It is well known that the set of separable pure states is measure zero in the set of pure states. We extend this fact and show that the set of rank-r separable states is measure 0 in the set of rank-r states provided r is less than ${\prod }_{i=1\mathrm{}}^p{n}_i+p-\sum n_i.$

The pump-probe excitation of a two-level dipolar $\left(d\ne 0\right)$ molecule, where the pump frequency is tuned to the energy level separation while the probe frequency is extremely small, is examined theoretically as an example of absolute phase control of excitation processes. The state populations depend on the probe field’s absolute carrier phase but are independent of the pump field’s absolute carrier phase. Interestingly, the absolute phase effects occur for pulse durations much longer and field intensities much weaker than those required to see such effects in single pulse excitation.

The g factor of Li-like ions is evaluated in the range $Z=6\mathrm{}–92.$ The interelectronic-interaction correction of first order in $1/Z$ is calculated to all orders in $\alpha Z$ within a rigorous QED approach. The higher-order interelectronic-interaction, the one-electron QED, and the nuclear recoil corrections are evaluated to lowest orders in $\alpha Z.\mathrm{}$ It is found that the uncertainty due to the nuclear size effect can be significantly reduced in a specific difference of the g factors of H- and Li-like ions.

The Mermin-Klyshko inequality for n spin-$\frac{1}{2}$ particles and two dichotomic observables is generalized to n spin-$s$ particles and two maximal observables. It is shown that some multiparty multilevel Greenberger-Horne-Zeilinger states [A. Cabello, Phys. Rev. A 63, 022104 (2001)] maximally violate this inequality for any $s.\mathrm{}$ For a fixed $n,$ the magnitude of the violation is constant for any $s,$ which provides a simple demonstration and generalizes the conclusion reached by Gisin and Peres for two spin-$s$ particles in the singlet state [Phys. Lett. A 162, 15 (1992)]. For a fixed $s,$ the violation grows exponentially with $n,$ which provides a generalization to any s of Mermin’s conclusion for n $\mathrm{spin}-\hspace{0.5em}\frac{1}{2}$ particles [Phys. Rev. Lett. 65, 1838 (1990)].

We represent both the states and the evolution of a quantum computer in phase space using the discrete Wigner function. We study properties of the phase space representation of quantum algorithms: apart from analyzing important examples, such as the Fourier transform and Grover’s search, we examine the conditions for the existence of a direct correspondence between quantum and classical evolutions in phase space. Finally, we describe how to measure directly the Wigner function in a given phase-space point by means of a tomographic method that, itself, can be interpreted as a simple quantum algorithm.

We show that by a suitable choice of a time-dependent Hamiltonian, Deutsch’s algorithm can be implemented by an adiabatic quantum computer. We extend our analysis to the Deutsch-Jozsa problem and estimate the required running time for both global and local adiabatic evolutions.

The fluorescence $\mathrm{KL}$ x-ray emission of argon induced by impact with monoenergetic photons was observed with a high-resolution reflecting-type bent crystal spectrometer. The measurements were performed at the European Synchrotron Radiation Facility in Grenoble, France. X-ray beam energies in the range $3455\hspace{0.5em}\mathrm{eV}<~ħ\omega <~6540\hspace{0.5em}\mathrm{eV}$ were used to investigate the $K+L$ double excitation from threshold to saturation. The excitation probabilities were deduced from the satellite-to-diagram line yield ratios $\mathrm{I}(KL-L_{2,3}L):\hspace{0.5em}\hspace{0.5em}\mathrm{I}(K-L_{2,3}).\mathrm{}$ Results were compared to predictions based on the model of Thomas and the one of Roy. From the comparison, parameters such as excitation onset energies, high-energy limit shake probabilities, and radius of the $2p$ orbital in ionic Ar were deduced. The satellite line shapes and their evolution with the excitation energy were analyzed with the aid of multiconfiguration Dirac-Fock calculations.

