EDITORIAL ADVISORY BOARD
G. S. Agarwal,
Ahmedabad, India
G. Agrawal,
Rochester, USA
T. Asakura,
Sapporo, Japan
A. Aspect,
Orsay, France
M.V Berry,
Bristol, England
A.T. Friberg,
Stockholm, Sweden
V L. Ginzburg,
Moscow, Russia
E Gori,
Rome, Italy
A. Kujawski,
Warsaw, Poland
L.M. Narducci,
Philadelphia, USA
J. Pefina,
Olomouc, Czech Republic
R. M. Sillitto,
Edinburgh, Scotland
H. Walther,
Garching, Germany
Preface This volume presents six review articles devoted to various topics of current interest both in classical and in quantum optics. The first article, by S.Ya. Kilin, entitled Quanta and information, is concerned with a multidisciplinary subject which involves optics, information theory, programming and discrete mathematics. It contains contributions from areas such as computing, teleportation, quantum cryptography and decoherence. The article presents an account of recent results obtained in this relatively new field. The second article. Optical solitons in periodic media with resonant and off-resonant nonlinearities, by G. Kurizki, A.E. Kozhekin, T. Opatrny and B. Malomed, reviews the properties of optical solitons in periodic nonlinear media. The emphasis is on solitons in periodically refractive media (Bragg gratings), incorporating a periodic set of thin layers of two-level systems resonantly interacting with the field. Such media support a variety of bright and dark 'gap solitons' propagating in the band gaps of the Bragg gratings, as well as their multi-dimensional analogs (light bullets). These novel gap solitons differ substantially from their counterparts in periodic media with either cubic or quadratic off-resonant nonlinearities. The article which follows, entitled Quantum Zeno and inverse quantum Zeno effects, by P. Facchi and S. Pascazio, deals with an effect and its inverse which is a manifestation of hindrance and enhancement, respectively, of the evolution of a quantum system by an external agent, such as a detection apparatus. The article includes some examples from quantum optics and quantum electrodynamics. The fourth article, by M.S. Soskin and M.V Vasnetsov, discusses the current status of a relatively new branch of physical optics, sometimes called singular optics. It is concerned with effects associated with phase singularities of wavefields. Wavefronts in the neighborhood of such points exhibit dislocations, optical vortices and other features which are not present in commonly encountered wavefields which have smooth wavefronts. The next article, by G. Jaeger and A.V Sergienko, presents a review of advances in two-photon interferometry and their relation to investigations of the foundations of quantum theory. A recent history of tests of Bell's inequality and the production of entangled photon pairs for testing it is given that illustrates the central role of spontaneous parametric down-conversion in current two-photon
vi
Preface
interferometry. Quantum imaging and quantum teleportation are used to illustrate the current power of entanglement in advanced quantum-optical applications. New, increasingly efficient sources of entangled photon states are described and the manner in which they will assure further progress in multiple-particle interferometry is discussed. Multiple-photon entanglement is shown to provide a new set of phenomena to be investigated in the future by multiple-photon interferometry. The concluding article, by R. Oron, N. Davidson, A.A. Friesem and E. Hasman, is concerned with transverse mode shaping and selection in laser resonators. It presents a review of recent investigations on the shaping and selection of laser modes, by the use of various elements that are inserted into laser resonators. Experimental techniques, as well as basic numerical and analytical methods, are presented. The qualities of the emerging beams, based on different criteria, are discussed, along with various applications for specially designed beams. I wish to use this opportunity to welcome three new members to the Editorial Advisory Board of Progress in Optics, namely Professor G. Agrawal (Rochester), Professor A.T. Friberg (Stockholm) and Professor L.M. Narducci (Philadelphia). Emil Wolf Department of Physics and Astronomy University of Rochester Rochester, New York 14627, USA July 2001
E. Wolf, Progress in Optics 42 © 2001 Elsevier Science B.V. All rights reserved
Chapter 1
Quanta and information by
Sergei Ya. Kilin Quantum Optics Lab, B.I. Stepanov Institute of Physics, National Academy of Sciences of Belarus, Francisc Skarina Avenue, 70, 220072, Minsk, Belarus
Contents
Page Introduction
3
§ 1. The quantum concept
3
§ 2.
Information
16
§ 3. Quantum information
30
§ 4.
The problem of decoherence
74
§ 5.
Conclusions
82
Acknowledgements
84
References
84
Introduction The first volume of the series Progress in Optics, launched and edited by Emil Wolf, contained a chapter by Dennis Gabor [1961] entitled "Light and Information". It was a record of his lecture presented at the University of Edinburgh in 1951. In an effort to answer the question of what information theory can contribute to the physics of light, Gabor suggested a program: "Information theory is of some heuristic use in physics, by asking the right sort of questions''. He also noted that information theory "prepares the mind for quantum theory", and that "in information theory we appear to have the right tool for introducing the quantum point of view in classical physics". These statements, which suggest a potentially useful program, sound very contemporary today. Before implementing the program, however, one important step should be taken, namely the need to realize that quantum objects are radically new objects to information theory, with new information resources, inaccessible for classical objects. In this chapter I will endeavor to show briefly how both "programs" - "Quantum Physics for Information Theory" and "Information Theory for Quantum Physics" - are working together to fiirther progress in understanding the world and to offer new practical applications in technology.
§ 1. The quantum concept The rapid development of quantum optics at the end of the twentieth century, has caused many individuals, not only specialists in quantum physics but also people working far fi-om this field, to appreciate once more the basic statements of quantum theory. Indeed, the abstract basic ideas of quantum physics, to which only a few specialists paid attention not long ago, are now important for almost everyone because of their new applications in technology and, primarily, in optical applications. Quantum computers, quantum teleportation and cryptography, observation and monitoring of single atoms, ions, and molecules, including biological molecules, all belong to the quantum world. This world is extremely difficult to explain in terms of the common classical world of
4
Quanta and information
[1, § 1
macroscopic physics. For its description, it requires a proper definition of quantum mechanics and quantum field theory. Quantum mechanics, which originated in the 1920s by the investigations of Niels Bohr (1885-1962), Erwin Schrodinger (1887-1961) and Werner Heisenberg (1901-1976), provided physicists with the recipes for calculating the energy states of atoms and molecules and the matrix elements of transitions between these states. However, in addition to this aspect which immediately found applications in practical physics, quantum mechanics contained the "ideological", philosophical aspect, which accounts for the odd nature of the quantum world and has remained almost unused until recent years. In the most complete and clear form, sometimes with deliberately paradoxical statements, this part of the quantum theory was presented by Erwin Schrodinger in his famous paper of 1935, which he classified as "a paper or a general confession" {Referat oder Generalbeichte). Using modem terminology, it examines one of the problems of quantum information, namely, what information about the states of quantum objects can we obtain and what happens to the quantum objects while we are obtaining this information? More than half of a century passed before the basic principles formulated by Schrodinger became necessary for an understanding of experiments with practical applications. The present chapter discusses several experiments of this kind. These are, first, experiments on quantum teleportation, quantum cryptography, and, second, quantum computers, which are expected to be extremely beneficial but difficult to construct. Some of the chapter is devoted to single material particles in quantum optics and the methods of their detection. These objects can serve as elements of quantum computers. In conclusion, I examine the problem of decoherence and possible solutions, which is crucial for quantum computation. First, this chapter discusses the language of quantum optics and the statements of quantum theory necessary for fiirther examination.
1.1. Schrodinger and his famous paper of 1935 On November 29, 1935, the journal Die Naturwissenschaften published Erwin Schrodinger's paper "Modem state of quantum mechanics". It was written during his compelled stay in Oxford (fig. 1), after he and Paul Dirac had been awarded the 1933 Nobel prize in physics. As Schrodinger mentioned, his work originated from the discussion started on May 15, 1935 by Albert Einstein, Boris Podolsky and Nathan Rosen in their paper "Can the Quantum Mechanical Description of Reality be Complete?", and continued in Niels Bohr's [1935] paper with the
1, § 1]
The quantum concept
'^w.
Fig. 1. Erwin Schrodinger was bom in Vienna, where he studied at first in a gymnasium, then in the university until graduation in 1910. Schrodinger started working in theoretical physics and soon became a professor in Breslau (now Wroclaw) and then in Zurich, where Einstein had worked earlier. In Zurich, Schrodinger published works that led him to a formulation of the basic equation of nonrelativistic quantum mechanics, the Schrodinger wave equation. For the development of quantum mechanics, Schrodinger together with Dirac was awarded a Nobel prize in 1933. In 1927, he was appointed to the chair of theoretical physics in Berlin, previously headed by Planck. When Hitler attained power, Schrodinger left fascist Germany and accepted an invitation to Oxford. In 1936 he returned to Austria for a short time and held a chair in Graz, but after the Anschluss he had to leave his country again. This time Schrodinger moved to Ireland, to the Institute of Fundamental Research in Dublin. In 1947 he finally returned to his homeland. His health was failing, however, and after a long illness he died in Vienna. [The photograph and biographical note are taken fi-om the anthology Zhizn' Nauki [Science life], edited by S.P. Kapitsa and published by Nauka (Kapitsa [1973]).]
same title. Despite the abstract and complicated style of Schrodinger's paper, its importance was soon realized by Russian scientists, and it was immediately translated into Russian and published in 1936. The English translation did not appear until 1980. In his paper, Schrodinger analyzed difficulties in the quantum-mechanical description of measurement procedures and formulated four basic principles. According to these principles, the states of quantum objects have the following properties: (1) Superposition: A quantum state is described by a linear superposition of the basic states. (2) Interference: The result of measurement depends on the relative phases of the amplitudes in this superposition.
6
Quanta and information
[1» § 1
(3) Entanglement: Complete information about the state of the whole system does not imply complete information about its parts. (4) Nondonability and uncertainty: An unknown quantum state can be neither cloned nor observed without being disturbed. I shall briefly comment on each of these statements. However, let us note first that, until recently, the third and fourth principles were almost unknown to most physicists and were discussed only in connection with the Einstein-PodolskyRosen (EPR) paradox and Bell's inequalities.
1.2. Quantum objects and their states 1.2.1. Superposition and the Schrodinger-cat paradox In contrast to a classical object, a quantum object has statistical origin. However, the probabilistic nature of a quantum object cannot be understood as a classical uncertainty connected, for instance, with incomplete knowledge about the object. For the description of a quantum object, the concept of state is used. By saying that an object is in a quantum state, we mean that there is a list (a catalog, in Schrodinger's language) or, similarly, a wave ftinction, a state vector, or a density matrix containing the information about the possible results of measurement on this object. In the general case, the results of measurement differ fi-om time to time even if the object is prepared in the same quantum state. Hence, the state vector should give statistical information, i.e., distribution functions for the results of measurement. As a simple example, consider the state vector for a system with two orthogonal basic states |1) and |2), e.g. energy states. The state of the object is described by the state vector (wave function) |^) = a|l)+^|2),
(1.1)
where a and /? are complex numbers. In other words, the total state is given by a linear superposition, and the squares of the absolute values of the amplitudes a and (3 are equal to the probabilities of finding the system in the corresponding states (|a|^ + |j8p = 1). As a result of measurement, the coherent superposition (1) is destroyed and reduced to a new state, which is determined by the type of measurement. For instance, an attempt to find the system in state 12) leads to its perturbation by the measurement device. At the moment of measurement, reduction (projection) takes place, |^)^|2)(2|^^)=^|2),
(1.2)
1, § 1]
The quantum concept
7
SO that after the measurement the system is driven into state |2) and the initial state is destroyed ^ A superposition state should be distinguished from a mixed state, which is described by the density matrix Pn,,x = | a | ' | l > ( l | + |i8|'|2)(2|.
(1.3)
In fact, state (1.3) is a classical state, since a system in a mixed state can be found either in state 11) or in state |2), whereas in the superposition state (1.1) the system can be simultaneously found in two states. This principal feature of a superposition state manifests itself in the interference terms of its density matrix p = | W ) ( i I / | = | a | 2 | i ) ( i | + ||3f|2)(2| + a r | l ) ( 2 | + a*/?|2>(l|.
(1.4)
To stress the unusual nature of superposition states, Schrodinger suggests an example disturbing to our common sense. Following Schrodinger, suppose that a steel chamber contains a flask with poison that can be broken by means of some mechanism triggered by the radioactive decay of a single atom. The box also contains a cat (initially alive), which can die as a result of the atom's decay. Similarly to the atom whose state is a superposition of the decayed and nondecayed states, the state of the cat is also given by the superposition of the states of an alive cat, 11), and a dead cat | i ) : | ^^) =L| T ) + ^ | i )• Since quantum superposition states are frequently observed for microscopic systems, such as atoms and molecules, but never observed for macroscopic systems, some effect must be destroying the Schrodinger-cat states for macroscopic systems. This effect, which is called decoherence, is considered below. Note that the problem of pertaining superposition (Schrodinger cat) states for mesoscopic systems is crucial, and its solution will give rise to many applications of quantum information. For further consideration, the superposition state is used to describe a singlephoton beam with a given wave vector, or a single-photon state 11 photon)The state of radiation with a given wave vector can be represented by a
^ Note that measurement, i.e., interaction with a macroscopic measurement device, is an irreversible process in principle. During this process, the state of the measured object changes (reduction takes place). Reduction, like other physical processes, has its own characteristic time scale, specific for each individual measurement. The process of reduction is very short, however, so the question of its internal dynamics, i.e., of the possibility to 'see it with one's own eyes', is usually ignored, although in some measurements, for instance in quantum tomography of ultrashort pulses, it is obviously of interest.
Quanta and information
|2>j|0)^=|W>
|l>l|l>«=|t^>
|i>t|o>«=|J>
[1, § 1
I0>j|2>„=|««>
l«>t|iL=l«> l«>l|0>«=|0>
Fig. 2. A light beam with a fixed wave vector is equivalent to two harmonic oscillators corresponding to two orthogonally polarized modes of the electromagnetic field. A single-photon state of this beam is given by a superposition of two energy-degenerate states of polarized photons | J) and | ^ ) . A twophoton state of this beam is generally a superposition of three energy-degenerate states, two of which represent pairs of photons with equal polarizations | I | ) and | ^ ^ ) , and the third representing a pair of orthogonally polarized photons \ll I) ^ I^FV)| n),\Rl)\
- > ^ |i?FH)| — ) ,
where \Ri) is the initial state of the cloning device and \Rv\), |^FH) are its final states after cloning photons with vertical and horizontal polarizations. In other words, instead of a single photon with a given polarization, we obtain two photons with the same polarization (fig. 2). However, if we try to clone
16
Quanta and information
[1? § 2
a photon with a polarization that is neither horizontal nor vertical, for instance, a II) +iS |7r (an effective nonlinear interaction length equal to unity) are |a/3)^., = v/A, |A,), m. + V^- |A^). I-/5),.
(3.20)
Therefore, depending on the ancillary field output (|jS)^ or |-/?)^), one can transform (encode) the input signal state |a) into even |A+}^ or odd |A_)^ coherent states. By means of two consecutive transformations realized on two (fig. 12a) or one (fig. 12b) nonlinear mirrors (X\ = X2 = ^ ) , the output of signal fields a and c will be conditioned by the output of the ancillary field:
|A„,>„ |A,,)^ + , y V V |A„_), |A,_>^) 1^), + ( Y Aa^Ay+ \^a-)a \^r+)c + y Aa+Ay- \K+)a \^r-)c 1 \~P)b
= H(l«)Jr). + l-«)J-y>c)l^>. + (Klr).-l-«)J-yUl-^>J. (3.21) These states of two signal fields are in an entangled state analogous to the polarization entangled states (1.8): |0^} when the ancillary field state is \P)b, or |0~) when the ancillary field state is hiS)^. Note that by introducing a phase shift (p = JT, WQ can obtain entangled states analogous to | ^ ^ ) (eq. 1.8c) and \^~) (eq. 1.8d). This example clearly demonstrates how initially independent particles become entangled and that the real price for entanglement is the use of nonlinear resources. Note also that the scheme presented in fig. 12b can generate a chain of entangled states, recently considered by Wootters [2000].
38
[1,§3
Quanta and information
|y)" (a)
(b)
Fig. 12. Scheme of the two consecutive transformations of a string of initially independent coherent field signals |a) • • • |y), realized by (a) two or (b) one nonlinear mirror(s) {X\^ Xi^ ^)- The output of signal fields a and c will be conditioned by the output of the ancillary field b in one of the entangled states (3.21). The round-trip time in the cavity of figure (b) should be equal to the delay time between two input pulses \a) and |y).
3.1.2. Communication of images by coherent states, image recognition and quantum limit of phase space partition 3.1.2.1. Quantum phase space partition theorem. The next example, that is of interest, is the coding and communication of images by coherent signals. The starting point for the coding can be the phase-space-partition-theorem, stating that for the partition of phase space of a harmonic oscillator on cells mn with areas S, the set of coherent states {amn), where Grtjn = mo)\ +no>i, (m,n = 0,1,...)
(3.22)
are the centers of cells with linear independent complex numbers a)\, (JOZ (Im(a;2 ^ i ) ^ 0)» is complete for S = Jt, and the set remains the same if one state is removed from it. Moreover, if S < Jt, the set is overcomplete and it remains the same when the finite number of states are removed. If *S' > ;r, the set is incomplete (Perelomov [1971], Bargmann, Butera, Girardello and Klauder [1971], Bacry, Grossman and Zak [1975]). The theorem conjectures that if Alice wants to encode an image by dividing it into N^ cells (pixels) (for certainty we will consider a square lattice and twograde images) and assigning black pixels to the presence of coherent signals in
1, § 3]
Quantum information
39
the corresponding cells of the phase plane, she can do this without any additional manipulation and resources by choosing a part of the phase plane of area ^ph = JTN^.
(3.23)
Is it possible, however, to squeeze the size ^ph below this value and still retain the ability of image recognition? The phase-space-partition-theorem permits this by discarding some states. Let us see how the problem is solved by means of quantum information methods. 3.1.2.2. Optimal encoding of images. Suppose the images to be encoded belong to the class of equally likely strings of coherent fields defined by the phase space lattice a^ = au = a{k + il)
(|A:|, |/| < L\
(3.24)
without the central pixel (0,0). The single particle density matrix of the lattice is
Ai) = i(El««))/2±jc^i-*V2V4 = x^/2(cosha^i COSa2)/2, (3.29) Ai3= (l-xTx^'^^^/^±x^^-*^/^)/4 = x^/^(sinha2±sina2)/2. The corresponding orthonormalized eigenvectors |Ao,2) = ((\a) + l-a)) ± (\ia) +
\-ia)))/y/X^2,
1^1,3) = ((l«) - |-«)) T (|i«) -
\-ia)))/^X;;,,
(3.30)
are generalized coherent states (Horoshko and Kilin [1997a]), which are also the eigenvalues of the operator exp(ijra+fl/2)a, and which have 4A:, 4A: + 1, 4A: + 2 and 4A: + 3 photons, respectively (k integer), distributed by the Poisson law. Because of the unlikely probabilities of those states, an economic code can be proposed for image transfer (Kilin, Mogilevtsev and Shatokhin [1999]). Note that this method of image encoding should not be confused with investigations on quantum traveling wave imaging (Sokolov, Kolobov and Lugiato [1999]). A much richer structure of the dependence of eigenvalues on pixel size |a|^ arises when the number of pixels is increased. This is illustrated in fig. 13,
1, § 3]
Quantum information
41
where the degeneracy of eigenvalues, that is, their clusterization, is evident for \a\ < 1 and \a\^ > Jt. In the region of \a\^ ^ jr, the eigenvalues become more uniformly distributed, showing in the near vicinity a universal structure ("the sea of information"), which undergoes minor changes with increasing number of pixels A^ in the image. The von Neumann entropy *SV(P(i)) of equally likely distributed images (3.25) per one pixel shows increasing compressibility with decreasing pixel area. An additional resource also appears for communication of images: Alice can send coherent signals without ordering them, but remembering that the distinguishability of signals depends on the distance between them, that is, on their scalar product |(a/ | ay)|. She and Bob can use a protocol by which AHce communicates an image block by block. In the blocks different elements are more distinguishable than the nearest neighbors in the lattice. As the simplest case, Alice can communicate pair-by-pair signals separated by 2a. That is why they become much more distinguishable. The mean value of mutual scalar products I (a/ I aj)\^ over the selected part of the phase plane x = Tr^(p2^)-p(i)/7V)/2
(3.31)
approaches fidelity (3.3) at p = p ' and iV > 1. An estimation of the number of qubits per pixel nia (3.18) with the use of criteria of distinguishability x instead of X points to possible gain for small values of |a| . The quantum data compression discussed in this section optimizes the use of one channel resource, the states of transmitted qubits, but it is possible to transmit an unknown quantum state with perfect fidelity without sending any qubits at all through a communication channel. This process known as quantum teleportation, uses a quantum-mechanical entanglement as a new physical resource. The process is realized by means of local quantum operations over entangled parts shared by Alice and Bob, and an additional classical communication channel between them. The name of this quantum channel is the LOCC-channel. 3.1.3. Quantum teleportation 3.1.3.1. Experimental quantum teleportation. Late in 1997 Anton Zeilinger and his colleagues in Innsbruck (Bouwmeester, Pan, Mattle, Eibl and Zeilinger [1997]) performed an experimental realization of teleportation, the dream of science fiction novelists. The term "teleportation" means that an object disappears at some place and reappears at another place, some distance apart.
42
Quanta and information
[1? § 3
Although the idea of quantum teleportation, that is, of transporting a quantum state from one object to another, had been suggested in 1993 by Charles Bennett and colleagues (Bennett, Brassard, Crepeau, Jozsa, Peres and Wootters [1993]), it was the Innsbruck experiment and other experiments following it (Boschi, Branca, De Martini, Hardy and Popescu [1998], Furusawa, Sorensen, Braunstein, Fuchs, Kimble and Polzik [1998]) that attracted public attention. From the classical viewpoint, teleportation means gaining all possible information about the properties of an object and transposing these properties onto the reconstructed object. This procedure is forbidden in the quantum world, however, because of the above-formulated postulates of projection and destruction of the state during measurement. Another method exists for passing a quantum state from one object to another. In brief, transmission of an unknown quantum state from Alice to Bob is performed as decribed below. Alice has a particle in some unknown quantum state | ^ ) . Teleportation means that Alice destroys the state | W) at her location but some particle at Bob's location is put into the same state ( | ^ ) ) . Neither Bob nor AHce obtains information about the state | W); moreover. Bob does not know that some state was teleported onto his particle. To tell Bob about the teleportation, Alice should use a classical information channel, in which the principal role is played by particles in entangled states. They provide the quantum information channel between Alice and Bob. Suppose that particle 1 (a photon) to be teleported by Alice is initially in the polarization state | ^ ) i = « ||)i +/? | ^ ) i (fig- 14). Alice is connected with Bob by means of photon pairs prepared by an EPR source in an entangled state I '^")23 = (11)2 | - ) 3 - | - ) 2 11)3)/^^-
(3-32)
Photons 2 are sent to Alice and photons 3 are sent to Bob. The joint state of photons 1 and 2 meeting at Alice's station is the product of | ^ ) i and | ^~)23' |^)i|^-)23-|^-)i2(a|-^)3+^||)3)/2+|¥^-)i2(-a|-)3+^II)3)/2 + |^")l2(-^|-)3 + C.|I)3)/2+|0-)i2(^|-)3 + a | I ) 3 ) A (3.33) Consider the wave function (3.33) for three particles, two belonging to Alice and one to Bob. If Alice projects the states of particles 1 and 2 onto the state |y^ )i2, the state of particle 3 at Bob's station is immediately reduced to the state of the first particle, 1^)3 = a l'^)^ + 1^ IDs- In other words, by measuring Bell states formed by mixing photons 1 and 2 on a beam splitter
43
Quantum information
1, § 3 ]
Classical information
->
Bob oo*o
n. in
^" >23-(|:)2l^)3-|^)2l^)3>/^|
= (a\^>,+/^\l>,)/^
Source of EPR photon pairs Fig. 14. Principal scheme of teleportation. Alice is going to transpose the state of particle 1 onto some particle at Bob's station. Alice and Bob obtain photons 2 and 3, which form an EPR pair in the entangled state |^}23- Alice performs the Bell state measurement over particles 1 and 2. This way she also projects the state of particle 3 at Bob's station. In one case of four, detectors Fl and F2 "click" simultaneously, so that Alice knows that the state of particle 3 becomes the same as the initial state of photon 1; that is, that teleportation of the state 1*^)1 occurs. Alice can tell Bob about this through the classical channel. Moreover, if Bob obtains the information through the classical channel and performs an additional unitary transformation over his particle, the state | ^) i will be teleported with 100% probability after each Bell state measurement performed by Alice.
and by registering the coincidences of photocounts from detectors Fl and F2, Alice performs an immediate reduction of photon 3 to the initial state of photon 1, namely, teleportation! Several features of quantum teleportation deserve additional comments. (1) The teleportation procedure does not violate the noncloning theorem for a single quantum object. As soon as Alice performs the Bell state measurement, photon 1 becomes a component of the polarization-entangled pair of photons 1 and 2. Hence, it is no longer an individual particle. Its initial state | ^^) 1 is destroyed. (2) Quantum information can be passed from photon 1 to photon 3 separated by any distance. At present, the largest achieved distance between entangled photons is about 10 kilometers. (3) At the moment of measurement, Alice is aware of the teleportation going on, whereas Bob is not. In fact, teleportation can occur without passing Bob any
(
A J^i^ilpi
Alice
Initial State . ^
•
A^ \ Pump
[1, §3
Quanta and information
44
f
Polarizer
^ fPI^ . ^...•*
R-"
yi
^
e Source of photon pairs
Bob Teleported
P3 ^ / V ?
state
7-V'"""^ii D2 Di
Fig. 15. Scheme of the quantum teleportation experiment (Bouwmeester, Pan, Mattle, Eibl and Zeihnger [1997]). Correlated photons 2 and 3 connecting Alice and Bob were produced by a nonlinear crystal via type-II parametric down-conversion from a UV femtosecond pulsed pump. The reflected pump-generated photon 1, whose state was to be teleported, and photon 4, which was used as a time reference. The Bell state measurement for photons 1 and 2 was performed by mixing them on a beam splitter and then registering by the detectors Fl and F2. The polarization properties of Bob's photon were analyzed by means of a polarizing beam splitter and two detectors Dl and D2.
information about it. Moreover, Alice may not know the state of photon 1 transmitted by her. (4) A classical information channel is required for informing Bob about the teleportation of the unknown state onto photon 3. (5) Suppose that Alice performs a complete Bell state measurement and identifies, in addition to the fermionic state, the three bosonic states, each occurring with a probability of 25%, and sends this information to Bob through the classical channel. Then, by means of an appropriate operation performed over photon 3, Bob can transform its state into the initial state of photon 1 for any result of Alice's measurement. If this procedure is omitted and Alice only projects for the fermionic state, teleportation occurs only in 25% of all trials. This fact has been demonstrated experimentally by Bouwmeester, Pan, Mattle, Eibl and Zeilinger [1997]. Their experimental scheme is shown in fig. 15. Correlated photons 2-3 connecting Alice with Bob were generated by way of type-II parametric down-
1, § 3]
Quantum information
45
conversion in a nonlinear crystal from a UV femtosecond pulsed pump. Photon 1, whose state was to be teleported, was generated from the reflected pump beam. The Bell state measurement fr)r photons 1 and 2 was performed by mixing these photons on a beam splitter and registering coincidences of photocounts from detectors Fl and F2. The polarization properties of Bob's photon were analyzed by means of a polarizing beam splitter and two detectors Dl and D2. Teleportation was experimentally demonstrated by registering coincidences of photocounts from detectors Fl and F2 and one of Bob's detectors (triple coincidences). Suppose that photon 1, which is to be teleported, is polarized at 45° and Bob's polarizing beam splitter is sending -45''-polarized light to detector Dl and +45''-polarized light to detector D2. Then the coincidence of photocounts from Fl and F2 means that photon 3 is polarized at +45*^, that is, a photocount comes from D2 and not from Dl. Hence, if triple coincidence counting rates (D1F1F2) and (D2F1F2) are registered as frmctions of the delay between photons 1 and 2, which is varied by shifting the mirror reflecting the pump, one should expect a gap with complete suppression of coincidences for (D1F1F2) and no dependence for (D2F1F2). Outside the teleportation domain, that is, for delays between photons 1 and 2 so large that these photons hit Fl and F2 independently, the probability of triple coincidences is constant and equal to 50% X 50% = 25% (one 50% is the coincidence probability of photons 1 and 2, the other 50% is the probability that photon 3, which in this case has no definite polarization, hits Dl or D2). The experimental data obtained by Bouwmeester, Pan, Mattle, Eibl and Zeilinger [1997] confirmed these predictions, both for the case of photon 1 polarized at +45"^ (fig. 16a,b) and for the case of photon 1 polarized at -45° (fig. 16c,d). Teleportation was also performed for photons in the superpositions of these polarization states: 0°, 90°, and circularly polarized photons. Note that, despite the relatively low efficiency of teleportation (one out of four attempts), the teleportation fidelity in the first Innsbruck experiment (Bouwmeester, Pan, Mattle, Eibl and Zeilinger [1997]) and in the next experiment (Pan, Bouwmeester, Weinfiirter and Zeilinger [1998]), demonstrating entanglement swapping (Zukowski, Zeilinger, Home and Ekert [1993]), that is, realizing the teleportation of a qubit which itself is still entangled to another one, was relatively high (-0.80) (Bouwmeester, Pan, Weinfiirter and Zeilinger [1999]). At almost the same time as the results of the Innsbruck experiment on teleporation were published, Boschi, Branca, De Martini, Hardy and Popescu [1998] demonstrated their results obtained in the Rome laboratory with a sophisticated scheme of conditioned measurements with pairs of polarizationentangled photons. They managed to realize a complete Bell state measurement
Quanta and information
46
+45°teleportation
-45° teleportation
400
400 c/) o o o
[1, § 3
200
200
CNJ i_
(D Q.
0 O C 0) •g
o c *o o
400
400
_0
200
200
Q.
-100
0
100
Time delay (^m)
-100
0
100
Time delay (|im)
Fig. 16. Triple coincidence counting rates, D1F1F2 (-45°) and D2F1F2 (+45°), as functions of the delay between photons 1 and 2. The delay is varied by moving the mirror reflecting the pump pulses. The teleported photon 1 is polarized at +45° (a,b) and at -45° (c,d).
with the assistance of local operations and a classical communication channel the attributes of a quantum teleportation - but they did not teleport a state of the third unknown object from Alice to Bob. Continuous variables entanglement of the two-mode squeezed state (1.9) with squeezing parameter exp(-2r) = 0.5 was used by Furusawa, Sorensen, Braunstein, Fuchs, Kimble and Polzik [1998] in Pasadena, California, for the first unconditional quantum teleportation of incoming coherent state |vin). The teleportation fidelity for the experiment F = |(Vin | Vin)P was above the classical boundary Fc\ = 0.5. The realization of quantum state teleportation opens up new possibilities for transmitting "fragile" superposition states for large distances without loss of coherence. Solving this problem is crucial for the development of quantum computers, quantum cryptography, and for increasing the communication channel capacity by means of the dense coding method (Bennett and Wiesner [1992]).
1, § 3]
Quantum information
47
The latter allows Alice to communicate to Bob a two-bit classical message by sending only one qubit through the channel, if they both have shared the entangled particles in advance. In this method, Alice codes the message in one of the Bell states (1.8) by a local unitary operation on her particle of the entangled pair, and sends the particle to Bob. Since Bob possesses both particles, he can distinguish among the four mutually orthogonal states. The method was experimentally demonstrated by Mattle, Weinfurter, Kwiat and Zeilinger [1996]. In addition, quantum teleportation is important in connection with some fundamental problems, such as, for instance, information exchange in complex, spatially separated molecular structures, including biological ones. As a first experimental result in this direction, an attempt at total quantum teleportation of the magnetic states of the hydrogen atom to the states of the chlorine atom within a single trichloroethylene molecule, performed by Nielsen, Knill and Laflamme [1998], should be mentioned. The importance of the paper by Bouwmeester, Pan, Mattle, Eibl and Zeilinger [1997] and subsequent experiments in Pasadena and Rome is clear because, since then, the information aspects of quantum mechanics have been treated not only as leading to "gedanken experiments" but also as "practically important". In addition, the teleportation experiments demonstrated that the classical interpretation of quantum mechanics, which is based on the notions of superposition and reduction and which so far predicted correctly the results of experiments, was confirmed once again. Since any quantum mechanical measurement fixes one of the possible realizations arising from the originally prepared state, Alice's measurements ensure that Bob obtains photon 3 in the original state of photon 1. This is only one of the possibilities that appear from the initial state of the three photons, two of which (2 and 3) are originally in an entangled state generated by a common source. Here, one should not forget that in quantum mechanics, the possibilities for arbitrary initial states are not necessarily described by positive probability distribution functions; that is, their description cannot be reduced to the classical probability theory. Of course, an alternative interpretation based on classical probabilities can be found for certain experiments, measurements, and states. At present, it is unclear, however, if this is interpretation possible in the general case. The state-of-the-art knowledge in this field is given in the review by Klyshko [1998]. It is also necessary to mention some new recent proposals and schemes for quantum teleportation: first, a proposal for teleportation of the wave function of a massive particle using entanglement between motional states of a collection of atoms trapped inside cavities and external propagating fields (Parkins and Kimble [1999]); second, a proposal for teleportation of an internal state of
48
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[1? § 3
an atom trapped in a cavity to a second atom trapped in a distant cavity by using the atom cavity entanglement and a projection type of measurement (Bose, Knight, Plenio and Vedral [1999]); note that atom-field entanglement is a source of a variety of unusual effects, such as the quantum instability predicted by Kilin and Shatokhin [1996, 1997a,b]; third, a protocol for swapping of continuous variable entanglement (Van Loock and Braunstein [2000]); fourth, a scheme for dense quantum coding for continuous variables (Braunstein and Kimble [1999]); fifth, a method of conditional photon-number measurement for the improvement of the fidelity of continuous variables teleportation (Opatrny, Kurizki and Welsch [1999]); the same purpose protocol based on a local quantum nondemolition (QND) measurement of the collective excitation number of several continuous variable entangled pairs was proposed by Duan, Griedke, Cirac and Zoller [1999]; and sixth, a new scheme and protocol for continuous variables teleportation based on the entanglement produced by the QND interaction (Horoshko and Kilin [2000]). 3.1.3.2. Quantum teleportation as a new class of physical communication channels and the problem of quantification of entanglement transformations. The successful realizations of quantum teleportation have clearly introduced a new class of physical communication channels: a multipartite quantum system transmitting information by means of entanglement, local quantum operations, and classical communication (LOCC). Here, local operations include any unitary transformations, additions of ancillas, projective measurements, and the discard of parts of the system, each performed by one party on his or her subsystem. Mathematically, the LOCC transformations can be represented as completely positive linear maps that do not increase the trace of the quantum channel density matrix
L{p) = Y,LipLl where the superoperators L/ = ^/ 0 5/ (g) C/ (8) • • • satisfy the relations Y^. L^Li ^ 1. For example, Alice performs a generalized measurement, described by the complete set of operators Ai (^fA'^Ai = 1), and sends the results to Bob, who performs an operation J5,, conditional on the result /. As a consequence of these actions the initial density operator of general system PAB is transformed to the density operator L(pAB) =
'^.BiAiPABAlBt.
The general properties of such LOCC transformations of entangled states have been the subject of extensive work in recent years. The problem was
1, § 3]
Quantum information
49
introduced by three papers (Bennett, Bernstein, Popescu and Schumacher [1996], Bennett, Brassard, Popescu, Schumacher, SmoHn and Wootters [1996], Bennett, DiVincenzo, Smolin and Wootters [1996]). The authors have studied entanglement distillation, solving the problem of transforming some given pure state into (approximate) EPR pairs in the asymptotic limit, where many identical copies of the pure state are initially available. The inverse procedure of entanglement formation solving the problem of transforming EPR pairs into many (approximate) copies of some given pure state, again in asymptotic limit, was also studied. In these investigations, the problem was also generalized to asymptotic and approximate transformations between mixed states and EPR pairs. An important result in this direction was obtained by Nielsen [1999], who proved the theorem {Nielsen s theorem) that any pure state 11/^) of a composite system AB transforms to another pure state \(p) using LOCC transformations if, and only if, the ordered set of the eigenvalues of Alice's initial density matrix PA (t/^) = Tr^ (|V^)(i/^|) is majorized (Marshall and Olkin [1979]) by the same set for the final density matrix p^ (0) = Tr^ (|0)(0|), that is, if for each k k
k
^A,(v;)^5]A,(0), where k\ > A2 > • • •. To prove the theorem, Nielsen used the Schmidt decomposition (Peres [1993]) of the pure state of a composite system
\x) = Yli^i\^^)\^B). where A/ > 0, J ] / ^ / ^ 1? ^"^^ VA) and Iz^) form an orthonormal basis for each subsystem. Jonathan and Plenio [1999a] and Vidal [1999] extended Nielsen's theorem to the case where the transformation for one pure state (say, nonmaximally entangled) to another (say, maximally entangled) need not be deterministic (less than 100%). The same authors (Jonathan and Plenio [1999b]), with the help of Nielsen's theorem, showed that in the case where the target states cannot be reached by LOCC starting from a particular initial state, the assistance of a distributed pair of auxiliary quantum systems can catalyze the transformation, and what is most surprising, these auxiUary quantum systems (catalysts) are left in exactly the same state and remain finally completely uncorrelated to the quantum system of interest. The suggestion was made to call this new class of LOCC transformation "entanglement-assisted local transformations".
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abbreviated to ELOCC. Eisert and Wilkens [1999] went further and studied the catalysis of entanglement manipulation for mixed bipartite states. The problem of entanglement identification for mixed states is of great importance because of the strong pressure of losses and decoherence processes, which transform pure maximally entangled states into partially entangled states of distant modes. As follows fi*om the previous consideration, entanglement arises when the state of a multiparticle system is nonseparable, that is, when it cannot be prepared by acting on the particles individually. Although in recent years important steps have been taken toward the understanding of this quantum resource, we do not know yet in detail how to classify and quantify entanglement for the general multiparticle states, including nonpure states. However, some important results have been reported. Mathematically entangled states are those which do not belong to the class of pure and mixed separable states. A pure state |t/;^^^ > is separable if it can be expressed as a tensor product of states of different parties:
A nonpure mixed state p^^B: correct 4 A and B create a code
t
/ ^-^ \ (exp'^ l-a;) (a,| + exp"*^ \at){-at\% This dependence describes a slow decrease in the amplitude at = a exp"^^""^ and a fast transition (with the rate t^^^^^ = 2y|ap) into a mixed state. In the general case of a nonlinear interaction between the oscillator a and the reservoir, relaxation is described by the kinetic equation p=\r{[A,pA^]
+ [Ap,A^]).
