Fully quantum mechanical moment of inertia of a mesoscopic ideal Bose gas

The superfluid fraction of an atomic cloud is defined using the cloud's response to a rotation of the external potential, i.e. the moment of inertia. A fully quantum mechanical calculation of this moment is based on the dispersion of L_z instead of quasi-classical averages. In this paper we derive analytical results for the moment of inertia of a small number of non-interacting Bosons using the canonical ensemble. The required symmetrized averages are obtained via a representation of the partition function by permutation cycles. Our results are useful to discriminate purely quantum statistical effects from interaction effects in studies of superfluidity and phase transitions in finite samples. Received 30 June 2000     

Binary mixtures of Bose-Einstein condensates: Phase dynamics and spatial dynamics

We investigate the relative phase coherence properties and the occurrence of demixing instabilities for two mutually interacting and time evolving Bose-Einstein condensates in traps. Our treatment naturally includes the additional decoherence effect due to fluctuations in the total number of particles. Analytical results are presented for the breathe-together solution, an extension of previously known scaling solution to the case of a binary mixture of condensates. When the three coupling constants describing the elastic interactions among the atoms in the two states are close to each other, a dramatic increase of the phase coherence time is predicted. Numerical results are presented for the parameters of the recent JILA experiments. Received 23 April 1999 and Received in final form 21 September 1999     

Splitting between quadrupole modes of dilute quantum gas in a two-dimensional anisotropic trap

We consider quadrupole excitations of quasi-two-dimensional interacting quantum gas in an anisotropic harmonic oscillator potential at zero temperature. Using the time-dependent variational approach, we calculate a few low-lying collective excitation frequencies of a two-dimensional anisotropic Bose gas. Within the energy weighted sum-rule approach, we derive a general dispersion relation of two quadrupole excitations of a two-dimensional deformed trapped quantum gas. This dispersion relation is valid for both statistics. We show that the quadrupole excitation frequencies obtained from both methods are exactly the same. Using this general dispersion relation, we also calculate the quadrupole frequencies of a two-dimensional unpolarized Fermi gas in an anisotropic trap. For both cases, we obtain analytic expressions for the quadrupole frequencies and the splitting between them for arbitrary value of trap deformation. This splitting decreases with increasing interaction strength for both statistics. For a two-dimensional anisotropic Fermi gas, the two quadrupole frequencies and the splitting between them become independent of the particle number within the Thomas-Fermi approach. Received 21 September 2001 and Received in final form 9 December 2001     

Phase dynamics of Bose-Einstein condensates: Losses versus revivals

In the absence of losses the phase of a Bose-Einstein condensate undergoes collapses and revivals in time due to elastic atomic interactions. As experiments necessarily involve inelastic collisions, we develop a model to describe the phase dynamics of the condensates in presence of collisional losses. We find that a few inelastic processes are sufficient to damp the revivals of the phase. For this reason the observability of phase revivals for present experimental conditions is limited to condensates with a few hundreds of atoms. Received: 23 February 1998 / Revised: 21 July 1998 / Accepted: 23 July 1998     

Ground state pressure and energy density of an interacting homogeneous Bose gas in two dimensions

We consider an interacting homogeneous Bose gas at zero temperature in two spatial dimensions. The properties of the system can be calculated as an expansion in powers of g, where g is the coupling constant. We calculate the ground state pressure and the ground state energy density to second order in the quantum loop expansion. The renormalization group is used to sum up leading and subleading logarithms from all orders in perturbation theory. In the dilute limit, the renormalization group improved pressure and energy density are expansions in powers of the T ^2B and T ^2Bln(T ^2B), respectively, where T ^2B is the two-body T-matrix. Received 19 April 2002 Published online 13 August 2002     

Bose-Einstein condensation in interacting gases

We study the occurrence of a Bose-Einstein transition in a dilute gas with repulsive interactions, starting from temperatures above the transition temperature. The formalism, based on the use of Ursell operators, allows us to evaluate the one-particle density operator with more flexibility than in mean-field theories, since it does not necessarily coincide with that of an ideal gas with adjustable parameters (chemical potential, etc.). In a first step, a simple approximation is used (Ursell-Dyson approximation), which allow us to recover results which are similar to those of the usual mean-field theories. In a second step, a more precise treatment of the correlations and velocity dependence of the populations in the system is elaborated. This introduces new physical effects, such as a change of the velocity profile just above the transition: the proportion of atoms with low velocities is higher than in an ideal gas. A consequence of this distortion is an increase of the critical temperature (at constant density) of the Bose gas, in agreement with those of recent path integral Monte-Carlo calculations for hard spheres. Received 13 November 1998     

