Seventy five years ago, three remarkable papers by Schrödinger, Kennard and Darwin were published. They were devoted to the evolution of Gaussian wave packets for an oscillator, a free particle and a particle moving in uniform constant electric and magnetic fields. From the contemporary point of view, these packets can be considered as prototypes of the coherent and squeezed states, which are, in a sense, the cornerstones of modern quantum optics. Moreover, these states are frequently used in many other areas, from solid state physics to cosmology. This paper gives a review of studies performed in the field of so-called `nonclassical states' (squeezed states are their simplest representatives) over the past seventy five years, both in quantum optics and in other branches of quantum physics. My starting point is to elucidate who introduced different concepts, notions and terms, when, and what were the initial motivations of the authors. Many new references have been found which enlarge the `standard citation package' used by some authors, recovering many undeservedly forgotten (or unnoticed) papers and names. Since it is practically impossible to cite several thousand publications, I have tried to include mainly references to papers introducing new types of quantum states and studying their properties, omitting many publications devoted to applications and to the methods of generation and experimental schemes, which can be found in other well known reviews. I also mainly concentrate on the initial period, which terminated approximately at the border between the end of the 1980s and the beginning of the 1990s, when several fundamental experiments on the generation of squeezed states were performed and the first conferences devoted to squeezed and `nonclassical' states commenced. The 1990s are described in a more `squeezed' manner: I have confined myself to references to papers where some new concepts have been introduced, and to the most recent reviews or papers with extensive bibliographical lists.
Seventy five years ago, three remarkable papers by Schrödinger, Kennard and Darwin were published. They were devoted to the evolution of Gaussian wave packets for an oscillator, a free particle and a particle moving in uniform constant electric and magnetic fields. From the contemporary point of view, these packets can be considered as prototypes of the coherent and squeezed states, which are, in a sense, the cornerstones of modern quantum optics. Moreover, these states are frequently used in many other areas, from solid state physics to cosmology. This paper gives a review of studies performed in the field of so-called nonclassical states' (squeezed states are their simplest representatives) over the past seventy five years, both in quantum optics and in other branches of quantum physics. My starting point is to elucidate who introduced different concepts, notions and terms, when, and what were the initial motivations of the authors. Many new references have been found which enlarge the standard citation package' used by some authors, recovering many undeservedly forgotten (or unnoticed) papers and names. Since it is practically impossible to cite several thousand publications, I have tried to include mainly references to papers introducing new types of quantum states and studying their properties, omitting many publications devoted to applications and to the methods of generation and experimental schemes, which can be found in other well known reviews. I also mainly concentrate on the initial period, which terminated approximately at the border between the end of the 1980s and the beginning of the 1990s, when several fundamental experiments on the generation of squeezed states were performed and the first conferences devoted to squeezed and nonclassical' states commenced. The 1990s are described in a more squeezed' manner: I have confined myself to references to papers where some new concepts have been introduced, and to the most recent reviews or papers with extensive bibliographical lists.