(4.6)
From this equation, in combination with the Hamiltonian (4.3), it follows that the relaxation of the oscillator a strongly depends on the form of the interaction A(a, a^). In fact, the eigenstates | W)A of the interaction operator
The problem of decoherence
l.§4]
11
Table 3 Various "system-reservoir" interactions for quantum reservoir engineering Type of interactions
"Pointer basis"
Stationary state
References
A = a + a^ ^ X
Coordinate eigenstates
Vacuum
Zurek [1981, 1982, 1991]
A = a^
Even and odd coherent states
Vacuum
Gerry and Hach [1993], Agarwal [1987]
A = {a + a)(a -a) Even and odd coherent states
Even and odd coherent states
Garraway, Knight [1994a,b], Filho and Vogel [1996], Poyatos, Cirac and ZoUer [1996]
A = a^a
Fock states
Vacuum
Poyatos, Cirac and Zoller [1996]
A = a{a^a - n)
Fock states
Fock states
Poyatos, Cirac and Zoller [1996]
A = e^^«^«fl
Yurke-Stoler superposition state
Vacuum
Horoshko and Kilin [1997a], Kilin, Horoshko and Shatokhin [1998], Kilin and Horoshko [1997]
A(a,a^) remain unperturbed by the interaction with the reservoir, and form the so-called "pointer basis" (Zurek [1991]), which determines the specific form of the relaxational evolution. Hence, by using various forms of the interaction operator A{a, a^), one can create various "pointer bases" and thus vary the relaxation process and, moreover, obtain various stationary states as a result of the relaxation. Several well-known examples of "quantum reservoir engineering" are given in table 3. In recent years, techniques have been realized to generate mesoscopic superpositions of motional states of trapped ions (IVlonroe, IVleekhof, King and Wineland [1996]) and of photon states in the context of the cavity QED (Brune, Hagley, Dreyer, Ivlaitre, IVIaali, Wunderlich, Raimond and Haroche [1996]), where decoherence through coupling to ambient reservoirs and the sensitivity of the rate of decoherence to the size of the superposition were observed. IVlyatt, King, Turchette, Sackett, Klelpinski, Itano, IVlonroe and Wineland [2000] went further, and studied the decoherence into an engineered quantum reservoir, using laser cooling techniques to generate an effectively zero-temperature bath. Note that by replacing the harmonic oscillator by a set of A^ two-level systems representing a quantum register, one can find a subspace of the register states that is completely orthogonal to the states of the reservoir. Such states will not be perturbed by the reservoir. Several special cases have been considered by Zanardi and Rasetti [1997]. Alternatively, a strong correlation exists between the states
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of a single two-level atom and the reservoir, for instance, radiation. Because of this correlation, the atom dynamics can be varied by changing the state of the field. Thus, if one of the reservoir modes (a resonance mode) is initially in the Yurke-Stoler state, |a) + i | - a ) , and the other reservoir modes are in the vacuum state, the entangled nature of joint states leads to the effect of quantum instability, which manifests itself in the exponential growth of the transition dipole moment of the atom (Kilin and Shatokhin [1996, 1997a]), instead of the usual Rabi oscillations. 4.2. Relaxation as a quantum stochastic process; purity of conditional states Relaxation of the oscillator a also can be considered as a result of averaging quantum stochastic processes of excitation transfer fi*om the oscillator a to the reservoir oscillators. In each process of this kind, such as, for instance, the escape of a photon fi-om a cavity, a quantum is passed from the oscillator a to the reservoir. An instant change results, which is a reduction of the state of the oscillator a. The absence of quantum exchange between the acts of reduction, which occur at random instants of time, does not mean that the state of a remains constant. In fact, the longer one waits for the next quantum to be emitted, the higher the probability that oscillator a will be in the ground state; hence, its amplitude should decrease during such periods. Such a sequence of reductions and intervals of nonunitary evolution is studied by the theory of continuous quantum measurements or quantum jumps (Davies [1976], Holevo [1982], Kilin [1990]). In the case of relaxation with linear interaction, this sequence of random events is described by the conditional state vector of the oscillator a after transmitting exactly n quanta to the reservoir at times ^1,^2? • •» ^n belonging to the interval [0, t) IV^cond(O) = y"S(t,tn)aS{tn,t„-i)a-'aS(tuO)\xl^(0)),
(4.7)
where S(ti,ti_i) = Qxp[-ya^a(ti ~ ti-\)/2] is the nonunitary operator of the evolution between two successive reductions at ^/_i and tf. Emission of quanta at times {ti} results in the reduction of the state. If |V^(0)) = |t/^+), this effect a (|a) ± exp^^ | - a ) ) = a (\a)
=F
exp^^ | - a ) )
(4.8)
increases the relative phase 6 by JT, but the state remains a pure superposition state. The nonunitary evolution S{ti,ti-\) between quantum emissions reduces the amplitude a exponentially, so that the conditional state IV^cond(O) = N(yay (|aexp-^^/2^ + (-l)'^ exp^^ | - aexp-^^^^))
(4.9)
1, § 4]
The problem of decoherence
79
remains pure throughout the evolution period, and its coherence is preserved. Conservation of purity for conditional states in the course of relaxation does not contradict the preceding consideration of the density matrix decoherence: if the conditional density matrix |V^cond(0)(V^cond(Ol is averaged over random realizations of quantum emissions, we immediately obtain result (4.5), which means that the information about the state of the system is partially lost. It is also evident that the first emission event occurring after the average waiting time equal to the decoherence time, t^l^^^ = 2 / |a|^, is sufficient to erase the quantum interference terms.
4.3. Error correction by means of feedback Relaxation considered as a quantum stochastic process also shows that although decoherence is a serious obstacle for quantum information processing, it can still be overcome. To correct errors and uncertainties caused by the interaction of the quantum object with the surroundings, it is not necessary to know the state of the surroundings. It is sufficient to control the times of quantum emissions from the object to the surroundings and to return the system after each reduction to its initial state by means of some unitary transformation (Horoshko and Kilin [1997a,b], VitaH, Tombesi and Milburn [1997] Kilin and Horoshko [1997, 1998], Kilin, Horoshko and Shatokhin [1998]). For the case of Yurke-Stoler coherent states |a) +i | - a ) , this protocol of error correction should be carried out by rotating the phase of the oscillator a by 180'' (Horoshko and Kilin [1997a]). Then the sequence of events in the quantum stochastic process would consist of alternating stages of nonunitary evolution (the absence of emissions), reduction, and phase variation, IV^cond(O) = y^'SiUtn) exp^^-'^aS{tn.tn-x) cxp^^^'^a-"exp^^«'^aS{hMn^^)(4.10) Due to the correcting procedure, which can be realized by the back action on the oscillator a (fig. 26), both the conditional and the unconditional states of the oscillator, obtained by averaging over random realizations of emissions, remain pure superpositions, \\l){t)) = (|aexp-^^/2^ + i |-aexp-^^^2)) / ^
^4^1)
In this case the only sign indicating the existence of relaxation is the exponential amplitude decay (energy relaxation).
80
Quanta and information Modulator
_
[1, § 4
Detector
Fig. 26. Slowing down decoherence by means of an error-correcting feedback. The intracavity field, initially in the Yurke-Stoler state, is continuously detected by a high-efificiency detector. Each photocount is converted into a signal on the phase modulator, which changes the phase of the field by JT. If this procedure is repeated continuously, the superposition state is preserved as long as some photons are in the cavity.
Note that the density matrix of state (4.11) satisfies an equation similar to eq. (4.4):
where the nonlinear interaction operators Ajt = exp'^^ "^ a and A^ = «+ exp~^^^ ^ belong to the class of generalized annihilation/creation operators A(p = exp^^^ ^ a, A'^ = a^ exp"*^'' ^, whose eigenvectors, which are generalized coherent states, have useful quantum properties (Kilin, Horoshko and Shatokhin [1998]). The experimental scheme for the proposed decoherence correction has been demonstrated (Kilin and Horoshko [1997,1998]) (fig. 26). The intercavity field, initially in the Yurke-Stoler state, is continuously registered by a high-efficiency detector. From each photocount of the detector, a signal is fed through a feedback to the phase modulator, which changes the field phase by jr. If this procedure is continued, the superposition state in the cavity is conserved as long as some photons are in the cavity. Suppression of decoherence by means of feedback is a universal method and can be applied to all systems with continuously controllable losses (local nodes of a quantum computer). At present, in addition to the work mentioned above, some other suggestions have been made along these lines (Vitali, Tombesi and Milbum [1997]). Agarwal [2000] demonstrated that considerable slowing down of decoherence can be achieved by fast frequency modulation of the system-heat-bath coupling. If the control of losses is difficult, as in the case of transmission through quantum channels, one should use quantum errorcorrecting methods based on duplicating transmitted qubits (Shor [1995], Ekert and Macchiavello [1996]). 4.4. Hamming code and quantum error corrections From previous examination, it can be concluded that decoherence must be
1, § 4]
The problem of decoherence
81
a unitary process that entangles a system of interest, say a qubit, with the environment. For example, the qubit states |0),|1) and the environment states (not generally orthogonal, not normalized) become entangled due to the interaction: the initial unentangled state |e/)|0) or |^/)|1) transforms like k-)|0) -^ M | 0 ) + |eoi)|l),
|e,)|l) ^ |6io)|0) + |en)|l).
(4.13)
The averaging over environment states transforms a pure initial state of the qubit to a mixture. Classical information theory can successfully both recognize and correct errors of that kind in a string of bits influenced by noise by means of error correcting codes (see § 2.2.2.3). The idea of adapting this method to the quantum situation independently led Shor [1995], Steane [1996a,b] and Calderbank and Shor [1996] to the powerful quantum error correction (QEC) method. The theory of QEC was further advanced by Ekert and Macchiavello [1996], Bennett, DiVincenzo, Smolin and Wootters [1996], and Knill and Laflamme [1997]. The paper by Bennett and colleagues describes the optimal 5-qubit code discovered, also independently, by Laflamme, Miquel, Paz and Zurek [1996]. Gottesman [1996] and Calderbank, Rains, Shor and Sloane [1997] discovered a general group-theoretical framework, introducing the important concept of the stabilizer, which also enabled many more codes to be found (Steane [1996c]). Quantum coding theory reached a further level of maturity with the discovery by Shor and Laflamme [1997] of a quantum analog to the Mac Williams identities of classical coding theory. The idea of QEC is the same as in a classical EC, namely, to encode information in distinguishable strings of bits. For example, instead of 1 qubit, information could be stored in 2 orthogonal superposition states composed of 7 qubits, each state in the superposition being taken from 2^ Hamming code symbols generated by the Gy matrix (eq. 2.21) |0)H = 10000000) + 11010101) + 10110011) + 11100110) + 10001111) + 11011010) + 10111100) + 11101001), |1)H = l l l l l l l l ) + 10101010) + 11001100) + 10011001) + 11110000) + 10100101) + 11000011) + 10010110). If one of any 7 qubits is changed due to the interaction with the environment (and this is the most probable case), then, according to eq. (2.22), one can localize this qubit without affecting the others by the syndrome extraction operation H^ acting on the environmentally influenced state of 7 qubits. The idea of QEC has opened up new possibilities on the way to xQdX fault-tolerant quantum computing (Steane [1998], Preskil [1998]).
82
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[1? § 5
At this point we can allow ourselves to imagine a picture of allegories with a big lake of quantum computational resources blocked by a dump, with the name of "decoherence". Three rivers supply the lake with their own resources: quantum physics provides an understanding of the entanglement and connection with experiments, information theory indicates that we can count information and supplies clever coding, and mathematics formulates the problems made intractable by means of modem computers. Some small holes in the dump give access to the lake resources: quantum error corrections, some proposals combining methods of solid state physics, single molecule spectroscopy, NMR techniques, cavity quantum electrodynamics, and new methods of field confinement. These small sources are transformed into the river which will possibly lead to a fixture low-cost quantum computer.
§ 5. Conclusions Despite the famous names and the long period separating contemporary physicistsfi-omthe basic paper by Schrodinger [1935] and the classical paper by Shannon [1948], the real development of quantum information, with its practical importance for human society, is being started only now. Quantum informatics is developing remarkably rapidly. The scientific race for new achievements in quantum information has involved, joined, and enriched several fields of science, such as discrete mathematics and quantum mechanics, computer science, and quantum optics. Moreover, it has given practical importance to studies that previously seemed to be farfi-ompractical applications, such as the investigation of single quantum objects. All this work stimulates such a rapid development of new approaches, methods, and materials that it is hardly possible to keep abreast of current publications. Some excellent review articles and introductory publications have appeared, reflecting a growing understanding in the field of quantum information (Lloyd [1993], DiVincenzo [1995a], Bennett [1995], Ekert and Jozsa [1996], Steane [1998]). Some special issues of journals are also important milestones in the stream of publications (J. Mod. Opt. 41 (1994) no. 12; 44 (1997) no. 11/12; Philos. Trans. R. Soc. London A 355 (1997) 2215-2416; Proc. R. Soc. London A 454 (1998) 257-482; Phys. Scripta T 76 (1998); Opt. Spectrosc. 87 (1999) no. 4/5). A usefiil source of information is provided by electronic publications and e-preprints available on the Internet earlier than the corresponding hard copies. However, this source of "quantum information" produces new papers incredibly rapidly. For example, the Los Alamos Archive www. l a n l . gov (quant-ph) has been publishing more
1, § 5]
Conclusions
83
than a hundred papers per month for the last two years. To those who are critically assessing this information boom, it may be interesting to read Shannon's warning, published in his short paper "The bandwagon" (Shannon [1956]), in which he notes that there is no single key for all secrets of nature. Even if we understand and adopt these critical warnings, however, we can state that unification of two previously separated fields, namely, quantum physics and classical information theory, has become a reality. It is hard to predict all possible resuhs of the unification. At present, in its initial stage, we can see that once again Nature is giving us a lesson, presenting a new physical resource that has demanded for its description notions and methods that were not available in the huge arsenal of mathematical methods. This resource of quantum entanglement revolutionizes our understanding of the world and opens a window for the unpredictable power of Hilbert space of distant quantum objects. This power promises to be the basis for the next generation of computers and will be able to solve many mathematical problems presently untouchable. In addition, the number theory has been enriched due to the introduction, from physics, of the notion of the qubit - a new measure of quantum information instead of the classical bit. This may possibly be the third revolution in number theory after P3^hagoras' adventure of the irrational numbers and the introduction of complex numbers by Gauss and his contemporaries, a revolution that concurs with Fourier's observation that the investigation of Nature is the richest source of mathematical adventures. To understand the importance of quantum entanglement and describe its potential, some concepts of information theory and the method of reasoning have been used, and an explanation of the wavefixnctionhas again been stressed. The information science language has become an important part of the description of quantum world objects. This kind of thinking and presentation of reality has revived old and fiindamental problems such as symmetries and separability of quantum systems. For example, the Pauli principle forbids using all power of Hilbert space of interacting and, therefore, separately nonperturbable, spins. We understood that entangled systems could serve as a new resource for information storage and handling, but at the same time, the quantum entanglement of all with all creates the primary obstacle to quantum computation, that of decoherence. The solution to the problem of quantum computers creation lies in our ability to find methods for the decoherence harnessing. It seems now that meeting this important challenge will require a major consolidated of effort. In conclusion, many indications have emerged that a new physical view on Nature is being formed. One useful point of view suggests that Nature communicates information encoded by means of a number of "languages", one
84
Quanta and information
[1
of which is quantum mechanics. It is the quantum information that tries to determine, as a branch of science, the structure of this language, which can help us to decode the messages of Nature and to adopt the power of quantum coding for the practical benefits to society.
Acknowledgements The author is grateful to Emil Wolf and Jan Perina for their suggestion to write this review; to PA. Apanasevich, D.B. Horoshko, VN. Shatokhin, A.P Nizovtsev, T.M. Maevskaya, D.S. Mogilevtsev, T.B. Karlovich, and V.A. Zaporozhchenko for their cooperation; and to H. Walther, P Berman, M. Raymer, G. Bjork, C. von Borczyskowski, and J. Wrachtrup for fiiiitful discussions. Gratefial acknowledgment is given for partial financial support fi-om the National Science Foundation, United States (grant NSF9414515 "Spectroscopy of single molecules"); Volkswagen Foundation (grant 1/72171 "Two-level systems in single-molecule spectroscopy"); International Association, European Communityx (grant 96 167 "Generation of single photons and quantum states synthesis"), and the National Research Council, United States (Twinning program "Quantum tomography and other reconstructive measurement methods in quantum optics").
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91
E. Wolf, Progress in Optics 42 © 2001 Elsevier Science B.V All rights reserved
Chapter 2
Optical solitons in periodic media with resonant and off-resonant nonlinearities by
Gershon Kurizki*, Alexander E. Kozhekin**, Tomas Opatrny*** Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel
and
Boris A. Malomed Department of Interdisciplinary Studies, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
* E-mail:
[email protected] ** Present address: Institute of Physics and Astronomy, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark *** Present address: Friedrich-Schiller-Universitat Jena, Theoretisch-Physikalisches Institut, Max Wien Platz 1, 07743 Jena, Germany and Palacky University, Faculty of Natural Sciences, Svobody 26, 77146 Olomouc, Czech Republic 93
Contents
Page § 1. Introduction
95
§ 2.
Solitons in Bragg gratings with cubic and quadratic nonlinearities
99
§ 3.
Self-induced transparency (SIT) in uniform media and thin
§ 4.
SIT in resonantly absorbing Bragg reflectors (RABR): the model
films
105 109
§ 5. Bright solitons in RABR
120
§ 6.
Dark solitons in RABR
129
§ 7.
Light bullets (spatiotemporal solitons)
136
§ 8.
Experimental prospects and conclusions
139
Abbreviations
142
Acknowledgments
142
References
142
94
§ 1. Introduction The study of light-matter interactions in dielectric structures with periodic modulation of the refractive index has developed into a vast research area. At the heart of this area is the interplay between Bragg reflections, which block the propagation of light in spectral bands known as photonic band gaps (PBGs), and the dynamical modifications of these reflections by nonlinear light-matter interactions (see bibliography compiled by Dowling and Everitt [2000]). Three- or two-dimensional (3D or 2D) PBGs are needed in order to extinguish spontaneous emission in all possible directions of propagation, which requires the nontrivial fabrication of 3D- or 2D-periodic photonic crystals (Yablonovitch [1987, 1993]). For controlling strictly unidirectional propagation, it is sufficient to resort to PBGs in one-dimensional (ID) periodic structures (Bragg reflectors or dielectric multilayer mirrors). Illumination of the periodic dielectric structure at a PBG frequency in the limit of vanishing nonlinearity leads to exponential decay of the incident field amplitude with penetration depth, at the expense of exponential growth of the back-scattered (Bragg-reflected) amplitude. However, this reflection may weaken or cease altogether, rendering the structure transparent, when the illumination intensity and the resulting nonlinearity modify the refractive index so as to shift (or even close down) the PBG. The pulsed mode of propagation in nonlinear periodic structures exhibits a variety of fimdamentally unique and technologically interesting regimes: nonlinear filtering, switching, and distributed-feedback amplification (Scalora, Dowling, Bowden and Bloemer [1994a,b]). Among these regimes, we have chosen here to concentrate on the intriguing solitary waves existing in PBGs, known as gap solitons (GS), and solitons propagating near PBGs. A GS is usually understood as a self-localized moving or standing (quiescent) bright region, where light is confined by Bragg reflections against a dark background. The soliton spectrum is tuned away from the Bragg resonance by the nonlinearity at sufficiently high field intensities. There is also considerable physical interest in finding a dark soliton (DS) in the vicinity of a PBG, i.e., a "hole" of a fixed shape in a continuous-wave (cw) background field of constant intensity (Kivshar and Luther-Davies [1998]). The first type of GS was predicted to exist in a Bragg grating filled with a 95
96
Optical solitons in periodic media
[2, § 1
Kerr medium, whose nonlinearity is cubic (Christodoulides and Joseph [1989], Aceves and Wabnitz [1989], Feng and Kneubuhl [1993]). Detailed theoretical studies of these Bragg-grating (Kerr-nonlinear) solitons (see De Sterke and Sipe [1994] for a review) were followed by their experimental observation (Eggleton, Slusher, De Sterke, Krug and Sipe [1996]) in a short (^-».
(2.4a)
,d£2_.d£2 i^-i^+^i+^3fr=0, d^ ^ dx
(2.4b)
2 l - ^ - q£2> + ^ - ^ 2 " + ^1^2 = 0.
(2.4c)
Here t and x are the propagation and transverse coordinates, respectively; the fields £\ and £2 are two components of the fundamental harmonic that are transformed into each other by resonant reflections on the ID Bragg grating, £3 is the second-harmonic component, and D is an effective diffraction coefficient for the second harmonic. The wave vectors ^1,2,3 of the three waves are related by the resonance condition, k\+k2 = k^,, the real parameter q accounts for a residual phase-mismatch. The configuration corresponding to this model assumes that the second harmonic propagates parallel to the Bragg grating (which has the form of the above-mentioned scores). It is therefore necessary to take into account the diffraction of this component, while for the two fundamental harmonics the effective diffraction induced by resonant Bragg scattering is much stronger than normal diffraction, which is neglected here. The soliton spectrum of this model is fairly rich. It contains not only fiindamental single-humped solitons but also their two-humped bound states.
2, § 3]
Self-induced transparency (SIT) in uniform media and thin films
105
some of which, as in the case of the four-wave model (2.3), may be dynamically stable (Mak, Malomed and Chu [1998b]). A rigorous stability analysis for various solitons in the model (2.4), based on computation of eigenvalues of the corresponding linearized equations, was performed by Schollmann and Mayer [2000]. This analysis has shown that some of these solitons, although quite stable in direct dynamical simulations, are subject to a very weak oscillatory instability, whereas other solitons in this model are stable in the rigorous sense. The three-wave model (2.4) possesses, besides the traditional GSs, numerous branches of embedded solitons: isolated solitary-wave solutions existing within the continuous spectrum, rather than inside the gap (Champneys and Malomed [2000]). Solutions of this kind appear also when the second-derivative terms are added to the generalized Thirring model (2.1). Finally, the four-wave model (2.3) with quadratic nonlinearity can be extended to the two- and three-dimensional cases, by adding transverse diffraction terms to each equation of the system. Physically, this generalization corresponds to spatiotemporal evolution of the fields in a two- or three-dimensional layered medium. Because, as is well known, quadratic nonlinearity does not give rise to wave collapse in any number of physical dimensions, the latter model can support stable spatiotemporal solitons, frequently called light bullets. Direct numerical simulations reported by He and Drummond [1998] have confirmed the existence of stable "bullets" in a multidimensional SHG medium embedded in a Bragg grating.
§ 3. Self-induced transparency (SIT) in uniform media and thin films 3.1. SIT in uniform media Self-induced transparency(SIT) is the solitary propagation of electromagnetic (EM) pulses in near-resonant atomic media, irrespective of the carrierfrequency detuning from resonance. This striking effect, which is of paramount importance in nonlinear optics, was discovered by McCall and Hahn [1969, 1970]. If the pulse duration is much shorter than the transition (spontaneousdecay) lifetime {Ti) and dephasing time {T2), then the leading edge of the pulse is absorbed, inverting the atomic population, while the remainder of the pulse causes atoms to emit stimulated light and thus return the energy to the field. When conditions for the process are met, it is found that a steady-state pulse envelope is established and then propagates without attenuation at a velocity that may be considerably less than the phase velocity of light in the medium.
106
Optical solitons in periodic media
[2, § 3
We start with the Hamiltonian for a single atom in the field, H=^^^E.d,
(3.1)
where w^\e){e\-\g)(g\
(3.2)
is the atomic inversion operator, COQ is the atomic transition firequency, \g) and \e) denote the atomic ground and excited states, respectively, E is the electric field vector, and d is the atomic dipole-moment operator. We take the projection on the field direction, so that E d = Ed, where d^^(p
+ P^),
(3.3)
pi being the dipole moment matrix element (chosen real) and P = 2\g){e\
(3.4)
is the atomic polarization operator. We express the electric field at a given point by means of the Rabifi-equencyQ as E=^{QQ"''^' + Q'e''^').
(3.5)
The Heisenberg equations of motion dA/dt = \/{\h) A,H , for the atomic polarization and inversion operators (3.4) and (3.2), yield the Bloch equations for their expectation values (c-numbers) P and w, respectively dtP{z, t) = w(z, t)Q -'\{ojo- con)P,
(3.6a)
S,w(z,0 = -J [ P * ( z , 0 ^ + c.c] .
(3.6b)
The Maxwell equations (Newell and Moloney [1992]) reduce in the rotatingwave and slow-varying approximations to
^ | 4 ) ^ = ro-P, nodz
(3.7)
at J
where
ro = " ^ J ^ ,
(3.8)
is the cooperative resonant absorption time, Qo being the TLS density (averaged over z), and no is the refraction index of the host media.
2, § 3]
Self-induced transparency (SIT) in uniform media and thin films
107
In the simplest case, when the driving field is in resonance with the atomic transition, a^o = ^c, the Bloch equations (3.6) can be easily integrated and the Maxwell equation (3.7) then reduces to the sine-Gordon equation FP-9
dm
sin a
(3.9)
for the "rotation angle", 0= /
QAt',
(3.10)
J-C
in terms of the dimensionless variables x = (t- noz/c)/ro and C = mz/cXo, This sine-Gordon equation is known to have solitary-wave solutions, for which the total area under the pulse is conserved and equal to In - the so-called pulsearea theorem by McCall and Hahn [1969, 1970]: 0(C,r) = (ro)-Uosech[i3(£-i;f)] ,
(3.11)
where the pulse width /? is an arbitrary real parameter uniquely defining the amplitude ^o = 2/^ and group velocity v = \/0^ of the soliton. Since its inception, SIT has become an active research area with many practical applications, for which we refer readers to excellent reviews by Lamb Jr [1971], Poluektov, Popov and Roitberg [1975], Maimistov, Basharov and Elyutin [1990] and references therein. In this section we will only briefly discuss results which are pertinent to the present review, such as SIT in thin films and collisions of counterpropagating SIT solitons. 3.2. SIT in th in films The interaction of light with a thin film of a nonlinear resonant medium located at the interface between two linear media has been described by Rupasov and Yudson [1982, 1987], who have shown that a nonlinear thin film of TLS can be a nearly ideal mirror for weak pulses, but transparent for pulses of sufficient intensity. The problem of light pulse transmission through the nonlinear medium boundary has been studied under conditions of coherent interaction with the matter. The system can be described by a set of nonlinear Maxwell-Bloch-like equations which effectively take the presence of the reflected wave into account by imposing boundary conditions on the electromagnetic fields at the interface. It has been shown (Rupasov and Yudson [1987]) that these equations are exactly
108
Optical solitons in periodic media
[2, § 3
integrable by the inverse scattering method, and 2jr-soHton-pulse transmission through the film has been studied. If the atomic density is such that on average there is more than one atom per cubic resonant wavelength, then near-dipole-dipole (NDD) interactions, or local-field effects, can no longer be ignored, contrary to the case of more dilute media. NDD effects necessitate a correction to the field that couples to an atom in terms of the incident field and volume polarization (Bowden, Postan and Inguva [1991], Scalora and Bowden [1995]). This effect can give rise to bistable optical transmission of ultrashort light pulses through a thin layer consisting of two-level atoms (Basharov [1988], Benedict, Malyshev, Trifonov and Zaitsev [1991]): the local-field correction leads to an inversion-dependent resonance frequency, and generates a new mechanism of nonlinear transparency. When the excitation frequency is somewhat larger than the original resonant frequency, the transmission of the layer exhibits a transient bistable behavior on the time scale of superradiance (Basharov [1988], Benedict, Malyshev, Trifonov and Zaitsev [1991]). It was shown that if an ultrashort pulse is allowed to interact with a thin film of optically dense two-level systems, the medium response is characterized by a rapid switching effect (Crenshaw, Scalora and Bowden [1992], Crenshaw and Bowden [1992]). This behavior is more remarkable than the response of conventional two-level systems, because the medium can only be found in one of two states: either fiilly inverted or in the ground state, depending (quasiperiodically) on the ratio between the peak field-strength and the NDD coupling strength. This feature was found to be impervious to changes in pulse shape, and to be independent of the pulse area (Crenshaw, Scalora and Bowden [1992]). Passage of light through a system of two thin TLS films of two-level atoms has been considered by Logvin and Samson [1992] and Logvin and Loiko [2000] who have shown that if the distance between the films is an integer multiple of the wavelength, then the system is bistable. Self-pulsations, i.e., periodically generated output, arise if an odd number of half-wavelengths can be fitted between the films and absorption in the medium is insignificant. In general, the dynamics admit both regular and chaotic regimes.
3.3. Collisions of counterpropagating SIT solitons Situations in which it is necessary to consider the interaction of incident (forward) and reflected (backward) light waves include: intrinsic optical bistability (Inguva and Bowden [1990]), dynamics of excitations in a cavity (Shaw and Shore [1990]) and collisions of counterpropagating SIT solitons
2, § 4]
SIT in RABR: model
109
(Afanas'ev, Volkov, Dritz and Samson [1990], Shaw and Shore [1991]). The field in such problems is represented as a superposition of forward- and backward-traveling waves. The atomic response to this field is determined by solving the Bloch equations (3.6) in the rotating-wave approximation. The population inversion w(z, t) and polarization P may be represented by a quasiFourier expansion over a succession of spatial harmonic carriers and slow varying envelopes, entangled in a fashion which leads to an infinite hierarchy of equations. The truncation of this hierarchy can only be justified by phenomenological arguments, such as atom movement in an active atomic gas. When the forward (F-) and backward (B-) wave pulses overlap in space and time, the resulting interference pattern of nodes modifies the atomic excitation pattern. The spatial quasi-Fourier expansion provides an efficient way of treating the spatial inhomogeneities of the response in those regions where the F- and Bpulses overlap, each successive Rabi cycle increasing the number of terms that contribute to the expansion (Shaw and Shore [1990]). Collisions of optical solitons produce observable effects on both the atoms and the pulses. The overlap of two counterpropagating pulses can produce an appreciable spatially localized inversion of the atomic population, thus causing optical solitons to lose energy. It was found that, whereas large-energy solitons passed freely through each other, solitons whose initial energy fell below a critical value were destroyed by collisions. In addition, the residual atomic dipole, created by the excitation, acts as a fiirther source of radiation. This radiation appears as an oscillating tail on the postcoUisional pulses and, over longer time scales, as fluorescence (Afanas'ev, Volkov, Dritz and Samson [1990], Shaw and Shore [1991]). § 4. SIT in resonantly absorbing Bragg reflectors (RABR): the model 4.1. Maxwell equations Let us assume (Kozhekin and Kurizki [1995], Kozhekin, Kurizki and Malomed [1998], Opatmy, Malomed and Kurizki [1999]) a one-dimensional (ID) periodic modulation of the linear refractive index n{z) along the z direction of the electromagnetic wave propagation (see fig. 3). The modulation can be written as the Fourier series n^(z) = nl[\+a\ cos(2A:cZ) + ^2 cos(4A:cz)+ • • •],
(4.1)
where n^, aj and kc are constants, and the medium is assumed to be infinite and homogeneous in the x and y directions.
Optical solitons in periodic media
no
[2. §4
Fig. 3. Schematic description of the periodic RABR and of the decomposition of the electric field into modes 2"+ and Z_. The shading represents regions with different index of refi-action; the darker the shading the larger n is. The black regions correspond to the TLS layers. The upper solid curve represents the electric field, the lower solid curves correspond to the components Re(2'+)cos^cZ and -\m{l_)smkcz; the dashed curves are the envelopes Re(2V) and -Im(2'_). The vertical dotted lines denote the positions of the TLS.
The periodic grating gives rise to photonic band gaps (PBGs) in the system's Unear spectrum, i.e., the medium is totally reflective for waves whose frequency is inside the gaps. The central frequency of the fundamental gap is 0)^ = kcc/no, c being the vacuum speed of light, and the gap edges are located at the frequencies 0)1,2 = CO, (1 ±
\a),
(4.2)
where a\ is the modulation depth from eq. (4.1). We further assume that very thin TLS layers (much thinner than l/A:,), whose resonance frequency o^o is close to the gap center H)^, are placed at the maxima of the modulated refraction index. In other words, the thin active layers are placed at the points Ziayer such that COS(A:c^layer) = ± 1 .
We shall study the propagation of electromagnetic waves with frequencies close to CL^C through the described medium. Let us write the Maxwell equation for one component of the field vector propagating in the z direction as
-n\z)
(4.3)
with the refraction index n modulated as in eq. (4.1), E being the electric field component and P„\ the nonlinear polarization. We use the substitution E= [£-F(z,0e'^»^ + £-B(2,0e-'''^]
(4.4)
SIT in RABR: model
4]
111
with 0)c satisfying the dispersion relation HQWC = Kc and £^ and £B denoting the forward and backward-propagating field components. We work in the slowly varying envelope approximation d^£ B,F d^£,B,F
df
d£ B,F dz
Z2j + A/4
(gP,.±i^cz\
— ^±iA:
SiO = 2\x-d\ni^){\-n\C)y
(5.10)
with
ia = i 2 x-s
(l-7^o)
x-n
+ (2710)-'In
tan
d-Y-j^
7^o + ^Jnl-n? n
(5.11)
and
nl = i
\(X^rj)(x-6)\
(5.12)
(note that 1Zl is positive under the conditions 5.6-5.9). It can be checked that this zero-velocity (ZV) gap soliton is always single-humped. Its amplitude can be found from eq. (5.11), 47^o *->max
r
-'
vU + ^l The polarization amplitude V is determined by S via eq. (5.2).
(5.13)
§5] 16 141
123
Bright solitons in RABR
(a)
x10"
^2\10
6
4l 2
C
50
100
150
-150
-100
-50
50
100
150
Fig. 6. Zero-velocity (RABR) solitons |2'+(f)p: (a) 5 = 0, ry = 0.9, x = -0.901 (divergent width and amplitude); (b) idem, but for / = 0.901 (divergent width and finite amplitude).
To calculate the electric field in the antisymmetric I- mode, we substitute I^ = iQ-'^^A(0
(5.14)
into eq. (4.21b) and obtain (5.15) which can be easily solved by the Fourier transform, once V(C) is known. An example of bright solitons is depicted in fig. 6. Note that, depending on the parameters rj, 6 and x, the main part of the soliton energy can be carried either by the 1+ or the 2^ mode. The most drastic difference of these new solitons firom the well-known SIT pulses is that the area of the ZV soliton is not restricted to 2jt, but, instead, may take an arbitrary value. As mentioned above, this basic new result shows that the Bragg reflector can enhance (by multiple reflections) the field coupling to the TLS, so as to make the pulse area effectively equivalent to In. In the limit of the small-amplitude and small-area solitons, T^Q ^) with some wavenumber p, in order to "push" the soliton. The results demonstrate that, at sufficiently small p, the "push" indeed produces a moving stable soliton (fig. 7a). However, if p is large enough, the multiplication by exp(i/^t) turns out to be a more violent perturbation, splitting the initial pulse into two solitons, one quiescent and one moving (fig. 7b). Another one-parameter subfamily of moving GS was found in the exact form of a phase-modulated 2:/r-soliton by Kozhekin and Kurizki [1995]: (5.18)
^+ = ^0 exp [i (/re - xr)] sech [jS (C - vr)],
where x is the detuning from the gap center, AQ is the amplitude of the solitary pulse, j8 its width and v its group velocity. Substituting djP from eq. (4.23a) into eq. (4.21a), we may express P in terms of 2V and the population inversion w. Then, upon eliminating P and using ansatz eq. (5.18), we can integrate eq. (4.23b) for the population inversion w, obtaining -1-
Alix-K/v) 2{d-r])
1 cosh^[iS(e-i;r)]'
(5.19)
Using these explicit expressions for P and w in eqs. (4.21a) and (4.23a), we reduce our system to a set of algebraic equations for the coefficients K, X that determine the spatial and temporal phase modulation, and the pulse width ^ as fiinctions of the velocity v\ 2{X~K/v)-(\-\/v'){8-r])
= (),
(5.20a)
{X-S){ P^v^ -l3^ + K^-x^ + ri^ + 2) + ip^v^ix - K/V) + 2{d -^) = 0,
(5.20b)
(/3V - ^ 2 + K^-X" + ri' + 2)-2{x
(5.20c)
+ S){x-^/v)
= 0.
126
Optical solitons in periodic media
[2, § 5
The soliton amplitude is then found to satisfy \AQ\ = 2^v, exactly as in the case of usual SIT (see § 3). This implies, by means of eq. (5.18), that the area under the 2V envelope is 2jt. Let us consider the most illustrative case, when the atomic resonance is exactly at the center of the optical gap, 6 = 0. Then the solutions for the above parameters are -
-
^1-3^^^
2u 1 -1;2 '
(5.21a)
2 I -u^
In the frame moving with the group velocity of the pulse, ^' = ^ - or, the temporal phase modulation will be (KU-X) ^, which is found from eq. (5.21) to be equal to -rjT. Since t] is the (dimensionless) "bare" gap width (see § 4), this means that the frequency is detuned in the moving frame exactly to the band-gap edge. The band-gap edge corresponds (by definition) to a standing wave, whence this result demonstrates that such a pulse is indeed a soliton, which does not disperse in its group-velocity frame. The allowed range of the solitary group velocities may be determined from eq. (5.21c) through the condition (3^ > 0 for a given rj. The same condition implies |?7| < rj^ax, where ,8.^(1-.^) '/max
(1+^2)2
'
^ •
^
It follows from eq. (5.22) that the condition for Ijt SIT gap soliton (5.18) is |ry| < 1, r/max = 1 corresponding to t; = l / \ / 3 . This condition means that the cooperative absorption length CTOMQ should be shorter than the reflection (attenuation) length in the gap Ac/{a\(ji)cnQ), i.e., that the incident light should be absorbed by the TLS before it is reflected by the Bragg structure. In addition, both these lengths should be much longer than the light wavelength for the weakreflection and slow-varying approximation to be valid. From eq. (4.21b) we find 1- = I+/u. The envelopes of both waves (forward and backward) propagate in the same direction; therefore the group-velocity of the backward wave is in the direction opposite to its phase-velocity! This is analogous to climbing a descending escalator. Analogously to Kerr-nonlinear gap solitons (§ 2), the real part of the nonlinear polarization ReP creates a traveling "defect" in the periodic Bragg reflector
2, § 5 ]
Bright solitons in RABR
127
Fig. 8. Dependence of the solitary pulse velocity (solid line) and amplitude (dashed line) in RABR on frequency detuning from the gap center for rj = OJ. The "bare" gap edge marked by dotted line.
structure which allows the propagation at band-gap frequencies. The real part of the nonlinear polarization is governed by the frequency detuning from the TLS resonance. Exactly on resonance (which we here take to coincide with the gap center) x = ^ = 0, RQP = 0, and our solutions (5.21) yield imaginary values of the velocity v and modulation coefficient K. The forward field envelope then decays with the same exponent as in the absence of TLS in the structure. Because of this mechanism, SIT exists only on one side of the band-gap center, depending on whether the TLS are in the region of the higher or the lower linear refractive index. This result may be understood as the addition of a nearresonant non-linear "refiractive index" to the modulated index of refraction of the gap structure. When this addition compensates the linear modulation, soliton propagation becomes possible (see fig. 3). On the "wrong" side of the band-gap center, soliton propagation is forbidden even in the allowed zone, because then the nonlinear polarization cannot compensate even for a very weak loss of the forward field due to reflection. The soliton amplitude and velocity dependence on frequency detuning fi:om the gap center (which coincides with atomic resonance) are illustrated in fig. 8. They demonstrate that forward soliton propagation is allowed well within the gap, for X satisfying (1 - ^J\ - r]^)/r] < x < (1 + \ / l - rj^)/rj. In addition to frequency detuning from resonance, the near-resonant GS possesses another unique feature: spatial self-phase modulation Kt, of both the forward and backward field components.
128
Optical solitons in periodic media
[2, §5
5.3. Numerical simulations To check the stability of the analytical solution eq. (5.18), as well as the possibility to launch a moving GS by the incident light field, numerical simulations of eqs. (4.6) were performed by Kozhekin and Kurizki [1995]. As the launching condition, the incident wave was taken in the form SY = A exp[ix(^-^)] /cosh|jS(r-/o)/ro] without a backward wave {£B = 0) at the boundary of the sample z = 0. By varying the detuning x and amplitude A we investigate the field evolution inside the structure. When these parameters are close to those allowed by eqs. (5.21) and (5.22), we observe the formation and lossless propagation of both forward and backward soliton-like pulses with amplitude ratios predicted by our solutions (fig. 9). By contrast, exponential decay of the forward pulse in the gap is numerically obtained in the absence ofTLS(fig. 10).
Fig. 9. Numerical simulations of the intensities of (a) "forward" and (b) "backward" waves in the RABR gap, when eqs. (5.21) and (5.22) are obeyed {r] = 0.7, group velocity v ~ 0.3).
Fig. 10. Numerical simulations of the intensities of "forward" waves in the RABR gap without TLS (same rj and incident pulse as in fig. 9).
2, §6]
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Dark solitons in RABR
The analysis surveyed in §§ 5.1-5.3 strongly suggests, but does not rigorously prove, that the solution subfamily (5.18) belongs to a far more general two-parameter family, whose other particular representatives are the exact ZV solitons (5.11) and the approximate small-amplitude solitons determined by eq. (5.17). 5.4. Collisions between gap solitons An issue of obvious interest is that of collisions between GSs moving at different velocities in RABR. In the asymptotic small-amplitude limit reducing to the NLS equation (5.17), the collision must be elastic. To get a more general insight, we simulated collisions between two solitons given by (5.18). The conclusion is that the collision is always inelastic, directly attesting to the nonintegrability of the model. Typical results are displayed in fig. 11, which demonstrates that the inelasticity may be strong, depending on the parameters.
T=5
^ ^
w
lU 5 n
' 7^: A
T=10
^
^ ]
'JK
T=0/^
:_
'-'^ A
:
!
"
A-25
: 1 ^ :
, - ^ - ^
^
^ 10
15
20
- ^ 25
30
Fig. 11. Typical example of inelastic collisions between the RABR solitons eq. (5.18) at (5 = 0 and r/ = 0.5, with the velocities (normalized to c) u\ = 0.6, U2 = -0.75.