Ground state and vortex states of bosons in an anisotropic trap: A variational approach

We propose a simple variational form of the wave function to describe the ground state and vortex states of a weakly interacting Bose gas in an anisotropic trap. The proposed wave function is valid for a wide range of the particle numbers in the trap. It also works well in the case of attractive interaction between the atoms. Further, it provides an easy and fast method to calculate the physical quantities of interest. The results compare very well with those obtained by purely numerical techniques. Using our wave function we have been able to verify, for the first time, the predicted behaviour of the aspect ratio. Received 7 December 1998 and Received in final form 4 February 1999     

Macroscopic oscillations between two weakly coupled Bose-Einstein condensates

We demonstrate, both from a theoretical and an experimental point of view, the possibility of realizing a weak coupling between two Bose-Einstein condensates trapped in different Zeeman states. The weak coupling drives macroscopic quantum oscillations between the condensate populations and the observed current-phase dynamics is described by generalized Josephson equations. In order to highlight the superfluid nature of the oscillations, we investigate the response of a ⁸⁷Rb non-condensate (thermal) gas in the same conditions, showing that the thermal oscillations damp more quickly than those of the condensate. Received 2 May 2002 / Received in final form 19 November 2002 Published online 6 March 2003 RID="a" ID="a"e-mail: smerzi@sissa.it     

Bose-Einstein transition in a dilute interacting gas

We study the effects of repulsive interactions on the critical density for the Bose-Einstein transition in a homogeneous dilute gas of bosons. First, we point out that the simple mean field approximation produces no change in the critical density, or critical temperature, and discuss the inadequacies of various contradictory results in the literature. Then, both within the frameworks of Ursell operators and of Green's functions, we derive self-consistent equations that include correlations in the system and predict the change of the critical density. We argue that the dominant contribution to this change can be obtained within classical field theory and show that the lowest order correction introduced by interactions is linear in the scattering length, a, with a positive coefficient. Finally, we calculate this coefficient within various approximations, and compare with various recent numerical estimates. Received 15 July 2001     

Evaporative cooling of an atomic beam

We present a theoretical analysis of the evaporative cooling of an atomic beam propagating in a magnetic guide. Cooling is provided by transverse evaporation. The atomic dynamics inside the guide is analyzed by solving the Boltzmann equation with two different approaches: an approximate analytical ansatz and a Monte-Carlo simulation. Within their domain of validity, these two methods are found to be in very good agreement with each other. They allow us to determine how the phase-space density and the flux of the beam vary along its direction of propagation. We find a significant increase for the phase-space density along the guide for realistic experimental parameters. By extrapolation, we estimate the length of the beam needed to reach quantum degeneracy. Received 24 September 1999     

Thermodynamics of Bose-condensed atomic hydrogen

We study the thermodynamics of the Bose-condensed atomic hydrogen confined in the Ioffe-Pritchard potential. Such a trapping potential, that models the magnetic trap used in recent experiments with hydrogen, is anharmonic and strongly anisotropic. We calculate the ground-state properties, the condensed and non-condensed fraction and the Bose-Einstein transition temperature. The thermodynamics of the system is strongly affected by the anharmonicity of this external trap. Finally, we consider the possibility to detect Josephson-like currents by creating a double-well barrier with a laser beam. Received 15 February 2000     

Axialisation, cooling and quenching of ions in a Penning trap

On the basis of a macroscopic ground state population it was argued recently that Bose-Einstein condensation should occur in a one-dimensional harmonic potential. We examine this situation by drawing analogies to bosons in a two-dimensional box, where the thermodynamic limit is well-defined. We show that in both systems although the ground state populations show sharp onsets at the critical temperature, the behaviour of the specific heat is analytic, which proves the absence of a phase transition in these systems. Received: 17 February 1997 / Revised: 3 September 1997 / Accepted: 13 October 1997     

