Atoms in Electromagnetic Fields

AI^WSIN ELECTROMAGNETJC World Scientific Series on Atomic, Molecular and Optical Physics - Vol. 3

Second Edition

FIELDS

SERIES ON ATOMIC, MOLECULAR AND OPTICAL PHYSICS Series Advisors: Raymond Y. Chiao, Steven Chu, C. Cohen-Tannoudji, S. Haroche, T. Y. Li, L. Mandel, F. de Martini, G. W. Series, Y. Z. Wang, E. Yablonovitch Published Vol. 1: Atoms in Electromagnetic Fields C. Cohen-Tannoudji Vol. 2: Semiconductor Quantum Dots LBanyai&S. W.Koch

AT^iSIN ELECTROMAGNETIC World Scientific Series on Atomic, Molecular and Optical Pfiysics - Vol. 3

Second Edition

FIELDS C. Cohen-Tannoudji College de France, Paris

World Scientific

N E W J E R S E Y • LONDON • SINGAPORE • B E I J I N G • S H A N G H A I • HONG KONG • TAIPEI • C H E N N A I

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The author and publisher would like to thank the following publishers of the various journals and books for their assistance and permission to include the selected reprints found in this volume: Academic des Sciences (C. R. Acad. Sci. (Paris)); American Institute of Physics (Phys. Today); American Physical Society (Phys. Rev., Phys. Rev. Lett.); EDP Sciences (Europhys. Lett., J. de Phys., J. Phys.); Elsevier Science Publishers (Phys. Lett., Frontiers in Laser Spectroscopy, Opt. Commun.); IEEE Intellectual Property Rights (IEEEJ. Quant. Electron.); Institute of Physics (J. Phys. B: Atom. Molec. Phys.); John Wiley & Sons, Inc. (Proc. Esfahan Symp., Proc. Int. Conf. Multiphoton Processes); The Nobel Foundation (Les Prix Nobel 1997); The Optical Society of America (J. Opt. Soc. Am.); The Royal Society (Phil. Trans. R. Soc. Lond. A); The Royal Swedish Academy of Sciences (Physica Scripta); Springer-Verlag (Metrologia, Laser Spectroscopy VIII, Z. Phys. D); Taylor & Francis Group (Cargese Lectures in Physics).

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L^C*a^t^aLc,

Co vic^t _ ~TX~~~-^~~ (fig. i) des deux composantes hyperfines de la transition 6'So — 6 ' P , de ""Hg. La figure i represente Ies niveaux d'energie; Ies nombres inscrits, Ies probabilites de transition optique qui ne sont autres que Ies A^. Les transitions virtuelles vers le niveau 6:1P1( F = 1/2, sont caracterisees par k — /cOi > o. D'apres (1,3), AE' est positif et le deplacement, indique sur la partie droite de la figure 1, se fait vers le haut. Par suite de la polarisation circulaire droite, Al/Si1/S = o et seul, le sous-niveau — 1/2 est deplace. Pour les transitions virtuelles vers le niveau 6 s Pi, F = 3/a, k — kai, et par suite AE', sont negatifs : le deplacement, indique sur la partie gauche

9 (

2

)