§ 6. Dark solitons in RABR 6.1. Existence conditions and the form of the soliton Dark solitons (DSs) in RABR have been studied by Opatrny, Malomed and Kurizki [1999]. They are obtained similarly to the bright ones, by solving
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Optical solitons in periodic media
[2, § 6
eq. (5.4) with the potential (5.5). The potential will give rise to DS's provided that it has two symmetric maxima (seefig.5). In this case the quadratic part of the potential is convex, i.e., \x\ < T], and the second (asymptotically linear) part of the expression (5.5) is concave, so that X > ^- From these two inequalities, a simple necessary restriction on the model's parameters follows, d, bit, r) = Z>oe' given in eqs. (5.6) and (5.7). From the discussion in § 5 it follows that the DS frequency band always coexists with one or two bands supporting the bright solitons. The special case when there are two bright-soliton bands coexisting with the DS band is singled out by the condition T]
< 6 < ri.
(6.12)
rj
One can readily check that the coexisting frequency bands supporting bright and dark solitons never overlap, i.e., quite naturally, the bright and dark solitons cannot have the same frequency. 6.5. Moving dark solitons Thus far we have considered only the quiescent DSs. A challenging question is whether they also have their moving counterparts. Adding the velocity parameter to the exact DS solution is not trivial, as the underlying equations (4.26a) and (4.26b) have no Galilean or Lorentzian invariance. The physical reason for this is the existence of the special (laboratory) reference frame, in which the Bragg grating is at rest. In principle it is possible, in analogy to the stationary solutions and eq. (5.1), to substitute functions of the argument (^ - vr) into the set (4.26a) and (4.26b), so as to obtain an ordinary differential equation. However, this would be a complicated complex nonlinear equation of the third order, containing all the lower-order derivatives, so that we would not be able to take advantage of the Newton-like structure, as in §§ 5 and 6. Though it is possible to solve such an equation numerically, it is more suitable to deal with the original set of partial differential equations, in order to better understand the nature of the evolution. In contrast to the case of bright solitons where the moving solutions can be found by multiplying, in the initial conditions, the quiescent DS by a factor
•6]
Dark solitons in RABR
135
Fig. 15. Moving dark soliton: the values of the parameters are the same as in fig. 12, the background phase-jump parameter (j) (see eqs. (6.13) and (6.14)) is 0 = - | j r . Dashed line: r = 0, continuous line: r = 600.
proportional to exp(i/ct) (see Kozhekin, Kurizki and Malomed [1998] and § 5), it has proven possible to generate stable moving DSs from the quiescent ones in a different way (Opatrny, Malomed and Kurizki [1999]). To this end, recall that a DS corresponds to a transition between two different values of the background cw field. The background field generally takes complex values (note the real values in the expressions (6.3) and (6.4) above are only our choices adopted for convenience). The quiescent DS corresponds to a transition between two background values with phases differing by Jt. A principal difference of the DSs in the present model from those in the NLS equation (Kivshar and Luther-Davies [1998]) is that here a moving DS is generated by introducing SL phase jump ^ n across the DS. Thus, one can take the initial condition for the system of equations (4.26a) and (4.26b) as X4C,0) =cos(^0)5q(t) + isin(|0)5M,
(6.13)
P = cos(^0)j9q(C) + isin(^0)j9M,
(6.14)
where <Sq and Pq are the (real) fimctions corresponding to the quiescent DS, «SM and Vu are given by eqs. (6.3) and (6.4), and (j) is the deviation from Jt of the background phase jump across DS. A typical result obtained by means of this modification of the initial state is displayed in fig. 15: the DS moves at a velocity that is proportional to (j). The resulting form of the moving DS is slightly different from that of the quiescent soliton. The moving DS appears to be stable over the entire simulation time.
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Optical solitons in periodic media
[2, § 7
§ 7. Light bullets (spatiotemporal solitons) A promising direction is the study of solitons in resonantly absorbing multidimensional (2D and 3D) media, in which the quasi-ID periodic structures can be realized as thin homogeneous layers set perpendicular to the direction of light propagation. In such media, spatiotemporal solitons, i.e., those localized in all dimensions, both transverse (spatial proper) and longitudinal (effectively, temporal), may exist. Spatiotemporal optical solitons or "light bullets" (LBs) in various nonlinear media are surveyed in § 1. Here we are concerned with LBs in RABRs that consist of thin TLS layers embedded in a 2D- or 3D-periodic dielectric medium. We will follow a recent analysis by Blaauboer, Kurizki and Malomed [2000], which extends an earlier prediction of stable LBs in uniform 2D and 3D SIT media (Blaauboer, Malomed and Kurizki [2000]). We start by considering a 2D SIT medium with a refractive index «(z,x) periodically modulated in the propagation direction z, which represents the quasione-dimensional Bragg grating. Light propagation in the medium is described by the lossless Maxwell-Bloch equations (Newell and Moloney [1992]):
dP ^-£w
=0,
(7.1b)
=0.
(7.1c)
OT
^-^^(£*P
+ P'S)
Here (as in § 4.1) £^ and P are the slowly varying amplitudes of the electric field and medium's polarization, w is the population inversion, C and x are longitudinal and transverse coordinates (measured in units of the resonant-absorption length), and r is time (measured in units of the input pulse duration). The Fresnel number, which governs the transverse diffraction in the 2D and 3D propagation, was incorporated into jc, and the detuning of the carrier frequency COQ from the central atomic-resonance fi-equency was absorbed into £ and P. To neglect the polarization dephasing and inversion decay, we assume pulse durations that are short on the time scale of the relaxation processes. Equations (7.1) are then compatible with the local constraint |Pp + w^ = 1, which represents the so-called Bloch-vector conservation. In a ID case, i.e., in the absence of the x-dependence and for «(z,x) = 1, eq. (7.1a) reduces to the sine-Gordon (SG) equation, which has a commonly known soliton solution, see eq. (3.11) and § 3. To search for LBs in a 2D medium subject to a resonant periodic longitudinal modulation, one may assume a periodic modulation of the refi-active index as
2, § 7]
Light bullets (spatiotemporal solitons)
137
per eq. (4.1). The RABR is then constructed by placing very thin layers (much thinner than X/k^) of two-level atoms, whose resonance frequency is close to (D^, at maxima of this modulated refractive index. The objective is to consider the propagation of an electromagnetic wave with a frequency close to o)^ through a 2D RABR. Due to the Bragg reflections, the electric field £ gets decomposed into forward- and backward-propagating components 8^ and £Q, which satisfy equations that are a straightforward generalization of the ID equations derived by Kozhekin and Kurizki [1995], Kozhekin, Kurizki and Malomed [1998], and Opatrny, Malomed and Kurizki [1999] (see also eqs. 4.21 and 4.23 in this review): ,d^l^
.d^I^ .d^I-
d^I^ .d^I+
d^I+ d^I-
d^I+ d^I-
.
.dP
,. „
^
,,,,
d^I-
2^
.dP
^
,^^^,
^ + {dP- 2^w = 0, ox ^ + UiiP or ^
+ i:+p*) = 0.
(7.2c) (7.2d)
Here I± is defined by eq. (4.9), and ?/ is a ratio of the resonant-absorption length in the two-level medium to the Bragg-reflection length, which was defined above by eq. (4.13). To construct an analytical approximation to the LB solutions, the starting point adopted by Blaauboer, Kurizki and Malomed [2000] is a subfamily of the exact ID soliton solutions to eqs. (7.2), which was found by Kozhekin and Kurizki [1995] (see also §5.2 in this review) and is given by eqs. (5.18) and (5.19). These solutions were taken with parameters satisfying eqs. (5.20):
K = -^8^
- r]\
x = 6.
(7.3b)
These solutions were chosen as a pattern to construct an approximate solution for LBs because the shape of the fields 1+ and I- in the solutions is similar to that of the SG soliton in the ID uniform SIT medium (see §3.1). Inspired by this analogy and by the fact that there exist LBs in the uniform 2D SIT medium which reduce to the SG solitons in the ID limit (Blaauboer, Malomed and Kurizki [2000]), one can search for an approximate LB solution to the 2D
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Optical solitons in periodic media
[2, § 7
equations (7.2), which also reduces to the exact soliton in ID. To this end, the following approximation was assumed: I^ = Aoy/sQcheiSQche2c'^''^-^'^^V'',
(7.4a)
^_ = I+/v,
(7.4b)
P = y/sQchO\ sech6)2 X |(tanh6)i +tanhe2) 6-rj ,r . -C^[(tanhei-tanh6)2) 4r/ 2(sech^6)i+sech^6)2)] ^ j1 w=
,1/2
[i-|/^IT '
(7.4c) e^i(K-t-xr)+iv
(^-^d)
with 6)i(r, C) = i3(C - i;r) + 00 + Cx, e2(^, C) = iS(C - ^^) + 6)o - Cc, the phase v and coefficients OQ and C being real constants, while the other parameters are defined by eqs. (7.3). The ansatz (7.4) satisfies eqs. (7.2a) and (7.2b) exactly, while eqs. (7.2d) are satisfied to order y/S/t]- IC^, which requires that y/d/rj- \C^ 1) reflectivities of the Bragg grating, provided that the detuning remains small with respect to the gap fi*equency. Comparison with numerical simulations of eqs. (7.2), using eq. (7.4) as an initial configuration (a finite-difference method, with Fourier transform scheme, described by Drummond [1983], was used), tests this analytical approximation and shows that it is indeed fairly close to a numerically exact solution; in particular, the shape of the bullet remains within 98% of its originally presumed shape after having propagated a large distance, as is shown in fig. 16. Three-dimensional LB solutions with axial symmetry have also been constructed in an approximate analytical form and succesfiilly tested in direct simulations, following a similar approach (Blaauboer, Kurizki and Malomed [2000]). Generally, they are not drastically different from their 2D counterparts described above. A challenging problem which remains to be considered is the construction of spinning light bullets in the 3D case (doughnut-shaped solitons, with a hole in the center, carrying an intrinsic angular momentum). Recently, spinning bullets were found by means of a sophisticated version of the variational approximation in a simpler 3D model, viz., the nonlinear Schrodinger equation with self-focusing cubic and self-defocusing quintic nonlinearities, by Desyatnikov, Maimistov
2, § 8]
Experimental prospects and conclusions
139
X
Fig. 16. The forward-propagating electric field of the two-dimensional "light bullet" in the Bragg reflector, \Sp\, vs. time r and transverse coordinate x, after having propagated the distance z = 1000. The parameters are rj = OA, 6 = 0.2, C = 0.1 and 0Q = -1000. The field is scaled by the constant /2/4ro/i«o-
and Malomed [2000]. Further direct simulations have demonstrated that these spinning bullets (unlike their zero-spin counterparts) are always subject to an azimuthal instability, that eventually splits them into a few moving zero-spin solitons, although the instability can sometimes be very weak (Mihalache, Mazilu, Crasovan, Malomed and Lederer [2000]). At present, it is not known whether spinning LBs can be completely stable in any 3D model. It is relevant to stress that two- (and three-) dimensional LB solutions of the variable-separated form, 2"+ ~ 2'_ ~ / ( r , £) • g(x), do not exist in the RABR model. Indeed, the substitution of this into eqs. (7.2a) and (7.2b) yields only a plane-wave solution of the form I± ^ exp (iAr -\- iBx), with constant A mdB.
§ 8. Experimental prospects and conclusions This review has focused on properties of solitons in RABR, combining a periodic refractive-index (Bragg) grating and a periodic set of thin active layers (consisting of two-level systems resonantly interacting with the field). It has been demonstrated that the RABR supports a vast family of bright gap solitons, whose properties differ substantially from their counterparts in periodic structures with either cubic or quadratic off-resonant nonlinearies reviewed in § 2. The same RABR can support, depending on the initial conditions, either dark or bright stable solitons, without any changes of the system parameters, which is a unique feature for nonlinear optical media (§§5, 6). Zero-velocity dark solitons can be found in an analytical form, as well as traveling dark
140
Optical solitons in periodic media
[2, § 8
solitons with a constant phase difference (^ :/r) of the background amplitudes across the soUton (§6). The latter property is a major difference with respect to dark solitons of the NLS equation, whose motion is supported by giving the background a nonzero wavenumber. Depending on the values of the parameters, the frequency band of the quiescent dark solitons coexists with one or two bands of the stable bright ones, without an overlap. Direct numerical simulations demonstrate that some darksoliton solutions are stable against arbitrary small perturbations, whereas others are unstable when they are close to the "dangerous" boundaries of their existence domain. A multidimensional version of the RABR model, corresponding to a periodic set of thin active layers placed at the maxima of the refractive index, which is modulated along the propagation direction of light has been considered too. It has been found to support stable propagation of spatiotemporal solitons in the form of two- and three-dimensional "light bullets" (LBs). The best prospect of realizing a RABR which is adequate for observing the solitons and light bullets discussed in §§5-7 is to use thin layers of rareearth ions (Greiner, Boggs, Loftus, Wang and Mossberg [1999]) embedded in a spatially periodic semiconductor structure (Khitrova, Gibbs, Jahnke, Kira and Koch [1999]). The two-level atoms in the layers should be rare-earth ions with the density of 10^^-10^^ cm~^, and large transition dipole moments. The parameter r/ can vary from 0 to 100 and the detuning is -10^^-10^^ s"^ Cryogenic conditions in such structures can strongly extend the dephasing time T2 and thus the soliton's or LB's lifetime, well into the \is range (Greiner, Boggs, Loftus, Wang and Mossberg [1999]), which would greatly facilitate the experiment. The construction of suitable structures constitutes a feasible experimental challenge. In a RABR with the transverse size of 10|im, LBs can be envisaged to be localized on the time and transverse-length scales, respectively, ^10"^^ s and 1 j^m. The incident pulse has uniform transverse intensity and the transverse diffraction is strong enough. One needs d^/hh^Xo < 1, where /abs, AQ and d are the resonant-absorption length, carrier wavelength, and the pulse diameter, respectively (Slusher [1974]). For /abs ^ 10"^ m and AQ ~ 10""^ m, one thus requires d < 10~^m, which implies that the transverse medium size L^ must be a few (uim. Effects of TLS dephasing and deexcitation in RABRs can be studied by substituting the values -\d - FiT^ for the frequency term -id in eqs. (4.47)(4.49) and the loss terms -riro(wo + 1) in eq. (4.50), -FxWi in eq. (4.51) and -r\(w2 - 2) in eq. (4.52). We have checked that these modifications
2, § 8]
Experimental prospects and conclusions
141
do not influence the qualitative behavior of the solutions on the time scale r < To < l/ri,2. Let us now discuss the experimental conditions for the realization of RABR solitons using quantum wells embedded in a semiconductor structure with periodically alternating linear index of refraction (Khitrova, Gibbs, Jahnke, Kira and Koch [1999]). We can assume the following values: the average refraction index is n^ ^ 3.6, the wavelength (in the medium) A ^ 232 nm, which corresponds to the angular frequency (Oc ^ 2.26 x 10^^ s"^ Excitons in quantum wells can, under certain conditions (such as low densities and proximity of the operating frequency to an excitonic resonance, see Khitrova, Gibbs, Jahnke, Kira and Koch [1999]) may be regarded as effective two-level systems (TLS's). We consider their surface density to be ^ 10^^-10^^ cm"^, which corresponds to a bulk density po ^ 10^^-10^^ cm~^. If we assume that the excitons are formed by electrons and holes displaced by ^ 1-10 nm, then the characteristic absorption time TQ defined in eq. (3.8) is TQ ^ 10"^^-10~^^ s, and the corresponding absorption length is CTQ/WO ^ 10-100 |im. The dephasing time for excitons discussed by Khitrova, Gibbs, Jahnke, Kira and Koch [1999] is l/r2 ^ 10~^^ s, which seems to be the chief limitation of the soliton lifetime for this system. The structures shown in figs. 6, 12, 13 and 15, occupying regions of approximately 100 absorption lengths, would require a device with a total width of approximately 1 mm to 1 cm, which corresponds to ?=^ 10^ to 10^ unit cells. The modulation of the refraction index can be as high as «i ?^ 0.3, so that the parameter rj (see eq. 4.13) can vary from 0 to 100. The unit of the dimensionless detuning 8 would represent a 10~^-10~^ fraction of the carrier frequency. The intensities of the applied laser field corresponding to 2"^- ^ 1 are then of the order 10^-10^ W/cm^. In the work by Khitrova, Gibbs, Jahnke, Kira and Koch [1999], the width of the active layers (quantum wells) is considered to be 5-20 nm, which corresponds to the parameter y^ (see eq. 4.29) in the range 10^^-2 x 10"^. In the simulations discussed by Opatrny, Malomed and Kurizki [1999], taking the largest of these values and the parameters as in fig. 12, i.e., t] = 0.6, 8 = - 2 , and x ^ 0.25, we have observed the time evolution of the system (4.36a)-(4.52). As the initial condition, both the DS solution corresponding to zero width of the active layers and the DS solution including the finite width correction, have been taken. In both cases, the evolution was quite regular over the observed time r ?^ 50, and the zero-width solution (with the quantities <S, V, A as given by eqs. 6.5, 5.2 and 5.15) started to change after r ^ 10. We can now sum up the discussion of experimental perspectives for the realization of RABR solitons: (a) The prospects appear to be good for
142
Optical solitons in periodic media
[2
gratings incorporating thin layers of rare-earth ions under cryogenic conditions, (b) The reahzation of these sohtons in excitonic superlattices would require much longer dephasing times than those currently achievable in such structures.
Abbreviations DS
dark soliton
GS
gap soliton
LB
light bullet
NDD
near-dipole-dipole
NLS
nonlinear Schrodinger
PBG
photonic band gap
QNLSE
quantum nonlinear Schrodinger equation
RABR
resonantly absorbing Bragg reflector
SHG
second-harmonic-generating
SIT
self-induced transparency
TLS
two-level system
ZV
zero-velocity
Acknowledgments We are grateful to M. Blaauboer for discussions and help. G.K. acknowledges the support of the EU (ATESIT) and US-Israel BSE TO. thanks the Deutsche Forschungsgemeinschaft for support. A.K. acknowledges support of the Thomas B. Thriges Center for Quantum Information.
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E. Wolf, Progress in Optics 42 © 2001 Elsevier Science B. V All rights reserved
Chapter 3
Quantum Zeno and inverse quantum Zeno effects by
Paolo Facchi and Saverio Pascazio Dipartimento di Fisica, Universitd di Bari and Istituto Nazionale di Fisica Nucleare, Sezione di Bari, 70126 Bari, Italy
147
Contents
Page § 1. Introduction
149
§ 2.
152
Two-level systems and Bloch vector
§ 3. Pulsed observation
155
§ 4.
Dynamical quantum Zeno effect
163
§ 5.
Continuous observation
167
§6.
Novel definition of quantum Zeno effect
173
§7.
Zeno effects in down-conversion processes
175
§ 8.
Genuine unstable systems and Zeno effects
191
§ 9.
Three-level system in a laser
field
199
§ 10. Concluding remarks
213
Acknowledgments
214
References
214
148
§ 1. Introduction Zeno and his master Parmenides lived about 2500 years ago in Elea, a small Italian town not far from Naples, in the Mediterranean region called "Magna Graecia". Parmenides, a profound and innovative philosopher, believed that senses are deceptive and our perception of reality in continuous change is an illusion. In his conception, there is a unique, indivisible Truth ("being") that does not undergo any change ("becoming") and cannot be decomposed into smaller entities. Zeno was Parmenides' most brilliant disciple and in order to support his master's ideas he would challenge the most "obvious" conclusions of common sense by putting forward many paradoxical examples. His captious arguments are ingenious and very famous. For example, he argued that Achilles, who starts running from point A, cannot reach a turtle that at the same time starts moving from point B, because when the former reaches B the latter has moved to C, and so on ad infinitum. Against the very idea of dividing an object into parts, he claimed that if a finite segment is made up of an infinite number of points then one runs into a contradictory conclusion: indeed, if a single point has a finite size, then the size of the segment is infinite; if, on the other hand, a single point has no size, their sum (the segment itself) cannot have a nonvanishing size. Zeno lacked the concept of infinitesimal. Modem infinitesimal calculus resolves the first paradox by introducing the concept of velocity as the derivative of position with respect to time: Achilles will reach the turtle in 2i finite time because it takes a very small time to cover a very small distance, where "very small" times and distances are infinitesimal of the same order. The solution of the second paradox is even subtler. An uncountable ensemble of points can have a nonvanishing size. At any rate, it is undeniable that his provocative arguments foreran very subtle concepts of infinitesimal calculus, such as derivatives, Riemann and Lebesgue measures. More to this, one should not forget that Zeno aimed at challenging the "obviousness" of common sense, in order to support the philosophy of his master Parmenides and bring to light the difficulties inherent in the very idea of "becoming". One of Zeno's paradoxes will be the object of the present investigation: A sped arrow never reaches its target, because at every instant of time, if we look at the arrow, we see that it occupies a portion of space equal to its own size. At any 149
150
Quantum Zeno effects
[3, § 1
given moment the arrow is therefore immobile, and by summing up many such "immobihties" it is clearly impossible, according to Zeno, to obtain motion. It is amusing that some quantum-mechanical states, under particular conditions, behave in a way that is reminiscent of this paradox. In this chapter we shall review the main features of the so-called quantum Zeno effect (von Neumann [1932], Beskow and Nilsson [1967], Khalfin [1968], Misra and Sudarshan [1977]). In very few words, the evolution of a quantummechanical state can be slowed down (or even halted in some limit) when very frequent measurements are performed on the system, in order to check whether it is still in its initial state: Zeno's quantum arrow (the wave function) does not move, if it is continuously observed. The interest in the quantum Zeno effect (QZE) has been revived, during the last decade, mainly because of some interesting proposals that made it liable to experimental investigation. Unlike previous studies, confined to a purely academic level, the investigation of the last few years has focused on practical experiments, possible applications, as well as theoretical implications and interpretative issues. In this chapter we shall first review the main features of the QZE and discuss some simple examples. We shall then concentrate our attention on more interesting physical situations and emphasize the occurrence of new physical phenomena, such as the "inverse" quantum Zeno effect. The quantum Zeno effect has been mainly investigated for oscillating systems (Cook [1988], Itano, Heinzen, Bollinger and Wineland [1990], Pascazio, Namiki, Badurek and Rauch [1993], Kwiat, Weinfurter, Herzog, Zeilinger and Kasevich [1995], Luis and Pefina [1996]), whose Poincare time is finite. However, the discussion cannot be limited to oscillating systems. New and somewhat unexpected phenomena are disclosed when one considers unstable systems, whose Poincare time is infinite (Bemardini, Maiani and Testa [1993], Facchi and Pascazio [1998], Maiani and Testa [1998], Joichi, Matsumoto and Yoshimura [1998], Alvarez-Estrada and Sanchez-Gomez [1999]). Unfortunately, in this case the analysis becomes more complicated and requires a quantum field theoretical framework. The QZE has recently become such a wide subject of investigation, that it is difficult to discuss all its multiple facets. In this chapter we shall therefore discuss only some of its aspects, by focusing our attention on quantum optics and quantum electrodynamics. As a general philosophy, we shall always start by considering simple physical systems and then extend our analysis to more complicated cases. As already emphasized, it would be wrong to limit the analysis to elementary situations (such as oscillating systems), because in doing so one would overlook a great deal of interesting physical effects. We will
3, § 1]
Introduction
151
therefore try to follow Einstein's precept: Things should be made as simple as possible, but not simpler. In this chapter we will often work in natural units {h = c= 1), but will put the physical constants back in the final formulas whenever it will be helpfiil to get a feeling for the numbers. After setting up the notation in § 2, we introduce in § 3 the fundamentals of the quantum Zeno and "inverse" quantum Zeno effect (IZE), by making use of elementary quantum-mechanical techniques. We shall first use the seminal formulation of QZE in terms of projection operators: This is the usual approach and makes use of what we might call a "pulsed" observation of the quantum state (Mihokova, Pascazio and Schulman [1997], Schulman [1998]). We then explain in § 4 that it is not necessary to use projection operators and nonunitary dynamics. A fiilly dynamical explanation of the QZE is possible, involving Hamiltonians and no projectors (Pascazio andNamiki [1994], Petrosky, Tasaki and Prigogine [1990]). In §5 we introduce the notion of "continuous" observation of the quantum state, e.g., performed by means of an intense field. Although this idea has been revived only recently (Mihokova, Pascazio and Schulman [1997], Schulman [1998]), it is contained, in embryo, in earlier papers (Kraus [1981], Peres [1980], Plenio, Knight and Thompson [1996]). This idea will lead us to a novel definition of QZE in § 6. We then discuss, in § 7, an interesting example of QZE in quantum optics, both with pulsed and continuous measurements. We look at a down-conversion process in a nonlinear crystal as a "decay" of a pump photon into a pair of signal and idler photons of lower frequency and study how the "decay" is modified by a measurement process of some sort. Interestingly, this system discloses the presence of an inverse Zeno effect. We shall see that by increasing the strength of the observation, the "decay" is sometimes accelerated rather than hindered. In § 8 we consider the QZE and IZE for bona fide unstable systems. This is a more complicated problem, because it requires the use of quantum field theoretical techniques. The study of a solvable (but significant) example enables us to understand the role played by the Weisskopf-Wigner approximation (Gamow [1928], Weisskopf and Wigner [1930a,b], Breit and Wigner [1936]) and the Fermi "golden rule" (Fermi [1932, 1950, I960]). Moreover, we shall see that for an unstable system, the form factors of the interaction play a fundamental role and determine the occurrence of a Zeno or an inverse Zeno regime, depending on the physical parameters describing the system. We finally investigate, in § 9, the intriguing possibility that the lifetime of an unstable quantum system be modified by the presence of a very intense electromagnetic field. We shall look at the temporal behavior of a three-level system (such as an atom or a molecule) illuminated by an intense laser field (Pascazio and Facchi [1999], Facchi and Pascazio [2000]) and see
152
Quantum Zeno effects
[3, § 2
that, for physically sensible values of the intensity of the laser, the decay can be enhanced. This will be interpreted as an inverse quantum Zeno effect. The choice of the subjects that appear in this review article reflects our own point of view on the quantum Zeno problem. This was inevitable and requires an apology. Some important aspects of this problem were either left out or only briefly mentioned. We regret, in particular, that some very important features of the temporal evolution, arisingfi-oma genuine quantum field theoretical analysis, are not even mentioned. Moreover, we will not explore other very important topics, such as neutron physics, irreversibility and deviationsfi*omMarkovianity. On the other hand, we intentionally did not discuss some academic issues of no practical interest. We shall not look in detail at the characteristics of the quantum-mechanical evolution law at short times (Beskow and Nilsson [1967], Khalfin [1968], Wilkinson, Bharucha, Fischer, Madison, Morrow, Niu, Sundaram and Raizen [1997]) and long times (Mandelstam and Tamm [1945], Fock and Krylov [1947], Hellund [1953], Namiki and Mugibayashi [1953], Khalfin [1957, 1958]). These are summarized in Nakazato, Namiki and Pascazio [1996] and will often be taken for granted. An excellent account of the most recent results on the QZE can be found in Home and Whitaker [1997] and Whitaker [2000]. Our attention will mainly be focused on quantum optics and quantum electrodynamics. Specific examples will be given particular importance and will therefore play a fiindamental role. We will not attempt to generalize, unless necessary. The leitmotif of this chapter is that the quantum Zeno effect is a dynamical phenomenon, that can be explained in terms of the Schrodinger equation, without making use of projection operators. We will implicitly assume, throughout our discussion, that a projection operator is just a shorthand notation that summarizes the effects of a much more complicated underlying dynamical process, involving a huge number of elementary quantummechanical systems (Namiki, Pascazio and Nakazato [1997]). This idea will constitute the "backbone" of our work. Only if this concept is fiilly elaborated and completely digested, can one realize that a broader definition of Zeno effect is required, that takes into account the very concept of continuous measurement, performed for example by a quantum field.
§ 2. Two-level systems and Bloch vector We start by considering a two-level system undergoing Rabi oscillations. This is the simplest nontrivial quantum-mechanical example, for it involves 2x2
3, § 2]
Two-level systems and Block vector
153
matrices and very simple algebra. One can think of an atom illuminated by a laser field whose frequency resonates with one of the atomic transitions, or a neutron spin in a magnetic field. We shall neglect the energy difference between the two states |ib). The (interaction) Hamiltonian reads /f, = D ( T , = 0 ( | + ) H + | - ) ( + | ) = ( ^ ^
^ ) ,
(2.1)
where ^ is a real number, Oj (j = 1,2,3) the Pauli matrices and
;.
H=;
(2.2)
are eigenstates of 03. We will use the above notation interchangeably. Let the initial state be
|V%) = I + ) = ( J ) ,
(2.3)
SO that the evolution yields \^l>t) = e-^^'\^Po) = cos(f20|+)-isin(i30|-) = ( _ f s i n ^ ^ ) •
^^.4)
In the following, we shall often make use of the rotating coordinates, introduced by Bloch [1946] and Rabi, Ramsey and Schwinger [1954], and of well-known computational techniques due to Feynman, Vernon and Hellwarth [1957]. In terms of the polarization (Bloch) vector R(t)={ilJt\o\ilJt)
= (RuR2,R3)',
(2.5)
where ^ denotes the transposed matrix, the Schrodinger equation reads ^ ( 0 = 2Qx R(tl
(2.6)
where Q = (Q,0,Of. The norm of the Bloch vector is preserved: \\R(t)\\ = 1,V^. See fig. 1.
(2.7)
Quantum Zeno effects
154
[3, §2
Fig. 1. The Poincare sphere and the Bloch vector.
The density matrix of a two-level system is expressed in terms of the Bloch vector according to the formula P=
P-+
P-
= ^(.l+R-a),
(2.8)
so that (2.9) where P± = p±± is the probability that the system is in level ± . Notice that Tr p = P+ + P. = I (normalization) and Tr{ pa) = R. Vice versa, the Bloch vector is readily expressed in terms of the density matrix:
/?,
=p^+p-^,
R2 = i ( p + - - p - + ) , R3 =p+^-p
(2.10)
^=P+-P-.
The level configuration and the dynamics of the oscillations are shown in fig. 2. Observe that the probability returns to its initial value after a time Tp = JT/Q. This is a very simple instance of Poincare recurrence time.
3, § 3 ]
Pulsed observation
l+>
155
0.5
Vtt/Tl
l-> Fig. 2. Rabi oscillations in a two-level system.
§ 3. Pulsed observation Let us introduce the fundamental features of the quantum Zeno effect. We shall follow the "historical" approach (von Neumann [1932], Beskow and Nilsson [1967], Khalfin [1968], Misra and Sudarshan [1977]), by considering "pulsed" measurements. The alternative notion of continuous measurement will be discussed in § 5. For the sake of simplicity we shall often refer to two-level systems. This will make our analysis more transparent. It goes without saying that more general and formal approaches are also possible (Nakazato, Namiki and Pascazio [1996], Home and Whitaker [1997]). 3.1. Survival probability under pulsed measurements We define the survival amplitude At) = {MH>t) = (V^ole-^^IV^o)
(3.1)
and the survival probability P{t)=\A(t)\^
= \{rPo\c-^'\tPo)\\
(3.2)
where H is the total Hamiltonian of the system. These quantities represent the amplitude and probability that a quantum system, initially prepared in state | V^o), is still in the same state at time t. An elementary expansion shows that the behavior of the survival probability at short times is quadratic P(t) = 1 -t^/rl + . . . ,
r~^ = {%\H^\%) - {^o\H\il^of.
(3.3)
For instance, with the Hamiltonian (2.1) one finds .4(0 = cos Qt,
(3.4)
156
Quantum Zeno effects
[3, § 3
P{t) = cos^ Qt,
(3.5)
rz
(3.6)
= Q-'.
The quantity iz is the "Zeno time" and seemingly yields a quantitative estimate of the short-time behavior. As we shall see in this chapter, this is a misleading estimate in many situations. Strictly speaking, Tz is simply the convexity of P(t) in the origin. Note that if one writes H = Ho+Hu
with Ho\m) = Wo\il^o), (V^ol^ilV^o) = 0,
(3.7)
the Zeno time reads T^'= {iPo\H^\%)-
(3.8)
Therefore iz depends only on the square of the off-diagonal part of the Hamiltonian. Let us now perform N measurements at time intervals r, in order to check whether the system is still in its initial state. The survival probability after the measurements reads
/'no=/'(rf=p(^y
N/ large large
-
^
//
*t
\ A'-^oo ^
)
(3.9) where ^ = A^r is the total duration of the experiment. The N ^^ oo limit was originally named limit of "continuous observation" and regarded as a paradoxical result (Misra and Sudarshan [1977]): Infinitely fi-equent measurements halt the quantum-mechanical evolution and "freeze" the system in its initial state. Zeno's quantum-mechanical arrow (the wave fiinction), sped by the Hamiltonian, does not move, if it is continuously observed. The investigation of the last few years has shown that the QZE is not paradoxical. Although the A/^ ^ oo limit must be considered as a mathematical abstraction (Ghirardi, Omero, Weber and Rimini [1979], Nakazato, Namiki, Pascazio and Ranch [1995], Venugopalan and Ghosh [1995], Pati [1996], Hradil, Nakazato, Namiki, Pascazio and Ranch [1998]), the evolution of a quantum system can indeed be slowed down for sufficiently large A^ (Itano, Heinzen, Bollinger and Wineland [1990], Petrosky, Tasaki and Prigogine [1990, 1991], Peres and Ron [1990], Ballentine [1991], Itano, Heinzen, Bollinger and Wineland [1991], Frerichs and Schenzle [1992], Inagaki, Namiki and Tajiri [1992], Home and Whitaker [1992, 1993], Pascazio, Namiki, Badurek and Ranch [1993], Blanchard and Jadczyk [1993], Altenmuller and Schenzle
3, §3]
Pulsed observation
157
Fig. 3. Evolution with frequent "pulsed" measurements: quantum Zeno effect. The dashed (solid) line is the survival probability without (with) measurements.
0(5x^)
Fig. 4. Short-time evolution of phase and probability.
[1994], Pascazio and Namiki [1994], Schulman, Ranfagni and Mugnai [1994], Berry [1995], Beige and Hegerfeldt [1996], Kofman and Kurizki [1996], Schulman [1997], Thun and Pefina [1998]). The Zeno evolution is shown in fig. 3. In a few words, the QZE is ascribable to the following mathematical properties of the Schrodinger equation. In a short time 8r ^ 1//V, the phase of the wave function evolves like 0((5r), while the probability changes by 0((5r^), so that
pW(Oc:^
•2x1 ^^ [\-0{\/N^)]
N^oo
(3.10)
This is sketched in fig. 4; it is a very general feature of the Schrodinger equation. 3.2. Quantum Zeno and Inverse quantum Zeno effects It is convenient to rewrite eq. (3.9) in the following way {t = Nr) P^''\t) = P(rf
= exp(7VlogP(r)) = exp(-7eff(r)0,
(3.11)
158
Quantum Zeno effects
[3, §3
0 n T2 Fig. 5. (a) Determination of r*. The solid line is the survival probability, the dashed line is the exponential e~'^^ and the dotted line is the asymptotic exponential Ze~^^ in eq. (3.17). (b) Quantum Zeno vs. inverse Zeno ("Heraclitus") effect. The dashed line represents a typical behavior of the survival probability P{t) when no measurement is performed: the short-time Zeno region is followed by an approximately exponential decay with a natural decay rate y. When measurements are performed at time intervals r, we get the effective decay rate YeffiT). The solid lines represent the survival probabilities, and the dotted lines their exponential interpolations, according to eq. (3.11). For Ti < r* < 12 the effective decay rate YeffirO [7eff(^2)] is smaller (QZE) [larger (IZE)] than the "natural" decay rate y. When r = r* one recovers the natural lifetime, according to eq. (3,15).
where we introduced an effective decay rate (3.12)
yeff(r) = - - l o g P ( r ) .
For instance, for times r such that P(T) :^ exp(-r^/r|) with good approximation, one easily checks that yeff is a linear function of r yeffCT") ^
for
0.
(3.13)
Notice that 7eff(^) in eq. (3.12) represents the effective decay rate of a system that evolves freely up to time r and is measured at time r. One expects to recover the "natural" decay rate y (if it exists), in agreement with the Fermi "golden" rule, for sufficiently long times, i.e., after the initial quadratic region is over yeff(r) •''2^' ' Y-
(3-14)
The quantitative meaning of the expression "long" in the above equation represents an interesting conceptual problem and will be tackled in § 8. Suffice it to say, at this stage, that TZ is not the right time scale. We now concentrate our attention on a truly unstable system, with decay rate y. We ask whether it is possible to find a finite time r* such that yeff(r*) = y.
(3.15)
If such a time exists, then by performing measurements at time intervals r* the system decays according to its "natural" lifetime, as if no measurements were performed. By eqs. (3.15) and (3.12) one gets P(T*) = e-'^^*,
(3.16)
i.e., r* is the intersection between the curves P(t) and e~^^ Figure 5 illustrates
3, § 3 ]
Pulsed observation
159
Fig. 6. Study of the case Z > \. The soHd Hne is the survival probability, the dashed line is the renormalized exponential e"^^ and the dotted line is the asymptotic exponential ^e~^^ (a) If P(0 and e"^^ do not intersect, then no finite solution r* exists, (b) If P{t) and e~^^ intersect, then a finite solution r* exists. (In this case there are always at least two intersections.)
an example in which such a time r* exists. By looking at this figure, it is evident that if r = Ti < r* one obtains a QZE. Vice versa, if r = r2 > r*, one obtains an inverse Zeno effect (IZE). In this sense, r* can be viewed as a transition time fi*om a quantum Zeno to an inverse Zeno effect. Paraphrasing Misra and Sudarshan (Misra and Sudarshan [1977]) we can say that r* determines the transition from Zeno (who argued that a sped arrow, if observed, does not move) to Heraclitus (who replied that ever5^hing flows). We shall see that in general it is not always possible to determine r*: eq. (3.15) may have no finite solutions. This will be thoroughly discussed in the following, but it is interesting to anticipate some general conclusions. As we shall see in §§ 7 and 8, for an unstable system and for sufficiently "long" times (the definition of "long" times will be sharpened later) the survival probability reads with very good approximation P{t) = \A(t)\^ c^ ZQ-
(3.17)
where Z, the intersection of the asymptotic exponential with the ^ = 0 axis, is the wave fixnction renormalization and is given by the square modulus of the residue of the pole of the propagator. We claim that a sufficient condition for the existence of a solution r* of eq. (3.15) is that Z < I. This is easily proved by graphical inspection. The case Z < I is shown in fig. 5a: P(t) and e~^^ must intersect, since according to (3.17) P(t) ^ ZQ~^^ for large t, and a finite solution r* can always be found. The other case, Z > 1, is shown in fig. 6. A solution may or may not exist, depending on the features of the model investigated. We shall come back to the Zeno-Heraclitus transition in §§ 7 and 8. The occurrence of an inverse Zeno effect has been discussed by several authors, in different contexts (Pascazio [1996], Schulman [1997], Pascazio and Facchi [1999], Kofinan and Kurizki [1999, 2000], Facchi and Pascazio [2000], Facchi, Nakazato and Pascazio [2001]). There are situations (e.g., oscillatory systems, whose Poincare time is finite) where y and Z cannot be defined. As we shall see, these cases require a different
160
Quantum Zeno effects
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treatment, for the very definition of Zeno effect becomes somewhat delicate. This will be discussed in §§ 6-8. 3.3. Pitfalls: "repopulation" and conceptual difficulties The quantum Zeno effect has become very popular during the last decade, mainly because of an interesting idea due to Cook (Cook [1988]), who proposed to test the QZE with a two-level system, and the subsequent experiment performed by Itano and collaborators (Itano, Heinzen, Bollinger and Wineland [1990]). This experiment provoked a very lively debate and was discussed by many authors (Petrosky, Tasaki and Prigogine [1990, 1991], Peres and Ron [1990], Ballentine [1991], Itano, Heinzen, Bollinger and Wineland [1991], Frerichs and Schenzle [1992], Inagaki, Namiki and Tajiri [1992], Home and Whitaker [1992, 1993], Blanchard and Jadczyk [1993], Pascazio, Namiki, Badurek and Ranch [1993], AltenmuUer and Schenzle [1994], Pascazio and Namiki [1994], Schulman, Ranfagni and Mugnai [1994], Berry [1995], Beige and Hegerfeldt [1996], Schulman [1997], Thun and Pefina [1998]). However, we shall follow here a different route: rather than analyzing Cook's proposal and the related experiment, we shall consider a physically equivalent situation that better suits our discussion and can be easily compared to the analysis of the following sections. The central mathematical quantity considered by Misra and Sudarshan (Misra and Sudarshan [1977]) is "the probability V(0,T;po) that no decay is found throughout the interval A = [0, T] when the initial state of the system was known to be po." (Italics in the original. Some symbols have been changed.) In the notation of § 3.1, this reads V(0,T;po)=\im P^^\Ty
(3.18)
N—^oc
Notice that the above-mentioned "survival probability" is the probability of finding the system in its initial state po at every measurement, during the interval A. This is a subtle point, as we shall see. Consider a three-level (atomic) system, shined by an rf field offi-equencyD, that provokes Rabi oscillations between levels |+) and | - ) . The equations of motion (2.6)-(2.7), with initial condition (in this section we omit the symbol ^ of vector Transposition) R(0) ~ (0,0,1) (only level |+) is initially populated), yield R(t) = (0, sin 2Qt, cos 2Qt).
