The exact explicit expression for the probability density $p_N(x)$ for a sum of $N$ random, arbitrary correlated summands is obtained. The expression is valid for any number $N$ and any distribution of the random summands. Most attention is paid to application of the developed approach to the case of independent and identically distributed summands. The obtained results reproduce all known exact solutions valid for the, so called, stable distributions of the summands. It is also shown that if the distribution is not stable, the profile of $p_N(x)$ may be divided into three parts, namely a core (small $x$), a tail (large $x$), and a crossover from the core to the tail (moderate $x$). The quantitative description of all three parts as well as that for the entire profile is obtained. A number of particular examples are considered in detail.

We show that a free quantum particle in two dimensions with zero angular momentum ($s$ wave) in the form of a ring-shaped wave packet feels an attraction towards the center of the ring, leading first to a contraction followed by an expansion. An experiment to demonstrate this effect is also outlined.

Previous results indicate that while chaos can lead to substantial entropy production, thereby maximizing dynamical entanglement, this still falls short of maximality. Random matrix theory modeling of composite quantum systems, investigated recently, entails a universal distribution of the eigenvalues of the reduced density matrices. We demonstrate that these distributions are realized in quantized chaotic systems by using a model of two coupled and kicked tops. We derive an explicit statistical universal bound on entanglement, which is also valid for the case of unequal dimensionality of the Hilbert spaces involved, and show that this describes well the bounds observed using composite quantized chaotic systems such as coupled tops.

We study the decay of a two-component Bose-Einstein condensate with negative effective interaction energy. With a decreasing atom number due to losses, the atom-atom interaction becomes less important and the system undergoes a transition from a bistable Josephson regime to the monostable Rabi regime, displaying oscillations in phase and number. We study the equations of motion and derive an analytical expression for the oscillation amplitude. A quantum trajectory simulation reveals that the classical description fails for low loss rates, as expected from analytical considerations. Observation of the proposed effect will provide evidence for negative effective interaction.

We consider a particle moving in a one-dimensional potential which has a symmetric deterministic part and a quenched random part. We study analytically the probability distributions of the local time (spent by the particle around its mean value) and the occupation time (spent above its mean value) within an observation time window of size $t$. In the absence of quenched randomness, these distributions have three typical asymptotic behaviors depending on whether the deterministic potential is unstable, stable, or flat. These asymptotic behaviors are shown to get drastically modified when the random part of the potential is switched on, leading to the loss of self-averaging and wide sample to sample fluctuations.

We derive $N$-particle Bell-type inequalities under the assumption of partial separability, i.e., that the $N$-particle system is composed of subsystems which may be correlated in any way (e.g., entangled) but which are uncorrelated with respect to each other. These inequalities provide, upon violation, experimentally accessible sufficient conditions for full $N$-particle entanglement, i.e., for $N$-particle entanglement that cannot be reduced to mixtures of states in which a smaller number of particles are entangled. The inequalities are shown to be maximally violated by the $N$-particle Greenberger-Horne-Zeilinger states.

We propose a simple scheme for generating rotating atomic clusters in an optical lattice which produces states with quantum Hall and spin liquid properties. As the rotation frequencies increase, the ground state of a rotating cluster of spin-1 Bose atoms undergoes a sequence of (spin and orbit) transitions, which terminates at an angular momentum $L^{*}$ substantially lower than that of the boson Laughlin state. The spin-orbit correlations reflect “fermionization” of bosons facilitated by their spin degrees of freedom. We also show that the density of an expanding group of clusters has a scaling form which reveals the quantum Hall and spin structure of a single cluster.

In the limit when the two-body scattering length $a$ is negative and much larger than the effective two-body interaction radius the contribution to the ground state energy due to the three-body correlations is given by the Efimov effect. For particular values of the diluteness parameter $\rho |a{|}^3$ the three-body contribution can become the dominant term of the energy density functional. Under these conditions both Bose-Einstein and Fermi-Dirac systems could become self-bound and either boson droplets or fermion “designer nuclei” of various sizes and densities could be manufactured.

We propose two experimentally feasible methods based on atom interferometry to measure the quantum state of the kicked rotor.

We experimentally demonstrate the fluctuation theorem, which predicts appreciable and measurable violations of the second law of thermodynamics for small systems over short time scales, by following the trajectory of a colloidal particle captured in an optical trap that is translated relative to surrounding water molecules. From each particle trajectory, we calculate the entropy production/consumption over the duration of the trajectory and determine the fraction of second law–defying trajectories. Our results show entropy consumption can occur over colloidal length and time scales.

Gaseous Bose-Einstein condensates of $2–3\times {10}^6{}^{23}\mathrm{Na}$ atoms were loaded into a microfabricated magnetic trap using optical tweezers. Subsequently, the condensates were released into a magnetic waveguide and propagated 12 mm. Single-mode propagation was observed along homogeneous segments of the waveguide. Inhomogeneities in the guiding potential arose from geometric deformations of the microfabricated wires and caused strong transverse excitations. Such deformations may restrict the waveguide physics that can be explored with propagating condensates. Finer perturbations to the guiding potential fragmented the condensate when it was brought closer to the surface.

