Two-photon processes are usually treated using effective-Hamiltonian approaches. In this paper the question of the validity of the effective Hamiltonian in the interaction of a single atom with a single-mode radiation field is considered, and conditions under which effective and microscopic Hamiltonians lead to identical results for the reduced density matrix for the field are derived. The conditions and limitations for the validity of the effective Hamiltonian are elucidated by considering the population inversion, photon distribution function, and squeezing in the two approaches.

We discuss the role of separatrix crossing in a classical Hamiltonian system for a description of transitions in the corresponding quantum system. We develop a simple picture of resonant trapping of classical states and its semiclassical analog. Its relevance is illustrated by numerical analysis and by pointing out its verification in microwave excitation of highly excited hydrogen.

The saturation properties of a single-mode laser are fully taken into account. The analytic expressions of the mean, variance, and skewness of the steady-state laser intensity are calculated. Compared with third-order laser theory and with experimental measurements, good agreement is obtained on the variance of the intensity, but noticeable deviations occur in the mean and skewness when the laser is operated far above threshold.

Stationary propagation of bright and dark surface waves at the nonlinear interface between a self-focusing medium and a self-defocusing medium is found possible within a certain range of the wave effective index ${\mathit{n}}_{\mathit{e}\mathit{f}\mathit{f}}$, depending on the linear refractive indices and nonlinear coefficients. In the low limits of the trapping region in terms of this ${\mathit{n}}_{\mathit{e}\mathit{f}\mathit{f}}$, each particular ${\mathit{n}}_{\mathit{e}\mathit{f}\mathit{f}}$ can be associated with two degenerate trapped mode patterns characterized by two different field profiles, whereas in the upper limits of the trapping region each ${\mathit{n}}_{\mathit{e}\mathit{f}\mathit{f}}$ corresponds to only one particular mode pattern. This contrasts with the bright or dark surface waves trapped at the interface between a linear medium and a nonlinear medium or between two self-focusing (self-defocusing) media where each particular ${\mathit{n}}_{\mathit{e}\mathit{f}\mathit{f}}$ pertains to only one mode pattern (for a nonlinear medium bounded by a linear medium) or two degenerate mode patterns (for two nonlinear media) over the whole trapping region.

Without making the nondepleted-pump approximation, we solve the problem of four-wave mixing (FWM) in a nondissipative ${\mathrm{\chi }}^{\mathrm{}\left(3\right)\mathrm{}}$ medium. Instead of describing the dynamics of FWM in terms of coupled wave amplitudes, we base our solution on canonical equations that describe the propagation of the field’s intensities. This structure clearly identifies the conservative exchange of energy in the FWM process. Consequently, analysis of the FWM process is reduced to a single propagation equation that describes the energy exchange between the pump and amplified waves. This equation yields an elliptic-integral solution. The conversion efficiency reduces to the simple analysis of a fourth-order polynomial, which we analyze to determine the conditions for optimization. The destructive influence of the optical Kerr effect on the phase-matching condition is shown to be eliminated by proper choice of nonzero initial wave-vector mismatch, dependent on the input intensity. The effect of the nonuniform transverse pump-beam intensity profile on the process of conversion is considered. The direction of the energy exchange is shown to be a periodic function of the propagation distance. The transverse intensity distribution of the generating waves provides the characteristic spatial structure (the set of coaxial rings with or without the central spot; the number of rings is explicitly determined by the length of interaction). Several methods of process optimization are discussed, and nonuniformity in the pump-beam profile is shown to be the main reason that complete energy transfer is not achieved.

The line shape of a pair of Doppler-broadened closely spaced transitions sharing a common lower level and under the effect of a standing-wave monochromatic radiation field is studied. The eight steady-state optical Bloch equations are solved analytically to obtain the polarization by retaining terms up to the third power in the electric-field amplitude. The absorption coefficient derived for the standing-wave case consists of two Lamb dips for the two allowed transitions at ${\mathrm{\omega }}_1$ and ${\mathrm{\omega }}_2$ and a crossover resonance dip at the intermediate frequency ${\mathrm{\omega }}_{\mathit{c}}$=(${\mathrm{\omega }}_1$+${\mathrm{\omega }}_2$)/2. The Lamp-dip peak frequencies show intensity-dependent shifts. Numerical computations are presented for different values of Rabi frequencies and relaxation linewidth parameters, which are small compared to the frequency difference Δ=${\mathrm{\omega }}_2$-${\mathrm{\omega }}_1$. The computed line shape and frequency shift are compared with the observed results in the Balmer ${\mathrm{H}}_{\mathrm{\alpha }}$ line of hydrogen and the infrared Zeeman spectra of methane.