(3.19)
3, § 3]
Pulsed observation
161
If the transition between the two levels is driven by an on-resonant Jt/2 pulse, of duration T=^,
(3.20)
one gets R{T) = (0,0,-1), so that only level |-) is populated at time T. Perform a measurement at time r = T/N = Jt/INQ, by shining on the system a very short "measurement" pulse, that provokes transitions from level |-) to a third level |M), followed by the rapid spontaneous emission of a photon. The measurement pulse "projects" the atom onto level |-) or |+) and "kills" the offdiagonal terms p ± ^ of the density matrix, while leaving unaltered its diagonal terms p±±, so that, from eq. (2.10),
Then the evolution restarts, always governed by eq. (2.6), but with the new initial condition R^^\ After N measurements, at time T = Nr = Jt/2Q, R(T) = (o, 0, cos^ - ) = R^"^^
(3.22)
and the probabilities that the atom is in level |+) or |-) read (see eq. 2.9) Vi^\T)
= 1 ( l + 7 ? f ) = 1 ( l + cos^ I ) ,
r(N)^T) = \ (l -Rf)
= \(\-
cos^ I ) ,
(3.23) (3.24)
respectively Since V^+\T) ^ 1 and V(_^\T) -> 0 as A^ -^ oo, this looks like a quantum Zeno effect. However, it is not the quantum Zeno effect a la Misra and Sudarshan: eq. (3.23) [(3.24)] expresses only the probability that the atom is in level |+) [|-)] at time T, after N measurements, independently of its past history. In particular, eqs. (3.23)-(3.24) take into account ihQpossibility that one level gets repopulated after the atom has made transitions to the other level. In order to shed light on this rather subtle point, let us look explicitly at the first two measurements. After the first measurement, by eq. (3.21), Rf = c4 + s4
c
V^^^ = 2s^c^
2
V- = 0
j,m = ,2
t=0
t = n/N
2/
t=2n/N
Fig. 7. Transition probabilities after the first two measurements for an oscillating system [s = sm(jt/2N) and c = cos(jt/2N)].
survival probability (3.18). The correct formula for the survival probability, in the present case, is obtained by considering only the « = 0 term in eq. (3.29):
PT\T) = cos^
27V'
(3.30)
This is a bona fide "survival probability", namely the probability that level |+) is populated at every measurement, at times nx = nT/N (n= 1,..., A^). The conclusions drawn in this section are always valid when the temporal behavior of the system under investigation is of the oscillatory type and no precautions are taken in order to prevent repopulation of the initial state (Nakazato, Namiki, Pascazio and Ranch [1996]). For instance, this problematic feature is present in the interesting proposal by Cook [1988] and the beautiful experiment by Itano, Heinzen, Bollinger and Wineland [1990]. On the other hand, no repopulation of the initial state takes place in other experiments involving neutron spin (Pascazio, Namiki, Badurek and Ranch [1993]) or photon polarization (Kwiat, Weinfurter, Herzog, Zeilinger and Kasevich [1995]). We have seen that P^f\T), in eq. (3.30), is a bona fide survival probability, but Vf^\T\ in eq. (3.23) is not (at least not according to Misra and Sudarshan's definition). However, both quantities tend to the same limiting value 1 as A/^ ^ oo and for large N the evolution is, in fact, hindered. We are therefore led to wonder whether it would not be meaningfixl to extend the notion of QZE beyond Misra and Sudarshan's definition of survival probability. This will be the subject of §6.
§ 4. Dynamical quantum Zeno effect In the usual formulation of QZE the measurement process is schematized by making use of projection operators a la von Neumann (Copenhagen
164
Quantum Zeno effects
[3, § 4
c+l+>®|l^)
(c+|+) + c_|-))®|l,)
7 C_|->®|1^ Fig. 8. The generalized spectral decomposition.
interpretation), without endeavoring to shed Hght on the underlying dynamics. However, a quantum-mechanical measurement is a very complicated physical process, taking place in a finite time and involving complex (macroscopic) physical systems. It is possible to give a dynamical explanation of the Zeno effect (Pascazio and Namiki [1994], Pascazio [1997]), that involves only the Schrodinger equation and makes no use of projection operators. Let us briefly sketch how this is accomplished by introducing the notion of "generalized spectral decomposition" (GSD). Consider again a two level system, prepared in a superposed state. A GSD is a d5mamical (Hamiltonian) process by which different states of the system become associated (entangled) with different external "channels" (e.g., different degrees of fi-eedom of a larger system). See fig. 8. One can think, for example, of a two-level atomic system getting entangled with different photon states of the electromagnetic field. The notion of "spectral decomposition" was introduced by Wigner [1963], who considered the Stem-Gerlach decomposition of an initial spin state, where each component of the spin becomes associated with a different wave packet. It is worth observing that the external channels the system gets entangled with need not be "external": for example, different wave packets of the system itself can act as "external" degrees of fi-eedom. A GSD is realized by the following Hamiltonian:
//GSD(0
= g{t) [|+)(+|c7^ + \-){-\Oy] Oa = g(t)H\
f \(t) dt = \jt, Jo
(4.1) where the interaction is switched on during the time interval [0, to], g is a. real
3, § 4]
Dynamical quantum Zeno effect
165
function, a^ = o^ (the index ^ = a,P,y labels the channel infig.8) and the effect of (7^ is defined by o,\0,) = \l,),
a^|l,.) = |0^>,
(4.2)
SO that if there is a "particle" in channel fi the operator a^ destroys it, while if there is no particle, a^ creates one. The effect of a^ (V/i) is therefore identical to that of the first Pauli matrix. We set [o^,Oy]=0.
(4.3)
The action of the Hamiltonian //QSD is ^GSD(c+|+)+c_|-))0|la,O^,Oy)a (c+|+>0|O«,l^,Oy>+c_|-)0|O«,O^,ly» (4.4) and consists in sending the |+) (|-)) state of the system in the upper (lower) channel infig.8, thus performing a GSD. In general, the only effect of a GSD is to set up a perfect correlation between the two states of the system and different external channels (namely, a univocal and unambiguous correspondence between different states of the system and different external channels). This is easily accomplished: the evolution engendered by i/osD can be explicitly calculated (Pascazio and Namiki [1994]) and the result is exp - i
['HGSD{t')dt' ( c ^ | + ) + c _ | - ) ) 0 | l « )
^^^^
Jo
= - i (c+|+) (8) |1^) +c_|-) (g) |ly))
(t > to),
where we suppressed all Os for simplicity. A projection operator represents an instantaneous measurement. This is clearly a very idealized situation that cannot correspond to a real physical process, taking place at a microscopic level. The problem is therefore to understand how we can simulate such an instantaneous and unphysical process in our analysis, that makes use only of unitary evolutions. We observe that, in general, a GSD must take place in a very short time. Obviously, the term "very short time" must be understood at a macroscopic level of description, because the time microscopically required to efficaciously perform a GSD can be very long. Therefore, if we restrict our analysis to a macroscopic level of description, we can describe an (almost) instantaneous GSD by means of the so-called impulse approximation to^O\ (4.6) / ''g(t)dt='.Jt, . , Jo which roughly amounts to setting g(t) -^ ^Jtb(t) as to -^ 0, where 6 is the Dirac fiinction/J'8(0 = 1. This is our alternative description of a von Neumann-like
166
Quantum Zeno effects
[3, § 4
instantaneous projection. It is a good approximation of the physical situation whenever ^ is much shorter than the characteristic time of the free evolution of the system under observation. By making repeated use of GSDs it is very simple to get quantum Zeno dynamics. A general proof is given by Pascazio and Namiki [1994] (a somewhat simpler version can be found in Pascazio [1997]), but here let us only sketch the main idea by looking at the example (2.1). The initial state (2.3), that we rewrite by including the external channel (wave packet) in the description, |^o) = | + ) 0 | l « ) ,
(4.7)
evolves after a short time r into state (2.4): 1^,) = e-'^'H^o) = [cos(Or)|+) -isin(r2T)|-)] 0 |1«).
(4.8)
The GSD then yields (for ^ < ^2"^) l^r + ro)=exp -i r HG^^{t')dt' \^r) Jo /o oc cos(Dr)|+) (g) \lp) -ism(QT)\-)
(4.9) (g) |ly).
apart from a phase factor. Observe that the quantum coherence is perfectly preserved, during this evolution. At the next "step" of the evolution, channels (3 and y become new incoming channels and the system evolves again under the action of//i for a time r and ^GSD for a time to. After N steps the final wave function reads ^^(r..o)) = n
| e x p [ - i 2 V S D ( ^ 0 c i / ' ] exp[-i//ir]| \Wo)
oc cos^(Or)|+) (g
'ft
(4.10)
\lf^)+0(N-^),
An)
where //QSQ is the Hamiltonian that performs a generalized spectral decomposition at the «th step and | iL^^) (all Os were suppressed) represent the wave packet traveling in channel j8 at step N. Note that N(T + to) is kept finite. The contribution of all the other channels is 0(A^"^): a QZE is obtained because the particle, initially in state (4.7), ends up with probability [1-0(1/A^^)]^ ~ 1-0(1/7V)
(4.11)
in state |+) 0 |1^ ^). The "external" degrees of freedom are irrelevant and can be traced out (or recombined with the initial one).
3, § 5]
Continuous observation
167
We would like to emphasize that the very dynamical mechanism leading to QZE is curious: QZE is obtained via repeated use of generalized spectral decompositions HGSD% ^ven though the interaction Hamiltonian Hi "attempts" to drive |+) into |-) for a finite time Ni. This is probably the reason why QZE is often considered a counterintuitive phenomenon.
§ 5. Continuous observation A projection a la von Neumann (von Neumann [1932]) is a handy way to "summarize" the complicated physical processes that take place during a quantum measurement. A measurement process is performed by an external (macroscopic) apparatus and involves dissipative effects, that imply an exchange of energy with and often a flow of probability towards the environment. The external system performing the observation need not be a bona fide detection system, namely a system that "clicks" or is endowed with a pointer. It is enough that the information on the state of the observed system be encoded in the state of the apparatus. For instance, a spontaneous emission process is often a very effective measurement process, for it is irreversible and leads to an entanglement of the state of the system (the emitting atom or molecule) with the state of the apparatus (the electromagnetic field). The von Neumann rules arise when one traces away the photonic state and is left with an incoherent superposition of atomic states. We shall now introduce several alternative descriptions of a measurement process and discuss the notion of continuous measurement. This is to be contrasted with the idea of pulsed measurements, discussed in §3. Both formulations lead to QZE. 5.1. Mimicking the projection with a non-Hermitian Hamiltonian It is useful for our discussion on the QZE and probably interesting on general grounds to see how the action of an external apparatus can be mimicked by a non-Hermitian Hamiltonian. Let us consider the following Hamiltonian:
^^^(S -i?r)^"'^^^^'''' ^ = (^'0,iK)\
(5.1)
that yields Rabi oscillations of frequency Q, but at the same time absorbs away the |-) component of the Hilbert space, performing in this way a "measurement".
168
Quantum Zeno effects
[3, §5
Pit) 1
, • • • ' • • " " " '
_i;;^= 10Q \
N.
/
l+>\\ Q.
0.5
/ ,/
\
Y=AV
0.5
Q^/TT
Fig. 9. Survival probability for a system undergoing Rabi oscillations in the presence of absorption {V = 0.4,2, \0Q). The gray line is the undisturbed evolution (V = 0).
Due to the non-Hermitian features of this description, probabihties are not conserved: we are concentrating our attention only on the |+) component. An elementary SU(2) manipulation yields the following evolution operator: h • iJ
^-m = ^-vt cosh(/zO - i ^ — sinh(/zO
(5.2)
where h = VV^ - Q^ and we supposed V > Q. Let the system be initially prepared in the state (2.3): the survival amplitude reads
=e
sinh(VF2-^2^)
cos h(VF2-r22/) + y/V^-Q^ -{V-\/V^-Q^)t
^1 1+
(5.3)
2
V
-{VWV^-Q^)t
The above results are exact and display some interesting and very general aspects of the quantum Zeno dynamics. The survival probability P{t) = \A(t)\^ is shown infig.9 for F = 0.4,2,10^. As expected, probability is (exponentially) absorbed away as ^ —» oo. However, as V increases, by using eq. (5.3), the survival probability reads
/'(0~(l + g)exp(-^r),
(5.4)
and the effective decay rate 7eff(F) = ^^/V becomes smaller, eventually halting the "decay" (absorption) of the initial state and yielding an interesting example
3, § 5]
Continuous observation
169
of QZE: a larger V entails a more "effective" measurement of the initial state. We emphasize that the expansion (5.4) becomes valid very quickly, on a time scale of order V'^. Notice that this example is not affected by the repopulation drawback described in § 3.3 (once the probability is absorbed away, it does not flow back to the initial state). 5.2. Coupling with aflat continuum We now show that the non-Hermitian Hamiltonian (5.1) can be obtained by considering the evolution engendered by a Hermitian Hamiltonian acting on a larger Hilbert space and then restricting the attention to the subspace spanned by {|+), | - ) } . Consider the Hamiltonian i / = 0(|+)(H + |-)(+|) + | d a ; H ^ ) H + y ^ / d a > ( | - ) H + |w)(-|), (5.5) which describes a two-level system coupled to the photon field in the rotatingwave approximation. The state of the system at time t can be written as \xl)t) =x(t)\-^)+y(t)\-)
+ Jdajz{(o,t)\a)),
(5.6)
and the Schrodinger equation reads ix(t) = Qy{tX iy(t) = Qx{t) + \hr-
/ dco z(ft), t),
(5.7)
i z(a;, t) = o)z{w, 0 + y ^ >'(^)By using the initial condition x(0) = 1 and ^(0) = Z{(D, 0) = 0 one obtains z(ca, t) = -i\hf- I dr e-^^('-"V(^) V In Jo
(5.8)
iy{t) = ^ ^ ( 0 - i ^ fdcofdr
(5.9)
and e-^"(^-^V(r) = Qx(t)-i^y(ty
Therefore z(a),t) disappears from the equations and we get two first-order differential equations for x and y. The only effect of the continuum is the
170
Quantum Zeno effects
[3, § 5
appearance of the imaginary frequency -iF/l. Incidentally, this is ascribable to the "flatness" of the continuum [there is no form factor or frequency cutoff in the last term of eq. (5.5)], which yields a purely exponential (Markovian) decay of>^(0. In conclusion, the dynamic in the subspace spanned by |+) and |-) reads ijc(0 = Qy{t\
{y{t) = -i^y-\-Qx(t).
(5.10)
Of course, this dynamic is not unitary, for probability flows out of the subspace, and is generated by the non-Hermitian Hamiltonian / / = f2(|+)(-| + | - > ( + | ) - i | | - ) ( - | .
(5.11)
This Hamiltonian is the same as (5.1) when one sets F = 4V. QZE is obtained by increasing F: a larger coupling to the environment leads to a more effective "continuous" observation on the system (quicker response of the apparatus), and as a consequence to a slower decay (QZE). The processes described in this section and the previous one can therefore be viewed as "continuous" measurements performed on the initial state. The non-Hermitian term - 2 i F is proportional to the decay rate F of state | - ) , quantitatively F = 4V. Therefore, state |-) is continuously monitored with a response time l/F: as soon as it becomes populated, it is detected within a time 1/r. The "strength" T = 4F of the observation can be compared to the frequency T~^ = (t/Ny^ of measurements in the "pulsed" formulation. Indeed, for large values of F one gets from eq. (5.4) 4Q^ 4 7eff(r)--- = ^ - , i
r^i
for
r-.oc,
(5.12)
which, compared with eq. (3.13), yields an interesting relation between continuous and pulsed measurements (Schulman [1998]) r - -
4 T
4N =_ .
(5.13)
t
5.3. Continuous Rabi observation The two previous examples might lead the reader to think that absorption and/or probability leakage to the environment (or in general to other degrees
3, § 5]
Continuous observation
171
of freedom) are fundamental requisites to obtain QZE. This expectation would be incorrect. Let us analyze a somewhat different situation by coupling one of the two levels of the system to a third one, which will play the role of a measuring apparatus. The (Hermitian) Hamiltonian is /fi = 0 ( | + ) H + |-)(+|) + ^ ( | - ) ( M | + |M>(-|)=
/O O \0
Q 0 ^
0\ K], 0/
(5.14)
where AT G R is the strength of the coupling to the new level M, and (+1 = (1,0,0),
(-1 = (0,1,0),
{M\ = (0,0,1).
(5.15)
This is probably the simplest way to include an "external" apparatus in our description: as soon as the system is in | - ) , it undergoes Rabi oscillations to \M). Similar examples were considered by Peres [1980] and Kraus [1981]. We expect level \M) to perform better as a measuring apparatus when the strength K of the coupling becomes larger. The above Hamiltonian is easily diagonalized. Its eigenvalues and eigenvectors are Ao = 0,
\uo) =
, ^^
^ 1 ^ 1 '
V2(K^^m
\
K
J
Let the initial state be
The evolution is easily computed:
(5.18) and the survival probability reads p(t)=
1
2
K^ + Q^ cos(y/K^ + Q^ t)
This is shown in fig. 10 forK=
1,3,9Q.
(5.19)
172
Quantum Zeno effects
[3, § 5
p(t) 1
^;-5!^-S^>^
^^^''~''^, F. Notice that two opposite tendencies compete in eqs. (7.40): an elliptic structure, leading to oscillatory behavior, governed by the coupling parameter /f, 'di = -K^Qi,
b = -K^b
(7.41)
and a hyperbolic structure, yielding exponential behavior, governed by the nonlinear parameter T, hs = r^as,
di = r^at.
(7.42)
The threshold between these two regimes occurs for F ^ K. The system of equations (7.40) is easily solved and the number of output signal photons, which is the same as the number of pump photons decays, reads (al(0«.(0> = ^ sin^ xt + ^ ( 1 - cosxtf, A
(7.43)
A
where x == V/c*^ ~ F^. Unlike the case of phase matched down-conversion (7.7), the exchange of energy between all modes now becomes periodical when K > F. As the linear coupling becomes stronger, the period of the oscillations gets shorter and the amplitude of the oscillations decreases as /c"^, namely {al(t)as(t))
F^ F^ T sin^ Kt+—r(l/C^
K^
4F^ Kt cos Ktf = —^ sin^ — /f^
2
( / c > F).
(7.44) For strong coupling the down-conversion process is completely frozen, the medium becomes effectively linear and the pump photons propagate through it without "decay". [In the regime of very large /c, however, the coupled modes theory breaks down and some other experimental realization of the Hamiltonian (7.39) should be found.] Notice that in this situation, even if t is increased, the number of down-converted photons is bounded [compare with the opposite
186
Quantum Zeno effects
[3, § 7
case (7.7)]. This is QZE in the following sense: by increasing the coupling with the auxiliary mode, a better "observation" of the idler mode (and therefore of the decay of the pump) is performed and the evolution is hindered. There is an intuitive explanation of this behavior: since the linear coupling changes the phases of the amplitudes of the interacting modes, the constructive interference yielding exponential increase of the converted energy (7.7) is destroyed and down-conversion is frozen (see § 7.5 in the following). In agreement with the final part of §5.3, by comparing eq. (7.44) with eqs. (7.27)-(7.29), we find that the linear coupling is effective as the square root of the number of pulsed measurements, namely
.-^-f.
,7.45,
Consider now the Hamiltonian (7.39) when K = 0, describing down-conversion with phase mismatch A. It is apparent that the coupling and the phase mismatch influence the down-conversion process in the same way. Indeed for large values of the phase mismatch A it is easy to find from eq. (7.34) that («t(?K(0) ~ ^
sin^ y
(A » n,
(7.46)
which is to be compared with eq. (7.44). The interesting interplay between coupling K and mismatch A will be investigated in the following subsection. 7.4. Competition between the coupling and the mismatch In the previous section we saw that the nonlinear interaction was affected by both linear coupling and phase mismatch in the same way. The effectiveness of the nonlinear process drops down under their action. In this section we show that when both disturbing elements are present in the dynamics of the downconversion process, the linear coupling can compensate for the phase mismatch and vice versa, so that the probability of emission of the signal and idler photons can almost return back to its undisturbed value. We start from the equations of motion generated by the full interaction Hamiltonian (7.39) Us = -iFa], at = -lAai - ira] - iKb,
(A^O,K^
0).
(7.47)
b = -iAb - iKai Although it is easy to write down the explicit solution of the system (7.47), we shall provide only a qualitative discussion of the solution. The main features are
3, § 7]
Zeno effects in down-conversion processes
187
then best demonstrated with the help of a figure. Eliminating idler and auxiliary mode variables fi-om eq. (7.47) we get a differential equation of the third order for the annihilation operator of the signal mode. Its characteristic polynomial (upon substitution as{t)=^ exp(-iAO) A^ + lAX^ + {A^ -K^ + r^)X + r^A,
K^O,
(IAS)
is a cubic polynomial in A with real coefficients. An oscillatory behavior of the signal mode occurs only provided the polynomial (7.48) has three real roots (casus irreducibilis), i.e., if its determinant D obeys the condition Z) < 0. Expanding the determinant in the small nonlinear coupling parameter F and keeping terms up to the second order in F we obtain D--—
[(K^ - A^f - (5A^ + 3K^)F^] ,
F T is of main interest here. Hence we can, eventually, drop F^ in eq. (7.50). The resulting intervals are hyperbohc behavior: oscillatory behavior:
K ^ {A- \[lF,A + \/lF), /r G (0, Zi - v ^ r ) U (z\ + \ / 2 r , oo).
n ^\\
The behavior of the mismatched down-conversion process is shown in fig. 14a for a particular choice of A. In absence of linear coupling the down-converted light shows oscillations and the overall effectiveness of the nonlinear process is small due to the presence of phase mismatch A. However, as we switch on the coupling between the idler and auxiliary mode, the situation changes. By increasing the strength K of the coupling the period of the oscillations gets longer and their amplitude larger. When K becomes larger than A - VlF, the oscillations are no longer seen and the intensity of the signal beam starts to grow monotonously. We can say that in this regime the initial nonlinear mismatch has been compensated by the coupling.
[3, §7
Quantum Zeno effects
(b)
(a)
Fig. 14. (a) Mean number of signal photons («s) behind the nonlinear medium as a function of interaction length t and strength K of the linear coupling. The nonlinear mismatch is Z\=10r. (b) Interplay between linear coupling and phase mismatch. The mean number of signal photons {«s) behind the nonlinear medium of length Ft = 1.5 is shown versus the strength K of the linear coupling and the nonlinear mismatch A. A significant production of signal photons, viewed as a "decay" of the initial state (vacuum), is a clear manifestation of an inverse Zeno effect.
The interplay between nonlinear mismatch and linear coupling is illustrated in fig. 14b. A significant production of signal photons is a clear manifestation of IZE. In accord with the observations of Luis and Sanchez-Soto [1998] and Thun and Pefina [1998], such an IZE occurs only if a substantial phase mismatch is introduced in the process of down-conversion. This is the condition (7.32) for having Z < 1 in the decay of the vacuum state. It is worth comparing the interesting behavior seen in fig. 14b with the Zeno and inverse Zeno effects in a sliced nonlinear crystal discussed in § 7.3. The coupling parameter K here plays a role similar to the number of slices A^, so that one can state again that K ~ \/N in the sense of § 5.
7.5. Dressed modes We now look for the modes dressed by the interaction K. This will provide an alternative interpretation and a more rigorous explanation of the result obtained above. Let us diagonalize the Hamiltonian (7.39) with respect to the linear coupling. It is easy to see that in terms of the dressed modes ^ (at + b)/V2,
d = {at - b)/V2,
(7.52)
3, § 7 ]
189
Zeno ejfects in down-conversion processes C
A +K
A - K
d
r~ Fig. 15. Energy scheme of a mismatched down-conversion process subject to linear coupling. The bottom solid lines denote a resonant process.
the Hamiltonian (7.39) reads
r
r
HK = 0)cC^c + o)dd^d+—i={alc^+asC)+ —=(ald^-\-asd), v2 v2 where the dressed energies are a)c = A + K,
cOd =
A-K.
(7.53)
(7.54)
The coupling of the idler mode at with the auxiliary mode b yields two dressed modes c and d that the pump photon can decay to. They are completely decoupled and due to their energy shift (7.54), exhibit a phase mismatch A±K, Since the phase mismatch effectively shortens the time during which a fixedphase relation holds between the interacting beams, the amount of converted energy is smaller than in the ideal case of perfectly phase-matched interaction, A = 0. A strong linear coupling then makes the subsequent emissions of converted photons interfere destructively and the nonlinear interaction is frozen. In this respect the disturbances caused by the coupling and by frequently repeated measurements are similar and we can interpret the phenomenon as a QZE. The energy scheme implied by the Hamiltonian (7.53) is shown in fig. 15. Under the influence of the coupling with the auxiliary mode b the mismatched downconversion splits into two dressed energy-shifted interactions. It is apparent that when K = ±A, one of the two interactions becomes resonant. The other one is "counter-rotating" and acquires a phase mismatch 2A, yielding oscillations. Also, the amplitude of such oscillations decreases as A~^ and the mode output becomes negligible compared to the other one. The use of the rotating wave
190
Quantum Zeno effects
[3, § 7
approximation in eq. (7.53) is fiilly justified in this case and the system is easily solved. The output signal intensity reads {al{t)as{t)) = sinh^ ( ^ M
(^ = =^^' ^^ > 1)
C^-^^)
(compare with eq. 7.7). The linear coupling to an auxiliary mode compensates for the phase mismatch up to a change in the effective nonlinear coupling strength
r -^ r/Vi. As a matter of fact, the condition K = ±A can also be interpreted as a condition for achieving the so-called quasi-phase-matching in the nonlinear process. A quasi-phase-matched regime of generation (Armstrong, Bloembergen, Ducuing and Pershan [1962], Fejer, Magel, Jundt and Byer [1992], Chirkin and Volkov [1998]) is usually forced by creating an artificial lattice inside a nonlinear medium, e.g., by periodic modulation of the nonlinear coupling coefficient. A periodic change of sign of F (rectangular modulation) yields the effective coupling strength F -^ 2F/jt, where, as before, F is the coupling strength of the phase-matched interaction. Thus the continuous "observation" of the idler mode even gives a slightly better enhancement of the decay rate than the most common quasi-phase-matching technique. To summarize, the statement "the down-conversion process is mismatched" means that the nonlinear process is out of resonance in the sense that the momentum of the decay products (signal and idler photons) differs from the momentum carried by the pump photon before the decay took place. When the linear interaction is switched on, the system gets dressed and the energy spectrum changes. A careful adjustment of the coupling strength K makes it possible to tune the nonlinear interaction back to resonance. In this way the probability of pump photon decay can be greatly enhanced. This occurs when K ~ ±A and explains why the inverse Zeno effect takes place along the lines K = ±A in fig. 14b. In some sense, on very general grounds, the Zeno effect is a consequence of the new dynamical features introduced by the coupling with an external agent that (through its interaction) "looks closely" at the system. When this interaction can be effectively described as a projection operator a la von Neumann, we obtain the usual formulation of the quantum Zeno effect in the limit of very frequent measurements. In general, the description in terms of projection operators may not apply, but the dynamics can be modified in such a way that an interpretation in terms of Zeno or inverse Zeno effect is appealing and intuitive. This is the main reason why we think that examples of the type analyzed in this chapter call for a broader definition of Zeno effects.
3, § 8]
Genuine unstable systems and Zeno effects
191
§ 8. Genuine unstable systems and Zeno effects We will now study the Zeno-inverse Zeno transition in greater detail, by making use of a quantum field theoreticalfi-amework,and discuss the primary role played by the form factors of the interaction. As usual, rather than analyzing the general case, we shall focus on simple examples. We generalize the two-level Hamiltonian (2.1) to N states \j) (J = 1,..., A/^) Qi 0
/ 0 Qi 7=1
0
0
0 /
0
0 (Oi
0
\ 0
0
(ON
\QN
(8.1)
and introduce different energies /COQ
' 7=1
(8.2) J
In order to obtain a truly unstable system we need a continuous spectrum, so we will consider the continuum limit of these Hamiltonians H = Ho-\-Hi = a;o|+)(+|+ / dw co\a)){a)\+ / dcog(a))(\+){(jD\-\-\a)){+\). (8.3) The transition to a quantum field theoretical framework is an important component of our analysis, as we shall see. As before, we take as the initial state IV^o) = |+). The interaction of this normalizable state with the continuum of states \co) is responsible for its decay and depends on the form factor g((o). We reobtain the physics of two-level systems in the limit g^((o) = Q^b(a)). The Fourier-Laplace transform of the survival amplitude for this model can be given a convenient analytic expression. Notice that the transform of the survival amplitude is the expectation value of the resolvent
A{E) = Jdt
c^'Ait) = (+1 j
At e^'^-'"'\+) = (+l;^r^l+). (8.4)
and is defined for ImE > 0. By using twice the operator identity 1 E—H
1 E — HQ
E
1 .. 1 -M — HQ E — H
(8.5)
192
Quantum Zeno effects
[3, § 8
one obtains
' ^ E^^^H,^ ^-A-H,-A-H,- ' — HQ E — HQ E — HQ E — HQ E — H
A{E) = (+1
E — Ho
E -COQ
+ E-^— /dw '^''"""" ME)- COQ J E -co
(8.6) In the above derivation we used the fact that Hi is completely off-diagonal in the eigenbasis of HQ, {|+), |a;)}, which is a resolution of the identity |+)(+|+ fd(jo\a)){co\ = l.
(8.7)
The algebraic equation (8.6) can be solved and gives
E-
(OQ- Z(E)
where the self-energy function ^{E) is related to the form factor g{coi) by a simple integration
Z ( £ ) = / d a , K l » ! = / d a , ^ . J E-w J E-o)
(8.9)
By inverting eq. (8.4) we finally get r Af
\
C
^(^) = / ^ ^'''AE) =^UE 7B 2 ^
271 J^
e~*^^
^ ^^ ^.„., E-COQ-
(8.10)
1(E)
the Bromwich path B being a horizontal line ImE = constant > 0 in the half plane of convergence of the Fourier-Laplace transform (upper-half plane). We consider now a particular case. Let the form factor be Lorentzian
g(co)=^J^^^.
(8.11)
This describes, for instance, an atom-field coupling in a cavity with high finesse mirrors (Lang, Scully and Lamb [1973], Ley and Loudon [1987], GeaBanacloche, Lu, Pedrotti, Prasad, Scully and Wodkiewicz [1990]). (Notice that
193
Genuine unstable systems and Zeno effects
3, § 8 ]
%
i-/A 0
cjo
2A
Fig. 16. (a) Form factor g^(co) and position of the initial state energy COQ. (b) Poles of the propagator in the complex £'-plane.
the Hamiltonian in this case is not lower bounded and we expect no deviations from exponential behavior at very large times.) In this case one easily obtains ^(^) = ^
,
(8.12)
whence the propagator iC^" + iA) (E - cooXE + iA) - Q^'
A{E)
(8.13)
has two poles in the lower-half energy plane (see fig. 16). Their values are El = (JOo+ A - i | ,
(8.14)
^2 = - A - i ( A - | ) ,
where ^^4+4^2^2+^2
A = — M _L M ^ 2 "^ 2
2a)i
with ^V
{y=^
g^ = cof) + 4Q^-A^
(8.15)
+4ft;2A2-g2
2
(Notice that g^ can be negative.) The survival amplitude reads ,, , A(t)
^1 + iA = -^
El
:p ,
— Q '
-E2
£"2 + i A - TT
EI-E2
:z7 ,
— Q '
a>o + A + i(A - 7/2) g
i(coo+A)t^-yt/2
(8.16)
0^0 + 2 A + i ( A - y)
A -17/2 _QiAt^-{A~y/2)t a>o + 2A + i(A - 7) = ( 1 - 7^) Q-^(^o+A)t^-yt/2
^
^Q^At^-{A-Y/2)t^
194
Quantum Zeno effects
[3, § 8
where I V wi i _ 2 ' ' ( £ , ) a)o + 2A + iiA-Y) is the residue of the pole E\ = a>o + A- iy/2 of the propagator. The survival probability reads P(t) = \Ait)\^ = Zexp(-yO + 2Re[7l*(l - TZ)e-*"*"'^^^"]exp(-AO +
\n\'cxp[-i2A-y)tl
(8.18) where Z = 11 - T^p is the wave function renormahzation. We now focus on the Zeno-inverse Zeno transition and the conditions for it to take place. The reader should refer to the discussion of § 3.2. For the sake of simplicity, we consider the weak coupling limit Q
(9.2)
kX *" = ^!5FS/"'""''"--•" 1) / / ^ (^o|2)(2| + Oo|3){3| + ^ k,X
a>,4,a,, + ^ ' ( 0 , , 4 J 1 ) ( 2 | + 0^^^^ k,?i
+ (^,„Ao<e'^'|l)(3| + $;,„aoe-«"|3)(l|), (9.5) where a prime means that the summation does not include (A:o,Ao) (due to hypothesis (9.3). In the above equations and henceforth, the vector |/;/2u) represents a state in which the atom is in state \i) and the electromagnetic field in a state with W^^A (k, A)-photons. We shall analyze the behavior of the system under the action of a continuous laser beam of high intensity. Under these conditions, level configurations similar to that of fig. 18 give rise to the phenomenon of induced transparency (Tewari and Agarwal [1986], Harris, Field and Imamoglu [1990], Boiler, Imamoglu and Harris [1991], Field, Hahn and Harris [1991], Zhu, Narducci and Scully [1995], Zhu and Scully [1996], Huang, Zhu, Zubairy and Scully [1996]), for laser beams of sufficiently high intensities. Our interest, however, will be focused on unstable initial states. We shall study the temporal behavior of level |2) when the system is shined by a continuous laser of intensity comparable to those used to obtain induced transparency. The operator
M-\2){2\+J2'ala,,,
(9.6)
k,X
satisfies [H,M] = 0,
(9.7)
which implies the conservation of the total number of photons plus the atomic excitation [Tamm-Dancoff approximation (Tamm [1945], Dancoff [1950])]. The Hilbert space splits therefore into sectors that are invariant under the action of the Hamiltonian. In our case, the system evolves in the subspace labeled by the eigenvalue J\f = I and the analysis can be restricted to this sector (Radmore and Knight [1982], Knight and Lauder [1990]).
202
Quantum Zeno effects
[3, § 9
9.2. Schrodinger equation and temporal evolution We will study the temporal evolution by solving the time-dependent Schrodinger equation i^|t/'(0>=i/(Ol'/'(0>,
(9.8)
where the states of the total system in the sector M = 1 read \xp(t))=x{t)\2-0) + Y^yuxm;
hx) + Y^z,,{t)^''"^'\3;
k,X
Ux)
(9.9)
k,X
and are normalized:
{w)m)) = \x{t)?+Y^\yutt+Y^\zutt = 1 (vo-
(9.10)
By inserting (9.9) in (9.8) one obtains the equations of motion \x(t) = Wox(t) + Yl
^xykxiO,
*"^ (9.11) ^hxit) = 0kxx(t) + Wkykxit) + aQ0koXo^kx(tX izkxit) = OoO^^ykxit) + cOkZkxit), where a dot denotes time derivative. At time ^ = 0 we prepare our system in the state \m))
= |2;0)
^
x(0) = 1, ykxiO) = 0, Zkx(0) = 0.
(9.12)
By Fourier-Laplace transforming the system of differential equations (9.11) and incorporating the initial conditions (9.12) the solution reads E-
COQ- I(B,
E)
with
^iB,E) = Y.\ oo), the matrix elements scale as follows }^:^^^,Y.j^^\t>i'^\^^s'(^ox\o)),
(9-18)
A
where Q is the solid angle. The (dimensionless) function x{^) and coupling constant g have the following general properties (Facchi and Pascazio [2000]) X {(O) ^ { g^
a
'.
^^ A
(9.19)
= a{coQ/AfJ^^^\
(9.20)
where j is the total angular momentum of the photon emitted in the 2 ^ 1 transition, =F represent electric and magnetic transitions, respectively, ^(> 1) is a constant, a the fine structure constant and A a natural cutoff (of the order of the inverse size of the emitting system, e.g., the Bohr radius for an atom), that can be explicitly evaluated and determines the range of the atomic or molecular form factor (Berestetskii, Lifshits and Pitaevskii [1982], Moses [1972a,b, 1973], Seke [1994a,b]). In order to scale the quantity B, we take the limit of a very large cavity, by keeping the density of Oo-photons in the cavity constant: K ^ cxo,
iVo ^ oo,
with
TVo — = WQ = const.,
(9.21)
and obtain from (9.17) B^ = noV\0koXo\' = {2jtyno\(pUko)\\
(9.22)
where q) = 0V^^^/(2jty^^ is the scaled matrix element of the 1-3 transition. If the 1-3 transition is of the dipole type, the above formula reads B^ = 2jtaQo\el,^'Xu\W
(9.23)
where xu is the dipole matrix element. In terms of laser power P and laser spot area A, eq. (9.23) reads P
X^
PX^
where P is expressed in watt, XL (laser wavelength) in |im, A in ^m^ and hPii, in eV In eq. (9.24) the quantity B is expressed in suitable units and can be
204
Quantum Zeno effects
[3, § 9
easily compared to a^o (the ratio B/w^ being the relevant quantity, as we shall see). For laser intensities that are routinely used in the study of electromagnetic induced transparency, the inverse quantum Zeno effect should be experimentally observable. For a quick comparison remember that B is just half the Rabi frequency of the resonant transition 1-3. 9.3. Laser off Let us first look at the case B = 0. The laser is off and we expect to recover the well-known physics of the spontaneous emission of a two-level system prepared in an excited state and coupled to the vacuum of the radiation field. In this case the self-energy function 1(0, E) reads, in the continuum limit (see eq. 8.9), 1(E) = g'cooq(E) = g'wo r do; 1 ^ , (9.25) Jo 1^ - (o where x is defined in (9.18). The function x(E) in eq. (9.13) (with ^ = 0) has a logarithmic branch cut, extending from 0 to +CXD, and no singularities on the first Riemann sheet (physical sheet) (Facchi and Pascazio [1998]). On the other hand, it has a simple pole on the second Riemann sheet, that is the solution of the equation E~coo-g^cooqu(E)
= 0,
(9.26)
where qu(E) = ^(£e-2^^0 = q(E) - 2Jiix\E)
(9.27)
is the determination oiq(E) on the second Riemann sheet. We note that g^q(E) is 0(g^), so that the pole can be found perturbatively. By expanding q\i(E) around 0)Q we get a power series, whose radius of convergence is Re = COQ because of the branch point at the origin. The circle of convergence lies half on the first Riemann sheet and half on the second sheet (fig. 19). The pole is well inside the convergence circle, because l^pde - (Oo\ '^ g^ojo "^ Re, and we can write ^poie = (Oo+g^cooqn((JOo-iO^)-^0(g'^) = a)o-^g^cooq(coo^iO^) +
0(g\ (9.28) because qu(E) is the analytic continuation of q(E) below the branch cut. By setting E^,,, = (Oo + A-i^,
(9.29)
3, § 9 ]
205
Three-level system in a laser field
,
I
\E
\
COo ^~-~.
I
n
X Z7
"A
\.
