressive and Eq. (11) is invalid. Thus, as atoms are added to the trap, the size of the cloud increases but the density is unchanged. We can obtain a more general calculation of the atomic distribution by numerically solving the equation of equilibrium in which a pressure gradient is balanced by the trapping and radiation forces. This is somewhat like the equation of hydrostatic equilibrium in stars, with the trapping force having replaced gravity.8 In our case the pressure gradient across the cloud is V P = TVn + nVT. For simplicity, we assume the temperature gradient across the cloud to be negligible, so for the atoms to be in mechanical equilibrium within the trapped cloud we have TVn(r) = (FA
+ FR- kr)n(r).
(12)
For the special case FA + F R - k r = 0 the density within the cloud is a constant, and we obtain Eq. (11). Taking the divergence of Eq. (12) and using Eqs. (4), (8), and (11) gives TV2 ln[n(r)] = 3k(n/n,.,
- 1).
(13)
949
We may then numerically integrate Eq. (13) for a cloud of
N atoms to find the spatial distribution of the atoms. For the calculation of the spatial distribution of the atoms in the cloud, we assumed strong damping, no convective motions, and no temperature gradients. We also ignored the standing-wave patterns of the lasers (except in the averaging for (UF) and (UL) as discussed below) and the spatial profiles of the lasers. We emphasize that we are not claiming a priori that these and other complications discussed below should (or should not) be significant. We are simply presenting a model that is made tractable by the neglect of them and then showing the successes (and failures) of this model in its explanation of the observations. The next stage of the calculation is to find (UF) and ( u L ) for the atoms in a laser field produced by three intersecting beams. The combination of beams creates a threedimensional standing-wave field
I(x,y, Z ) = 61,
COS'
2TX 2 T y 2rrz -+ - h
h
+
h
+4
where 4 is an arbitrary phase factor. We may visualize this expression for the intensity by first considering the result of two counterpropagating beams of opposite circular polarization. This results in a local linear polarization that traces out a helix in space. Equation (14) follows, then, if we add the light from the two other directions. We obtain the spatially averaged cross section (UL) by averaging the cross section uL = v0[l + Z(x, y, z)/Zs + ~(A/AN)']-~ over a wavelength in an arbitrary direction, where uois the unsaturated, resonant cross section, Is is the saturation intensity, A = W L - oois the trap laser detuning, and A N is the natural linewidth of the transition. We find that
We used this spatially averaged cross section, since this was the approach used in the calculation of (up) in Ref. 9. Note that there are other effects that are, in principle, important in calculating (uF)and (uL), but we neglected them in the spirit of simplification discussed above. These include magnetic field and Doppler shifts, the polarization and coherence properties of the light, local optical pumping of the atoms, and Zeeman precession of the atoms in the magnetic fields. The spatially averaged cross section (UF) for absorption of the reradiated light is different, in general, from that for the incident laser light, (uL). This is due to a change in the frequency properties of the scattered light produced by the ac Stark effectg (in our experiments the average light intensity is 6-12 times the saturation intensity). Calculating (uF)/(uL) exactly is difficult for optically trapped cesium because of the many levels involved, each of which is shifted by the light field. For the purposes of this work we crudely estimated (uF)/(uL), using the emission and absorption spectra for a two-level atom in a monochromatic one-dimensional standing-wave field. These spectra are illustrated in Fig. 3. The scatteredradiation spectrum contains an elastic component at the frequency of the incident photons and an inelastic component, which is the result of the ac Stark shift by the time-
950
Sesko et al.
J. Opt. SOC. Am. B/Vol. 8, No. 5iMay 1991
-ydr/dt. Note that the tangential force k ‘ 2 x r plus the damping force leads to a net orbital drift velocity. The existence of a small ball of atoms within the rings (Fig. 1) suggests that a force (aq,N/r2)? due to radiation pressure from a central cloud of q,N atoms needs to be added as well as a radiation pressure force [ q , N ln(2qrN/7r)/ 2ar2]?on an atom in the ring due to the q,N other atoms Adding these, we obtain in the ring ( a = (uL)(uF)Z/4m). the equation of motion for the trapped atom:
40 -
30 -
m-d ‘r = -kr dt2
- y-dr - k‘2
x r
dt
+ -?. (YN‘
(16)
r2
Here N ’ is the effective number of atoms that provide a repulsive radiation rapping force on the atom and is given by
-1.0
-7.5
-5.0
-2.5
0.0
2.5
5.0
7.5
A/A, Fig. 3. Absorption (solid curve) and emission (dotted-dashed curve) profiles calculated for a two-level atom in a onedimensional standing-wave field. The curves are for an average intensity of 12 mW/cmz and a laser detuning of -1.5A~. The emission profile is in arbitrary units. The zero-width elastic emission peak (which constitutes 50% of the total) is represented by the spike at - 1 . 5 A ~ . dependent fields. This gives the familiar Mollow triplet for the emitted light. The absorption profile is strongly peaked near the blue component of this Mollow triplet. This causes the average absorption cross section for the emitted light, (up), to be greater than that for the incident light, (u~); this critical difference is what causes the combination of attenuation and radiation trapping forces to give a net repulsive force between the atoms. Taking the convolution of the emission and absorption spectra in Fig. 3 to calculate (uF), we find that ( u p ) / ( u L ) = 1.2 for our experimental parameters (a detuning of -1.5AN and a total average intensity 61, = 12 mW/cm2, where AN = 5 MHz is the natural linewidth). Because (uF) > (uL), Eq. (11)applies, and we expect the maximum density to be We emphasize that although a numlimited to n c n-. ber of approximations and simplifications were made in the calculation of ( u ~and ) (uL), the basic physics of the expansion of the cloud relies only on the relation (uF) > (L). As long as the approximations do not change this relation, the cloud expands, subject to a maximum density given by Eq. (11).
B. Formation of Stable Orbits The discussion in Subsection 2.A is valid for any number of atoms as long as the optical thickness is small and the pairs of trapping beams are exactly overlapping. However, a misalignment of the retroreflected trapping beams can put a torque on the cloud of atoms. As Subsection 4.D describes, we observe abrupt transitions t o orbiting rings of atoms surrounding a central nucleus when the number of atoms exceeds a critical value. The formation of the stable orbits can be most easily understood by a consideration of the motion of atoms in the presence of a misalignment of the laser beams in the x-y plane (Fig. 4). Such a misalignment produces a local imbalance between the beams, and thus a force that may be represented phenomenologically by F, = k’& x r. The optical trap also provides a restoring force -kr and a damping force
In order to compare theory and experiment precisely, we must consider the spatial dependence of the trapping beams, which gives an r dependence to k , k’,and ‘y. However, it is illustrative to ignore this spatial dependence for the moment so that Eq. (16)can be solved analytically. By trying the oscillatory solution r = R(32 cos ot + 9 sin w t ) , we find that stable circular orbits exist at a radius
R
=
[a’/(k-
mo2)11/3,
(18)
with orbital frequency o = k’/y Thus we obtain orbital trajectories at a radius R as long as the condition o < (k/m)”’ is satisfied. This condition on the orbital frequency is violated if the torque on the cloud is too high or the damping is too small. In these cases the atoms fly out of the trap rather than orbit. Equation (18) shows that if the radiation pressure is increased (larger a“), the radius of the rings increases. The orbital radius also increases when the spring constant of the trap is decreased
Fig. 4. Geometry of the misalignment in the horizontal plane that gives rise to the orbital modes. The bold boundary and the central ball show the positions of the atoms. The arrow indicates the direction of the orbit. The radiation pressure from the central core is also illustrated.
286 Sesko et al.
Vol. 8, No. 5iMay 199115. Opt. SOC.Am. B
or the frequency of rotation increases. If the radiation pressure term is not included in Eq. (16), we find that there are no stable orbiting solutions. We performed a more general calculation of the orbits than one would obtain by including the r dependence of k , k ' , and y. Rather than using Eq. (16)to calculate this, it is more natural for one to consider the force due t o four separate laser beams. We start by considering the spontaneous force Fxz on an atom for a single beam propagating in the positive or negative x direction, where the Gaussian shape of the beams, the saturation effects, the Zeeman shifts, and the Doppler shifts are included. This force is assumed t o be given by
951
.... 0.2 -
T
g ,,
0.0
-
x
-0.2
-
/ I
-0.4
where Zs is the saturation intensity (IS= 1 mW/cm'), Z(r) is the total local laser intensity, A = O L - w o is the detuning of the laser from line center, oB' is the Zeeman shift in frequency per unit length, and A is the wavelength of the light. This equation for the force is what one obtains for a two-level atom, adding an ad hoc Zeeman shift and assuming that the intensities can be summed to determine the saturation. Although this assumed form is not exactly correct, we believe that it is a good approximation, which contains the relevant physics. The Gaussian intensity profile for the laser beams propagating in the - t x direction is given by
where s is the displacement of the beams with respect to each other and w is the full width a t half-maximum (FWHM) of the Gaussian beam profile. By summing the forces in the forward and reverse directions and adding a radiation trapping force
I
0.0
-0.2
-0.4
I
I
0.2
0.4
(4
x Fig. 5. Trajectories of an atom numerically calculated from our model. The trajectories are shown for ten initial positions and velocities of the atom. The trap parameters were chosen t o be the same as for the conditions in Fig. l(e) (A = - 1 . 5 A ~ ,I/Is = 12, B' = 15 G/cm, w = 6 mm, and s = 1.5 mm). The frequency of the orbit is 82 Hz.
We first numerically solve the equations of motion for a radiation trapping force corresponding to an effective number of atoms, N ' = 5 x lo7. Figure 5 shows the solutions to the equations of motion for several arbitrary starting positions and velocities of the atoms. The model is seen to predict that the atoms orbit in a ring that closely resembles the photographs in Fig. 1. The orbit has a mean radius of 2.5 mm and a rotational frequency of 82 Hz. For N ' = 9 x lo7 the equations of motion predict that the atoms slowly spiral out of the trap, and for larger N ' the atoms are rapidly ejected from the trap. If we examine the case of no radiation trapping force ( N ' = 0), we find that the atoms spiral into the center of the trap. Thus the radiation trapping force is necessary to an explanation of the existence of the orbits.
we obtain a net force
Fx
= FRx
+ ThANz, ~
A Is
{
(g+ - g-) 1 X
Z(r) 4[A2 + ( w B ' x + ux/A)'] ++
Is Z(r) 4[A2 + {I+%+
AN2 (UB'X
+ FyyI,
A.v4
AN'
(23)
where Fr is given by the transposition of x and y and the substitution of uy for ux in Eqs. (21) and (22). These equations of motion for a single atom are solved numerically for the conditions A = -1.5A~,Z(r) = 12 mW/cm', and s = 1.5 mm. The value of W B ' = doB/dr is difficult to calculate, since the atoms are distributed among the various m levels. However, from our measurements of the spring constant and using the correct limits in Eq. (22), we derive a value of W E ' = 10 MHz/cm.
uX/A)
AN'
+ ux/A)'] - 64A'(o~'x + U X / A ) ~'
The totaI force on the atom is found by adding the forces in the x and y directions and is thus
F = FX2
A ( w ~ ' x+
3.
I
'
(22)
EXPERIMENTAL APPARATUS
To produce a cloud of trapped cesium atoms, we started with a thermal atomic beam with an average velocity of 250 m/s. These atoms were slowed to a few meters per second by counterpropagating laser light. The slowly moving atoms then drifted into the spontaneous force Zeeman-shift trap described below and were held there for times on the order of 100 s. The trap cooled the atoms so that their average velocity was approximately 20 cm/s in the nonorbiting mode and compressed the atoms to densities of as much as 10l1~ m - ~We . studied the cloud of trapped atoms by observing the total fluorescence with a
287 952
J. Opt. SOC.Am. BIVol. 8,No. 5/May 1991
Sesko et al.
/I
TRAP LASER
HYPERFINE PUMPING ANTI-HELMHO
Fig. 6. Schematic of the experimental apparatus. The third trapping beam, perpendicular to the page, is not shown.
photodiode and the spatial distribution of the fluorescence with a calibrated charge-coupled device camera. A schematic of the experimental apparatus is shown in Fig. 6. A beam of cesium atoms effused from an oven at one end of the vacuum chamber. A capillary array was used as an output nozzle for the oven to create a large flux of cesium atoms." These atoms were then slowed by a frequency-chirped diode laser entering from the opposite (trap) end of the chamber. The distance from the oven t o the trap was approximately 90 cm. This stopping laser drove the 6S1,2,F=4 + 6P312,p Z 5 resonance transition. A second laser beam, which overlapped the first, excited the 6Sl,z,p=3-+ 6P3,2,~=4 transition to ensure that atoms were not lost to the F = 3 ground state. The frequencies of these two lasers were swept from 500 MHz below the atomic transition to within a few megahertz of the transition in approximately 15 ms. At the end of each chirp the stopping laser was quickly shifted out of resonance, and the F = 3 state depletion laser remained at the same frequency that it had possessed at the end of the ramp. It took atoms 35 ms after the end of the chirp to drift into the trap, after which the 15-ms cooling chirp was repeated. The frequency of the stopping laser at the end of the chirp determined the final drift velocity of the atoms. The number of atoms loaded per chirp was largest when the velocity was approximately 8 m/s. We were able to load approximately 2 x lo6 atoms per chirp into the trap under these conditions. The total number of atoms in the trap was determined by the balance between the loading rate and the collisional loss rate. When no loading was desired, shutters blocked the cesium and stopping laser beams.
The trap was similar to that described in Refs. 1 and 2. The trapping light was produced by a third diode laser tuned below the 6Sliz,F=4 -+ 6P3/2,F = 5 resonance transition. Data were taken over the range of detunings from 5 to 15 MHz. The output from this laser was passed through an optical isolator and was spatially filtered. It was then carefully collimated and split into three circularly polarized beams of 6-mm diameter (Gaussian halfpower widths). These intersected orthogonally at the center of a pair of anti-Helmholtz coils with a 25-mm radius and a 32-mm separation. The beams were then reflected on themselves with the opposite polarization. The magnetic field produced by the coils was 0 at the center of the trap and had a vertical field gradient (5-20 G/cm) two times larger than the horizontal field gradient. The center of the trap was positioned 1 cm below the stopping laser so that the slowing process would not eject the previously trapped atoms. A fourth laser tuned to the 6Sl,z,F=3-+ 6P3/z F=4 transition pumped atoms out of the F = 3 ground state in the trap. All lasers used in this experiment were stabilized diode lasers with long-term stabilities and linewidths of well under 1 MHz. We frequency-stabilized the stopping and trapping lasers by optically locking them to an interferometer cavity." This was done by our sending approximately 10% of the laser light t o a 5-cm Fabry-Perot interferometer located a few centimeters from the laser. The cavity was tilted slightly so that the beam entered at an angle relative to the cavity axis. The cavity returned of the order of 1%of this light to the laser and caused the laser to lock to the resonant frequency of the cavity. The two other lasers, which were used for hyperfine pumping,
288 Sesko et a1
were stabilized by feeding back light from a grating in the Littrow configuration. The front surface of these laser diodes was antireflection coated so that the grating served as a tuning element for the laser frequency and as one of the end mirrors of the laser cavity. The frequencies of all four lasers were monitored with small cesium saturatedabsorption spectrometers. The absorption signals were used to derive error signals, which were fed back electronically to control the laser frequencies by changing the laser currents and/or cavity lengths. In this experiment we observed that the shape and the density of the cloud of atoms depended sensitively on the alignment of the trapping beams. We had also discovered, in a previous experiment5 on collisional loss of atoms from the trap, that the depth of the trap was sensitive to the alignment. For a reproducible experiment, we found that care must be taken to obtain a symmetric trapping potential. We could obtain consistent results only if we carefully centered all the spatially filtered trapping beams on the magnetic field zero while the vacuum chamber was open. In addition, the Earth's magnetic field was zeroed t o 0.01 G so that the field gradient of the trap was radially symmetric. Before these precautions were taken, we observed a large variety of density variations and unpredictable cloud shapes within the trap. We note that motions of atoms in traps with misalignment were briefly described in Ref. 2, but it is difficult to compare our results with those because of the brevity of that discussion.
4.
