We study the role played by nonlinearities in the formation of optical vortices when the boundary conditions determine the pattern formation in a Fabry-Pérot resonator. Particular care is taken to avoid trivial vortices coming from particular Gaussian modes or from the cavity alignment. We show that, while the existence of optical vortices is a generic linear property of the superpositions of Gaussian modes, their actual appearance in the transmitted patterns is controlled by the nonlinearity itself.

We outline an analytical framework for the treatment of radial Rydberg wave packets produced by short laser pulses in the absence of external electric and magnetic fields. Wave packets of this type are localized in the radial coordinates and have p-state angular distributions. We argue that they can be described by a particular analytical class of squeezed states, called radial squeezed states. For hydrogenic Rydberg atoms, we discuss the time evolution of the corresponding hydrogenic radial squeezed states. They are found to undergo decoherence and collapse, followed by fractional and full revivals. We also present their uncertainty product and uncertainty ratio as functions of time. Our results show that hydrogenic radial squeezed states provide a suitable analytical description of hydrogenic Rydberg atoms excited by short-pulsed laser fields.

Production of sum-frequency radiation and ‘‘even’’ harmonics from the mixed 1ω (1053 nm) and 2ω (527 nm) field is observed and compared to measurements of the relative conversion efficiency of the independent fields. By controlling the relative polarization of the two fields, we can control the mixing efficiency and produce coherent extreme ultraviolet (XUV) radiation polarized orthogonally to the strong driving field. Extension to produce XUV radiation of arbitrary polarization is discussed.

We propose the use of spectrally colored rather than white (i.e., incoherent) population pumping in lasing-without-inversion schemes. We show that colored pumping can lead to a significant enhancement in gain by presenting explicit results for the gain in a three-level ladder system. These results are obtained from Monte Carlo simulations of the coupled density-matrix equations, which, for a colored pump, acquire the structure of Langevin equations.

Photodissociation probabilities for coherent and incoherent laser sources are described, using number- and coherent-state representations of the light field. Previous calculations of ${\mathrm{H}}_2^{+}$ photodissociation in strong laser fields are used to calculate probabilities, for coherent and incoherent sources, that show increasing differences at lower light intensities.

It is proposed that each of several analytical techniques used to evaluate the path integral when a spinless particle encounters an infinite potential barrier relies on more constraints than is necessary. We review the image, forbidden-paths, and eigenfunction-expansion methods, and the infinitesimal-propagator method is developed herein. It is then shown that the latter two methods contain assumptions leading to extra terms in the path-integral integrand which serve only to aid in the analytical evaluation of the path integral but which do not affect the outcome of that evaluation. The analytical arguments are further confirmed by a recently developed numerical method that, for a limited number of cases, is a direct evaluation of the path integral.

A relative-state formulation of quantum systems is presented in terms of relative-coordinate states, relative-number states, and relative-energy states. The relative-coordinate states are used to describe quantum systems in position and momentum representations. The probability distribution is calculated in terms of the relative-coordinate states and is shown to be equivalent to the functional definition of the quantum probability in phase space. It is shown that a quantum-mechanical phase operator can be constructed in terms of the relative-number states without the well-known difficulties. The results are compared with those obtained by the Pegg-Barnett phase-operator formalism and the relations to various other phase-operator methods are also discussed. The energy-measurement and energy-probability distributions are discussed in terms of the relative-energy states. Furthermore, a relative-state formulation is developed in the Liouville space. A phase representation in the Liouville space is introduced to investigate the time evolution of quantum coherence. In the Liouville space a time operator is defined as a canonical conjugate of the time-evolution generator, but not the Hamiltonian energy operator. The relation to the internal time presented by Prigogine and Misra is discussed.

The problem of finding eigenvalues of the static nonlinear Schrödinger equation with a potential is numerically investigated. The change of eigenfunctions resulting from the transformation of a potential well into a potential hill is studied. Unlike the linear Schrödinger equation, continuous and square-integrable solutions exist, not only for potential wells, but also for potential hills. For potential hills there may exist a few different eigenfunctions with the same number of nodes, whereas eigenfunctions for potential wells are single valued. Moreover, regions of stability are discovered where a continuum of eigenfunctions exist. In these regions, eigenfunctions may continuously transform one into another in certain energy intervals. The possible practical use of these solutions is discussed.

Quantum cellular automata (QCA) have been introduced [G. Grössing and A. Zeilinger, Complex Syst. 2, 197 (1988); 2, 611 (1988)] as n-dimensional arrays of discrete sites characterized by a complex number whose absolute square lies between 0 and 1 such that each site represents a quantum-mechanical probability amplitude. The evolutions of one-dimensional QCA with a local (i.e., nearest-neighbor) interaction and with periodic boundary conditions have been studied in some detail. In this paper we present a thorough mathematical analysis of one-dimensional QCA, and we particularly emphasize the effects of what we term nonlocal computation: because of the conservation of the total probability for each time step, information of the global array spread out over generally nonlocal distances must be conveyed for each time step to each local site via the normalization procedure. With the aid of a mathematical description of QCA evolution the following, previously observed phenomena can be explained: (i) the asymptotic appearance of plane-wave patterns after a characteristic transient phase, (ii) the dependence of the period of the patterns on the input parameters for both small and large values of the couplings between the sites, and (iii) the role of the initial values of the cells. Finally, the effects of nonlocal computation are made visible with the simulation of a ‘‘double-slit-like’’ experiment on a QCA grid.