•vil \ Fig. 19. Cut and pole in the £'-plane {B = 0) and convergence circle for the expansion of ^{E) around E = COQ.I and II are the first and second Riemann sheets, respectively. The pole is on the second Riemann sheet, at a distance 0{g^)firomCL>O-
one obtains from eq. (9.25)
7 = 2jtg'(Da\(o^) + 0{g%
A = g'cooP r JQ
doj ^ ^ ^ + 0{g\ (OQ-CO
(9.30) which are the Fermi "golden rule" and the second-order correction to the energy of level 12) (see eqs. 8.19). The Weisskopf-Wigner approximation consists in neglecting all branch cut contributions and approximating the self-energy function with a constant (its value in the pole), that is (9.31)
x(E) = E-COQ-
1(E)
E~COO-
2'ii(£'pole)
E - £'pole
where in the last equality we used the pole equation (9.26). This yields a purely exponential behavior, x(t) = exp(-L£'poie05 without short-time (and long-time) corrections. As is well known, the latter are all contained in the neglected branch cut contribution. 9.4. Laser on We now turn our attention to the situation with the laser switched on (B ^ 0) and tuned at the 1-3 transition frequency QQ. The self-energy fiinction I(B,E)
206
Quantum Zeno effects
[3, § 9
in (9.16) depends on B and can be written in terms of the self-energy function 1{E) in absence of the laser field (eq. 9.25), by making use of the following remarkable property:
^(i..«=ii:fcp(^-j-^.^-^) ,,,
X - -^
-
-
/
(9.32)
= \{1{E-B) + I{E + B)\. Notice, incidentally, that in the continuum limit {V ^ oo), due to the above formula, I(B,E) scales just like 1(E). The position of the pole £'poie (and as a consequence the lifetime IE = y~^ = -l/2Im£'poie) depends on the value of ^. There are now two branch cuts in the complex E plane, due to the two terms in eq. (9.32). They lie over the real axis, along [-B, +oo) and [+^, +oo). The pole satisfies the equation E-(Oo-I(B,E) = 0,
(9.33)
where I(B,E) is of order g^, as before, and can again be expanded in a power series around E = (OQ, in order to find the pole perturbatively. However, this time one has to choose the right determination of the fiinction I(B,E). Two cases are possible: (a) The branch point +B is situated at the lefi; of COQ, SO that (Oo lies on both cuts; see fig. 20a. (b) The branch point +B is situated at the right of c^o, so that WQ lies only on the upper branch cut; seefig.20b. We notice that in the latter case (B > COQ) a number of additional effects should be considered. Multi-photon processes would take place, the other atomic levels would start to play an important role and our approach (3-level atom in the rotating wave approximation) would no longer be completely justified. Notice also that our approximation still applies for values ofB that are of the same order of magnitude as those utilized in the electromagnetic-induced transparency. In this case the influence of the other atomic levels can be taken into account and does not modify the main conclusions (Facchi and Pascazio [2000]). In case (a), i.e., for B < COQ, the pole is on the third Riemann sheet (under both cuts) and the power series converges in a circle lying half on thefirstand half on the third Riemann sheet, within a convergence radius Re = COQ - B, which decreases as B increases [fig. 20a]. On the other hand, in case (b), i.e., for B > (OQ, the pole is on the second Riemann sheet (under the upper cut only) and the power series converges in a circle lying half on the first and half on the second Riemann sheet, within a convergence radius Re = B - CJOQ, which increases with B (fig. 20b).
3, § 9 ]
207
Three-level system in a laser field
0)„ -\-B
|H)1L-
\+B [II
ZZ'l-nilc
III
(a)
(b)
Fig. 20. Cuts and pole in the £^-plane (B ^ 0) and convergence circle for the expansion of I(B, E) around E = (DQ.1,11 and III are the first, second and third Riemann sheets, respectively, (a) B < COQ. (b) B > coQ.ln both cases, the pole is at a distance 0(g^)fi*omCOQ.
In either cases we obtain, for \Epo\Q - a>o| < ^c = |^ - ^o|, ^poie = (Oo+\ [^((JOo +B + /0+) + I{m
-B + /0+)] + 0(g4)
(9.34) = 0^0+ \g^(JOo [q{0)Q + ^ + /0+) + q{(OQ -B + /0+)] + 0{g^). We write, as in eq. (9.29), .reff(5) ^pole = ^ 0 + A{B) - i
(9.35)
Substituting (9.25) into (9.34) and taking the imaginary part, one obtains the following expression for the decay rate yeff(5) = V a > o [x\oj^ +B) + x\o)^ -B)d(coo - B)] + 0(g'),
(9.36)
which yields, by (9.30), _ ^^ x\coo +B) + x\o)o - B)d{wo - B) + 0(g4). (9.37) =y 2x\coo) Equation (9.37) expresses the "new" lifetime yeff (^)~^ when the system is bathed in an intense laser field B, in terms of the "ordinary" lifetime y"^ when there is no laser field. By taking into account the general behavior (9.19) of the matrix elements X^(^) and substituting into (9.37), one gets to 0(g'^) YMB)
y.ff{B)
i+—
+
1
B \^^'^^
e{a)Q-B)
(B < A),
COoJ
(9.38) where =F refers to 1-2 transitions of electric and magnetic type, respectively. Observe that, since A ^ inverse Bohr radius, the case B < (OQ 1 (1-2 transitions of electric quadrupole, magnetic dipole or higher), the decay rate /eff (^) increases with B, so that the lifetime yeff(^)"^ decreases as B is increased. Since B is the strength of the observation performed by the laser beam on level |2), this is an IZE, for decay is enhanced by observation. 7eff(5)/7
leAB)/j 4
eff(i^)/7 2\
/
3 1
nl 0
^/^'o 0.4
0.8
0
/
8
2 1
12
_
^ B/uo
0
3 = 1
/
4
0.4 0.8 J = 2
0-"""^
0
0.4
B/uJo 0.8
J = 3
Fig. 21. The decay rate yeffW versus B, for electric transitions withy = 1,2,3; XeffW is in units y and 5 is in units COQ. Notice the different scales on the vertical axis.
As already emphasized, eq. (9.38) is valid for B A, by (9.19) and (9.37), one gets to 0(g^)
7eff ( ^ )
^
7
X\B)
2 x\m)
oc (B/A)-
(B^A)
(9.39)
This result is similar to that obtained by Mihokova, Pascazio and Schulman [1997]. If such high values ofB were experimentally obtainable, the decay would be considerably hindered (QZE). A final remark is now in order. If one would use the Weisskopf-Wigner approximation (9.31) in eq. (9.32), in order to evaluate the new lifetime, by setting 1(E) = 2'(£'poie) = const, one would obtain I{B,E) = 1(E) = 2'(£'poie), i.e., no ^-dependence. Therefore, the effect we are discussing is ultimately due to the nonexponential contributions arising from the cut. In particular, viewed from the perspective of the time domain, this effect is ascribable to the quadratic short-time behavior of the |2) -^ |1) decay.
Three-level system in a laser field
9]
209
9.5. Photon spectrum, dressed states and induced transparency It is interesting to look at the spectrum of the emitted photons. It is easy to check that, in the Weisskopf-Wigner approximation, the survival probability \x{t)\^ decreases exponentially with time. In this approximation, for any value of 5, the spectrum of the emitted photons is Lorentzian. The proof is straightforward and is given in Facchi and Pascazio [2000]. One finds that, for B = 0, the probability to emit a photon in the range (a>, O) + do;) reads (9.40) where WQ = WQ + ^{E) and 1
(9.41)
/ L ( a > ; 7 ) = 0)2 + y 2 / 4 -
On the other hand, when B ^0 one gets: dPB = g^o)oX^{o))\ [Mco -oJo-B;
yeff(^)) +fL((o -(bo+B; y^B))] do;.
(9.42) The emission probability is given by the sum of two Lorentzians, centered in WQ ± B. We see that the emission probability of a photon of frequency a)o + ^ (d)o - B) increases (decreases) with B (fig. 22). The linewidths are modified according to eq. (9.38). When B reaches the "threshold" value COQ, only the photon of higher fi-equency (a)o +B) is emitted (with increasing probability vs. B). dP/duj
dPslduj
0.5
1.5
2
(a) Fig. 22. The spectrum (9.42) of the emitted photons. The height of the Lorentzians is proportional to the matrix element X^i^J^) (dashed line). We chose an electric quadmpole transition, withy = 2 and Y = 10~^a)o, and used arbitrary units on the vertical axis, (a) 5 = 0; (b) 5 = COQ/S; note that, from eq. (9.38), 7eff (^) = (28/25)y.
210
Quantum Zeno effects
[3, § 9
Photons of different frequencies are therefore emitted at different rates. In order to understand better the features of the emission, let us look at the dressed states of the system. For simplicity, since the average number TVO of ^O-photons in the total volume V can be considered very large, we consider number (rather than coherent) states of the electromagnetic field. Henceforth, the vector \i\nkx,M{)) represents an atom in state |/), with riux (A:, A)-photons and MQ laser photons. The Hamiltonian (9.1) becomes H ~ (yo|2)(2| +Do|3)(3| + ^ a > , 4 , a u + $ ] ' (^hxa\,\\){2\ + k,X
r,,a,x\2){\\)
k,X
+ ( ^ ^ o A o < A o | l ) ( 3 | + ^;oAo^AoAo|3)(l|),
(9.43) where a prime means that the summation does not include (A:o,Ao) (due to hypothesis 9.3). Besides (9.6), there is now another conserved quantity: indeed the operator •A/'o = |3)(3|+aUa*oAo
(9-44)
satisfies [H,Mo] = [^fo,^f]-0.
(9.45)
In this case, the system evolves in the subspace labeled by the two eigenvalues J\f = 1 and J\fo= No, whose states read |i/;(0)=x(0|2;0,iVo) + ^ W ( 0 | l ; U A , ^ o ) + ^ W ) | 3 ; U A , A ^ o - l ) . k^
k,X
(9.46) By using the Hamiltonian (9.43) and the states (9.46) and identifying A/Q with ^^0 = I aoP of §9.1, the Schrodinger equation yields again the equations of motion (9.11), obtained by assuming a coherent state for the laser mode. Our analysis is therefore independent of the statistics of the driving field, provided it is sufficiently intense, and the (convenient) use of number states is completely justified. Energy conservation implies that if there are two emitted photons with different energies (see eq. 9.42), there are two levels of different energies to which the atom can decay. This can be seen by considering the laser-dressed (Fano) atomic states (Fano [1961], Cohen-Tannoudji and Reynaud [1977a-c], Yoo and Eberly [1985]). The shift of the dressed states can be obtained directly
3, § 9]
Three-level system in a laser
field
211
from the Hamiltonian (9.43). In the sector A/Q = NQ, the operator A/Q is proportional to the unit operator, the constant of proportionality being its eigenvalue. Hence one can write the Hamiltonian in the following form H = H- QoJVo + QoNo,
(9.47)
which, by the setting Ei -\-NoQo = 0, reads H = Ho-\- ifint = a;o|2)(2| + Y^'cokal^aki + ^ ' (^^k?.al^\l){2\ + (l>l^akx\2){l\) k,l
k,l
+ (^*«Ao«Lll>(3| +
^'
+fi 0
|2>—
ii> "
-fi
(a)
(h)
Fig. 23. Shift of the dressed states |+) and |-) vs. B. (a) For B < (JOQ there are two decay channels, with y_ > y^. (b) For 5 > o^o, level |+) is above level |2) and only the y_ decay channel remains.
the well-known Autler-Townes doublet (Autler and Townes [1955], Townes and Schawlow [1975]). Therefore, by applying the Fermi golden rule, the decay rates into the dressed states read
r.-2^,W^^^,
y--2.,W^^^,
(9.53)
and the total decay rate of state |2) is given by their sum 7eff(^)=y^ + 7-,
(9.54)
which yields (9.36). One sees why there is a threshold at 5 = WQ- For B < COQ, the energies of both dressed states |±) are lower than that of the initial state |2) (fig. 23a). The decay rate y_ increases with B, whereas y+ decreases with B; their sum y increases with B. These two decays (and their lifetimes) could be easily distinguished by selecting the frequencies of the emitted photons, e.g. by means of filters. On the other hand, when B > COQ, the energy of the dressed state |+) is larger than that of state |2) and this decay channel disappears (fig. 23b). Finally, let us emphasize that if state |2) were below state 11), our system would become a three-level system in a ladder configuration, and the shift of the dressed states would give rise to electromagnetically induced transparency (Tewari and Agarwal [1986], Harris, Field and Imamoglu [1990], Boiler, Imamoglu and Harris [1991], Field, Hahn and Harris [1991]). The situation we consider and the laser power required to bring these effects to light are therefore similar to those used in induced transparency.
3, § 10]
Concluding remarks
213
For physically sensible values of the intensity of the laser field, the decay of level |2) is faster when the laser is present. Equations (9.37)-(9.38) (valid to 4th order in the coupling constant) express the new lifetime as a function of the "natural" one and other parameters characterizing the physical system. The initial state decays to the laser-dressed states with different lifetimes, yielding an IZE.
§ 10. Concluding remarks The usual formulation of quantum Zeno effect in terms of repeated ("pulsed") measurements a la von Neumann is a very effective one. It motivated quite a few theoretical proposals and experiments and provoked very interesting discussions on their physical meaning. In general, the quantum Zeno effect is a straightforward consequence of the new dynamical features introduced by a series of measurement processes. In turn, a measurement process is due to the coupling with an external apparatus that, after interacting with the system, gets entangled with it. It is then very natural to think that a quantum Zeno effect can also be obtained if the (Hamiltonian) dynamics is such that the interaction takes a sort of "close look" at the system. When such an interaction can be effectively described as a projection operator a la von Neumann, we obtain the usual formulation of the quantum Zeno effect in the limit of very frequent measurements. Otherwise, if the description in terms of projection operators does not apply, but one can still properly think in terms of a "continuous gaze" at the system, an explanation in terms of Zeno can still be very appealing and intuitive. These considerations and the diverse examples analyzed in this chapter motivated us to interpret several physical phenomena as quantum Zeno or inverse quantum Zeno effects. We believe that this approach is prolific. Not only does it often yield a simple intuitive picture of the dynamical features of the system, it also enables one to look at these dynamical features from a different, new perspective. The very concept of inverse Zeno effect is a good example. Other examples are the phenomena discussed in §§7 and 9. The underlying idea is that coupling the system to an "observer" (like a laser) can sometimes enhance the evolution. This is close to Heraclitus' viewpoint, who used to argue (against Zeno and Parmenides) that ever3^hing flows. The physical features of the dynamical evolution laws have profound implications (Prigogine [1980]) and always provide matter for thoughts. In this way, one even finds links with instability (Facchi, Nakazato, Pascazio, Pefina and Rehacek [2001]), chaos (Facchi, Pascazio and
214
Quantum Zeno effects
[3
Scardicchio [1999], Kaulakys and Gontis [1997]) and geometrical phases (Berry and Klein [1996], Facchi, Klein, Pascazio and Schulman [1999]). The very fact that these links may not always be obvious is in itself a motivation to pursue the investigation in this direction.
Acknowledgments It is a pleasure to thank the many colleagues who have collaborated with us during the last few years on the topics discussed in this chapter. We would like to mention in particular G. Badurek, Z. Hradil, A.G. Klein, H. Nakazato, M. Namiki, J. Pefina, H. Ranch, J. Rehacek, A. Scardicchio and L.S. Schulman. We owe them much of our own comprehension of the diverse phenomena known as Zeno effects.
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E. Wolf, Progress in Optics 42 © 2001 Elsevier Science B.V All rights reserved
Chapter 4
Singular optics by
M.S. Soskin and M.V Vasnetsov Institute of Physics, National Academy of Sciences of Ukraine, Prospect Nauki 46, Kiev, 03650 Ukraine
219
Contents
Page § 1.
1. Introduction
221
§ 2. Anticipations of singular optics
223
§ 3. Wave-front dislocations - phase defects
226
§ 4.
Circular and linear edge dislocations
231
§ 5.
Screw wave-front dislocation - axial OV
239
§ 6.
Reflection, refraction, interference and diffraction of OVs . . .
244
§ 7.
Topology of wave fronts and vortex trajectories
249
§ 8.
Gouy phase shift in singular optics
258
§9.
Statistics of phase dislocations
261
§ 10. Optical vortices in frequency conversion processes
263
§ 11. AppUcations
268
§ 12. Conclusions
271
References
272
220
§ 1. 1. Introduction What is singular optics? This new branch of modern physical optics deals with a wide class of effects associated with phase singularities in wave fields, as well as with the topology of wave fronts. Although phase is an auxiliary fiinction in electromagnetic field description, it is very usefiil because it gives a visual perception of wave propagation and transformation along its path. An important relevant conception is a wave front, or a surface of equal phase, usually associated with a crest of a wave, where the field strength attains its highest value. The wave fronts follow each other with spatial separation of one wavelength, and between two neighboring wave fronts there are two surfaces where field strength becomes zero, and one surface where the field reaches minimum (negative) value (trough). This perfect regular motion, being true for a plane wave, can sometimes be violated for real waves. In brief, phase of a wave can experience a ";r-jump", corresponding to a step on half of a wavelength in a wave train, producing a phase defect of a wave front along a continuous line in space. For instance, some physical reasons can be responsible for local retardation or acceleration of the phase velocity across a wave front. Resulting wave-front bending can lead to a tear of the wave front, and the phase becomes indeterminate, or singular along the tear. The necessary condition for a phase singularity to appear is the vanishing of the field amplitude. M. Berry has recently emphasized that the study of wave singularities in optics and physics in general started in "the miraculous 1830s" (Berry [2000]). In 1838 Airy showed that the rainbow is the caustic where light rays are gathered. He recognized also that the infinite brightness of caustic predicted by ray optics is softened to finite value because of the wave origin of light. Whewell in 1833 discovered the phase singularities in the tide waves (for more details see Berry [1981]). Hamilton in 1832 discovered the third type of singularities, namely polarization light singularities, as the "effect of conical refi-action" (see Born and Wolf [1999]). The general types of polarization singularities were analyzed later by Nye (see Nye [1999] and references therein). More recently the results of optical singularities investigations are treated in the Singular Optics, a branch of modern optics studying new important features of light, which are absent in the traditional optics of waves with smooth wave 221
222
Singular optics
[4, § 1
fronts (Soskin and Vasnetsov [1999a,b]). More precisely, there are three levels of optical singularities: (i) ray singularities (caustics) considered as optical catastrophes, (ii) singularities of plane polarized waves (scalar fields), (iii) polarization singularities of vector light fields. An important step to understanding light singularities was made when it was recognized that light flow could create vortices (Braunbek and Laukien [1952], Boivin, Dow and Wolf [1967], Nye and Berry [1974], Gullet, Gil and Rocca [1989]). In general, vortices are inherent to any wave phenomena, including even complex probability wave ftinction in quantum mechanics, e.g. encountered in connection with the Dirac monopole (Dirac [1931]) and the Aharonov-Bohm effect (Aharonov and Bohm [1959], Berry, Ghambers, Lange, Upstill and Walmsley [1980]). Vortices exist in various physical systems from superconductors (de Gennes [1989]) and superfluids (Donnely [1991]) up to cosmic strings (Vilenkin and Shellard [1994]). In the case of phase singularities in light, much of the revived interest has come from the ability of lasers to generate and easily manipulate a great variety of optical fields. The aim of this report is to collect a "harvest" of the results. By the end of the 20-century a few review papers and monographs appeared. Namely, the linear "catastrophe" optics was developed in a comprehensive way by Wright [1979], Berry and Upstill [1980] and summarized by Nye [1999]. The nonlinear optical catastrophe was observed recently during self-action of an elliptical Gaussian beam focused into a Kerr-like medium (Deykoon, Soskin and Swartzlander [1999]). Some important topics of singular optics where considered in a monograph "Optical Vortices" (Vasnetsov and Staliunas [1999]). The optical vortex solitons in nonlinear media where considered by Kivshar and LutherDavies [1998] and Akhmediev and Ankiewicz [1997], and vortices in nonlinear systems by Pismen [1999]. The presence of orbital angular momentum, one of the most specific features of optical vortices, was reviewed recently by Allen, Padgett and Babiker [1999]. Due to the large scope of research in the area, we shall restrict our review of singular optics to systematic exposition of its physical background and to the new features absent in light fields with a smooth wave front. Propagation of a singular laser beam through free space and/or nonlinear media will be considered. The rapidly developing field of nonlinear optics is accompanied by studies of transverse pattern formation in wide-aperture nonlinear optical systems (Rosanov [1996]). Full overview of pattern formation and competition in active and passive optical systems was given recently by Arecchi, Boccaletti and Ramazza [1999], where phase singularities were also briefly considered. Some aspects of phase
4, § 2]
Anticipations of singular optics
223
singularities in laser systems were considered by Weiss and Vilaseca [1991] and in an overview "Solitons and vortices in lasers" by Weiss, Vaupel, Staliunas, Slekys and Taranenko [1999].
§ 2. Anticipations of singular optics The history of optics embraces geometrical optics, according to which light propagates along straight lines (rays) and wave optics, which explains interference of light beams and diffraction, elucidating light penetration into a shadow area. Wave optics teaches us that light energyflow,associated with the Poynting vector, does not follow a straight line. In any case, the flow in a free space should be laminar, as it is in inviscid fluid. The question is, under which circumstances does turbulence in a light flow appear, or is it possible to create at least one isolated "optical vortex"? At first glance, the answer is negative. However, the nature of light brings more surprises than one might expect. It appears that the possibility of a local backward light flow has been first noted by Ignatowskii [1919] who studied a field structure in a focal plane of a focusing lens. This fact was analyzed later in detail by Richards and Wolf [1959], who computed the Poynting vector distribution in the focal plane of an aplanatic system. The energy flow was found to have vortices around certain lines in the focal plane (Boivin, Dow and Wolf [1967]) (fig. 1). The important feature of the light vortex is that the field amplitude vanishes on the axis of the vortex. Another way which has led to the discovery of an "optical vortex" was shown by Braunbek and Laukien [1952] (fig. 2). The vortices were found in an interference field produced by an incident plane wave and its reflection from a semi-infinite perfectly reflecting half-plane screen. Due to the presence of the screen edge, the standing-wave field in fi-ont of the screen was slightly modulated. Instead of the zero-amplitude nodal interference planes only lines parallel to the edge appeared where the field amplitude vanished. Again, the lines of zero amplitude were the axes of vortices. The two examples illustrated in figures 1 and 2 reveal a possibility of generation of vortices in a monochromatic light wave. As was pointed out by Sommerfeld [1950], a deviation from monochromaticity violates the periodic wave front sequence in a traveling light wave. Phase defects appear at the points where the beating of waves with slight differences in frequencies gives a zero value for the field amplitude. Around zero-amplitude point the crest of the wave transforms to the trough, and the phase becomes undetermined, or singular. An important feature of a phase defect is that its center, i.e. zero-amplitude point.
224
[4, §2
Singular optics
'15
-i.Q
iM m
ifi
^.i
'OS
Fig. 1. Flow lines of the time-averaged Poynting vector, showing vortex behavior around the phase singularity at the focal plane (Boivin, Dow and Wolf [1967], fig. 1).
••m'i
i
71 / '
It
/ / ; M
I "lip'' /
///;,
/ / / /11 k
///I I
'illi,.,
ll'll'l 'iiiiin'i'.iil
Fig. 2. Flow lines of the time-averaged Poynting vector at the vicinity of a perfectly conducting halfplane, illuminated by a plane wave (Braunbek and Laukien [1952], fig. 3).
4, § 2]
Anticipations of singular optics
225
belongs to a continuous line in space, where the amplitude vanishes. The phase circulates around the line, creating a vortex. However, Sommerfeld concluded: "due to zero value of amplitude, the influence of these points is no stronger than other points of variable amplitude". The essential role of phase singularities has been recognised only after pubHcation of the seminal paper of Nye and Berry [1974], who introduced a new concept into wave theory based on phase singularities in a wave field as a new class of objects in optics, and more generally in electromagnetic waves. In analogy with defects in crystals, wave-front dislocations were introduced. Furthermore, Berry [1981] concluded that phase singularities are the most remarkable features of wave fronts. To our knowledge, Bryngdahl [1973], Bryngdahl and Lee [1974] and Lee [1978] first reported the idea of artificial introduction of phase singularities into a smooth wave-front beam in connection with the radial-fringe interferogram formation. The phase structure of Laguerre-Gaussian (LG) "doughnut" modes of laser emission, which possess zero value of intensity on the axis, was the point of investigation of Vaughan and Willetts [1979]. The phase singularities have been considered as exotic objects until Zel'dovich with co-workers found phase dislocations in speckle fields (Baranova, Zel'dovich, Mamaev, Pilipetskii and Shkunov [1981]). The speckle structure of coherent light scattered by a rough surface was observed still in 1962 immediately after the development of the first lasers, namely a cw He-Ne laser (Rigden and Gordon [1962]). These speckles, or local intensity maxima, arise from constructive interference of partial waves with random phases. It was shown that there are also points of fiilly destructive interference with zero amplitude, and consequently with appearance of phase singularities in the form of screw wave-front dislocations (Baranova, Zel'dovich, Mamaev, Pilipetskii and Shkunov [1982]). They noted a "sea" of optical vortices and also established the important features of screw wave-front dislocations in a speckle field, in which amplitude distribution is a smooth random function of coordinates with the following properties: (i) the average number of "positive" and "negative" screw dislocations per unit area of a beam cross-section is equal; (ii) continuous zero-amplitude lines possess snake-like structures and extend along the beam propagation axis (Baranova, Mamaev, Pilipetskii, Shkunov and Zel'dovich [1983]). An intriguing term "optical vortex" (OV) was introduced by Gullet, Gil and Rocca [1989], stimulating a wave of new researches. Sometimes the terms "phase dislocations", "phase singularity lines" and "optical vortices" are used as equivalent (Berry and Dennis [2000b]). In our
226
Singular optics
[4, § 3
opinion, they are not all totally identical but rather complement each other. Indeed, a phase dislocation is the loci of the zero amplitude that can be as a surface, as a line. The case of a line of zero field amplitude is a phase singularity line. The optical vortex has a complicated structure with a dark core (zero-amplitude axis) with phase circulation around it. The screw wave-front dislocation appears as a helicoidal wave-front structure around the dislocation line. By the end of the 1980s, a "critical mass" in the field of phase singularities of wave fronts for singular optics was attained.
§ 3. Wave-front dislocations - phase defects Nye and Berry [1974] introduced the term "wave-front dislocations" as close analog of those found in crystals. Let us examine this analogy deeper. According to how the Burgers vector is arranged relative to the direction of the dislocation line, dislocations may be of edge, screw or mixed edge-screw type (Nye and Berry [1974]). Figure 3 gives a comparison between two main types of defects in a periodic structure in a crystal and in a wave-front sequence. Burgers [1939] showed that a contour formed from the main translation vectors of a lattice embracing any point is closed in an ideal crystal, but in a defect crystal the contour embracing a dislocation is torn. The additional vector connecting the end point to the start point is called the Burgers vector. In the case of edge dislocation (fig. 3a) the Burgers vector is perpendicular to the dislocation line, in the case of screw dislocation (fig. 3b) it is parallel to the dislocation line. Let us determine how the phase grows progressively with the optical path, say along z-axis. For a plane wave the phase 0, which depends both on time t and distance z, appears as 0(z,t) = kz-(ot,
(3.1)
where k is the wavenumber, co is the light frequency, and co/k = c is the speed of light. This choice is rather arbitrary, and we can determine the phase to grow progressively with time, but we prefer expression (3.1) for convenience. For more complicated beams, the phase should be also dependent on transverse coordinates. Fortunately, often it is possible to separate terms responsible for the wave-front shape ("transversal" phase) and for propagation along the beam axis ("longitudinal" phase). A family of lines of equal phase (contour phase map) infig.3c is a momentary cross-section of the phase of a wave propagating along the z-axis, with phase
4, § 3 ]
Wave-front dislocations - phase defects
111
Fig. 3. Comparison between edge and screw dislocations in a crystal (a,b) and in a wave front (c, d). Edge dislocation axis, shown in (a) as a cross, is perpendicular to the picture plane. Burgers contour ABCDEF embracing a screw dislocation is shown (b). Burgers vector FA connecting points F and A is parallel to the dislocation line, the sign of dislocation is positive, (c) Edge dislocation of a wave front, (d) Helical wave front with an axial OV possessing unity charge. The vector FA is the analog of the Burgers vector, the sign is positive.
interval Ji/A, Wave fronts (crests) are shown by a thick line, trough contour is indicated by a dashed line. We can attribute for selected continuous wave-front surface (crest of a wave) zero phase value, as shown in fig. 3c. Phase grows along the z-axis, with a correspondence Ijt in phase -^ one wavelength in space. Very similar to the edge dislocation in a crystal, a wave-front dislocation can appear as an edge of an extra sheet between neighbor wave fronts, as fig. 3c shows. The end point of this extra sheet in the XZ cross-section is depicted as O. Here the crest of the wave gradually transforms to the trough, and the field amplitude necessary attains zero value at this point. Looking on the left side of the wave-front sequence, we detect one full wavelength, while on the right side two wavelengths could be found.
228
Singular optics
[4, § 3
At a saddle point S, two troughs are shown to meet together. In a half period of time oscillation, two crests (which are normally separated by one wavelength in a regular wave) will meet here. To understand the physical reason for this rather strange behavior, let us have a look at a separation between equiphase lines in fig. 3c. It is seen with the naked eye that the lines are compressed on the right side of the edge dislocation, where an extra-sheet of the waveft-ontappears. This is evidence that the phase velocity is somewhat slower here, and higher on the other side of the dislocation. An analogue of screw dislocation in a crystal (fig. 3b), wave-fi'ont screw dislocation can also exist. Figure 3d is a sketch of helicoid-shape wave front with a "Burgers contour" ABCDEF, where the "Burgers vector" length is a wavelength (Nye [1997]). Of course, we can choose any other plane to make a section of a running wave and obtain a contour phase map. However, the picture which occurs in any section perpendicular to the z-axis will show nothing but parallel lines, the same for any section perpendicular to the x-axis. These pictures are not very informative. The best choice is to use a plane, which is transpierced by the dislocation line. In this plane, we can easily recognize the phase singularity manifesting itself as a point radiating equiphase lines (or, equivalently, equiphase lines terminate there), see e.g. a detail view of the equiphase structure around the dislocation line (point O in fig. 3c). One round-trip around point O by any closed contour will change the phase on 2jr. A wave dislocation can be defined in terms of an integral around a circuit that contains within an isolated dislocation line (Nye and Berry [1974]):
/
d 0 = 2mjz,
(3.2)
where the integer m, which may be positive or negative, is the winding number, or the charge of a dislocation. For a monochromatic wave a dislocation is stationary in space, forming an isolated interference fringe. However, in contrast to the stationary dislocations in crystals, wave-fi-ont dislocations are dynamic objects due to the continuous motion of light and therefore phase variation. We shall see further how the phase circulates around a dislocation line, producing an "optical vortex". To begin analysis, we have to check how the used field description fits our task. First, we shall restrict ourselves within a frame of scalar field, i.e., linear field polarization. The oscillations of the electric field are assumed to occur in a plane, which contains the 7-axis, as is shown in fig. 4. Three optical rays are shown, OA, OB and OC: OA is of arbitrary direction, OB is in YZ plane, OC is
Wave-front dislocations - phase defects
4, § 3 ]
229
,'AEy
Fig. 4. Electric field of a spherical wave emitted by a point source located at the coordinate origin O. All three polarization components Ey, E^ and E^ are present in observation point A for a wave with the plane polarization.
in xz-plane. The electric field component in the transversal electromagnetic wave is perpendicular to the direction of propagation, therefore we can consider E{C) to have the only Ey component, E(B) will have both Ey and E^ components, and finally E{A) has all three E^, Ey and Ez components. The amplitude Ey amounts to E cos Yy, Ez=E sin y^ cos YX, Ex=E sin Yy sin Yx- The paraxial approximation eliminates both Ex and E^ components, assuming the angles Yy ^^^ Yx to be small. The only Ey=E component is used for the field description. We start with the scalar wave equation written in cylindrical coordinates p, q), z and time t to describe a light wave propagating along the z-direction: dE\
1 d^E
d^E _ 1 d^E
pdp where E(p,q),z,t) the frequency co:
(3.3)
is the strength of the electric field, oscillating in time with
E{p, (p,z, 0 = E(p, (p,z) Qxpi-icot).
(3.4)
The wave oscillation in space can be represented by introducing the wave number k, resembling propagation of a plane wave (eq. 3.1): E(p, cp,z) = E{p, (p,z)exp(i^z).
(3.5)
where the amplitude E(p, cp,z) at the right-hand side of eq. (3.5) is a "slowly varying" function of p, q) and z. The essence of the paraxial approximation
230
Singular optics
[4, § 3
consists in neglecting the second derivative of the slowly varying amplitude, and results in the final equation \ d ( dE\
1 d^E
^.BE
^
^_^^
In a physical sense, the paraxial approximation is based on an assumption that the wave amplitude varies very little over a distance of the order of a wavelength. In the further analysis we shall use the solution without any azimuthal dependence, therefore the second term in eq. (3.6) vanishes. The merit of eq. (3.6) is the analytical solution in a form of the Gaussian beam, £'(p,z) = £G — exp (
r ) ^^P f i^7^ ~ i ^^^tan — ) ,
(3.7)
where EQ is the amplitude parameter, k = 2jt/X is the wavenumber, A is the wavelength, WQ is the beam waist parameter, with associated Rayleigh range, ZR = kwl/2, where the transversal beam dimension w = wo(l +Z^/ZR)^^^ enlarges in ^/2 times with respect to the waist. The radius of the wave-front curvature is R = z(\ -^z\/z^). The phase term includes the longitudinal phase fe, transversal phase kp^/lR and the Gouy phase shift (Siegman [1986]) appears as 0 = kz+^-
arctan (—\
2R
.
(3.8)
\z^J
Let us examine solution (3.7). First, the presence of the Gouy phase shift arctan(z/zR) influences the phase velocity of the Gaussian beam (Siegman [1986]). On its way from the waist (z = 0) to the far field (z^oo) the beam experiences an additional phase shift -nil with respect to a plane wave, and the corresponding shift amounts to -;r/4 on a distance equal to the Rayleigh range. This phase shift can be interpreted as a small acceleration of the beam in the region near its waist, where its transversal dimension is compressed. The effect of the Gouy phase shift is considered in § 8. Second, the beam has a variable curvature of the wave front, being a function of z. At the waist, the wave front is plane ( 0 = 0) due to the fact that 7^(0)—>oc. Near the waist (z 1, the following wave fi-ont at the distance of one wavelength is enclosed within the preceding one, which has pitch |/|A. As a result, a |/|-start helicoid is built, with the distance between neighboring 2«;r-phase surfaces equal to the wavelength with small deviations caused by the change of the curvature and the Gouy phase shift.
4, §5]
241
Screw wave-front dislocation — axial OV -;=r (-111
>
{i
__ -• /=0
/ = T/8
/ = T/4
/ = 3T/8
r = T/2
Fig. 10. Contour lines of a momentary electric field strength distribution across the beam crosssection.
Transverse coordinate p Fig. 11. Amplitude and intensity distribution of OV beam (LGi mode) (a); intensity distribution in a gray scale (b); wave-front shape (c).
Why is the hehcoidal wave-front beam a vortex? Let us examine how the field ampHtude varies with the time in any given cross-section of the beam (z = const). Figure 10 shows the distribution of the electric field strength in a cross-section of the beam (LGQ mode) at different instants of time, with an interval of 1/8 time period T of the wave oscillation. At half of the period, the field distribution makes a half-turn around the beam axis. The time averaging over a period results in a "doughnut" with zero field value on the axis (fig. lla,b). Rotation of the field around the axis produces the vortex, and the combination of the field circulation and of the longitudinal propagation of the wave results in the helicoidal wave-front structure (fig. lie). Another consequence of these phenomena is the existence of the orbital angular momentum (OAM) in the beam. The origin of the orbital angular momentum for a beam with an axial OV can be explained by a simple consideration. As the wave front has a helicoidal shape, the Poynting vector 5 ( p , ^),z), which is perpendicular to the wave-front surface, has a nonzero, tangential component S(p{p, q),z) at each point (fig. 1 Ic). The value of this component can be found in the paraxial approximation from analysis of the wave-front geometry. Figure 12 schematically shows a narrow
242
Singular optics
[4, § 5
f Z
Pd
0 is no longer a single LG mode. As we see, the topological charge of the SHG beam is dictated by the charge of the input fundamental harmonic. The same result was obtained for sum and difference frequency mixing (Berzanskis, Matijosius, Piskarskas, Smilgevisius and Stabinis [1997, 1998]). It was shown that total topological charge of pumping beams conserved during any parametric scattering. The beam walk-off essentially changes the dynamics of OVs' interaction during sum-frequency mixing. As a result, the higher-order vortex decays into single-charged vortices with the same sign of topological charge aligned perpendicular to the walk-off direction. The three-wave mixing of OV beams with moderate power in an x^^"^ medium is also defined at negligible depletion by their topological charges only. Much more interesting is the evolution of OV beams during SHG when an input OV beam is mixed with a Gaussian second-harmonic seed beam (Petrov and Torner [1998], Petrov, Molina-Terriza and Torner [1999]). It was shown that the qualitative behavior of the combined beam formed by the mutual coherent seeded and generated second-harmonic beams during propagation inside the quadratic medium. The two pairs of the second-harmonic OVs that appeared, have had zero net charge consistent with predictions of the theory. It is remarkable that the OVs did not form any type of solitary waves. Nontrivial dynamics of the multicharged OV was predicted for SHG with a seed beam (Molina-Terriza and Torner [2000]). The vortex streets exist under the combined effects of diffiraction and Poynting vector walk-off in SHG (Molina-Terriza, Torner and Petrov [1999]). SHG of intense OV beams is accompanied by new nonlinear phenomena due to the appearance of /^^^ nonlinearity in a quadratic medium. It was shown by numerical simulations that second-harmonic waves with higher-order OVs are parametrically azimuthal unstable and decay into set of stable bright solitary waves both in types I and II SHG schemes (Torner and Petrov [1997], Torres, Soto-Crespo, Tomer and Petrov [1998a,b]). This opens the possibility of producing a new class of optical devices that can potentially process
266
Singular optics
[4, § 10
information by mixing of topological charges and to form certain patterns of bright soliton spots. What is the form of the 0AM conservation law during transformation of OV beams to bright solitons? The answer was given by Firth and Skryabin [1997] and Skryabin and Firth [1998] for the case of axial OV transformations in self-focusing saturable Kerr-like and quadratic nonlinear media. It was shown that when the OV is broken into a family of bright filaments due to azimuthal modulation instability they move off along straight-line trajectories tangentially to the initial ring structure without any spiraling due to orbital angular momentum conservation. To our knowledge, this important conclusion is not yet proven. Self-action of laser beams in nonlinear media (Askaryan [1962]) leads to formation of solitons (Boardman and Xie [1998]) including OV ("black") solitons in self-focusing media (Snyder, Poladian and Mitchell [1992]). OV solitons were realized first in the self-focusing Kerr-like media (Swartzlander, Andersen, Regan, Yin and Kaplan [1991]) and then in the photorefi-active crystal SBN (Duree, Morin, Salamo, Segev, Crosignani, Di Porto, Sharp and Yariv [1995]). OV solitons in this crystal were investigated in detail (see review by Mamaev and Zozulya [1999] and references therein). Nonlinear transformation of an OV beam into a soliton was observed in a LiNbOs crystal (Chen, Segev, Wilson, Miller and Maker [1997]). OV solitons were obtained and investigated thoroughly in saturable atomic rubidium vapor (Kivshar, Nepomnyashchy, Tikhonenko, Christou and Luther-Davies [2000] and references therein). The breakup of vortex beams in self-focusing nonlinear media to bright solitons is quite complicated. Actually, it is known that the vortex can disappear in only two possible processes: annihilation in collision with an opposite-sign vortex with the same absolute value of topological charge, or disappearance on the border of the beam. Therefore, a fi-eely propagated axial OV beam has to conserve the topological charge and zero-amplitude trajectory even while breaking up into several bright solitons. The unusual structure of the vortex wave front in this case has not yet been considered. The reconfigurable self-induced waveguides are very promising for photonics applications. It was shown that they can be realized due to "cascade nonlinearities" first considered by Karamzin and Sukhorukov [1976] and revisited in the middle of the ninety's (Torruellas, Wang, Hagan, VanStryland and Stegeman [1995]). It was demonstrated that an intense (above some threshold value) smooth wave-front pump beam focused into a quadratic medium results in the formation of two-dimensional spatial solitary waves. It happens due to the mutual trapping of fiindamental and generated second-harmonic fields. This
4, § 10]
Optical vortices in frequency conversion processes
267
Strong cascading nonlinear coupling counteracts both diffraction and walk-off of the created beams. Dark OV solitons in quadratic media without exact phase matching are stabilized by incoherent coupling between the harmonics due to the selfdefocusing Kerr effect for both harmonics (Alexander, Buryak and Kivshar [1998]). This interplay between diffraction and parametric coupling of the harmonics field carrying OVs leads to formation of a new class of solitons parametric vortex solitons. A general approach (Alexander, Kivshar, Buryak and Sammur [2000]) predicts two novel types of vortex solitons: (i) a "ring-vortex'' soliton, which is a vortex in a harmonic field that guides a ring-like localized mode of the fi^ndamental frequency field, and (ii) a "halo-vortex'\ consisting of a two-component vortex surrounded by a bright ring of its harmonic field, which appears as a result of a third-harmonic generation in a medium with defocusing Kerr nonlinearity. Quite nontrivial are the parametric down-conversion processes with OV beams. It is known (Bloembergen [1965], Boyd [1992]) that they are the result of threewave nonlinear coupling of pump, signal and idler waves with frequencies cO/, wave vectors hi and phases 0/. These interactions obey the energy conservation law 0^1 + 0^2 = CO3 and phase-matching conditions k\+k2 = k^, or (Pi + ^2 ^ ^3These conditions have to be supplemented with the conservation law of 0AM l\h + hh = l^h, which leads automatically to conservation of the OV topological charge l\ -V h = h- At last, the phase terms Of contain the azimuthal phase term //(p. The clearest case is the situation when both pump and signal beams exist at the input of a nonlinear medium. The OV properties in three-wave nonlinear coupling were first investigated theoretically for the degenerate case when signal and idler waves are identical (Staliunas [1992]). The first experiments with a singular signal beam and an ordinary pump beam were performed by Berzanskis, Matijosius, Piskarskas, Smilgevisius and Stabinis [1997]. The "spontaneous parametric fluorescence", or parametric scattering (Klyshko [1988]), when only one pump beam is launched into a nonlinear medium, is quite nontrivial. In this case, both signal and idler waves build up from quantum noise. Di Trapani, Berzanskis, Mirandi, Sapone and Chinaglia [1998] have shown both experimentally and numerically that Bessel-like JQ beams and Bessel-like vortices are generated in a traveling-wave optical parametric amplifier with a ringshape gain angular spectrum. These results are of fundamental importance and show that the vortex structure is characteristic of vacuum quantum noise as well. The situation of only one OV pump beam launched into nonlinear medium was investigated for a cw laser beam (wavelength 532 nm) transmitted through
268
Singular optics
[4, § 11
the lithium triborat crystal with types I and II critical schemes (Arlt, Dholakia, Allen and Padgett [1999]). It was observed that the down-converted signal and idler beams are spatially incoherent and that the OAM is not conserved "within the classical wave fields". This important result finds a natural explanation in spatial uncorrelated phases 3
«*
il--
w,
Nonlinear /?, =* 100% crystal /?a^lOO% ^3=0%
S
w, -
K, (high but < 1 ) Rj (high but < 1 ) R,= 0%
Fig. 1. Optical parametric oscillator (OPO).
