A single beam of excited two-level atoms couples two micromasers in a series. It is shown that in the absence of dissipation the possible steady states of their fields are superpositions, of two-field trapping number states. Which one is realized depends upon the initial state of the fields and the interaction parameters, g’τ’ and g’’τ’’, of the two cavities. A large number of these trapping states are pure quantum states, some of them showing entanglement of the two nonlocal micromaser fields of the form ‖N,N+M〉±‖N+M,N〉. Here, N and N+M are arbitrary trapping numbers that belong to disconnected blocks of the photon number space and M specifies the order of correlation between the fields. The time evolution of the system toward steady states is investigated numerically, mainly concentrating on the production of pure entangled trapping states of the form discussed earlier. We describe a special procedure to amplify a number state, ‖N,N〉, into such states that is based on conditions regarding the interaction parameters. In principle, N and M can be made arbitrarily large resulting in a steady-state nonlocal quantum superposition of distinct macroscopical fields (nonlocal ‘‘Schrödinger cat’’). We also present a solution of the standard master equation of a damped harmonic oscillator at finite temperature and apply it to study the effect of dissipation on the production of entangled trapping states at regular and Poissonian pump statistics.

It is found that although the entanglement does not survive at a steady state it can build up in the short-time transient regime when cavity losses and the number of thermal photons are not too large. In this small-loss (large ${\mathit{N}}_{\mathrm{ex}}$) regime the quantum correlation between the micromaser fields decays into a steady-state classical superposition at a rate that depends on the cavity lifetime and the difference (or order of correlation), M, between the superposed photon numbers. In the large-loss (small ${\mathit{N}}_{\mathrm{ex}}$) regime, however, no transient correlation can be produced and the photon statistics spreads out towards a vacuum for increasing losses. The system undergoes a transition from an uncorrelated to a correlated behavior when the pumping parametrized by ${\mathit{N}}_{\mathrm{ex}}$ exceeds the threshold between the large- and small-loss regimes. Thermal photons enhance the decay of the correlation and by coupling the disconnected blocks of the photon number space they populate the trapping number states of adjacent blocks. However, it is shown that the production of transient entanglement is not significantly affected by either thermal radiation or pumping fluctuations. The experimental realization of the entanglement of nonlocal micromaser fields employing these two-field trapping states of small photon numbers is shown to be feasible in the short-time transient regime using the presently available high-Q cavities and low temperatures, and could be possible on the macroscopic scale in the near future.

Two lossless micromasers are coupled in a series by the common pumping beam of two-level atoms the states of which are measured conditionally after the second cavity. Pure evolutions of the two fields are studied starting from uncorrelated coherent states for the four measurement schemes denoted by a-M’M’’-a, b-M’M’’-b, a-M’M’’-b, and b-M’M’’-a indicating the state of each atom, ‖a〉 or ‖b〉, before and after the two maser cavities, M’ and M’’. It is shown that energy-preserving schemes (first two above) produce a two-dimensional set of distinct Fock states at a steady state under the envelope of the initial amplitude distribution of the fields. Since the initial fields were uncorrelated the generated ones will be uncorrelated, too. In the case of energy-transferring schemes (second two) the system makes transitions between correlated and uncorrelated regimes. Nonlocal superpositions reminiscent of the form of ‖N,N+M〉+‖N+M,N〉 can be generated at an optimum number of atoms as a result of two coexisting trapping mechanisms. This is a transient entanglement since it is destroyed by the atoms to follow due to the trapping effects themselves. However, we also show that by switching from any of the two energy-transferring schemes to any of the two preserving ones the transient correlation produced by the former scheme can be frozen into a steady state by the latter one. In the absence of dissipations this combination of schemes can generate steady-state coherent superpositions of arbitrary number states of two nonlocal fields (nonlocal Schrödinger cats) at reasonably high detection probabilities of the conditioned atomic states.

