The question of how many entangled or, respectively, separable states there are in the set of all quantum states is considered. We propose a natural measure in the space of density matrices $\varrho$ describing $N$-dimensional quantum systems. We prove that, under this measure, the set of separable states possesses a nonzero volume. Analytical lower and upper bounds of this volume are also derived for $N=2\mathrm{}\times 2$ and $N=2\mathrm{}\times 3$ cases. Finally, numerical Monte Carlo calculations allow us to estimate the volume of separable states, providing numerical evidence that it decreases exponentially with the dimension of the composite system. We have also analyzed a conditional measure of separability under the condition of fixed purity. Our results display a clear dualism between purity and separability: entanglement is typical of pure states, while separability is connected with quantum mixtures. In particular, states of sufficiently low purity are necessarily separable.

Quantum robots and their interactions with environments of quantum systems are described, and their study justified. A quantum robot is a mobile quantum system that includes an on-board quantum computer and needed ancillary systems. Quantum robots carry out tasks whose goals include specified changes in the state of the environment, or carrying out measurements on the environment. Each task is a sequence of alternating computation and action phases. Computation phase activites include determination of the action to be carried out in the next phase, and recording of information on neighborhood environmental system states. Action phase activities include motion of the quantum robot and changes in the neighborhood environment system states. Models of quantum robots and their interactions with environments are described using discrete space and time. A unitary step operator $T$ that gives the single time step dynamics is associated with each task. ${T=T}_a{+T}_c$ is a sum of action phase and computation phase step operators. Conditions that $T_a$ and $T_c$ should satisfy are given along with a description of the evolution as a sum over paths of completed phase input and output states. A simple example of a task—carrying out a measurement on a very simple environment—is analyzed in detail. A decision tree for the task is presented and discussed in terms of the sums over phase paths. It is seen that no definite times or durations are associated with the phase steps in the tree, and that the tree describes the successive phase steps in each path in the sum over phase paths.

I report two general methods to construct quantum convolutional codes for $N$-state quantum systems. Using these general methods, I construct a quantum convolutional code of rate 1/4, which can correct one quantum error for every eight consecutive quantum registers.

The properties of a rf quadrupole trap, the elliptical ion trap, are derived. Elliptical traps can confine large numbers of ions in the Lamb-Dicke regime due to a hitherto unrecognized mechanism unique to one-dimensional Coulomb crystals, implicit in the theories of Dubin and Schiffer. This follows from a linear crystal stability condition, which uniquely relates the crystal size to ellipticity, and a micromotion relation, which reveals a 1/5-root dependence on the number of trapped ions. Elliptical traps offer several advantages over linear traps in the Cirac-Zoller model of quantum computing, both for initial tests and as a potential method of creating a full-scale quantum computer. Numerical solutions of a one-electrode structure show that microscopic elliptical traps, each containing $\approx 100$ ions, can be constructed at a density of 100 traps/${\mathrm{cm}}^2,$ making possible arrays containing $>{10}^6$ ions in the Lamb-Dicke regime for precision spectroscopy or quantum computation.

Many interesting computational problems can be reformulated in terms of decision trees. A natural classical algorithm is to then run a random walk on the tree, starting at the root, to see if the tree contains a node $n$ level from the root. We devise a quantum-mechanical algorithm that evolves a state, initially localized at the root, through the tree. We prove that if the classical strategy succeeds in reaching level $n$ in time polynomial in $n,$ then so does the quantum algorithm. Moreover, we find examples of trees for which the classical algorithm requires time exponential in $n,$ but for which the quantum algorithm succeeds in polynomial time. The examples we have so far, however, could also be solved in polynomial time by different classical algorithms.

Classical trajectory calculations are used to study the origin of chiral dynamics in the dissociation of structurally chiral molecules. A ratchet torsional potential about the chemical bond to be broken produces a net preference for one sense of product rotation about the recoil velocity vector. The dynamical chirality of these helical product trajectories is quantified using body-fixed bipolar moments that can be measured in gas-phase-photodissociation experiments. It is shown that the chiral dynamics is generated in the exit channel as the dissociation dynamics becomes irreversible, and is greatly accentuated by a late barrier to the dissociation reaction.

