We discuss the effect of Doppler broadening on the gain and refractive index experienced by an optical probe when passing through a collection of atoms in the absence of a population inversion. We calculate the gain-absorption spectrum of the medium when the atoms have a Maxwell velocity distribution and compare the atomic response when the pump and probe beams are arranged in copropagating and counterpropagating configurations. The results in these two cases can be qualitatively very different from each other, as we show with the help of numerical calculations. A physical interpretation and the conditions for the validity of the results are provided.

The parametric amplifier is usually studied under the assumption that the pump mode is in a large-amplitude coherent state. We consider what happens when this is no longer the case. We first derive some exact relations between the number fluctuations of the pump and signal, the pump and the idler, and also the signal and the idler. It is then determined how much phase noise can be tolerated in the initial pump state if two-mode squeezing is to be produced in the signal-idler field. The relation between the rotational symmetry in phase space of the initial pump state and that of the signal-idler subsystem is determined. We find that the sum phase distribution of the signal and idler has twice the rotational symmetry as that of the initial pump state. Finally, we generalize the parametric approximation to treat pump states which are not coherent states. We examine the effect of a phase-squeezed pump state and find that the pump squeezing can actually decrease the two-mode squeezing of the signal-idler field while enhancing a form of higher-order squeezing, sum squeezing, for these fields.

We present experimental and theoretical investigations of the ground-state levels of the Cs cycling transition. We show that in such multilevel nondegenerate systems, the populations of the extreme Zeeman sublevels, under optical pumping, can have asymmetric (‘‘dispersive’’) line shapes as a function of laser frequency. A density-matrix model which takes into account optical, Zeeman, and hyperfine coherences, and allows for a fluctuating laser field, correctly predicts the experimental observations. A comparison of the density-matrix model with a rate-equation model demonstrates the role of coherences in the population distribution. Investigation of the effect theoretically and experimentally for σ and π laser polarizations is presented in the paper.

We analyze the inversionless gain in a three-level ladder system by replacing the usual coherent coupling field with an incoherent field. Surprisingly, it is found that one can obtain inversionless amplification of a weak probe even in the absence of a coherent field in the model. We conclude that gain is determined by the ensemble average of the product of the two-photon coherence and the ‘‘effective Rabi frequency’’ of the field. Thus, even though the incoherent pump reduces the two-photon coherence, gain can be restored by choosing sufficiently high strengths of the incoherent field.

We develop a formalism for studying pulse propagation in the vicinity of a two-photon resonance, in the spectral region where the effects of two-photon absorption are not overwhelming. The formalism predicts self-steepening and asymmetric spectral broadening in the transmitted pulse, as previously observed in experiments and numerical simulations involving atomic vapors and semiconductors.

We show that sub-Poisson photoelectrons can result from the finite relaxation time of the fundamental photoelectron-emitting entities, which could be individual atoms, or valence-band electrons, etc., depending on the type of the detectors. After identifying the precise quantum-mechanical criterion for a light without the intrinsic antibunching property, it is demonstrated that the variance to mean ratio of photoelectron counts within a fixed time period can be reduced to as small as 1/2, even for such light. The condition for strong fluctuation reduction is found to be the saturation of light absorption. Semiconductor photodetectors are considered in relative details, and their photoelectron statistics are expressed in terms of the intrinsic material parameters.

A comprehensive model of intense laser light propagation in a fully saturable medium is presented. This model employs the Maxwell-Bloch equations for a two-level atomic system. Light propagating through a medium creates intensity-dependent index of refraction gradients that reduce to a Kerr nonlinearity only for a weak field. When the laser detuning is positive, self-induced index gradients cause the beam to self-focus. Stationary filaments occur when self-focusing exactly balances diffraction. There exists a unique set of two-dimensional fundamental (and higher mode) solutions for which the filament transverse profile (not Gaussian in general) depends on the following quantities: density of saturable absorbers, laser detuning from resonance, laser power, and natural linewidth. (In contrast the Kerr solution does not have a unique filament diameter.) Emphasis will be placed on propagation of Gaussian incident beams; deviations from the stationary solution in either shape or power generally give rise to oscillating spatial patterns, including donutlike profiles, or beam breakup.

