This Comment discusses a paper by ], a Comment thereon by ], and an Erratum to that Comment []. An important further misprint beyond those cited by Sims and Hagstrom is identified, and misprints in the prior Comment (and not corrected in its Erratum) are pointed out. Calculations are performed to confirm that, with the corrections found to date, all Remiddi’s integral formulas are now in a consistent, correct form.

Bound $4d$ inner-shell excited states have been observed as $4d^{10}5s^{2}5p^5\to 4d^{9}5s^{2}5p^6{}^{2}D_{5∕2,3∕2}$ Feshbach resonances in the photodetachment spectrum of ${\text{Te}}^{-}$ utilizing the merged-beam technique at the storage ring ASTRID. The strong binding of these core-excited levels, 2.95 and $1.47\text{eV}$ respectively, is attributed to the extra stability of the full $5p$ shell. The role of the present data as a prototype spectrum for negative ions having an outer shell with a single vacancy in the initial state is discussed.

We investigate the gauge invariance of the leading logarithmic radiative correction to the two-photon decay width in hydrogenlike atoms. It is shown that an effective treatment of the correction using a Lamb-shift “potential” leads to equivalent results in both the length and velocity gauges provided all relevant correction terms are taken into account. Specifically, the relevant radiative corrections are related to the energies that enter into the propagator denominators, to the Hamiltonian, to the wave functions, and to the energy conservation condition that holds between the two photons; the form of all of these effects is different in the two gauges, but the final result is shown to be gauge invariant, as it should be. Although the actual calculation only involves integrations over nonrelativistic hydrogenic Green functions, the derivation of the leading logarithmic correction can be regarded as slightly more complex than that of other typical logarithmic terms. The dominant radiative correction to the $2S$ two-photon decay width is found to be $-2.020536(\alpha ∕\pi )\left(Z\alpha {)}^2\ln \right[\left(Z\alpha {)}^{-2}\right]$ in units of the leading nonrelativistic expression. This result is in agreement with a length-gauge calculation [S. G. Karshenboim and V. G. Ivanov, e-print physics∕9702027], where the coefficient was given as $-2.025\left(1\right)$.

We investigate the average bipartite entanglement, over all possible divisions of a multipartite system, as a useful measure of multipartite entanglement. We expose a connection between such measures and quantum-error-correcting codes by deriving a formula relating the weight distribution of the code to the average entanglement of encoded states. The multipartite entangling power of quantum evolutions is also investigated.

Entanglement is defined for each vector subspace of the tensor product of two finite-dimensional Hilbert spaces, by applying the notion of operator entanglement to the projection operator onto that subspace. The operator Schmidt decomposition of the projection operator defines a string of Schmidt coefficients for each subspace, and this string is assumed to characterize its entanglement, so that a first subspace is more entangled than a second, if the Schmidt string of the second majorizes the Schmidt string of the first. The idea is applied to the antisymmetric and symmetric tensor products of a finite-dimensional Hilbert space with itself, and also to the tensor product of an angular momentum $j$ with a spin $1∕2$. When adapted to the subspaces of states of the nonrelativistic hydrogen atom with definite total angular momentum (orbital plus spin), within the space of bound states with a given total energy, this leads to a complete ordering of those subspaces by their Schmidt strings.

As a continuation of our previous work [Phys. Rev. A 68, 012504 (2003)] an accurate study of the lowest $1{\sigma }_g$ and the low-lying excited $1{\sigma }_u$, $2{\sigma }_g$, $1{\pi }_{u,g}$, $1{\delta }_{g,u}$ electronic states of the molecular ion $\mathrm{H}_2{}^{+}$ is made. Since the parallel configuration where the molecular axis coincides with the magnetic field direction is optimal, this is the only configuration which is considered. The variational method is applied and the same trial function is used for different magnetic fields. The magnetic field ranges from ${10}^9$ to $4.414\times {10}^{13}\mathrm{G}$, where nonrelativistic considerations are justified. Particular attention is paid to the $1{\sigma }_u$ state which was studied for an arbitrary inclination. For this state a one-parameter vector potential is used which is then variationally optimized.

The narrowing of electron and ion wave packets in the process of photoionization is investigated, with the electron-ion recoil taken fully into account. Packet localization of this type is directly related to entanglement in the joint quantum state of the electron and ion, and to Einstein-Podolsky-Rosen localization. Experimental observation of such packet-narrowing effects is suggested via coincidence registration by two detectors, with a fixed position of one and varying position of the other. A similar effect, typically with an enhanced degree of entanglement, is shown to occur in the case of photodissociation of molecules.

