The Wigner Centennial Conference was held in Pécs, Hungary, on 8–12 July 2002. Eugene Paul Wigner was born in Budapest on 17 November 1902 and left us on 1 January 1995 in Princeton, USA. Numerous other conferences and conference sessions were also held during 2002 to commemorate the centennial year of his birth, because so many wished to pay tribute to him. It would, of course, take a major international effort to review thoroughly all the contributions Wigner made in all branches of physics and for the cause of world peace. The purpose of the Wigner Centennial Conference was to enrich and enhance the research lines initiated by Wigner. The conference was particularly interested in assembling young researchers who will develop and expand those research lines in the future. Indeed, there were many papers of current interest, including symmetry problems in quantum mechanics and quantum field theory, group theoretical issues, foundations of quantum mechanics, nuclear physics, chemical physics, the phase-space formulation of quantum mechanics, as well as quantum computing, information and entanglement problems. In 1932 Wigner published a seminal paper entitled 'On the quantum correction for thermodynamic equations' ( Phys. Rev. 40 749--759) in which he introduced a fundamental tool for quantum mechanics known these days as the Wigner function. The Wigner function has a long history, but began to gain its major strength in quantum optics during the 1970s. Since then, it has become the primary scientific language for squeezed states of light dealing with multi-photon coherent states. The Wigner function is also the basic language for transition from classical to quantum mechanics through phase-space. It therefore plays a major role in detecting quantum effects in processes routinely regarded as classical. Indeed, most optical instruments have been based on classical optics; it is, however, a great challenge to find quantum effects in those devices. This itself is an important subject. Wigner initially formulated the Wigner function to understand thermodynamic effects in physical systems, and thus it plays an important role in studying entropy in measurement processes, statistical mechanics and chemical physics. Physicists do not seem to fully appreciate the fact that Wigner functions can be used for the purpose of group representations. For instance, the single-mode and two-mode squeezed states are isomorphic to the Lorentz groups O(2,1) and O(3,2), respectively. This can be done very cleanly within the framework of the Wigner phase-space approach. By combining two of Wigner's main research areas, namely group theory and the Wigner function, we can construct a very rich field of physics. This is a future possibility for today's young physicists. Turning to computing, a man or woman is born with ten fingers, which constitute a natural computer based on decimal numbers. Indeed, the Chinese developed an abacus for dealing with decimal numbers. Slide rules perform additions, but they can do multiplications in the logarithmic scale. In the 1940s, another Hungarian scientist named John von Neumann observed that vacuum tubes could perform 'yes or no' logic. His observation led to the computer age in which we live. It is interesting to note that both Wigner and von Neumann came from the same high school in Hungary, the Budapest Evangelikus Gimnazium. Likewise, Eugene Wigner began to worry about the language that atoms speak even before the present form of quantum mechanics was formulated. He published in 1929 a book entitled 'Group Theory and its Applications to Quantum Mechanics of Atomic Spectra'. This is, in fact, the first book on quantum computing. Physicists do not yet seem to appreciate this aspect of Wigner's contribution, but eventually they will. If we are seriously interested in building quantum computers, we should use atoms. We should then develop mathematical algorithms to use group theory of atomic spectra for numerical computation. This is what Wigner left to us as a homework. The Wigner Centennial Conference was dedicated to Wigner's future rather than to his past. Indeed, we are very happy to note that many young physicists presented their new research results at this conference. We regret that this special issue cannot include all the subjects covered at the Conference. We are, however, very fortunate to have so many excellent papers on Wigner functions, quantum optics and quantum information, as well as quantum computing. The special issue is not meant to be the conference proceedings. The papers included here have been refereed according to the standards of Journal of Optics B: Quantum and Semiclassical Optics and publication was not restricted to the participants of the Conference. This is indeed a special issue dedicated to the above-mentioned subjects.
The Wigner Centennial Conference was held in Pécs, Hungary, on 8–12 July 2002. Eugene Paul Wigner was born in Budapest on 17 November 1902 and left us on 1 January 1995 in Princeton, USA. Numerous other conferences and conference sessions were also held during 2002 to commemorate the centennial year of his birth, because so many wished to pay tribute to him. It would, of course, take a major international effort to review thoroughly all the contributions Wigner made in all branches of physics and for the cause of world peace. The purpose of the Wigner Centennial Conference was to enrich and enhance the research lines initiated by Wigner. The conference was particularly interested in assembling young researchers who will develop and expand those research lines in the future. Indeed, there were many papers of current interest, including symmetry problems in quantum mechanics and quantum field theory, group theoretical issues, foundations of quantum mechanics, nuclear physics, chemical physics, the phase-space formulation of quantum mechanics, as well as quantum computing, information and entanglement problems. In 1932 Wigner published a seminal paper entitled 'On the quantum correction for thermodynamic equations' ( Phys. Rev. 40 749--759) in which he introduced a fundamental tool for quantum mechanics known these days as the Wigner function. The Wigner function has a long history, but began to gain its major strength in quantum optics during the 1970s. Since then, it has become the primary scientific language for squeezed states of light dealing with multi-photon coherent states. The Wigner function is also the basic language for transition from classical to quantum mechanics through phase-space. It therefore plays a major role in detecting quantum effects in processes routinely regarded as classical. Indeed, most optical instruments have been based on classical optics; it is, however, a great challenge to find quantum effects in those devices. This itself is an important subject. Wigner initially formulated the Wigner function to understand thermodynamic effects in physical systems, and thus it plays an important role in studying entropy in measurement processes, statistical mechanics