Table of contents

Cavity QED interactions of light and matter have been investigated in a wide range of systems covering the spectrum from microwaves to optical frequencies, using media as diverse as single atoms and semiconductors. Impressive progress has been achieved technologically as well as conceptually. This topical issue of Journal of Optics B: Quantum and Semiclassical Optics is intended to provide a comprehensive account of the current state of the art of cavity QED by uniting contributions from researchers active across this field.

As Guest Editors of this topical issue, we invite manuscripts on current theoretical and experimental work on any aspects of cavity QED. The topics to be covered will include, but are not limited to:

•Cavity QED in optical microcavities •Semiconductor cavity QED •Quantum dot cavity QED •Rydberg atoms in microwave cavities •Photonic crystal cavity QED •Microsphere resonators •Microlasers and micromasers •Microdroplets •Dielectric cavity QED •Cavity QED-based quantum information processing •Quantum state engineering in cavities

The DEADLINE for submission of contributions is 31 October 2003 to allow the topical issue to appear in about February 2004.

All papers will be peer-reviewed in accordance with the normal refereeing procedures and standards ofJournal of Optics B: Quantum and Semiclassical Optics.

Advice on publishing your work in the journal may be found at www.iop.org/journals/authors/jopb. Submissions should ideally be in either standard LaTeX form or Microsoft Word.

There are no page charges for publication. In addition to the usual 50 free reprints, the corresponding author of each paper published will receive a complimentary copy of the topical issue.

Contributions to the topical issue should if possible be submitted electronically atwww.iop.org/journals/jopb. or by e-mail to jopb@iop.org.

Authors unable to submit online or by e-mail may send hard copy contributions (enclosing the electronic code) to: Dr Claire Bedrock (Publisher), Journal of Optics B: Quantum and Semiclassical Optics, Institute of Physics Publishing, Dirac House, Temple Back, Bristol BS1 6BE, UK.

All contributions should be accompanied by a readme file or covering letter, quoting `JOPB topical issue – Cavity QED', giving the postal and e-mail addresses for correspondence. Any subsequent change of address should be notified to the publishing office.

We look forward to receiving your contribution to this topical issue.

A topical issue of Journal of Optics B: Quantum and Semiclassical Optics will be devoted to recent advances in optical solitons. The topics to be covered will include, but are not limited to:

•Properties, control and dynamics of temporal solitons •Properties, control and dynamics of spatial solitons •Cavity solitons in passive and active resonators •Three-dimensional spatial solitons •Dark, bright, grey solitons; interface dynamics •Compound or vector solitons; incoherent solitons •Light and matter solitons in BEC •Nonlinear localized structures in microstructured and nanostructured materials (photonic crystals, etc) •Angular momentum effects associated with localized light structures; vortex solitons •Quantum effects associated with localized light structures •Interaction of solitons with atoms and other media •Applications of optical solitons

The DEADLINE for submission of contributions is 31 July 2003 to allow the topical issue to appear in about February 2004.

All papers will be peer-reviewed in accordance with the normal refereeing procedures and standards ofJournal of Optics B: Quantum and Semiclassical Optics.

Advice on publishing your work in the journal may be found at www.iop.org/journals/authors/jopb. Submissions should ideally be in either standard LaTeX form or Microsoft Word.

There are no page charges for publication. In addition to the usual 50 free reprints, the corresponding author of each paper published will receive a complimentary copy of the topical issue.

Contributions to the topical issue should if possible be submitted electronically atwww.iop.org/journals/jopb. or by e-mail to jopb@iop.org.

Authors unable to submit online or by e-mail may send hard copy contributions (enclosing the electronic code) to: Dr Claire Bedrock (Publisher), Journal of Optics B: Quantum and Semiclassical Optics, Institute of Physics Publishing, Dirac House, Temple Back, Bristol BS1 6BE, UK.

All contributions should be accompanied by a readme file or covering letter, quoting `JOPB topical issue – Optical Solitons', giving the postal and e-mail addresses for correspondence. Any subsequent change of address should be notified to the publishing office.

We look forward to receiving your contribution to this topical issue.

