Ford, Lewis, and O’Connell [Phys. Rev. A 64, 032101 (2001)] have recently discussed a thought experiment in which a Brownian particle is subjected to a double-slit measurement . Analyzing the decay of the emerging interference pattern, they derive a decoherence rate that is much faster than previous results and even persists in the limit of vanishing dissipation. This result is based on the definition of a certain attenuation factor, which they analyze for short times. In this note, we point out that this attenuation factor captures the physics of decoherence only for times larger than a certain time $t_{\mathrm{mix}}$, which is the time it takes until the two emerging wave packets begin to overlap. Therefore, the strategy of Ford et al. of extracting the decoherence time from the regime $t<t_{\mathrm{mix}}$ is in our opinion not meaningful. If one analyzes the attenuation factor for $t>t_{\mathrm{mix}}$, one recovers familiar behavior for the decoherence time; in particular, no decoherence is seen in the absence of dissipation. We confirm the latter conclusion by calculating the off-diagonal elements of the reduced density matrix.

While agreeing with our exact expression for the time dependence of the motion of a free particle in an initial superposition state, corresponding to two identical Gaussians separated by a distance $d$, at temperature $T$, Gobert et al., in the preceding Comment [Phys. Rev. A70, 026101 (2004)], dispute our conclusions on decoherence time scales. However, the parameters they used to generate their figures are outside the regime of validity of our interpretation of the results and, moreover, are not of physical interest in that they correspond to $T\approx 0$. The point is that in their figures they have chosen the thermal de Broglie wavelength ${\lambda }_{th}=\hslash \sqrt{mkT}$ to be equal to slit spacing $d$, whereas we have clearly stated [in the paragraph preceding Eq. (21) of our paper] that decoherence occurs and that our expression for the decoherence time applies only in the limit where $d$ is large compared not only with the slit width $\sigma$ but also with the thermal de Broglie wavelength, $d⪢{\lambda }_{th},\sigma$.

Why do we not see large macroscopic objects in entangled states? There are two ways to approach this question. The first is dynamic. The coupling of a large object to its environment cause any entanglement to decrease considerably. The second approach, which is discussed in this paper, puts the stress on the difficulty of observeing a large-scale entanglement. As the number of particles $n$ grows we need an ever more precise knowledge of the state and an ever more carefully designed experiment, in order to recognize entanglement. To develop this point we consider a family of observables, called witnesses, which are designed to detect entanglement. A witness $W$ distinguishes all the separable (unentangled) states from some entangled states. If we normalize the witness $W$ to satisfy $\mid \mathrm{tr}\left(W\rho \right)\mid ⩽1$ for all separable states $\rho$, then the efficiency of $W$ depends on the size of its maximal eigenvalue in absolute value; that is, its operator norm $\parallel W\parallel$. It is known that there are witnesses on the space of $n$ qubits for which $\parallel W\parallel$ is exponential in $n$. However, we conjecture that for a large majority of $n$-qubit witnesses $\parallel W\parallel ⩽O\left(\sqrt{n\log n}\right)$. Thus, in a nonideal measurement, which includes errors, the largest eigenvalue of a typical witness lies below the threshold of detection. We prove this conjecture for the family of extremal witnesses introduced by Werner and Wolf [Phys. Rev. A 64, 032112 (2001)].

We investigate the quantum theory of closed systems based on the linear positivity decoherence condition of Goldstein and Page. The objective of any quantum theory of a closed system, most generally the universe, is the prediction of probabilities for the individual members of sets of alternative coarse-grained histories of the system. Quantum interference between members of a set of alternative histories is an obstacle to assigning probabilities that are consistent with the rules of probability theory. A quantum theory of closed systems therefore requires two elements: (1) a condition specifying which sets of histories may be assigned probabilities and (2) a rule for those probabilities. The linear positivity condition of Goldstein and Page is the weakest of the general conditions proposed so far. Its general properties relating to exact probability sum rules, time neutrality, and conservation laws are explored. Its inconsistency with the usual notion of independent subsystems in quantum mechanics is reviewed. Its relation to the stronger condition of medium decoherence necessary for classicality is discussed. The linear positivity of histories in a number of simple model systems is investigated with the aim of exhibiting linearly positive sets of histories that are not decoherent. The utility of extending the notion of probability to include values outside the range of 0–1 is described. Alternatives with such virtual probabilities cannot be measured or recorded, but can be used in the intermediate steps of calculations of real probabilities. Extended probabilities give a simple and general way of formulating quantum theory. The various decoherence conditions are compared in terms of their utility for characterizing classicality and the role they might play in further generalizations of quantum mechanics.