This very weak OPO spontaneous noise occupied a very broad spectral range, from near the blue pump frequency through to the infrared absorption band. A corresponding spatial distribution of different frequencies followed the wellknown and simple phase-matching conditions of nonlinear optical systems (see §2). This effect had several names at the time: "parametric fluorescence", "parametric luminescence", "spontaneous parametric scattering", and "splitting". The existence of such a process follows from the quantum consideration of a parametric amplifier developed by Louisell, Yariv and Siegman [1961]. The Hamiltonian for the down-conversion process is given by H, = \ j dvP.Ep{rj)=\ j dvxflE,{rJ)E2{rj)Ep,
(1.1)
where P is the nonlinear polarization induced in the medium by the pump field E. The polarization is defined in terms of the second-order dielectric susceptibility of the medium Xup^ coupling the pump field to the two output fields E\ and E2. The field annihilation operators for photons at two output frequencies a)\ and 0)2 can be written as ai(t)
=
e^-ift^iV '^'\axQ coshg^ + ie '^4o sinhgO.
(1.2)
a2(t) = e ''^\a2o coshgt + ie '^ajo sinhgO. where g is a parametric amplification coefficient proportional to the secondorder susceptibility, the crystal length and the pump field amplitude, ato and AJQ are the initial operator values, and (/) is determined by the pump wave phase. Accordingly, the average number of photons per mode in the outputfieldsni(t) and ^2(0 is n\(t)= (a\(t)a\(t)\
= n \ocosh^ gt-\-(I+n2o)smh^
gt, (1.3)
«2(0 = ( 4 ( 0 ^ 2 ( 0 / = n2o cosh^ gt + (I +/7io)sinh^g^
282
Multi-photon quantum interferometty
[5, § 1
where «io and «2o are the inputs into the n\{t) and «2(0 fields, respectively. This describes a two-component gain feature having the odd property that the " 1 " in the second terms means that there is nonzero output, even when both input fields are zero. This "extra" one photon per mode - due to vacuum fluctuations - can be viewed as stimulating spontaneous down-conversion. The practical theory describing the generation of such radiation was analyzed in detail in 1967-1968 by Mollow and Glauber [1967], Giallorenzi and Tang [1968, 1969] and Klyshko [1967]. The next step in the theory was to analyze the statistics of photons appearing in such spontaneous conversion of one photon into a pair. This was done by Zel'dovich and Klyshko [1969], and Mollow in 1969 (and later treated in detail by Mollow [1973] and Kleinman [1968]), demonstrating the existence of very strong correlations between these photons in space, time and fi-equency. Bumham and Weinberg [1970] first demonstrated the unique and explicitly nonclassical features of states of two-photons generated in the spontaneous regime fi-om the parametric amplifier. Quantum correlations involving twophotons were exploited again 10 years later in experimental work by Malygin, Penin and Sergienko [1981a,b]. Because of a very active research program at the University of Rochester led by Mandel, and the work of Alley at the University of Maryland, the use of highly correlated pairs of photons for the explicit demonstration of Bell inequality violations has become popular and convenient since the mid-1980s. The contemporary name for the process of generating these states, "spontaneous parametric down-conversion" (SPDC), has become widely accepted in the research community, and new, high-intensity sources of SPDC have been developed (see § 3). A number of excellent reviews on the topic of two-photon quantum interference exist. Among the most comprehensive recent reviews covering the topic of entangled-photon interference are Quantum Optics and the Fundamentals of Physics by Perina, Hradil and Jurco [1994], Optical Coherence and Quantum Optics by Mandel and Wolf [1995], Hariharan and Sanders [1996], Quantum Optics by Scully and Zubairy [1997], and The Physics of Quantum Information by Bouwmeester, Ekert and Zeilinger [2000]. Though quantum optics has always kept the attention of the physics community, these reviews have mainly covered the subjects of quantum coherence, squeezed states, quantum non-demolition measurement and, most recently, quantum information. The main goal of this review is to exhibit several different contemporary trends in the development of entangled-photon interferometry using SPDC. We shall concentrate mainly on developments in the area of experimental two-photon interferometry, which has received a significant boost recently due to the importance of the properties
5, § 2]
Two-photon interferometry with type-Iphase-matched SPDC
283
of quantum entanglement in such exciting, but still relatively young areas as quantum teleportation, quantum cryptography and quantum computing (see also § 3 of ch. 1 in this volume).
§ 2. Two-photon interferometry with type-I phase-matched SPDC Spontaneous parametric down-conversion (SPDC) of one photon into a pair is said to be of one two types, type I or type II, depending on whether the two photons of the down-conversion pair have the same polarization or orthogonal polarizations. The two photons of a pair can also leave the down-converting medium either in the same direction or in different directions, the collinear and noncollinear cases respectively. A medium is required for down-conversion, as conservation laws exclude the decay of one photon into a pair in vacuum. The medium is usually some sort of birefringent crystal, such as potassium dihydrogen phosphate (KDP), having a x^^^ optical nonlinearity Upon striking such a nonlinear crystal there is a small probability (on the order of 10~^) that an incident pump photon will be down-converted into a two-photon (see fig. 2). If down-conversion occurs, these conserved quantities are carried into that of the resulting photon pair under the constraints of their respective conservation laws, with the result that the phases of the corresponding wavefunctions match, in accordance with the relations 0)1 + (D2 = (Dp,
ki+k2=kp,
(2.1)
known as the "phase-matching" conditions, where the kt and (Ot are momenta and firequencies for the three waves involved. The individual photons (here labeled / = 1,2) are often arbitrarily called "signal" and "idler", for historical reasons. When the two photons of a pair have different momenta or energies, entanglement will arise in SPDC, provided that the alternatives are in principle experimentally indistinguishable. The two-photon state produced in type-I down-conversion can be written \^)=
da;i (picoi, COQ - coi) \a)i) \(0o - (Oi),
(2.2)
Jo where (l)(a)i,(Oo - (Oi) is the frequency density and the two photons leave the nonlinear medium with the same polarization, orthogonal to the polarization of the pump beam photons. Down-conversion photons are thus produced in two
284
Multi-photon quantum interferometry
[5, §2
Pump bdam
Fig. 2. Spontaneous parametric down-conversion (Saleh [1998]).
thick spectral cones, one for each photon, within which two-photons appear each as a pair of photons on opposite sides of the pump-beam direction (see fig. 2). In a pioneering experiment in the mid-1980s, Hong, Ou and Mandel [1987] created noncollinear, type-I phase-matched SPDC photon pairs in KDP crystal using an ultraviolet continuous-wave (cw) laser pump beam (see fig. 3). These photon pairs were directed to a movable beamsplitter by two mirrors, so that the two resulting spatially superposed beams impinged on two photodetectors Di and D2. Filters placed in the apparatus determined the fi-equency spread of the down-converted photons. This experiment empirically demonstrated the strong temporal correlation of the two-photons. The correlation fiinction for two-photons is g(t) = G(tyG(0). In the experiment G{t) = Jdt(t>[(coo/2) + (jOi,(a)o/2) - a)\], where the down-converted light was frequency-degenerate, so that 0 peaked at a){ = JCDQ = (O2, with COQ = 351.1 nm; g was nearly Gaussian in o) with a bandwidth Aco. The probability of joint detection of the two photons of the pair at Di and D2, at times t and t-\-T respectively, in such an experiment is given by Pl2(0 = ^ E[-\t)E^{^(t-\-T)E^^\t^r)E\'^
(O)
= K\G(0)\' {T'\g(T)\' + R'\g(2AT - T)p - R T \ g \ x ) g { 2 A r - r) + c.c.]} , (2.3) where the £, are the electric fields at detectors D/, and ^ is a constant characterizing the detectors. In the Hong-Ou-Mandel (HOM) experiment, the coincidence rate for photon joint detection at D\ and D2 was studied as the beamsplitter (BS) was translated vertically from its central location by small distances c 5r, giving rise to optical path differences for the two outgoing beams. With R/T = 0.95, the corresponding joint count rate A^^ exhibited a sharp dip.
5, §2]
Two-photon interferometry
with type-I phase-matched
285
SPDC
1 Amp. 1 T*" Counter j Disc. 1 V
U-
Coincidence Counter
—¥-
PDF 11/23+ A
Amp.
& [
Counter
\ Disc. 1
Fig. 3. Hong-Ou-Mandel interferometer (Hong, Ou and Mandel [1987]).
^
260
2B0
300
320
340
380
Position of beam splitter (yum) Fig. 4. Hong-Ou-Mandel dip (Hong, Ou and Mandel [1987]).
near the time difference 8r, having a width determined by the length of the wavepacket (or, equivalently, the coherence time) of the two-photons. This nonclassical coincidence dip was seen to fall to a few percent from the maximum value (see fig. 4), whereas classical optics predicts a visibility that cannot exceed 0.5 (IVlandel [1983]) and Bell-type inequality violations can be obtained once coincidence visibilities exceed 71% (see, for example Tittel, Brendel, Gisin and Zbinden [1999]). Such a dip - hereafter referred to as the "Hong-Ou-]VIandel dip" - also provided for an empirical measure of the time intervals between the two photon arrivals with sub-picosecond precision. Unlike methods requiring the observation of second-order (i.e., single-photon) interference, this technique does not require keeping path differences stable to within a fraction of a wavelength. In 1988, a similar arrangement and light frequency was used by Ou and IVIandel [1988a,b] to demonstrate the violation of Bell's inequality by six standard deviations, in addition to disagreement with classical optical
286
Multi-photon quantum interferometry j/
[5, §2
/ 3 - B a B 2 0 4 gnd
PRISM 2
PRISM I 4"^
150 PS 7 0 P P S / NO-YAG LASER )
M - MIRROR N D - N O FILTERS L - LENS P - PIN H O L E B - 5 0 - 5 0 BEAM S P L I T T E R A - GLAN-THOMPSON POLARIZATION ANALYSER F - NARROW BAND SPECTRAL FILTER D - DETECTOR
,e3 Fig. 5. Shih-Alley experiment (Shih and Alley [1988]).
predictions. In that experiment, the idler photon was rotated by 90 degrees in one beam before reaching the beamsphtter, and polarizers were placed before D\ and D2 at angles 0\ and 62, respectively, to obtain count rates corresponding to the joint probabilities of the left-hand side of Bell's inequality. Taking into account alignment imperfections, the observed joint probabilities were found in this experiment to be in agreement with the quantum mechanical predictions and in violation of the CHSH inequality. According to quantum theory, the choices 0i=jr/8, 02 = ^ / 4 , 0 / = 3jr/8, 02^ = 0, for example, yield 5" = \K{y/2-\)>Q, where 5" < 0 is the Clauser-Home variant of the inequality (Clauser and Home [1974]). The corresponding two-photon interference visibility was empirically found to be F = 0.76. That same year, Shih and Alley [1988] used a similar experimental arrangement, but replaced the cw pump laser with a pulsed laser operating at 266 nm (fig. 5), to demonstrate a three-standard-deviation Bell-type inequality violation. In particular, it was found that d = \ [R^{\JI)-R^iljt)]/RQ\ = 0.34 ± 0.03 > \, where 5 ^ ^ is the Freedman-Clauser variant of the inequality (Freedman and Clauser [1972]). Furthermore, the results were in good agreement with the quantum-mechanical prediction of 6 = ^\/2 = 0.35. In a variation on the same experimental arrangement. Rarity and Tapster obtained a coincidence dip by translating right-angle prisms, instead of fixed mirrors, placed in the beam paths before the beamsplitter. They next explored the frequency non-degenerate case of SPDC to obtain an interferogram exhibiting additional oscillations (Rarity and Tapster [1990a]). The time resolution was improved to approximately 40 fs and the observed visibility reached V = 0.84.
2]
287
Two-photon interferometry with type-I phase-matched SPDC
Fig. 6. Rarity-Tapster experiment (Rarity and Tapster [1990b]).
During the same period, after proposals by Home, Pykacz, Shimony, Zeilinger and Zukowski (Home and Zeilinger [1985], Zukowski and Pykacz [1988], Home, Shimony and Zeilinger [1989]), Rarity and Tapster [1990b] used a modified arrangement involving two beamsplitters and two balanced MachZehnder interferometers to test Bell's inequality. In this case, the variable of the state entanglement was momentum-direction and phase-shifting elements were placed in space-like separated locations (see fig. 6). The measured value for the left-hand side of the CHSH inequality using this arrangement was found to reach S = 22\ at an interference visibility oiV = 0.78, amounting to an inequality violation by 10 standard deviations. SPDC had also previously been used for similar experiments by Ou and Mandel [1988b] using polarization variables. A different interferometric arrangement having two spatially separated, unbalanced Mach-Zehnder interferometers, each involving a phase shift 0/ (/ = 1,2) between the long and short beam paths, was also proposed by Franson [1989], in order to test a Bell-type inequality for position and energy without the involvement of polarization variables or polarizers. This latter sort of experiment was carried out by Franson [1991] (see fig. 7). The interferometer was pumped by a cw laser that produced energy-degenerate two-photons by SPDC. Brendel, Mohler and Martienssen [1992], Kwiat, Steinberg and Chiao [1993] and Shih, Sergienko and Rubin [1993] carried out similar experiments, though with somewhat different arrangements. The initial two-photon state for such experiments can be written \n>) = \ (|5)i \S), - e'2
- W Atom j-4 F2
»
Mi
L2
Fig. 7. Franson interferometer (Franson [1991]).
where S and L refer to the temporal position corresponding to short and long optical path lengths, respectively. By using a sufficiently large path-length difference between long and short options, the last two terms may be neglected an entangled two-photon state results. In the above experiments, the difference of optical paths in the two interferometers, AL, satisfies the requirement cT^oh = - ] ^ Vx[F^'^ = +, r^^^ = - ] + Pr[f^^) = +, f^^^ = - ] ,
(2.5)
where F and T refer to the special cases of pure frequency and pure time measurements, and the superscripts each refer to one of the two photons, arbitrarily labeled 1 and 2. There followed a truly remarkable violation of local realism by roughly 40 standard deviations. This result was achieved in an experiment by Torgerson, Branning, Monken and Mandel [1995] (see fig. 8). Motivated by the ambiguous results of Bell-type inequality tests in which two photons pass through QWPs before reaching polarization analyzers, these workers obtained tremendous inequality violations that removed any lingering questions about nature's ability to violate such inequalities.
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[5, § 3
In late 1994, an argument was presented to the effect that many experiments involving SPDC cannot be used to properly test Bell-type inequalities because the states they utilized are in fact product states (De Caro and Garuccio [1994]). That is, only by post-selecting from the full ensemble of down-conversion events that one-half of events in which joint-detections occur, can Bell tests be simulated. In particular, such a method appears invalid because the intrinsic efficiency of detection required for loophole-free tests is 67% (Kwiat [1995], Kwiat, Eberhard, Steinberg and Chiao [1994], Eberhard [1993]). However, it was subsequently pointed out that even type-I phase-matched down-conversion sources can be configured so as to produce genuine entanglement without the need for post-selection (Kwiat [1995]). This concern can be completely avoided by using type-II phase-matched down-conversion sources that produce a state truly entangled in regard to polarization. Furthermore, the CHSH inequality may be slightly modified so as to allow the use of the full ensemble in a valid Belltype inequality test. § 3. Two-photon interferometry with type-II SPDC In the case of type-II spontaneous parametric down-conversion (SPDC), the two photons of each down-conversion pair have orthogonal rather than identical polarizations. This allows the entanglement of their states to involve polarization in addition to those other quantities potentially involved in the type-I case. This sort of entanglement, including multiple degrees of freedom, has been referred to as "hyper-entanglement" (Kwiat [1997]). In the type-II case, if the two photons of a pair leave the down-converting medium in different directions, i.e., noncollinearly, their entanglement will involve both directions - as it is not possible to identify which photon went in each direction - and polarizations. Moreover, for a nearly monochromatic, continuous-wave laser pump any sort of down-conversion pair entanglement will involve energy, yielding hyperentanglement with three relevant quantities. Such states are generally given by H^) = \ I Jo
do)(l)(co, COQ - (o) \co) \a)o - co)
• (I*,) 1*0 + exp[i0] 1*2) 1*2)) (Ie) |o) + |o) |e)), where the orthogonal polarizations of the down-conversion photons are labeled "e" and "o", according to their orientation relative to the polarizations associated with the extraordinary and ordinary axes of the nonlinear crystal used for downconversion. Unlike the case of type-I phase-matched down-conversion, the two
5, §3]
Two-photon interferometry with type-II SPDC
/ ^
291
Entangled-state emission directions
ordinary Fig. 9. SPDC under type-II phase-matching conditions (Kwiat [1997]).
V—B
351.1 nm
BBO Type li
^
Ar laser
No I Fig. 10. Bell inequality tests using type-II phase-matched two-photons (Kiess, Shih, Sergienko and Alley [1993]).
down-conversion light cones are not concentric about the direction of the pump beam (see fig. 9, and contrast with fig. 2). The new ingredient in the type-II case (eq. 3.1), compared with the type-I case (eq. 2.2), is the involvement of polarization in the entanglement. Entangled states of this kind were used by Kiess, Shih, Sergienko and Alley [1993] to find CHSH inequality violation by 22 standard deviations. In that experiment, a 351.1 nm cw laser pump was used to produce two-photons in BBO crystal at 702.2 nm. These collinear-photon pairs were deflected by a nonpolarizing beamsplitter to two Glan-Thompson polarization analyzers followed by photodetectors, and the resulting coincidence detections were studied (see fig. 10).
292
Multi-photon quantum interferometry
[5, §3
2500
a tn
2000
R
§ S3: 3 (O
1500 1000
5
33
o
?
P -100
-50
0
SO
Optical Delay Al -15
too
150
(|xm)
-10 -5 0 5 10 Number of Quartz Plates Inserted
15
Fig. 11. Polarization two-photon coincidences varying optical delay (Rubin, Klyshko, Shih and Sergienko [1994]).
Shortly thereafter, a comprehensive theoretical treatment of these type-II phase-matched two-photons was given by Rubin, Klyshko, Shih and Sergienko [1994]. A review of several experiments done at the University of MarylandBaltimore County verifying this treatment was presented therein. Quantum beating between polarizations was also observed as absolute polarizations were varied while relative polarization was kept orthogonal (see fig. 11). A similar experimental arrangement was then used to demonstrate the violation of two Bell-type inequalities, one for polarization and one for spacetime, in a single experimental arrangement (Pittman, Shih, Sergienko and Rubin [1995]). In order to test the latter, EPR states were produced by probabilityamplitude cancellation. The experimental arrangement was similar to that of fig. 10, but included also a large quartz polarization delay line and a number of thinner reorientable birefiingent quartz plates placed before the predetector polarization analyzers. Two optical paths to each detector were thus created, so that a two-photon state of the form W=A{XuX2)-A{Y,,Y2) was created, where 1 and 2 label the fast-axis path and the slow-axis path respectively, analogously to the short and long paths of the Franson interferometer, and X and Y indicate two orthogonal linear polarizations. Notably different from the Franson interferometer, however, is that the entangled state here arises from probability-amplitude cancellation rather than from the use of a short coincidence counting time window. In the position test, by activating two spacelike separated Pockels cells, a coincidence counting
5, § 3]
Two-photon interferometry with type-II SPDC
293
Fig. 12. High-intensity two-photon source (Kwiat, Mattle, Weinfurter, Zeihnger, Sergienko and Shih [1995]).
rate R^ = i^o [1-cos(ft>iZ\i - 0)2^2)] was found, where the A are the total optical delay between the optical paths of the two detectors, and (J)\ and CO2 are the signal and idler frequencies. An inequality violation of more than 14 standard deviations was achieved. Similarly, a test in polarization was made by rotating polarization analyzers behind each Pockels cell with coincidence counting rate Rc{(p), where (j) is the difference in polarization analyzer angles at counters 1 and 2, such that 8 = \ [i?c(^^)-^c(|^)]/^o| = 0.309 zb 0.009 > \. A violation of the constraints of local hidden variables theory by more than six standard deviations was observed. In 1995, a new high-intensity, type-II phase-matched SPDC two-photon source was developed in order to take full advantage of two-photon entanglement involving polarization. Two-photons were produced noncollinearly and directly, i.e., without the use of extra beamsplitters or mirrors previously required to emulate entanglement post-selectively (see fig. 12) (Kwiat, Mattle, Weinfurter, Zeilinger, Sergienko and Shih [1995]). This source allowed the observation of CHSH inequality violations by more than 100 standard deviations in less than 5 minutes. Furthermore, all four polarization Bell-states |if'±> = i ( | H , V ) ± | V , H ) ) ,
|0±> = i(|H,H>±|V,V)),
(3.2)
were readily produced. The use of a half-wave plate (HWP) allowed for polarization flipping between ordinary and extraordinary, that is H and V, states. It thus allowed for the exchange of states |*^~) and | ^ " ) , and states | ^ ^ ) and |0+). Similarly, a birefi-ingent phase-shifter allowed for a sign change between two-photon joint amplitudes, so that an exchange between two-photon states |^+) and | ^ ~ ) , and between |(P+) and |cp"), was also accompHshed. Bell-type inequalities were tested using all four Bell states, with significant violations in each case. In addition to the problem of creating high-intensity sources two-photons with entanglement involving polarization, there have been other difficulties associated
294
Multi-photon quantum interferometry
[5, § 3
with entangled optical states. First, long crystals capable of producing entangled states with two polarizations give rise to nontrivial walk-offs. This problem can arise in the form of spatial walk-off: a photon of one polarization moves more quickly through the crystal than the other (yielding longitudinal walk-off) and, though they will leave the crystal collinearly, they can move in different directions while within the crystal (transverse walk-off). For sufficiently short crystals, one can completely compensate for the walk-off, as interference occurs pairwise between processes where the photon pair is created at equal distances but on opposite sides of the crystal central axis. This is accomplished by the introduction in each of the two photon paths of a similar crystal half as long (or in one path and of identical length) after polarization rotation of the photons. This makes the polarization that was previously fast the slow polarization, and vice versa (Rubin, Klyshko, Shih and Sergienko [1994], Kwiat, Mattle, Weinfurter, Zeilinger, Sergienko and Shih [1995]). Similarly, optimal transverse walk-off compensation is accomplished. However, for a sufficiently long crystal, the o and e rays may separate by more than the coherence length of the pump photons, making complete compensation impossible. After the Hong-Ou-Mandel (fig. 3) and Shih-Alley (fig. 5) experiments, it was often intuitively believed that the two-photon interference could be understood in terms of the simultaneous arrival - and hence possible interaction of the two photons of each pair at the common beamsplitter. This is incidental, however. The essential requirement is the equality of optical path length to within the coherence length of the photons, resulting in in-principle indistinguishability. Type-II phase-matched two-photons provided an opportunity to demonstrate this. Pittman, Strekalov, Migdall, Rubin, Sergienko and Shih used collinear type-II phase-matched SPDC in a similar arrangement to observe two-photon interference, where the two photons of each pair were made to reach the common beamsplitter at times greater than the coherence length of their 702.2 nm photons yet still yield two-photon interference (Pittman, Strekalov, Migdall, Rubin, Sergienko and Shih [1996]) (see fig. 13). This provided a counterexample to the intuitive, local picture of some local influence at a common beamsplitter "telling them" which way to travel afterward. First, a phase shifter (r^^) was placed in the path of the signal photon. Then, since that alone could eliminate the indistinguishability of the two-photon alternatives necessary for coincidence interference, "postponed compensation" was used, the leading photon was delayed for Ti^ = 2r«v after the beamsplitter. Thus the arrival of the photons at the two detectors was accomplished in exactly the same order and time difference, successfiilly restoring indistinguishability of detection events, as can be clearly seen in a space-time portrayal of alternative events (see
5, § 3 ]
Two-photon interferometry with type-II SPDC
295
Fig. 13. Schematic of postponed compensation demonstration (Pittman, Strekalov, Migdall, Rubin, Sergienko and Shih [1996]). time
tirfie
space space (a) Balanced HOM/SA Interferometer time
^tim.e
/El
Ek
i^x /
1 ^ \ / « space spac (b) PostpK)ned Compensation Experiment Fig. 14. Space-time diagram of restored indistinguishability (Pittman, Strekalov, Migdall, Rubin, Sergienko and Shih [1996]).
fig. 14). In fig. 13, the delay, state and path labels are identical, allowing for direct comparison with those of fig. 14, if one reorients the apparatus schematic so that the down-conversion crystal is placed at the bottom with its output directed upward. Such an apparatus later proved useful for high-precision polarization mode dispersion measurements (see below).
296
[5, § 3
Multi-photon quantum interferometry 351 nm
detector 1
Pockels cell
collection lens
zr—=" BBO
quartz compensator
quartz rod --r-^-L ^
analyzer
fjiter detector 2
Fig. 15. Apparatus for postselection-free Bell-inequality test energy (Strekalov, Pittman, Sergienko, Shih and Kwiat [1996]).
In 1996, another step exhibiting the nonlocal character of two-photon quantum interference was taken when the new high-intensity type-II phase-matched SPDC source was used by some of the same investigators to make a post-selectionfree test of a Bell inequality with entanglement involving energy (Strekalov, Pittman, Sergienko, Shih and Kwiat [1996]). Unlike the experiment of Franson (fig. 7), where a short-duration time window was used to post-select the coincidence alternatives of interest, this experimental arrangement avoided unwanted alternatives by design: the short-long and long-short alternatives were engineered out. Noncollinear beams of 702.2 nm-photon pairs were created in a symmetrical configuration and passed through a quartz compensator, quartz compensator rods, Pockels cells and polarization analyzers (see fig. 15). The quantum state of two-photons that emerged fi-om the birefiingent rods along the two propagation directions was |
(3.4)
The phase 0 is adjustable by changing crystal tilt, by phase shifting one of the output beams or adjusting the phase relation between horizontal and vertical polarization components of the pump state. A two-photon interference visibility of V = 0.996 was thus achieved. In another significant step forward, Zeilinger et al. tested a Bell-type inequality under strict locality conditions, in order to close one loophole of previous Bell tests, since it is conceivable that polarizers might somehow communicate their settings to one another before two-photons reach them (Weihs, Jennewein, Simon, Weinfiarter and Zeilinger [1998]). In this they surpassed the previous attempt of Aspect, Dalibard and Roger [1982], whose polarizer orientations had only been rapidly, periodically changed during the photons' flight from their source to polarizers separated by a distance of 12 m, which accordingly suffered from what has come to be known as the "periodicity loophole" (Shimony [1990], Weihs, Weinfurter and Zeilinger [1997]).
5, § 3]
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299
Zeilinger et al. separated their polarizers by 400 m, thereby allowing them a full I3jiis to make ultrafast physically random (as opposed to physically deterministic pseudorandom) polarizer orientations, as well as to independently register measurement results from each wing of the apparatus. At the end of each cycle of the experiment, the two sets of data were brought together to view the correlations. 97% coincidence interference visibility was obtained, and the CHSH inequality was violated by 30 standard deviations. There remains only the "detection loophole", to such tests after this work. This final loophole can be blocked when photodetection efficiencies can be improved beyond 0.841 (Shimony [1990]). As a practical application of two-photon interferometry, two-photon interference patterns similar to the HOM dip (see fig. 4) have proven usefiil for measuring polarization mode dispersion (PMD), the difference in propagation rate between two polarization modes in a birefringent medium (see, for example, Dauler, Jaeger, Muller, Migdall and Sergienko [1999]). PMD is important, among other reasons, in understanding propagation of polarized light in optical fibers, such as has been proposed for the purposes of quantum cryptography (see below). The extreme constraint on the simultaneity of the creation of the two photons of a down-conversion pair allows for the high resolution achieved using such a method. The PMD can be determined with sub-femtosecond resolution by studying the effect of dispersive media on this interference feature. An important advantage of this technique, relative to some non-white light interferometric methods, is that it determines the optical delay absolutely, as opposed to simply measuring the delay modulo a wavelength. The PMD is directly determined from the temporal shifi; of the HOM-type interference feature produced by the insertion of a birefringent sample into the interferometer (see fig. 18). Two ways of producing a coincidence event are arranged so that they cannot be distinguished (even in principle). A differential delay line is used to delay one polarization relative to the other. The coincidence rate from spatially separated detectors is recorded as this delay line is varied. When the two photons are separated at the beamsplitter by more than their coherence time the two coincidence events can be distinguished, so no interference is possible; the total coincidence rate is simply the sum of the two individual rates. When the two photons reach the beamsplitter to within their coherence time, however, destructive interference occurs, as the detector polarizers are oriented at 45 degrees and 135 degrees. The two types of coincidence, the first photon produced in the e polarization and the second in the o polarization, or vice versa, become indistinguishable. The temporal correlations are limited by the length
300
[5, § 3
Multi-photon quantum interferometry
\>>
d Easer
BBO Crystal ^•^rrr 351 nm mirror 2 photons (702 nm)
Quartz wedgei
y r 3 0 fs/mm
AA filter EI" beam < splitter
Coinc,
lAT" Optical Delay
/ ^Coinc.
Coincidence circuit Fig. 18. Apparatus for high-precision PMD measurements (Dauler, Jaeger, Muller, Migdall and Sergienko [1999]).
of the down-conversion crystal, and a triangular-shaped interference feature is seen. This occurs because the effect is to convolve two rectangular two-photon wavefiinctions. The shift of the center of this interference feature is identical to the PMD of the sample. The uncertainty limit of the method was determined by how well the center of that feature was determinable. This was found to be as low as 0.15fs. Most recently, the two-photon interferometer has been modified to produce a modified interferogram, with additional "internal fiinging" (see fig. 19) (Branning, Migdall and Sergienko [2000]). This feature of "fiinging in the HOM dip" is introduced by moving the additional variable delay line of the first arrangement for PMD measurement after the first beamsplitter (see fig. 20). Using this improved technique allows one to measure the PMD with a precision of 8 attoseconds. In the quantum-informational context, decoherence-fi-ee subspaces within multiple-photon Hilbert spaces have been a subject of interest. They could be
5, §3]
ffl.
Two-photon interferometry with type-II SPDC
©Ulfl
i3
301
302 Multi-photon quantum interferometry
O O
m §
TS
5
[5, § 3
5, § 3]
Two-photon interferometry with type-II SPDC
303
Birefringent Crystals Down-Conversion Crystal (BBO) Pump{UV)
\
-^
^
* ^ Phase Tuner Correlated Pairs Detectors
Fig. 21. Preserving a two-photon decoherence-free polarization subspace (Berglimd [2000]).
useful, for example, in the redundant coding of quantum information (Ekert, Palma and Suominen [2000]). One could, for example, encode the states |6) , 11) as follows: |6) = |0,1),
|1) = |1,0),
(3.5)
where information-bearing states built from the double eigenstate are chosen not to be susceptible to decoherence in a way that those constructed using the single eigenstate might be. The decoherence-free (DF) subspaces of the Hilbert space of hyperentangled polarization states have recently begun to be studied (Berglund [2000] and Kwiat, Berglund, Altepeter and White [2000]). In particular, it has been shown that, while the energy correlations required by down-conversion phase-matching conditions can render two-photons susceptible to decoherence under the influence of an environment where frequencypolarization coupling is present, the DF subspace can be readily preserved (fig. 21). By appropriately symmetrizing the induced phase errors for the antisymmetric polarization state | V^~), its decoherence-free character can be demonstrated (Zanardi [1997]). The Bell states |t/;±) = ^ ( | H , V ) ± | y H ) ) ,
|^±) = ^ ( | H , H ) ± | V , V ) ) ,
(3.6)
are initially considered, with identical birefringent crystals placed across both paths L and R; denoting the thickness of the nonlinear crystal as C, the phase difference between the two arms will be co^^. The off-diagonal elements of the density matrix for It/;"^) approach zero as the crystal thickness surpasses the coherence length of the down-conversion photons, which is proportional to c/bcD, with b(D the width of the frequency spectrum. In this configuration, \(j)^) do not undergo decoherence, while |i/;=^) do.
304
Multi-photon quantum interferometry
[5, § 3
In particular, at the analyzer, |0=^) is transformed to
which has no effect on the magnitude of off-diagonal density matrix elements. However, these states do not generate a DFS, since in the left/right-circular polarization basis they are written |0±) = i ( | L , R ) ± | R , L ) ) ,
(3.8)
and thus can lose phase information in crystals with eigenmodes |L),|R). By contrast, the antisymmetric state is rotationally invariant, so there is hope for recovering its DFS. By rotating the nonlinear crystal in one arm by 90 degrees, the states 10"^) and |i/;^) will be seen to decohere, while the state |i/^~) will not: its state at the analyzers will be |t/;'-) = ^ ( | H , V ) - e x p [ i ^ ^ ] |V,H)).
(3.9)
Quantifying the fidelity of the transmission process, F = Tr(pinPout) or, for a mixed input state, F = [Tr(y^y^pout v0in)] » decoherence-free subspaces will have F = 1, which is the case for |t/;~). The currently most advanced form of quantum information experimentation is taking place in quantum cryptography - more precisely, quantum key distribution (QKD). QKD is the distribution of a secret key (bit sequence) between two interested parties, usually called Alice and Bob. This key can be used to encrypt and decrypt secret messages using the safe one-time pad method of encryption. The security of QKD is not based on complexity, but on quantum mechanics, since it is generally not possible to measure an unknown quantum system without altering it. Any eavesdropping introduces physical errors in the transmitted data (see also §3.1.4 of ch. 1 in this volume). The basic QKD protocols are the BB84 scheme (Bennett and Brassard [1984]) and the Ekert scheme (Ekert [1991]). BB84 uses single photons transmitted from sender (Alice) to receiver (Bob), which are prepared at random in four partly orthogonal polarization states: 0, 45, 90 and 135 degrees. When an eavesdropper. Eve, tries to obtain information about the polarization, she introduces observable bit errors, which Alice and Bob can detect by comparing a random subset of the generated keys. The Ekert protocol uses entangled pairs and a Belltype inequality. In that scheme, both Alice and Bob receive one particle of
5, § 3]
Two-photon interferometry with type-II SPDC
305
the entangled pair. They perform measurements along at least three different directions on each side, where measurements along parallel axes are used for key generation, and those along oblique angles are used for security verification. Several innovative experiments have been made using entangled photon pairs to implement quantum cryptography in the recent period, 1999-2000 (Sergienko, Atatiire, Walton, Jaeger, Saleh and Teich [1999], Jennewein, Simon, Weihs, Weinfurter and Zeilinger [2000] and Tittel, Brendel, Zbinden and Gisin [2000]). Quantum cryptography experiments have had two principal implementations: weak coherent state realizations of QKD and those using two-photons. The latter approach made use of the nonlocal character of polarization Bell states generated by spontaneous parametric down-conversion. The strong correlation of photon pairs, entangled in both energy-time and momentum-space, eliminates the problem of excess photons faced by the coherent-state approach, where the exact number of photons actually injected is uncertain. In the entangled-photon technique, one of the pair of entangled photons is measured by the sender, confirming for the sender that the state is the appropriate one. It has thus become the favored experimental technique. The first of the recent innovative experiments using SPDC demonstrated a more flexible and robust method of quantum secure key distribution with type-II phase-matched two-photons, in an improved configuration (Sergienko, Atatiire, Walton, Jaeger, Saleh and Teich [1999]). The high contrast and stability of the fourth-order quantum interference, along with the available knowledge of the exact number of photons present in the quantum communication channel, clearly show the performance of EPR-state-based quantum key distribution to be superior to the coherent-state-based technique. The entangled-photon technique had previously used type-I phase-matched pairs and, as a result, suffered from low visibility (only up to 85%) and poor stability of the intensity interferometer. This has primarily been due to the need in previous experiments for the synchronous manipulation of interferometers well separated in space. The intervention of any classical measurement apparatus (eavesdropping) will cause an immediate reduction of the visibility to 70.7%, so high visibility is required to ensure key security. Only an undisturbed EPR state can produce 100% interference visibility. Previous attempts to demonstrate the feasibility of quantum key distribution using EPR photons had failed to attain the high-visibility coincidences. A double, strongly unbalanced, distributed polarization intensity interferometer was used to avoid the simultaneous spatial manipulation that compromised previous attempts. A frequency-doubled femtosecond Ti:sapphire laser was used to generate 80-fs pulses at 541.5 nm that were sent through a 0.1-mm-thick BBO crystal, oriented so as to yield collinearly propagating type-II phase-matched EPR pairs. The
306
Multi-photon quantum interferometry
I
[5, §3 Alice
M*
Ar* Laser
ThSapphire A. = 830 nm Variable polarization delay line BBO crystal Type-il
Bs
i\\
' ri
APD Detector 1
Analyzer-modulator 450(-45P.|/)B;|/)PJS>B
\Uf>A
\I)P.\I)B
LU UL
^ / Alice
Bob
Fig. 24. Time-energy entanglement quantum key distribution scheme (Tittel, Brendel, Zbinden and Gisin [2000]).
apparatus. Key distribution works as follows. The polarizations of the photons are randomly modulated by switching each analyzer-modulator in the rectilinear basis (45'' and SIS*'), providing O*' or 90'' relative phase shift between them. In order to fiilly complete the procedure of quantum key distribution, it would also be necessary to randomly switch the polarization parameters of the two-photon entangled state between two nonorthogonal polarization bases, such as rectilinear and circular polarization. This could be accomplished using fast Pockels-cell polarization rotators. These sets of randomly selected angles force the mutual measurements by Alice and Bob to be destructive (a binary "0") or constructive (a binary "1") with a 50-50 probability, depending on the mutual orientation of the modulators on both sides. Communications between Alice and Bob, which give the set of polarizer orientations selected during each measurement but not the measurement outcomes themselves, are then to be sent over a public classical communication channel. Other protocols may be devised to endow this configuration with the full security that has been added to other configurations. A second experiment uses a scheme that combines using photon pairs and energy-time entanglement (Tittel, Brendel, Zbinden and Gisin [2000]). This scheme realizes the initial concept of using photon-pair correlations (Ekert [1991]) for QKD (fig. 24). However, it implements Bell states, and the robustness of energy-time entanglement allows the information produced using this second method to be preserved over long distances. In this scheme, a light pulse sent at time to enters an initial interferometer imposing a large path length difference
5, § 3]
Two-photon interferometry with type-II SPDC
309
relative to the pulse length. The pulse is split in two, so that the subpulses leave time-separated but with a definite phase difference. The 655 nm, 80 MHz pulses entered a down-converting (KnbOa) crystal creating two-photons described by |t/ and (O, on photon B. When the polarization was parallel to the analyzer axis the result was 1; with polarization orthogonal
310
Multi-photon quantum interferometry
Source
Alice Detectors
-1
Electro Optic Modulator
Photdn>^
Clock
Photon a
Random Number Generator
-r-\-^... T
Bob Electro Optic Modulator
Optical Flber^
Polarizer. &Po\a^
[5,
Classical Communication
Detectors
«\ arizer^ [ Polarizer Random Number . ^ j | - - | R b | Generator | Clock
......... ^r^
Fig. 25. Realization of two-photon QKD over a long distance (Jennewein, Simon, Weihs, Weinfiirter and Zeilinger [2000]).
to the analyzer axis the result was - 1 . By assuming premeasurement values for properties along x, u and v and perfect anticorrelation of measurements along parallel axes, the probabilities for obtaining 1 on both sides obey the inequality P++( X, ^) -^P++(^, 0)) -/7++( X, (o) ^ 0.