We use a nonperturbative method to solve the time-dependent Schrödinger equation for an electronic state of an atom subject to a very intense (>${10}^{13}$ W/${\mathrm{cm}}^2$) laser. The oscillating, time-dependent dipole induced by the laser serves as a source for the photoemission. Calculations of single-atom photospectra reveal peaks at the odd harmonics of the incident laser field superimposed on a broad continuous background. We discuss a series of calculations for the hydrogen atom and a short-range Yukawa potential containing a variable number of bound states. At intensities up to a few times ${10}^{13}$ W/${\mathrm{cm}}^2$ at 1064 nm, we achieve stable, converged spectra that agree very well with previously published results. As the intensity increases to ${10}^{14}$ W/${\mathrm{cm}}^2$, the ionization rate increases to about 1% of the laser frequency, and converged results become extremely difficult to obtain, even for impractically large integration volumes. These difficulties are caused by a rising background due to electron density far from the nucleus and the increasing importance of the interaction of the wave function with the edges of the grid. We discuss the implications of our findings for calculations at high intensity and suggest alternative ways to calculate harmonic emission rates in the strong-ionization regime.

We show that the problem of a Λ-type (and also a V-type) three-level single-mode laser (maser) is isomorphic to that of a two-level single-mode laser (maser) under the conditions of energy-level degeneracy and equal decay rates. This correspondence reveals that lasing without population inversion can occur in the Λ-type system with or without an initial atomic coherence, and also gives immediately the master equation of the laser (maser) field. We analyze in detail the operational and noise properties of a Λ-type laser using a Fokker-Planck approach. We also present a vector model to visualize the population trapping and the absorption and emission processes in such a laser (maser) system.

We study theoretically the electromagnetic field emitted by atoms prepared in linear superpositions of several internal states, each of which is attached to a different center-of-mass wave packet by the preparation process. It is shown that the motion and mutual separation of the wave packets can be monitored either by observing the coherent spontaneous emission in a heterodyne experiment or by measuring the energy-absorption rate from a weak probe-laser beam. We also show that a three-level system involving three wave packets coherently emits photons, forming a linear superposition of states of opposite wave vectors. A heterodyne scheme for detecting the photons in this state is proposed.

The phase properties of fields generated from the vacuum in the m-photon down-conversion process with quantum pumping are studied from the point of view of the Hermitian phase formalism. The joint phase distribution P(${\mathrm{\theta }}_{\mathit{a}}$,${\mathrm{\theta }}_{\mathit{b}}$) as well as the marginal phase distribution P(${\mathrm{\theta }}_{\mathit{a}}$) for the signal mode are derived and illustrated graphically for m=2, 3, and 4. The relationship between these phase distributions and the ‘‘classical’’ distributions obtained by integrating the Q function is established. It is shown that the classical phase distribution is a result of an averaging procedure that leads to a broadening of the phase distribution.

We consider here the problem of quantum noise in light produced by an optical parametric oscillator operating below threshold inside a single-sided cavity. By employing the correct boundary conditions on quantum fields at the output mirror, a procedure that is valid for arbitrary mirror transmission, we derive exact expressions for quantum-noise reduction for arbitrary pump phase and output-mirror coupling. We find that for a given output-mirror power reflectance R, squeezing of quantum noise in the intracavity field increases to a final fractional value equal to R/(1+R) as the pump intensity is increased and the oscillation threshold is approached. The output field, on the other hand, exhibits perfect squeezing of noise in one quadrature of its central Fourier component at threshold regardless of the output-mirror coupling. The present approach, which is based on boundary conditions, is quite general and applicable to any matter-field interaction problem inside a cavity.

A globally coupled class-B laser array with incoherent feedback is proposed for exploring the complex dynamics of dynamical systems with the highest connectivity. This feedback shows a fundamental characteristic, information lag, and results in the general features of synchronization, attractor fission, and attractor fusion processes. The common characteristics shared by different clusters in the segregation process are found. The effect of multiple different time delays on synchronization is also investigated.

The collapse-and-revival phenomenon in the population inversion and the time evolution of the quasiprobability Q function of an atom undergoing a two-photon process are investigated when the cavity is supposed to be filled with a Kerr-like medium. An exact expression for the atomic population inversion has been given for a field initially in the coherent state. The closed form of this expression as well as that of the Q function are obtained under the high-field approximation. Also, the tendency toward inhibited decay of the excited state has been explicitly shown. This study reveals how the simultaneous occurrence of two nonlinear processes affects the atomic dynamics.

A symmetry analysis of degenerate four-wave-mixing equations in photorefractive crystals is carried out. Using underlying SU symmetries, a systematic derivation of conserved quantities is performed, and a method of integration of the equations is introduced. Five conserved quantities are found, suggesting that the initial four complex equations can be expressed in terms of three real quantities (Euler angles or other). However, due to the form of the equations, only one independent variable is found necessary to solve the problem.