The kinetic energy spectra of ${\mathrm{D}}^{+}$ ions resulting from the sequential double ionization of ${\mathrm{D}}_2$ molecules in an intense few-cycle laser field are calculated. The ${\mathrm{D}}_2$ molecule is first ionized at its equilibrium distance, followed by further ionization of $\mathrm{D}_2{}^{+}$ at larger internuclear separations. The time evolution of the nuclear wave packet is calculated including loss from ionization by the time-dependent laser field. The calculated ${\mathrm{D}}^{+}$ energy spectra are compared with the recent experiment, and the time duration between the two ionization events is deduced to subfemtosecond accuracy.

Bell inequalities for position measurements are derived using the bits of the binary expansion of position-measurement results. Violations of these inequalities are obtained from the output state of the nondegenerate optical parametric amplifier.

Corrections of order ${\alpha }^3{\left(Z\alpha \right)}^{5}m$ to the Lamb shift and corrections of order ${\alpha }^3\left(Z\alpha \right)E_F$ to the hyperfine splitting, generated by insertion of the three-loop one-particle reducible diagrams with radiative photons in the electron line, are calculated. The calculations are performed in the Yennie gauge.

The interaction of an atomic gas confined inside a cavity containing a strong electromagnetic field is numerically and theoretically investigated in a regime where recoil effects are not negligible. The spontaneous appearance of a density grating (atomic bunching) accompanied by the onset of a coherent, back-propagating electromagnetic wave is found to be ruled by a continuous phase transition. Numerical tests allow us to convincingly prove that the transition is steered by the appearence of a periodic atomic density modulation. Consideration of different experimental relaxation mechanisms induces us to analyze the problem in nearly analytic form, in the large detuning limit, using both a Vlasov approach and a Fokker-Planck description. The application of our predictions to recent experimental findings, reported by Kruse et al. [Phys. Rev. Lett., 91 183601 (2003)], yields a semiquantitative agreement with the observations.

Optimal identification (OI) is a recently developed procedure for extracting information about quantum Hamiltonians from experimental data. It employs techniques from coherent learning control to drive the quantum system such that dynamical measurements provide maximal information about its Hamiltonian. OI is an optimal procedure as initially presented; however, the data inversion component is computationally expensive. Here, we demonstrate that highly efficient global, nonlinear, map-facilitated inversion procedures can be combined with the OI concept to make it more suitable for laboratory implementation. A simulation of map-facilitated OI illustrates how the input-output maps can greatly accelerate the data inversion process.

We propose a scheme for creating atomic coherent superposition states via the technique of stimulated Raman adiabatic passage in a $\Lambda$-type system where the final state has twofold levels. With the application of a control field, it is found that the presence of double dark states leads to two arbitrary coherent superposition states with equal amplitude but inverse relative phases, even though the condition of multiphoton resonance is not met. In particular, two orthogonal maximal coherent superposition states are created when the control field is resonant with the transition of the twofold levels. This scheme can also be extended to manifold $\Lambda$-type systems.

O’Dell et al. in recent work [Phys. Rev. Lett. 90, 110402 (2003)] have proposed that laser-induced dipole-dipole interactions may drive a gaseous Bose-Einstein condensate to an instability against a periodic density modulation. We discuss the possibility that the instability is preceded by a first-order quantum phase transition to a modulated state.

We investigate the time evolution of some models with $N$ spins and pairwise couplings, for the case of large $N$, in order to compare evolution times with “speed-limit” minima derived in the literature. Both in a (symmetric) case with couplings of the same strength between each pair and in a case of broken symmetry, the times necessary for evolution to a state in which the simplest initial state has evolved into a nearly orthogonal state are proportional to $1∕N$, as is the speed limit time. However, the coefficient in the broken symmetry case comes much closer to the speed limit value. Introducing a different criterion for evolution speed, based on macroscopic changes in occupation, we find a corresponding enhancement in rates in the asymmetric case as compared to the symmetric case.

It is not always possible to distinguish multipartite orthogonal states if only local operations and classical communication (LOCC) are allowed. We prove that we cannot distinguish the states of an unextendible product basis by LOCC even with infinite resources (infinite-dimensional ancillas, infinite number of operations). Moreover we give a necessary and sufficient condition for the LOCC distinguishability of a complete product basis.

In this paper we investigate the effect of dephasing on proposed quantum gates for the solid-state Kane quantum computing architecture. Using a simple model of the decoherence, we find that the typical error in a controlled-NOT gate is $8.3\times {10}^{-5}$. We also compute the fidelities of $Z$, $X$, swap, and controlled $Z$ operations under a variety of dephasing rates. We show that these numerical results are comparable with the error threshold required for fault tolerant quantum computation.

