The Friedel sum rule for Dirac fermions in arbitrary space dimensions is given. The density oscillation of the fermions is calculated. From the sum rule the Levinson theorem is proven to be determined by the phase shift at the Fermi energy and the critical point of zero momentum. Since the Friedel sum has been of major importance to understand the impurity scattering in many body systems, the present result is helpful in studying the general property of Dirac fermions in the presence of a barrier in any dimensional systems.

Robust quantum computation with $d$-level quantum systems (qudits) poses two requirements: fast, parallel quantum gates and high-fidelity two-qudit gates. We first describe how to implement parallel single-qudit operations. It is by now well known that any single-qudit unitary can be decomposed into a sequence of Givens rotations on two-dimensional subspaces of the qudit state space. Using a coupling graph to represent physically allowed couplings between pairs of qudit states, we then show that the logical depth (time) of the parallel gate sequence is equal to the height of an associated tree. The implementation of a given unitary can then optimize the tradeoff between gate time and resources used. These ideas are illustrated for qudits encoded in the ground hyperfine states of the alkali-metal atoms ${}^{87}\mathrm{Rb}$ and ${}^{133}\mathrm{Cs}$. Second, we provide a protocol for implementing parallelized nonlocal two-qudit gates using the assistance of entangled qubit pairs. Using known protocols for qubit entanglement purification, this offers the possibility of high-fidelity two-qudit gates.

We report on the calculated elastic differential, elastic integral, momentum transfer, and excitation cross sections for the low-energy electron-${\mathrm{NH}}_3$ scattering using the $R$-matrix method. Elastic differential and momentum transfer cross sections are obtained by summing over rotationally elastic and rotationally inelastic cross sections for rotor states up to $J=4$. The excitation cross sections of the first four low-lying electronically excited states from the ground state of ${\mathrm{NH}}_3$, at its equilibrium geometry, are presented. These excited states have symmetries $a{}^{3}A_1$, $A{}^{1}A_1$, $b{}^{3}E$, and $B{}^{1}E$. The set of self-consistent-field molecular orbitals is obtained by optimizing these on the first excited state ${}^{3}A_1$. Configuration-interaction (CI) wave functions are used to represent the target states. In our CI model, we kept the core two electrons frozen in doubly occupied molecular orbital $1a_1$, and the remaining eight electrons moved freely among the five molecular orbitals $2a_1$, $3a_1$, $1e$, $4a_1$, $2e$. With this CI model, we obtain good agreement for the vertical spectrum of excited states with the experimental values. The Born approximation is employed to account for higher partial waves excluded in the $R$-matrix method to evaluate elastic cross sections. Cross sections are reported for the electron-impact energy range 0.025–$20\mathrm{eV}$.

We report an ab initio investigation of the ionization of an oriented molecule $\left(\mathrm{H}_2{}^{+}\right)$ by intense linearly polarized laser pulses, using initial states (i.e., the active orbitals) having various symmetries and electron distributions. We show that the orientation dependence of the total (nondifferential) ionization probability of a molecule is very sensitive to the electron distribution within the active orbital. This sensitivity is strong enough that by collecting the ionization probability for various molecular orientations relative to the laser polarization, one can infer the electron distribution within the active orbital. The ionization of oriented molecules by intense low frequency fields can be a tool for imaging electron densities in active molecular orbitals.

We present the microscopic theory for the BEC-BCS crossover of an atomic Fermi gas in an optical lattice, showing that the Feshbach resonance underlying the crossover in principle induces strong multiband effects. Nevertheless, the BEC-BCS crossover itself can be described by a single-band model since it occurs at magnetic fields that are relatively far away from the Feshbach resonance. A criterion is proposed for the latter, which is obeyed by most known Feshbach resonances in ultracold atomic gases.

We demonstrate transport and evaporative cooling of several atomic clouds in a chain of magnetic Ioffe-Pritchard traps moving at a low speed $(<1\mathrm{m}∕\mathrm{s})$. The trapping scheme relies on the use of a magnetic guide for transverse confinement and of magnets fixed on a conveyor belt for longitudinal trapping. This experiment introduces a different approach for parallelizing the production of Bose-Einstein condensates as well as for the realization of a continuous atom laser.

We study a ring cavity filled with a slab of a right-handed material and a slab of a left-handed material. Both layers are assumed to be nonlinear Kerr media. First, we derive a model for the propagation of light in a left-handed material. By constructing a mean-field model, we show that the sign of diffraction can be made either positive or negative in this resonator, depending on the thicknesses of the layers. Subsequently, we demonstrate that the dynamical behavior of the modulation instability is strongly affected by the sign of the diffraction coefficient. Finally, we study the dissipative structures in this resonator and reveal the predominance of a two-dimensional up-switching process over the formation of spatially periodic structures, leading to the truncation of the homogeneous hysteresis cycle.