RESULTS
We now discuss our experimental observations and compare them with our models. First we determined the shape of the trap potential and measured the temperature of the trapped atoms. We then measured the dependence of the cloud shape and size on the number of atoms in the trap. Finally, we studied the atoms' behavior in the orbital mode. This included observing the shape of the orbits, the orbital frequencies, and the transitions between modes. 1.50
-E -E
I
I
I
I
I
I
0.25
0.50
0.75
1.00
1.25
1.50
1.25
1.00
c
C a,
E
0.75
a,
u 0 -
2
0.50
.-
n 0.25
0.00
0.00
953
Vol. 8, No. 51May 199115. Opt. SOC.Am. B
1.75
I (mW/crnz) Fig. 7. Displacement of a small ball of atoms versus intensity of the pushing beam (circles). The solid-line fit shows that the force is harmonic and gives a spring constant of 6 K/cmz. The laser detuning was - M A N , the total intensity was I/Is = 12, and the magnetic field gradient was 15 G/cm. The pushing beam was at the same frequency as the trap laser.
.-. .= u)
24.0
3 3r
$
20.0
.4 L
0
16.0
a,
u
t
C
8
12.0
VI
aJ L
0
2
8.0
LL
4.0
I 0.00
I
0.01
I
0.02
I
0.03
I
0.04
I 0.05
t (sec) Fig. 8. The dots show the TOF spectrum, after the trap light is turned off and the fluorescence is measured, of a probe beam 4 mm to the side of the cloud. The solid curve is a fit to the data for a Maxwell-Boltzmann distribution (T = 265 pK) that heavily weights the rising edge of the TOF spectrum. A. Trap Potential We measured the trap potential by observing the displacement of a small ball of trapped atoms when they were pushed by an additional laser, as was done in Ref. 2. This pushing laser was split off from the trapping laser beam and was linearly polarized. The plot of displacement versus pushing intensity is shown in Fig. 7. The fit indicates that the trap potential was harmonic out to a 1.5-mm radius. The spring constant was 6 K/cmZfor a detuning of -1.5AN and a magnetic field gradient of 15 G/cm. We were unable to measure the trap potential beyond 1.5 mm because at larger radii it was difficult to displace the small ball of atoms in a line.
B. Temperature of Atoms To measure the average temperature of the trapped atoms, we passed a 0.5-mm-diameter probe beam 4 mm from the side of the cloud. We then observed a time-of-flight (TOF) spectrum" after the trap light was rapidly shifted out of resonance. This shift effectively turns off the optical trap quickly, and we accomplished it by switching the current on the trapping laser so that the laser frequency was moved many gigahertz from the atomic transition. The switching time with this technique was well under 10 p s . Approximately 0.5 ms later, all the lasers, except for the probe laser and pumping laser 1, were mechanically shuttered to reduce the scattered light in the vacuum chamber. We then obtained the TOF spectrum from the fluorescence of the probe laser. Figure 8 shows the average of 16 TOF spectra. In calculating the theoretical fit in Fig. 8, we assumed a Maxwell-Boltzmann distribution of velocities and heavily weighted the rising edge of the TOF spectrum. The fit gives a temperature of 265 pK for a cloud with approximately 5 x lo6 atoms. The poor fit shows that either there is a spatially dependent temperature in the cloud or the velocity distribution is non-Maxwellian. The plot of temperature (as defined by the above arbitrary procedure) versus cloud diameter is shown in Fig. 9(a) for two detunings. In both cases the tempera-
289 954
J. Opt. SOC.Am. B/Vol. 8, No. 5/May 1991
Sesko et al.
A=-1.5AN
0
0.0
0
A=-2.OAN
0.8
1.6
3.2
2.4
4.0
FWHM (mm)
(4 2.5
I
I
in the trap and on how the trapping beams were aligned. These modes are described in detail here and in Subsection 4.D. The ideal-gas mode occurred when the number of atoms was fewer than approximately 80,000. The atoms formed a small sphere with a diameter of approximately 0.2 mm. We found the distribution of the atoms by digitizing the image of the cloud and assuming that the observed light intensity was directly proportional t o the number of atoms along the line of sight. The density distribution of the atoms is shown in Fig. lO(a) and clearly matches the Gaussian dependence that one would expect for an ideal gas in a harmonic potential. The density increase was proportional to the number of atoms as atoms were added, but the diameter remained unchanged. Using the measured spring constant, we derived a temperature from the fit to the distribution. We found a temperature of 300 p K for the atoms, which agrees well with the TOF temperature measurement.
I
150.0
I
I
0.1
0.2
I
I
I
0.3
0.4
0.5
-
2.0
v1
125.0
%'
5 P
3
1.5
r
&
3 +
100.0
L
-
+
P W
D
r P
L
2
75.0
a,
:
U
50.0
0.5
0 u
ar L
0.0 -16.0
I
-12.0
I
-8.0
2 25.0 -
I
-4.0
0.0
Detuning (MHz) (b) Fig. 9. Measured temperature of the atoms versw the diameter (FWHM) of the cloud for detunings of - M A N (filled circles) and -2.5AN (open circles). (b) Measured temperature of the atoms versus detuning for a 1.5-mm-diameter cloud. The total laser intensity is 12 mW/cmz, and the magnetic field gradient is 15 G/cm.
LL
0.0
0.0
' (mm) (4
0.6
c n
c
*
ture increases with increasing numbers of trapped atoms. We speculate that this heating arises from radiation trapping. From Fig. 3 we note that since the blue-shifted light is preferentially absorbed, the light that escapes the cloud is red shifted on average from the incident trapping light. The excess energy may then be converted to kinetic energy of the atoms. We did not investigate in detail how this occurs. Figure 9(b) shows the plot of measured temperature versus detuning for clouds of a fixed diameter (1.5 mm). No significant temperature changes were observed when the magnetic field gradient was changed or when the intensities of the lasers were changed. We emphasize that, although the temperature rises with increasing numbers of atoms, the temperature rise would need to be orders of magnitude larger to explain the observed expansion of the cloud. C. Dependence of Cloud Shape on Number of Atoms Three well-defined spatial modes of the trapped atoms were observed. These depended on the number of atoms
1000
3
P
--
800
-
600
D L
0
(u
V
C
$
400
0 (u L
0
3
200
L L
nn _.I 4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
I (mm) ( b) Fig. 10. Top view of fluorescence profile of the cloud. (a) Data for the ideal-gas mode (circles) with a fit (solid curve) for a Gaussian distribution. (b) Data for the static mode (circles) with a fit (solid curve) assuming that we are looking along the minor axis of a ellipsoid of constant density. The fit gives a ratio of 1.5 for the ellipsoidal axes.
290 Sesko et al.
1.6
Vol. 8, No. 5/May 199115. Opt. Soc. Am. B
I
I
'
0.0 0.0
I
I
I
I
I
10.0
0.0
1.0
2.0
1
I
3.0
I
20.0 I
4.0
II 5.0
Number of atoms (X107) Fig. 11. Plot of the diameter (FWHM) of the cloud of atoms as a function of the number of atoms in the cloud (circles). For the main figure the magnetic field gradient is 9 G/cm, and for the inset it is 16.5 G/cm. The laser detuning is - M A N , and the total laser intensity is 12 mW/cmz. The solid curves show the predictions of the mode that are described in the text.
When the number of atoms was increased past 80,000, the cold atoms no longer behaved as an ideal gas and the presence of a strong long-range force was evident. This static mode was characterized by an elliptically symmetric cloud, the diameter of which increased with increasing numbers of atoms. One distinctive feature of this regime was that the density did not increase as we added more atoms to the trap. Instead of following the Gaussian distribution of the ideal-gas mode, the atoms were distributed fairly uniformly, as Fig. 10(b) illustrates. We obtained these data by looking through the cloud along the axis with the greater (ax) magnetic field gradient. The solid curve shows the calculated distribution of fluorescence for an ellipsoidal cloud of atoms with an aspect ratio of 1.5 and constant density. The cloud's minor axis was along the direction of the higher magnetic field gradient. By looking along a major axis, we found that the aspect ratio varied from approximately 1 for a 1-mm diameter (FWHM) cloud to 2 for a 4-mm cloud. We studied the growth of the cloud by measuring the number of atoms versus the major-axis diameter of the cloud. We performed this measurement by filling the trap with as many atoms as possible and then turning off the loading. With a charge-coupled device camera we recorded the evolution of the fluorescence as the atoms were lost from the trap and stored this record on video tape. We calibrated the video signal by also measuring the total fluorescence with a photodiode. To analyze these data, we digitized the frames of the video and stored them for analysis. We determined the total fluorescence in each image to obtain the number of atoms, and we calculated the FWHM diameter of the image. The results are shown in Fig. 11. Note that if the cloud were an ideal gas, the temperature would have t o be of the order of 50 mK to explain the cloud's expansion. This supports our earlier claim that the expansion is not due to a rise in temperature. We obtained the fit (solid curve) to the data (circles) in Fig. 11 by integrating Eq. (13), given N atoms, t o predict the FWHM diameter of the cloud. We assumed the temperature dependence shown in Fig. 9(a) for a detuning of
955
- M A N , but the results are insensitive to this parameter. The only free parameter in this calculation is the ratio (uF)/(uL). A value of (uF)/(uL) = 1.3 gives the solid curve shown in Fig. 11. This is greater than the value of (uF)/(uL) = 1.2 given by the theory, discussed in Section 2, for a two-level atom in a one-dimensional standing-wave field. However, it is not surprising that our cesium atoms with 20 relevant levels in a three-dimensional field differed from the two-level case by this amount. The inset of Fig. 11 shows that our theoretical model for the expansion of the cloud deviates from the data at diameters greater than 1.5 mm. In fact, these data show that the density reached a maximum and then started to decrease as we added more atoms. However, the assumption of a harmonic potential may break down in this regime. Also, the cloud stability at such large sizes was quite sensitive to the alignment and the spatial quality of the trapping beams. Other effects that are not considered in this model but that may be important for clouds of this size are magnetic field broadening, multiple levels of the atom, polarization of the scattered light, and multiple (>2) scattering of the light. Taking these effects into account makes the equation of equilibrium for large clouds quite complex. We were able to obtain as many as 4 x 10' trapped atoms in this static mode if the trapping laser beams were retroreflected ( e l mrad). However, if the return beams were slightly misaligned in the horizontal plane, dramatic and abrupt transitions to orbiting clumps or rings occurred at approximately 10' atoms. D. Rotating Clumps and Rings We now describe the observed spatial distributions and orbital motions for the third mode of the trapped atoms, which we call the orbital mode. This mode is characterized by atoms orbiting a central ball and abrupt transitions between different orbital modes and between static and orbital modes. We studied how the atoms' motion depends on a variety of parameters. Here we compare the data with the numerical model described above. This model explains much of the equilibrium behavior; however, it does not explain the clumping or any of the transition dynamics that are 0b~erved.l~ Photographs of the orbital modes are shown in Fig. 1 [Figs. l(b)-l(e) show the top view, and Fig. l(f) shows the side view]. We first observed a ring surrounding a dense central ball of atoms (henceforth known as the nucleus), as Fig. l(b) shows. By strobing the camera, we then found that the ring was actually a clump of atoms rotating counterclockwise around the nucleus, as Fig. l(c) shows. Note that there is a tenuous connection between the nucleus and the clump and that the nucleus has an asymmetry that rotates with the clump. The orbital radius of these clumps was insensitive to the trapping laser intensity, the detuning, and the magnetic field gradient. The rotational frequency did depend on these parameters, which we discuss below. We also observed clumps rotating at smaller radii, as Fig. l(d) shows. These smaller orbits occurred when fewer atoms were in the trap. The rotating clump had as many as three different radii with abrupt transitions between them as the number of atoms changed. The different orbital radii depended on the number of trapped atoms and varied in size by a factor of 3. Within each of the intermediate stages the radius of the orbit slowly increased as we added more atoms until
291 J. Opt. SOC.Am. B/Vol. 8, No. 5/May 1991
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10'
10"
z
Id 10'
lo3
0.0
30.0
60.0
90.0
120.0
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TIME (sec) Fig. 12. Fluorescence from a cloud of trapped atoms as a function of time after the loading is terminated. The main figure shows the decay of atoms from the static mode into the ideal-gas mode. The inset shows the decay from an orbitingmode through an abrupt transition (indicated by the arrow) to the static mode.
the cloud suddenly jumped to the next stage. The rotating clump in these intermediate stages was not so extended as it was for the larger radii shown in Fig. l(c), and there was a large number of atoms connecting the clump t o the nucleus. The size of the clump took up approximately one quarter of the area shown in the unstrobed image. Another orbital shape that we observed was a continuous ring surrounding a nucleus [Fig. l(e)]. We assumed that the atoms were still orbiting in this mode but that no clumping of the atoms occurred. Unlike in the rotating clump case, the radius of the continuous ring increased by a factor of 2 as we increased the detuning or decreased the magnetic field gradient. The rectangular shape of the ring was formed by the intersection geometry of the laser beams, with the corners of the ring corresponding to the centers of the beams. The atoms lay in a horizontal plane approximately 0.5 mm thick [Fig. l(f)]. Approximately 80% of the atoms were in the outer ring, with the remaining 20% contained in the nucleus. The formation of these orbital modes depended critically on the alignment of the trapping beams in the horizontal plane. The beams had a Gaussian width of 6 mm, and the return beams were misaligned horizontally by 4-8 mrad (1-2 mm) at the trap region. It was observed that the direction of rotation corresponded to the torque produced by the misalignment (Fig. 4). If the vertical trapping beam was misaligned, the plane of the rotation could be tilted by as much as 20", but no stable orbits were observed in the vertical direction. We believe that this was due t o an asymmetry in the trapping force caused by the vertical magnetic field gradient's being two times larger than that of the horizontal. The shape of the rings depended on the alignment and could be changed by tilting the retroreflection mirrors. The rings had stable orbits at radii from 2 to 3 mm. No rings were observed at radii < 2 mm. At radii > 3 mm the ring would form for only a few tenths of a second, then become unstable, and the atoms would fly out of the trap. The degree of misalignment in the horizontal plane and the number of atoms determined which of the orbital
modes occurred. When enough atoms were loaded into the trap, the cloud typically switched into the rotating clump for misalignments between 4 and 6 mrad and into the continuous ring for those between 6 and 8 mrad. If the light beams were misaligned even further, the atoms continuously loaded into the ring mode for any number of atoms, and for large misalignments we could even form a ring with no visible nucleus. In Ref. 3 we reported that a nucleus of atoms always exists, but we can now make rings without nuclei by a suitable adjustment of the various trap parameters. We also studied how the trap loss rate depended on the mode. This was done by observation of the time evolution of the cloud when no additional atoms were being loaded. Another study of interest was to see how long various cloud modes survived as the number of atoms decreased owing to atomic collisions. These data are shown in Fig. 12. The total fluorescence from the cloud, which we assumed to be proportional to the number of atoms, indicates that there were initially 4 x lo7atoms in the orbital mode (inset of Fig. 12) and that the fluorescence decreased exponentially with a time constant of T = 9 s. With 1.5 x l o 7 atoms remaining, the cloud abruptly jumped into the static mode, and the number of atoms continued to decrease exponentially with time (T = 10 s). This transition is indicated by the arrow in the figure. The main plot in Fig. 12 shows a decay of fluorescence from a different run starting with 4 x lo6 atoms in the static mode. In this run the initial time constant was approximately 17 s rather than the 10 s shown in the inset, owing to a lower vacuum pressure in the chamber. As the cloud started to change from the static mode to the ideal-gas mode after approximately 30 s, the loss rate increased. We believe that this was due to an increase in losses resulting from collisions between the trapped atoms, as Ref. 4 describes. At this point the density had reached its maximum value. Finally, after so many atoms were lost that the density was much lower (=70 s), the fluorescence decayed with a time constant of T = 60 s. This loss rate was due entirely to the collisions with the 2 x 10-l' Torr of background gas in the vacuum chamber. The rotational frequencies of the clumps were monitored either by strobing the image viewed by the camera or by observing the sinusoidal modulation of the fluorescence by a photodiode [Fig. 13(a)]. The peak-to-peak amplitude of this modulation was from 10% to 50% of the total fluorescence. Although it is obvious that the fluorescence from a limited portion of the orbit should vary, it is somewhat surprising that the total fluorescence should show such large modulations. The frequencies of rotation were between 80 and 130 Hz for the larger radii as we changed the trap parameters and were approximately 15% higher for the smaller orbits. As Fig. 14 shows, the orbital frequency depended on the detuning of the trapping laser (A), the magnetic field gradient (I?'), and the intensity of the trapping light ( I ) . Changing any of these parameters so as to increase the spring constant of the trap caused the rotation frequency t o increase. The circles in Fig. 14 show the data points, and the pluses show the results of our model when we use Eq. (22). The experimentally observed orbital radius varied by less than 10% over the range of detunings, magnetic field gradients, and intensities used. In the calculation of the frequency as
292 Vol. 8, No. 5/May 1991/J. Opt. SOC. Am. B
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-
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(b) Fig. 13. (a) Time dependence of total fluorescence for the rotating clump without modulation of the magnetic field. (b) Beat signal of total fluorescence produced after the transition is induced to the rotating clump mode. The orbital frequency was 112 Hz, and the magnetic field was modulated at 119 Hz. The peak-to-peakmodulation was approximately 40% of the total fluorescence for both (a) and (b).