do la figure i, se fait vors le bas et cst trois fois plus grand pour lo sousniveau + 1/2 quo pour le sous-niveau — 1/2 (rapport des probabilities do transition). Ces deux effets conduisent done tous deux a augmenter la separation energetique entre les deux sous-niveaux i 1/2 de l'etat fondamental. Les conclusions sont inversees si au lieu d'operer en excitation IJ+, on opere en n". Les transitions virtuelles vers les autres niveaux excites ne deplaccnt pas la raie de resonance magnetique car k — k« et AE' ne varient plus alors d'une composante hyperfine a 1'autre et, d'apres les regies de somme sur les A^i, les deux sous-niveaux ± 1/2 sont deplaces de la memo quantite. L'experience a ete realisee sur le montage de Cagnac (~). Des courbes de resonance magnetique de 1'isotope ""'Hg pompe optiquement par une lampe a 20l Hg ont ete tracees suivant les techniques habituelles. On operait a une frequence fixe de 5 kHz obtenue par demultiplication a partir d'un quartz de 100 kHz. Dans l'axe du champ magnetique et dans le sens oppose a celui du premier faisceau orientateur, nous avons dispose un deuxieme faisceau : la lumiere issue d'une lampe remplie de J '"Hg est concentree sur la cellule de resonance apres avoir traverse successivement un filtre a '"'Hg (constitue par une cellule cubique de 4 cm de cote remplie de cet isotope), un nicol et une lame quart-d'onde. Le role du filtre a 1!1"Hg est d'absorber toute fraction de la lumiere du deuxieme faisceau susceptible de provoquer des transitions reelles dans la cellule de resonance. L'isotope ->01Hg a ete choisi pour remplir la lampe du deuxieme faisceau excitateur car on doit obtenir ainsi un deplacement partieulierement fort. On peut en effet, d'apres (I, 3), etudier comment varie AE' en fonction de l'ecart entre le centre k de la raie excitatrice et le centre ka de la raie d'absorption. En comparant AE' a l'elargissement lv 0 que creerait la memo intensite excitatrice si elle etait centree en k0 et non plus en k, on troiivc que AE'/Av0, nul pour k = k0, croit pour atteindre un maximum de Fordre de 1 pour k — ka ~ A (A est la largeur de la raie excitatrice), puis decroit comme A//c — /c0 pour k — /c0 ^> A. D'apres la figure 2, la composante 5/2 de 3l"Hg est a une largeur Doppler de la composante 1/2 de ''"Hg, ce qui donne l'effet optimal; la composante 3/2 agit dans le meme sens; la composante 1/2 dans le sens contraire, mais beaucoup plus faiblement parce qu'elle est trop proche de la composante 3/2 de 191)Hg et quo son intensite dans la lampe est cinq fois plus faible que celle de la composante 5/2. La figure 3 montre un exemple dos courbes experimentales obtenues. La courbe de resonance magnetique du centre est prise avec le deuxieme faisceau masque, celles de gauche (et de di'oite) en presence du deuxieme faisceau polarise a+ (et a~). Le deplacement vaut environ o,4 Hz. II a le bon signe : comme nous operons a frequence fixe, un deplacement vers les differences d'energie plus grandes correspond a un deplacement vers

]0 ( 3 ) lcs champs plus bas. II change do signc quand on passe do n¥ a ?'. Nous avons egalement verifie qu'il n'y avait plus do deplacement quand on enlevait la lame quart-d'onde, e'est-a-dire quand on eclairait avec un melange de r:+ et n~. Enfin nous nous sommes assures que le deplacement etait proportionnel a l'intensite lumineuse.

Fig. 3.

On remarque sur la figure 3 que l'intensite de la resonance est plus faible lorsque los deux faisceaux ont des polarisations opposees : a+ pour le premier, n~ pour le deuxieme. En effet, dans ce cas, les atomes sont accumules par le premier faisceau dans le sous-niveau + 1/2 et la vapeur est particulierement absorbante pour toute fraction du deuxieme faisceau susceptible de provoquer des transitions reelles et que le filtre a ' " H g n'absorberait pas entiei-ement. D'ou un affaiblissement de la detection de la resonance. Nous avons verifie cette hypothese en operant avec des filtres a ' ""Hg mo ins absorbants, ce qui augmente la dissymetrie. D'autre part, nous avons inverse le sens de la dissymetrie en pompant avec le premier faisceau en n" et non plus en n*. L'experience decrite dans cette Communication fait appel a la polarisation de la lumiere pour deplacer de fagon differente deux sous-niveaux Zeeman d'un memo nivean hyperfin de 1'etat fondamental (inegalite des A^). On peut imaginer d'atitres experiences ou il n'est pas besoin de faire appel a cette propriete et ou l'ecart des deux niveaux est suffisamment grand pour que AE' varie de 1'un a l'autre (cas des deux niveaux hyperfins d'un atomc alcalin dans l'etat fondamental). (*) Seance du 9 Janvier 19C1. (') J.-P. BARHAT et C. COHEN-TANNOUDJI, Comptes rendux, 252, 1961, p. g3 (I); 252, 1961, p. a55 (II). (-) B. CAGNAC et J. BDOSSEL, Comples rendus, 249, jip)FFT^ RUV) ] • (1. 12) *.« As in this rotation "Y,*' is simply multiplied by e'"v, we have

SHIFTS...

