We introduce a separability criterion based on the positive map $\Gamma :\overrightarrow{\rho }\left(\mathrm{Tr}\rho \right)-\rho ,$ where $\rho$ is a trace-class Hermitian operator. Any separable state is mapped by the tensor product of $\Gamma$ and the identity into a non-negative operator, which provides a simple necessary condition for separability. This condition is generally not sufficient because it is vulnerable to the dilution of entanglement. In the special case where one subsystem is a quantum bit, $\Gamma$ reduces to time reversal, so that this separability condition is equivalent to partial transposition. It is therefore also sufficient for $2\times 2$ and $2\times 3$ systems. Finally, a simple connection between this map for two qubits and complex conjugation in the “magic” basis [Phys. Rev. Lett. 78, 5022 (1997)] is displayed.

Entanglement between three or more parties exhibits a realm of properties unknown to two-party states. Bipartite states are easily classified using the Schmidt decomposition. The Schmidt coefficients of a bipartite pure state encompass all the nonlocal properties of the state and can be “seen” by looking at one party’s density matrix only. Pure states of three and more parties, however, lack such a simple form. They have more invariants under local unitary transformations than any one party can “see” on their subsystem. These “hidden nonlocalities” will allow us to exhibit a class of multipartite states that cannot be distinguished from each other by any party. Generalizing a result of Bennett, Popescu, Rohrlich, Smolin, and Thapliyal, and using a recent result by Nielsen, we will show that these states cannot be transformed into each other by local actions and classical communication. Furthermore, we will use an orthogonal subset of such states to hint at applications to cryptography and illustrate an extension to quantum secret sharing [using recently suggested $\mathrm{}\left(\right(n,k\left)\right)$-threshold schemes].

Nonlinear interactions between two photons are required for the construction of optical quantum logic gates, but those interactions are normally very weak due to the small magnitude of the electric field associated with a single photon. We note that exchange interactions can have a large effect even when there is no physical interaction between two particles, and we exploit this property for the construction of optical quantum logic gates. We show that the probability of there being two virtually-excited atoms in a medium can be a factor of 2 larger when two nonresonant photons propagate through the same medium as compared to the case in which they propagate through two separate media, in analogy with photon bunching. As a result, the application of one or more laser pulses will produce a nonlinear phase shift that can be used to construct an XOR quantum logic gate. This provides an example of a quantum control process in which one photon can control the state of another photon even when there is no sequence of physical interactions linking the two photons. From a classical point of view, it is not possible to identify a path for the flow of information or a specific cause for the outcome of the control process.

We show how the partial entanglement inherent in a two-mode squeezed vacuum state admits two different teleportation protocols. These two protocols refer to the different kinds of joint measurements that may be made by the sender. One protocol is the recently implemented quadrature phase approach of Braunstein and Kimble [Phys. Rev. Lett. $80,$ 869 (1998)]. The other is based on recognizing that a two-mode squeezed vacuum state is also entangled with respect to photon-number difference and phase sum. We show that this protocol can also realize teleportation; however, limitations can arise due to the fact that the photon-number spectrum is bounded from below by zero. Our examples show that a given entanglement resource may admit more than a single teleportation protocol and the question then arises as to what is the optimum protocol in the general case.

A scheme is proposed for coupling the vibrational motion of atoms trapped at distant sites of a dilute optical lattice. The coupling is provided by the backaction of the atomic motion on the lattice potential wells, which is strongly enhanced by recycling the lattice beams in a high finesse optical ring resonator. The resulting phononlike excitations are characterized and their frequency spectrum is calculated. How this coupling can be utilized for implementation of quantum logic gates in an optical lattice is discussed. A concrete experimental scenario in a rubidium lattice is illustrated.

We have measured the relative transition probabilities of about $100\hspace{0.5em}3s-3p$ and $3p-3d$ lines of neutral fluorine in the visible and near-infrared spectrum with a wall-stabilized high-current arc, which is operated under conditions very close to partial local thermodynamic equilibrium. The set of measured lines includes about 40 intersystem transitions. Our data have been placed on an absolute scale by normalizing several strong transitions to the results of the OPACITY Project calculations, which are expected to be quite accurate for such transitions. We estimate that the uncertainties of our absolute transition probability values are in the ±15% to ±20% range, while the uncertainties of the relative values do not exceed 6%, except for very weak transitions. Our results indicate that especially for $3p-3d$ transitions appreciable departures from LS coupling are encountered. Comparisons with other results indicate that earlier experimental data agree well with ours on a relative scale, but need to be renormalized.