We propose the creation of a molecular Bose-Einstein condensate by loading an atomic condensate into an optical lattice and driving it into a Mott insulator with exactly two atoms per site. Molecules in a Mott insulator state are then created under well defined conditions by photoassociation with essentially unit efficiency. Finally, the Mott insulator is melted and a superfluid state of the molecules is created. We study the dynamics of this process and photoassociation of tightly trapped atoms.

In the evolution of distributions localized around classical equilibrium points, the quantum-classical correspondence breaks down at a time, the so-called quantum breaking, or Ehrenfest time, which is related to the minimal separation of the quantum levels in proximity of the classical equilibrium energy. By studying one-dimensional systems with single- and double-well polynomial potentials, we find that the Ehrenfest time diverges logarithmically with the inverse of the Planck constant whenever the equilibrium point is exponentially unstable. In all the other cases, we have a power law divergence with the exponent determined by the degree of the potential near the equilibrium point.

A Bose-Einstein condensate trapped in a two-dimensional optical lattice exhibits an abrupt transition manifested by the macroscopic wave function changing character from spatially localized to extended. Resulting from a bifurcation, this irreversible transition takes place as the interwell potential barrier is adiabatically decreased below a critical value. This is in sharp contrast to the corresponding one-dimensional case where such a bifurcation is absent and the extent of a localized mode is continuously tunable. We demonstrate how these phenomena can be experimentally explored.

We demonstrate that quantum fluctuations suppress Bose-Einstein condensation of quasi-two-dimensional bosons in a rapidly rotating trap. Our conclusions rest in part on the derivation of an exact expression for the boson action in terms of vortex position coordinates, and in part on a solution of the weakly interacting boson Bogoliubov equations, which simplify in the rapid-rotation limit. We obtain analytic expressions for the collective-excitation dispersion, which is quadratic rather than linear. Our estimates for the boson filling factor at which the vortex lattice melts are consistent with recent exact-diagonalization calculations.

We obtain the exact probability $\exp [-LF(\{\rho \left(x\right)\}\left)\right]$ of finding a macroscopic density profile $\rho \left(x\right)$ in the stationary nonequilibrium state of an open driven diffusive system, when the size of the system $L\to \infty$. $F$, which plays the role of a nonequilibrium free energy, has a very different structure from that found in the purely diffusive case. As there, $F$ is nonlocal, but the shocks and dynamic phase transitions of the driven system are reflected in nonconvexity of $F$, in discontinuities in its second derivatives, and in non-Gaussian fluctuations in the steady state.

The Mermin-Ho and Anderson-Toulouse coreless vortices are demonstrated to be thermodynamically stable in ferromagnetic $F=1\mathrm{}$ spinor Bose-Einstein condensates under rotation. We have carried out extensive calculations of the Gross-Pitaevskii equations by assuming uniform density along the $z$ axis and comparing the energies of other competing non-axis-symmetric or singular vortices. The phase diagram is thereby established in a plane of the rotation drive vs the total magnetization. Their stability is also checked by calculating collective modes based on the Bogoliubov equations.

By colliding two Bose-Einstein condensates, we have observed strong bosonic stimulation of the elastic scattering process. When a weak input beam was applied as a seed, it was amplified by a factor of 20. This large gain atomic four-wave mixing resulted in the generation of two macroscopically occupied pair-correlated atomic beams.

An initially stable ${}^{85}\mathrm{Rb}$ Bose-Einstein condensate (BEC) was subjected to a carefully controlled magnetic field pulse near a Feshbach resonance. This pulse probed the strongly interacting regime for the BEC, with the diluteness parameter $\left({\mathrm{na}}^3\right)$ ranging from 0.01 to 0.5. Condensate number loss resulted from the pulse, and for triangular pulses shorter than 1 ms, decreasing the pulse length actually increased the loss, until very short time scales ( $\sim 10\mu \mathrm{s}$) were reached. The observed time dependence is very different from that expected in traditional inelastic loss processes, suggesting the presence of new microscopic BEC physics.

A systematic perturbation scheme is developed for approximate solutions to the time-dependent Schrödinger equation with a space-adiabatic Hamiltonian. For a particular isolated energy band, the basic approach is to separate kinematics from dynamics. The kinematics is defined through a subspace of the full Hilbert space for which transitions to other band subspaces are suppressed to all orders, and the dynamics operates in that subspace in terms of an effective intraband Hamiltonian. As novel applications, we discuss the Born-Oppenheimer theory to second order and derive for the first time the nonperturbative definition of the $g$ factor of the electron within nonrelativistic quantum electrodynamics.