We review experiments performed at the National Institute of Standards and Technology on entanglement, Bell's inequality and decoherence-free subspaces (DFSs) in a quantum register of trapped ⁹Be⁺ ions. The group of Dr David Wineland has demonstrated entanglement of up to four ions using the technique of Mølmer and Sørensen. This method produces the state (|↓↓ ⟩ + |↑↑ ⟩)/√2 for two ions and the state (|↓↓↓↓ ⟩ + |↑↑↑↑ ⟩)/√2 for four ions. The entanglement was generated deterministically in each shot of the experiment. Measurements on the two-ion entangled state violate Bell's inequality at the 8σ level. Because of the high detector efficiency of the apparatus, this experiment closes the detector loophole for Bell's inequality measurements for the first time. This measurement is also the first violation of Bell's inequality by massive particles that does not implicitly assume results from quantum mechanics. The group also demonstrated measurement of an interferometric phase with precision better than the shot-noise limit using a two-ion entangled state. A large-scale version of this scheme could improve the signal-to-noise ratio of atomic clocks by orders of magnitude. Further experiments demonstrated reversible encoding of an arbitrary qubit, originally contained in one ion, into a DFS of two ions. The DFS-encoded qubit resists applied collective dephasing noise and retains coherence under ambient conditions 3.6 times longer than does an unencoded qubit. The encoding method, which uses single-ion gates and the two-ion entangling gate, demonstrates all the elements required for two-qubit universal quantum logic. Finally, we describe an architecture for a large-scale ion trap quantum computer. By performing logic gates on small numbers of ions trapped in separate regions of the array, we take advantage of existing techniques for manipulating small trapped-ion quantum registers while enabling massively parallel gate operation. Encoding the quantum information in the DFS removes decoherence associated with ion transport and imperfect clock synchronization.

Formation of complicated emission patterns consisting of many transverse modes and associated intensity pulsations at beat frequencies between some pairs of transverse eigenmodes in microchip solid-state lasers with laser-diode asymmetric end-pumping are reviewed. The dependence of billiard-like transverse patterns on pump power and crystal rotation (i.e. kaleidoscopic patterns) was demonstrated in a 0.3 mm thick thin-slice LiNdP₄O₁₂ laser with sheet-like end-pumping. Pump-power-dependent high-speed self-pulsations were observed. The asymmetric optical confinement resulted in the formation of transverse patterns which were totally different from normal Hermite–Gaussian resonator modes. The interference among pairs of non-orthogonal transverse eigenmode fields, whose energy levels exhibited avoided crossing with increasing pump power, was shown to result in high-speed intensity modulations. A good numerical reproduction of the observed high-speed modulations was obtained with model equations.

We investigate analytically and numerically the role of diffraction in the operation of a broad-area inversionless laser in a cascade three-level configuration. Through a linear stability analysis of the trivial non-lasing solution and numerical integration of the corresponding Maxwell–Schrödinger equations, we show that off-axis emission allows stationary inversionless lasing over a cavity detuning range much larger than in small-aspect-ratio cavities and in conventionally inverted three-level lasers. In addition, we investigate inversionless lasing in a self-pulsing regime in the presence of diffraction, which leads to rich spatiotemporal dynamics.

We present an experimental scheme of preparing the three-photon W-state in this paper. This scheme is based on a linear optical element, four polarization-entangled photons produced in spontaneous parametric down-conversion and single-photon detection. Effects of the imperfection of optical elements on nonlocality are also discussed.

Nonclassical properties such as the autocorrelation function, quadrature and amplitude-squared squeezing, the phase properties in the Pegg–Barnett formalism, the Husimi–Kano Q function and the Wigner–Moyal (W) function of a wider class of nonlinear coherent states, are calculated and discussed in this paper.

We show that a nonlinear phase shift of π can be obtained by using a single two-level atom in a one-sided cavity with negligible losses. This result implies that the use of a one-sided cavity can significantly improve the π/18 phase shift previously observed by Turchette et al (1995 Phys. Rev. Lett.75 4710).

We investigate polarization effects of vertical cavity surface emitting lasers under optical injection of circularly polarized light. A theoretical model is used to examine the effects of the coupling between circularly polarized light and electron spin. We find that emission can occur in various polarization states depending on the cavity parameters and the power of the optical pump.

The probability representation of angular momentum states and the connection of the representation with the formalism of the star-product quantization procedure are reviewed. The Schrödinger equation of a general time-dependent Hamiltonian, linear in angular momentum operators, and the evolution equation for the density operator in the probability representation are solved analytically by means of the formalism of linear time-dependent constants of motion. These analytical solutions define wavefunctions and tomograms of states (generic Dicke states), which contain the atomic coherent states as a particular case. The statistical properties of these new states are also evaluated. General forms of analytically solvable Hamiltonians are established in terms of the Euler angle parametrization of the three-dimensional rotations.