(3.11)
The quantum-mechanical prediction for arbitrary analyzer settings a with Alice and P with Bob given the linear polarization singlet Bell state W' is QM
(a,l3)=\sm\a-P).
(3.12)
Maximum violation of the inequality is thus obtained for x ^ -30'', i/; = O"", w = 30'', when the l.h.s. reaches -1/8. To send the quantum key Alice and Bob randomly change their analyzer settings: Alice between -30'' and O'', Bob between 0" and 30". Four combinations of analyzer settings can thus occur: the three oblique settings allow a test of Wigner's inequality, the remaining combination of parallel settings allows key generation using perfect anticorrelations. When the probabilities violated Wigner's inequality, then the generated key was taken to be secured. The second QKD realization of this experiment implemented a variant of the BB84 protocol with entangled photons, with the same |V^")-state polarizationentangled photon pairs approximating the single-photon realization of BB84. Alice and Bob randomly changed their polarizer settings between 0" and 45°. They observed perfect anticorrelations whenever their analyzers were parallel. They obtained identical keys by simply inverting all resulting bit values. Whenever Alice made a measurement on photon A, photon B was projected into the orthogonal state that was analyzed by Bob, or vice versa. After the initial bit distribution, key security could be checked by classically comparing a small subset of their keys to check the security via the error rate.
5, § 4]
Higher multiple-photon entanglement
311
The nonlinear crystal used was again BBO, producing polarization-entangled photon pairs at a wavelength of 702 nm from cw pump light of 351 nm at a power of 350 mW. The photons were each coupled into 500 m long optical fibers and transmitted to "Alice" and "Bob", respectively, who were separated by 360 m. Wollaston polarizing beamsplitters were used as polarization analyzers. The users generated raw keys at rates of 400-800 bps, with bit error rates of approximately 3%.
§ 4. Higher multiple-photon entanglement The entanglement of three or more photons has been a subject of great interest since the proposals in 1989 and 1990 of Greenberger, Home, Zeilinger and Shimony to test locality, reality and completeness assumptions of EPR using entangled three-particle states (Greenberger, Home and Zeilinger [1989], Greenberger, Home, Shimony and Zeilinger [1990], Bernstein, Greenberger, Home and Zeilinger [1993], Klyshko [1993], Aravind [1997]). Bell's inequality had provided a test of these assumptions using statistical correlations, with the most striking results involving Bell states. The GHZ theorem provided a test involving perfect correlations without the use of inequalities, through the use of "GHZ states". The GHZ states can be written as | 0 ± ) = -^(|H)|H)|H)±|V)|V>|V)), |
r Fig. 7. Resonator configuration with a flat output coupler, a diffractive back mirror, and an additional intra-cavity diffractive element. (From Leger, Chen and Dai [1994].)
As for the laser resonator with a GPM, the length of the resonator with a diffractive mirror should be comparable to the Rayleigh distance. Thus, relatively long resonators are required. Leger, Chen and Dai [1994] proposed and demonstrated the insertion of an additional intra-cavity diffractive grating, which allows for high modal discrimination, yet with shorter resonator lengths. A typical resonator configuration is shown in fig. 7. The output mirror is simply flat, the back mirror is a diffractive mirror, and the internal diffractive phase grating is placed approximately in the middle of the resonator. The diameters of two apertures in the resonator are so chosen that negligible loss is introduced to the fundamental mode, but high losses to other modes. Typically, the internal diffractive element is a sinusoidal phase grating of the form exp[iwsin(2:/r/gx], where m is the modulation index and/g is the spatial fi-equency of the grating. Figure 8 shows the calculated modal threshold gain (given by l/|y|^) for the second-order mode in a specific resonator geometry, whereas for the fiandamental mode it is nearly unity; for higher-order modes the threshold gain would of course be higher, so it need not be considered. As evident, the modal
6, §3]
Intra-cavity elements and resonator configurations
0
10
20
339
30
Grating frequency [1/mm]
Fig. 8. Calculated modal threshold gain for the second order mode, for a laser with an internal diflfractive siriusoidal grating. (From Leger, Chen and Dai [1994].)
threshold gain, which indicates the modal discrimination, is low for very low and very high grating frequencies, but reaches a maximum of approximately 2.5 at a grating frequency/g ?^ 6 mm"^. This could be understood by considering the field at the diffractive mirror. This field consists of a multiplicity of near-field patterns resulting from several different orders of the internal grating that are separated by Az2/g. For very low grating frequencies (typically/g < 3 mm~^), the patterns greatly overlap, leading to relatively low mode discrimination. For very high grating frequencies (typically/g > 10 mm"^), there is little, if any, overlap, leading to relatively low mode overlap between the diffraction patterns, again leading to very low mode discrimination. For moderate grating frequencies (typically 3mm~^ :|"-l'!'i";;".l";.'g*l
(b)
Fig. 25. Near- and far-field intensity distributions emerging fi-om a slab waveguide CO2 laser, operating with the 12th-order mode, with an intra-cavity wire grid. (From Morley, Yelden, Baker and Hall [1995].)
[1993]. Here the spatial filter next to the output coupler was simply a circular aperture, whereas the spatial filter next to the back mirror had concentric rings, corresponding to the zeros of the Airy pattern. Experimental results with a pulsed Nd:YAG laser, operating with such square and circular intensity distributions with output energies of 200 mJ, were obtained. Bourliaguet, Mugnier, Kermene, Barthelemy and Froehly [1999] showed that the performance of a pulsed optical parametric oscillator (OPO) could be improved by intra-cavity spatial filtering. Specifically, a five-fold increase of brightness with respect to the multimode operation was demonstrated when applying an intracavity two-dimensional wire grid designed to form a few lobes in the far field. Le Gall and Bourdet [1994] also investigated a Fourier resonator configuration in which an internal spatial filter coupled the phases of an array of CO2 waveguide lasers. Fourier resonators were also investigated by Wolff, Messerschmidt and Fouckhardt [1999] for selecting high-order modes in broad area lasers. Abramski, Baker, Colley and Hall [1992] exploited a one-dimensional wire grid in a slab waveguide CO2 laser in order to select a single high-order mode. The wire grid spacing d was designed to match the periodicity of the desired mode. In principle, this by itself could lead to high modal discrimination, but in practice the alignment tolerances cannot be met, so excessive losses are introduced. The losses can be significantly reduced by resorting to a resonator with intra-cavity coherent self-imaging, based on the Talbot effect. Specifically, the resonator length L was chosen to match the Talbot length, namely, L = \p(flX, where ;? is a small integer that corresponds to the number of imaging planes in a round-trip. Such a self-imaging Talbot effect is particularly advantageous in a waveguide laser where the boundaries reflect the light, leading to a "kaleidoscope" effect, in which a much larger periodic structure is more efficiently self-imaged. The modal properties of such a slab waveguide CO2 laser were experimentally investigated by Morley, Yelden, Baker and Hall [1995], and their results are presented in fig. 25. It shows the near- and far-field intensity
356
Transverse mode shaping and selection
[6, § 3
distributions cross-sections, when the laser operated with the 12th-order mode. As evident, the 12 lobes in the near-field pattern transformed into two main lobes (of opposite phase) in the far field, indicating that the laser operates with a single mode. The wires were 75 |Jim thick, and output powers of up to 65 W were obtained with the single high-order mode, compared to 90 W power for the multimode operation, in a resonator length of 25.4 cm. Self-imaging resonators based on the Talbot effect were also applied for mode selection in waveguide lasers (Baneiji, Davies and Jenkins [1997]), to coherently lock arrays of diode lasers (e.g., Jansen, Yang, Ou, Botez, Wilcox and Mawst [1989]), and for phase matching the modes in waveguide CO2 lasers with intracavity binary phase elements (Glova, Elkin, Lysikov and Napartovich [1996]). Similarly, Tang, Xin and Ochkin [1998] replaced the back mirror in a CO2 laser with a reflective binary phase element to obtain high-output powers. In these cases, either in-phase or anti-phase operation resulted in either one high central lobe or two main lobes in the far field.
3.6. Polarization-selective resonators The light emerging ft-om most resonators is either linearly polarized or unpolarized. Linear-polarization operation is typically obtained by either inserting into the resonator a Brewster window or other polarization selective elements (such as birefringent crystals, polarizers or polarizing beam splitters), or by a polarizationsensitive pumping system (such as RF-excited slab lasers). Unpolarized light is simply obtained where there are no polarization-sensitive elements in the resonator. Also, circularly polarized light can be obtained by inserting a quarterwave (A/4) plate into the resonator (see for example Trobs, Balmer and Graf [2000]). In all the above, the light polarization is uniform across the entire laser output beam. In this subsection, we present laser resonator configurations in which the polarization in different parts of the output beam can be varied, namely a laser output beam with space-variant polarization. Space-variant polarization, such as azimuthal and radial polarizations, results in completely symmetric laser beams that can be exploited in various applications. Such polarizations have been obtained, outside the laser resonator, either by transmitting a linearly polarized laser beam through a twisted nematic liquid crystal (Stadler and Schadt [1996]) or by combining two linearly polarized laser output beams interferometrically (Tidwell, Kim and Kimura [1993]). Azimuthal and radial polarizations have also been obtained by inserting polarization-selective elements into the laser resonator. Pohl [1972] inserted a
6, § 3]
357
Intra-cavity elements and resonator configurations Output coupler
Aperture + stop Calcite
Gain medium (Ruby)
Back mirror
Output beam
C>
U
-Telescope>J -
Q-^switch
Fig. 26. Resonator configuration for selecting an azimuthally polarized mode. (From Pohl [1972].)
birefringent calcite crystal, in which the principal axis was along the z-axis (z cut), into a pulsed ruby laser in order to discriminate between azimuthal and radial polarizations. The resonator configuration is shown in fig. 26. The calcite crystal was inserted inside a two-lens telescope arrangement, so as to increase the divergence of the mode inside the crystal and thereby also the polarization discrimination. Specifically, due to different angles of refraction, the diameter of the azimuthally polarized mode differed from that of the radially polarized beam; so discrimination and selection of an azimuthally polarized mode were obtained by inserting an aperture and a stop with the appropriate diameters. Wynne [1974] generalized this method and showed experimentally, with a wavelength-tunable dye laser, that it is possible to select either the azimuthally or the radially polarized mode. This was achieved by controlling the telescope length and location, so in a certain range of telescope lengths and locations, the azimuthally polarized mode is stable whereas the radially polarized mode is unstable or vice versa. Mushiake, Matsumura and Nakajima [1972] used a conical intra-cavity element to select a radially polarized mode. The conical element introduced low reflection losses to the radially polarized mode but high reflection losses to the azimuthally polarized mode. This method is somewhat similar to applying a Brewster window for obtaining a linear polarization. Similarly, Tovar [1998] suggested using complex Brewster-like windows, of either conical or helical shape, to select radially or azimuthally polarized modes. Nesterov, Niziev and Yakunin [1999] replaced one of the mirrors of a highpower CO2 laser by a sub-wavelength diffractive element. This element consisted of either concentric circles (for selecting azimuthal polarization) or straight lines through a central spot (for selecting radial polarization) to obtain different reflectivities for the azimuthal and radial polarizations. Experimentally, high output power of 1.8 kW was obtained, but the polarization purity was relatively low, with mixed transverse mode operation. Liu, Gu and Yang [1999] analyzed a
358
Transverse mode shaping and selection
[6, § 3
00^® (b)
Fig. 27. Coherent superposition of two orthogonally polarized TEMQI modes to form azimuthally and radially polarized modes: (a) azimuthally {0) polarized doughnut mode; (b) radially (r) polarized doughnut mode.
resonator configuration, into which two sub-wavelength diffractive elements were incorporated, to obtain a different fiindamental mode pattern for two different polarizations. Oron, Blit, Davidson, Friesem, Bomzon and Hasman [2000] presented a method for efficiently obtaining an essentially pure either azimuthally or radially polarized beam directly from a laser. It is based on the selection and coherent summation of two linearly polarized transverse modes that exist inside the laser resonator; specifically, two orthogonally polarized TEMQI modes. The coherent summation of TEMoi(.r) and TEMoi(v) Laguerre-Gaussian modes (or TEMio and TEMoi Hermite-Gaussian modes), having orthogonally linear polarizations, leads to the formation of either an azimuthally or radially polarized mode, whose vectorial field distributions have the form Azimuthal:
£(r, 6) = yEon,)(r, d)-xEoiiy)(r, 6) = 0Eop2exp(-p/2%
Radial:
E(r, 6) = xEo^.^ir, 6) + j^oiCv)(^, 8) = r^op2 exp(-p/2),
where 0 and r are unit vectors in the azimuthal and radial directions, respectively. This coherent summation is illustrated in fig. 27. Figure 27a depicts an azimuthally polarized mode, obtained by a coherent summation of a >^-polarized TEMoi(x) mode and an x-polarized TEMoi(j;) mode, whereas fig. 27b depicts a radially polarized mode, obtained by a coherent summation of an x-polarized TEMoi(jc) mode and a >^-polarized TEMoi(v) mode. The laser resonator configuration in which specific transverse modes are selected and coherently summed is schematically shown in fig. 28. Here, the
6, §3]
Intra-cauity elements and resonator configurations
359
X polarization Combined DPE
0 «» T^
•
y polarization
•
Birefringent beam displacer
I Back mirror
\
Gain medium
Aperture Alignment plate
Output coupler
Fig. 28. Laser resonator configuration with a discontinuous phase element (DPE) for forming azimuthally or radially polarized beam. (From Oron, Blit, Davidson, Friesem, Bomzon and Hasman [2000].)
light propagating inside the laser is split and displaced by means of a birefringent beam displacer to obtain two separate paths with orthogonally polarized light. A differently oriented discontinuous phase element (DPE) is inserted in each path, adjacent to the back mirror, to select the TEMoi mode. Specifically, one of these modes is TEMoi(x), and the other is TEMoi(3;). In practice, the two DPEs can be fabricated on the same substrate. In order to add the two modes coherently with the appropriate phase between them, an additional aligning plate is inserted into one of the paths (in the region after separation), so as to control the optical path by slightly tilting the window. Note that exact phase locking between the two orthogonal modes is obtained by a small coupling between them; the alignment plate brings the two modes close enough to allow this locking to occur. At the back mirror, two spatially separated TEMoi modes evolve, each with a different linear polarization. However, as a result of the coherent summation of these two modes, a circularly symmetric doughnut-shaped beam emerges from the output coupler. This approach was verified experimentally with a continuous-wave lamppumped Nd:YAG laser into which were inserted a calcite crystal as the birefringent beam displacer, two DPEs for selecting the orthogonally polarized TEMoi modes, and an alignment plate to adjust the phase between the two orthogonally polarized TEMQI modes. The calcite crystal was 4 cm long, so the two orthogonally polarized light paths were displaced 4 mm apart. The phase elements were aligned to obtain two orthogonal TEMQI modes. The alignment plate was simply a flat-fused silica window with antireflection layers on both faces. To ensure that the beam emerging from the laser is indeed azimuthally or
360
Transverse mode shaping and selection
[6, § 3
Fig. 29. Experimental intensity distributions of an azimuthally polarized beam that emerges from an NdiYAG laser: (a) directly from the laser with no external elements; (b) after passing a horizontal A/4 plate and a polarizer oriented at 45 degrees; (c) after passing a polarizer oriented in the horizontal direction; (d) after passing a polarizer oriented at 45 degrees; (e) after passing a polarizer oriented in the vertical direction. (From Oron, Blit, Davidson, Friesem, Bomzon and Hasman [2000].)
radially polarized, it was passed through a linear polarizer at 45 degrees. Then, the alignment plate was tilted until the intensity distribution after the polarizer had two lobes perpendicular (for azimuthally polarized) or parallel (for radially polarized) to the polarization direction. This indicated that the orthogonal TEMQI modes add coherently. Some results for an azimuthally polarized beam are shown in figs. 29 and 30. Figure 29 shows the intensity distributions, detected with a CCD camera, that emerge fi"om an NdiYAG laser, which emits an azimuthally polarized beam. Figure 29a shows the near-field intensity distribution of the azimuthally polarized beam, emerging directly fi-om the laser. Here the doughnut shape is evident. In order to determine the polarization of the output beam, four additional intensity distributions were detected. These are shown in figs. 29b-e. Figure 29b shows the intensity distribution of the emerging beam after passing through a quarter wave plate, whose main axis was oriented in the horizontal direction, and a polarizer oriented at 45 degrees. Here, the nearly doughnut-shaped intensity distribution (with approximately half the power) indicates that the polarization of the original beam is linear at each point. Figures 29c-e show the intensity distributions of the beam emerging fi-om the laser, after passing a single linear polarizer oriented at different orientations. Figure 29c shows the intensity distribution with the polarizer oriented in the horizontal direction, fig. 29d that in the diagonal (45 degrees) direction and fig. 29e in the vertical direction. At these three
3]
361
Intra-cavity elements and resonator configurations
•v
t
'^ f ^ '^ >^ A
^
f ^
^
^
1,
(16)
370
Transverse mode shaping and selection !
1
[6, §4
"
S
j .
1/
W^PSx^ \
1 \//
MDFv 1
\f"^^% f
1
1
1
„,...
i.„ 1
0.5
0
Fig. 33. Calculation of entropy, MDF and ln(A/^), for a superposition of the two lowest-order modes, as a function of the portion of the lowest-order mode CQ. The entropy was calculated for a total number of 100 photons. (From Graf and Balmer [1996].)
where «mode is the number of photons in the resonator mode. For a constant number of photons «tot = X]mode«mode, the maximal entropy is obtained when the photons are equally distributed among all the modes, whereas the minimal entropy is obtained when the laser operates with a single mode. Moreover, the second law of thermodynamics (dS ^ 0) implies that photons can only be transferred from mode 1 to mode 2 if «i >«2, so a transformation from multimode operation to single-mode operation is not possible. Also, the entropy of a single high-order mode is equal to that of a single Gaussian mode. Thus, it is possible thermodynamically to transform a high-order mode into a Gaussian beam without losses. The entropy in eq. (16) depends on the total number of photons. Similarly, it is possible to define the mode distribution frmction (MDF) that does not depend on the number of photons. This MDF was referred to as the information entropy by Bastiaans [1986], namely ^^^
/ ^ ^mode ^^ ^mode? mode
^mode
^mode
(17)
«tot
The entropy and MDF were compared to the M^ value for a laser operating with the two lowest-order modes. The results, as a function of the relative number of photons in each mode, are presented in fig. 33. Here, both the entropy and the MDF reach a maximum (poorest beam quality) when the photons are equally divided between the two modes, namely e{) = e\ = \, whereas the M^ value decreases monotonically with EQ. Thermodynamically, it is possible to reduce the M^ value from that shown by point 1 to that shown by point 2 since the entropy of these two states is the same.
6, § 4]
Properties of the laser output beams
371
It should be noted that the coherence properties, the MDF and the entropy depend significantly on the modal structure of the beam. This modal structure can be evaluated from intensity distribution cross-section measurements (see for example Cutolo, Isernia, Izzo, Pierri and Zeni [1995] or Santarsiero, Gori, Borghi and Guattari [1999]) or coherence measurements (Warnky, Anderson and Klein [2000]). Thus, the MDF and entropy can be measured experimentally. Also, the entropy is related to the possible brightness improvement of a beam. The brightness is inversely proportional to the M^ value, thus, a brightness improvement is concomitantly obtained with the reduction of the M^ value. Note that there are two possible orthogonal polarization states, and the above discussion is valid for each of them.
4.2. Intensity and phase distributions In this subsection, we consider the field distributions of beams that emerge from laser resonators. Properties of such field distributions along with methods to distinguish between them are presented. Moreover, methods to improve the focusability of beams having specified field distributions are demonstrated both theoretically and experimentally. 4.2.1. Uniform phase distribution Beams with a uniform phase distribution can be shaped or transformed using various techniques (see for example Bryngdahl [1974] and Davidson, Friesem and Hasman [1992]). The most widespread laser output beam with uniform phase is the Gaussian beam, in which the transverse intensity distribution is maintained while propagating, leading to simple propagation properties. Such Gaussian beams can be readily obtained fi:om lasers operating with only the ftindamental TEMQO mode. For other beams with uniform phase, the transverse intensity distribution is changed during propagation, and their M^ value is greater than unity. Their propagation properties depend on their intensity distributions in the near field. For example, the propagation properties of super-Gaussian beams, which can be obtained from laser resonators with intra-cavity diffractive elements or GPMs, were analyzed by Parent, Morin and Lavigne [1992]. 4.2.2. Binary phase distribution Beams emerging from a laser operating with a high-order Hermite-Gaussian mode or a high-order degenerate (non-helical) Laguerre-Gaussian mode have
372
Transverse mode shaping and selection
[6, § 4
binary phase distributions, which consist of lobes and rings, where neighboring lobes or rings have opposite phases {n phase shift). Casperson [1976] compensated for the phase differences between neighboring lobes or rings, by letting the output beam pass through a binary phase element with Jt phase shifts in proper locations. This increased the peak intensity and power in the main lobe of the farfield intensity distribution, implying a better beam quality. Yet, Siegman [1993] calculated that the beam quality, in terms of M^ is not improved but remains the same. Indeed, he concluded that binary phase plates cannot improve the M^ value. The contradiction between the two approaches results from different criteria for beam quality. The beam quality in accordance to percentage of power in the main lobe criterion is hardly affected by low-power side-lobes, whereas in accordance to the M^ criterion the side-lobes contribute significantly to the improvement in the M^ value. Optimized binary phase-compensating elements were tested experimentally by Casperson [1977] and Casperson, Kincheloe and Stafsudd [1977] with HeNe and CO2 lasers, yielding improvement in the peak power and percentage of power in the main lobe, in agreement with predictions. Lescroart and Bourdet [1995] analyzed binary phase-compensating elements for improving the far-field characteristics of an array of waveguide lasers and determined the trade-off between a main lobe with high peak but with side-lobes to that of main lobes of lower power concomitant with very low side-lobes. Lapucci and Ciofini [1999] optimized the design of a binary phase-compensating element for narrow annular laser sources. Note that, in a laser configuration in which a DPE is inserted next to the output coupler, as shown in fig. 12 (see sect. 3.3), there is no need for a phase-compensating element, since the mode-selecting DPE acts also as a phasecompensating element. Baker, Hall, Hornby, Morley, Taghizadeh and Yelden [1996] showed that by introducing a binary phase-compensating element, the beam emerging from a waveguide laser operating with a high-order antisymmetric mode is transformed so that the far-field distribution consists of a high-intensity main lobe and lowintensity side-lobes. Moreover, by resorting to spatial filtering in the far-field, the side-lobes were eliminated, thereby significantly improving the M^ value with a relatively small decrease in power. Specifically, an original beam with M J =21.7 was transformed into a beam with M^ close to unity with an efficiency of 59%. 4.23. Helical phase distribution In general, the intensity distribution of a helical laser beam is the same as that of a doughnut-shaped laser beam, but their field distributions are distinctly different.
6, § 4]
Properties of the laser output beams
373
Fig. 34. Experimental interference fringe patterns: (a) Gaussian beam; (b) lowest-order helical beam; (c) lowest-order helical beam of opposite helicity. (From Harris, Hill, Tapster and Vaughan [1994].)
Specifically, the doughnut-shaped laser beams are composed of an incoherent superposition of two TEMQ/ modes. For example, when the two field distributions of the TEMoi(x) and TEMQ 1(3;) modes in eq. (6) are added incoherently, they form a hybrid mode whose intensity distribution is doughnut-shaped. On the other hand, when they are added coherently with the appropriate phase, they form a pure helical mode. Several techniques were developed to distinguish between helical and doughnut beams, and between helical beams of opposite helicity. In one technique, the determination whether a beam is helical, having a phase of exp(i0), is done by examining the interference of the beam with its mirror image or with a reference beam (see Indebetouw [1993], Harris, Hill and Vaughan [1994], or Harris, Hill, Tapster and Vaughan [1994]). Examples of such interference patterns are shown in fig. 34. As evident, the helicity can be easily obtained from the fringe pattern. Alternatively, one can let the emerging beam pass through another SPE. An SPE having a phase of exp(-i/0) will focus the helical beam to obtain a main lobe with a high central peak intensity, whereas one having a phase of exp(+i/0) will diverge it fiirther away from the center. This property is unique to the helical beams formed by the TEMo,+/ modes. For the beams formed by the hybrid mode, either one of these two SPEs will focus the hybrid beam to a main lobe with a high central peak intensity, since all parts of the beam are approximately in phase. Experimental results for the helical beams formed by the TEMo,+i mode are shown in fig. 35, along with those predicted for hybrid and helical beams. Figure 35a shows the cross-sections of the far-field intensity distributions with the first phase-correcting SPE having a phase of exp(-i0). As evident, there is a main lobe with a high central peak intensity and very low side-lobes, in agreement with those predicted for a helical beam, while the incoherent hybrid beam has more power spreading. Figure 35b shows the corresponding far-field cross-sections of the intensity distributions with the second SPE of exp(i0). Here the energy spreads out from the center to form an annular shape, as expected for a helical beam. However, for a hybrid beam, no spreading should occur, and there
[6, § 4
Transverse mode shaping and selection
374
Fig. 35. Experimental and calculated far-field intensity distribution cross sections with an additional transmittive SPE: (a) SPE of exp(-i0); (b) SPE of exp(+i0) (dashed lines, experimental results; sohd lines, calculated results for the coherent-helical; dotted lines, calculated results for the incoherenthybrid). (From Oron, Davidson, Friesem and Hasman [2000b].)
Still is one main central lobe. These results clearly indicate that the emerging beam is indeed helical. An interesting property of helical beams is that their M^ value can be significantly improved. Oron, Davidson, Friesem and Hasman [2000a] showed that continuous-spiral phase elements can improve the M^ value of helical beams. Specifically, a single high-order helical beam was transformed into a nearly Gaussian beam. An arrangement for transforming the helical output beam into a nearly Gaussian beam is shown schematically in fig. 36. A helical TEMo,+/ beam, with a field distribution given by eq. (1), emerges from the laser in which a reflective SPE is inserted. The beam is collimated by a cylindrical lens, and its M^ value is 1 + /. In the optical mode converter, the collimated beam first passes through a transmissive SPE, which introduces a phase of exp(-i/0), thereby modifying the helical-phase distribution into a uniform distribution yielding £opl'l''L]/'(p)exp(-p/2). Laser resonator
Optical mode converter Transmissive SPE
U
-^u—>\
Spatial filter (Back mirror)
Fig. 36. Basic configuration of a laser resonator that yields a high-order helical mode and an optical mode converter that yields a nearly Gaussian mode. (From Oron, Davidson, Friesem and Hasman [2000a].)
6, § 4]
Properties of the laser output beams
375
Table 1 Initial and final M^ values and transformation efficiency r/, for a laser operating with either the fiindamental mode or high-order helical modes Mode
Initial M^
Final M^
Transformation efficiency r]
TEMoo
1
1
TEMo, + i
2
1.036
94%
TEMo, + 2
3
1.06
87%
TEMo, + 3
4
1.07
80%
TEM0. + 4
5
1.07
74%
100%
Analysis based on Fourier transformation of the near field and the secondorder moments reveal that the phase modification with the external SPE reduces the M^ value significantly, fi-om 1+/ to (1+/)^^^. This result is in contrast with that obtained for a laser operating with degenerate modes, where a correcting binary-phase plate can improve the peak power of the far-field intensity distribution, but not the M^ value. Moreover, the phase modification significantly changes the far-field intensity distribution, yielding a high central lobe and low ring-shaped side-lobes that contain only a small portion of the total power (e.g., 6% for a laser operating with the TEMo,±i modes). Thus, by exploiting a simple spatial filter (e.g., a circular aperture), it is possible to obtain a further significant improvement in the M^ value. Specifically, a nearly Gaussian beam, withM^ near 1 (theoretically 1.036 for the TEMo,+i mode), with only a small decrease in output power is obtained. Table 1 shows the calculated initial and final (after spatial filtering) M^ values, as well as the transformation efficiency ry, denoting the percentage of power in the main lobe, for a laser operating with either the fundamental mode or in high-order helical modes. Note that the transformation efficiency decreases as the order of the mode increases. The configuration shown in fig. 36 was tested with a linearly polarized CO2 laser in which a reflective SPE replaced the usual back mirror. The SPE was designed to ensure that the laser operated with the helical TEMo,+i mode, as described in sect. 3.4. The optical mode converter contained a transmissive SPE formed on zinc selenide substrate, a telescope configuration of two lenses the first (/i =50 cm) placed 50 cm from the SPE and the second (/2 = 25cm) 75 cm from the first - and a spatial filter in the form of a circular aperture. The intensity distributions were detected at the spatial filter plane and the output plane with a pyroelectric camera. The results are presented in figs. 37 and 38. Figure 37 shows the detected intensity distributions, along
376
Transverse mode shaping and selection
[6,
Fig. 37. Detected intensity distributions and experimental and calculated intensity cross sections at the spatial-filter plane: (a) without SPEs; and (b) with a transmissive SPE. (Solid lines, calculated; dashed lines, experimental). (From Oron, Davidson, Friesem and Hasman [2000a].)
with calculated and experimental intensity cross-sections at the spatial filter plane. Figure 37a shows the intensity distribution and cross-sections without the transmissive SPE. Thus, the usual nearly doughnut-shaped distribution of a helical beam whose phase was not compensated by the transmissive SPE is obtained. Figure 37b shows the intensity distribution and cross-sections when the transmissive SPE was inserted. As is evident, there is a high central peak with low side-lobes that are removed by spatial filtering, yielding a nearly Gaussian beam. Moreover, the detected intensity distribution is narrower than that obtained with no SPE, indicating the improvement of M^. Figure 38 shows photographs of the detected intensity distributions along with calculated and experimental intensity cross-sections at the output of the optical mode converter. Here, the calculated results were obtained by Fourier transformation of the field distribution in the spatial filter plane. Figure 38a shows the intensity distribution and cross-sections at the output plane, when the mode converter includes the SPE but no spatial filter. This is simply an image of the doughnut-shaped helical beam from the laser, whose intensity distribution results from a TEMo,+i mode. The SPE in this case does not affect the intensity distribution at the output plane but only its phase. Figure 38b shows the detected intensity distribution and cross-sections at the output plane with both the SPE and the spatial filter in the mode converter. As predicted, the intensity distribution has a Gaussian shape. In this case the efficiency r] was 85%, which is somewhat lower than the calculated limit of 94%. The M^ value of this beam was measured to be better than 1.1, as expected.
6, §4]
Properties of the laser output beams
?>11
Fig. 38. Detected intensity distributions and calculated and experimental intensity cross sections at the output of the optical mode converter: (a) without a spatial filter; and (b) with a spatial filter (solid lines, calculated; dashed lines, experimental). (From Oron, Davidson, Friesem and Hasman [2000a].)