A review of the varieties of optical solitons and their possible interactions, combined with the requirements for a robust digital logic gate motivate the use of (3+1)-dimensional optical solitons (light bullets) as information carriers and soliton dragging gates as switches. Soliton dragging is the asymmetric interaction between two initially overlapping, orthogonally polarized solitons propagating at different angles so that a weak signal soliton can drag a strong pump out of a spatial aperture, thereby implementing a phase-insensitive, high-contrast, logical switch with gain. Light bullets may be an ideal choice for use in these soliton dragging gates but are unstable in Kerr media, but stable (for sufficient pulse energy) in materials with physically reasonable saturating or negative ${\mathit{n}}_4$${\mathit{I}}^2$ nonlinearities. An efficient technique for the propagation of spherically symmetric (3+1)-dimensional field envelopes is developed and used to verify the theoretical stability predictions. A split-step numerical algorithm that models the propagation and phase-independent interaction of arbitrary (3+1)-dimensional, vector e.m. fields in anisotropic media with up to sixth-order tensor nonlinearities is developed and used to demonstrate the features of the gates. NOT and single-stage, two- and four-input NOR light-bullet dragging logic gates are simulated and their performance over a range of operating parameters is presented. It is shown that, with material parameters in the range of those currently available from highly nonlinear organic crystals, high-contrast, all-optical, soliton logic gates with a clock rate greater than 1 THz, latency of a few picoseconds, and switching energy of 25 pJ may be possible.

The master equation for the field in the two-mode Λ laser with atoms injected in a superposition of their states is presented, and lasing without population inversion is studied from this vantage. An explicit expression for the steady-state photon distribution is derived under the conditions of detailed balance and no injected coherence. We study the photon statistics, as well as the decay of off-diagonal density-matrix elements and the linewidths of the two laser modes. The effect of an injected atomic coherence on the amplification of light is examined. It is shown that the interference terms in the master equation cancel the absorption terms if the field in each mode is in a coherent state and the conditions for atomic trapping are fulfilled. Numerical calculations demonstrate the buildup of lasing without inversion from the vacuum state of the field.

An index relation dim ker ${\mathit{a}}^{\mathrm{°}}$a-dim ker ${\mathrm{aa}}^{°}$=1 is satisfied by the creation and annihilation operators ${\mathit{a}}^{\mathrm{°}}$ and a of a harmonic oscillator, where dim ker ${\mathit{a}}^{\mathrm{°}}$a, for example, stands for the number of normalizable basis vectors ${\mathit{u}}_{\mathit{n}}$ that satisfy ${\mathit{a}}^{\mathrm{°}}$${\mathit{au}}_{\mathit{n}}$=0. A Hermitian phase operator, which inevitably leads to dim ker ${\mathit{a}}^{\mathrm{°}}$a-dim ker ${\mathrm{aa}}^{°}$=0, cannot be consistently defined. If one considers an (s+1)-dimensional truncated theory, a Hermitian phase operator of Pegg and Barnett [Europhys. Lett. 6, 483 (1988)] which carries a vanishing index can be defined. However, for arbitrarily large s, we show that the vanishing index of the Hermitian phase operator of Pegg and Barnett causes a substantial deviation from minimum uncertainty in a characteristically quantum domain with small average photon numbers. We also mention an interesting analogy between the present problem and the chiral anomaly in gauge theory which is related to the Atiyah-Singer index theorem. It is suggested that the phase operator problem related to the above analytic index may be regarded as an alternative class of quantum anomaly. From an anomaly viewpoint, it is not surprising that the phase operator of Susskind and Glogower [Physics 1, 49 (1964)], which carries a unit index, leads to an anomalous identity and an anomalous commutator.

Numerical simulations of an extended sample of three-level atoms with slow dipole dephasing exhibit a superfluorescent pulse shortly after incoherent pumping begins. The operator Maxwell-Bloch equations are simulated by classical Langevin equations. The counterpropagating fields are solved for in the (one-dimensional) limit of the Fresnel number near unity, with open-end boundary conditions. An initial, ‘‘quiet’’ buildup of the inversion is followed by superfluorescence. Oscillation of the populations, coherences, and fields is observed. The superfluorescence delay time is compared with analytical predictions made in the linear regime. A transition to amplified spontaneous emission is found as the pump rate is decreased.

