A detailed calculation of the $O\left(m{\alpha }^6\right)$ corrections to the energy levels of positronium for the orbital angular momentum number $l>0$ is presented. The formulas obtained are in agreement with the known results for the particular case of $l=1$. The complete analytical formulas to $O\left(m{\alpha }^6\right)$ are given for an arbitrary $l>0$ state, as well as numerical values of the energy levels $3{}^{2S+1}D_J$.

Multipartite Bell-type inequalities are derived for general systems. They involve up to eight observables with arbitrary spectra on each site. These inequalities are closely related to the algebras of quaternions and octonions.

Depolarizing maps acting on an $N$-dimensional system are completely positive maps resulting into compression of the Bloch “ball” along $N^2-1$ polarization directions. In the qubit case these maps are a convex sum of four extremal maps and form a simplex in the space of compression coefficients along the three polarization directions. We calculate the compression domain for three- and four-level systems. For a three-level system the region has curved surfaces, but it is a simplex for a four-level system. We conjecture that it is a simplex in the case of $2^n$-level systems.

The extension of the notion of quantum fidelity from the state-space to the operator level can be used to study environment-induced decoherence. The state-dependent operator fidelity susceptibility (OFS), the leading-order term for slightly different operator parameters, is shown to have a nontrivial behavior when the environment is at critical points. Two different contributions to the OFS are identified which have distinct physical origins and temporal dependence. Exact results are obtained for the finite-temperature decoherence caused by a bath described by the Ising model in a transverse field.

The $1s^{2}2p_{3∕2}–1s^{2}2s_{1∕2}$ x-ray transition in the Li-like ${\mathrm{Pb}}^{79+}$ has been observed using the Tokyo electron beam ion trap at the University of Electro-Communications. The contribution of QED $(-24.99\pm 0.10\mathrm{eV})$ to the $1s^{2}2p_{3∕2}–1s^{2}2s_{1∕2}$ transition in ${}^{208}\mathrm{Pb}^{79+}$ is determined and compared with different calculations. The measurement was made by calibrating the Pb line using the Lyman-$\alpha$ lines of the H-like $S$ and the $1s4p{}^{1}P_1–1s^2{}^{1}S_0$ transition in the He-like $P$.

Variational calculations for the energy levels of doubly excited $3dnf$ $\left({}^{1,3}D^{\circ }\right)$ states $(n=4–12)$ for helium have been performed using the extended Hylleraas basis set. No previous calculations for the energy eigen-values for the $3dnf$ $\left({}^{1,3}D^{\circ }\right)$ states for $(n=7–12)$ exist. Prediction of two-electron–one-photon peaks at 12.220, 12.649, 12.859, 12.978, 13.054, 13.110, 13.141, 13.168 and $13.213\mathrm{eV}$ corresponding to the transitions $2p^2$ $\left({}^{3}P^{\mathrm{e}}\right)\to 3dnf$ $\left({}^{3}D^{\circ }\right)$ $(n=4–12)$, respectively, is reported here. Highly precise energy eigenvalues of doubly excited $2pnd$ states $\left({}^{1,3}D^{\circ }\right)$ $(n=3–8)$ for helium are also reported and the upper bound energies are the lowest yet obtained. The effective quantum numbers $\left(n^{*}\right)$ of the above mentioned states are calculated using quantum defect theory.

We present an analytic calculation for linear susceptibility in pump-probe spectroscopy for a Doppler broadened atomic vapor. The analytic form of the population of each magnetic sublevel of a ${}^{87}\mathrm{Rb}$ atom, illuminated by a weak circularly polarized pump beam, was calculated for the lowest order of laser intensity and used in the calculation for the susceptibility of a counterpropagating (copropagating) probe beam. The obtained analytic results for the susceptibility can be used for calculating the absorption or the refractive index of the probe beam as functions of pump beam diameter and intensity.