With this special issue, Journal of Optics B: Quantum and Semiclassical Optics contributes to the celebration of the World Year of Physics held in recognition of five brilliant papers written by Albert Einstein in 1905. There is no need to explain to the readers of this journal the content and importance of these papers, which are cornerstones of modern physics. The 51 contributions in this special issue represent current trends in quantum optics —100 years after the concept of light quanta was introduced. At first glance, in his famous papers of 1905, Einstein treated quite independent subjects—special relativity, the nature and statistical properties of light, electrodynamics of moving bodies and Brownian motion. We now know that all these phenomena are deeply related, and these relations are clearly shown in many papers in this issue. Most of the papers are based on the talks and poster contributions from participants of the 9th International Conference on Squeezed States and Uncertainty Relations (ICSSUR'05), which took place in Besancon, France, 2–6 May, 2005. This was the continuation of a series of meetings, originating with the first workshops organized by Professor Y S Kim at the University of Maryland, College Park, USA, in 1991 and by Professor V I Man'ko at the Lebedev Physical Institute, Moscow in 1992. One of the main topics of ICSSUR'05 and this special issue is the theory and applications of squeezed states and their generalizations. At first glance, one could think that this subject has no relation to Einstein's papers. However, this is not true: the theory of squeezed states is deeply related to special relativity, as far as it is based on the representations of the Lorentz group (see the paper by Kim Y S and Noz M E, S458–S467), which also links the current concepts of entanglement and decoherence with Lorentz-covariance. Besides, studies of the different quantum states of light imply, after all, the study of photon (or photo-electron) statistics and fluctuations of the electromagnetic field, whose importance was first emphasized by Einstein in 1905. The squeezed states can also be considered as a generalization of the concept of coherent states, which turned out to be one of the most important theoretical tools for solving the numerous problems of quantum optics. It seems highly symbolical that the printed version of this special issue will appear in the same month when one of the prominent creators of the theory of coherent states and modern quantum optics—Professor Roy J Glauber—will receive his Nobel Prize in Stockholm. ICSSUR'05 was opened by the invited talk of R J Glauber, `What makes a quantum jump?', and we take great pleasure in congratulating him on this well deserved award. We are sure that all participants of ICSSUR'05 and all readers of this special issue share our feelings. Two other Nobel Prize winners of 2005—Professor J L Hall and Professor T W H\ansch—also made great contributions to quantum optics. In particular, in 1986, J L Hall with collaborators, performed the first experiments on the generation of squeezed states by parametric down conversion, having obtained squeezing at the 50\% level (Wu L A, Kimble H J, Hall J L and Wu H 1986 Phys. Rev. Lett. 57 2520). Another area, which has attracted the attention of many researchers in the past decade and which is well represented in this special issue, is related to the problems of quantum correlations, entanglement and quantum nonlocality. It is also connected with the name of Einstein due to his famous `EPR' paper of 1935 written together with Podolsky and Rosen. For several decades this was an area of `thought experiments' only, but now this field is becoming a new part of physics, known as `quantum information'. The reader can find several papers which introduce new concepts in this area, such as applications of the Galois algebras and discrete Wigner functions. Solutions of different problems of the interaction between light and matter (which also take their origin in Einstein's paper of 1905), stationary and nonstationary Casimir effect, decoherence, new forms of uncertainty relations and their experimental verification, etc, can also be found in this issue. Many other contributions will be published in another special issue of the International Journal of Modern Physics B entitled `Quantum Information in Modern Optics'. This special issue is also the last issue of Journal of Optics B: Quantum and Semiclassical Optics. For the past 15 years this journal and its predecessors—Quantum Optics and Quantum and Semiclassical Optics—gained great respect among the quantum optics community. Many breakthrough papers were published in its pages during this period (see, for example, Schrade G, Man'ko V I, Schleich W P and Glauber R J 1995 Wigner Functions in the Paul trap Quantum Semiclass. Opt. 7 307). Since 1999, Journal of Optics B: Quantum and Semiclassical Optics has published a special issue for each ICSSUR meeting. This is the fourth issue of this series. We would like to thank Institute of Physics Publishing and the staff of Journal of Optics B: Quantum and Semiclassical Optics for providing the opportunity to pursue this programme, hoping that such a cooperation will continue in the future. We would also like to thank the many colleagues, who served as referees and whose efforts helped immensely in the preparation of this issue at such a high standard. The 10th ICSSUR conference will be organized for 2007 in Bradford, UK, by Professor A Vourdas. We invite readers to join us in two years.
