Three fermions in a box at the unitary limit: universality in a lattice model

We consider three fermions with two spin components interacting on a lattice model with an infinite scattering length. Low lying eigenenergies in a cubic box with periodic boundary conditions, and for a zero total momentum, are calculated numerically for decreasing values of the lattice period. The results are compared to the predictions of the zero range Bethe-Peierls model in continuous space, where the interaction is replaced by contact conditions. The numerical computation, combined with analytical arguments, shows the absence of negative energy solution, and a rapid convergence of the lattice model towards the Bethe-Peierls model for a vanishing lattice period. This establishes for this system the universality of the zero interaction range limit.Comment: 6 page

Limits of sympathetic cooling of fermions by zero temperature bosons due to particle losses

It has been suggested by Timmermans [Phys. Rev. Lett. {\bf 87}, 240403 (2001)] that loss of fermions in a degenerate system causes strong heating. We address the fundamental limit imposed by this loss on the temperature that may be obtained by sympathetic cooling of fermions by bosons. Both a quantum Boltzmann equation and a quantum Boltzmann \emph{master} equation are used to study the evolution of the occupation number distribution. It is shown that, in the thermodynamic limit, the Fermi gas cools to a minimal temperature $k_{{\rm B}}T/\mu\propto(\gamma_{{\rm loss}}/\gamma_{{\rm coll}})^{0.44}$, where $\gamma_{{\rm loss}}$ is a constant loss rate, $\gamma_{{\rm coll}}$ is the bare fermion--boson collision rate not including the reduction due to Fermi statistics, and $\mu\sim k_{{\rm B}}T_{{\rm F}}$ is the chemical potential. It is demonstrated that, beyond the thermodynamic limit, the discrete nature of the momentum spectrum of the system can block cooling. The unusual non-thermal nature of the number distribution is illustrated from several points of view: the Fermi surface is distorted, and in the region of zero momentum the number distribution can descend to values significantly less than unity. Our model explicitly depends on a constant evaporation rate, the value of which can strongly affect the minimum temperature.Comment: 14 pages, 7 figures. Phys. Rev. A in pres

Stability analysis of sonic horizons in Bose-Einstein condensates

We examine the linear stability of various configurations in Bose-Einstein condensates with sonic horizons. These configurations are chosen in analogy with gravitational systems with a black hole horizon, a white hole horizon and a combination of both. We discuss the role of different boundary conditions in this stability analysis, paying special attention to their meaning in gravitational terms. We highlight that the stability of a given configuration, not only depends on its specific geometry, but especially on these boundary conditions. Under boundary conditions directly extrapolated from those in standard General Relativity, black hole configurations, white hole configurations and the combination of both into a black hole--white hole configuration are shown to be stable. However, we show that under other (less stringent) boundary conditions, configurations with a single black hole horizon remain stable, whereas white hole and black hole--white hole configurations develop instabilities associated to the presence of the sonic horizons.Comment: 14 pages, 7 figures (reduced resolution

Anomalous spatial diffusion and multifractality in optical lattices

Transport of cold atoms in shallow optical lattices is characterized by slow, nonstationary momentum relaxation. We here develop a projector operator method able to derive in this case a generalized Smoluchowski equation for the position variable. We show that this explicitly non-Markovian equation can be written as a systematic expansion involving higher-order derivatives. We use the latter to compute arbitrary moments of the spatial distribution and analyze their multifractal properties.Comment: 5 pages, 3 figure

Quasicondensation reexamined

We study in detail the effect of quasicondensation. We show that this effect is strictly related to dimensionality of the system. It is present in one dimensional systems independently of interactions - exists in repulsive, attractive or in non-interacting Bose gas in some range of temperatures below characteristic temperature of the quantum degeneracy. Based on this observation we analyze the quasicondensation in terms of a ratio of the two largest eigenvalues of the single particle density matrix for the ideal gas. We show that in the thermodynamic limit in higher dimensions the second largest eigenvalue vanishes (as compared to the first one) with total number of particles as $\simeq N^{-\gamma}$ whereas goes to zero only logarithmically in one dimension. We also study the effect of quasicondensation for various geometries of the system: from quasi-1D elongated one, through spherically symmetric 3D case to quasi-2D pancake-like geometry

Quantum fluctuations in coupled dark solitons in trapped Bose-Einstein condensates

We show that the quantum fluctuations associated with the Bogoliubov quasiparticle vacuum can be strongly concentrated inside dark solitons in a trapped Bose Einstein condensate. We identify a finite number of anomalous modes that are responsible for such quantum phenomena. The fluctuations in these anomalous modes correspond to the `zero-point' oscillations in coupled dark solitons.Comment: 4 pages, 3 figure

Non-diffusive phase spreading of a Bose-Einstein condensate at finite temperature

We show that the phase of a condensate in a finite temperature gas spreads linearly in time at long times rather than in a diffusive way. This result is supported by classical field simulations, and analytical calculations which are generalized to the quantum case under the assumption of quantum ergodicity in the system. This super-diffusive behavior is intimately related to conservation of energy during the free evolution of the system and to fluctuations of energy in the prepared initial state.Comment: 16 pages, 7 figure

Dynamics of a Bose-Einstein Condensate in an Anharmonic Trap

We present a theoretical model to describe the dynamics of Bose-Einstein condensates in anharmonic trapping potentials. To first approximation the center-of-mass motion is separated from the internal condensate dynamics and the problem is reduced to the well known scaling solutions for the Thomas-Fermi radii. We discuss the validity of this approach and analyze the model for an anharmonic waveguide geometry which was recently realized in an experiment \cite{Ott2002c}

Images of the Dark Soliton in a Depleted Condensate

The dark soliton created in a Bose-Einstein condensate becomes grey in course of time evolution because its notch fills up with depleted atoms. This is the result of quantum mechanical calculations which describes output of many experimental repetitions of creation of the stationary soliton, and its time evolution terminated by a destructive density measurement. However, such a description is not suitable to predict the outcome of a single realization of the experiment were two extreme scenarios and many combinations thereof are possible: one will see (1) a displaced dark soliton without any atoms in the notch, but with a randomly displaced position, or (2) a grey soliton with a fixed position, but a random number of atoms filling its notch. In either case the average over many realizations will reproduce the mentioned quantum mechanical result. In this paper we use N-particle wavefunctions, which follow from the number-conserving Bogoliubov theory, to settle this issue.Comment: 8 pages, 6 figures, references added in version accepted for publication in J. Phys.

The N boson time dependent problem: an exact approach with stochastic wave functions

We present a numerically tractable method to solve exactly the evolution of a N boson system with binary interactions. The density operator of the system rho is obtained as the stochastic average of particular operators |Psi_1><Psi_2| of the system. The states |Psi_{1,2}> are either Fock states |N:phi_{1,2}> or coherent states |coh:phi_{1,2}> with each particle in the state phi_{1,2}. We determine the conditions on the evolution of phi_{1,2} -which involves a stochastic element- under which we recover the exact evolution of rho. We discuss various possible implementations of these conditions. The well known positive P-representation arises as a particular case of the coherent state ansatz. We treat numerically two examples: a two-mode system and a one-dimensional harmonically confined gas. These examples, together with an analytical estimate of the noise, show that the Fock state ansatz is the most promising one in terms of precision and stability of the numerical solution.Comment: 21 pages, 5 figures, submitted to Phys.Rev.