Electromagnetic Casimir energies are a quantum effect proportional to $ħ.$ We show that in certain cases one can obtain an exact semiclassical expression for them that depends only on periodic orbits of the associated classical problem. A great merit of the approach is that infinities never appear if one considers only periodic orbits that make contact with the boundary surface. This notion is made more precise by classifying the closed orbits in a phase space with boundaries and identifying the classes that contribute to Casimir effects. A semiclassical evaluation of the path integral gives a systematic expansion of the Casimir energy in terms of the lengths of classical periodic orbits. For some simple geometries the semiclassical expansion can be summed and explicitly shown to reproduce known results. This is the case, for example, for the force per unit area between parallel plates at a separation small compared to their linear dimensions. A more interesting example for our purposes is the closely related problem of the force on a conducting sphere arbitrarily close to a conducting wall. We provide a rigorous proof of Derjaguin’s result for the leading contribution to the force. The semiclassical approach, which has never been truly exploited in Casimir studies, is relatively simple and transparent, and should have a wide range of applications. The methods presented, however, do not apply to cases where diffraction is important; diffraction can, in principle, also be described within this semiclassical approach, but its implementation presents some technical problems. In cases where diffraction is important, conventional methods of calculating the Casimir energy may often be simpler.

The influence of the nuclear size and shape on quantum electrodynamic corrections to electron binding energies in heavy hydrogenlike ions is evaluated. For the self-energy, numerical values for the energy shift that take into account the size of the nucleus are given for ${1S}_{1/2}$-, ${2S}_{1/2}$-, and ${2P}_{1/2}$-states in the $Z$ range from 26 to 110. Simple parametrizations are derived for the dependence of the self-energy and the vacuum polarization on the nuclear radius, which allow corrections to be made for small deviations from the currently known nuclear radii. The influence of nuclear size on second-order quantum electrodynamic corrections is examined for the two-loop vacuum-polarization contributions. The contribution is found to be experimentally indistinguishable from other contributions to the energy shift which are of nonquantum electrodynamic origin.

We present a fully relativistic extension of the optimized-potential method (ROPM), including the transverse electron-electron interaction and vacuum corrections. Using perturbation theory on the basis of the Kohn-Sham Hamiltonian an exact representation of the relativistic exchange-correlation energy $E_{\mathrm{xc}}$ in terms of Kohn-Sham orbitals and eigenvalues is derived. The most simple, viable approximation to this $E_{\mathrm{xc}}$ is obtained by a second-order expansion in powers of $e^2,$ which leads to a Møller-Plesset-type correlation functional $E_{\mathrm{c}}^{\mathrm{}\left(2\right)\mathrm{}}.$ Due to this origin $E_{\mathrm{c}}^{\mathrm{}\left(2\right)\mathrm{}}$ allows a first-principles, seamless description of long-range dispersive forces. The ROPM integral equation that determines the full exchange-correlation four potential $v_{\mathrm{xc}}^{\mu }$ is presented, and specified in detail for $E_{\mathrm{c}}^{\mathrm{}\left(2\right)\mathrm{}}.$ We also analyze the Krieger-Li-Iafrate (KLI) approximation to the exact ROPM integral equation, pointing out an inherent ambiguity of the KLI approximation which arises for eigenvalue-dependent $E_{\mathrm{xc}}.$ The gauge properties of $E_{\mathrm{xc}}$ and the ROPM integral equation are discussed by examining the transversality of the Kohn-Sham current-current response function. It is demonstrated that due to the multiplicative nature of the total effective potential the density functional definition of the no-pair transverse exchange energy guarantees gauge invariance, in contrast to the relativistic Hartree-Fock scheme. On the other hand, the correlation energy is gauge dependent as soon as the no-pair approximation is applied. In addition, we show that the no-pair approximation automatically implies a definite intrinsic gauge for the spatial components of $v_{\mathrm{xc}}^{\mu }.$ The significance of the self-consistent treatment of the transverse interaction for heavy atoms is investigated numerically within the exchange-only limit. By comparing self-consistent with first-order perturbative inclusion of the transverse exchange it is shown that second-order transverse corrections cannot be neglected in calculations of ground state or inner-shell transition energies of heavy atoms, if one aims at spectroscopic accuracy. It is furthermore found that the Breit approximation for the full transverse interaction is not as accurate for the exchange potential as it is for the exchange energy. Finally, the KLI approximation is examined numerically, thereby resolving the ambiguity for the case of the transverse exchange.

We analyze the gauge dependence of the relativistic electron gas correlation energy $E_{\mathrm{c}}$ resulting from the no-pair approximation. In particular, we evaluate the relativistic no-pair Lindhard function as the basic ingredient of the random-phase approximation (RPA) for $E_{\mathrm{c}}.$ The resulting gauge-dependent no-pair RPA is compared with a gauge-invariant counterpart. The implications for relativistic density functionals are examined by applying the local-density approximation based on each of these forms of $E_{\mathrm{c}}$ to atoms. It is found that the gauge dependence of $E_{\mathrm{c}}$ is irrelevant on the overall level of accuracy obtained with present density functionals.