Quantum-information theory predicts that when the transmission resource is doubled in quantum channels, the amount of information transmitted can be increased more than twice by quantum-channel coding technique, whereas the increase is at most twice in classical information theory. This remarkable feature, the superadditive quantum-coding gain, can be implemented by appropriate choices of code words and corresponding quantum decoding which requires a collective quantum measurement. Recently, an experimental demonstration was reported [M. Fujiwara et al., Phys. Rev. Lett. 90, 167906 (2003)]. The purpose of this paper is to describe our experiment in detail. Particularly, a design strategy of quantum-collective decoding in physical quantum circuits is emphasized. We also address the practical implication of the gain on communication performance by introducing the quantum-classical hybrid coding scheme. We show how the superadditive quantum-coding gain, even in a small code length, can boost the communication performance of conventional coding techniques.

We investigate the atomic four-level $N$ configuration both analytically and numerically, for various pump and probe intensities, with and without transfer of coherence (TOC) and Doppler broadening, and compare the results obtained to those of realistic atomic systems. We find that TOC affects the whole spectrum, in addition to producing an electromagnetically induced absorption (EIA) peak at line center. We show that the EIA peak splits as the pump intensity increases. These results are compared with those of realistic systems. When the pump is ${\sigma }^{+}$ polarized and the probe is $\pi$ polarized, the results are similar to those of the $N$ configuration. When the pump and probe polarizations are both linear with perpendicular polarizations, various $N$-like subsystems contribute to the spectrum. Consequently, the splitting of the EIA peak only occurs at very high pump intensities. We also discuss the influence of the probe on the pump absorption and refraction and find that both the pump and probe show EIA peaks when the pump intensity is low, and complementary behavior when the pump is intense. At both low and high pump intensity, the pump and probe dispersions are of opposite sign.

Searches for permanent electric-dipole moments (EDM) of atoms provide important constraints on competing extensions to the standard model of elementary particles. Recently proposed experiment with liquid ${}^{129}\mathrm{Xe}$ [M.V. Romalis and M.P. Ledbetter, Phys. Rev. Lett. 87, 067601 (2001)] may significantly improve present limits on the EDMs. To interpret experimental data in terms of CP-violating sources, one must relate measured atomic EDM to various model interactions via electronic-structure calculations. Here we study density dependence of atomic EDMs. The analysis is carried out in the framework of the cell model of the liquid coupled with relativistic atomic-structure calculations. We find that compared to an isolated atom, the EDM of an atom of liquid Xe is suppressed by about $40\%$.

Quantum logic operations can be performed using linear optical elements, additional photons (ancilla photons), and postselection based on measurements made on the ancilla. Here we describe a method for generating the required entangled state of $n$ ancilla photons using elementary logic gates and postselection. This approach is capable of generating the ancilla states required for either the original proposal by Knill, Laflamme, and Milburn [E. Knill, R. Laflamme, and G. J. Milburn, Nature 409, 46 (2001)] or those required for the more general high-fidelity approach [J. D. Franson, M. M. Donegan, M. J. Fitch, B. C. Jacobs, and T. B. Pittman, Phys. Rev. Lett. 89, 137901 (2002)]. We also show that the entangled ancilla photons could be generated using a series of quantum wells coupled with tunnel junctions.

In density functional theory, the adiabatic coupling may be treated perturbatively. This Görling-Levy perturbation theory can be applied to both nondegenerate and degenerate states. Formulas are given to second order. Some excitations are shown to be poorly described to first order (i.e., in exact exchange), no matter how weakly correlated the system is. Results are demonstrated on a simple model system, two fermions in a one-dimensional parabolic well with a contact interaction.

Energies of $n{s}_{1∕2}\left(n=6\text{−}9\right)$, $n{p}_j\left(n=6\text{−}8\right)$, $n{d}_j\left(n=6\text{−}7\right)$, and $5f_j$ states in neutral Au and Au-like ions with nuclear charges $Z=80–83$ are calculated using relativistic many-body perturbation theory. Reduced matrix elements, oscillator strengths, transition rates and lifetimes are determined for the 30 possible $n{l}_j-n^{\prime }{l}_{j^{\prime }}^{\prime }$ electric-dipole transitions. Results for a limited number states $n{s}_{1∕2}$, $n{p}_j\left(n=6–7\right)$ and $6d_j$ are obtained in the relativistic single-double (SD) approximation, where single and double excitations of Dirac-Fock wave functions are included to all orders in perturbation theory. Using SD wave functions, accurate values are obtained for energies of the eight lowest states and for the fourteen possible electric-dipole matrix elements between these states. With the aid of the SD wave functions, we also determine transition rates and oscillator strengths for the fourteen transitions together with lifetimes of $6p_j$, $7p_j$, and $6d_j$ levels. We investigate the hyperfine structure in $\text{Hg}\text{II}$ and $\text{Tl}\text{III}$. The hyperfine $A$ values are determined for $6s_{1∕2}$ and $6p_j$ states in ${}^{199}\mathrm{Hg}^{+}$, ${}^{201}\mathrm{Hg}^{+}$, and ${}^{205}\mathrm{Tl}^{2+}$ isotopes. These calculations provide a theoretical benchmark for comparison with experiment and theory.