Auger spectroscopy has been used to measure the fraction of metastable $1s2s{}^{3}S$ ions in fast He-like B, C, N, O, and F beams produced in collisions with thin carbon foils as a function of both the incident energy in the range of $0.5–2\hspace{0.5em}\mathrm{MeV}/u$ and the foil thickness in the range of $1–5\hspace{0.5em}\mu {\mathrm{g}/\mathrm{c}\mathrm{m}}^2.$ The method used for determination of the metastable fraction is based on measurements of the electron emission from the doubly excited states of Li-like ions formed in collisions of primary beams with hydrogen and helium targets. Some differences were observed both in the energy dependence and the absolute value of metastable fractions for different beams. In particular, the metastable content in ${\mathrm{C}}^{4+}$ ions produced in carbon foils was found to be significantly lower than that of other investigated beams. The observed deviation has been explained on the basis of K-vacancy sharing, which is known to have the highest probability for symmetric collisions. A model recently proposed for the calculation of the metastable fraction in He-like beams has been used to predict the metastable fraction for ${\mathrm{B}}^{3+}$ ions leading to a good agreement with the experiment.

We show that the dipole selection rules for the high-order-harmonic-generation spectra of atoms interacting with two laser beams break down unless the polarization of each one of the beams is orthogonal to the wave vector of the other one. In a general two-beam setup, the interaction of the electrons with the magnetic components of the electromagnetic waves leads to sometimes very efficient generation of the harmonics that are symmetry forbidden within the dipole approximation, e.g., even harmonics. Our symmetry analysis and model calculations elucidate the essential differences between several schemes that have been recently proposed for the generation of circularly polarized high-order harmonics.

A spectral fitting method for solving the time-dependent Schrödinger equation has been developed and applied to the atom in intense laser fields. This method allows us to obtain a highly accurate time-dependent wave function with a contribution from the high-order term of $▵t.\mathrm{}$ Moreover, the time-dependent wave function is determined on a small number of discrete mesh points, thus making calculations simple and accurate. This method is illustrated by computing wave functions and harmonic generation spectra of a model atom in laser fields.

When the dipole-dipole energy transfer, $\mathrm{Rb}\hspace{0.5em}{25s}_{1/2}+\mathrm{Rb}\hspace{0.5em}{33s}_{1/2}\to \mathrm{Rb}\hspace{0.5em}{24p}_{1/2}+\mathrm{Rb}\hspace{0.5em}{34p}_{3/2},$ resonant at fields of 3.0 and 3.4 V/cm, is observed in a cold Rydberg gas, the resonances are broader than expected for this dipole-dipole interaction alone. Using a variant of the Ramsey interference method, in which the system is detuned from the resonance, we observe a density-dependent dephasing which we attribute to the inhomogeneities in the always resonant dipole-dipole excitation exchange interactions $\mathrm{Rb}\hspace{0.5em}{25s}_{1/2}+\mathrm{Rb}\hspace{0.5em}{24p}_{1/2}\to \mathrm{Rb}\hspace{0.5em}{24p}_{1/2}+\mathrm{Rb}\hspace{0.5em}{25s}_{1/2}$ and $\mathrm{Rb}\hspace{0.5em}{33s}_{1/2}+\mathrm{Rb}\hspace{0.5em}{34p}_{3/2}\to \mathrm{Rb}\hspace{0.5em}{34p}_{3/2}+\mathrm{Rb}\hspace{0.5em}{33s}_{1/2}.$ Since the dephasing rate is comparable to the observed widths, it appears that all of these interactions contribute to the observed linewidths.

We investigate the spatiotemporal dynamics of Bose-Einstein condensates in optical lattices that have a linear-or a circular-chain configuration with the tunneling couplings between nearest-neighbor lattice sites. A discrete nonlinear Schrödinger equation has been solved for various initial conditions and for a definite range of repulsive and attractive interatomic interactions. It is shown that the diversity of the spatiotemporal dynamics of the atomic population distribution such as a macroscopic self-trapping, bright and dark solitons, and symmetry breaking is derived from the positive and negative interatomic interactions. For the circular-chain configuration, two types of rotational modes are obtained as we introduce a definite relation for the initial phase conditions.

Starting from the first principles of nonrelativistic QED we have derived the system of Maxwell-Schrödinger equations which can be used for theoretical description of atom optical phenomena at high densities of atoms and high intensities of the laser radiation. The role of multiple atomic transitions between ground and excited states in atom optics has been investigated. Nonlinear optical properties of interacting Bose gases are studied and an equation for the refractive index has been derived. We have investigated the role of light-induced and collisional nonlinearities in the diffraction of an ultracold atomic beam from an intense standing laser wave in the Raman-Nath and in the Bragg regimes.