Approximate energy eigenvalues for the double-well quartic oscillator are calculated using the phase-integral method of Fröman and Fröman up to the fifth-order approximation. By means of appropriate transformations, the integrals appearing in the quantization condition are expressed in terms of integrals solved in previous papers. Different approximations are discussed. In general, the results are very good.

A beam of particles obeying the Schrödinger equation is split into two parts; the two parts are passed through identical rectangular waveguides and then brought back together at the position of the detector where they interfere constructively. If the boundaries of one of the two waveguides starts moving while the other remains stationary, the waves arriving at the detector will have different frequencies and the absolute value of the total wave function oscillates in time and shows the presence of quantum beats. For different types of motion in which the initial and the final positions of the walls are the same, the result of calculation shows that the beat frequency is not very sensitive to the details of the motion of the wall.

A canonical generator for the conformal transformation of the second quantized Schrödinger field in the grand canonical ensemble is obtained. Using this generator, the finite-temperature generalized Ward-Takahashi relation for Matsubara Green functions formulated by the present author [Ann. Phys. (N.Y.) 173, 226 (1987)] is extended to include conformal transformations. The form of the generator is shown to be similar to the electromagnetic interaction Hamiltonian of the charged Schrödinger particle.

Electron movement in a spatially periodic high-frequency electric field is considered. If the spatial period is small enough and the frequency is sufficiently high, the effect of this field on the charged particle is in a sense equivalent to the effect of a time-independent effective potential that is a sequence of deep dynamic quantum wells where the electrons are confined. Also, a physical model is proposed which could be described in terms of such a Hamiltonian.

We show how a two-body potential may be uniquely determined from n-body spectra in cases where the hypercentral approximation is valid. We illustrate this by considering a harmonic-oscillator potential which has been altered by changing the energy or normalization constant of the ground state of the n-body system and finding how this modifies the two-body potential. It is shown that with an increasing number of particles the spectrum must be known more precisely to obtain the two-body potential to the same degree of accuracy.

We present the Bohm theory of hydrogenlike atoms, the measurement of orbital angular momentum, and Einstein-Podolsky-Rosen angular momentum correlations. We use the illustrations to discuss the arguments of von Neumann [described in M. Jammer, The Philosophy of Quantum Mechanics (Wiley, New York, 1974)] and Kochen and Specker [J. Math. Mech. 17, 59 (1967)] against hidden variables.

Starting from a potential with a continuum of energy eigenstates, we show how the methods of supersymmetric quantum mechanics can be used to generate families of potentials with bound states in the continuum (BIC’s). We also find the corresponding wave functions. Our method preserves the spectrum of the original potential except it adds these discrete BIC’s at selected energies. Specifically, we compute and graph potentials which have bound states in the continuum starting from a null potential representing a free particle and from both the attractive and the repulsive Coulomb potentials.

The exact rule describing the transformation of spectra of multilevel quantum systems perturbed by a random matrix has been found with the help of the all-order perturbation theory. For systems with a large number of levels this rule has a simple form, resembling the second-order perturbation theory. It immediately gives the well-known semicircle Wigner distribution of the density of states and the double-hump distribution for binary alloys. The spectra of small atomic or molecular clusters resulting from the perturbation of a small number of identical levels are also given among some other examples. Discontinuities of the derivatives of the energy spectra and a nonexponential time evolution are shown to be the typical features of large multilevel systems perturbed by a random matrix.

The laser rf double-resonance technique in a Paul trap has been used to make high-precision measurements of the hyperfine structure (hfs) of the 4${\mathit{f}}^7$${(}^8$${\mathit{S}}_{7/2}$)6s ${}_{}{}^9{}_{}{}^{}$${\mathit{S}}_4$ ground state in ${}_{}{}^{151,153}{}_{}{}^{}\mathrm{Eu}_{}^{+}{}_{}{}^{}$. We have obtained linewidths below 1 kHz for the magnetic-field-dependent rf transitions at frequencies between 1.7 and 10 GHz. After corrections for the second-order hfs we obtain values for the magnetic-dipole, electric-quadrupole, magnetic-octupole, and electric-hexadecapole interaction constants.

We use the B-spline approximation to solve the Brueckner equation. The Brueckner equation is obtained from Dyson equation for the one-particle propagator by cutting the kernel at second order in the electron-electron interaction. Thus the Brueckner approximation is a straightforward extension of the frozen-core Dirac-Fock approximation to include second-order correlations in a self-consistent way. We also give the explicit expression for the gauge-invariant transition in the Brueckner approximation. The numerical results of energies and transition amplitudes for alkali-metal atoms in the Brueckner approximation differ from experimental values by percentages ranging from 0.1% to 2%, increasing from Li to Cs. These results are comparable to existing accurate theoretical values.

The problem of the Schrödinger equation for two electrons (interacting with Coulomb potentials) in an external harmonic-oscillator potential is revisited and shown to be solvable analytically for a particular, denumerably infinite set of oscillator frequencies. Solutions are given for ground and excited states in the singlet and triplet spin configurations.