An experimental and theoretical study of laser sideband cooling of a trapped ion in a radiofrequency trap is presented. The influence of the micromotion in the time-dependent potential of a Paul trap can lead to a counterintuitive situation, where sideband cooling is possible for positive values of the laser detuning, i.e., for a laser frequency that is higher than the resonance frequency of the ion at rest. The cooling rate and the capture range are calculated in a semiclassical model. Experimental results of laser sideband cooling at positive detuning are demonstrated with a single trapped ${\mathrm{In}}^{+}$ ion stored in a miniature Paul trap.

We consider a three-dimensional weakly interacting Bose gas in the fluctuation region (and its vicinity) of the normal-fluid–superfluid phase transition point. We establish relations between basic thermodynamic functions: density $n(T,\mu )$, superfluid density $n_s(T,\mu )$, and condensate density $n_{\text{cnd}}(T,\mu )$. Being universal for all weakly interacting ${\mid \psi \mid }^4$ systems, these relations are obtained from Monte Carlo simulations of the classical ${\mid \psi \mid }^4$ model on a lattice. Comparing with the mean-field results yields a quantitative estimate of the fluctuation region size. Away from the fluctuation region, on the superfluid side, all the data perfectly agree with the predictions of the quasicondensate mean-field theory. This demonstrates that the only effect of the leading above-the-mean-field corrections in the condensate-based treatments is to replace the condensate density with the quasicondensate one in all local thermodynamic relations. Surprisingly, we find that a significant fraction of the density profile of a loosely trapped atomic gas might correspond to the fluctuation region.

The interference of photons with the same polarization generated by spontaneous parametric down conversion in a Hong-Ou-Mandel interferometer, after one of them passed through a symmetric cavity, is demonstrated experimentally. In the regime where the cavity length is larger than the one-photon coherence length, photon coalescence and anticoalescence interference is observed. We show that by changing the cavity length, coincidence peaks can be transformed in dips and vice versa, even though the coherence length of the photon wave packet is smaller than the cavity length. The photon pair coalescence or anticoalescence at the beam splitter can be controlled by changing the length of the cavity.

Stochastic Schrödinger equations for quantum trajectories offer an alternative and sometimes superior approach to the study of open quantum system dynamics. Here we show that recently established convolutionless non-Markovian stochastic Schrödinger equations may serve as a powerful tool for the derivation of convolutionless master equations for non-Markovian open quantum systems. The most interesting example is quantum Brownian motion (QBM) of a harmonic oscillator coupled to a heat bath of oscillators, one of the most employed exactly soluble models of open system dynamics. We show explicitly how to establish the direct connection between the exact convolutionless master equation of QBM and the corresponding convolutionless exact stochastic Schrödinger equation.

Continuous observation of a quantum system yields a measurement record that faithfully reproduces the classically predicted trajectory provided that the measurement is sufficiently strong to localize the state in phase space but weak enough that quantum backaction noise is negligible. We investigate the conditions under which classical dynamics emerges, via a continuous position measurement, for a particle moving in a harmonic well with its position coupled to internal spin. As a consequence of this coupling, we find that classical dynamics emerges only when the position and spin actions are both large compared to $\hslash$. These conditions are quantified by placing bounds on the size of the covariance matrix which describes the delocalized quantum coherence over extended regions of phase space. From this result, it follows that a mixed quantum-classical regime (where one subsystem can be treated classically and the other not) does not exist for a continuously observed spin-$\frac{1}{2}$ particle. When the conditions for classicality are satisfied (in the large-spin limit), the quantum trajectories reproduce both the classical periodic orbits as well as the classically chaotic phase space regions. As a quantitative test of this convergence, we compute the largest Lyapunov exponent directly from the measured quantum trajectories and show that it agrees with the classical value.

The minimum requirements for entanglement detection are discussed for a spin chain in which the spins cannot be individually accessed. The methods presented detect entangled states close to a cluster state and a many-body singlet state, and seem to be viable for experimental realization in optical lattices of two-state bosonic atoms. The entanglement criteria are based on entanglement witnesses and on the uncertainty of collective observables.

Improving perturbation theory via a variational optimization has generally produced in higher orders an embarrassingly large set of solutions, most of them unphysical (complex). We introduce an extension of the optimized perturbation method which leads to a drastic reduction of the number of acceptable solutions. The properties of this method are studied and it is then applied to the calculation of relevant quantities in different ${\varphi }^4$ models, such as the anharmonic oscillator energy levels and the critical Bose-Einstein condensation temperature shift $\Delta T_c$ recently investigated by various authors. Our present estimates of $\Delta T_c$, incorporating the most recently available six and seven loop perturbative information, are in excellent agreement with all the available lattice numerical simulations. This represents a very substantial improvement over previous treatments.