a function of the magnetic field gradient and the laser intensity, the number of atoms composing the nucleus i n our model was adjusted to keep the orbital radius constant because we saw no change i n the experimental radius. The calculated radius had little sensitivity to the laser detuning, so that no adjustment was needed in the calculation of values of orbital frequency versus detuning. The calculated values show excellent agreement with all t h e experimental data, considering the simplicity of this model. One of the notable features that this model does not explain is the observed abrupt transitions between modes. These transitions occurred when the number of atoms exceeded a particular critical value, and they lasted roughly 20 ms. The transition to the largest rotating clump mode involved usually several intermediate transitions to rotating clumps with smaller orbital radii, as we discussed above. In the case of the continuous ring the atoms always jumped directly into this mode from the static mode, with no intermediate stages. We discovered that we could induce the transitions from the static mode to rotating clumps at fewer than the usual number of atoms (=los) by modulating the magnetic field gradient at a frequency close (within 8 Hz) to the rotational frequency. In this case the number of atoms needed
30.0
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Fig. 14. Dependence of the rotational frequency of the clump on (a) the laser detuning (B’ = 12.5 G/cm, I = 12 mW/cmz)),(b) the magnetic field gradient (A = -1.5A~,I = 12 mW/cmz),and (c) the total average intensity (A = - 2 . 0 A ~ B’ , = 15 G/cm). The circles are the experimental data, and the pluses give the results of our numerical model.
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J. Opt. SOC.Am. B/Vol. 8, No. 5/May 1991
for the transition (=lo7) depended on the amplitude and the detuning of the magnetic field modulation from the orbital resonance. Another amusing phenomenon was the following: after the cloud had made a transition, the fluorescence was modulated at the beat frequency between the two frequencies. This is shown in Fig. 13(b). Profound hysteresis effects were observed in the transition between the static and the orbital modes. If the number of atoms was slowly increased to lo', the cloud suddenly jumped from the static mode to the rotating clump, and the trap lost approximately half of its atoms in the succeeding 50-100 msec. This orbital mode remained stable at 5 x lo7 atoms until there was presumably a small fluctuation such as a change in the loading rate. Then the cloud switched back to the static mode without losing any atoms. It was necessary that the number of atoms build back up to 10' before it could again jump into the orbital mode. Thus the orbital modes can be self-sustaining at lower numbers of atoms than those required for a transition. This implies that when the atoms are in a different mode, the cloud obeys a different equation of equilibrium. We occasionally observed a different hysteresis in the transition between the static and continuous ring orbital modes. The cycle started in the static mode with 10' atoms. Subsequently, 80% of these atoms jumped into a continuous ring, while 20% remained in the nucleus. In the succeeding couple of seconds most of the atoms in the ring transferred back into the nucleus. When this occurred, the nucleus grew in size and the radius of the ring decreased. Finally, with approximately 70% of the initial atoms back in the nucleus and 30% in the ring, the cloud collapsed back into the static mode. This transition occurred just before the outer diameter of the nucleus and the inner diameter of the ring merged. During this process the total number of atoms in the trap remained relatively constant. The cloud repeatedly switched between the two modes with an oscillation period of a few seconds. In addition t o the abrupt collective transitions there are a variety of other features of the orbital mode that cannot be described by the simple model. These include the clumping of the orbiting atoms, the number of atoms that appear in the core nucleus versus that in the ring, and the discrete radii of the clumps. The formation of clumps with discrete radii and the interesting transition dynamics may indicate the presence of some nonlinear interaction. A number of effects that we neglected could lead to such nonlinearities. For example, the radiation trapping and attenuation forces can change locally because of the local variation in the density or the polarization of the atoms. It is possible that this dependence of the equation of equilibrium on density could cause a rapid buildup of the number of orbiting atoms and also the forces keeping them there. The possible nonlinear processes inside a dense cloud of low-temperature atoms are not well understood and certainly deserve study.
Sesko et al.
5. CONCLUSION We observed that the motions of atoms in a spontaneous force optical trap have a n unexpectedly rich behavior. They act collectively at surprisingly low densities and make abrupt transitions. This behavior arises from the coupling between the atoms owing to the multiple scattering of photons and is enhanced by the change in the frequency properties of the scattered light. This force is a key element in the equation of equilibrium for optically trapped atoms and limits the obtainable density. A simple model was presented that explains the growth of the cloud and the formation of the circular rings. However, it does not give an explanation of the formation of the clumps or the dynamics of the transitions between different distributions.
ACKNOWLEDGMENTS This work was supported by the U.S. Office of Naval Research and the National Science Foundation. We are pleased to acknowledge the valuable contributions made by C. Monroe and W Swann to this work. The authors are also with the Department of Physics, University of Colorado, Boulder, Colorado 80309-0440.
REFERENCES AND NOTES 1 D. Pritchard, E. Raab, V. Bagnato, C. Wieman, and R. Watts, Phys. Rev. Lett. 57, 310 (1986). 2 . E. L. Raab, M. G. Prentiss, A. E. Cable, S. Chu, and D. E. Pritchard, Phys. Rev. Lett. 59, 2631 (1987). 3. T. Walker, D. Sesko, and C. Wieman, Phys. Rev. Lett. 64,408 (1990). 4. A. Gallagher and D. Pritchard, Phys. Rev. Lett. 63, 957 (1989). 5. D. Sesko, T. Walker, C. Monroe, A. Gallagher, and C. Wieman, Phys. Rev. Lett. 63, 961 (1989); M. Prentiss, A. Cable, J. Bjorkholm, S. Chu, E. Roab, and D. Pritchard, Opt. Lett. 13,452 (1988). 6. J. Dalibard, Opt. Commun. 68, 203 (1988). 7. A. P. Kazantsev, G. I. Surdutovich, D. 0. Chudesnikov, and V. P. Yakovlev, J. Opt. SOC.Am. B 6, 2130 (1989). 8. L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon, Oxford, 1989), p. 6. 9. B. R. Mollow, Phys. Rev. 188, 1969 (1969); Phvs. Rev. A 5, 2217 (1972); D. A. Holm, M. Sargent 111, and S . Stenholm, J. Opt. SOC.Am. B 2, 1456 (1985). 10. S. Gilbert, Ph.D. dissertation (University of Michigan, Ann Arbor, Mich., 1984). 11. B. Dahamani, L. Hollberg, and R. Drullenger, Opt. Lett. 12, 876 (1988). 12. P. D. Lett, R. N. Watts, C. I. Westbrook, W D. Phillips, P. L. Gould, and H. J. Metcalf, Phys. Rev. Lett. 61, 169 (1988); P. D. Lett, W D. Phillips, S. L. Rolston, C. E. Tanner, R. N. Watts, and C. I. Westbrook, J. Opt. SOC.Am. B 6, 2084 (1989). 13. R. Sinclair (Division of Physics, National Science Foundation, Washington, D.C. 20550, personal communication) h a s pointed out that the clumping is similar in character to the negative-mass instability that is observed in electron plasmas. We are now investigating this analogy.
VOLLMF.
6 5 , N U M B € [ < 13
PHYSICAL REVIEW LETTERS
24 S E P T E M B E R 1990
Very Cold Trapped Atoms in a Vapor Cell C. Monroe, W. Swann, H. Robinson,'J' and C. \Vieman Joinr In.ctrrure Jor Laborator) A.rrroph>,sirs, 1'nii.er.Yiiy of' Colorado and ;I'aiional lnsrirure oJ.5randard.r and Terhnologj,, and Deparrnienr of Phj,.tics. L ' n i w r s i r j , of- Colorado. Bortldcr. Colorado 80309-0440 (Reccived 3 I M a y 1990)
We have produced a very cold sample of spin-polarired trapprd atoms. The technique used dramatically simplifies the production of laser-cooled atoms. I n this experiment. I .8 x lo7 neutral cesium atoms were optically captured directly from a low-pressure vapor i n ii small glass cell. We then cooled the < I - m m ' cloud o f trapped atoms and loaded i t into a lowfield magnetic t r a p in the same cell. The magnetically trapped atoms had an effective temperature as low as 1 . 1 I 0 . 2 p K , which is the lowest kinetic temperature ever observed a n d f a r colder t h a n any previous sample of trapped atoms. PACS numbers: 32.8O.Pj. 4 2 . 5 U . V k
I n recent years there has been dramatic progress in the use of laser light to cool' and trap' neutral atoms. We and other^^,^ have previously reported using optical traps to produce cold atomic samples with high densities and optical thicknesses beyond that attainable in most beams. Such samples would be ideal for a variety of experiments which are currently done with atomic beams. However, in previous optical traps, a substantial vacuum apparatus was required to slow an atomic beam prior to trapping. The ability to trap atoms directly and efficiently from a room-temperature vapor means that for many experiments, this apparatus can be replaced with a small cell. Such an optically trapped sample is useful for many applications, but it has some inherent limitations; the atomic spins are randomly oriented, perturbing light fields must be present, and it is difficult to achieve temperatures lower than 300 p K . W e have overcome these limitations by loading the optically trapped atoms into a magnetostatic trap. Because the atoms are very cold when first loaded, we can trap them with relatively small magnetic fields. By properly cooling the atoms before turning on the magnetic trap we have produced a sample which is more than 100 times colder than any previously trapped neutral atoms. In our experiment the atoms are initially captured using the Zeeman-shift spontaneous-force optical trap and used by us in a (ZOT) reported first by R a a b er variety of recent This trap uses the light pressure from six orthogonal intersecting laser beams. A weak magnetic-field gradient acts to regulate the light pressure in conjunction with the laser frequency to produce a damped harmonic potential. In previous work we found that the trap could capture cesium atoms from a cooled beam with speeds up to = 12 m/s and substantial numbers of atoms from an uncooled beam. Similar observations were made by Cable, Prentiss, and Bigelow for s o d i u m 6 This suggested the possibility of capturing atoms directly from a room-temperature vapor. T h e number of atoms contained in a trap which is in a
dilute vapor is determined by the balance between the capture rate into the trap and the loss rate from the trap. Given the maximum capture speed of the trap, [-'., the number of atoms per second entering the trap volume with low enough velocities to be captured is easily calculated.' This rate is R = ~ . S ~ V " ' I ~ , P ( ~ / ~where ~ T ) "n ' , is the density of cesium atoms, m is the mass, T is the temperature, and V is the trapping volume (about 0.1 cm' for our trap). For cesium at a temperature of 300 K, approximately one atom in lo4 is slow enough to be captured. T h e loss rate from the trap, I/r (r=lifetime), is primarily due to collisions with atoms in the vapor. I f we assume the density of noncesium atoms is negligible, then l / r = n a ( 3 k T / m ) " ' , where CT is the cross section for an atom in the vapor to eject a trapped atom. T h e number of atoms in the trap, N, is then given by the solution to the simple rate equation d N l d r = R - N / r . Assuming N(r=O) =0, we obtain N ( t ) =N,(I - e -''r), where the steady-state number N, is given by N , =R r = I
'
( V '/'/a)~,P(rn/2 k T ) .
(I
1
Note that N , is independent of pressure. However, the lifetime r , which is also the time constant for filling the trap, does depend on pressure. Because N , is very sensitive to t'c, we should note which parameters of the ZOT determine r 3 X lo' atoms) have lower densities than small clouds. This is consistent with the data of Sesko et al. [19], and inconsistent with Eq. (7). We believe that the source of this discrepancy is the fact that Eq. (7) was derived with the assumption that the photons are scattered no more than twice. However, with more than 3 X lo7 atoms and the density predicted by Eq. (7), the optical thickness would be large enough for a significant number of photons to scatter more than twice in the cloud of trapped atoms before escaping. This would increase the radiation-trapping force, causing the cloud to expand and the density t o drop below the value given by Eq. (7). If the cloud expands too far, however, the optical thickness will begin to decrease, thereby reducing the radiation-trapping force. This leads to a balance of forces that maintains the density of a large cloud at an optical thickness where, on the average, each photon absorbed from the laser beam will, after reemission, scatter no more than once on its way out of the cloud. This means that when the number of atoms is greater than 3 X lo7 the diameter of the cloud will go as the square root of the number of atoms in the cloud rather than as the cube root as Eq. (7)predicts. In Fig. 10 we show that this dependence is very close t o what was reported in Ref. [19]. Our current data are similar but covet a smaller range. VII. CONCLUSION
We have reported results of a study of optical trapping in a vapor-cell Zeeman optical trap. We have developed a model that agrees with our observations of how the number of atoms in the trap varies with the intensity of the trapping beams, the size of the trapping beams, the detuning of the laser, and the strength of the magnetic field. Our model includes the one-dimensional slowing force on an atom moving through the trap, and the three-dimensional effects of the magnetic field. We are
31 9 4090
K. LINDQUIST, M. STEPHENS, AND C. WIEMAN
46
rameters. W e observed small increases in density with magnetic field, laser intensity, a n d detuning.
confident that we can accurately predict numbers of atoms for any reasonable t r a p parameters. In addition, we have shown that for many conditions it is possible t o reliably calculate dependences using a simple onedimensional, two-level model that neglects the magnetic field. This allows one t o estimate the capture rates a n d numbers of trapped atoms quite easily. Attempts to increase the number of trapped atoms by frequency chirping or by bandwidth broadening t h e laser were not successful, a n d we can now explain why these techniques d o not work. Finally, we have described the results of measurements of the t r a p density a s a function of the t r a p pa-
This work was supported by the office of Naval Research and the National Science Foundation. We would like t o thank E. Cornell and C. Monroe for useful discussions. W e are also grateful to K. E. Gibble, S . Kasapi, a n d S. C h u for providing us with their data prior t o publication, and for permission t o reproduce it.
[ I ] J . Opt. SOC.Am. B 6 122) (1989) Special Issue on laser cooling and trapping. [2] D. Grison, B. Lounis, C. Salomon, J. Y . Courtois, and G. Grynberg, Europhys. Lett. 15, 149 (1991). [3] J . W. R. Tabosa, G. Chen, Z . Hu, R . B. Lee, and H. J . Kimble, Phys. Rev. Lett. 66, 3245 (1991). [4] A. Gallagher and D. Pritchard, Phys. Rev. Lett. 63, 957 (1989). [5] D. Sesko, T. Walker, C. Monroe, A. Gallagher, and C. Wieman, Phys. Rev. Lett. 63, 961 (1989). [6] P. D. Lett, P. S. Jessen, W. D. Phillips, S. L. Rolston, C. I. Westbrook, and P. L. Gould, Phys. Rev. Lett. 67, 2139 (1991). [7] P. S. Julienne and J. Vigue, Phys. Rev. A 44,4464 (1991). [8] C. Monroe, H. Robinson, and C. Wieman, Opt. Lett. 16, 50 (1991). [9] S. L. Rolston and W. D. Phillips, Proc. IEEE 79, 943 (1991). [lo] S. Freedman and K. Coulter (private communication). [ l l ] C. Wieman, C. Monroe, and E. Cornell, in TenthInternational Conference on Laser Spectroscopy, edited by M. Ducloy, E. Giacobino, and G. Camy (World Scientific, Singapore, 19921, p. 77. [I21 E. Cornell, C. Monroe, and C. Wieman, Phys. Rev. Lett. 67,2439 (1991).