We describe a proposal to probe the quantum tunnelling mechanism of an individual ion trapped in a double-well electromagnetic potential. The time evolution of the probability of fluorescence measurement of the electronic ground state is employed to characterize the single-particle tunnelling mechanism. The proposed scheme can be used to implement quantum information devices.

We study theoretically the polarization dynamics of a new type of quantum oscillator that is based on the two-photon stimulated emission process in the presence of a magnetic field of arbitrary orientation. Both cases of cascade (small intermediate-state atomic detuning) and two-photon (large atomic detuning) lasers are considered. The primary goal of this work is to investigate the origin of recently observed polarization instabilities in a two-photon laser (Pfister et al 2001 Phys. Rev. Lett.86 4512) using a highly simplified model. It is found that the two-photon laser can emit linearly polarized radiation with its plane of polarization orthogonal to the direction of the magnetic field at small magnetic field strengths. It can also emit elliptically polarized radiation over a large range of magnetic field strengths and orientations. When the magnetic field deviates from a direction perpendicular to the laser cavity axis periodic instabilities can appear through a Hopf bifurcation. This dynamic regime could have contributed to the polarization instabilities observed in the experiment.

We compare the original formalism in which the travelling-wave setup of the experiment due to Zou et al (1991 Phys. Rev. Lett.67 318) was analysed with a simple quantum optical approach that was later used for the same purpose. The analyses, both original and subsequent, included the quantum correlation between the signal beams. Whereas the original work spoke of a single photon per beam, the subsequent papers generalized to many photons per mode. We try to show that, in the original theory as well, one may speak of many photons per beam, without changing the relationship with the experiment. We conclude that the original analysis represents multiphoton stationary fields by interval restrictions comprising single photons.

We report the realization of a light source specifically designed for the generation of bright continuous-variable entangled beams and for Heisenberg-limited inteferometry. The source is a nondegenerate, single-mode, continuous-wave optical parametric oscillator in Na:KTP, operated at frequency degeneracy and just above threshold, which is also of interest for the study of critical fluctuations at the transition point. The residual frequency-difference jitter is ±150 kHz for a 3 MHz cold cavity half width at half maximum. We observe 4 dB of photon-number-difference squeezing at 200 kHz. The Na:KTP crystal is noncritically phase matched for a 532 nm pump and polarization crosstalk is therefore practically nonexistent.

We propose an experiment for obtaining about 0.4% inversionless gain at the sidebands of resonance in cold free ⁸⁷Rb atoms. This gain is of the same order of magnitude as the one already observed near resonance (Kitching and Hollberg 1999 Phys. Rev. A 59 4685).

We study theoretically stimulated Raman scattering (SRS) in a nonlinear dielectric microcavity and compare SRS thresholds for the cavity and the bulk material it is made of. We show that cavity SRS enhancement results solely from the intensity build up in the cavity and from the differences of the SRS dynamics in free and confined space. There is no significant modification of the Raman gain due to cavity QED effects. We show that the SRS threshold depends significantly on the nature of the dominating cavity decay as well as on the coupling technique with the cavity used for SRS measurements.

In this article we propose a very simple scheme for detecting the state and the degree of entanglement of two modes of radiation using a beam splitter. We find that using this device we can tell whether the state coming out of a certain apparatus is maximally entangled, by measuring the intensity of the radiation of one of the modes. This result is independent of the transmittivity of the beam splitter. In some cases it is also possible to determine the state exactly, by measuring the dispersion of one of the modes.

For a tripartite entangled system, based on the newly constructed tripartite Einstein–Podolsky–Rosen entangled state, we find the entangled state representation of the Wigner operator. The corresponding Wigner function then leads us to obtain the physical marginal distributions. We emphasize that, for an entangled-particle system, the physical meaning of the Wigner distribution function should lie in that its marginal distributions give the probability of finding the particles in an entangled way.

We introduce a simple model for electromagnetically induced transparency in which all fields are treated quantum mechanically. We study a system of three separated atoms at fixed positions in a one-dimensional multimode optical cavity. The first atom serves as the source for a single spontaneously emitted photon; the photon scatters from a three-level 'Λ'-configuration atom which interacts with an additional single-mode field coupling two of the atomic levels; the third atom serves as a detector of the total transmitted field. We find an analytical solution for the quantum dynamics. From the quantum amplitude describing the excitation of the detector atom we extract information that provides exact single-photon analogues to wave delays predicted by semi-classical theories. We also find complementary information in the expectation value of the electric field intensity operator.