We investigate the effects of rotation on the excited bands of a tight-binding lattice, focusing particularly on the first excited $\left(p\right)$ band. Both the on-site energies and the hopping between lattice sites are modified by the effective magnetic field created by rotation, causing a nontrivial splitting and magnetic fine structure of the $p$ band. We show that Peierls substitution can be modified to describe $p$ band under rotation, and use this method to derive an effective Hamiltonian. We compare the spectrum of the effective Hamiltonian with a first-principles calculation of the magnetic band structure and find excellent agreement, confirming the validity of our approach. We also discuss the on-site interaction terms for bosons and argue that many-particle phenomena in a rotating $p$ band can be investigated starting from this effective Hamiltonian.

We present exact numerical solutions in the form of spatially localized three-dimensional (3D) nonrotating and rotating (azimuthon) multipole solitons in the Bose-Einstein condensate (BEC) confined by a parabolic trap. We numerically show that the 3D azimuthon solutions exist as a continuous family parametrized by the angular velocity (or, equivalently, the modulational depth). By a linear stability analysis we show that 3D azimuthons with a sufficiently large phase modulational depth can be stable. The results are confirmed by direct numerical simulations of the Gross-Pitaevskii equation.

Solitons and necklaces in the first band gap of a two-dimensional optically induced honeycomb defocusing photonic lattice are theoretically considered. It is shown that dipoles, soliton necklaces, and vortex necklaces exist and may possess regions of stable propagation through a photorefractive crystal. Most of the configurations disappear in bifurcations close to the upper edge of the first band. Solutions associated with such bifurcations are also numerically examined, and it is found that they are often asymmetric and more exotic. The dynamics of the relevant unstable structures are also examined through direct numerical simulations revealing either breathing oscillations or, in some cases, destruction of the original wave form.

The local complementation rule is applied for continuous-variable (CV) graph states in the paper, which is an elementary graph transformation rule and successive application of which generates the orbit of any graph states. The corresponding local Gaussian transformations of local complementation for four-mode unweighted graph states were found, which do not mirror the form of the local Clifford unitary of qubit exactly. This work is an important step to characterize the local Gaussian equivalence classes of CV graph states.

The stationary and normally incident electromagnetic modes in Fibonacci lattices with generating layers of positive and negative indices of refraction are calculated by a transfer-matrix technique. It is shown that the condition for constructive interference of reflected waves is fulfilled when the ratio of optical paths in positive and negative media are given by the golden ratio. Furthermore, in the long-wavelength limit, it is demonstrated that the edges of the $⟨n⟩=0$ gap are the frequencies satisfying the conditions $⟨ϵ⟩=0$ and $⟨\mu ⟩=0$.

We have developed a theory of a Josephson junction formed by two tunnel-coupled Bose-Einstein condensates in a double-well potential in the regime of strong atom-atom interaction for an arbitrary total number $N$ of bosons in the condensates. The tunnel resonances in the junction are shown to be periodically spaced by the interaction energy, forming a single-atom staircase sensitive to the parity of $N$ even for large $N$. One of the manifestations of the staircase structure is the periodic modulation with the bias energy of the visibility of the interference pattern in lattices of junctions.

The Hartman effect is analyzed in both the position and momentum representations of the problem. The importance of Wigner tunneling and deep tunneling is singled out. It is shown quantitatively how the barrier acts as a filter for low momenta (quantum speed up) as the width increases, and a detailed mechanism is proposed. Superluminal transmission is also discussed.

We propose a scheme for deterministically generating three-dimensional (3D) entanglement between two distant five-level atoms based on the dispersive atom-field interaction. In our scheme, the two atoms are trapped separately in two spatially separated optical cavities coupled by an optical fiber. To check the experimental feasibility of our scheme, we numerically simulated effects of the atomic spontaneous decay and photon leakage out of the fiber, and the numerical simulation shows that those effects can be suppressed by appropriately choosing the frequency detuning of atom-field and the coupling intensity of cavity fiber, respectively. We also discussed the influence of photon leakage out of the cavities, and the strictly numerical simulation shows our proposal is good enough to demonstrate the generation of 3D entanglement with high fidelity and within the current experimental technology.