With this special issue, Journal of Optics B: Quantum and Semiclassical Optics contributes to the celebration of the World Year of Physics held in recognition of five brilliant papers written by Albert Einstein in 1905. There is no need to explain to the readers of this journal the content and importance of these papers, which are cornerstones of modern physics. The 51 contributions in this special issue represent current trends in quantum optics —100 years after the concept of light quanta was introduced. At first glance, in his famous papers of 1905, Einstein treated quite independent subjects—special relativity, the nature and statistical properties of light, electrodynamics of moving bodies and Brownian motion. We now know that all these phenomena are deeply related, and these relations are clearly shown in many papers in this issue. Most of the papers are based on the talks and poster contributions from participants of the 9th International Conference on Squeezed States and Uncertainty Relations (ICSSUR'05), which took place in Besançon, France, 2–6 May, 2005. This was the continuation of a series of meetings, originating with the first workshops organized by Professor Y S Kim at the University of Maryland, College Park, USA, in 1991 and by Professor V I Man'ko at the Lebedev Physical Institute, Moscow in 1992. One of the main topics of ICSSUR'05 and this special issue is the theory and applications of squeezed states and their generalizations. At first glance, one could think that this subject has no relation to Einstein's papers. However, this is not true: the theory of squeezed states is deeply related to special relativity, as far as it is based on the representations of the Lorentz group (see the paper by Kim Y S and Noz M E, S458–S467), which also links the current concepts of entanglement and decoherence with Lorentz-covariance. Besides, studies of the different quantum states of light imply, after all, the study of photon (or photo-electron) statistics and fluctuations of the electromagnetic field, whose importance was first emphasized by Einstein in 1905. The squeezed states can also be considered as a generalization of the concept of coherent states, which turned out to be one of the most important theoretical tools for solving the numerous problems of quantum optics. It seems highly symbolical that the printed version of this special issue will appear in the same month when one of the prominent creators of the theory of coherent states and modern quantum optics—Professor Roy J Glauber—will receive his Nobel Prize in Stockholm. ICSSUR'05 was opened by the invited talk of R J Glauber, `What makes a quantum jump?', and we take great pleasure in congratulating him on this well deserved award. We are sure that all participants of ICSSUR'05 and all readers of this special issue share our feelings. Two other Nobel Prize winners of 2005—Professor J L Hall and Professor T W Hänsch—also made great contributions to quantum optics. In particular, in 1986, J L Hall with collaborators, performed the first experiments on the generation of squeezed states by parametric down conversion, having obtained squeezing at the 50\% level (Wu L A, Kimble H J, Hall J L and Wu H 1986 Phys. Rev. Lett. 57 2520). Another area, which has attracted the attention of many researchers in the past decade and which is well represented in this special issue, is related to the problems of quantum correlations, entanglement and quantum nonlocality. It is also connected with the name of Einstein due to his famous `EPR' paper of 1935 written together with Podolsky and Rosen. For several decades this was an area of `thought experiments' only, but now this field is becoming a new part of physics, known as `quantum information'. The reader can find several papers which introduce new concepts in this area, such as applications of the Galois algebras and discrete Wigner functions. Solutions of different problems of the interaction between light and matter (which also take their origin in Einstein's paper of 1905), stationary and nonstationary Casimir effect, decoherence, new forms of uncertainty relations and their experimental verification, etc, can also be found in this issue. Many other contributions will be published in another special issue of the International Journal of Modern Physics B entitled `Quantum Information in Modern Optics'. This special issue is also the last issue of Journal of Optics B: Quantum and Semiclassical Optics. For the past 15 years this journal and its predecessors—Quantum Optics and Quantum and Semiclassical Optics—gained great respect among the quantum optics community. Many breakthrough papers were published in its pages during this period (see, for example, Schrade G, Man'ko V I, Schleich W P and Glauber R J 1995 Wigner Functions in the Paul trap Quantum Semiclass. Opt. 7 307). Since 1999, Journal of Optics B: Quantum and Semiclassical Optics has published a special issue for each ICSSUR meeting. This is the fourth issue of this series. We would like to thank Institute of Physics Publishing and the staff of Journal of Optics B: Quantum and Semiclassical Optics for providing the opportunity to pursue this programme, hoping that such a cooperation will continue in the future. We would also like to thank the many colleagues, who served as referees and whose efforts helped immensely in the preparation of this issue at such a high standard. The 10th ICSSUR conference will be organized for 2007 in Bradford, UK, by Professor A Vourdas. We invite readers to join us in two years.