Using a unique method, we create coherent states in an atom by periodically (∼100–200 MHz) applying a low-amplitude, ∼5-V/cm, ns electric-field pulse that drives the atoms to a realm where many stationary states lie within a very narrow energy range. Using a cw laser we excite a coherent superposition of these coherent states and probe the attributes of such a state in the frequency domain (rather than in the time domain). We show that such coherent superpositions exhibit localized electronic probability densities and, therefore, qualify as wave packets.

We examine photoassociation by using a quasicontinuum to describe the colliding atoms. The quasicontinuum system is analyzed using methods adapted from the theory of laser spectroscopy and quantum optics, and a continuum limit is then taken. In a degenerate gas the equilibrium probability of photoassociation may be close to unity. In the continuum limit, for a thermal atomic sample, the stimulated Raman adiabatic passage (STIRAP) mechanism cannot be employed to eliminate unwanted spontaneous transitions.

A nonperturbative analytical semiclassical approach describing the interaction of a quantum system with strong oscillating fields is presented, including the limit where the external high-frequency field destroys the classical trajectories of a field-free system. Applied to ionization of a Rydberg atom, our approach allows us to describe the so-called “interference” and “adiabatic” mechanisms of laser-induced stabilization of atomic Rydberg states in a unified way.

A Raman photoassociation process can be used to form translationally cold molecules from a Bose-Einstein condensate. Coupled-channels scattering calculations show that the rate coefficient for coherent production of molecules in specific target states, $>{10}^{-10}\hspace{0.5em}{\mathrm{cm}}^3/\mathrm{s},$ can greatly exceed the rate coefficient for heating and loss channels due to excited-state spontaneous decay. This implies the feasibility of rapid coherent conversion of atom pairs in a condensate to molecules on a time scale short compared to the trap oscillation time.

We discuss a superfluid phase transition in a trapped neutral-atom Fermi gas. We consider the case where the critical temperature greatly exceeds the spacing between the trap levels and derive the corresponding Ginzburg-Landau equation. The latter turns out to be analogous to the equation for the condensate wave function in a trapped Bose gas. The analysis of its solution provides us with the value of the critical temperature $T_c$ and with the spatial and temperature dependence of the order parameter in the vicinity of the phase transition point.

We introduce an ansatz that permits a treatment of the effects of Doppler broadening in the theory of intense steady-state pulse pairs propagating in three-level (Λ or V) media. We have derived analytic solutions for pulse amplitudes, for group and phase velocities, and for the probability amplitudes of the atoms in the transmission medium. The solutions are different for the Λ and V cases.

We establish a nonlinear optical cavity to be excitable. Excitability in this system is shown to originate from the combined dynamical effects of nonlinear intracavity field saturation and temperature-dependent absorption in the medium on two different time scales. The model may be experimentally realized using optical bistable devices with possible useful applications.

A measuring apparatus is described by quantum mechanics while it interacts with the quantum system under observation and then it must be given a classical description so that the result of the measurement appears as objective reality. Alternatively, the apparatus may always be treated by quantum mechanics and be measured by a second apparatus, which has such a dual description. This article examines whether these two different descriptions are mutually consistent. It is shown that if the dynamical variable used in the first apparatus is represented by an operator of the Weyl-Wigner type (for example, if it is a linear coordinate), then the conversion from quantum to classical terminology does not affect the final result. However, if the first apparatus encodes the measurement in a different type of operator (e.g., the phase operator), the two methods of calculation may give different results.

I utilize the Caves-Milburn model for continuous position measurements to formulate a broadband version of the standard quantum limit (SQL) for monitoring the position of a free mass and illustrate the use of Kalman filtering to recover the SQL for estimating a weak classical force that acts on a quantum-mechanical test particle under continuous observation. These derivations are intended to clarify the interpretation of SQL’s in the context of broadband quantum measurement, with particular attention paid to the question of how it might be possible to verify that a given laboratory measurement does indeed achieve backaction-limited sensitivity. The method used to analyze force detection may be extended to the case of weak classical forces with arbitrary time dependence, including nonstationary and impulsive signals.

Simple Hartree-type equations lead to dynamics of a subsystem that is not completely positive in the sense accepted in mathematical literature. In the linear case this would imply that negative probabilities have to appear for some system that contains the subsystem in question. In the nonlinear case this does not happen because the mathematical definition is physically unfitting as shown in a concrete example.