[I31 This idea has been suggested by many people. The first, to our knowledge, was D. Pritchard in about 1985. Also see Ref. [15]. [I41 M. Zhu, C. W. Oates, and J. L. Hall, Phys. Rev. Lett. 67, 46 (1991). [ 1 5 ] K . E. Gibble, S. Kasapi, and S. Chu, Opt. Lett. 17, 526 ( 1 992). [I61 C. Monroe, W. Swann, H. Robinson, and C. Wieman, Phys. Rev. Lett. 65, 1571 (1990). [17] E. L. Raab, M. Prentiss, A. Cable, S. Chu, and D. Pritchard, Phys. Rev. Lett. 59,2631 11987). [18] C. Wieman and L. Hollberg, Rev. Sci. Instrum. 62, 1 (1991). [I91 D. W. Sesko, T. G. Walker, and C. E. Wieman, J . Opt. SOC.Am. B 8,946 11991); T. G. Walker, D. W. Sesko, and C. E. Wieman, Phys. Rev. Lett. 64,408 (1990). [20] E. Cornell, C. Monroe, and C. Wieman (unpublished). [21] H . Metcalf, J. Opt. SOC.Am. B 6,2206 (1989). (221 R. N. Watts and C. E. Wieman, Opt. Lett. 11, 291 (1986). [23] W. Ertmer, R. Blatt, J. L. Hall, and M. Zhu, Phys. Rev. Lett. 54, 996 (1985). [24] A. S. Parkins and P. Zoller, Phys. Rev. A 45, R6161 (1992). [25] A. S. Parkins and P. Zoller, Phys. Rev. A 45, 6522 (1992). [26] M. Zhu and J. L. Hall (private communication).
ACKNOWLEDGMENTS
THE LOW (TEMPERATURE) ROAD TOWARD BOSE-EINSTEIN CONDENSATION 1N OPTICALLY AND MAGNETICALLY TRAPPED CESIUM ATOMS
Chris Monroe, Eric Cornell and Carl Wieman Joint Institute for Laboratory Astrophysics University of Colorado and National Institute of Standards and Technology and Department of Physics, University of Colorado Boulder, CO 80309-0440
1. INTRODUCTION There is considerable reason to pursue the extreme limits of low temperature and high density that can be achieved in an atomic vapor. First there is the intrinsic interest in the trapping and cooling of atoms, and the behavior of atoms in new regimes of temperature. Much
of this is discussed extensively elsewhere in this volume. Second, many interesting applications of laser cooled and trapped atoms require very cold high density samples. For example, cooled and trapped atoms are ideal for spectroscopy of weak transitions, such as those involved in
measurement of parity nonconservation in atoms. Clearly, colder and denser samples will enhance any such experiment with trapped atoms. Finally, the ultimate limit of high density and low temperature is of great interest in its own right. This is Bose Einstein Condensation (BEC)
of a trapped vapor sample of bosonic atoms. The ability, in principle, to cross the phase boundary for BEC at several widely spaced points in the temperature-density plane offers the tantalizing prospect of studying the phase transition while varying the relative significance of
interparticle interactions. Although these interactions may well prevent BEC, it will be very interesting to see exactly what will happen when one reaches the necessary condition of the deBroglie wavelength of the atoms exceeding the interparticle spacing. Over the past several years we have been working to obtain higher density and lower 320
32 1
temperature samples of atomic cesium in pursuit of this limit. In the course of this work we have found many interesting new phenomena which have acted as barriers, and we have developed new technologies which have provided ways around these barriers or have generally aided our efforts to cool and trap atoms. In this paper we describe this work and discuss the radiation catalyzed collisions which lead to losses from optical traps, the "radiation trapping repulsion" which limits density in an optical trap, the vapor cell optical trap which allows one to obtain trapped atoms very easily, the loading of optically trapped and cooled samples into magnetic traps, and finally the ac magnetic trap. 11. THE ZEEMAN-SHIFT OPTICAL TRAP
The starting point for all of this work is the Zeeman shift spontaneous force optical trap (ZOT) first demonstrated by Raab et al.'
In our initial studies of collisions and collective
behavior of atoms in an optical trap, a beam of cesium atoms is effused from an oven and passed down a 1 m tube into a UHV chamber (lo-'' TOK). As the atoms speed down the tube they are slowed by a counterpropagating laser beam. In the UHV chamber, the atoms are trapped in a
ZOT formed by having three perpendicular laser beams which are reflected back on themselves. The beams are all circularly polarized, with the polarizations of the reflected and incident beams being opposite. They have Gaussian profiles with diameters of about 0.5 cm, and are tuned to
the 6SF=4to 6P,,*
F=5
transition of cesium. Magnetic field coils, 3 cm in diameter and arranged
in an anti-Helmholtz configuration, provide a field that is zero at the middle of the intersection
of the six beams, and has a longitudinal gradient of about 10 G/cm. In addition to the main trap laser beams, there are beams to insure the depletion of the F=3 hyperfine ground state in the trap. Thus, all the trapped atoms are in the F = 4 ground state. All the laser light in these experiments is provided by diode lasers which are frequency stabilized using optical feedback. The fluorescence from the trapped atoms is observed with both a photodiode, which monitors the total fluorescence, and a charged-coupled-device television camera which shows the size and shape of the trapped atomic cloud. We have discovered that the size and shape (but not the density) of the trapped atom cloud is strongly dependent on the number of atoms.* When the number is less than about 10s atoms, the cloud is a constant diameter ellipsoid with the Gaussian distribution one expects for an ideal
322
gas in a three dimensional harmonic well. Only in this regime can we accurately measure the density -- a necessary condition for collision studies. Therefore, all our collision studies have been carried out with less than 5 X 1 0 4 trapped atoms.
We have studied the low temperature collisions between the trapped atoms by observing the loss rate of the atoms from the optical trap.3 Assuming a loss rate of the form dn/dt = -an
- @n2, where n is the density of atoms in the trap, we find the deviation from a pure exponential decay indicates a trap loss mechanism due to collisions between trapped atoms. Similar observations of collisional trap loss have been obtained previously by Prentiss et aL4 The quantity of interest, 8, is the rate constant for the low temperature collision between trapped atoms; a is the coefficient that describes the traditional collisions with the room temperature background gas in the vacuum chamber. We studied how /3 depends on the intensity of the trapping laser beams.3 Changing the intensity varies the fraction of the atoms that arc in the excited state and also varies the trap depth. In Fig. 1, we show the dependence of /3 on the total intensity of all trap laser beams for
a trap laser detuning one linewidth to the red (5 MHz). As the intensity is decreased from 4 mW/cm2, 13 increases dramatically. We interpret this as due to hyperfine-changing collisions between ground-state atoms. Two atoms acquire velocities of 5 m/s if they collide and one changes from the 6s F-4 hyperfine state to the lower
F=3 state. For high trap laser intensities, atoms with this velocity cannot escape, but at lower intensities the trap is too weak to hold them. We also find that if the beams are misaligned slightly, a similar large-/3 loss rate is observed for all intensities. We estimate that hyperfinechanging collisions between ground-state atoms due to the van der Waals interaction will give a @ in the range lo-'' to lo-'' cm3/s, which is consistent with the observed low intensity value.
Above 4 mW/cm2, 8 appears to increase linearly with intensity. We believe that this is due to the low temperature energy transfer collisions of the type discussed by Gallagher and Pritchard' and by Julienne in this volume. In this mechanism, two atoms, one excited, are accelerated toward each other via the -1/? ground-excited potential. However, the excited atom spontaneously decays during the collision and leaves the trap due to its large accrued kinetic energy. At sufficient intensities or detunings, this loss rate can dominate the loss rate due to background collisions, thereby giving an effective ceiling to the density of trapped atoms.
323
We have studied in some detail the variations in the size and shape of the cloud of trapped atoms with the number of atoms it
contain^.^ We find the atoms behave in a highly collective
manner, and this behavior arises because of a previously overlooked long-range interaction between the atoms. This interaction is negligible at room temperature or for any temperature at which an atomic vapor has previously been studied. However, because of the low temperatures and high densities in a ZOT, it becomes the dominant force in determining the behavior of the
atom cloud, and leads to dramatic "phase-like" transitions. Three separate well-defined distributions or "modes" of the trapped atoms were observed, depending on the number of atoms in the trap and on how we aligned the trapping beams. The "ideal-gas" mode discussed earlier occurred when the number of trapped atoms was less than
105. The atoms formed a small sphere with a constant diameter of approximately 0.2 mm. In this regime, the density distribution of the sphere was a Gaussian, as expected for a damped harmonic potential, and increased linearly in proportion to the number of atoms. The atoms in this case behaved just as one would expect for an ideal gas with the measured temperature and trap potential. When the number of atoms was increased past = 105, the behavior deviated dramatically from that of an ideal gas, and long-range repulsions between the atoms became apparent. In the "static" mode the diameter of the cloud smoothly increased with increasing numbers of atoms. Instead of a Gaussian distribution, the atoms in the trap were distributed fairly uniformly. To study the growth of the cloud, we measured the number of atoms versus the diameter of the cloud using a calibrated CCD camera. The results are shown in Fig. 2. To fully understand the growth mechanism, the temperature as a function of diameter was also measured. For a detuning of 7.5 MHz, the temperature increased from the asymptotic value of 0.3 up to 1.0 mK as the cloud grew to a 3 mm diameter. If the cloud were an ideal gas, the temperature increase would need to be orders of magnitude higher to explain this expansion. One distinctive feature of this
regime is that the density did not increase as we added more atoms to the trap. It remained nearly constant with some decrease at the largest numbers of atoms.
When all the trapping beams were reflected exactly back on themselves, we obtained a maximum of 3.0 x lo8atoms in this "static" distribution. In contrast, if the beams were slightly misaligned in the horizontal plane, we observed unexpected and dramatic changes in the
324
distribution of the atoms. The atoms would collectively and abruptly jump into a third mode when the cloud contained approximately lo8 atoms. We call this the "orbital" mode. The number of atoms needed for a transition depended on the degree of misalignment.
The expansion of the cloud with increasing numbers of atoms and the various collective rotational behaviors clearly show that some long-range interatomic force dominates the behavior of the atomic cloud. Since normal l/r6 or 1/? interatomic forces are negligible for the atom densities in the trap (lO1o-lO" ~ m - ~other ) , forces must be at work. In particular, since the optical depth for the trap lasers is on the order of 0.1 for our atom clouds (3 or more for the probe absorption at the resonance peak), forces resulting from attenuation of the lasers or radiation trapping can be important. The attenuation force is due to the intensity gradients produced by absorption of the trapping lasers, and has been discussed by Dalibard6 in the context of optical molasses. This force tends to compress the atomic cloud and so cannot explain
our observations. We have demonstrated that radiation trapping produces a repulsive force between atoms which is larger than this attenuation force and thus causes the atomic cloud to expand.* It is this force that limits the density of our optical trap to lO''/cm3. 111. THE VAPOR CELL TRAP
During the course of this work, we discovered that the depth of the ZOT is large enough that a significant number of atoms could be captured directly from an uncooled beam. Similar results have also been obtained with sodium.'
We have extended this idea to build such an
optical trap directly in a small vapor ce1L8 We have been able to use this trap to produce clouds
of trapped atoms containing = 2 lo8 ~ atoms with temperatures of a few hundred microkelvins. This now makes it possible to have cooled and trapped atoms with an extremely simple and inexpensive apparatus. This is a significant technological breakthrough because these atoms can then be used for a variety of other experiments. The trap is made with beams from two diode lasers as described in Sec. 11. The only difference is that now the region of intersection of the beams is inside the center of a small glass vapor cell containing approximately lo'* TOKof cesium. The geometric details of the cell and the optical quality of the windows is unimportant. In fact, one can send the beams directly through the curved walls of a glass tube and still obtain very successful atom trapping. Small anti-Helmholtz coils outside the cell provide the necessary
325
magnetic field gradient. The number of atoms contained in such a trap in a cell is determined by the balance between the capture rate into the trap and the loss rate from the trap. Given the maximum capture speed of the trap, v,, the number of atoms per second from the vapor entering the trap volume with low enough velocities to be captured is easily calculated. This rate, R = 1 . 5 ~ */2nV2’3~,4(m/2kT)3/2, where n is the density of cesium atoms, m is the mass, T is the temperature, and V is the trapping volume (about 1 cm3 for our trap). For cesium at a temperature of 300 K, approximately one atom in 7000 is slow enough to be captured. The loss rate from the trap, 1/7 lifetime), is primarily due to collisions with the fast atoms in the vapor. If we assume the density of noncesium atoms is negligible, then 1/7 = na(8T/rm)”* atoms/s where
(T
is the cross section for an atom in the vapor to eject a trapped atom. The
number of atoms in the trap, N, is then given by the solution to the simple rate equation dN/dt = R - NIT. Assuming N(t=O) = 0, we obtain N(t) = N,(l-e-t’q, where the steady-state number, N,, is given by
(-)
N, = 3 vm v:($ 4
.
0
Note that N, is independent of the density of atoms in the cell (i.e. pressure). However, the lifetime 7 , which is also the time constant for filling the trap, does depend on pressure. Because N, is very sensitive to v,, we should note which parameters of the ZOT determine v,. The trap is an overdamped system, so the primary stopping force comes from the damping
force, which arises from the imbalance in the radiation pressure due to the differential Doppler shift between the two counterpropagatingbeams. The force decreases rapidly with velocity once the Doppler shift gets so large that the atom is out of resonance with both beams. Since the frequency of lhc laser is typically one linewidth to the red of the atomic resonance, v, = where
is the velocity at which the Doppler shift equals the natural linewidth
ZrX,
r (5 MHz for
our transition). For cesium, 2rX is 10 m/s. We have constructed several cells which have produced similar results. Cells have been made out of fused silica and of standard stainless steel UHV equipment. It is necessary to have a small (2 Pls) ion pump on the system to keep the non-cesium pressure in the cell less than
326
TOK. After construction the cells are evacuated and baked under vacuum. A fraction of a gram
of cesium is then distilled into the cell and it is sealed off, To maintain the desired cesium pressure of 10-7-10-9 Torr in the cell, a small "cold finger" can be used. This cold finger is maintained at about -25°C to control the vapor pressure of the cesium in the entire cell. Alternatively, the cesium well can be valved off to limit the vapor pressure of cesium in the cell. When the trapping laser was tuned between 1 and 15 MHz to the red of the 6S,,,
to
6P3,2,F=5transition (-8 MHz was optimal), a bright cloud of trapped atoms with a volume < 4 mm3 appeared in the center of the cell. By measuring the fluorescence we determined that the
cloud contained as many as 2 x lo8 atoms, in agreement with the value predicted by Eq. (1) for v, = 15 m/s. As we observed in the work just discussed, the density of atoms in the cloud was
limited by radiation trapping to about 10" atoms/cm3, and the temperature was about 300 pK. The trapped atom cloud was more than lo00 times brighter than the background fluorescence in the cell at a cesium pressure of 6
X
TOK. The growth in the number of atoms with time
agreed well with the predicted form. The lifetime of atoms in the trap was 1 s at a pressure of
6x
TOK. The steady-state number of atoms in the trap changed by less than 30% as the
cesium pressure was varied between
TOK (7 = 0.06 s) and - 5 ~ 1 0 - "Torr (7=10 s).
Similar traps in small vapor cells have been demonstrated in sodium9 as well as rubidium." Results were comparable to those we obtained with cesium. Although the temperature of a large optically trapped sample is typically 300 pK, the sample can be cooled much further by rapidly turning off the ZOT magnetic fields and detuning
the laser further to the red by several linewidths. Three dimensional polarization-gradient cooling forces reduce the temperature to near the recoil temperature (2 pK) in under 1 ms." The temperature is measured by loading the atoms into a magnetic trap, as discussed below. This "sub-Doppler molasses" cooling mechanism is described extensively in this volume. The drawback to optical molasses however, is that the atoms are no longer trapped. Sub-Doppler temperatures in a ZOT have been attained,'* but at the expense of very low densities.