This special issue of Journal of Optics B: Quantum and Semiclassical Optics is composed mainly of extended versions of talks and papers presented at the Eighth International Conference on Squeezed States and Uncertainty Relations held in Puebla, Mexico on 9–13 June 2003. The Conference was hosted by Instituto de Astrofísica, Óptica y Electrónica, and the Universidad Nacional Autónoma de México. This series of meetings began at the University of Maryland, College Park, USA, in March 1991. The second and third workshops were organized by the Lebedev Physical Institute in Moscow, Russia, in 1992 and by the University of Maryland Baltimore County, USA, in 1993, respectively. Afterwards, it was decided that the workshop series should be held every two years. Thus the fourth meeting took place at the University of Shanxi in China and was supported by the International Union of Pure and Applied Physics (IUPAP). The next three meetings in 1997, 1999 and 2001 were held in Lake Balatonfüred, Hungary, in Naples, Italy, and in Boston, USA, respectively. All of them were sponsored by IUPAP. The ninth workshop will take place in Besançon, France, in 2005. The conference has now become one of the major international meetings on quantum optics and the foundations of quantum mechanics, where most of the active research groups throughout the world present their new results. Accordingly this conference has been able to align itself to the current trend in quantum optics and quantum mechanics. The Puebla meeting covered most extensively the following areas: quantum measurements, quantum computing and information theory, trapped atoms and degenerate gases, and the generation and characterization of quantum states of light. The meeting also covered squeeze-like transformations in areas other than quantum optics, such as atomic physics, nuclear physics, statistical physics and relativity, as well as optical devices. There were many new participants at this meeting, particularly from Latin American countries including, of course, Mexico. There were many talks on the subjects traditionally covered in this conference series, including quantum fluctuations, different forms of squeezing, unlike kinds of nonclassical states of light, and distinct representations of the quantum superposition principle, such as even and odd coherent states. The entanglement phenomenon, frequently in the form of the EPR paradox, is responsible for the main advantages of quantum engineering compared with classical methods. Even though entanglement has been known since the early days of quantum mechanics, its properties, such as the most appropriate entanglement measures, are still under current investigation. The phenomena of dissipations and decoherence of the initial pure states are very important because the fast decoherence can destroy all the advantages of quantum processes in teleportation, quantum computing and image processing. Due to this, methods of controlling the decoherence, such as by the use of different kinds of nonlinearities and deformations, are also under study. From the very beginning of quantum mechanics, the uncertainty relations were basic inequalities distinguishing the classical and quantum worlds. Among the theoretical methods for quantum optics and quantum mechanics, this conference covered phase space and group representations, such as the Wigner and probability distribution functions, which provide an alternative approach to the Schr\"odinger or Heisenberg picture. Different forms of probability representations of quantum states are important tools to be applied in studying various quantum phenomena, such as quantum interference, decoherence and quantum tomography. They have been established also as a very useful tool in all branches of classical optics. From the mathematical point of view, it is well known that the coherent and squeezed states are representations of the Lorentz group. It was noted throughout the conference that another form of the Lorentz group, namely, the 2 x 2 representation of the SL(2, c ) group, is becoming more prominent while providing the mathematical basis for the Poincaré sphere, entanglement, qubits and decoherence, as well as classical ray optics traditionally based on 2 x 2 `ABCD' matrices. The contributions of this special issue cover the most recent trends in all areas of quantum optics and the foundations of quantum mechanics.