4.2.4. Several transverse modes When the laser operates with multiple modes, i.e., fundamental and higher order modes, the emerging beam quality is relatively poor and is mainly determined by the highest-order mode. In such lasers the phase distribution of the output beam is random, and little, if anything, can be done to improve the quality of the beam. When the laser operates with a single high-order mode, the emerging beam quality is still inferior to that from a laser operating with the fundamental mode, because the intensity distribution and the divergence of the beam are relatively large. Yet, a beam which originates from a laser operating with a single highorder mode has well-defined amplitude and phase distributions, so in accordance to entropy, it is allowed thermodynamically to efficiently transform it into a nearly Gaussian beam (see sect. 4.1.3). A laser may also operate with only a few modes, where most of the modes between the fundamental and the highest-order modes are not present. Here again the phase distribution is undefined at any point of the beam emerging from the laser. Yet, Oron, Davidson, Friesem and Hasman [2001] demonstrated that beam quality could be improved in a laser operating with a limited number of modes N, much smaller than A/^T, namely. N
0A 305-337
VOLUME 12 (1974) 1 2 3 4 5 6
Self-focusing, self-trapping, and self-phase modulation of laser beams, O. Svelto Self-induced transparency, R.E. Slusher Modulation techniques in spectrometry, M. Harwit, JA. Decker Jr Interaction of light with monomolecular dye layers, K.H. Drexhage The phase transition concept and coherence in atomic emission, R. Graham Beam-foil spectroscopy, S. Bashkin
1-51 53-100 101-162 163-232 233-286 287-344
VOLUME 13 (1976) 1
On the validity of Kirchhoff's law of heat radiation for a body in a nonequilibrium environment, H.P. Baltes 2 The case for and against semiclassical radiation theory, L. Mandel 3 Objective and subjective spherical aberration measurements of the human eye, WM. Rosenblum, J.L. Christensen 4 Interferometric testing of smooth surfaces, G Schulz, J. Schwider
1-- 25 27- 68 69- 91 93--167
408
Contents of previous volumes
Self-focusing of laser beams in plasmas and semiconductors, M.S. Sodha, A.K. Ghatak, V.K. Tripathi Aplanatism and isoplanatism, W.T. Welford
169-265 267-292
VOLUME 14 (1976) 1 2 3 4 5 6 7
The statistics of speckle patterns, J.C. Dainty High-resolution techniques in optical astronomy, A. Labeyrie Relaxation phenomena in rare-earth luminescence, L.A. Riseberg, M.J. Weber The ultrafast optical Kerr shutter, M.A. Duguay Holographic diffraction gratings, G. Schmahl, D. Rudolph Photoemission, RJ. Vernier Optical fibre waveguides - a review, P.J.B. Clarricoats
1- 46 47- 87 89-159 161-193 195-244 245-325 321^02
VOLUME 15 (1977) 1 2 3 4 5
Theory of optical parametric amplification and oscillation, W Brunner, H. Paul Optical properties of thin metal films, P. Rouard, A. Meessen Projection-type holography, T. Okoshi Quasi-optical techniques of radio astronomy, T. W. Cole Foundations of the macroscopic electromagnetic theory of dielectric media, J. Van Kranendonk, J.E. Sipe
1- 75 71-131 139-185 187-244 245-350
VOLUME 16 (1978) 1 2 3 4 5
Laser selective photophysics and photochemistry, VS. Letokhov Recent advances in phase profiles generation, J.J. Clair, C.I. Abitbol Computer-generated holograms: techniques and applications, W-H. Lee Speckle interferometry, A.E. Ennos Deformation invariant, space-variant optical pattern recognition, D. Casasent, D. Psaltis 6 Light emission fi-om high-current surface-spark discharges, R.E. Beverly III 7 Semiclassical radiation theory within a quantum-mechanicalfi^amework,I.R. Senitzky
1- 69 71-117 119-232 233-288 289-356 357-411 413^48
VOLUME 17 (1980) 1 Heterodyne holographic interferometry, R. Ddndliker 2 Doppler-fi-ee multiphoton spectroscopy, E. Giacobino, B. Cagnac 3 The mutual dependence between coherence properties of light and nonlinear optical processes, M. Schubert, B. Wilhelmi 4 Michelson stellar interferometry, W.J. Tango, R.Q. Twiss 5 Self-focusing media with variable index of reft-action, A.L. Mikaelian
1-84 85-161 163-238 239-277 279-345
VOLUME 18 (1980) 1 Graded index optical waveguides: a review, A. Ghatak, K. Thyagarajan 2 Photocount statistics of radiation propagating through random and nonlinear media, J Pefina
1-126 127-203
Contents of previous volumes 3 4
409
Strong fluctuations in light propagation in a randomly inhomogeneous medium, V.I. Tatarskii, VU. Zavorotnyi 204-256 Catastrophe optics: morphologies of caustics and their diffraction patterns, M. V. Berry, C. Upstill 257-346
VOLUME 19 (1981) 1 2 3 4 5
Theory of intensity dependent resonance light scattering and resonance fluorescence, B.R. Mollow 1- 43 Surface and size effects on the light scattering spectra of solids, D.L. Mills, K.R. Subbaswamy 45-137 Light scattering spectroscopy of surface electromagnetic waves in solids, ^. Ushioda 139-210 Principles of optical data-processing,//.J ^M/^erwecA: 211-280 The effects of atmospheric turbulence in optical astronomy, F. Roddier 281-376 VOLUME 20 (1983)
1 2 3 4 5
Some new optical designs for ultra-violet bidimensional detection of astronomical objects, G. Courtis, P. Cruvellier, M. Detaille, M. Saisse 1-61 Shaping and analysis of picosecond light pulses, C Froehly, B. Colombeau, M. Vampouille 63—153 Multi-photon scattering molecular spectroscopy, S. Kielich 155-261 Colour holography, P. Hariharan 263-324 Generation of tunable coherent vacuum-ultraviolet radiation, W. Jamroz, B.P. Stoicheff 325-380
VOLUME 21 (1984) 1 2 3 4 5
Rigorous vector theories of diffraction gratings, D. Maystre Theory of optical bistability, Z.^. Lt/g/a^o The Radon transform and its applications, H.H. Barrett Zone plate coded imaging: theory and applications, N.M. Ceglio, D. W. Sweeney Fluctuations, instabilities and chaos in the laser-driven nonlinear ring cavity, J.C. Englund, R.R. Snapp, W.C. Schieve
1- 67 69-216 217-286 287-354 355-428
VOLUME 22 (1985) 1 Optical and electronic processing of medical images, D. Malacara 2 Quantum fluctuations in vision, M.A. Bouman, W.A. Van De Grind, P. Zuidema 3 Spectral and temporal fluctuations of broad-band laser radiation, A. V. Masalov 4 Holographic methods of plasma diagnostics, G.V Ostrovskaya, Yu.I. Ostrovsky 5 Fringe formations in deformation and vibration measurements using laser light, /. Yamaguchi 6 Wave propagation in random media: a systems approach, R.L. Fante
1- 76 77-144 145-196 197-270 271-340 341-398
VOLUME 23 (1986) Analytical techniques for multiple scattering from rough surfaces, J.A. DeSanto, G.S. Brown Paraxial theory in optical design in terms of Gaussian brackets, K. Tanaka Optical films produced by ion-based techniques, P.J. Martin, R.P Netterfield
1- 62 63-111 113-182
410
Contents of previous volumes
4 Electron holography, A. Tonomura 5 Principles of optical processing with partially coherent light, F.T.S. Yu
183-220 221-275
VOLUME 24 (1987) 1 2 3 4 5
Micro Fresnel lenses, H. Nishihara, T. Suhara Dephasing-induced coherent phenomena, L Rothberg Interferometry with lasers, P. Harihamn Unstable resonator modes, K.E. Oughstun Information processing with spatially incoherent light, /. Glaser
1- 37 39-101 103-164 165-387 389-509
VOLUME 25 (1988) Dynamical instabilities and pulsations in lasers, N.B. Abraham, P. Mandel, L.M. Narducci Coherence in semiconductor lasers, M Ohtsu, T. Tako Principles and design of optical arrays, Wang Shaomin, L. Ronchi Aspheric surfaces, G. Schulz
1-190 191-278 279-348 349-415
VOLUME 26 (1988) 1 2 3 4 5
Photon bunching and antibunching, M.C. Teich, B.E.A. Saleh Nonlinear optics of liquid crystals, /. C. Khoo Single-longitudinal-mode semiconductor lasers, G.P Agrawal Rays and caustics as physical objects, Yu.A. Kravtsov Phase-measurement interferometry techniques, K. Creath
1-104 105-161 163-225 227-348 349-393
VOLUME 27 (1989) 1 The self-imaging phenomenon and its applications, K. Patorski 2 Axicons and meso-optical imaging devices, L.M. Soroko 3 Nonimaging optics for flux concentration, I.M. Bassett, W.T. Welford, R. Winston 4 Nonlinear wave propagation in planar structures, D. Mihalache, M. Bertolotti, C. Sibilia 5 Generalized holography with application to inverse scattering and inverse source problems, R.P Porter
1-108 109-160 161-226 227-313 315-397
VOLUME 28 (1990) 1 Digital holography - computer-generated holograms, O. Bryngdahl, F. Wyrowski 2 Quantum mechanical limit in optical precision measurement and communication, Y. Yamamoto, S. Machida, S. Saito, N. Imoto, T. Yanagawa, M. Kitagawa, G. Bjork 3 The quantum coherence properties of stimulated Raman scattering, M.G. Raymer, LA. Walmsley 4 Advanced evaluation techniques in interferometry, J. Schwider 5 Quantum jumps, i?./CooA:
1- 86 87-179 181-270 271-359 361-416
Contents of previous volumes
411
VOLUME 29 (1991) 1 Optical waveguide diflfraction gratings: coupling between guided modes, D.G. Hall 1-63 2 Enhanced backscattering in optics, YuM. Bambanenkov, Yu.A. Kravtsov, V.D. Ozrin, A.L Saichev 65-197 3 Generation and propagation of ultrashort optical pulses, LP. Christov 199-291 4 Triple-correlation imaging in optical astronomy, G. Weigelt 293-319 5 Nonlinear optics in composite materials. 1. Semiconductor and metal crystallites in dielectrics, C. Flytzanis, E Hache, M.C. Klein, D. Ricard, Ph. Roussignol 321-411 VOLUME 30 (1992) 1 2 3 4 5
Quantum fluctuations in optical systems, S. Reynaud, A. Heidmann, E. Giacobino, C. Eabre 1- 85 Correlation holographic and speckle interferometry, Yu.I. Ostrovsky, V.P Shchepinov 87-135 Localization of waves in media with one-dimensional disorder, V.D. Ereilikher, S.A. Gredeskul 137-203 Theoretical foundation of optical-soliton concept in fibers, Y Kodama, A. Hasegawa 205-259 Cavity quantum optics and the quantum measurement process, P Meystre 261-355 VOLUME 31 (1993)
1 2 3 4 5 6
Atoms in strong fields: photoionization and chaos, PW. Milonni, B. Sundaram Light diffraction by relief gratings: a macroscopic and microscopic view, E. Popov Optical amplifiers, N.K. Dutta, J.R. Simpson Adaptive multilayer optical networks, D. Psaltis, Y Qiao Optical atoms, R.J.C. Spreeuw, J.P Woerdman Theory of Compton free electron lasers, G. Dattoli, L. Giannessi, A. Renieri, A. Torre
1-137 139-187 189-226 227-261 263-319 321^12
VOLUME 32 (1993) 1 Guided-wave optics on silicon: physics, technology and status, B.P Pal 1- 59 2 Optical neural networks: architecture, design and models, ET.S. Yu 61-144 3 The theory of optimal methods for localization of objects in pictures, L.P Yaroslavsky 145-201 4 Wave propagation theories in random media based on the path-integral approach, M.I. Charnotskii, J. Gozani, V.I. Tatarskii, VU. Zavorotny 203-266 5 Radiation by uniformly moving sources. Vavilov-Cherenkov effect, Doppler effect in a medium, transition radiation and associated phenomena, V.L. Ginzburg 267-312 6 Nonlinear processes in atoms and in weakly relativistic plasmas, G. Mainfray, C. Manus 313-361 VOLUME 33 (1994) 1 The imbedding method in statistical boundary-value wave problems, V.I. Klyatskin 2 Quantum statistics of dissipative nonlinear oscillators, V Pefinovd, A. Luks 3 Gap solitons, CM. De Sterke, J.E. Sipe 4 Direct spatial reconstruction of optical phase from phase-modulated images, VI Vlad, D. Malacara 5 Imaging through turbulence in the atmosphere, M.J. Beran, J. Oz-Vogt 6 Digital halftoning: synthesis of binary images, O. Bryngdahl, T. Scheermesser, E Wyrowski
1-127 129-202 203-260 261-317 319-388 389^63
412
Contents of previous volumes VOLUME 34 (1995)
1 2 3 4 5
Quantum interference, superposition states of light, and nonclassical effects, V Buzek, P.L. Knight 1-158 Wave propagation in inhomogeneous media: phase-shift approach, L.P. Presnyakov 159-181 The statistics of dynamic speckles, T. Okamoto, T. Asakura 183-248 Scattering of light from multilayer systems with rough boundaries, /. Ohlidal, K. Navrdtil, M. Ohlidal 249-331 Random walk and diffusion-like models of photon migration in turbid media, A.H. Gandjbakhche, G.H. Weiss 333^02 VOLUME 35 (1996)
1 Transverse patterns in wide-aperture nonlinear optical systems, N.N. Rosanov 2 Optical spectroscopy of single molecules in solids, M. Orrit, J. Bernard, R. Brown, B. Lounis 3 Interferometric multispectral imaging, K. Itoh 4 Interferometric methods for artwork diagnostics, D. Paoletti, G. Schirripa Spagnolo 5 Coherent population trapping in laser spectroscopy, E. Arimondo 6 Quantum phase properties of nonlinear optical phenomena, R. Tanas, A. Miranowicz, Ts. Gantsog
1-60 61-144 145-196 197-255 257-354 355^46
VOLUME 36 (1996) 1 Nonlinear propagation of strong laser pulses in chalcogenide glass films, V. Chumash, I. Cojocaru, E. Fazio, E Michelotti, M. Bertolotti 2 Quantum phenomena in optical interferometry, P. Hariharan, B.C. Sanders 3 Super-resolution by data inversion, M Bertero, C. De Mol 4 Radiative transfer: new aspects of the old theory, Yu.A. Kravtsov, L.A. Apresyan 5 Photon wave function, /. Bialynicki-Birula
\- ATI 49-128 129-178 179-244 245-294
VOLUME 37 (1997) 1 The Wigner distribution fimction in optics and optoelectronics, D. Dragoman 2 Dispersion relations and phase retrieval in optical spectroscopy, K.-E. Peiponen, E.M. Vartiainen, T. Asakura 3 Spectra of molecular scattering of light, I.L. Fabelinskii 4 Soliton communication systems, R.-J. Essiambre, G.P Agrawal 5 Local fields in linear and nonlinear optics of mesoscopic systems, O. Keller 6 Tunneling times and superluminality, R. Y. Chiao, A.M. Steinberg
1- 56 57- 94 95-184 185-256 257-343 345-405
VOLUME 38 (1998) 1 Nonlinear optics of stratified media, S. Dutta Gupta 2 Optical aspects of interferometric gravitational-wave detectors, P. Hello 3 Thermal properties of vertical-cavity surface-emitting semiconductor lasers, W.Nakwaski, M. Osinski 4 Fractional transformations in optics, A. W. Lohmann, D. Mendlovic, Z. Zalevsky 5 Pattern recognition with nonlinear techniques in the Fourier domain, B. Javidi, J.L. Horner 6 Free-space optical digital computing and interconnection, J. Jahns
1- 84 85-164 165-262 263-342 343-^18 419-513
Contents of previous volumes
413
VOLUME 39 (1999) 1 Theory and applications of complex rays, Yu.A. Kravtsov, G. W. Forbes, A.A. Asatryan 1- 62 2 Homodyne detection and quantum-state reconstruction, D.-G. Welsch, W. Vogel, T.Opatrny 63-211 3 Scattering of light in the eikonal approximation, S.K. Sharma, D.J. Somerford 213-290 4 The orbital angular momentum of light, L. Allen, M.J. Padgett, M. Babiker 291-372 5 The optical Kerr effect and quantum optics in fibers, A. Sizmann, G. Leuchs 373-469 VOLUME 40 (2000) 1 Polarimetric optical fibers and sensors, T.R. Wolinski 2 Digital optical computing, J TflmV/fl, Z/c/zzoA^fl 3 Continuous measurements in quantum optics, V. Pefinovd, A. Luks 4 Optical systems with improved resolving power, Z. Zalevsky, D. Mendlovic, A.W.Lohmann 5 Diffractive optics: electromagnetic approach, J. Turunen, M. Kuittinen, F. Wyrowski 6 Spectroscopy in polychromatic fields, Z Ficek and H.S. Freedhoff
1- 75 11-114 115-269 271-341 343-388 389-441
VOLUME 41 (2000) 1 Nonlinear optics in microspheres, M.H. Fields, J. Popp, R.K. Chang 2 Principles of optical disk data storage, J. Carriere, R. Narayan, W.-H. Yeh, C Peng, P. Khulbe, L. Li, R. Anderson, J. Choi, M. Mansuripur 3 EUipsometry of thin film systems, /. Ohlidal, D. Franta 4 Optical true-time delay control systems for wideband phased array antennas, R.T. Chen, Z. Fu 5 Quantum statistics of nonlinear optical couplers, J. Pefina Jr, J. Pefina 6 Quantum phase difference, phase measurements and Stokes operators, A. Luis, L.L. Sdnchez-Soto 7 Optical solitons in media with a quadratic nonlinearity, C. Etrich, F Lederer, B.A. Malomed, T. Peschel, U. Peschel
1- 95 97-179 181-282 283-358 359-417 419-479 483-567
Cumulative index - Volumes 1-42*
Abeles, E: Methods for determining optical parameters of thin films Abella, I.D.: Echoes at optical fi-equencies Abitbol, C.I., see Clair, J.J. Abraham, N.B., R Mandel, L.M. Narducci: Dynamical instabilities and pulsations in lasers Agarwal, G.S.: Master equation methods in quantum optics Agranovich, VM., VL. Ginzburg: Crystal optics with spatial dispersion Agrawal, G.P.: Single-longitudinal-mode semiconductor lasers Agrawal, G.R, see Essiambre, R.-J. Allen, L., D.G.C. Jones: Mode locking in gas lasers Allen, L., M.J. Padgett, M. Babiker: The orbital angular momentum of light Ammann, E.O.: Synthesis of optical birefi^ingent networks Anderson, R., see Carriere, J. Apresyan, L.A., see Kravtsov, Yu.A. Arimondo, E.: Coherent population trapping in laser spectroscopy Armstrong, J.A., A.W. Smith: Experimental studies of intensity fluctuations in lasers Amaud, J.A.: Hamiltonian theory of beam mode propagation Asakura, T, see Okamoto, T. Asakura, T., see Peiponen, K.-E. Asatryan, A.A., see Kravtsov, Yu.A. Babiker, M., see Allen, L. Baltes, H.P.: On the validity of Kirchhoff's law of heat radiation for a body in a nonequilibrium environment Barabanenkov, Yu.N., Yu.A. Kravtsov, YD. Ozrin, A.I. Saichev: Enhanced backscattering in optics Barakat, R.: The intensity distribution and total illumination of aberration-fi:ee diffraction images Barrett, H.H.: The Radon transform and its applications Bashkin, S.: Beam-foil spectroscopy Bassett, I.M., W.T Welford, R. Winston: Nonimaging optics for flux concentration Beckmatm, P.: Scattering of light by rough surfaces Beran, M.J., J. Oz-Vogt: Imaging through turbulence in the atmosphere Bernard, J., see Orrit, M.
^ Volumes I-XL were previously distinguished by roman rather than by arable numerals. 415
2, 249 7, 139 16, 71 25, 11, 9, 26, 37, 9, 39, 9, 41, 36, 35, 6, 11, 34, 37, 39,
1 1 235 163 185 179 291 123 97 179 257 211 247 183 57 1
39, 291 13,
1
29, 65 1, 21, 12, 27, 6, 33, 35,
67 217 287 161 53 319 61
416
Cumulative index - Volumes 1-42
Berry, M.V, C. Upstill: Catastrophe optics: morphologies of caustics and their diffraction patterns Bertero, M., C. De Mol: Super-resolution by data inversion Bertolotti, M., see Mihalache, D. Bertolotti, M., see Chumash, V Beverly III, R.E.: Light emission from high-current surface-spark discharges Bialynicki-Birula, I.: Photon wave function Bjork, G., see Yamamoto, Y. Bloom, A.L.: Gas lasers and their application to precise length measurements Bouman, M.A., W.A. Van De Grind, P. Zuidema: Quantum fluctuations in vision Bousquet, P., see Rouard, P. Brown, G.S., see DeSanto, J.A. Brown, R., see Orrit, M. Brunner, W., H. Paul: Theory of optical parametric amplification and oscillation Bryngdahl, O.: AppHcations of shearing interferometry Bryngdahl, O.: Evanescent waves in optical imaging Bryngdahl, O., F. Wyrowski: Digital holography - computer-generated holograms Bryngdahl, O., T. Scheermesser, F. Wyrowski: Digital halftoning: synthesis of binary images Burch, J.M.: The metrological applications of diffraction gratings Butterweck, H.J.: Principles of optical data-processing Buzek, V, PL. Knight: Quantum interference, superposition states of light, and nonclassical effects Cagnac, B., see Giacobino, E. Carriere, J., R. Narayan, W.-H. Yeh, C. Peng, P Khulbe, L. Li, R. Anderson, J. Choi, M. Mansuripur: Principles of optical disk data storage Casasent, D , D. Psaltis: Deformation invariant, space-variant optical pattern recognition Ceglio, N.M., D.W. Sweeney: Zone plate coded imaging: theory and applications Chang, R.K., see Fields, M.H. Chamotskii, M.I., J. Gozani, VI. Tatarskii, VU. Zavorotny: Wave propagation theories in random media based on the path-integral approach Chen, R.T, Z. Fu: Optical true-time delay control systems for wideband phased array antennas Chiao, R.Y, A.M. Steinberg: Tunneling times and superluminality Choi, J., see Carriere, J. Christensen, J.L., see Rosenblum, WM. Christov, LP: Generation and propagation of ultrashort optical pulses Chumash, V, I. Cojocaru, E. Fazio, F. Michelotti, M. Bertolotti: Nonlinear propagation of strong laser pulses in chalcogenide glass films Clair, J.L, C.I. Abitbol: Recent advances in phase profiles generation Clarricoats, P.LB.: Optical fibre waveguides - a review Cohen-Tannoudji, C , A. Kastler: Optical pumping Cojocaru, \., see Chumash, V Cole, T.W: Quasi-optical techniques of radio astronomy Colombeau, B., see Froehly, C. Cook, R.J.: Quantum jumps Courtes, G., P. Cruvellier, M. Detaille, M. Saisse: Some new optical designs for ultraviolet bidimensional detection of astronomical objects Creath, K.: Phase-measurement interferometry techniques Crewe, A.V: Production of electron probes using a field emission source
18, 257 36, 129 27, 227 36, 1 16, 357 36, 245 28, 87 9, 1 22, 77 4, 145 23, 1 35, 61 15, 1 4, 37 11, 167 28, 1 33, 389 2, 73 19, 211 34,
1
17, 85 41, 97 16, 289 21, 287 41, 1 32, 203 41, 37, 41, 13, 29,
283 345 97 69 199
36, 16, 14, 5, 36, 15, 20, 28,
1 71 327 1 1 187 63 361
20, 1 26, 349 11, 223
All
Cumulative index - Volumes 1-42 Cruvellier, P., see Courtes, G. Cummins, H.Z., H.L. Swimiey: Light beating spectroscopy
20, 1 8, 133
Dainty, J.C: The statistics of speckle patterns Dandliker, R.: Heterodyne holographic interferometry DattoH, G., L. Giannessi, A. Renieri, A. Torre: Theory of Compton free electron lasers Davidson, N., see Oron, R. De Mol, C , see Bertero, M. De Sterke, CM., J.E. Sipe: Gap solitons Decker Jr, J.A., see Harwit, M. Delano, E., R.J. Pegis: Methods of synthesis for dielectric multilayer filters Demaria, A.J.: Picosecond laser pulses DeSanto, J.A., G.S. Brown: Analytical techniques for multiple scattering from rough surfaces Detaille, M., see Courtes, G. Dexter, D.L., see Smith, D.Y. Dragoman, D.: The Wigner distribution fimction in optics and optoelectronics Drexhage, K.H.: Interaction of light with monomolecular dye layers Duguay, M.A.: The ultrafast optical Kerr shutter Dutta, N.K., J.R. Simpson: Optical amplifiers Dutta Gupta, S.: Nonlinear optics of stratified media
14, 17, 31, 42, 36, 33, 12, 7, 9,
1 1 321 325 129 203 101 67 31
23, 20, 10, 37, 12, 14, 31, 38,
1 1 165 1 163 161 189 1
Eberly, J.H.: Interaction of very intense light with free electrons Englund, J.C, R.R. Snapp, W.C. Schieve: Fluctuations, instabilities and chaos in the laser-driven nonlinear ring cavity Ennos, A.E.: Speckle interferometry Essiambre, R.-J., G.P. Agrawal: Soliton communication systems Etrich, C , F. Lederer, B.A. Malomed, T. Peschel, U. Peschel: Optical solitons in media with a quadratic nonlinearity Fabelinskii, I.L.: Spectra of molecular scattering of light Fabre, C , see Reynaud, S. Facchi, P., S. Pascazio: Quantum Zeno and inverse quantum Zeno effects Fante, R.L.: Wave propagation in random media: a systems approach Fazio, E., see Chumash, V Ficek, Z. and H.S. Freedhoff: Spectroscopy in polychromatic fields Fields, M.H., J. Popp, R.K. Chang: Nonlinear optics in microspheres Fiorentini, A.: Dynamic characteristics of visual processes Flytzanis, C , F. Hache, M.C. Klein, D. Ricard, Ph. Roussignol: Nonlinear optics in composite materials. 1. Semiconductor and metal crystallites in dielectrics Focke, J.: Higher order aberration theory Forbes, G.W., see Kravtsov, Yu.A. Frangon, M., S. Mallick: Measurement of the second order degree of coherence Franta, D., see Ohlidal, I. Freedhoff, H.S., see Ficek, Z. Freilikher, VD., S.A. Gredeskul: Localization of waves in media with one-dimensional disorder Frieden, B.R.: Evaluation, design and extrapolation methods for optical signals, based on use of the prolate fiinctions Friesem, A.A., see Oron, R.
7, 359 21, 355 16, 233 37, 185 41, 483 37, 30, 41, 22, 36, 40, 41, 1,
95 1 \A1 341 1 389 1 253
29, 4, 39, 6, 41, 40,
321 1 1 71 181 389
30, 137 9, 311 42, 325
418
Cumulative index - Volumes 1-42
Froehly, C , B. Colombeau, M. Vampouille: Shaping and analysis of picosecond light pulses Fry, G.A.: The optical performance of the human eye Fu, Z., see Chen, R.T. Gabor, D.: Light and information Gamo, H.: Matrix treatment of partial coherence Gandjbakhche, A.H., G.H. Weiss: Random walk and diffusion-like models of photon migration in turbid media Gantsog, Ts., see Tanas, R. Ghatak, A., K. Thyagarajan: Graded index optical waveguides: a review Ghatak, A.K., see Sodha, M.S. Giacobino, E., B. Cagnac: Doppler-free multiphoton spectroscopy Giacobino, E., see Reynaud, S. Giannessi, L., see Dattoli, G. Ginzburg, VL., see Agranovich, VM. Ginzburg, VL.: Radiation by uniformly moving sources. Vavilov-Cherenkov effect, Doppler effect in a medium, transition radiation and associated phenomena Giovanelli, R.G.: Diffusion through non-uniform media Glaser, L: Information processing with spatially incoherent light Gniadek, K., J. Petykiewicz: Applications of optical methods in the diffraction theory of elastic waves Goodman, J.W.: Synthetic-aperture optics Gozani, J., see Chamotskii, M.I. Graham, R.: The phase transition concept and coherence in atomic emission Gredeskul, S.A., see Freilikher, YD.
20, 63 8, 51 41, 283 1, 109 3, 187 34, 35, 18, 13, 17, 30, 31, 9,
333 355 1 169 85 1 321 235
32, 267 2, 109 24, 389 9, 8, 32, 12, 30,
281 1 203 233 137
Hache, R, see Flytzanis, C. Hall, D.G.: Optical waveguide diffraction gratings: coupling between guided modes Hariharan, P.: Colour holography Hariharan, P.: Interferometry with lasers Hariharan, P., B.C. Sanders: Quantum phenomena in optical interferometry Harwit, M., J.A. Decker Jr: Modulation techniques in spectrometry Hasegawa, A., see Kodama, Y. Hasman, E., see Oron, R. Heidmann, A., see Reynaud, S. Hello, P.: Optical aspects of interferometric gravitational-wave detectors Helstrom, C.W.: Quantum detection theory Herriot, D.R.: Some applications of lasers to interferometry Homer, J.L., see Javidi, B. Huang, T.S.: Bandwidth compression of optical images
29, 29, 20, 24, 36, 12, 30, 42, 30, 38, 10, 6, 38, 10,
321 1 263 103 49 101 205 325 1 85 289 171 343 1
Ichioka, Y., see Tanida, J. Imoto, N., see Yamamoto, Y Itoh, K.: Interferometric multispectral imaging
40, 77 28, 87 35, 145
Jacobsson, R.: Light reflection ft-om films of continuously varying refractive index Jacquinot, P., B. Roizen-Dossier: Apodisation Jaeger, G., A.V Sergienko: Multi-photon quantum interferometry Jahns, X: Free-space optical digital computing and interconnection Jamroz, W., B.P. Stoicheff: Generation of tunable coherent vacuum-ultraviolet radiation
5, 3, 42, 38, 20,
247 29 277 419 325
419
Cumulative index - Volumes 1-42 Javidi, B., XL. Horner: Pattern recognition with nonlinear techniques in the Fourier domain Jones, D.G.C., see Allen, L.
38, 343
9, 179
Kastler, A., see Cohen-Tannoudji, C. Keller, O.: Local fields in linear and nonlinear optics of mesoscopic systems Khoo, I.e.: Nonlinear optics of liquid crystals Khulbe, P., see Carriere, J. Kielich, S.: Multi-photon scattering molecular spectroscopy Kilin, S.Ya.: Quanta and information Kinosita, K.: Surface deterioration of optical glasses Kitagawa, M., see Yamamoto, Y. Klein, M.C., see Flytzanis, C. Klyatskin, VI.: The imbedding method in statistical boundary-value wave problems Knight, PL., see Buzek, V Kodama, Y, A. Hasegawa: Theoretical foundation of optical-soliton concept in fibers Koppelman, G.: Multiple-beam interference and natural modes in open resonators Kottler, R: The elements of radiative transfer Kottler, R: Diffraction at a black screen, Part I: Kirchhoff's theory Kottler, R: Diffraction at a black screen. Part II: electromagnetic theory Kozhekin, A.E., see Kurizki, G. Kravtsov, Yu.A.: Rays and caustics as physical objects Kravtsov, Yu.A., see Barabanenkov, Yu.N. Kravtsov, Yu.A., L.A. Apresyan: Radiative transfer: new aspects of the old theory Kravtsov, Yu.A., G.W. Rorbes, A.A. Asatryan: Theory and applications of complex rays Kubota, H.: Interference color Kuittinen, M., see Turunen, J. Kurizki, G., A.E. Kozhekin, T. Opatrny, B.A. Malomed: Optical solitons in periodic media with resonant and off-resonant nonlinearities
5, 1 37 257 26, 105 41, 97 20, 155 42 1 4, 85 28, 87 29, 321 33 1 34 1 30 205 7, 1 3, 1 4 281 6 331 42 93 26, 227 29 65 36 179 39, 1 1, 211 40 343
Labeyrie, A.: High-resolution techniques in optical astronomy Lean, E.G.: Interaction of light and acoustic surface waves Lederer, R, see Etrich, C. Lee, W.-H.: Computer-generated holograms: techniques and applications Leith, E.N., J. Upatnieks: Recent advances in holography Letokhov, VS.: Laser selective photophysics and photochemistry Leuchs, G., see Sizmann, A. Levi, L.: Vision in communication Li, L., see Carriere, J. Lipson, H., C.A. Taylor: X-ray crystal-structure determination as a branch of physical optics Lohmann, A.W., D. Mendlovic, Z. Zalevsky: Rractional transformations in optics Lohmann, A.W., see Zalevsky, Z. Lounis, B., see Orrit, M. Lugiato, L.A.: Theory of optical bistability Luis, A., L.L. Sanchez-Soto: Quantum phase difference, phase measurements and Stokes operators Luks, A., see Pefinova, V Luks, A., see Pefinova, V
14 11 41 16 6 16 39 8 41
47 123 483 119 1 1 373 343 97
5 38 40 35 21
287 263 271 61 69
Machida, S., see Yamamoto, Y Mainfray, G., C. Manus: Nonlinear processes in atoms and in weakly relativistic plasmas
28 87 32 313
42 93
41 419 33 129 40 115
420
Cumulative index - Volumes 1-42
Malacara, D.: Optical and electronic processing of medical images Malacara, D., see Vlad, VI. Mallick, S., see Frangon, M. Malomed, B.A., see Etrich, C. Malomed, B.A., see Kurizki, G. Mandel, L.: Fluctuations of light beams Mandel, L.: The case for and against semiclassical radiation theory Mandel, P., see Abraham, N.B. Mansuripur, M., see Carriere, J. Manus, C , see Mainfray, G. Marchand, E.W.: Gradient index lenses Martin, P.J., R.P. Netterfield: Optical films produced by ion-based techniques Masalov, A.V: Spectral and temporal fluctuations of broad-band laser radiation Maystre, D.: Rigorous vector theories of diffraction gratings Meessen, A., see Rouard, P. Mehta, C.L.: Theory of photoelectron counting Mendlovic, D., see Lohmann, A.W. Mendlovic, D., see Zalevsky, Z. Meystre, P.: Cavity quantum optics and the quantum measurement process Michelotti, F, see Chumash, V Mihalache, D., M. Bertolotti, C. Sibilia: Nonlinear wave propagation in planar structures Mikaelian, A.L., M.L. Ter-Mikaelian: Quasi-classical theory of laser radiation Mikaelian, A.L.: Self-focusing media with variable index of refraction Mills, D.L., K.R. Subbaswamy: Surface and size effects on the light scattering spectra of solids Milonni, PW., B. Sundaram: Atoms in strong fields: photoionization and chaos Miranowicz, A., see Tanas, R. Miyamoto, K.: Wave optics and geometrical optics in optical design MoUow, B.R.: Theory of intensity dependent resonance light scattering and resonance fluorescence Murata, K.: Instruments for the measuring of optical transfer functions Musset, A., A. Thelen: Multilayer antireflection coatings
22, 1 33, 261 6, 71 41, 483 42, 93 2, 181 13, 27 25, 1 41, 97 32, 313 11, 305 23, 113 22, 145 21, 1 15, 77 8, 373 38, 263 40, 271 30, 261
36,
1
27, 227 7, 231 17, 279 19, 45 31, 1 35, 355 1, 31 19, 1 5, 199 8, 201
Nakwaski, W., M. Osihski: Thermal properties of vertical-cavity surface-emitting semiconductor lasers Narayan, R., see Carriere, J. Narducci, L.M., see Abraham, N.B. Navratil, K., see Ohlidal, I. Netterfield, R.P, see Martin, PJ. Nishihara, H., T. Suhara: Micro Fresnel lenses
38, 41, 25, 34, 23, 24,
165 97 1 249 113 1
Ohlidal, I., K. Navratil, M. Ohlidal: Scattering of light from multilayer systems with rough boundaries Ohlidal, I., D. Franta: Ellipsometry of thin film systems Ohlidal, M., see Ohlidal, I. Ohtsu, M., T Tako: Coherence in semiconductor lasers Okamoto, T, T. Asakura: The statistics of dynamic speckles Okoshi, T: Projection-type holography Ooue, S.: The photographic image Opatmy, T, see Welsch, D.-G. Opatrny, T, see Kurizki, G.
34, 41, 34, 25, 34, 15, 7, 39, 42,
249 181 249 191 183 139 299 63 93
421
Cumulative index - Volumes 1-42 Oron, R., N. Davidson, A.A. Friesem, E. Hasman: Transverse mode shaping and selection in laser resonators Orrit, M., J. Bernard, R. Brown, B. Loimis: Optical spectroscopy of single molecules in solids Osinski, M., see Nakwaski, W. Ostrovskaya, G.V, Yu.I. Ostrovsky: Holographic methods of plasma diagnostics Ostrovsky, Yu.L, see Ostrovskaya, G.V. Ostrovsky, Yu.L, VP. Shchepinov: Correlation holographic and speckle interferometry Oughstun, K.E.: Unstable resonator modes Oz-Vogt, J., see Beran, M.J. Ozrin, VD., see Barabanenkov, Yu.N.
42, 325
35, 61 38, 165 22, 197 22 197 30 87 24 165 33, 319 29, 65 291 1 197 147 1 1 1 67
Padgett, M.J., see Allen, L. Pal, B.P.: Guided-wave optics on silicon: physics, technology and status Paoletti, D., G. Schirripa Spagnolo: Interferometric methods for artwork diagnostics Pascazio, ^., see Facchi, P. Patorski, K.: The self-imaging phenomenon and its applications Paul, H., see Brunner, W. Pegis, R.J.: The modern development of Hamiltonian optics Pegis, R.J., see Delano, E. Peiponen, K.-E., E.M. Vartiainen, T. Asakura: Dispersion relations and phase retrieval in optical spectroscopy Peng, C , see Carriere, J. Pefina, X: Photocount statistics of radiation propagating through random and nonlinear media Pefina, J., see Pefina Jr, J. Pefina Jr, J., J. Pefina: Quantum statistics of nonlinear optical couplers Pehnova, V, A. Luks: Quantum statistics of dissipative nonlinear oscillators Pefinova, V, A. Luks: Continuous measurements in quantum optics Pershan, PS.: Non-linear optics Peschel, T., see Etrich, C. Peschel, U., see Etrich, C. Petykiewicz, J., see Gniadek, K. Picht, J.: The wave of a moving classical electron Popov, E.: Light diffraction by relief gratings: a macroscopic and microscopic view Popp, J., see Fields, M.H. Porter, R.P.: Generalized holography with application to inverse scattering and inverse source problems Presnyakov, L.P.: Wave propagation in inhomogeneous media: phase-shift approach Psaltis, D., see Casasent, D. Psaltis, D., Y Qiao: Adaptive multilayer optical networks
39 32 35 42 27 15 1 7
Qiao, Y, see Psaltis, D.
31, 227
Raymer, M.G., LA. Walmsley: The quantum coherence properties of stimulated Raman scattering Renieri, A., see Dattoli, G. Reynaud, S., A. Heidmann, E. Giacobino, C. Fabre: Quantum fluctuations in optical systems Ricard, D., see Flytzanis, C. Riseberg, L.A., M.J. Weber: Relaxation phenomena in rare-earth luminescence
37 57 41 97 18 41 41 33 40 5 41 41 9 5 31 41
127 359 359 129 115 83 483 483 281 351 139 1
27 34 16 31
315 159 289 227
28, 181 31, 321 30, 1 29, 321 14, 89
422
Cumulative index - Volumes 1-42
Risken, H.: Statistical properties of laser light Roddier, E: The effects of atmospheric turbulence in optical astronomy Roizen-Dossier, B., see Jacquinot, P. Ronchi, L., see Wang Shaomin Rosanov, N.N.: Transverse patterns in wide-aperture nonlinear optical systems Rosenblum, W.M., XL. Christensen: Objective and subjective spherical aberration measurements of the human eye Rothberg, L.: Dephasing-induced coherent phenomena Rouard, P., P. Bousquet: Optical constants of thin films Rouard, P., A. Meessen: Optical properties of thin metal films Roussignol, Ph., see Flytzanis, C. Rubinowicz, A.: The Miyamoto-Wolf diffraction wave Rudolph, D., see Schmahl, G.
8, 239 19, 281 3, 29 25, 279 35, 1
Saichev, A.I., see Barabanenkov, Yu.N. Saisse, M., see Courtes, G. Saito, S., see Yamamoto, Y. Sakai, H., see Vanasse, G.A. Saleh, B.E.A., see Teich, M.C. Sanchez-Soto, L.L., see Luis, A. Sanders, B.C., see Hariharan, P. Scheermesser, T, see Bryngdahl, O. Schieve, W.C., see Englund, J.C. Schirripa Spagnolo, G., see Paoletti, D. Schmahl, G., D. Rudolph: Holographic diffraction gratings Schubert, M., B. Wilhelmi: The mutual dependence between coherence properties of light and nonlinear optical processes Schulz, G., J. Schwider: Interferometric testing of smooth surfaces Schulz, G.: Aspheric surfaces Schwider, J., see Schulz, G. Schwider, J.: Advanced evaluation techniques in interferometry Scully, M.O., K.G. Whitney: Tools of theoretical quantum optics Senitzky, LR.: Semiclassical radiation theory within a quantum-mechanical fi-amework Sergienko, A.V, see Jaeger, G. Sharma, S.K., D.J. Somerford: Scattering of light in the eikonal approximation Shchepinov, VP, see Ostrovsky, Yu.I. Sibilia, C , see Mihalache, D. Simpson, J.R., see Dutta, N.K. Sipe, J.E., see Van Kranendonk, J. Sipe, J.E., see De Sterke, CM. Sittig, E.K.: Elastooptic light modulation and deflection Sizmann, A., G. Leuchs: The optical Kerr effect and quantum optics in fibers Slusher, R.E.: Self-induced transparency Smith, A.W, see Armstrong, J.A. Smith, D.Y, D.L. Dexter: Optical absorption strength of defects in insulators Smith, R.W: The use of image tubes as shutters Snapp, R.R., see Englund, J.C. Sodha, M.S., A.K. Ghatak, VK. Tripathi: Self-focusing of laser beams in plasmas and semiconductors Somerford, D.J., see Sharma, S.K. Soroko, L.M.: Axicons and meso-optical imaging devices
29, 65 20, 1 28, 87 6, 259 26, 1 41, 419 36, 49 33, 389 21, 355 35, 197 14, 195
13, 69 24, 39 4, 145 15, 77 29, 321 4, 199 14, 195
17, 163 13, 93 25, 349 13, 93 28, 271 10, 89 16, 413 42, 277 39, 213 30, 87 27, 227 31, 189 15, 245 33, 203 10, 229 39, 373 12, 53 6, 211 10, 165 10, 45 21, 355 13, 169 39, 213 27, 109
Cumulative index - Volumes 1-42
423
Soskin, M.S., M.V Vasnetsov: Singular optics Spreeuw, R.J.C., J.P. Woerdman: Optical atoms Steel, W.H.: Two-beam interferometry Steinberg, A.M., see Chiao, R.Y. Stoicheff, B.P., see Jamroz, W. Strohbehn, J.W.: Optical propagation through the turbulent atmosphere Stroke, G.W.: Ruling, testing and use of optical gratings for high-resolution spectroscopy Subbaswamy, K.R., see Mills, D.L. Suhara, T., see Nishihara, H. Sundaram, B., see Milonni, RW. Svelto, O.: Self-focusing, self-trapping, and self-phase modulation of laser beams Sweeney, D.W, see Ceglio, N.M. Swinney, H.L., see Cummins, H.Z,
42, 31, 5, 37, 20, 9,
219 263 145 345 325 73
Tako, T, see Ohtsu, M. Tanaka, K.: Paraxial theory in optical design in terms of Gaussian brackets Tanas, R., A. Miranowicz, Ts. Gantsog: Quantum phase properties of nonlinear optical phenomena Tango, W.J., R.Q. Twiss: Michelson stellar interferometry Tanida, J., Y. Ichioka: Digital optical computing Tatarskii, VI., VU. Zavorotnyi: Strong fluctuations in light propagation in a randomly inhomogeneous medium Tatarskii, VI., see Chamotskii, M.I. Taylor, C.A., see Lipson, H. Teich, M.C., B.E.A. Saleh: Photon bunching and antibunching Ter-MikaeHan, M.L., see Mikaelian, A.L. Thelen, A., see Musset, A. Thompson, B.J.: Image formation with partially coherent hght Thyagarajan, K., see Ghatak, A. Tonomura, A.: Electron holography Torre, A., see Dattoli, G. Tripathi, VK., see Sodha, M.S. Tsujiuchi, I : Correction of optical images by compensation of aberrations and by spatial frequency filtering Turunen, J., M. Kuittinen, R Wyrowski: Diffractive optics: electromagnetic approach Twiss, R.Q., see Tango, W.J.
25, 191 23, 63
2, 1 19, 45 24, 1 31, 1 12, 1 21, 287 8, 133
35, 355 17, 239 40, 77 18, 32, 5, 26, 7, 8, 7, 18, 23, 31, 13,
204 203 287 1 231 201 169 1 183 321 169
2, 131 40, 343 17, 239
Upatnieks, X, see Leith, E.N. Upstill, C , see Berry, M.V Ushioda, S.: Light scattering spectroscopy of surface electromagnetic waves in solids
6, 1 18, 257 19, 139
Vampouille, M., see Froehly, C. Van De Grind, W.A., see Bouman, M.A. Van Heel, A.C.S.: Modem alignment devices Van Kranendonk, X, XE. Sipe: Foundations of the macroscopic electromagnetic theory of dielectric media Vanasse, G.A., H. Sakai: Fourier spectroscopy Vartiainen, E.M., see Peiponen, K.-E. Vasnetsov, M.V, see Soskin, M.S. Vernier, P.X: Photoemission
20, 63 22, 77 1, 289 15, 6, 37, 42, 14,
245 259 57 219 245
424
Cumulative index - Volumes 1-42
Vlad, VI., D. Malacara: Direct spatial reconstruction of optical phase from phasemodulated images Vogel, W., 5e^ Welsch, D.-G. Walmsley, LA., see Raymer, M.G. Wang Shaomin, L. Ronchi: Principles and design of optical arrays Weber, M.J., see Riseberg, L.A. Weigelt, G.: Triple-correlation imaging in optical astronomy Weiss, G.H., see Gandjbakhche, A.H. Welford, WT: Aberration theory of gratings and grating mountings Welford, W.T.: Aplanatism and isoplanatism Welford, WT, see Bassett, I.M. Welsch, D.-G., W Vogel, T. Opatmy: Homodyne detection and quantum-state reconstruction Whitney, K.G., see Scully, M.O. Wilhelmi, B., see Schubert, M. Winston, R., see Bassett, I.M. Woerdman, J.R, see Spreeuw, R.J.C. Wolihski, T.R.: Polarimetric optical fibers and sensors Wolter, H.: On basic analogies and principal differences between optical and electronic information Wynne, C.G.: Field correctors for astronomical telescopes Wyrowski, F., see Bryngdahl, O. Wyrowski, F, see Bryngdahl, O. Wyrowski, F, see Turunen, J. Yamaguchi, I.: Fringe formations in deformation and vibration measurements using laser light Yamaji, K.: Design of zoom lenses Yamamoto, T.: Coherence theory of source-size compensation in interference microscopy Yamamoto, Y, S. Machida, S. Saito, N. Imoto, T. Yanagawa, M. Kitagawa, G. Bjork: Quantum mechanical limit in optical precision measurement and communication Yanagawa, T., see Yamamoto, Y Yaroslavsky, L.P.: The theory of optimal methods for localization of objects in pictures Yeh, W.-H., see Carriere, J. Yoshinaga, H.: Recent developments in far infrared spectroscopic techniques Yu, F.TS.: Principles of optical processing with partially coherent light Yu, F.TS.: Optical neural networks: architecture, design and models Zalevsky, Z., see Lohmann, A.W ^ Zalevsky, Z., D. Mendlovic, A.W. Lohmann: Optical systems with improved resolving power Zavorotny, VU., see Chamotskii, M.I. Zavorotnyi, VU., see Tatarskii, V.I. Zuidema, P., see Bouman, M.A.
33, 261 39, 63 28, 25, 14, 29, 34, 4, 13, 27,
181 279 89 293 333 241 267 161
39, 10, 17, 27, 31, 40,
63 89 163 161 263 1
1, 10, 28, 33, 40,
155 137 1 389 343
22, 271 6, 105 8, 295 28, 28, 32, 41, 11, 23, 32,
87 87 145 97 77 221 61
^^^ ^^^ 40, 32, 18, 22,
271 203 204 77