The q-analog coherent states ‖z${\mathrm{〉}}_{\mathit{q}}$ are used to identify some of the canonical physical properties of the single-mode q-analog quantized radiation field in the ‖z${\mathrm{〉}}_{\mathit{q}}$ ‘‘classical limit’’ where ‖z‖ is large. In this quantum-optics-like limit, the fractional uncertainties of most physical quantities (momentum, position, amplitude, phase) which characterize the quantum field are shown to be O(1), and only vanish as O(1/‖z‖) when q=1. In contrast to this more-quantum-like behavior for q≠1, the fractional uncertainties do still approach zero for the usual number operator, N, and the N-Hamiltonian ${\mathit{H}}_{\mathit{N}}$≡ħω(N+1/2) which describes a free q-boson gas. An empirical signature for q-boson counting statistics is that (ΔN${)}^2$/〈N〉→0 as ‖z‖→∞. Properties of the q-analog generalizations of the phase operators of Susskind and Glogower (SG) and of the phase operator φ${\mathrm{^}}_{\mathit{q}}$ of Pegg and Barnett are investigated. In contrast to the manifest q deformed properties of SG operators for moderate ‖z${\mathrm{\Vert }}^2$, the ‘‘Hermitian’’ phase operator φ${\mathrm{^}}_{\mathit{q}}$ still exhibits almost normal classical behavior in the ‖z${\mathrm{〉}}_{\mathit{q}}$ basis. In particular, the conventional (approximate) number-phase uncertainty relation ΔNΔφ${\mathrm{^}}_{\mathit{q}}$≥1/2 and approximate commutation relation [N,φ${\mathrm{^}}_{\mathit{q}}$]=i are found to follow for the single-mode q-analog quantized field. So N and φ${\mathrm{^}}_{\mathit{q}}$ are almost canonically conjugate operators in the ‖z${\mathrm{〉}}_{\mathit{q}}$ classical limit. The ‖z${\mathrm{〉}}_{\mathit{q}}$ coherent states minimize this uncertainty relation for moderate ‖z${\mathrm{\Vert }}^2$. q-analog generalizations of the P, Q, and W phase-space representations are treated in the Appendix.

The interaction of a bichromatic quantized field with three-level Λ-type atoms is analyzed on the basis of c-number Langevin equations. In the presence of ground-state coherence one particular combination of field modes is selectively absorbed leading to a correlation of the output fields. This correlation manifests itself in a matching of Fourier components of the field modes and in a strong reduction of the diffusion of the difference phase. This correlation phenomenon is studied for the case of externally generated coherence as well as for coherence produced by population trapping in conjunction with a nonadiabatic atomic response.

We show that an optical cavity made of mirrors coated on a piezoelectric crystal can provide a quantum-nondemolition measurement of the intensity of a light beam. Due to radiation pressure, the crystal length is sensitive to the light intensity and can be measured by an electric circuit. The intensity fluctuations are not modified by reflection on such a cavity and the quantum correlations between these fluctuations and the output current of the electric circuit can be made very large even at nonzero temperature.

Approximate analytical expressions for the time evolution of the photon-number statistics and squeezing in the thermal Jaynes-Cummings model with a two-level atom being initially in a coherent superposition of its ground and excited states are presented.

A quantum-nondemolition (QND) measurement is one which does not disturb the quantity it measures. Any measurement can be regarded as a QND measurement followed by an additional back-action which may disturb that quantity. By feeding back the measurement result to control the system dynamics, this back-action can always be eliminated in principle. A practical device based on homodyne measurement of the x quadrature of a cavity mode is investigated. First, a ${\mathrm{\chi }}^{\mathrm{}\left(2\right)\mathrm{}}$ crystal is inserted in the cavity so that it becomes a degenerate parametric oscillator at threshold. Then the photocurrent is used to control coherent driving of the cavity, so as to give positive feedback with unit gain. This feedback, combined with the nonlinearity, turns the homodyne measurement into a QND measurement of x. The device can also be used to measure the x quadrature of a traveling wave, and gives near-perfect QND correlations over a large bandwidth. However, the quality of the measurement is badly degraded by even slightly inefficient photodetectors.

A unitary superoperator that represents a quantum phase is introduced by means of a density-matrix formalism of quantum mechanics. In this formalism, a quantum phase can be treated without using any of the additional techniques that are indispensable for defining a unitary phase operator when conventional phase operator methods are used. Furthermore, a phase-space distribution function for number and phase variables is introduced by means of the eigenoperator of the phase superoperator, and the basic properties of the distribution function are investigated.

We apply Monte Carlo wave-function-simulation methods to study the effects of dissipation on a recently proposed technique to generate entangled states of relevance for quantum teleportation. We find that a high degree of quantum entanglement can be preserved for damping rates achievable with presently available cavities. We further determine the effects of dissipation in an atom tagging scheme relevant for quantum complementarity experiments.

Numerical scattering experiments in a medium characterized by a simple saturable refractive index demonstrate the particlelike behavior of three-dimensional bright optical envelope solitons (‘‘light bullets’’) for the situation where the colliding bullets initially have a π-phase difference and are on the positive-slope branch of the energy curve. By varying the initial bullet velocities, bullet energy content, and the impact parameter, the nature of the repulsive interaction potential (i.e., the force law) is ascertained to be a Yukawa potential whose decay constant agrees quite well with that obtained by assuming that each light bullet ‘‘sees’’ only the ‘‘tail’’ of the other bullet. The scattering data scale correctly with velocity and one finds that the bullet energy is the natural analog of particle mass.

Fluctuations in photon number and quantum phase and in quadratures of the displaced Kerr states have been studied for different values of Kerr evolution parameter. The dynamics of phase distributions, taking into account that of the preferred phase, have resulted in well centered peaks in the phase interval. The role of the preferred phase in the distributions of quadratures has been considered, leading to approximate identification of conjugate quadratures with the number-phase-operator pair. Nonclassical properties have been demonstrated in the case of noise-minimum displaced Kerr states. The necessary displacement parameter has been represented as a center of a crescent-shaped phase region and expressed in terms of the lowest moments of the ${\mathrm{\Phi }}_{\mathit{s}\mathit{c}\mathit{r}\mathit{A}}$ quasidistribution.