Dilute Bose gas: short-range particle correlations and ultraviolet divergence

The modified Bogoliubov model where the primordial interaction is replaced by the t matrix is reinvestigated. It is shown to provide a negative value of the kinetic energy for a strongly interacting dilute Bose gas, contrary to the original Bogoliubov model. To clear up the origin of this failure, the correct values of the kinetic and interaction energies of a dilute Bose gas are calculated. It is demonstrated that both the problem of the negative kinetic energy and the ultraviolet divergence, dating back to the well-known paper of Lee, Yang and Huang, is connected with an inadequate picture of the short-range boson correlations. These correlations are reconsidered within the thermodynamically consistent model proposed earlier by the present authors. Found results are in absolute agreement with the data of the Monte-Carlo calculations for the hard-sphere Bose gas. Received 10 February 2000 and Received in final form 28 November 2000     

Matter-wave amplification and collective internal excitations in a trapped Bose gas

Received: 23 May 1997/Revised version: 29 July 1997     

Coherent population trapping of Bose-Einstein condensates: Detection of phase diffusion

Two Bose-Einstein condensates in different Zeeman sublevels can be decoupled from driving light fields in coherent population trapping. A condensate pair with a deterministic entanglement and a controllable value of the relative phase may be prepared by selecting the phase difference between the coherent light fields. The rate of the condensate phase diffusion may be determined from the two-photon resonant absorption of radiation. Received: 29 June 1998 / Revised: 10 October 1998 / Accepted: 19 October 1998     

Canonical and microcanonical ensemble approaches to Bose-Einstein condensation: The thermodynamics of particles in harmonic traps

The thermodynamic properties of bosons moving in a harmonic trap in an arbitrary number of dimensions are investigated in the grand canonical, canonical and microcanonical ensembles by applying combinatorial techniques developed earlier in statistical nuclear fragmentation models. Thermodynamic functions such as the energy and specific heat are computed exactly in these ensembles. The occupation of the ground or condensed state is also obtained exactly, and signals clearly the phase transition. The application of these techniques to fermionic systems is also briefly discussed. Received 18 August 1998 and Received in final form 14 October 1998     

Instanton approach to Josephson tunneling between trapped condensates

An instanton method is proposed to investigate the quantum tunneling between two weakly-linked Bose-Einstein condensates confined in double-well potential traps. We point out some intrinsic pathologies in the earlier treatments of other authors and make an effort to go beyond these very simple zero order models. The tunneling amplitude may be calculated in the Thomas-Fermi approximation and beyond it; we find it depends on the number of the trapped atoms, through the chemical potential. Some suggestions are given for the observation of the Josephson oscillation and the MQST. Received 29 June 2001 and Received in final form 17 September 2001     

Production of a bose einstein condensate of metastable helium atoms

We recently observed a Bose-Einstein condensate in a dilute gas of ⁴He in the 2³S₁ metastable state. In this article, we describe the successive experimental steps which led to the Bose-Einstein transition at 4.7 μK: loading of a large number of atoms in a MOT, efficient transfer into a magnetic Ioffé-Pritchard trap, and optimization of the evaporative cooling ramp. Quantitative measurements are also given for the rates of elastic and inelastic collisions, both above and below the transition. Received 15 October 2001     

Coherent control of the self-trapping transition

We discuss the dynamics of two weakly coupled Bose-Einstein condensates in a double-well potential, contrasting the mean-field picture to the exact N-particle evolution. On the mean-field level, a self-trapping transition occurs when the scaled interaction strength exceeds a critical value; this transition essentially persists in small condensates comprising about 1000 atoms. When the double-well is modulated periodically in time, Floquet-type solutions to the nonlinear Schr?dinger equation take over the role of the stationary mean-field states. These nonlinear Floquet states can be classified as “unbalanced” or “balanced”, depending on whether or not they entail long-time confinement of most particles to one well. Since the emergence of unbalanced Floquet states depends on the amplitude and frequency of the modulating force, we predict that the onset of self-trapping can efficiently be controlled by varying these parameters. This prediction is verified numerically by both mean-field and N-particle calculations. Received 5 November 2000 and Received in final form 16 February 2001