We present theoretical descriptions of Sisyphus cooling in a two-dimensional laser configuration, based on semiclassical as well as quantum approaches. A detailed comparison of these various approaches and a discussion of their range of validity are given. Features of the cooled atomic distribution such as channeling of the atoms along specific lines in momentum space and the observability of tunneling induced resonances in the population of the ground state of the atomic motion in the optical potential wells are discussed.

The cavity field spectra for various initial fields of a nondegenerate two-photon Jaynes-Cummings Model are investigated. It is shown that the two-peak structure due to the resonant interaction between a single two-level atom and two different mode fields inside a high-Q cavity persists for coherent and thermal cases. Moreover, coherent sidebands appear for a coherent field whose widths are governed by the field statistics. However, squeezed vacuum fields exhibit rather different features showing the classical resonance fluorescence spectra at large n¯ (even in two-photon transitions).

The theory of the two-photon state generated by type-II optical parametric down-conversion is studied with emphasis on the space-time and polarization entanglement of the photons. Several experiments are reviewed that demonstrate various aspects of the quantum nature of this state. The theory of a different type of two-photon interferometer is presented.

In this paper, we present a density-matrix formalism for treating second-order sum-frequency generation (SFG) and difference-frequency generation (DFG). The theory can treat both steady-state and time-resolved SFG and DFG. However, only the steady-state SFG will be described in this paper. As a practical application, we show how the theory can be applied to study the infrared-visible SFG. The band-shape function of SFG is derived, which consists of both real and imaginary parts. The real part of the SFG band-shape function is related to the infrared spectral band-shape function and the imaginary part is related to the real part by Kramers-Krönig relations. The temperature effect on SFG is taken into account in our expression of the SFG band-shape function. We show that for a vibrational mode to be SFG active in infrared-visible SFG measurements, it has to be both infrared and Raman active.

We examine the solitary-wave or soliton states in the nonlinear process involving waves of different frequencies interacting through the third-order nonlinear susceptibility in a uniform self-focusing medium. For the wave-mixing case (involving pump, Stokes, and anti-Stokes waves), it is found that there can be three solitary-wave states associated with each frequency shift below a critical frequency shift (Δω=Δ${\mathrm{\omega }}_{\mathit{c}1}$), one of which is stable and the other two unstable. However, above this critical frequency shift but below another critical frequency shift Δ${\mathrm{\omega }}_{\mathit{c}2}$ (<Δ${\mathrm{\omega }}_{\mathit{c}1}$) each frequency shift corresponds to one stable solitary-wave state. In the case of the third-harmonic generation, we have also found a solitary-wave solution. Unfortunately, the solitary-wave state in this case is unstable due to the presence of the complex growth rate of the linearized equations.

By using a modal approach, the distributed coupling of two inhomogeneous, transient waves with opposite angles of incidence into nonlinear waveguides is studied. We show that cross modulation and the coherent interaction via a dynamic grating, nonlinearily induced by the excited counterpropagating leaky waves, provide a longitudinal feedback. This feedback is the necessary prerequisite for bistability to occur. We show that bistability may even appear when one incident beam is weak. Consequently, modulation of the strong signal beam by a weak control pulse can be achieved. Furthermore, we demonstrate that the symmetry of the input signal can be broken beyond a certain intensity and that set-reset flip-flop operation with finite beams is feasible. We demonstrate that the mechanism responsible for this flip-flop operation differs considerably from that acting in the plane-wave case. The optical nonlinearities may be caused by either virtual or real carrier excitation, which models the behavior of semiconductors as well as nonlinear polymers or organic dyes well below or close to a resonance, respectively. For real carrier excitation the interplay of carrier relaxation, diffusion, and the propagation of the excited fields is taken into account.

Phase singularities (topological charges, dislocations, defects, vortices, etc.), which may be either positive or negative in sign, are found in many different types of wave fields. We show that on every zero crossing of the real or imaginary part of the wave field, adjacent singularities must be of opposite sign. We also show that this ‘‘sign principle,’’ which is unaffected by boundaries, leads to the surprising result that for a given set of zero crossings, fixing the sign of any given singularity automatically fixes the signs of all other singularities in the wave field. We show further how the sign of the first singularity created during the evolution of a wave field determines the sign of all subsequent singularities and that this first singularity places additional constraints on the future development of the wave function. We show also that the sign principle constrains how contours of equal phase may thread through the wave field from one singularity to another. We illustrate these various principles using a computer simulation that generates a random Gaussian wave field.