To summarize the performance of optical trapping and cooling, we have produced a sample of 2 X lo8 cesium atoms at a density of 5 X 10'0/cm3 with a temperature of 2 pK. The sample is produced in a small vapor cell, ideal for performing further experiments on the atoms.13
327
IV. THE dc MAGNETIC TRAP
Although optical trapping and cooling represents a very efficient method of collecting atoms and cooling them, optically trapped samples are not polarized and the density and temperature are limited by interactions with the light as discussed above. Furthermore, it is not possible to maintain high densities while obtaining the very low temperatures possible in optical molasses
because there is no confining force in the molasses. Magnetic trapping is a way to avoid these limitations because it provides confinement without heating. Furthermore it allows cooling far below the limits obtainable in optical molasses. The photon recoil energy, which is the primary
limit for optical cooling, is no obstacle to evaporative cooling in a magnetic trap. Spin-polarized hydrogen in a magnetic trap has been evaporatively cooled (allowing the highest energy atoms
a temperature 13 times below the photon recoil limit,14 and to a density orders of magnitude larger than the present limit for an optical trap. to escape the trap) to
To this end, we have used the optical trap to load a magnetostatic trap in the same cell.
To do this, we first turn on the optical trap, then we turn off all the magnetic fields and detune the laser further red for a few milliseconds. At the end of this time, we have a dense, although untrapped sample, which has a temperature of a few microkelvins. We next use the light to pump the atoms to the m F = 4 state and then quickly (500 ps) turn on the magnetic fields to create the magnetic trap. In the magnetostatic trap or bowl, the atoms are held by magnetic
w.3)and gravitational forces at the same location as when they are in the optical trap.
In this
way we have produced a very cold, 100%polarized trapped sample which is not subjected to the perturbing fields of the laser.
Because of the low temperatures, the magnetic fields needed for confinement are quite modest (tens of G k m ) . To trap the I F,m,
>
I
= 4,4 > weak-field-seeking state, four vertical
bars, each carrying a current with direction opposing that of its nearest neighbors, are placed around the cell. In the horizontal (x,y) plane, they provide a quadrupole field distribution having zero field in the center along the vertical (z) axis. In addition, a horizontal coil below the atoms gives a vertical bias field as well as a vertical field gradient."
This vertical field gradient
(aB,/az) supplies the levitating force to support the atom against gravity. Thus, as the atom searches for a local minimum of
13 I , only small changes occur
in the direction of
preventing nonadiabatic spin-flip transitions to other mF states ("Majorana" transitions).
3,
328
To study the magnetically trapped atoms, the magnetic fields were quickly (500 ps) switched off after a variable time delay, and the laser beams quickly turned back on. A television camera recorded the image of the cloud as the laser came back on. By varying the time delay we could observe the shape of the atom cloud and the number of atoms as a function of time in the magnetic trap. Essentially all the optically trapped atoms could be loaded into the magnetic trap, and the density could be as high as in the optical trap. In general, the cloud was not in equilibrium when first loaded into the magnetic bowl. When it is loaded into the bowl off center, secular oscillations or orbits of the cloud are observed with typical periods of 2 - 5 Hz. Even when it is centered, radial oscillations are observed as the atoms slosh back and forth in the bowl. We are presently studying the thermal equilibration of the atoms in the magnetic bowl. This equilibration time depends on the elastic scattering cross section of the atoms at a temperature of a few microkelvins. Obviously, this is a completely unexplored regime and any
data on it will be of interest. Although the atoms are not in thermal equilibrium, we can measure an effective temperature that corresponds to the time averaged kinetic energy. This is done by loading the atoms in a magnetic trap with a small spring constant and observing how far they expand. Taking r to be the radius of the half height point of the spatial distribution at the time of largest spatial extent, the effective temperature is then 0.25 @/K,
where k is the spring constant and
K
is Boltzmann's constant. This technique is valid when the spring constant is small enough that r is several times larger than the initial radius of the cloud. We find that the temperature of the cloud is different along the different axes (which have different spring constants) and depends on the length of time the optical pumping light is on. Neither of these results is surprising since the optical pumping light heats the sample in an anisotropic manner. With no optical pumping light we obtain T, = 1.4 & 0.25 pK and Tx,y= 1.1
0.25 pK. These are the coldest kinetic
temperatures ever observed, and are more than two orders of magnitude colder than any previous sample of trapped atoms. With no optical pumping we have about 1/4 as many atoms in the magnetic trap as we had initially in the optical trap. With enough optical pumping to have 50% of the atoms in the magnetic trap, T, rises to about 2.9 pK and Tx,yis about 1.4 pK. The differences in the temperature in the different directions, and the dependence on the optical pumping is presently being studied.
329
V. THE ac MAGNETIC TRAP Traditional (dc) magnetic traps have a significant limitation themselves, namely spin-flip collisions. To date, all magnetic traps have been designed to work only for weak-field seekers, that is, for atoms or neutrons whose magnetic moment is anti-aligned with the magnetic field. These atoms are confined to a local minimum in the magnitude of a static magnetic field. Unfortunately, because weak-field-seeking states are not the lowest energy configuration within the confining magnetic field, the atoms are susceptible to two-body collisions that change the
hyperfine or Zeeman level of one or both of the colliding atoms. The energy released in these collisions heats the trapped sample and the spin-flipped atoms leave the trap. The loss rate due
to these exothermic collisions scales as the density squared and thus provides an effective ceiling
to the attainable density, and indirectly limits temperature as well? However, if atoms were trapped in the lowest energy spin state (magnetic moments aligned with the magnetic field) spinflip collisions would be endothermic and hence greatly suppressed at low temperatures. Unfortunately, such "strong-field-seeking" atoms can not be confined in a static configuration of magnetic fields, because Maxwell's equations do not allow a local maximum in magnetic field magnitude.
However there are time-varying field configurations which provide stable
confinement. Originally proposed for hydrogen atoms,I7 our trap operates on the same dynamical principle as the Paul trap" for ions: Near the center of the trap, the potential is an axially symmetric quadrupole, oscillating at frequency 0: 4 = -p.B = A(z2-p2/2)cos nt. During each oscillation, the atoms are first confined axially and expelled radially, and then confined radially and expelled axially. For a range of values of the amplitude A, the net force averaged over a
cycle of Lhc: oscillation is inward; the result is stable confinement in all three dimensions. For easily obtainable oscillating magnetic fields, the trap is extremely shallow (tens of microkelvins). However, using laser cooling we are able to produce atomic samples which are colder than this, and load them into the trap."
The loading procedure has a number of steps, as shown in Table I. First, cesium atoms are collected and cooled in an optical trap exactly as done for loading the dc magnetic trap. The cold atoms are then launched into a differentially pumped vacuum region which contains the ac magnetic trap, as shown in Fig. 3. To reduce the spreading of the atoms they are magnetically
330
focused as they move between the two traps. When they reach the magnetic trap the atoms are optically pumped into the state that is trapped. This approach allows multiple bunches from the optical trap to be transferred to the magnetic trap, thereby directly increasing the density.
To toss the atoms up into the magnetic trap region, we use the technique of moving optical molasses2': we shift the frequency of the downward-angled molasses beams (Fig. 1) 1.7 MHz
to the red, and the upward-angled beams 1.7 MHz to the blue, relative to the frequency of the horizontal beams. The standing-wave pattern at the intersection of the six laser beams now becomes a walking-wave pattern. To achieve the necessary acceleration (200 g), while having the lowest possible final velocity spread, we vary the central laser frequency over time during the launch (Table I). After the atoms are accelerated, the molasses beams are shut off abruptly and a 2 ms pulse of a+ polarized light optically pumps the atoms to the F=4, mf=4 state. The optical pumping beam is tuned to the F=4 + F'=4 transition to minimize heating. The tossed atoms begin their ascent with an internal velocity spread of about 3 cm/s rms. In the 130 ms it takes to reach the site of the magnetic trap, even this small velocity spread would result in a 100-fold decrease in cloud density. We avoid much of this decrease by
focusing the atoms with magnetic lenses. Approximately half way (in time) along their route, the atoms pass through two pulses of magnetic field, each around 3 ms long, separated by 8 ms. These pulses provide an impulse that reverses all three components of the velocity. During the first focus pulse, the curvature of the magnetic field is such that the atoms are focused axially and defocused radially. During the second focus pulse, the curvature of the lenses is reversed.
We adjust the timing and strength of the pulses (that is, the positions and effective focal lengths of the two magnetic lenses), in order to bring the atoms together to a focus axially and radially at the magnetic trap center. As shown in Fig. 3, the magnetic trap is created by a combination of ac and dc coils and
is located 14 cm above the optical trap. The 60 Hz ac field is generated by two pairs of coils which are arranged to produce maximum field curvature with relatively small oscillating field at trap center. The dc coil produces a field at trap center of 250 G, a gradient of 31 G/cm in the vertical direction and very little curvature. This static gradient exactly balances the force of gravity, which the magnetodynamic forces are too weak to counteract. The peak amplitude of the ac component is 100 G at the center, with a curvature of 875 G/cm2 in the axial direction
33 1
and of 440 G/cm2 in the radial direction. To make contact with the RF ion trap literature we note that the forces in our trap correspond to Paul trap parameters
= 0 and qz = 0.40, where
a, and qz are stability parameters, defined in Ref. 18. Inserting the atoms into the magnetic trap requires some way of dissipating their energy. This is accomplished by optically pumping the atoms from the F=4 to the F=3 ground states. The trap will hold atoms in the F=3, m,=3 state, but the atoms rising upward from the optical trap are in the F=4, mr=4 state. Because this is a weak-field-seeking state, F=4 atoms approaching the magnetic trap are not only pulled downward by gravity but also repelled downward by the dc field gradient. The initial centersf-mass velocity of the atom cloud is adjusted so that this combination of forces brings the cloud to a stop just at the center of the trap. (The large timedependent fringing fields from the trap make it somewhat difficult to achieve a stationary tightly focused cloud at exactly the correct position and time.) The stationary atoms are then illuminated with light linearly polarized along the magnetic field and tuned to the F=4 4
F'=4 transition. This optically pumps the atoms to the trapped, F=3, mf=3 ground state.
The F=3 atoms already loaded in the trap are transparent to the light used in this final optical pumping. The timing of the final optical pumping relative to the phase of the ac fields is critical.
The micromotion (the small-amplitude, high-frequency oscillation induced by the ac forces") of the weak-field seeking atoms just arriving in the trap has opposite phase from the micromotion
of the trapped atoms. Unless the pumping occurs when the micromotion velocity is zero, the atoms are strongly heated. While the atoms are in the magnetic trap, they do not interact with laser beams. To observe the trapped atoms, we abruptly turn off all the magnetic fields and then illuminate the trap region with a probe beam containing laser light which excites both F=3
-,F'=4
and F=4
--+
F'=5
transitions. The resulting fluorescence from the atoms is imaged by a video camera and recorded on tape. Subsequent image processing reveals the spatial distribution and center-of-mass location of the atom cloud. Because the measurement is destructive -- the photons from the probe beam blow the atoms away -- time evolution of the trapped atoms can be studied only by repeating the load-probe cycle with varying time delays. We study the magnetic trap by observing the shape and motion of the cloud of trapped atoms. Pulses of magnetic field from supplementary coils give the cloud any desired initial
332
center-of-mass motion. From the evolution of the cloud motion we determine the frequency of the guiding-center oscillation in the effective potential. For an oscillation amplitude of 0.1 cm
or less, we masure an axial frequency of 8.5(4) Hz and a radial frequency of 4.5(1.0) Hz.
These agree with calculated'* frequencies of 8.5 and 4.25 Hz. The overall depth of the trapping potential is difficult to calculate or even to define precisely. The depth of the trap is determined mainly by the behavior of the trap@ atoms away from the center of the trap. In those areas the field becomes anharmonic and the approximation that the ac fields simply provide an effective dc conservative potential is not valid. The harmonic region is quite small unless there is an axial dc bias field that is much stronger than the radial and axial components of the ac field everywhere in the trapping region. We confirmed experimentally that increasing the bias field improved the stability of the trap. We have used a computer simulation to understand the behavior of the trap in the regions where the effective potential approximation is not valid. This simulation calculated the motion of clouds of noninteracting atoms with various temperatures in the trap. For clouds inserted with temperatures of 9 p K or less, nearly all the atoms remain trapped and the final spatial extent of the cloud is consistent with the initial velocity spread and a harmonic potential of the calculated spring constant. For clouds with higher initial temperatures, the final spatial distribution of trapped atoms is largely independent of initial temperature. The high-energy fraction of the atoms leaves the trap immediately, and the atoms remaining in the trap have an rms diameter of about 0.2 cm and an rms velocity of 5 cm/s (equivalent to a temperature of about 12 pK). These simulation results match our experimental observations that a fraction of the atoms focused up into the magnetic trap region leave the trap within 0.15 s, and that those remaining form a ball about 0.2 cm in diameter. We conjecture, then, that the rms velocity of our trapped atoms is about 5 cm/s and that the depth of the trap is somewhat deeper than 12 pK. The trap has a I/e lifetime which is typically about 5 s, as shown in Fig. 4. We believe that this is due entirely to collisions with the 1 x
Ton of residual background gas. When we improved
the vacuum by cooling some of the metal surfaces of the upper vacuum chamber with liquid nitrogen, the lifetime nearly doubled. The advantage of multiple loading is apparent in Fig. 4. By accumulating a number of loads, we were able to collect nearly five times as many atoms as from a single load. The factor
333
of 5 increase agrees well with our calculated value and is determined by the ratio of magnetic trap lifetime to optical trap fill time, by perturbations from the fringing fields of the magnetic focus, and by unintentional excitation of the trapped atoms during the initial optical pumping. Without great additional technical effort, this ratio could be a hundred or larger. As the density increases, viscous heating due to the micromotion may become a problem."
We have not yet
seen any sign of such heating, and a simple estimate2' of the heating rate leads us to believe that, with suitable choice of experimental parameters, evaporative cooling can overwhelm the
viscous heating. VI. EVAPORATIVE COOLING AND ADIABATIC COMPRESSION: THE FINISH LINE
To cool magnetically trapped atoms below the recoil limit and finally reach BEC, we plan Because hotter atoms in the to evaporatively cool the sample, as has been done in h~dr0gen.l~ trap experience trajectories through larger magnetic fields, these atoms can be selectively ejected by judicious tuning of RF fields.22
However, effective evaporation requires rapid
thermalization. The high velocity tail of the velocity distribution must be populated by elastic collisions in a time much shorter than the lifetime of the trapped sample. We discuss now a strategy for increasing the rate of desirable, evaporationenabling elastic collisions relative to trapdepleting background collisions. Optical trapping and cooling can
produce a cloud of lo8 cesium atoms with a density of 10" atoms/cm3 and an rms internal velocity of 2 cm/s. There is a large theoretical uncertainty in the low velocity s-wave cross section 0, but for the velocities and densities just mentioned, even the most optimistic estimates do not put the collision rate, nav, very much larger than typical background collision rates of perhaps 0%.
The elastic collision rate can be increased many-fold, however, by adiabatically
compressing the sample in a magnetic trap. The procedure would work as follows: Atoms are optically trapped and cooled and loaded into a magnetic trap as discussed above. The confining magnetic potential is harmonic and nearly spherically symmetric, with the optimal initial trap given simply by the ratio of the atom cloud's initial velocity spread to its initial frequency ainit radial spread (thus preventing entropy-increasing initial collective internal motion in the cloud). The curvature of the magnetic field may then be adiabatically increased. The compression conserves phase space volume (Sr)3(6v)3for the cloud, so that the atoms remain at a constant
334
distance from the BEC transition in phase-space density during the compression. But the coordinate-space density and the rms velocity spread of the atoms increase during the compression, so that the elastic collision rate nav increases by (ofiml/oinil>*. This will likely allow effective forced evaporation to reach the conditions necessary for BEC. VII. CONCLUSION
We have presented a progression of optical and magnetic technology for possibly attaining Rose-Einstein Condensation in a vapor of cesium. The Zeeman-tuned Optical Trap is a very efficient capture device, and is the starting point of the progression. However, optical traps cannot seem to provide the density and temperature necessary for BEC due to the presence of the laser field. Therefore, we have cooled an optically trapped sample of cesium at 10**/cm3 to
2 pK and loaded the atoms into a dc magnetic trap. But exothermic collisional processes are expected to provide a barrier preventing BEC in such a magnetic trap. So we have alternatively loaded the optically trapped and cooled sample into an ac magnetic trap, which traps atoms at their lowest energy, and is immune to any exothermic collisions. To increase the phase space density further in the magnetic trap, we propose to adiabatically squeeze and evaporate the atoms. This "road" offers an appealing way to achieve the phase space density necessary for BoseEinstein condensation in a dilute gas of neutral atoms. This work was supported by the Office of Naval Research and the National Science Foundation. We acknowledge T. Walker and D. Sesko for much of the initial research on limits
of an optical trap.