The diffracted near field distribution from an LP₀₁ mode in a hollow optical fibre was recently calculated using a scalar model based on the weakly waveguiding approximation (Yoo et al1999 J. Opt. B: Quantum Semiclass. Opt.1 364). It showed a dominant Gaussian-like distribution with an increased axial intensity in the central region (not a doughnut-like distribution), so the diffracted output beam from the hollow fibre cannot be used to form an atomic funnel. Using exact solutions of the Maxwell equations based on a vector model, however, we calculate the electric field and intensity distributions of the HE₁₁ mode in the same hollow fibre and study the diffracted near- and far-field distributions of the HE₁₁-mode output beam under the Fresnel approximation. We analyse and compare the differences between the output beams from the HE₁₁ and LP₀₁ modes. Our study shows that both the near- and far-field intensity distributions of the HE₁₁-mode output beam are doughnut-like and can be used to form a simple atomic funnel. However, it is not suitable to use the weakly waveguiding approximation to calculate the diffracted near-field distribution of the hollow fibre due to the greater refractive-index difference between the hollow region (n₀ = 1) and the core (n₁ = 1.45 or 1.5). Finally, the 3D intensity distribution of the HE₁₁-mode output beam is modelled and the corresponding optical potentials for cold atoms are calculated. Some potential applications of the HE₁₁-mode output beam in an atomic guide and funnel are briefly discussed.

A quantum code based on decoherence-free subspace is considered. We investigate a special interaction—exchange interaction in the quantum system—and show that quantum states encoded by the quantum code are free of the exchange interaction. On the basis of this quantum code, universal quantum computation can be implemented by switching 'off and on' exchange interaction.

We carry out a theoretical study of the collective spontaneous emission (superradiance) from an ultrathin film comprised of three-level atoms with V configuration of the operating transitions. As the thickness of the system is small compared to the emission wavelength inside the film, the local-field correction to the averaged Maxwell field is relevant. We show that the interplay between the low-frequency quantum coherence within the subspace of the upper doublet states and the local-field correction may drastically affect the branching ratio of the operating transitions. This effect may be used for controlling the emission process by varying the doublet splitting and the amount of low-frequency coherence.

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.

A system of coherently driven two-level atoms is analysed in the presence of two independent stochastic perturbations: one due to collisions and a second due to phase fluctuations of the driving field. The behaviour of the quantum interference induced by the collisional noise is considered in detail. The quantum-trajectory method is utilized to reveal the phase correlations between the dressed states involved in the interfering transition channels. It is shown that the quantum interference induced by the collisional noise is remarkably robust against phase noise. This effect is due to the fact that the phase noise, similarly to collisions, stabilizes the phase difference between the dressed states.

The Wigner quasiprobability function for the superposition of squeezed displaced Fock states is reviewed. The phase distribution obtained by integrating the Wigner function over the radial variable is studied and compared with the Pegg and Barnett phase distribution. The Wigner phase distributions for some values of parameters are illustrated.

A unitary transformation of the N-ion Jaynes–Cummings Hamiltonian is proposed. It is shown that any approximate expression of the evolution operator associated with the transformed Hamiltonian retains its validity independent of the intensity of the external driving field. In particular, using the rotating wave approximation, one obtains a solution for the N-ion Jaynes–Cummings model which improves the standard rotating wave approximation solution.

The neutron interferometer set-up S18 at the ILL reactor has been used for the first neutron quantum state reconstruction experiment. By a simultaneous measurement of the coherence function and of the modulated momentum distribution behind the interferometer, the Wigner function of various quantum states can be reconstructed. This is equivalent to knowledge of the particle wavefunction. Its spatial dimension follows from the measurement of the coherence function, which is the autocorrelation function of the wavefunction and the shape in momentum space follows from momentum post-selection experiments. Non-orthogonal traces through the (x, k)-phase space have to be measured for an unambiguous reconstruction. Nearly classical (coherent) and non-classical states in the form of Schrödinger cat-like states have been identified. Comparisons of calculated and measured Wigner functions will be shown.