We present a theoretical study of three- and four-beam configurations that generalize, in the two-dimensional (2D) and three-dimensional (3D) cases, the well-known 1D lin ⊥ lin (two counterpropagating beams having crossed linear polarizations) and magnetic-assisted Sisyphus effect cooling schemes. It is shown how the Bravais lattice is determined from the propagation directions of the laser beams while the basis is associated with the polarizations of the incident waves. The optical potentials are calculated for a ${\mathit{J}}_{\mathit{g}}$=1/2→${\mathit{J}}_{\mathit{e}}$=3/2 transition for a variety of angular and polarization configurations and some of the characteristics of the atomic motion inside a single micrometer-sized potential well are predicted.

The effects of pulse shapes with simple phase and frequency modulation (phase proportional to amplitude or intensity, frequency proportional to intensity) are explored numerically. Many such shapes can be generated by combining self-phase-modulation with modern laser pulse-shaping capabilities (which are more flexible for amplitude modulation than for frequency modulation). Even though all of these wave forms begin and end at the same frequency (and hence violate the mathematical conditions for adiabatic inversion), we show that they can generate adiabatic population inversion or suppress intramolecular energy redistribution.

A set of coupled-mode equations is developed to describe the multiwave diffraction of light in two-dimensional nonlinear superlattices, and is solved by numerical methods. At low incident intensities, the solution is time-independent and shows that bistable behavior may appear in the incident-diffracted relations. With an increasing incident intensity, the solution becomes unstable and eventually turns chaotic through the route of intermittance. The threshold intensity for chaos varies with the index-modulation strengths of the superlattice. If the relaxation time of the Kerr-form nonlinearity exceeds the transmission time, only stable solutions are obtained.

The stability of vector solitary waves in birefringent nonlinear dispersive media is considered. It is shown that the vector solitary waves characterized by two peak intensities in one or both polarization components, such as those reported by Christodoulidis and Joseph [Opt. Lett. 13, 53 (1988)] and Tranik and Sipe [Phys. Rev. A 38, 2011 (1988)], are unstable. The growth rate, a measure of the degree of the instability, is calculated. The possible existence of a class of vector periodic solutions is also predicted.

A quantum system described by a two-dimensional harmonic oscillator with a varying mass parameter plus the projection of the angular momentum in the z direction with a time-dependent mode coupling constant is considered to study the two-mode squeezing and photon generation phenomena. For some specific time dependences of the oscillator mass and an arbitrary function of time for the coupling constant exact results on the time evolution of the density matrix and the Wigner function are presented. An analytical expression of the photon distribution is obtained explicitly in terms of Hermite polynomials of two variables, and it shows the effect of oscillations characteristic of squeezing and correlation of the photon quadratures.

We report on the observations of a nonlinear optical response in the radiation pulses transmitted by a multiple-quantum-well GaAs/${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As sample with a ridged waveguide configuration obtained by chemical etching. The nonlinear response is investigated through optical hysteresis obtained by plotting the transmitted power versus the input power. The physical conditions are complementary to those that characterize previous experiments in the field of optical bistability in multiple-quantum-well structures. The numerical simulations, obtained using the first-principles approach previously developed by some of us [G. P. Bava, F. Castelli, P. Debernardi, and L. A. Lugiato, Phys. Rev. A 45, 5180 (1992)], are in satisfactory qualitative and semiquantitative agreement with the experimental data. The numerical simulations also prove that optical bistability could be observed in ridged waveguide samples using cw laser sources.

We investigate the nonclassical properties of the single-mode binomial states of the quantized electromagnetic field. We concentrate our analysis on the fact that the binomial states interpolate between the coherent states and the number states, depending on the values of the parameters involved. We discuss their statistical properties, such as squeezing (second and fourth order) and sub-Poissonian character. We show how the transition between those two fundamentally different states occurs, employing quasiprobability distributions in phase space, and we provide, at the same time, an interesting picture for the origin of second-order quadrature squeezing. We also discuss the phase properties of the binomial states using the Hermitian-phase-operator formalism.