335
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13. C. Monroe, H. Robinson, and C. Wieman, Opt. Lett.,
fi, 50 (1991)
14. J. Doyle, J. Sandberg, I. Yu,C. Cesar, D.Kleppner, and T.Greytak, Phys. Rev. Lett, 67, 603 (1991). 15. D. Pritchard, Phys. Rev. Lett. 51, 1336 (1983). This work demonstrates the use of this trap design to hold neutral sodium atoms.
16. N. Masuhara et al., Phys. Rev. Lett. 61, 935 (1988). 17. R.V.E. Lovelace, C. Mehanian, T.J.Tommila, and D.M. Lee, Nature R.V.E. Lovelace and T.J.Tommila, Phys. Rev. A s, 3597 (1987). 18. F.G. Major and H.G. Dehmelt, Phys. Rev.
m,30 (1985);
m, 91 (1968).
19. E. Cornell, C. Monroe, and C. Wieman, Phys. Rev. Lett. 68, 2439 (1991). 20.
M. Kasevich, D.S.Weiss, E. Riis, K. Moler, S. Kasapi, and S. Chu, Phys. Rev. Lett. f& 2296 (1991); A. Clarion, C. Salomon, S. Guellati, and W.D. Phillips, Europhys. Lett. 16, 165 (1991).
21.
W. Ketterle, private communication (1991).
22.
D. Pritchard, Proc. 1lth Int'l. Conf. on Atomic Physics, eds. S. Haroche, J. Gay, and G. Grynberg (Singapore: World Scientific, 1989) pp. 179-97.
337
Table 1. Timing and laser detuning during one cycle of the multiple loading sequence.
Relative time (ms)
Laser detuning, function
-500
to -2
-7 MHz. Atoms accumulate in optical trap.
-1
to 0
-30 MHz. Molasses cooling to 2 pK.
0
to 0.4
-7 MHz, and offset A ramped from 0 to 1.7 MHz. Atoms accelerate.
0.4
to 0.8
-30 MHz, and offset A constant at 1.7 MHz. Atoms cool to 4 pK in moving frame.
0.8
to 130
Main beams blocked. Optical trap off. Atoms rise to magnetic trap.
to 4
-250 MHz. Initial optical pumping.
2
60 to 62
Magnetic focusing pulse.
67
to 70
Magnetic focusing pulse.
to 130
-310 MHz. Final optical pumping as atoms reach magnetic trap.
128
338
Figure Captions.
Figure 1.
Dependence of @ on the total intensity in all the trap laser beams. The solid line indicates the prediction of the Gallagher-Pritchard model (Ref. 3, and would be a straight line with linear scales.
Figure 2.
Plot of the diameter (FWHM) of the cloud of atoms as a function of the number
of atoms contained in the cloud. For the full figure the magnetic field gradient is 9 G/cm, and 16.5 G/cm for the inset. The laser detuning is -7.5 MHz, and the total laser intensity is 12 mW/cm2. The solid lines show the predictions of the
model presented in Ref. 2. Figure 3.
Schematic of the apparatus, showing the upper and lower vacuum chambers, the configuration of key trapping and focusing coils, the paths of the initial optical pumping (IOP) and final optical pumping (FOP) laser beams, and the paths of four of the six laser beams defining the Zeeman-shift optical trap (ZOT) and the molasses. Frequencies of the molasses laser beams during the launch are shown
in parentheses. Not shown: the paths of two horizontal laser beams which are perpendicular to the plane of the paper at the ZOT, the magnetic coils for tuning the optical trap, and a variety of shim magnetic coils. Figure 4.
Total fluorescence from the magnetically trapped atoms as a function of time. A new bunch was loaded every 0.65 s, for the first 10 s.
Dashed line is an
exponential decay fit to the data, with 4.9 s l/e time. The y-axis has been calibrated to indicate the total number of trapped atoms.
339
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VOLUME70, NUMBER4
PHYSICAL REVIEW LETTERS
25 JANUARY1993
Measurement of Cs-Cs Elastic Scattering at T = 3 0 p K C. R. Monroe, E. A. Cornell, C. A. Sackett, C. J. Myatt, and C. E. Wieman Joint Institute for Laboratory Astrophysics, National Institute of Standards and Technology and Uniuersity of Colorado, and the Department of Physics, Uniuersiry of Colorado. Boulder. Colorado 80309-0440 (Received 9 October 1992)
We have measured the elastic collision cross section for spin polarized atomic cesium. Neutral cesium atoms are optically cooled, then loaded into a dc magnetic trap. We infer t h e scattering rate from the rate a t which anisotropies in the initial energy distribution are observed to relax. The cross section for F - 3 , m F = - 3 on F = 3 , m F = - 3 is 1.5(4)x cm2, and is independent of temperature from 30 to 250 p K , This determination clarifies the technical requirements for attaining Bose-Einstein condensation in a magnetically trapped Cs vapor. We also study heating due to glancing collisions with 300 K background Cs atoms PACS numbers: 34.40.+n, 05.30.Jp, 32.80.Pj. 34.50.--s
Recent advances in optical trapping and cooling techniques have allowed collision studies in an entirely new regime of temperatures. In this regime of mK or even colder temperatures, collisions take on unusual characteristics, and often have cross sections many orders of magnitude different from that observed at previously attainable temperatures. Studies of ultralow-temperature collisions have investigated several inelastic processes which cause loss from optical traps: two-photon quasimolecular photoionization 111, photon-assisted collisions [2,31, and hyperfine-state changing collisions [2,41. All of these processes are inelastic, and while they involve low initial energies for the atoms involved, the final states have kinetic energies of 1 K or greater. It appears that this allows these processes to be treated semiclassically, although the exact accuracy of such treatments is currently a topic of debate. I n the work presented here, we examine elastic collisions in which the initial and final energies are both much less than 1 mK. In order to study such low-energy collisions we have developed techniques for observing collisions in very cold samples of magnetically trapped atoms. From these measurements, we deduce the s-wave collision cross section. The elastic scattering cross section is of particular interest because of its importance in determining the experimental feasibility of achieving Bose-Einstein condensation (BEC) in a cesium vapor. As demonstrated in experiments with spin polarized hydrogen 151, evaporative cooling of magnetically trapped atoms is a promising strategy for achieving the temperature and density necessary for the BEC phase transition, because evaporative cooling does not have the density and temperature limitations encountered in optical cooling and trapping [2,61. To study the very low energy elastic collisions, we first accumulate Cs atoms in a Zeeman-shift optical trap (ZOT) i n a low pressure cell [71. The atoms are then optically cooled and spin polarized in the F = 3 , m p = - 3 state in preparation for magnetic trapping. After all laser light is shut o f f ,magnetic fields are turned on around the atoms in situ 171, thereby trapping them. In this work we use a type of dc magnetic trap which 414
has been previously proposed but, to our knowledge, never used to trap neutral atoms. The field coils consist of wire wound like the seams of a baseball [Sl. The 2.5-cm radius “baseball” coil consists of nine turns of No. 3 Cu capillary tubing, and is cooled by flowing water through the tubing. The field at the center of the trap is aligned with gravity. To counteract the force of gravity, a vertical mapiletic field gradient of -31 G/cm is generated with an additional pair of circular coils which sit above and below the baseball coils. The superposition of the gravity-canceling fields and the baseball coil fields breaks the cylindrical symmetry of the baseball potential. Although the coil geometry is novel, the fields produced near the center of the trap are the same as for the loffe coil configuration; see Refs. 171 and [Sl. The baseball coil geometry has a number of desirable characteristics. It has a nonzero minimum in the magnetic field, allows excellent optical access, and provides strong curvatures for a given electrical power. The trap depth and oscillation frequencies of the atoms in the harmonic magnetic trapping potential are related to the current applied through the coil. With 30 A through the coil, the trap has a central field of 50 G with oscillation frequencies of ( v x , v y , v , ) = ( 4 , 8 , 4 ) Hz for atoms in the F=3, m,==-3 trapped state. At 300 A, the central field is 500 G with oscillation frequencies of (19,20,1 1 ) Hz. When the atoms are first loaded into the magnetic trap from the optical trap the fields are set to give oscillation frequencies ( v , , v y , v , ) = ( 1 0 . 5 , 1 2 . 5 , 6 . 5 ) Hz. The temperature and density of the magnetically trapped cloud can then be increased by ramping up the strength of the trap around the atoms. We typically load 10’ atoms with an initial average density of 7 x 109/cm3 and initial temperature of 30 p K . The temperature is higher than typical optical cooling temperatures due to heating from optical pumping and the addition of magnetic potential energy when the trap is suddenly turned on. As the atoms are compressed in the magnetic trap, the density and temperature can be continuously increased up to 10”/cm3 and 80 p K , respectively. The lifetime of the atoms in the magnetic trap is limited by collisions with room-tem-
@ I993 The American Physical Society
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344 VOLUME 70, NUMBER 4
PHYSICAL REVIEW LETTERS
25 JANUARY 1993
-
perature background gas. The I/e decay time depends on the pressure and composition of the residual gas, and is typically about 1 5 s. W e have also determined that the trap lifetime is independent of density to within o u r uncertainty, which indicates that the spin-flip relaxation cm3/s. rate is less than 5 x For the higher temperature measurements, the atoms are transferred to a separate chamber [91 where they arc magnetically trapped. I n the double chamber setup, heating i n the transfer process kept attainable temperatures over I00 p K . While the atoms are in the magnetic trap, they do not interact with laser beams and therefore are invisible. T o observe the trapped atoms, we abruptly turn off all magnetic fields and illuminate the atoms with a 0.5-ms pulse of resonant laser light. The resulting fluorescence from the atoms is imaged onto a video camera and a photodiode, revealing both the spatial distribution and total number of atoms. Because the measurement is destructive-the photon pressure in the probe beam blasts away the atoms-time evolution of the trapped atoms is studied by repeating the measurement and changing the amount of time the atoms spend in the magnetic trap before the illumination pulse. Shot-to-shot reproducibility is fairly good- measured size, for instance, reproduces to a few percent. There is always some initial expansion-contraction oscillations and center-of-mass sloshing of the cloud, but these decay away in about 2 s (about twenty oscillations) due to small anharmonicities in the trapping potential. The distribution of atoms in the cloud can be characterized by three temperatures [lo], T,, T,, and T,, where T, is defined in terms of the spring constant ki and the mean-square extent of the cloud in each direction keTi ~ 0 . 5 k i ( r : ) .Because of asymmetries in the loading process, the three temperatures are initially unequal. Perhaps because the oscillation frequencies in the three different directions are well separated, residual anharmonic terms in the trapping potential are not observed to cause the three temperatures to equilibrate, a t least not on the time scale of our experiments. Time evolution of the three temperatures is almost purely due to collisional processes. The time dependence of the individual one-dimensional cloud sizes (Fig. 1) shows two main effects. T h e first, visible only a t high densities of trapped atoms, is that the sizes tend to be mixed. This is due to the effect we are trying to study, low-temperature elastic collisions between trapped cesium atoms bringing the sample into thermal equilibrium. The observed cross-dimensional equilibration rate can, with further analysis, be turned into an elastic collision cross section. Unfortunately, there is a second effect, visible in both high and low density clouds-the sizes in all three directions tend to i n crease uniformly because of elastic collisions occurring between trapped atoms and atoms in the 300-K residual
vapor i n o u r vacuum chamber. Usually, such a collision imparts a substantial amount of energy to the trapped atom, ejecting it cleanly from the trap with no effect on the remaining trapped atoms. But on occasion (in the event of a glancing angle collision), the background atom imparts such a small amount of energy to the trapped atom that the latter remains in the trap, although with increased energy. The glancing collisions with background atoms make our study more difficult, but are of some interest i n their own right [ I I ] . The collisional energies are very high and hundreds of partial waves are involved, yet a classical treatment of the collisions yields incorrect results. I n the extreme glancing collision regime, the momentum transferred can be so small that the associated de Broglie wavelength can be large compared to the classical impact parameter, meaning that diffraction effects become significant. Both trap loss and heating rates vary proportionally with cesium pressure in the background gas, except that, a t very low Cs pressures, the heating rate goes nearly to zero while the loss rate approaches = 0.06 s - ’ . W e attribute this residual loss rate with very little accompanying heating to residual amounts of helium gas i n the system. Because helium is much lighter than cesium, the diffractive regime begins a t a much higher energy transfer, and thus the probability of a He-Cs collision transferring a small enough amount of energy to heat but not to eject an atom is very small [121. In any case, the heating is evidently isotropic and during data analysis can be separated from the interdimensional equilibration arising from intratrap collisions. T o increase the initial temperature anisotropy, and thus to bring out the observable signature of temperature equalizing collisions, we heat one of the dimensions of the trapped atom distribution using a parametric drive. The trapping field is modulated a t twice the y-dimension harmonic frequency. The width of the parametric excitation resonance is about 1 Hz, so this parametric drive does not heat the other two dimensions. After the drive has been turned off, we wait for an interval of time f , and then illuminate the sample and measure the distribution of atoms in the cloud. To simplify the d a t a analysis and interpretation, only very small times f are considered, such that rmixf . x..::>...
1
-
\
: - - -’-I: 5=:= --=:-1_-.
,
,
,1
-
- . .-..--.-..--.-..--.- ..-. ..-. -.--..- -..-.
2
-YvL--.,,-.-.-.--
3
4
,
.-4 I
5
time (sec)
Fig. 8. Percentage of Cs atoms that have been released into the cell that will be trapped as a function of time. Trace 1 shows the capture efficiency for a wall with a stick time of 1.4 ms, an outgassing of 1.8 lO-’Ton cm3/(cm2 s), and a reaction rate with the Cs of 0.065 cm3/(cm’s). Trace 2 is the same as 1 but with a stick time of only 0.045 ms (this trace represents Dryfilm). Trace 3 is the same as 2 but the stick time if 0 s. Trace 4 is the same as 3 but the reaction rate is 0. Trace 5 is the same as 4 but the outgassing is reduckd by a factor of 60. For all of these, the capture velocity is 32 m/s (corresponding to 200 rtlW/beam and 2 cm beams) and the cell has a surface area of 75 cm2 and a volume bf 30 cm3.