The Wigner problem, i.e. the investigation of general quantum mechanical commutation relations consistent with the Heisenberg evolution equations of a given shape, is studied. We follow a recently proposed generalization of this idea within which the classical analogy is postulated only for the shape of the time evolution equations and not for a Hamiltonian itself. This links our investigation to the problem of alternative Hamiltonians of classical mechanics and to canonically inequivalent phase-space descriptions of physical systems governed by the same Newton equations of motion. Warned that the time evolution and the other symmetry generators may be given ambiguously even in the formalism of classical mechanics, we do not a priori assume the shape of their quantum analogues. Instead we only require that the set of basic algebraic relations, which quantum mechanical observables are to obey, has a Lie algebra structure. Such a requirement appears to be sufficient to find solutions for simple oscillator-like dynamics. New algebras of quantum mechanical observables are not constructed as a linear envelope of the Heisenberg algebra, and their representations reflect physical results unexpected in the framework of the canonical approach. We illustrate our approach in detail for the example of the one-dimensional harmonic oscillator using the representation of the generalized coherent states.

The theory of Lie groups and representations was developed by Lie and followers to a degree of quasi-perfection, in the years 1870–1930. The main topological features of compact simple Lie groups were elucidated in the 1940s by Hopf and colleagues. The exceptional groups were studied by Chevalley and colleagues in 1949–57. Torsion in the exceptional groups was considered by Toda and colleagues in the 1980s. However, one can still ask some questions to which the answer is either incomplete or absent. We would like to raise and discuss some of these in this paper.

In the quantum theory of measurement, the positive operator-valued measure (POVM) is an important concept, and its implementation can be useful. A POVM consists of a set of non-negative quantum-mechanical Hermitian operators that add up to the identity. The probability that a quantum system is in a particular state is given by the expectation value of the POVM operator corresponding to that state. Following a brief review of the mathematics and mention of the history of POVMs in quantum theory, a particular implementation of a POVM for use in the measurement of nonorthogonal photon polarization states is reviewed. The implementation consists simply of a Wollaston prism, a mirror, two beam splitters, a polarization rotator and three phototubes arranged in an interferometric configuration, and it is shown analytically that the device faithfully represents the POVM. Based on Neumark's extension theorem, the two-dimensional Hilbert space of the POVM implementation can be embedded in the three-dimensional Hilbert space of an ordinary projective-valued measure. Also, analytical expressions are given for the maximum Renyi information loss from the device to a disturbing probe, and for the error and inconclusive rates induced by the probe. Various aspects of the problem of probe optimization are elaborated.

We modify some aspects of the continuous photodetection theory proposed by Srinivas and Davies (SD) (1981 Opt. Acta28 981), which describes the non-unitary evolution of a quantum field state subjected to a continuous photocount measurement. In order to remedy inconsistencies that appear in their approach, we redefine the 'annihilation' and 'creation' operators that enter in the photocount super-operators. We show that this new approach not only still satisfies all the requirements for a consistent photocount theory according to SD precepts, but also avoids some weird result appearing when previous definitions are used.

The general idea of a stochastic gauge representation is introduced and compared with more traditional phase-space expansions, like the Wigner expansion. Stochastic gauges can be used to obtain an infinite class of positive-definite stochastic time-evolution equations, equivalent to master equations, for many systems including quantum time evolution. The method is illustrated with a variety of simple examples ranging from astrophysical molecular hydrogen production, through to the topical problem of Bose–Einstein condensation in an optical trap and the resulting quantum dynamics.

We analyse the coherence properties of neutron wavepackets, after they have interacted with a phase shifter undergoing different kinds of statistical fluctuation. We give a quantitative (and operational) definition of decoherence and compare it to the standard deviation of the distribution of the phase shifts. We find that in some cases the neutron ensemble is more coherent, even though it has interacted with a wider (i.e. more disordered) distribution of shifts. This feature is independent of the particular definition of decoherence: this is shown by proposing and discussing an alternative definition, based on the Wigner function, that displays a similar behaviour. We briefly discuss the notion of entropy of the shifts and find that, in general, it does not correspond to that of decoherence of the neutron.

This paper presents a first step towards combining two well-established methods used in semiconductor physics—semiconductor Bloch equations and the Wigner approach to quantum transport. This combination provides the possibility of including spontaneous emission, i.e., the spontaneous recombination of excited electron–hole pairs in semiconductors, into the Wigner approach, which so far has been used only for systems with fixed particle number. The theory is presented and first numerical results for a three-dimensional system are shown.