We study Clary’s theory for an electron in the field of a dipolar molecule. We find for a molecule of rotor constant $B$ and dipole moment $\mu$ that Clary’s “rotationally adiabatic” molecular potential ${ϵ}_{ad}\left(r\right)$ behaves like $-64r^4∕3{\mu }^4$ near $r=0$ and like $-{\mu }^2∕6B{r}^4$ at infinity. We show, using general results about the spectrum of the Schrödinger Hamiltonian, that the salient features of the electron bound states spectrum, i.e., the existence of only a few weakly bound states, are a consequence of these asymptotic formulas. We argue that the long-range $(1∕r^4)$ part of the interaction may be interpreted as resulting from the angle-averaged molecular dipole potential at some rotation temperature. Our results should be contrasted with those of Camblong et al. Phys. Rev. Lett. 87, 220402 (2001), who claim that these features arise from the renormalization of the static point dipole interaction.

A semiclassical model is used to investigate oscillations of atomic fermions in a combined magnetic trap and one-dimensional optical lattice potential following axial displacement of the trap. The oscillations are shown to be strongly suppressed on the scale of the displacement, and to have a characteristic small amplitude, damped behavior in the collisionless regime. The presence of a separatrix in the semiclassical Brillouin-zone phase space is predicted and shown to produce strongly asymmetric phase-space distributions as a function of the trap displacement.

The Gross-Pitaevskii equation for a Bose-Einstein condensate confined in an elongated cigar-shaped trap is reduced to an effective system of nonlinear equations depending on only one space coordinate along the trap axis. The radial distribution of the condensate density and its radial velocity are approximated by Gaussian functions with real and imaginary exponents, respectively, with parameters depending on the axial coordinate and time. The effective one-dimensional system is applied to a description of the ground state of the condensate, to dark and bright solitons, to the sound and radial compression waves propagating in a dense condensate, and to weakly nonlinear waves in repulsive condensate. In the low-density limit our results reproduce the known formulas. In the high-density case our description of solitons goes beyond the standard approach based on the nonlinear Schrödinger equation. The dispersion relations for the sound and radial compression waves are obtained in a wide region of values of the condensate density. The Korteweg–de Vries equation for weakly nonlinear waves is derived and the existence of bright solitons on a constant background is predicted for a dense enough condensate with a repulsive interaction between the atoms.

We present a method of manipulating the Raman process by using a coherent control field which leads to splitting of the Raman gain peak into a doublet and anomalous dispersion in the region between the gain peaks. We show how the region of almost no Raman gain and strong anomalous dispersion is ideally suited for producing superluminal propagation. In particular, we show that the group index for a ${}^{23}\mathrm{Na}$ condensate could be in the range $-{10}^3$ to $-{10}^4$.

Correlated excitations in a plane circular configuration of identical atoms with parallel dipole moments are investigated. The collective energy eigenstates, which are formally identical to Frenkel excitons, can be computed together with their level shifts and decay rates by decomposing the atomic state space into carrier spaces for the irreducible representations of the symmetry group ${\mathbb{Z}}_N$ of the circle. It is shown that the index $p$ of these representations can be used as a quantum number analogously to the orbital angular momentum quantum number $l$ in hydrogenlike systems. Just as the hydrogen $s$ states are the only electronic wave functions which can occupy the central region of the Coulomb potential, the quasiparticle corresponding to a collective excitation of the atoms in the circle can occupy the central atom only for vanishing ${\mathbb{Z}}_N$ quantum number $p$. If a central atom is present, the $p=0$ state splits into two and shows level crossing at certain radii; in the regions between these radii, damped quantum beats between two “extreme” $p=0$ configurations occur. The physical mechanisms behind super- and subradiance at a given radius are discussed. It is shown that, beyond a certain critical number of atoms in the circle, the lifetime of the maximally subradiant state increases exponentially with the number of atoms in the configuration, making the system a natural candidate for a single-photon trap.

A quantum degenerate, dilute gas mixture of bosonic and fermionic atoms was produced using ${}^{87}\mathrm{Rb}$ and ${}^{40}\mathrm{K}$. The onset of degeneracy was confirmed by observing the spatial distribution of the gases after time-of-flight expansion. Furthermore, the magnitude of the interspecies scattering length between the doubly spin-polarized states of ${}^{87}\mathrm{Rb}$ and ${}^{40}\mathrm{K}$, $\mid a_{\mathrm{RbK}}\mid$, was determined from cross-dimensional thermal relaxation. The uncertainty in this collision measurement was greatly reduced by taking the ratio of interspecies and intraspecies relaxation rates, yielding $\mid a_{\mathrm{RbK}}\mid =250\pm 30a_0$, which is a lower value than what was reported in [M. Modugno et al., Phys. Rev. A 68, 043626 (2003)]. Using the value for $\mid a_{\mathrm{RbK}}\mid$ reported here, current $T=0$ theory would predict a threshold for mechanical instability that is inconsistent with the experimentally observed onset for sudden loss of fermions in [G. Modugno et al., Science 297, 2240 (2002)].