The $J=\frac{3}{2}\leftarrow \frac{1}{2}$ fine-structure transition in atomic ${}^{27}\mathrm{Al}$ in its ground ${}^{2}P$ state has been detected in the laboratory by far-infrared laser magnetic resonance. The fine-structure interval has been measured accurately as 3359.6228(17) GHz or 112.064 95(6) ${\mathrm{cm}}^{\mathrm{-}1}$. The spectra show the hyperfine structure associated with a nuclear spin of $\frac{5}{2}$ for ${}^{27}\mathrm{Al}.$ The splittings are well reproduced by the hyperfine parameters determined earlier by atomic beam methods. The analysis of the laser magnetic-resonance spectra casts doubt on the published value for the Zeeman parameter $g_{3/2}$ also from the atomic beam measurements. Although the value determined in this paper is intrinsically much less accurate, it is in better agreement with the value obtained by a completely independent theoretical calculation.

A special application of the equal-factor method is used to reformulate perturbation theory. We derive formulas for determining the $m\mathrm{th}$-order equal factors $\{f_n^{\mathrm{}\left(m\right)\mathrm{}}\}$ and the corresponding $m\mathrm{th}$-order energy corrections $\{E_n^{\mathrm{}\left(m\right)\mathrm{}}\}.$ A set of the $m\mathrm{th}$-order perturbation energy levels $\{E_n^{\mathrm{}\left(m\right)\mathrm{}}\}$ thus may be calculated to arbitrarily high order by self-consistent equations. We have applied this so-called equal-factor perturbation to hydrogenlike ions in a potential field $-\delta {/r}^2.$ Our results illustrate important advantages in aspects both of practice and theory. Work is presently underway to introduce the equal-factor assumption to perturbation theory for reformulation of the principle of the first and second kind of balance theory.

We give a density-matrix account of an atom in a dilute vapor which is subject to a magnetic field and through which is propagating plane-polarized monochromatic light with frequency in the vicinity of a transition between levels of total angular momenta $J=0\mathrm{}$ and 1, where the degeneracy may be in either the upper or lower level. The master equation approach allows saturation effects to be treated in a nonperturbative manner, and collisions are included within the context of the impact approximation. The particular case of the Voigt effect in a $\left(\overrightarrow{1}0\right)$ transition (lower level $J=1\mathrm{}$) is considered in detail, and contrasted with that in $\left(\overrightarrow{0}1\right),$ already published; in the absence of collisions, the $\left(\overrightarrow{1}0\right)$ rotation spectrum is of Lorentzian form at all intensities, while if collisional relaxation is much faster than radiative decay, it takes the same form as for the $\left(\overrightarrow{0}1\right)$ case. The rotation produced by a magnetic field of arbitrary direction and the effects of atomic motion are also discussed.

We report here the finding that, to obtain reliable lifetimes in highly ionized atoms measured by beam foil excitation, the conditions must be set so that the charge state studied is well below the average charge of the foil emergent beam. If this condition is not fulfilled, the decay curves may be significantly distorted by blending with close-lying satellite lines. We present evidence to show that failures to follow this criterion have led to serious underestimates in reported lifetimes in isoelectronic line strength studies. We also suggest guidelines for avoiding these distortions in future measurements.

Degenerate perturbation theory is used to study dipole susceptibilities of an excited helium atom in an external electric field. The dependence of the perturbed energy of levels in atoms on fine-structure effects and on the higher-order Stark effect is investigated. Numerical results have been obtained for the $\mathrm{}\left(1s3p\right)\mathrm{}{}^3{P}_0$ and $\mathrm{}\left(1s3p\right)\mathrm{}{}^3{P}_2$ states of helium. The magnitude of the electric field and the energy separation at the $0^{-}\times 0^{-}$ anticrossing are calculated. Calculations of polarizabilities and hyperpolarizabilities are carried out using sums of oscillator strengths and, alternatively, with the excited electron Green function. An estimate is given based on the model potential method for the contribution of an infinite series over the bound states, including the integral over the continuum, for second- and higher-order matrix elements. The relativistic approach for evaluating reduced dipole matrix elements based on the relativistic no-pair Hamiltonian and including both the Coulomb and Breit interactions (configuration-interaction method) is analyzed.

Transition rates, oscillator strengths, and line strengths are calculated for the 49 possible electric-dipole transitions between the eight even-parity ${2s2p}^2$ states and the seven odd-parity ${2s}^{2}2p$ and ${2p}^3$ states in boronlike ions with nuclear charges ranging from $Z=6\mathrm{}$ to 100. Relativistic many-body perturbation theory (MBPT), including the Breit interaction, is used to evaluate retarded $E_1$ matrix elements in length and velocity forms. The calculations start from a ${1s}^2$ Dirac-Fock potential. First-order MBPT is used to obtain intermediate coupling coefficients and second-order MBPT is used to calculate transition matrix elements. Contributions from negative-energy states are included in the second-order $E_1$ matrix elements to ensure gauge independence of transition amplitudes. The transition energies used in the calculation of oscillator strengths and transition rates are from second-order MBPT. Transition rates, line strengths, and oscillator strengths are compared with critically evaluated experimental values and with results from other recent calculations. Lifetimes of the five possible odd-parity upper levels and the five possible even-parity upper levels are given for $Z=6\mathrm{}–100$. Trends of the transition rates as functions of Z are illustrated graphically for selected transitions.