A new approach to deformation quantization on the cylinder considered as phase space is presented. The method is based on the standard Moyal formalism for ² adapted to S¹ × by the Weil–Brezin–Zak transformation. The results are compared with other solutions of this problem presented by Kasperkovitz and Peev (1994 Ann. Phys., NY230 21) and by Plebański and collaborators (2000 Acta Phys. Pol. B 31 561). The equivalence of these three methods is proved.

We propose an algebraic solution for a wide class of Lindblad-type master equations. Examples of dissipation in free field evolution, field evolution in the Kerr medium, two-photon field dissipation, atomic dissipation and two-mode field dissipation are given.

A definition of entanglement in terms of local measurements is discussed. Namely, the maximum entanglement corresponds to the states that cause the highest level of quantum fluctuations in all local measurements determined by the dynamic symmetry group of the system. A number of examples illustrating this definition is considered.

One of potential applications of quantum processors should be a simulation of dynamics of quantum systems. In this paper we analyse whether it is possible to simulate a paradigmatic model of an exponential decay of a two-level system using a programmable quantum gate array. That is, we would like to simulate the decay of a two-level system in such a way that the parameters of the decay itself are encoded in the state of the ancilla that serves as a program for the programmable gate array. We show that it is impossible to simulate the exponential decay precisely with the finite-dimensional programmable gate array. On the other hand, we present a very simple model of a Markovian decay process that can be efficiently simulated on a simple programmable gate array. We compare this Markovian process with the exponential decay.

We propose a tomographic approach to the study of quantum nonlocality in continuously variable quantum systems. On one hand we derive a Bell-like inequality for measured tomograms, and on the other we introduce pseudospin operators whose statistics can be inferred from the data characterizing the reconstructed state, thus giving the possibility of using standard Bell inequalities. Illuminating examples are also discussed.

One of the most prominent quasiprobability functions in quantum mechanics is the Wigner function, which gives the correct marginal probability functions if integrated over position or momentum. Here we depart from the definition of the position–momentum Wigner function to, by analogy, construct a number–phase Wigner function that, if summed over photon numbers, gives the correct phase distribution and, if integrated over phase, gives the correct photon distribution.

We present Wigner and Husimi functions for the stationary states of the quartic oscillator and the more general double-well potential, paying particular attention to the corresponding classical structure. We find that the qualitative behaviour of both these functions depends strongly on the height of the potential barrier, and that the Husimi function has vestiges of the classical trajectories even for states below this barrier. The zero-point energy is also seen to play an important role.

We first review the usefulness of the Wigner distribution functions (WDFs), associated with Lindblad and pre-master equations, for analysing a host of problems in quantum optics where dissipation plays a major role, an arena where weak coupling and long-time approximations are valid. However, we also show their limitations for the discussion of decoherence, which is generally a short-time phenomenon with decay rates typically much smaller than typical dissipative decay rates. We discuss two approaches to the problem both of which use a quantum Langevin equation as a starting-point: (a) use of a reduced WDF but in the context of an exact master equation (b) use of a WDF for the complete system corresponding to entanglement at all times.

Continuous variable remote state preparation and teleportation are analysed using Wigner functions in phase space. We suggest a remote squeezed state preparation scheme between two parties sharing an entangled twin beam, where homodyne detection on one beam is used as a conditional source of squeezing for the other beam. The scheme also works with noisy measurements, and provides squeezing if the homodyne quantum efficiency is larger than 50%. The phase space approach is shown to provide a convenient framework to describe teleportation as a generalized conditional measurement, and to evaluate relevant degrading effects, such the finite amount of entanglement, the losses along the line and the nonunit quantum efficiency at the sender location.

In this work we construct approximate coherent states for the Morse potential using a method inspired by the f-oscillator formalism (Man'ko et al 1996 Proc. 4th Wigner Symp. ed M Natig, Atakishiyev, T H Seligman and K B Wolf (Singapore: World Scientific) p 421). We make even and odd combinations of these states and evaluate the temporal evolution of the position operator and its dispersion as a function of time when the states evolve under a nonlinear Morse Hamiltonian.