The response of a nonlinear oscillator to an external electromagnetic field is studied in the quantum regime. It is demonstrated that bistability occurs when the shift of the resonance frequency, induced by the external field and modified by fluctuations (quantum and thermal), surpasses the natural linewidth. This theory can be applied to the study of the nonlinear response, near resonance, of a single electron, trapped in a magnetic field, to an external electromagnetic field. It is shown that, even at low quantum levels of the oscillator, bistability can be observed due to the interplay between the small relativistic nonlinearity, the external electromagnetic field, and the heat bath.

We consider a parametrically driven Kerr medium in which the pumping may be sinusoidally varied. It has been previously found that this system exhibits coherent tunneling between two fixed points which can be either enhanced or suppressed by altering the driving frequency and strength. We numerically investigate the dynamics when damping is included. This is done both by solving a master equation and using the quantum-trajectory method. In the latter case it is also possible to model the result of a continuous heterodyne measurement of the cavity output. The dissipation destroys the coherences which give rise to the tunneling, causing the sinusoidal oscillation of the mean to give way to a stochastic jumping between the fixed points, manifested as a random telegraph signal. In the quantum-trajectory picture we show that the coherences responsible for tunneling are an exponentially decreasing function of the signal-to-noise ratio for heterodyne measurements. However, evidence of both the bare tunneling rate and the driving modified tunneling rate are still apparent in the random telegraph signal.

We investigate the spectrum of resonance fluorescence of laser-driven three-level systems in the V or $\Lambda$ configuration where one transition is metastable. If the laser driving the weak transition has a sufficiently small Rabi frequency we find an extremely narrow structure in the resonance fluorescence spectrum of the strong transition. Analytical expressions approximating the spectrum are given and the properties of the spectrum are discussed. With the help of a simplified model of resonance fluorescence emitted on the strong transition we are able to show that the origin of the narrow peak can be traced back to electron shelving, i.e., to the existence of light and dark periods in the resonance fluorescence emitted on the strong transition.

We use a tunable quantum interference in type-II parametric down-conversion to construct two-photon quantum states that exhibit less-than-maximal, and tunable, violations of Bell-type inequalities. These states, no longer singlet-state analogs, possess an interference term of controllable magnitude that we adjust by tuning the measurement bandwidth. We show violations of two Bell-type inequalities in polarization variables, one more general than the other. We use these tunable violations to probe the threshold degree of interference that must be present to generate a violation, using an appropriate quantitative figure of merit.

The problem of an electron bound by a Coulomb field of strength $\alpha =Z{\alpha }^{\prime }\left({\alpha }^{\prime }\approx {137}^{-1\mathrm{}}\right)$ is considered in the free-particle representation of the Dirac equation in momentum space. It is shown explicitly how $O\left({\alpha }^5\right)$ contributions to the energies for all $s$-wave states that come from the relativistic free-particle kinetic energy and from the modified Coulomb interaction are canceled by a contribution from electron-positron pair formation. The relativistic correction to the momentum-space Schrödinger wave function is also shown to remove an unwanted $O\left({\alpha }^6\ln \alpha \right)$ energy dependence that arises in lowest-order perturbation theory. The work shows how to perform more accurate calculations in semirelativistic quasipotential approaches that are commonly used in quark models.

Exact Schrödinger wave functions of a time-dependent harmonic oscillator are found in analytically closed forms for the eigenstates of the generalized invariant and the instantaneous Hamiltonian. The cyclic initial state (CIS) and corresponding nonadiabatic Berry phase are also found exactly for a $\tau$-periodic Hamiltonian. There may exist $N\tau$-periodic CISs and corresponding Berry phases, but the cases with unstable classical motions do not have CISs in which cases Berry phases do not exist.

Allowing for the inclusion of the parity operator, it is possible to construct a model of an oscillator whose Hamiltonian admits an exact square root, which is rather different from the conventional approach based on creation and annihilation operators. We outline such a model, the method of solution, and some generalizations.

Two corrections to the Lamb shift, induced by the polarization operator insertions in the external photon lines, are calculated.

The nonadiabatic ground state for the helium atom is obtained with the hyperspherical adiabatic approach. Potential curves, nonadiabatic couplings, and channel functions are calculated by a numerically exact procedure based on the analytical expansion of the channel functions. The coupled radial equations are solved by standard techniques. The convergence of the procedure is investigated as nonadiabatic couplings are systematically introduced. The inclusion of 11 potential curves and channel functions gives a ground-state energy that differs from the best variational calculation by 0.1 parts per million.