While the linearity of the Schrödinger equation and the superposition principle are fundamental to quantum mechanics, so are the backaction of measurements and the resulting nonlinearity. It is remarkable, therefore, that the wave equation of systems in continuous interaction with some reservoir, which may be a measuring device, can be cast into a linear form, even after the degrees of freedom of the reservoir have been eliminated. The superposition principle still holds for the stochastic wave function of the observed system and exact analytical solutions are possible in sufficiently simple cases. We discuss here the coupling to Markovian reservoirs appropriate for homodyne, heterodyne, and photon counting measurements. For these we present a derivation of the linear stochastic wave equation from first principles and analyze its physical content.

We consider the quantum and classical noise limits to position measurement and force detection by an atomic force microscope (AFM) with an optical readout of cantilever position. We model this by treating the cantilever as a perfectly reflecting mirror for a highly damped optical cavity. There are three sources of noise: the shot noise in the output laser measurement, the thermal noise in the cantilever, and the measurement back-action noise. This last source of noise becomes large for good measurements, measurements for which there is a high correlation between the output phase of the light and the changing position of the cantilever. The back action simultaneously drives a diffusion process in momentum and diagonalizes the cantilever state in the position basis. This latter result is ‘‘state reduction.’’ Explicit expressions for the rate of the measurement back action in terms of the device parameters are given. We also calculate the signal-to-noise ratio in the limit of a bad cavity. A comparison to recent experiments suggests that current AFMs are not quantum limited.

An operational method is presented that can avoid some of the difficulties associated with numerically simulating the Langevin operator equations for a dissipative quantum optical system. The approach is based on a set of quantum-classical correspondence rules that relate atomic operators in the Heisenberg picture to their corresponding double-dimensioned classical atomic variables. With this approach, the Langevin operator equations can be transformed into a doubled set of classical stochastic differential equations (SDE’s). The relationship to the quantum regression theory up to the two-time correlation function and similarities to the positive P representation are discussed. As an illustrative example, we show that both the resonance fluorescence spectrum (Mollow spectrum) as well as the absorption spectrum for a two-level atom interacting with a near-resonance field can be simulated from the doubled set of classical SDE’s without explicit reference to the quantum regression theorem.

The dynamics and thermodynamics of a quantum time-dependent field coupled to a two-level system, well known as the Jaynes-Cummings Hamiltonian, is studied, using the maximum entropy principle. In the framework of this approach we found three different infinite sets of relevant operators that describe the dynamics of the system for any temporal dependence. These sets of relevant operators are connected by isomorphisms, which allow us to consider the case of mixed initial conditions. A consisent set of initial conditions is established using the maximum entropy principle density operator, obtaining restrictions to the physically feasible initial conditions of the system. The behavior of the population inversion is shown for different time dependencies of the Hamiltonian and initial conditions. For the time-independent case, an explicit solution for the population inversion in terms of the relevant operators of one of the sets is given. It is also shown how the well known formulas for the population inversion are recovered for the special cases where the initial conditions correspond to a pure, coherent, and thermal field.

We investigate two schemes for the efficient conversion of coherent input light into bright-squeezed output light. Both schemes utilize strong signal and weak probe fields, interacting with three-level ladder-configuration atoms inside optical cavities. The schemes differ in the resonance requirements of the cavities and produce noise suppression for quite different tuning regimes. Quantum-noise reduction is a consequence of the dressing of the atoms with two coherent fields. By tuning the probe light in the right fashion, spontaneous emission from the excited state can be made to counteract signal-light intensity fluctuations.

A general analysis of an oscillator with periodic frequency modulation is given. It is shown that squeezing and excitation energy exponentially grow with the number of modulation cycles under the condition of parametric instability. This condition yields a prescription for maximized squeezing when the frequency is periodically swept in an adiabatic fashion, with an abrupt return to the initial frequency at the end of each period. The type of modulation considered is shown to have remarkably broad instability domains near arbitrarily high ratios of the oscillator period to the modulation cycle duration. This property stands in striking contrast to the rapid narrowing of the squeezing domains with the ratio of the pump frequency to that of the signal in existing parametric processes. We discuss a possible realization of the proposed scheme, based on frequency modulation of a cavity mode in the microwave domain by a periodic train of optical pulses, and show that extremely strong squeezing is feasible under rather moderate requirements.