Let $\mathcal{H}$ be the state space of a quantum computer coupled with the environment by a set of error operators spanning a Lie algebra $\mathcal{L}.$ Suppose $\mathcal{L}$ admits an error-avoiding quantum code, i.e., a subspace $\mathcal{C}\subset \mathcal{H}$ annihilated by $\mathcal{L}.$ We show that a universal set of gates over $\mathcal{C}$ is obtained by any generic pair of $\mathcal{L}$-invariant gates. Such gates—if not available from the outset—can be obtained by resorting to a symmetrization with respect to the group generated by $\mathcal{L}.$ Any computation can then be performed completely within the coding decoherence-free subspace.

It is shown how to calculate the total energy, the single-particle energies, and the current density of an interacting current-carrying many-body system, from solutions of the Kohn-Sham equation of conventional density-functional theory. On the conceptual side, this procedure, which applies both in the presence and absence of external magnetic fields, sheds light on the role of orbital magnetism in density-functional theory and the interrelation of conventional and current-density functional theory. On the practical side it provides a computationally simple many-body approach to the interplay of orbital magnetism and electron-electron interactions in atoms, molecules, and solids.

Cu atoms have been prepared in the Ni-like Cu ${3d}^9{4s}^2$ state by uv-laser pumping. In contrast to the ground-state atoms, the Cu ${3d}^9{4s}^2$ atoms exhibit a prominent asymmetric resonance in the ${3d}^{-1}$ partial cross section at the $3p$ threshold. A close similarity to the corresponding resonance in the spectrum of Ni ${3d}^{9}4s$ atoms is demonstrated.

The position of the maximum in longitudinal electron momentum distributions, for He and Ne single ionization by proton impact, has been studied as a function of projectile velocity using the continuum-distorted-wave–eikonal-initial-state model. At intermediate to high energies the position of the maximum is determined by low-energy electron emission and it can be related to the soft-collision peak. At intermediate to low energies, the position of the maximum depends on the interplay between soft-collision and cusp electrons, which produces a linear dependence of its position as a function of the projectile velocity.

We report the magneto-optical trapping of ytterbium atoms using the ${6s}^2{}^1{S}_0-6s6p{}^3{P}_1$ intercombination transition. The magneto-optical trap is continuously loaded from an atomic beam decelerated by a Zeeman-tuned atomic-beam slower operated at the ${6s}^2{}^1{S}_0-6s6p{}^1{P}_1$ transition. Among seven stable isotopes of ytterbium, six isotopes $\mathrm{of}{\hspace{0.5em}}^{170}\mathrm{Y}\mathrm{b},$ ${}^{171}\mathrm{Yb},$ ${}^{172}\mathrm{Yb},$ ${}^{173}\mathrm{Yb},$ ${}^{174}\mathrm{Yb},$ and ${}^{176}\mathrm{Yb}$ are successfully trapped. The number and density of the trapped atoms are about ${10}^8$ and ${10}^{11}\hspace{0.5em}{\mathrm{cm}}^{-3},$ respectively. The temperature of atoms is estimated to be about $20\hspace{0.5em}\mu \mathrm{K}$ from the time-of-flight measurement. The decay time of the trapped atoms after stopping the loading of atoms is about 3 s.

The problem of strong-field photoionization of a three-dimensional Rydberg hydrogen atom is solved numerically. The results are compared with those of the newest version of an analytical theory based on the quasiclassical approach. The probabilities of ionization calculated numerically and analytically are found to agree perfectly. The results obtained confirm the existence of strong-field interference stabilization interpreted in terms of Raman Λ-type transitions via the continuum.

This paper reports observations of Ramsey fringes on a 10.6-$\mu \mathrm{m}$ two-photon transition of ${\mathrm{SF}}_6.$ We used two interaction zones separated by 8 cm and observed fringes of frequency 2.3 kHz. The experimental apparatus might be adapted to increase the resolution by an order of magnitude. The present 10-$\mu \mathrm{m}$ secondary frequency standards based on the ${\mathrm{CO}}_2$ laser locked onto a saturated absorption resonance of ${\mathrm{OsO}}_4$ reach an accuracy of a few ${10}^{-13}.$ This work contributes to the current attempts to build a new generation of frequency standards with a potential accuracy in the ${10}^{-14}–{10}^{-15}$ range.

We have studied the effect of generative-pulse amplitude, spatial profile, and temporal character as well as medium optical thickness on the power of photon and stimulated echoes. Working in atomic Yb vapor, photon echoes are observed that, absent material relaxation, are more powerful than the first generative pulse, thereby exceeding typically expected photon-echo powers by approximately two orders of magnitude. Factors crucial to the generation of powerful echoes identified through Maxwell-Bloch simulation are experimentally confirmed.