The scaling of the capacitance with radius is explored in detail for neutral atoms, and it is found that they behave much like macroscopic spherical capacitors. The quantum capacitances of atoms scale as a linear function of the mean radii of their highest occupied orbitals. The slopes of the linear scaling lines include a dimensionless constant of proportionality $\kappa$ that is somewhat analogous to a dielectric constant, but for individual atoms. The slope and $\kappa$ assume discrete values characteristic of elements in different regions of the periodic table. These observations provide a different, electrostatics-based way of understanding the periodic behavior of the elements.

We present a family of three-qubit quantum states with a basic local hidden-variable model. Any von Neumann measurement can be described by a local model for these states. We show that some of these states are genuine three-partite entangled and also distillable. The generalization for larger dimensions or higher number of parties is also discussed. As a by-product, we present symmetric extensions of two-qubit Werner states.

We focus on the nonlocality concerning local copying and local discrimination, especially for a set of orthogonal maximally entangled states in any prime dimensional system, as a study of nonlocality of a set of states. As a result, for such a set, we completely characterize deterministic local copiability and show that local copying is more difficult than local discrimination.

We examine the implications of several recently derived conditions [Hillery and Zubairy, Phys. Rev. Lett. 96, 050503 (2006)] for determining when a two-mode state is entangled. We first find examples of non-Gaussian states that satisfy these conditions. We then apply the entanglement conditions to the study of several linear devices, the beam splitter, the parametric amplifier, and the linear phase-insensitive amplifier. For the first two, we find conditions on the input states that guarantee that the output states are entangled. For the linear amplifier, we determine in the limit of high and no gain, when an entangled input leads to an entangled output. Finally, we show how application of two two-mode entanglement conditions to a three-mode state can serve as a test of genuine three-mode entanglement.

We present an improved ab initio time-dependent density-functional theory (TDDFT) approach to electronic excitations. A conventional TDDFT scheme within the local-density approximation (LDA) inaccurately predicts Rydberg and charge-transfer excitation energies, mainly because the electron-hole $\left(e\text{−}h\right)$ interaction is inappropriately described in these excitations, as can be found by analyzing the linear response formula [M. Petersilka, U. J. Gossmann, and E. K. U. Gross, Phys. Rev. Lett. 76, 1212 (1996)]. When the formula is averaged over the electron occupation, the inappropriate $e\text{−}h$ interaction within LDA is corrected to become explicitly similar to that of the exact exchange system. As anticipated from the similarity, our proposed scheme of modified linear response greatly improves the prediction of the problematic excitations, which are exemplified for typical molecules.

The density-matrix theory, based on Dirac’s relativistic equation, is applied for studying the entanglement between the photoelectron and residual ion in the course of the photoionization of atoms and ions. In particular, emphasis is placed on deriving the final-state density matrix of the overall system “$\text{photoion}+\text{electron}$,” including interelectronic effects and the higher multipoles of the radiation field. This final-state density matrix enables one immediately to analyze the change of entanglement as a function of the energy, angle and the polarization of the incoming light. Detailed computations have been carried out for the $5s$ photoionization of neutral strontium, leading to a photoion in a $5s{}^{2}S$ $J_f=1∕2$ level. It is found that the photoion-electron entanglement decreases significantly near the ionization threshold and that, in general, it depends on both the photon energy and angle. The possibility to extract photoion-electron pairs with a well-defined degree of entanglement may have far-reaching consequences for quantum information and elsewhere.

Elastic and inelastic collisions between $\mathrm{NH}$ molecules in the $(N=0,J=1)$ state were analyzed. These analyses showed that the elastic collision is caused mainly by electric dipole-induced dipole interaction, and the inelastic collision is caused mainly by the magnetic dipole-dipole interaction. The collision cross sections obtained by the quantum and classical methods are roughly in good agreement in the kinetic-energy region of $5–500\mathrm{mK}$. Moreover, the ratio of the inelastic-collision rate to the elastic-collision rate is lower than 1/200. These results show that the $\mathrm{NH}$ molecule is advantageous to perform evaporative cooling.

This work reports a theoretical analysis of the $\mathrm{Li}(2s\to 2p)$ photoabsorption spectra when the lithium atoms are evolving in a ground helium buffer gas. The influence of temperature on the far-line wing spectra is particularly examined by adopting classical and full quantum-mechanical approaches. The computations show the appearance, above approximately $1000\mathrm{K}$, of satellite structures in the blue wing around the wavelength $536\mathrm{nm}$. The data obtained in the range of temperature $500–3000\mathrm{K}$ agree very well with those already calculated with different theoretical methods.