971

It looks like the Stark Hamiltonian describing the second-order effect on the ground state produced by a fictitious static field E/, parallel to the polarization vector. 3CC (F) removes only partially the Zeeman degeneracy. The energy shift is the same for the m and the -m sublevels: um = b[3n? - F(F + 1)] .

(1.18)

This is a direct consequence of the invariance of the light beam in a plane reflection. c. F > j . Nonpolarized beam. A nonpolarized beam (intensity *) is a superposition of two incoherent beams of equal intensities i * , linearly polarized at right angle to each other. If the beam propagates along the z direction, the corresponding effective Hamiltonian is 3Ce(F) = | 6 [ 3 F / - F ( F + l ) ] + l 6 [ 3 F / - F ( F + l)] , (1.19) which can be expressed as (1.20)

Ke(F)=-ib[3Ff-F(F+l)].

This result can also be obtained from the invariance properties of the light beam under rotations around the z axis and xOz plane reflection. d. F > | . Circularly polarized beam. B t is parallel to the z axis. The rotational invariance of Bl around Oz gives Xe (F) = c^(F) FFT(OV +c^> (F) = aF,+b[3F,z-F{F

FFT^>

+ l)].

(1.21)

The circularly polarized light beam is not invariant any more under a plane reflection and its effect inside a level F>j is described by a fictitious magThe invariance requirement Ke(F) =3Ce (F) implies netic field Hf and a fictitious electric field Ef, c'J" {F) = ciQk> (F)e'iJ* ) (F)s-'«'] "T™.

(1.13)

16

972

C.

C O H E N - T A N N O U D J I AND J .

i s split into a a* and a o~ component, located on / of the vapor. either side of the absorption line An example is shown on Fig. 2. (A similar situation exists for the <J> =§- component of 201Hg excited by 204Hg and for the

. The expression (1.1) of We{F) can be transformed in the following way: K (F) = AB' IF) P , ( ejf • 5 ) ( S 0 P.)( e x • D) PF . We then use the relation

(1< 2 5 )

i-wti P2 line of a 85Rb lamp. As shown in Fig. 3, the e5Rb hyperfine lines lie just

between the two hyperfine components of 87Rb. As a consequence of the small mismatch k -kFt, the wings of the nr lines usually contain resonant wavelengths, which produce a broadening of the ground state. This resonant light is suppressed by a filter, filled with the same element as the resonance cell, and placed in front of it. The choice of the filter temperature results from a compromise: at high temperature, the filter absorbs also a fraction of the nr light and diminishes the magnitude of the shift; at a too low temperature, the resonant wavelengths are not absorbed and the ground-state sublevels are broadened. We operate near the temperature which optimizes the ratio of the shift to the width of the sublevels. To obtain large shifts, a high intensity is required for the nr beam. The size of the discharge lamps is rather large (disks of 3 cm in diameter for mercury lamps, 5 cm for Rb lamps). They are filled with neon or argon buffer gas and excited with a powerful microwave generator (more than 150 W). Polarizers are avoided if possible. For 199Hg the arrangement, described in Sec. I B 2 e , is used. In the case of 201Hg, fictitious electric fields are produced by nonpolarized beams instead of linearly polarized ones. We use circular polarizers only for the experiments with Rb. The magnitude of the shift is found to depend critically on the operating conditions of the lamps. From one day to another, fluctuations of the order of 10% are observed on the absolute value of the shift. But during a day, it was possible to keep its value constant within a few percent.

FIG. 3. Hyperfine components of the £>2 line of 87Rb and 85Rb (the hfs in the excited state is negligible compared to the Doppler width).

A diagram of the setup for the experiment on mercury is given in Fig. 4, and that for the experi-

A. Experimental Setup

1. Nonresonant Light Beam

2. Other Parts of Setup

18 974

C.

COHEN-TANNOUDJI

AND J .