The Pólya number characterizes the recurrence of a random walk. We apply the generalization of this concept to quantum walks [M. Štefaňák et al., Phys. Rev. Lett. 100, 020501 (2008)] which is based on a specific measurement scheme. The Pólya number of a quantum walk depends, in general, on the choice of the coin and the initial coin state, in contrast to classical random walks where the lattice dimension uniquely determines it. We analyze several examples to depict the variety of possible recurrence properties. First, we show that for the class of quantum walks driven by Hadamard tensor-product coins, the Pólya number is independent of the initial conditions and the actual coin operators, thus resembling the property of the classical walks. We provide an estimation of the Pólya number for this class of quantum walks. Second, we examine the two-dimensional Grover walk, which exhibits localization and thus is recurrent, except for a particular initial state for which the walk is transient. We generalize the Grover walk to show that one can construct in arbitrary dimensions a quantum walk which is recurrent. This is in great contrast with classical walks which are recurrent only for the dimensions $d=1,2$. Finally, we analyze the recurrence of the 2D Fourier walk. This quantum walk is recurrent except for a two-dimensional subspace of the initial states. We provide an estimation of the Pólya number in its dependence on the initial state.

We report the experimental characterization of axialization—a method of reducing the magnetron motion of a small number of ions stored in a Penning trap. This is an important step in the investigation of the suitability of Penning traps for quantum-information processing. The magnetron motion was coupled to the laser-cooled modified cyclotron motion by the application of a near-resonant oscillating quadrupole potential (the “axialization drive”). Measurement of cooling rates of the radial motions of the ions showed an order-of-magnitude increase in the damping rate of the magnetron motion with the axialization drive applied. The experimental results are in good qualitative agreement with a recent theoretical study. In particular, a classical avoided crossing was observed in the motional frequencies as the axialization drive frequency was swept through the optimum value, proving that axialization is indeed a resonant effect.

We present time-resolved measurements of the laser-induced fluorescence from Cs atoms and ${\mathrm{Cs}}^{*}{\mathrm{He}}_N$ exciplexes in a solid ${}^4\mathrm{He}$ matrix. The linear ${\mathrm{Cs}}^{*}{\mathrm{He}}_{N=2}$ exciplex is strongly quenched by a radiationless transformation into a ring-shaped ${\mathrm{Cs}}^{*}{\mathrm{He}}_{N=6,7}$ exciplex, which decays radiatively with a lifetime of $88\pm 2\mathrm{ns}$. The analysis of this lifetime provides information on the formation rate of ${\mathrm{Cs}}^{*}{\mathrm{He}}_{N=6,7}$ and on perturbations of the electronic states of the Cs atom by the bound helium atoms.

The formalism of complex rotation of the radial coordinate is studied in the context of time-dependent systems. The applicability of this method is discussed and illustrated with numerical examples involving atoms exposed to electromagnetic field pulses. Complex rotation proves to be an efficient tool to obtain ionization probabilities and rates. Although, in principle, any information about the system may be obtained from the rotated wave function by transforming it back to its unrotated form, a good description of the ionized part of the wave function is generally subject to numerical challenges. It is, however, found that the combination of complex rotation and Floquet formalism offers an alternative and promising possibility to retrieve the physical information.

Based on a second-quantized representation of the nonrelativistic Hamiltonian of a system of $N$ electrons with pairwise Coulomb interactions, we demonstrate the exact statistical reduction of the $N$-body problem to a three-dimensional Schrödinger equation for the motion of a single active electron with all other $N-1$ electrons acting as spectators. As a by-product, three-dimensional Schrödinger equations for the ground and excited states of two-electron atoms and ions are derived and the dynamical role of Pauli’s exclusion principle is established. The classical limit $\hslash \to 0$ of the quantal all-electron equations is examined, and the Thomas-Fermi equation including the Amaldi correction is obtained.