365
M. Stephens
212
5.
ef
al. / Optimizing capture in optical traps
The performance of Dryfilm and Pyrex
A number of experimenters have studied wall coatings that preserve spin polarization of optically pumped alkali atoms [ 15- 191. Since the probability that an atom will depolarize at the surface is directly proportional to the amount of time it spends on the surface [15], the surfaces that work well in optical pumping experiments are surfaces with low adsorption energies. In particular, it has been found that paraffin and an organosilicon polymer, often called "Dryfilm", both work well. We chose to test Dryfilm since we expected it to have a lower vapor pressure than paraffin. We have compared Dryfilm-coated Pyrex to uncoated Pyrex. All of our tests have been with 133Cs(the stable isotope). We measured the adsorption energy of Cs on Dryfilm and Cs on Pyrex in two ways. First, we made a direct measurement by measuring the change in Cs vapor as a function of the temperature of the walls (fig. 9). We cooled the cell to liquid
-
+
-
+ 4-
c
I
< 5?
m
m
+
-
-
m
m m
+ +
m
m
+
+
m
+
+
5.5 -
m
m
m
+
m
m 1.o
1
42.0
43.6
1
'
~
1
1
-
1
45.2 46.8 l/kT (ev**(-1))
1
1
1
'
48.4
1
1
'
'
50.0
Fig. 9. The change in Cs vapor density as a function of wall temperature; n is the measured density at that temperature, nO is the maximum density (defined by the number of atoms initially released into the cell). The slope of the line is the adsorption energy. The asterisks are Cs on Pyrex and the crosses are Cs on Dryfilm. A leastsquares fit provides &, = 0.44 eV for Cs on Dryfilm and E, = 0.53 eV for Cs on Pyrex.
nitrogen temperatures, let in a small amount of Cs which initially completely adsorbed to the wall, and then monitored the Cs vapor density as the cell warmed up. We measured an adsorptioh energy of 0.44 eV for Cs on Dryfilm and 0.53 eV for Cs on Pyrex. Second, we measured the characteristic fill time of the cell. If initially
366
M . Stephens et al. / Optimizing capture in optical traps
213
no Cs is in the cell, and then a valve with a conductance C is opened between a Cs reservoir and the cell, the Cs density in the cell will fill exponentially with a time constant
The stick time, and therefore the adsorption energy, can be inferred from the fill time. The fill time measured for Dryfilm-coated Pyrex was 0.108 f 0.02 s, which s and Ea 5 0.44 eV. The fill time for the uncoated implies z, = 4.5 x f 4.5 x s Pyrex cell was measured to be 1.1 rfi 0.3 s, which implies 2, = 1.4 x f6x and 0.51 < E , < 0.54 eV, in good agreement with the temperature measurements. The primary source of uncertainty for the fill time measurements was the conductance of the valve connecting the cell to the Cs reservoir. The sticking properties of the Dryfilm-coated Pyrex fulfills the requirements for sticking time; however, the Cs does react with the Dryfilm. The reactions have two effects. The first is to cause the permanent loss of trappable Cs, and the second is to increase the outgassing of the Dryfilm. The first effect is not a major difficulty: the Dryfilm can be “cured” by prolonged (several days) exposure to Cs. After the Torr vapor of Cs for a few days, the reaction Dryfilm has been exposed to a rate is 0.065 cm3/(cm2s). This is a reaction time of around 6 s for a cell with a surface area of 80 cm2 and a volume of 30 em3. The capture time for the same size cell (assuming a capture velocity of 32 m/s, i.e. a Ti : sapphire laser and 2 cm beam diameter) is roughly 3 s. This means one would not lose many atoms to the wall reactions during the trapping process. The second effect, however, is large enough to become the dominant effect on trapping efficiency. After exposure to Cs, the Torr cm3/(cm2s). Because of this relatively Dryfilm outgases at around 2 x large outgassing rate, the maximum number of atoms obtained with a Dryfilmcoated cell will be 5-15% of the number released into the cell (see fig. 8). The outgassing is increased by the reactions of the Cs with the Dryfilm [20]. Some possible reactions are shown in fig. 10 [21]. The Cs is probably reacting with the oxygen in the Dryfilm polymer. Although 5-15% of the atoms may be enough for the experiments planned, we will explore other coatings, for example diamond, Langmuir-Blodgett films, and thiolates [22] in the hope of further improving our trapping efficiency.
6.
Conclusions To most efficiently capture atoms in a MOT, one must increase the capture
rate with high laser power and large trapping beams. Wall-coated vapor cells show promise for minimizing atom- wall interactions. We have measured the adsorption energy of Cs on Dryfilm to be 0.44 eV, This corresponds to a 45 ps stick time. Reactions of the Cs with the Dryfilm increase the outgassing of the Dryfilm so that
367
M . Stephens et al. 1 Optimizing capture in oprical traps
214
CH3
I I 0 I CH3-SiI
CH3-Si-
OH
CH3
73Fig. 10. (a) A properly formed Dryfilm polymer. The S i - 0 backbone is shielded from the Cs by methyl groups. (b, c) Some ways in which th epolymer can form improperly. The Cs can react with the oxygen and hydroxide radicals [21].
loss from the trap due to collisions with background gas is now the dominant factor limiting trapping efficiency. We currently believe that with a Dryfilm-coated cell, we will be able to trap 5-15% of the atoms released into the trapping cell.
Acknowledgetnent This work was supported by the Office of Naval Research and the National Science Foundation.
References 111 K.E.Gibble, S. Kasapi and S. Chu., Opt. Lett. 17(1992)526. [2] b.W. Sesko, T.G. Walker and C.E.Wieman, J. Opt. SOC. Am. B8(1991)946. [3] C. Monroe, W. Swann, H. Robinson and C. Wiemafl, Phys. Rev. Lett. 65(1990)1571.
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215
C. Wieman, C. Monroe and E. Cornell, 10th I n f . Conf. on Laser Spectroscopy, eds. M. Ducloy, E. Giacobino and G. Camy (World Scientific, Singapore, 1992) p. 77. H. Gould, private communication. S . Freedman and K. Coulter, private communication. D. Grison, B. b u n i s , C. Salomon. J.Y. Courtois and G. Grynberg, Europhys. Lett. 15(1991)149; J.W.R. Tabosa, G. Chen, Z. Hu, R.B. Lee and H.J. Kimble, Phys. Rev. Lett. 66(1991)3245. A. Gallagher and D. Pritchard, Phys. Rev. Lett. 63(1989)957; D. Sesko, T. Walker, C. Monroe, A. Gallagher and C. Wieman, Phys. Rev. Lett. 63(1989)961. K. Lindquist. M. Stephens and C. Wieman, Phys. Rev. A46(1992)4082. E.L. Raab, M. hentiss, A. Cable, S. Chu and D. Pritchard, Phys. Rev. Lett. 59(1987)2631. A.M. Steane, M. Chowdhury and C.J. Foot, J. Opt. SOC. Am. B9(1992)2142. J.H. de Boer, in: The Dynamical Character of Adsorption (Clarendon Press, Oxford, 1978) p. 23. F. Reif, in: Fundmntols of Statistical and Thermal Physics (McCraw-Hill, New York, 1965) p. 273. R.G. Brewer, J. Chem. Phys. 38(1963)3015. M.A.Bouchiat and J. Brossel, Phys. Rev. 147(1966)41. J.C. Camparo, 1. Chem. Phys. 86(1987)1533. L. Young, R.J. Holt, M.C. Green and R.S. Kowalczyk, Nucl. Instr. Meth. B24(1987)963. H. Goldenberg, D. Kleppner and N. Ramsay, Phys. Rev. 123(1961)530. D.R. Swenson and L.W. Anderson, Nucl. Instr. Meth. B29(1988)627. J.C. Camparo, R.P. Frueholz and B. Jaduszliwer, I. Appl. Phys. 62(1987)676. D.W. Sindorf and G.E. Maciel, J. Am. Chem. SOC.105(1983)3767. R.G. Nuzzi, L.H. Dubois and D.L. Allara. J. Am. Chem. SOC.112(1990)558.
VOLUME 72, N U M B E R24
PHYSICAL REVIEW LETTERS
13 J U N E1994
High Collection Efficiency in a Laser Trap M. Stephens and C. Wieman Joinr Instirure f o r Laborarory Astrophysics, Uniilersity of Colorado, Boulder, Colorado 80309-0440 (Received 14 October 1993) We present the results of the first study of the fundamental processes governing collection efficiency of neutral atoms in a vapor cell laser trap. The experimental test of our model of the collection process closely agrees with our predictions based on the measured properties of the laser beams and cell. This study has led to the development of special vapor cell wall coatings. We have demonstrated a collection efficiency of 6% for cesium and predict more than 50% could be achieved under optimum conditions. This work develops the concepts for and demonstrates the feasibility of new experiments using shortlived radioactive isotopes. P A C S numbers: 32.80.Pj. 42.50.Vk
There are many important experiments that would be possible if one had a dense vapor of short-lived radioactive isotopes free from perturbations from the container walls. One obvious example is high precision atomic spectroscopy [I]; however, there are a number of more exotic and interesting experiments such as the study of ,i3 decay from polarized samples 121, the search for electric dipole moments [31, and the investigation of atomic parity nonconservation in ways that cannot be done with available isotopes [41. These exotic experiments require dense, polarized samples. Over the years there have been numerous, often heroic, efforts to achieve this goal with very limited success [I]. A large number of experiments have been carried out on alkali isotopes obtained by depositing radioactive ions onto metal foils that are heated to quickly release the volatile neutral atoms 111. However, because of the small numbers of ions initially available, all of these experiments have been limited by low densities and short interaction times ( 1 msec). Here we demonstrate how laser cooling and trapping can be used to efficiently collect small numbers of atoms in a dense (10"/cm3), easily polarizable sample that is held away from the perturbing effects of walls, making a host of new experiments possible. Atoms can be optically trapped from a slowed atomic beam [51 or from an atomic vapor [6,71. Although there are very little data on the collection efficiency from a slowed beam, measured efficiencies have always been low [81. We have chosen to optimize the collection from a vapor. The collection efficiency depends on the geometry of the vapor cell, the properties of the laser beams, and the properties of the cell walls. We have developed and tested a model of how the collection efficiency varies with these properties. Efficient collection of small samples of atoms requires several changes from the usual vapor cell trap. In a standard cell trap [61, atoms from the low-velocity tail of the Maxwell-Boltzman distribution are collected from a dilute vapor that is constantly replenished from a reservoir. Nonalkali background gases are continuously pumped away to prevent them from knocking the trapped atoms
out of the trap. Some of the alkali vapor is also pumped away or sticks to the walls, but i t is replenished from the reservoir. However, when collecting atoms from a limited source, it is necessary to prevent their loss to the pump or to the walls. The cell walls must be a material that the atoms will not stick to, and it is necessary to stop pumping on the cell while trapping the atoms. Without continuous pumping, the density of background gases in the cell builds up over time. This buildup and the corresponding increase in the loss rate from the trap due to collisions with the background gases must be minimized. We have analyzed the situation of suddenly introducing N o atoms into a cell and attempting to trap as many as possible. The number of atoms which are trapped as a function of time depends on the capture rate, R, and the loss rate, 11.5 [71, according to
where N is the number of trapped atoms. Unlike in Ref. 171, R and I/r are time dependent. The capture rate is given by
R(0
---
1
1':
?rL2n(t), (2) 4& .dl where L'th and n are the thermal velocity and the density, respectively, of untrapped atoms, and L', is the maximum velocity an atom can have and be captured. This maximum velocity depends primarily on the laser beam Gaussian width, L, and intensity, I, and can be calculated
171. The untrapped alkali density, unlike the normal vapor pressure, depends on the interactions between the atoms and the walls: (3)
Here A is the surface area of the cell, V is the volume of the cell, r is the effective pumping speed of the wall per cm2 due to permanent chemical reactions between the atoms and the wall, and rs-IO-'Zexp(E,/kT) sec is the
003 1 -9007/94/72(24)/3787(4)$06.00
@ 1994 The American Physical Society
369
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V O L U M E 72, N U M B E R24
13 J U N E 1994
P H Y S I C A L REVIEW LETTERS
average time an atom sticks to the cell wall, where E, is the wall-alkali binding energy “91. The loss rate of atoms from the trap depends on the pressure P (in torr). 1/T
=3.3x 10‘6PabL’h,
(4) c
where ‘Tb is the cross section for the nonalkali background gas to knock an atom from the trap, and I‘b is the thermal velocity of the gas. We have measured C s h [ ’ b - 1 0 - ~ crn3/sec for the background gases produced in o u r coated cell. We can neglect collisional loss due to the alkali background because it has a much lower density. The nonalkali pressure will initially be P o = Q A / S , where S is the pumping speed a t the cell and Q is the outgassing of the walls. After the pump is closed off, P rises linearly with time, P = P o + Q A t / V. With Eqs. (1)-(4) we can calculate the number of trapped atoms as a function of time after closing off the pump and releasing the atoms into the cell. The collection efficiency is the ratio of the maximum number of atoms trapped to the number released into the cell. One can see from Eq. ( 3 ) that preventing the sticking time, r S , from impacting the capture rate requires r,4.0
SC-I7
on Pyrex on stainless steel
0.025 5.0
Oxysilanes
on Pyrex on stainless steel
0.1 0.8
lO1"/s. When operated at lower power the LVIS flux drops in proportion to the reduced capture rate for the MOT. However, the beam collimation and velocity remain nearly the same so LVIS would still produce a nice beam with low power diode lasers. Finally, w e observed a much higher peak flux when the LVIS system was operated in a pulsed mode. In these geometries, the VCMOT empties in 50 ms, providing nearly the same time-averaged flux but 10 times the peak flux and brightness. The flux and flux density depend on the collimation angle and geometry of the LVIS setup in the manner predicted above. Although we achieved 70% extraction efficiency when 0 = 36 mrad, we only achieved a trans-
3334
14 OCTOBER 1996
fer efficiency of 20% when N = 5 mrad. Figure 5 shows the tradeoff between flux density and collimation of the atomic beam. Recycling causes the flux density (number s-' cm') at the detection region to increase while the divergence decreases. Total flux decreases beyond 4.5 cm because for a collimation this tight, the atomic beam transverse temperature becomes comparable to the temperature of atoms in the VCMOT. With tight collimations, an atom must make many more attempts to be successfiilly transferred into the atomic beam. This makes r c / r , larger, decreasing the flux as predicted. The simplicity, brightness, and versatility of LVIS will make it useful in a wide range of applications. This work was supported by the National Science Foundation, the Office of Naval Research, and the National Institute of Standards and Technology.
*Permanent address: Physics Department, Michigan Technological University, Houghton, MI 4993 1. iPermanent address: Meadowlark Optics, 7460 Weld County Road 1, Longmont, CO 80504-9470. [I] K. Cibble and S. Chu, Phys. Rev. Lett. 70, 1771 (1993). [ 2 ] D. W. Keith et al., Phys. Rev. Lett. 66, 2693 (1991). [3] M. J . Renn er al., Phys. Rev. A 53, R648 (1996). [4] M. H. Anderson ef a/., Science 269, 198 (1 995). [5] Z-T. Lu et a/., Phys. Rev. Lett. 72, 3791 (1994). [6] M. Stephens and C. Wieman, Phys. Rev. Lett. 72, 3787 (1994). [7] T. E. Barrett et al., Phys. Rev. Lett. 67, 3483 (1991). [8] W. D. Phillips and H. Metcalf, Phys. Rev. Lett. 48, 596 (1982). [9] W. Ertmer er al., Phys. Rev. Lett. 54, 996 (1985). [lo] M. Zhu, C. W. Oates, and J. S. Hall, Phys. Rev. Lett. 67, 46 (1991). [ l l ] W. Ketterle et al., Phys. Rev. Lett. 69, 2483 (1992). [12] E. Riis et al., Phys. Rev. Lett. 64, 1658 (1990). [13] A. Scholz et nl., Opt. Commun. 111, 155 (1994). [I41 J . Yu et nl., Opt. Commun. 112, 136 (1994). [15] C. Monroe et al., Phys. Rev. Lett. 65, 1571 (1990). [ 161 K. Lindquist, M. Stephens, and C. Wieman, Phys. Rev. A 46, 4082 (1 992). [ 171 C. Monroe, Ph. D. thesis, University of Colorado, Boulder, CO, 1993. [ 181 A hole was drilled in the quarter-wave plate, and then the back surface was coated with a reflecting layer of gold.