Necessary and sufficient conditions are derived that allow one to completely characterize the nonclassicality of an arbitrary quantum state of a harmonic oscillator. Starting from a theorem by Bochner we derive an infinite hierarchy of conditions for the characteristic functions of the quantum state. Following the hierarchy step by step, the nonclassicality can be classified with respect to increasing orders. The full hierarchy of conditions is completely equivalent to the most commonly accepted criterion for nonclassicality: the failure of the Glauber–Sudarshan P-function to be a probability density. Our reformulated version of the criterion is directly related to measurable quantities that can be observed in phase-sensitive measurements such as homodyne detection. It directly generalizes a criterion for nonclassicality (Vogel 2000 Phys. Rev. Lett.84 1849) that has already been applied in experiments (Lvovsky and Shapiro 2002 Phys. Rev. A 65 033830). Examples of phase-insensitive quantum states are considered that exhibit no first-order nonclassical effects, but that are nonclassical in second order.

When Fermi's golden rule (FGR) is studied in the Wigner representation, the transition rate from an initial pure state or from an initial thermal distribution into a quasicontinuum manifold of degenerate states is given by an overlap integral of Wigner functions in phase space. In the semiclassical limit the transition rate is obtained by integrating over the regions in phase space where the energy difference between the initial and final potential surfaces is equal to the available energy. The integral is weighted by the initial probability density to be at that phase-space region. The classical limit of FGR is thus both simple and intuitive. In one dimension a relation to the Landau–Zener–Stuckelberg formula is established. The multi-dimensional case is considered by induction, proving that for separable multi-dimensional systems deviations of the logarithm of the transition rate from its classical limit scale at worst linearly with the dimension.

We investigate systems of identical bosons with the focus on two-body correlations. We apply a Faddeev type of decomposition of the wavefunction. At large scattering length a series of spatially extended many-body bound states appears, analogously to the three-body Efimov states. We discuss a recombination process with these many-body Efimov states as intermediate states.

In a double-loop neutron interferometer two and three beams are superposed behind the first and second loop respectively. The wavefunctions in different arms of the interferometer are discussed using wavepackets. Various correlations can be defined. Both the momentum and the space distributions which are influenced by phase shifters are calculated. Those distributions are marginal distributions of the corresponding Wigner functions. While the momentum distribution can be measured by means of post-selection, the space distribution is directly accessible via the visibility which is expressed by an appropriate correlation function. It is shown that the space distribution behind the first loop can be represented by the visibility which can be determined by measuring neutron intensities behind the second loop.

The structure and dynamics of an n-particle system are described with coupled nonlinear Heisenberg commutator equations where the nonlinear terms are generated by the two-body interaction that excites the reference vacuum via particle–particle and particle–hole excitations. Nonperturbative solutions of the system are obtained with the use of the dynamic linearization approximation and cluster transformation coefficients. The dynamic linearization approximation converts the commutator chain into an eigenvalue problem. The cluster coefficients factorize the matrix elements of the (n) particles or particle–hole systems in terms of the matrix elements of the (n − 1)-systems coupled to particle–particle, particle–hole and hole–hole bosons. Group properties of the particle–particle, particle–hole and hole–hole permutation groups simplify the calculation of these coefficients. The particle–particle vacuum excitations generate superconductive diagrams in the dynamics of three-quark systems. Applications of the model to fermionic and bosonic systems are discussed.

On the basis of the Hamiltonian form of the Klein–Gordon equation for a charged scalar particle field introduced by Feshbach and Villars, the gauge-invariant 2 × 2 Wigner matrix has been constructed whose diagonal elements describe positive and negative charge densities and whose off-diagonal elements correspond to cross-densities in phase space. The system of coupled transport equations has been derived in the case of interaction with an arbitrary external electromagnetic field. A gauge-independent generalization of the free particle representation due to Feshbach and Villars is given, and on the basis of it both the nonrelativistic and the classical limits of the relativistic quantum Boltzmann–Vlasov equation are discussed. In the nonrelativistic limit (p/mc → 0) the set of equations of motion decouple to two independent quantum transport equations describing the dynamics of oppositely charged positon and negaton densities. In the classical limit (ℏ → 0) two relativistic Boltzmann–Vlasov equations result for the diagonal positon and negaton densities. Even though the Planck constant ℏ is absent from the latter equations, the real part of the positon–negaton cross-density does not vanish.

In this paper we discuss the excitation of a quantum harmonic system, such as a particle bound in a harmonic oscillator potential, by means of dipole and quadrupole electric fields, focusing on effects of their interference. We obtain an exact solution to this problem with methods that date back to the ideas of Wigner. We discuss the rich class of dynamic polarization effects that are involved in such excitation processes and which are relevant to stopping theory, and electromagnetic excitation of atomic and nuclear systems.