DUPONT-ROC

!5

can be reconstructed. In this section, we describe different methods for measuring the Bohr frequencies of atoms irradiated by a nr light beam. These measurements a r e performed in zero or nonzero external magnetic field. Various optical pumping techniques are used: transients, level crossings, different types of resonances, etc. Before describing these methods, let us briefly recall a few results concerning the evolution of the density matrix in an optical-pumping experiment. FIG. 5. Setup for experiments on Rb.

ment on rubidium in Fig. 5. Besides the nonresonant shifting beam Bu a second light beam Bz is needed. Bz is resonant and its intensity is sufficiently weak so as not to broaden the ground state. Bz optically pumps the atoms and allows the measurement of their Bohr frequencies in the presence of B,. The resonances a r e detected by monitoring the absorption of Bz by the resonance cell. In the experiment on Hg (Fig. 4), the stray light due to the reflection of the intense B^ light on the walls of the cell i s important and produces an appreciable noise compared to the weak signal detected on Bz. We eliminate it by collecting the transmitted light on a second cell C' which absorbs only the resonant light coming from B 2 (the nr light coming from B1 goes through). A photomultiplier PM measures the light reemitted by c' at right angles. The Hg or Rb atoms are contained in cells .„ , , ... ... mu -J^ c^ , , with long relaxation times. The width of the levels is then much smaller than the shift produced by By. For mercury isotopes, the fused quartz cells a r e heated to about 300 °C in order to increase the r e laxation time. 2 2 Paraffin-wall-coated cells are used for 87 Rb. 23 The currently obtained results concerning the shifts 6 '/2TT and the widths T1/2v are summarized in Table II. Such long relaxation times imply a high sensitivity to the magnetic noise present in the laboratory. ... . ,. , . , , „ _ j.. We remove it by using magnetic shields. For the . . .? i XT tA mercury experiments, a three-layer Netic and ., ,, , , . . . . ,_•• t Conetic shield, with a total dynamic shielding fac. „„ . ' . . . ,., ,. t. . tor of 60 is sufficient (the gyromagnetic ratio is a . _ .. • i TJV, , nuclear one). For the experiments on Rb, a large five-layer mumetal shield provides a good protection with a dynamic shielding factor of 10 4 -10 5 , depending on the direction. Inside the shield, three orthogonal sets of Helmholtz coils can produce fields in all the directions and are used to compensate for the residual field. B. Experimental Procedures

L

of Dengity

Matyix

Atoms in the ground state are described by the density matrix a. The diagonal element a,„ =< a la I a > is the population of the a sublevel; a a 6 =< a la IP > is the "coherence" between the a and P sublevels. The evolution of a is due to three processes: effect of the ground-state Hamiltonian (including Ke), relaxation, and optical pumping, a. Effect of ground-state Hamiltonian. The total Hamiltonian X i s JC = K + >C ?C« is the effective Hamiltonian describing the effeet of the nr light irradiation, and Km is the Zeeman Hamiltonian in an applied external magnetic field, The corresponding rate of variation of a is ^ u> ~aTf~ a=-z[7C, a ] . (2.1) T, If

, , .. . . , , „ ... I a ) are the eigenstates of JC, with energy ua, . we have d ~fa <Jas= - i^aea0Le

(2-2)

,

^"^ w « ~ ^s • (2- 3 ) populations of the I a > states do not change; coherence aaB evolves at the Bohr frequency ,^

T h e t n e

^ "

b. Relaxation. The atoms are thermalized by . \ various relaxation processes (collision on the walls, r ; essentially). The corresponding evolution of the L. , .,.**. ± * ,• J* density matrix is described by a set of linear diff . J ferential equations, which may be written formally ^ ' J

TABLE n

E x p e r i m e n t a l r e s u l t s concerning

t l n g 6, i n z e r o f i e l d a n d t h e w i d t h r

the split-

of the levels.

= Atom I9S>Hg 201rTp.

The only measurable quantities are the differences ^0.8= B « - ^s between the Zeeman energy levels; CDaS/2ir is a Bohr frequency of the system. When all the <jL>aB are known, the energy-level pattern

EvoluHm

s 87Rb

State 6's o (/=i) fi^S

(T= ~ )

° 2 52Si/2ir = 2 F=1

6'/2ir(Hz) 5 ^

15 10

T'/2w(Hz) 0.3 0

?

S/V 16 -i c;

3

5

2 3

4 5

-

19 5.