and chemical physics. Physicists do not seem to fully appreciate the fact that Wigner functions can be used for the purpose of group representations. For instance, the single-mode and two-mode squeezed states are isomorphic to the Lorentz groups O(2,1) and O(3,2), respectively. This can be done very cleanly within the framework of the Wigner phase-space approach. By combining two of Wigner's main research areas, namely group theory and the Wigner function, we can construct a very rich field of physics. This is a future possibility for today's young physicists. Turning to computing, a man or woman is born with ten fingers, which constitute a natural computer based on decimal numbers. Indeed, the Chinese developed an abacus for dealing with decimal numbers. Slide rules perform additions, but they can do multiplications in the logarithmic scale. In the 1940s, another Hungarian scientist named John von Neumann observed that vacuum tubes could perform 'yes or no' logic. His observation led to the computer age in which we live. It is interesting to note that both Wigner and von Neumann came from the same high school in Hungary, the Budapest Evangelikus Gimnazium. Likewise, Eugene Wigner began to worry about the language that atoms speak even before the present form of quantum mechanics was formulated. He published in 1929 a book entitled 'Group Theory and its Applications to Quantum Mechanics of Atomic Spectra'. This is, in fact, the first book on quantum computing. Physicists do not yet seem to appreciate this aspect of Wigner's contribution, but eventually they will. If we are seriously interested in building quantum computers, we should use atoms. We should then develop mathematical algorithms to use group theory of atomic spectra for numerical computation. This is what Wigner left to us as a homework. The Wigner Centennial Conference was dedicated to Wigner's future rather than to his past. Indeed, we are very happy to note that many young physicists presented their new research results at this conference. We regret that this special issue cannot include all the subjects covered at the Conference. We are, however, very fortunate to have so many excellent papers on Wigner functions, quantum optics and quantum information, as well as quantum computing. The special issue is not meant to be the conference proceedings. The papers included here have been refereed according to the standards of Journal of Optics B: Quantum and Semiclassical Optics and publication was not restricted to the participants of the Conference. This is indeed a special issue dedicated to the above-mentioned subjects.
This paper starts with the transition from classical physics to quantum mechanics which was greatly aided by the concept of phase space. The role of canonical transformations in quantum mechanics is addressed. The Wigner phase-space distribution function is then defined which arises from the formulation of the density matrix, followed by the harmonic oscillator in phase space. Coherent and one- and two-mode squeezed states of light as well as the squeezed vacuum are discussed in the phase-space picture. Attention is also drawn to the fact that squeezed states naturally generate entanglement between the two-modes. Coupled harmonic oscillators are also elucidated in connection with the Wigner phase space. It will be noted that the phase-space picture of quantum mechanics has become an important scientific language for the rapidly expanding field of quantum optics. Here, we mainly focus on the simplest form of the Wigner function, which finds application in many branches of quantum mechanics. We make use of several symmetry groups such as Lorentz groups, the symplectic group in two and four dimensions, and the Euclidean group. The decoherence problem of an optical field is examined through a reformulation of the Poincaré sphere as a further illustration of the density matrix.
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.
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.
The Wigner Symposia series emerged from an International Symposium On SpaceTime Symmetries, held in College Park, Maryland, 1988 organized by Y.S. Kim and W. Zachary and commemorating the 50'th anniversary of Wigner's fundamental paper On the Unitary Representation of the Inhomogenous Lorentz Group. That symposium covered various areas of physics, of theoretical and mathematical physics to which Eugene Paul Wigner has contributed. It has shown how new and fruitful ideas connected with symmetries for a mathematical modeling of complex physical systems developed under the influence of Wigners work. Following the idea of the symposia to discuss new developments in physics centered around fundamental questions topics with emphasis on theoretical and mathematical physics were chosen and new frontiers like neuronal networks and quantum holography, whose origins can be traced back to Wigner's work, where included. The aim was to show the unity of different parts in physics and to demonstrate the power of arguments which are directly or indirectly related with mathematical realizations of symmetries. (1) Foundations of Quantum Mechanics; (2) General group Theoretical and Quantization Methods; (3) Coherent States; (4) Berry Phases; (5) Phases Space and Wigner Distributions; (6) Applications of Quantum Mechanics From Channel Space to Wigner Crystal; (7) Quantum Fields and Particles; (8) C-Algebraic and Methods; (9) Differential Geometric Methods: and (10) Nonlinear Partial Differential Equations, Dynamical Systems and Neural Networks.
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.
The Wigner function was formulated in 1932 by Eugene Paul Wigner, at a time when quantum mechanics was in its infancy. In doing so, he brought phase space representations into quantum mechanics. However, its unique nature also made it very interesting for classical approaches and for identifying the deviations from classical behavior and the entanglement that can occur in quantum systems. What stands out, though, is the feature to experimentally reconstruct the Wigner function, which provides far more information on the system than can be obtained by any other quantum approach. This feature is particularly important for the field of quantum information processing and quantum physics. However, the Wigner function finds wide-ranging use cases in other dominant and highly active fields as well, such as in quantum electronics—to model the electron transport, in quantum chemistry—to calculate the static and dynamical properties of many-body quantum systems, and in signal processing—to investigate waves passing through certain media. What is peculiar in recent years is a strong increase in applying it: Although originally formulated 86 years ago, only today the full potential of the Wigner function—both in ability and diversity—begins to surface. This review, as well as a growing, dedicated Wigner community, is a testament to this development and gives a broad and concise overview of recent advancements in different fields.