Analytic representations based on generalized coherent states are studied. The growth of the analytic functions is intimately connected to the completeness of the sequences of these generalized coherent states. The least density that such sequences must have in order to be overcomplete is calculated. The results generalize known results on the completeness of von Neumann lattices for the standard coherent states to other sets of coherent states.

The impact of an anti-unitary symmetry on the spectrum of non-Hermitian operators is studied. Wigner's normal form of an anti-unitary operator accounts for the spectral properties of non-Hermitian, -symmetric Hamiltonians. The occurrence of either single real or complex conjugate pairs of eigenvalues follows from this theory. The corresponding energy eigenstates span either one-or two-dimensional irreducible representations of the symmetry . In this framework, the concept of a spontaneously broken -symmetry is not needed.

Previously, an explicit solution for the time evolution of the Wigner function was presented in terms of auxiliary phase space coordinates which obey simple equations that are analogous with, but not identical to, the classical equations of motion. They can be solved easily, and their solutions can be used to construct the time evolution of the Wigner function. In this paper, the usefulness of this explicit solution is demonstrated by solving a numerical example in which the Wigner function has strong spatial and temporal variations as well as regions with negative values. It is found that the explicit solution gives a correct description of the time evolution of the Wigner function. We examine next the pseudoparticle method which uses classical trajectories to evolve the Wigner function. We find that the lowest-order pseudoparticle approximation reproduces the general features of the time evolution, but there are deviations. We show how these deviations can be systematically reduced by including higher-order correction terms in powers of ℏ².

The two types of SU(1, 1) coherent states of Barut and Girardello and of Perelomov are dual in a sense that the operators in the eigenvalue equation and in the exponentials which create these types of coherent states from the lowest eigenstate (vacuum) form an asymmetric Heisenberg–Weyl algebra. A new type of SU(1, 1) coherent states which takes on an intermediate position between the two dual types of SU(1, 1) coherent states already mentioned is established and investigated. In this new type, the duality of the operators becomes a self-duality and the corresponding operators form a usual Heisenberg–Weyl algebra. Properties of the different SU(1, 1) coherent states are investigated for the realization of SU(1, 1) by one-dimensional quantum-mechanical potential problems leading to a quadratic law of energy-level spacing. Coherent SU(1, 1) phase states are discussed for this realization of SU(1, 1). It is shown that no wavepackets corresponding to any of the SU(1, 1) coherent states can exactly preserve their shape during time evolution.

We study how entanglement between an open system and a reservoir is established. The system is considered to be a qubit, while the reservoir is modelled as a collection of qubits. The system and the reservoir qubits interact via a sequence of partial-swap operations. This processes is called quantum homogenization since at the output the system as well as all reservoir qubits are in states that are, in a limit sense, equal to the original state of the reservoir qubits. We show that in this process the Coffman–Kundu–Wootters inequalities are saturated. This means that no intrinsic multi-partite entanglement is created.

Several of the imponderables associated with the Copenhagen interpretation of quantum mechanics have alternative, parallel explanations in terms of nonlinear dynamics and chaos. For example, an exponential law for processes such as radioactive decay can be obtained by iterating unimodal maps operating in chaotic domains, utilizing their extreme sensitivity to initial conditions. Such alternative explanations raise the question of the possibility of underlying nonlinearities in quantum mechanics. This is a 'from the bottom up', empirical approach to the question and will be compared with other, more theoretical, 'from the top down' investigations into the possibility of nonlinear extensions to quantum mechanics. Nonlinear dynamics, although deterministic, in its chaotic extreme provides only statistical information about the outcomes of state-space orbits. Thus in principle it provides the determinism so dear to Einstein; yet in practice it exhibits the probabilistic behaviour of Bohr. Perhaps Einstein and Bohr could both have been right.

We study the dynamics of a complex open quantum many-body system. The coupling to external degrees of freedom can be viewed as a coupling to a radiation field, to continuum states or to a measuring apparatus. This perturbation is treated in terms of an effective non-Hermitian Hamiltonian. The influence of such coupling on the properties of the many-body dynamics is discussed, with emphasis on new effects related to dynamical segregation of fast and slow decays and the phase transition to Dicke superradiance. Relations to quantum optics, the continuum shell model, the theory of measurement, quantum chaos, percolation theory and quantum reactions are stressed.