EXPERIMENTAL

STUDY OF Z E E M A N

d is a l i n e a r o p e r a t o r in the Liouville s p a c e . We assume that uaB«kB® (kB Boltzmann constant _. > F1C fi , .,., • ® temperature), so that the thermal equilibrium is , r^/«> l , , , i o=l [£>(l)=0]. In general, several relaxation time . 1 ... . , constants appear in the evolution of o (eigenvalues , _. T j . . , . , .. , „ . ,. . of £>). In order to simplify the following discussion, ' , ,, , , , f we will assume that all these time constants are , , , equal. A more realistic calculation can be per, , , , formed. The results a r e qualitatively the same, but the algebra is more complicated. Thus Eq. (2. 4) becomes ^ (o) are of the order . _ , . , .;' . are used to determine the position of the crossing of T, (pumping time associated1 with Bt). We asF r o m ^ p o s i t i o n of ^ level.crossi sume a weak pumping (1/r ). Accordingly, reS onances, the splitting in zero field is deduced 4. , c t «. r, the broadening of the levels due to the pumping ... ,, , ' . i u t l . ,f , . ± j. j ,. x with the help of the theoretical form of the Zeebeam is small, and the orientation and alignment ,. . ,.,„ . ,. ' . man diagram. A different way to detect the same are weak (a- 1 very small). We will therefore „ . ... , ,. . , . . .. . . ., , • \ fi,u effect with a better signal to noise ratio is described approximate (&. b) by in gec n B 3 T h e g e l e v e l . c r o s s i n g resonances in 1 ei ,„ ,,•, the ground state are similar to the well-known rf(1) dt Tb "Franken resonances" 24 observed on the fluorescent light emitted from two crossing excited sublevels. "V describes the state of an initially disoriented T h e s p l i t t i n g i n z e r o f i e l d c a n a l s o b e determined atom after an optical pumping cycle. iThe replacem o r e d i r e c t l y b y a transient experiment. The ment of (2. 6) by (2. 7) implies that this atom will a t o m g a r e t r a n s v e r s e l y p Unl p e d in zero field, the n r b e a m B^ b e i n g o f f g ^ ^ a ] I t h e w ^ a r e zerQ> be thermalized before undergoing another pumping cycle. ] The total population of the ground state is w e h a v e i n s t e a d y . s t a t e conditions constant, so that T r r a , = 0. (2.8) ^(0)= ( l / r ' T , ) « a + l . (2.13) ex

If a has only diagonal matrix elements, optical pumping is said to be "longitudinal. " If ' ff has also nondiagonal matrix elements, we have ''transverse" optical pumping which introduces "coherence" between energy sublevels. d. Master equation. It can be shown that the total rate of variation of a is simply d da, d (a, d(3,

dt°=dl

°+di

° + Tt °>

. r,n n _ , . v „/>.. =-,[3Cfa]+r'(l-a)+(l/r,)»a. This gives for the evolution of aaB d , 1 ex , dJ(JM=( r +iwc,ti)a)- ^ eitff r " T'TP a T' + i<j)ae, in the following way:

n

Q

^ B (0=K s (0)-ff £ , s (=o)]e- r ' t e- i ™-«'

+

\^\

a. s (=o).

(2.15) Each coherence undergoes a damped oscillation at its Bohr frequency. H r ' « c o a 6 , i. e., if B, has a sufficient intensity to produce splittings larger than r i n zerQ j . ^ several oscillations at the frequen. cy 0)^3 can be detected on the absorbed light, with an appreciable amplitude since a o6 (=°)^0.

20 976

C. COHEN-TANNOUDJI AND J. 3. Resonance Methods

must be replaced by (2.16) "ffaa="° r a8 +°%o%eiat+'*<JiJ)e-i"t and aa$ undergoes a forced oscillation at the frequency . The pumping rate is modulated at the frequency u/2n (the initial polarization is , . . . , , . , T „ ,„ , n ex restored after half a turn). In Eq. (2.11), "aai

oS~

DUPONT-ROC

(2 17)

When u, is swept around o,Bft I a a6 1 undergoes a

resonant variation centered at «o= « . . . The pumping efficiency and a aB are both modulated at the frequency » r"), by means of an rf field Hi coso,/ parallel to Ho.

= ua(H0)+ga cojcosoi/ ,

(2.20)

where 0)!= - yH^ Consequently, the rate of variation of aaB is d , , . , 1 el ^ « f l - " l r +*w«s("o)+^«s