Coherent states (CS) of the harmonic oscillator (also called canonical CS) were introduced in 1926 by Schrödinger in answer to a remark by Lorentz on the classical interpretation of the wave function. They were rediscovered in the early 1960s, first (somewhat implicitly) by Klauder in the context of a novel representation of quantum states, then by Glauber and Sudarshan for the description of coherence in lasers. Since then, CS have grown into an extremely rich domain that pervades almost every corner of physics and have also led to the development of several flourishing topics in mathematics. Along the way, a number of review articles have appeared in the literature, devoted to CS, notably the 1985 reprint volume of Klauder and Skagerstam [1], the 1990 review paper by Zhang et al [2], the 1993 Oak Ridge Conference [3] and the 1995 review paper by Ali et al [4]. Textbooks also have been published, among which one might mention the ground breaking text of Perelomov [5] focusing on the group-theoretical aspects, that of Ali et al [6] 1 analyzing systematically the mathematical structure beyond the group-theoretical approach and also the relation to wavelet analysis, that of Dodonov and Man'ko [7] mostly devoted to quantum optics, that of Gazeau [8] more oriented towards the physical, probabilistic and quantization aspects, and finally the very recent one by Combescure and Robert [9]. In retrospect, one can see that the development of CS has gone through a two-phase transition. First, the (simultaneous) discovery in 1972 by Gilmore and Perelomov that CS were rooted in group theory, then the realization that CS can be defined in a purely algebraic way, as an eigenvalue problem or by a series expansion (Malkin and Man'ko 1969, Barut and Girardello 1971, Gazeau and Klauder 1999; references to the original articles may be found in the textbooks quoted above). Both facts resulted in an explosive expansion of the CS literature. We thought, therefore, that the time was ripe to devote a special issue of Journal of Physics A: Mathematical and Theoretical to CS. However, because of limitations of space and time, it would have been impossible to get a fully representative cross-section of papers, covering all the different facets of the subject. Consequently, we have selected 37 articles, including some by a few of the originators of the field. We thank all the authors for submitting their up-to-date thoughts on this fascinating subject. The contents of this special issue are subdivided into five categories: (1) review papers; (2) physics-oriented CS; (3) physics and quantum information; (4) mathematics, general topics; and (5) mathematics, particular problems. (1) Review papers We start with five review papers. The first paper, by Klauder, surveys the many possible applications of affine variables, both in classical and quantum physics. The second, by Sanders, proposes a grand tour of entangled CS, which are present in many fields, such as quantum optics, quantum information processing, etc. The next paper, by Rowe, surveys the field of vector CS and the attendant group representation problems (including induced representations). Then Oriti et al describe a particular class of CS relevant to (loop) quantum gravity. Finally, Combescure and Robert present a comprehensive review of fermionic CS, including all mathematical details. (2) Physics-oriented CS The six contributions in this section deal with specific physical problems: (i) Dajka–Luczka study Gazeau–Klauder cat states associated with a nonlinear Kerr oscillator, instead of the usual canonical CS leading to Schrödinger cat states; (ii) Angelova et al discuss squeezed CS associated with a 1D Morse potential, used in molecular physics; (iii) Bagrov et al study CS in a magnetic solenoid field and prove their completeness; (iv) Blasone–Jizba treat Nambu–Goldstone dynamics in spontaneously broken symmetries, using CS functional integrals; (v) Calixto et al describe accelerated relativistic particles in the context of spontaneous breakdown of conformal SU(2,2) symmetry, using SU(2,2) CS; and (vi) Mortazavi–Tavassoly study f -deformed charge CS and their physical properties (nonclassical features, sub-Poissonian statistical behavior, etc). (3) Physics and quantum information The second group of physically related CS contains four contributions with a distinct quantum information theoretic flavor. First, Thilagam describes the dynamical behavior of entanglement of a pair of qubits (excitons), using a CS basis. Next, Lavoie–de Guise study SU(3) intelligent states (i.e., minimal uncertainty states), of interest in the quantum information community. Then Muñoz constructs discrete CS for n qubits. Finally, Wagner–Kendon explore the continuous variable Deutsch–Jozsa algorithm known in quantum computing in a discrete formulation. (4) Mathematics, general topics In this subgroup, there are eight papers dealing with general properties of CS, independently of any particular system or application. A whole series discusses the interaction between CS and various mathematical objects: pseudodifferential operators and Weyl calculus (Unterberger); induced representations of the affine