VOLUME 79, NUMBER6
PHYSICAL REVIEW LETTERS
1 1 AUGUST 1997
Efficient Collection of 221Frinto a Vapor Cell Magneto-optical Trap Z.-T. Lu, K. L. Corwin, K. R. Vogel, and C. E. Wieman JILA, National Institute of Standards and Technolog,v and University of Coloivdo, Bozilder, Colorado 80309-0440, and Phj)sics Departnieiit, IJnivei-sity qf Colorado. Boulder, Colorudo 80309-0440
T. P. Dinneen, J. Maddi, and Harvey Could Lait~renccBerltelev National Laboratory, Universit,v of California, Mail Stop 71-259. Berkeley, California 94720 (Reccived 23 April 1997) We have efficiently loaded a vapor cell magneto-optical trap from an orthotropic source of '"Fr with a trapping efficiency of 56( lo)'?& A novel detection scheme allowed us to measure 900 trapped atoms with a signal to noise ratio of -60 in 1 sec. We have measured the energies and the hypefine constants of the 7'P1/2 and 7'P3/2 states. [SO03 I-9007(97)03811-81 PACS numbers: 32.80.Pj, 32.10.Fii
the trap volume (region of laser beam overlap) to the cell There has recently been considerable activity in the surface area. To minimize the loss rate from the cell, the field of laser trapping of short-lived radioactive atoms. opening through which the radioactive atoms enter the cell While a wide range of isotopes is being pursued, laser trapping of "Na [I], 37,38K[2], 79Rb [3], and 209,210,211Frmust be small enough to minimize the leak rate, and the loss of atoms via adsorption to the glass walls must be sig[4] atoms has been experimentally realized. Efficient opnificantly reduced. The latter is accomplished by coating tical trapping is essential for creating large samples of rare the glass surfaces with dryfilm coatings made of siliconatoms. Such samples are very appealing for tests of the standard model including atomic parity nonconservation based hydrocarbon polymers, which are then "cured" by exposure to alkali vapor [ 121. (PNC), the electric dipole moment (EDM), and p decay The analysis of the capture process differs from that of [l]. Francium, which has no long-lived isotopes, is particularly interesting for these tests, because calculations a conventional VCMOT. I n a highly efficient VCMOT, predict PNC amplitudes and EDM enhancements to be the number of atoms in the vapor and the number of atoms 10 times larger in Fr than Cs [5,6]. In this Letter we in the trap are so strongly coupled that the vapor density demonstrate efficient trapping of '"Fr. The general apcannot be considered constant. The time evolution of proach should work as well with any alkali isotope, and such a coupled system of N f atoms in the trap and Nu should make tests of the standard model possible in rare atoms in the vapor depends on three rates: L , the loading rate of atoms from the vapor to the trap; C, the loss rate trapped atoms. of atoms from the trap to the vapor due to collisions with Various techniques have been developed to collect vapor atoms; and W , the loss rate of atoms from the vapor atoms into traps [7], but since short-lived radioactive atoms are available only in limited quantities, improving to the cell walls or out of the cell. In the case where a the optical trap collection efficiency is a central issue. constant flux, I , of atoms enter the cell, the dependence can be described by two coupled differential equations: The highest efficiency yet demonstrated used a vapor cell magneto-optical trap (VCMOT) [8] in a glass cell coated with dryfilm. Using coated cells, Stephens et al. [9] and CN, LN,, Guckert et al. [ 101 have demonstrated, respectively, 6% and 20% collection efficiencies of stable cesium. Similar d ( N v ) = C N , - LN, - W N , + I techniques have been applied to trapping radioactive dt K, Rb, and Fr atoms [2-41, but with far lower trap The time evolution of the number of trapped atoms efficiencies. We have created a highly efficient (56%) follows a double-exponential function, '"Fr VCMOT in a coated cell and used it in spectroscopic measurements on *"Fr. A conventional VCMOT [8] traps a very small fraction of the available atoms. To obtain high collection C + W and klk? = W C . N f ( t = where kl + k2 = L efficiency, one must raise the collection rate and lower 0 ) and N,(t = 0) fully determine a1 and a2. We define the vapor loss rate. The dependence and optimization of trap efficiency (7) as the probability of trapping an atom collection rate on various trap parameters has been previthat has entered the cell [ 131, ously investigated [ 111. To have a high collection rate, L CNf a trap should have large, high-power laser beams. The (CN, I ) ' = (L + W ) trapping cell should be designed to maximize the ratio of
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The "'Fr nuclei are produced in the decay chain 229Th(t1/2= 7300 yr)
u
-
775
'--Ra(tl/2
=
15 d)
22sA~(tl/2 10 d)
a 221
y
Fr(tl/?
=
4.9 min).
In order to produce a portable '*'Fr source with shortlived radioactivity, 2 2 5 Awas ~ first extracted out of a longlived 2'9Th sample, and deposited onto a small piece of platinum using one of two methods. Results reported here were obtained from a sample of "'Ac chemically extracted from a "'Th solution [ 141 and electroplated onto a Pt ribbon. In our preliminary measurements, we used 2 2 5 Aimplanted ~ onto Pt via electrostatic collection [15]. The Pt ribbon was then placed in the cavity of an orthotropic oven [15] where '"Fr daughters were continuously produced and distilled into a collimated atomic beam. At the beginning of the experiment, the oven was loaded with 50 p C i 2 2 5 A ~which , produced 2 X lo6 s-' of *"Fr. The full divergence angle of the atomic beam was measured to be 180 mrad. By counting the cy particles from the decay of the ?"Fr, we measured an atomic **'Fr flux of 3.8 X lo4 s-', or about 2% of the "'Fr atom production rate inside the oven. Thirty days later, at the end of the experiment, the 221Frflux was 5.1 X lo3 s-I, reflecting a decrease matching the natural decay of 2 2 5 A ~ . A schematic of the apparatus is shown in Fig. 1. Both the francium oven and the vapor cell were situated inside a chamber where the vacuum was maintained at 2 X lo-* Tom even when the oven was heated to the operating temperature of 1050 "C. The cell was a quartz glass cube (4.4 cm inside dimension) whose top lid could be opened and closed via a mechanical feedthrough. Following the recipe developed by Stephens et al. [12], the cell walls were coated with a short-chain dryfilm called SC-77 (Silar Laboratories), and then cured by opening the cell lid and
maintaining a Rb vapor in the cell at about 2 X 1 O r 7 Torr for 10 h. The Fr atoms from the oven entered the cell through a 2 mm diameter hole at a lower edge of the cell. Over several days, the coating performance deteriorated, reducing the number of trapped Fr atoms by a factor of 3. This damage to the coatings could be attributed to heat or material evaporating from the oven, and was repaired by providing a continuous, low level curing with - 1 X l o p 8 Torr of rubidium in the cell. Up to 1 W of light from a Ti:sapphire ring laser was tuned to the D l line of Fr at 7 18 nm for trapping. The laser frequency was locked to the side of a Doppler-broadened I? absorption peak [ 161 that conveniently covered the frequency of the 7S1/2, F = 3 7P3/2, F = 4 cycling transition (Fig. 2). In addition, an SDL 100 mW diode laser at 8 17 nm was tuned to the D I line to pump atoms out ofthe 7SI/2,F = 2 state. The frequency of the diode laser was tuned to the 7S1/2,F = 2 7P1p,F = 3 transition and locked to a Fabry-Perot cavity, which was in turn actively stabilized to the rubidium 0 2 line. We assessed the performance of the wall coatings and found the correct laser frequencies by observing fluorescence from the room-temperature Fr vapor. We used an optical-optical double resonance technique to separate the fluorescence from light scattered off the cell walls. For this technique, a beam from the Tksapphire laser was sent through the center of the cell, while its frequency scanned over one Doppler width (300 MHz) about the 7S1/2, F = 3 7P312, F = 4 cycling transition. A 60 mW beam from the 817 nm diode laser, chopped at 100 Hz, copropagated and overlapped the Tksapphire beam through the cell. The diode laser frequency was tuned close to the 7S112,F = 2 7Pl/2, F = 3 transition to pump atoms from the 7S1/2, F = 2 to the 7Sl/2, F = 3 state. The population of the 7Sl/2, F = 3 state was therefore fully modulated for the group of atoms in resonance with both laser beams. Because this was a narrow velocity group, the resulting resonance feature was sub-Doppler (50 MHz
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-
F=4-
316MHz
18.6GHz/
7Pg/2
751/2
2
FIG. 2. The atomic level diagram of 22'Fr. In our I2'Fr trap, the trap laser (718 nm) was tuned to -30 MHz below the 7S1/2, F = 3 7P3/2rF = 4 cycling transition, and the repump laser (817 nm) was tuned to the 7s1/2, F = 2 7P1/2, F = 3 transition. To detect the trapped atoms, a 718 nm pump beam (2 mm diameter, 50 mW/cm'), tuned to the 7S,/2, F =3 7P3/2, F = 3 transition, was chopped to modulate the 8 17 nm fluorescence from the trap. -+
FIG. 1. A diagram of the 22' Fr trap setup. Both the oven and the cell were situated inside a vacuum chamber with 10 cm diameter windows. The same oven and cell assembly was used for the "'Fr vapor fluorescence measurements.
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FWHM). The resulting modulated fluorescence at 718 nm generated near the center of the cell was imaged onto a low noise photodiode through a 7 18 n m interference filter and demodulated with a lock-in amplifier. T ~ Uwe S were capable of detecting as few as 600 atoms in the entire cell, or equivalently 1 Fr atom in resonance in the viewing region, with a signal to noise ratio (SNR) of l/&. By monitoring the exponential buildup of the number after the cell lid was closed, we determined the loss rate of Fr atoms from the vapor (W) to be 1.59(9) s - ' , and a constant flux of 2.3 X lo3 s-' Fr atoms entering the cell. We then trapped the Fr atoms in a VCMOT using 4 cm diameter laser beams. These beams provided a sixbeam-total intensity of up to 110 mW/cm2 in the cell. In addition, a 60 mW, 4 cm diameter beam from the diode laser at 8 17 nin was sent through the cell four times along two normal axes of the cube cell. A set of anti-Helmholtz coils generated the MOT quadrupole field gradient of -7 G/cm. The laser light scattered from the six cell walls made detection of the 718 n m fluorescence from the trapped Fr atoms impossible. Instead, to further avoid scattered light and thereby increase our sensitivity, we imaged the 817 nm repump fluorescence froin the trapped atoms onto a photodetector. This light was observed through a cell window not illuminated by any repump beams and through a 8 17 nm bandpass interference filter. The 817 nm fluorescence was modulated using a small, frequency-shifted 7 18 nm pump beam that was chopped and retroreflected through the trap center (see Fig. 1). This beam increased the fractional population in the 7s1/2, F = 2 state to 70% without affecting the trap loading rate. This detection scheme allowed us to detect as few as 15 trapped Fr atoms with a SNR of 1 1 6 in the presence of large amounts of scattered light. The trap contained an estimated 900 atoms, but this is a lower limit assuming that both the 718 nm pump and the repump lasers were tuned to resonance. This trapped atom sample could be maintained for many hours. Figure 3 shows a trap loading curve fitted to the doubleexponential function predicted by our model. The fit gives the trap rate constants, from which we calculate the trapping efficiency to be 7 = L / ( L W ) = 56(10)%. As a check, 7 = C N r / ( C N r + I) gives a lower limit of 30% assuming the repump laser is tuned to resonance, but can be as high as 50% when plausible detunings are assumed in calculating N t . The atomic fluxes at various stages of the experiment are listed in Table I, showing a total efficiency of 0.4% from production to trapping. Engineering improvements in the oven and in the coupling into the cell would significantly improve the total efficiency. An orthotropic source has operated with a 15% efficiency, and 50% (limited by diffusion of Fr into the oven walls) is theoretically possible [ 151. We have made spectroscopic measurements on **'Fr using our apparatus. In the thermal vapor, we measured
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Time(sec) FIG. 3. The number of atoms loaded into a "'Fr trap vs time. In a trap with a high collection efficiency, the loading signal is expected to be a double-exponential function. Because the signal was filtered by a low-pass nctwork with a time constant T = 100 ins, it was fitted with the function 01[1 - exp(-klt)] + b2[1 exp(-k~t)] ~ [ b l k lt h?k?][I - e x p ( - t / ~ ) ] . The best fitting results were kl = 4.8(1.0) s-' and k2 = 0.57(23) s-I. Combining these rates with the measured result that on average a Fr atom stayed in the vapor for l / W = 0.63(4) s, we derived that, on average, a Fr atom stayed in the trap for l / C = 0.58(27) s and it takes I/L = 0.48(19) s to load a Fr atom from the vapor into the trap. These rates are consistent with those found in Rb traps, and imply a trap efficiency of L / ( L + W ) = 56(10)%. -
the hyperfine splittings of the 7 P 3 p and 7 P 1 p levels. We observed the individual hyperfine transitions by scanning the 718 nm laser, while the 817 nm diode laser was set (to either the F = 2 -, F' = 3 or F = 2 + F' = 2 transition) to select one velocity group of atoms. We obtained the splittings by measuring the frequency differences between each hyperfine line (Table 11) using a high-resolution A meter 1171. By checking against known splittings and wavelengths in rubidium, we found the accuracy of the A meter to be 3 MHz or less when measuring small differences, but 50 MHz when measuring absolute frequency. This is because uncertainties in the refractive index of air and laser beam alignment are largely canceled in a frequency difference measurement. Using the trapped atoms as a frequency reference, we have measured the wave numbers of the 0 1 and 0 2 transitions of **'Fr (Table 11). While the wave number of the D 1transition measured here is in agreement with the number measured by the ISOLDE Collaboration [ 181, our value of the 0 2 transition is lower by 3 ~ T D As . a check on our calibration. we measured the wave number of the TABLE I. Various rates, fluxes, and efficiencies on Dec. 1718, when the source strength was 9 pCi. Total efficiency is 0.4%.
(1)
(2) (3) (4)
Stage
22'Fr Flux (s-')
Produced in oven Exit oven Enter cell Collected into trap
3.3 x 105 7 x lo3 2.3 x 103
"1.3 x 103
Efficiency 2% of (1) 33% of (2) 56% of (3)
'Deduced from flux entering cell and trap efficiency.
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PHYSICAL REVIEW LETTERS
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TABLE II. The hyperfine constants and wave numbers of 221Fr. The wave numbers of the previous work were derived from the published wave numbers of '"Fr and isotope shifts [ 181. ISOLDE [IS] A (7f'3/2) MHz B (7P3p) MHz A I PI/?) M T z v (01)cinv (02) c m - '
65.4(2.9) -259(16) 808(12) 12 236.6601(20) 13 923.2 1 1 8(20)
Current work
66.5(0.9) - 260(4.8)
81 l.O(l.3) 12 236.6579( 1 7) 13923.2041(17)
f2 line that is only 0.5 GHz away from the Lll trapping transition and found an excellent agreement (difference = c n - l ) with the I? atlas [16]. l(3) X We have demonstrated and analyzed the dynamics of a highly efficient collection of short-lived radioactive alkali atoms in an optical trap. The techniques used here, including the optimized VCMOT, the orthotropic source, and sensitive detection, can be easily applied to other alkalis as well [19]. Future applications will likely employ immediate, efficient transfer of trapped atoms out of the cell into a much longer-lived trap [20]. By combining a stronger and currently available 221Frsource (flux - 106/s) with a double-MOT system (trap lifetime - I O2 s), a sample of I Ox trapped 22'Fratoms could be prepared for the next generation of high precision spectroscopy measurements to test fundamental symmetries. We thank the following: Lane Bray and Tom Tenforde for supplying 225Ac,Ken Gregorich for electroplating the 2 2 5 A ~Albert , Ghiorso for showing us how to work with Fr, Michelle Stephens for building the lasers and teaching us the art of cell coatings, and Dave Alchenberg for the design and construction of much of our vacuum system. This work was supported by NSF, ONR, and the Office of Energy Research, Office of Basic Energy Sciences, of the U.S. DOE under Contract No. DE-AC03-76SF00098.
[I] [2] [3] [4]
Z.-T. Lu et d., Phys. Rev. Lett. 72, 3791 (1994). J . A . Behr et u / . , Phys. Rev. Lctt. 79, 375 (1997). C . Cwinner er a/.. Phys. Rev. Lett. 72, 3795 (1994). J. E. Sinisariati e f al., Phys. Rev. Lett. 76, 3522 (1996); L. A . Orozco (private communication). [5] V. A . Dzuba, V. V. Flanibaum, and 0.P. Sushko\', Phys. Rev. A 51, 3454 (1995). [6] P. G . H. Sandars, Phys. Lctt. 22, 290 (1966). [7] W.D. Phillips and H. Mctcalf, Phys. Rev. Lett. 48, 596 (1982); W. Erlmer et NI., Phys. Rev. Lett. 54, 996 ( I 9S5). [S] C. Monroe e f al., Phys. Rev. Lctt. 65, 1571 (1990). [9] M. Stephens and C. Wienian, Phys. Rev. Lett. 7 2 , 3787 (1994). [ l o ] R. Guckert et LII., Nucl. Instrum. Methods Phys. Rcs., Sect. B 126, 383 (1997). [ 1 11 K. Lindquisl, M. Stephens, and C. Wicman, Phys. Rev. A 46, 4082 (1992). [I21 M. Stephens, R. Rhodes, and C. Wieman, J. Appl. Phys. 76, 3479 (1994). [I31 Reference [9] defines a recapture efficiency which is in general lower than 7 , but equal to 7 in the limit of a small collisional loss rate (C