A generalized second-order Fokker-Planck equation is derived for a single two-level atom in a cavity driven by an external classical field without using system size expansion or any truncation. Effects of detuning, as well as atomic and cavity decays, are incorporated in this Fokker-Planck equation. This equation is used to study atomic inversion and the second-order intensity correlation function of light in the bad-cavity limit by adiabatically eliminating the field variables. Several novel features in atomic inversion and intensity correlations that arise due to atomic and cavity detuning are discussed. Curves are presented to illustrate the behavior.

Statistical properties of second-harmonic photons generated by a single-mode laser operating near threshold are investigated by means of photoelectric counting measurements. It is found that second-harmonic photons display superthermal fluctuations below threshold where pump statistics are thermal. Above threshold, as the pump field becomes coherent, second-harmonic photons approach Poisson statistics. These measurements, interpreted as two-photon statistics of the fundamental beam, provide experimental evidence for the dependence of the fluctuation properties of a light beam on the process of detection.

We present a nonrelativistic Hamiltonian of the interaction between a moving mirror and radiation pressure. This Hamiltonian is derived directly from the equation of motion of a moving mirror, and the wave equation with time-varying boundary conditions. We discuss the canonical quantization of both the field and the motion of the mirror.

We have observed direct time domain exciton-polariton oscillating emission from a single-quantum-well GaAs microcavity. Observed oscillation periods of 21.7 ps were in good agreement with the spectral splitting of 0.12 nm. Such observations are consistent with the view that recently observed thresholdless coherent emission is not resonant Rayleigh scattering but rather exciton-polariton emission.

A compact and transparent plane-wave expansion of the complete Green function for an absorbing multilayered system is derived. The plane-wave solutions involved match, in the appropriate limit, the space-mode functions recently employed for quantization of the radiation field in a planar cavity and make this Green function particularly convenient for consideration of optical processes in realistic cavities and multilayers in general. This is illustrated by considering classically light scattering on a molecule in a planar cavity. It is demonstrated that the cross sections derived, besides generally describing light scattering on molecules embedded in a multilayer, correctly reproduce the effects of field confinement on light scattering in a planar cavity as predicted recently by the quantum-mechanical theory of Cairo et al. [Phys. Rev. Lett. 70, 1413 (1993)]. It is also pointed out that scattering into waves guided by the cavity lossy (metallic) mirrors may become an effective process in low-order microcavities. As a by product, a classical approach to spontaneous emission in absorbing planar cavities emerges from these considerations.

We show that the injection of an external field in a laser helps to improve amplitude squeezing and sub-Poissonian light generation close to threshold, but spoils quantum noise reduction for very high pumping rates. A general nonlinear theory is developed for the resonant case (no detunings), without any assumption on the relative magnitudes of field and atomic decay constants, thus encompassing a large class of lasers. Several possible pumping schemes are accommodated, ranging from regular to Poissonian. We carry out a detailed stability analysis of the general equations of motion, relating it to the noise-reduction problem. Both the photon-number variance inside the cavity and the homodyne spectra of the output are discussed.

We present an entropic description of quantum-mechanical states based on an operational approach to a phase-space measurement. We give a simple phase-space interpretation of sampling entropies in terms of which we derive very general entropic uncertainty relations reflecting the degree of the phase-space uncertainty of the quantum-mechanical state in the given measurement (i.e., for a given ‘‘quantum-ruler’’ state). We relate the sampling entropy to the von Neumann and Shannon entropy and show that the Wehrl entropy represents a particular example of a sampling entropy when the quantum ruler is represented by coherent states.

We use the operational phase-space distributions and sampling entropies developed in the preceding paper [V. Bužek, C. H. Keitel, and P. L. Knight, Phys. Rev. A 51, 2575 (1995)] to discuss the nature of quantum interference between components of superpositions of states. We show how the Wehrl entropy, a special case of the sampling entropy, is a useful discriminator between different kinds of superpositions and of statistical mixtures, and is determined essentially by the coherent-state content. Apart from interference terms, this content is given by the quantum uncertainty of a single coherent state and the classical contribution to the number of coherent states necessary to tile the dominant phase-space ‘‘patch’’ representing the quantum state of interest. We illustrate these ideas using nonclassical superpositions of coherent states, where interference modifies the phase-space distributions, and show how these features are sensitive to dissipation.

We demonstrate a simple, fully analytic method of calculating the amount of radiation emitted by optical solitons perturbed by higher-order dispersion effects in fibers and find good agreement with numerical results. It is pointed out that this radiation mechanism is analogous to the well-known Cherenkov radiation processes in nonlinear optics.