The ionization distances $R_c^I$ as well as the ionization rates and eigenenergies of one-electron multiply charged Rydberg ions (core charge $Z⪢1$, principal quantum number $n⪢1$) approaching solid surfaces are calculated. Within the framework of a nonperturbative étalon equation method (EEM), these quantities are obtained simultaneously. The complex energy eigenvalue problem for the decaying eigenstates is solved within the critical region $R\approx R_c\approx R_c^I$ of the ion-surface distances $R$. This region is characterized by the energy terms localized in the vicinity of the top of an effective potential barrier, created between the ion and polarized solid. We take into account that the parabolic symmetry is preserved for $R\approx R_c$ and that the parabolic quantum numbers can be taken as approximate but sufficiently good quantum numbers. The parabolic rates, energies, and corresponding ionization distances are presented in relatively simple analytical forms. The ionization distances are compared with the results of a classical overbarrier model. Comparison of the obtained energies and rates with the available theoretical predictions of the coupled angular mode method shows good agreement. The use of the EEM for an estimation of the upper limit of the first neutralization distance in the subsequent neutralization cascade is briefly discussed.

We present a comprehensive analysis of the Landau-Zener tunneling of a nonlinear three-level system in a linearly sweeping external field. We find the presence of nonzero tunneling probability in the adiabatic limit (i.e., very slowly sweeping field) even for the situation that the nonlinear term is very small and the energy levels keep the same topological structure as that of the linear case. In particular, the tunneling is irregular with showing an unresolved sensitivity on the sweeping rate. For the case of fast-sweeping fields, we derive an analytic expression for the tunneling probability with stationary phase approximation and show that the nonlinearity can dramatically influence the tunneling probability when the nonlinear “internal field” resonate with the external field. We also discuss the asymmetry of the tunneling probability induced by the nonlinearity. Physics behind the above phenomena is revealed and possible application of our model to triple-well trapped Bose-Einstein condensate is discussed.

We investigate the quantum many-body dynamics of dissociation of a Bose-Einstein condensate of molecular dimers into pairs of constituent bosonic atoms and analyze the resulting atom-atom correlations. The quantum fields of both the molecules and atoms are simulated from first principles in three dimensions using the positive-$P$ representation method. This allows us to provide an exact treatment of the molecular field depletion and $s$-wave scattering interactions between the particles, as well as to extend the analysis to nonuniform systems. In the simplest uniform case, we find that the major source of atom-atom decorrelation is atom-atom recombination which produces molecules outside the initially occupied condensate mode. The unwanted molecules are formed from dissociated atom pairs with nonopposite momenta. The net effect of this process—which becomes increasingly significant for dissociation durations corresponding to more than about 40% conversion—is to reduce the atom-atom correlations. In addition, for nonuniform systems we find that mode mixing due to inhomogeneity can result in further degradation of the correlation signal. We characterize the correlation strength via the degree of squeezing of particle number-difference fluctuations in a certain momentum-space volume and show that the correlation strength can be increased if the signals are binned into larger counting volumes.

We investigate, both theoretically and experimentally, the phenomenon of polarization rotation of a weak, linearly polarized optical (probe) field in an atomic system with multiple three-level electromagnetically induced transparency (EIT) subsystems. The polarization rotation angle can be controlled by a circularly polarized coupling beam, which breaks the symmetry in number of EIT subsystems seen by the left and right circularly polarized components of the weak probe beam. A large polarization rotation angle (up to 45°) has been achieved with a coupling beam power of only $15\mathrm{mW}$. Detailed theoretical analyses including different transition probabilities in different transitions and Doppler-broadening are presented and the results are in good agreements with the experimentally measured results.

An infrared frequency standard based on the NH $\mid n_v=0,N_0=0,J=1,F_1=3∕2,F=5∕2,M_F=5∕2⟩\to \mid n_v=1,N_0=0,J=1,F_1=3∕2,F=5∕2,M_F=5∕2⟩$ transition $\left(3\mu \mathrm{m}\right)$ is proposed. There is no Zeeman shift with this transition frequency, because the Zeeman coefficient is not dependent on the vibrational state. NH molecules, precooled by collisions with liquid ${}^3\mathrm{He}$ vapor, are trapped by an inhomogeneous magnetic field. The temperature of the trapped NH molecules holds the potential to be reduced down to several $\mu \mathrm{K}$ by evaporative cooling. The uncertainty of the clock transition can potentially be reduced to lower than ${10}^{-17}$.