Adiabatic inversion in two- and three-level systems excited by stochastic fields is investigated. The stochastic electric fields, which are called the chaotic field and the phase-diffusion field, can invert the population of a two-level atom. In a three-level system, we compare the method of sweeping the laser frequency with that of the resonance excitation by lasers that have bandwidths appropriate to the width of Doppler broadening. When the Doppler shift is taken into account, bandwidths of the electric fields are effective to ionize atoms efficiently. When sweeping the first exciting laser frequency, the delay of the second laser pulse increases the ionization probability. If the half width at half maximum of Doppler broadening is narrow and two lasers irradiate atoms simultaneously, the finite bandwidths of the lasers ionize more atoms than sweeping the laser frequency. On the other hand, if the Doppler broadening is large or the second laser irradiates the atoms properly after the first laser, sweeping the first laser frequency is efficient for ionization. Sweeping the laser frequency is not suitable for the system where the intermediate state’s spontaneous-decay constant is short. For that system it is efficient to ionize the atoms by lasers that have the finite bandwidth appropriate for the Doppler broadening.

Exact expressions are obtained for the density matrix in the operator form and for the quasiprobabilities for the system of an atom undergoing Raman transitions between two degenerate levels on interaction with a quantized field in a lossy cavity driven by an external field. The model is generalized to include the effects of atomic dephasing collisions. The spectral properties of the radiation are discussed, and the equation of motion of the density matrix is derived in the presence of collisions and an external drive. The equation is found to be of second order with respect to time in the absence of the driving field and also analytically solvable.

The dissociation of a diatomic molecule via the electromagnetic excitation of its vibrations is analyzed by a direct solution of the Schrödinger equation in the energy representation of the uncoupled molecule fully including the effects of the continuum. Using a Morse potential, all dipole matrix elements are obtained analytically. We give exact results for the bound-continuum matrix elements and semiclassical results for the continuum-continuum matrix elements corrected for the diverging density of states near the dissociation energy. We compute dissociation probabilities and the form of the excited wave packets in the bound and the unbound subspaces of the uncoupled molecule. The influence of a smooth switch-on and switch-off of the electromagnetic field is also analyzed. Our results are compared with and interpreted by the help of a simple Poincaré map (‘‘Morse map’’) derived in earlier work.

We consider a two-channel Λ-type three-level atom interacting simultaneously with two quantized cavity fields and a classically described cw laser field, all of different frequencies. The effective interactions are of the Raman type, in which the two cavity fields are regarded as Stokes and anti-Stokes modes, and the classical field acts as the pump mode, so that both the pump-Stokes and the pump–anti-Stokes transitions are included. We present the analytic solution of the system evolution operator. We also study the time dependence of the atomic inversion and the statistical properties of the cavity fields. We show that strongly correlated photons can be generated in our model.

The interaction between electromagnetic radiation and two three-level atoms is investigated. The atom-atom separation is assumed to be smaller than the corresponding mean resonance wavelength. The total radiance rate of the atomic system is calculated as a function of time. Explicit results are given for several different initial states of the atomic system. Some of them exhibit superradiance, and some act as photon-trapping states. Instead of exponential decay, the oscillations appear in the time evolution of the radiation rate due to the correlation between atoms. The decay constants and coupling decay constants show strong effects on the emission rates. Completely differing from two identical double-level atoms, the cooperative effects always exist in the three-level atomic system except for the case in which the coupling decay constants ${\gamma }_{12}$ and ${\gamma }_{21}$ are zero.

This paper presents a correspondence between the annihilation a^ and creation a${\mathrm{^}}^{\mathrm{°}}$ operators and two independent complex variables α and β, which makes possible the definition of positive nondiagonal quasiprobabilities for any Cahill-Glauber s order. A generalized version of the Cahill and Glauber s-ordered displacement operator displacing the annihilation and creation operators by α and β is defined. Corresponding to this n/Inondiagonal displacement, a nondiagonal ordering operator is introduced so that a map of s-ordered operators into c numbers a^→α, a${\mathrm{^}}^{\mathrm{°}}$→β can be defined. The Drummond-Gardiner projector and the Cahill-Glauber diagonal ordering operator are obtained as particular cases. In order to use the nondiagonal correspondence in the determination of quantum expectation values of observables, several families of quasiprobabilities are defined, generalizing Drummond-Gardiner (normal order and nondiagonal) positive-P-function and Cahill-Glauber (s-order and diagonal) quasiprobabilities. It is demonstrated that these nondiagonal quasiprobabilities exist as well-behaved functions where at least one is non-negative, for any state and any order. The time evolution of these quasiprobabilities is discussed both in the Schrödinger and in the Heisenberg pictures. In the Heisenberg picture, a method for obtaining c-number stochastic differential equations (SDE’s) directly from operator equations using the s-order nondiagonal correspondence is described. The main difference between this method and the Langevin approach is that in the latter a diagonal correspondence is used, leading eventually to wrong results. The use of these SDE’s to solve quantum optical problems is discussed, and an application to the nonlinearly damped degenerate parametric oscillator is made. In order to obtain the SDE, various (inequivalent) truncation schemes are necessary.