group and intertwining operators (Elmabrok–Hutnik); measure-free CS and reproducing kernels (Horzela–Szafraniec); extremal POV measures (Heinosaari–Pellonpää); Hilbert W*-modules (Bhattacharyya–Roy); Toeplitz operators (Hutníková–Hutník); and operator localization and homogeneous structure of nilpotent Lie groups (Kisil). In addition, Balazs et al consider multipliers for continuous frames, including CS or wavelet frames. (5) Mathematics, particular problems The second group of mathematically oriented papers contains 14 contributions, devoted to CS in particular systems. We start with a paper by Gilmore, which explores the (sometimes chaotic) evolution of atomic CS under a time-periodic driving field, using sphere maps S 2 → S 2 . Next, we include a paper on CS on the 2-sphere in a magnetic field (Hall–Mitchell); a paper on CS for a quantum particle on a Möbius strip (Cirilo–Lombardo); a discussion of quantization on the circle (Chadzitaskos et al ); SUSY CS for Pöschl–Teller potentials (Bergeron et al ); generalized Bargmann functions and von Neumann lattices (Vourdas et al ); partial reconstruction for a finite CS system, using the Fock–Bargmann representation (Calixto et al ); phase operators for SU(3) irreps, thus for finite quantum systems (de Guise); semiclassical CS in periodic potentials (Carles–Sparber); complexified CS with non-Hermitian Hamiltonians (Graefe–Schubert); minimal uncertainty states in the context of (semisimple) group representation theory (Oszmaniec); localization operators in the time-frequency domain, i.e., in Gabor analysis (Muzhikyan–Avanesyan); and, finally, two papers about fermionic CS (Daoud–Kibler and Trifonov). This brief description illustrates perfectly the extreme versatility of the CS concept. As already stressed, coherent states constitute nowadays a flourishing research topic, with applications to a wide spectrum of domains. Indeed, CS are everywhere in physics: condensed matter physics, atomic physics, nuclear and particle physics, quantum optics, dynamics—both quantum and classical potentials—quantum gravity, quantization and quantum information theory. On the other hand, CS have grown into a fully-fledged domain in mathematics, incorporating many tools such as group representations, POV measures, frames, holomorphic functions, orthogonal polynomials and so on. Interestingly enough, the majority of contributions to this special issue (22 out of 37) are mathematically minded, demonstrating the widespread interest CS have generated in various areas of mathematics. A third field related to CS (but almost not represented in the present collection) is signal processing. Indeed both Gabor analysis and wavelet analysis derive in the first place from CS theory, namely, CS associated to the Weyl–Heisenberg and the ax + b group, respectively. Here too, a tremendous development has taken place in recent years, another testimony to the richness of the notion of CS. We leave it to the jury of public opinion to judge whether the call for a special issue of the journal, devoted to coherent states, has been justified. References [1] Klauder J R and Skagerstam B S 1985 Coherent States—Applications in Physics and Mathematical Physics (Singapore: World Scientific) [2] Zhang W-M, Feng D H and Gilmore R 1990 Coherent states: theory and some applications Rev. Mod. Phys. 62 867–927 [3] Feng D H, Klauder J R and Strayer M (ed) 1994 Coherent States: Past, Present and Future (Singapore: World Scientific) [4] Ali S T, Antoine J-P, Gazeau J-P and Mueller U A 1995 Coherent states and their generalizations: a mathematical overview Rev. Math. Phys. 7 1013–104 [5] Perelomov A M 1986 Generalized Coherent States and Their Applications (New York: Springer) [6] Ali S T, Antoine J-P and Gazeau J-P 2000 Coherent States, Wavelets and Their Generalizations (New York: Springer) [7] Dodonov V V and Man'ko V I (ed) 2003 Theory of Nonclassical States of Light (London: Taylor & Francis) [8] Gazeau J-P 2009 Coherent States in Quantum Physics (Berlin: Wiley) [9] Combescure M and Robert D 2012 Coherent States and Applications in Mathematical Physics (New York: Springer) 1 A second edition of that volume is in preparation.
The primary aim of the present paper is to attract the attention of particle physicists to new developments in studying squeezed and correlated states of the electromagnetic field as well as of those working on the latest topic to new findings about multiplicity distributions in quantum chromodynamics. The new types of nonclassical states used in quantum optics as squeezed states, correlated states, even and odd coherent states (Schrödinger cat states) for one--mode and multimode interaction are reviewed. Their distribution functions are analyzed according to the method used first for multiplicity distributions in high energy particle interactions. The phenomenon of oscillations of particle distribution functions of the squeezed fields is described and confronted to the phenomenon of oscillations of cumulant moments of some distributions for squeezed and correlated field states. Possible extension of the method to fields different from the electromagnetic field (gluons, pions, etc.) is speculated.