We investigate theoretically the transition temperature of a weakly interacting $F=1\mathrm{}$ spinor Bose-Einstein condensate (BEC) formed in an optical trap. The Hartree-Fock-Bogoliubov formalism for the spinor BEC is developed based on the mean-field theory for inhomogeneous systems at finite temperatures. The shift of the transition temperature due to finite-size effects and the two-body interaction is obtained simultaneously by analytically solving the coupled eigenvalue equations for quasiparticle excitations and amplitudes under Hartee-Fock-Bogoliubov and other relevant approximations. When applying this to the recent experiment of the all-optical formation of a spinor BEC with a condensate of $3.5\times {10}^4$ ${}^{87}\mathrm{Rb}$ atoms [Phys. Rev. Lett. 87, 010404 (2001)], our calculations show that the condensation occurs at a temperature about 177 nK. An analytic result for the shift of the transition temperature in the presence of a weak magnetic field is also derived.

A single two-level atom in a coherently driven cavity and damped by a broadband squeezed vacuum centered about the atomic transition frequency is studied. A second-order Fokker-Planck equation for this system is obtained without using system size expansion. Effects of detunings and cavity decays are also incorporated in the Fokker-Planck equation. This equation is used to study atomic inversion, fluorescent spectrum, and the intensity correlations of the transmitted and fluorescent photons in the bad-cavity limit. Several interesting effects in the atomic inversion, spectrum, and intensity correlations due to the squeezed vacuum are presented. These results are also compared with an atom that is damped by a thermal reservoir.

The intensity fluctuations of a three-level laser are known to drop below shot noise in the presence of depletion of the ground state. We study the fluctuations of the laser output as a function of the parameter $\varepsilon ,$ defined as the fraction of atoms needed for the laser to operate. For a sufficiently small number of active atoms, the value of $\varepsilon ,$ and thereby the ground-state depletion, can be appreciable even for modest pumping. This suggests that the intensity fluctuations in the laser output would decrease below shot noise as the number of active atoms is reduced. A microscopic approach that uses a quantum-trajectory method and a macroscopic approach using semiclassical rate equations both show, however, that rather than decreasing, the intensity fluctuations actually increase with $\varepsilon .$ We find that the fluctuations of the output are determined by the dependence of the cycle time of the atoms on the number of photons in the laser mode.

The dispersion theory of meromorphic total reflectivity is considered by dealing with the poles of the complex reflectivity function in the upper half of the complex angular frequency plane. Modified Kramers-Kronig relations and corresponding static limit sum rules are derived for the degenerate third-order nonlinear reflectivity. A high-frequency sum rule is also suggested.

A method for Einstein-Brillouin-Keller (EBK) semiclassical wave functions is applied to the case where the quantizing torus is close to a chain of resonance islands, and standard EBK wave functions are distorted by the resonance islands. The method tested here is a modification of a recent “direct trajectory” implementation of the standard EBK method. The phase factor of the semiclassical wave function is obtained exactly as by Yang and Kellman [Phys. Rev. A 62, 022105 (2000); 65, 034103 (2002)], by running a single trajectory on the quantizing torus. However, rather than obtaining the amplitude factor from this trajectory, the method here uses an ensemble of trajectories uniformly distributed in a region enclosing an area ħ around the torus. The agreement between the resulting semiclassical wave function and the quantum wave function is excellent.

We propose a scheme to generate nonclassical states of a quantum system, which is composed of the one-dimensional trapped-ion motion and a single-cavity field mode. We show that two-mode SU(2), entangled coherent states, two-mode squeezed vacuum states, and their superposition can be generated. If the vibration mode and the cavity mode are used to represent separately a qubit, a quantum phase gate can be implemented.

Cold collisions of ground-state oxygen molecules with helium have been investigated in a wide range of cold-collision energies (from $1\mu \mathrm{K}$ up to 10 K) treating the oxygen molecule first as a rigid rotor and then introducing the vibrational degree of freedom. The comparison between the two models shows that at low energies the rigid-rotor approximation is very accurate and able to describe all the dynamical features of the system. The comparison between the two models has also been extended to cases where the interaction potential $\mathrm{He}-{\mathrm{O}}_2$ is made artificially stronger. In this case vibration can perturb rate constants, but fine tuning the rigid-rotor potential can alleviate the discrepancies between the two models.