Bose–Einstein condensation of a relativistic Bose gas in a harmonic potential

Using semiclassical method, Bose–Einstein condensation (BEC) of a relativistic ideal Bose gas (RIBG) with and without antibosons in the three-dimensional (3D) harmonic potential is investigated. Analytical expressions for the BEC transition temperature, condensate fraction, specific heat and entropy of the system are obtained. Relativistic effects on the properties of the system are discussed and it is found that the relativistic effect decreases the transition temperature T_c but enlarges the gap of specific heat at T_c. We also study the influence of antibosons on a RIBG. Comparing with the system without antibosons, the system with antibosons has a higher transition temperature and a lower Helmholtz free energy. It implies that the system with antibosons is more stable.     

Spinor Bose–Einstein condensates

An overview of the physics of spinor and dipolar Bose–Einstein condensates (BECs) is given. Mean-field ground states, Bogoliubov spectra, and many-body ground and excited states of spinor BECs are discussed. Properties of spin-polarized dipolar BECs and those of spinor–dipolar BECs are reviewed. Some of the unique features of the vortices in spinor BECs such as fractional vortices and non-Abelian vortices are delineated. The symmetry of the order parameter is classified using group theory, and various topological excitations are investigated based on homotopy theory. Some of the more recent developments in a spinor BEC are discussed.     

Bose–Einstein condensate on a persistent-supercurrent atom chip

A Bose–Einstein condensate was achieved in a stable magnetic trap on a persistent-supercurrent atom chip with a superconducting closed-loop circuit. We determined precisely the shape of the magnetic trapping potential by systematically controlling the persistent supercurrent. The condensation was verified by time-of-flight imaging and by atom number decay measurements. The measured decay rates agreed quantitatively with numerical simulations on the three-body loss process assuming all of the atoms to be a condensate. We also discuss the feasibility of creating a quasi-one-dimensional Bose gas on our atom chip.     

Solitons in Bose–Einstein condensates

The Gross–Pitaevskii equation (GPE) describing the evolution of the Bose–Einstein condensate (BEC) order parameter for weakly interacting bosons supports dark solitons for repulsive interactions and bright solitons for attractive interactions. After a brief introduction to BEC and a general review of GPE solitons, we present our results on solitons that arise in the BEC of hard-core bosons, which is a system with strongly repulsive interactions. For a given background density, this system is found to support both a dark soliton and an antidark soliton (i.e., a bright soliton on a pedestal) for the density profile. When the background has more (less) holes than particles, the dark (antidark) soliton solution dies down as its velocity approaches the sound velocity of the system, while the antidark (dark) soliton persists all the way up to the sound velocity. This persistence is in contrast to the behaviour of the GPE dark soliton, which dies down at the Bogoliubov sound velocity. The energy–momentum dispersion relation for the solitons is shown to be similar to the exact quantum low-lying excitation spectrum found by Lieb for bosons with a delta-function interaction.     

The forces on a single interacting Bose–Einstein condensate

Using double parabola approximation for a single Bose–Einstein condensate confined between double slabs we proved that in grand canonical ensemble (GCE) the ground state with Robin boundary condition (BC) is favored, whereas in canonical ensemble (CE) our system undergoes from ground state with Robin BC to the one with Dirichlet BC in small-L region and vice versa for large-L region and phase transition in space of the ground state is the first order. The surface tension force and Casimir force are also considered in both CE and GCE in detail.     

Dipolar Bose–Einstein condensate soliton on a two-dimensional optical lattice

Using a three-dimensional mean-field model we study one-dimensional dipolar Bose–Einstein condensate (BEC) solitons on a weak two-dimensional (2D) square and triangular optical lattice (OL) potentials placed perpendicular to the polarization direction. The stabilization against collapse and expansion of these solitons for a fixed dipolar interaction and a fixed number of atoms is possible for short-range atomic interaction lying between two critical limits. The solitons collapse below the lower limit and escapes to infinity above the upper limit. One can also stabilize identical tiny BEC solitons arranged on the 2D square OL sites forming a stable 2D array of interacting droplets when the OL sites are filled with a filling factor of 1/2 or less. Such an array is unstable when the filling factor is made more than 1/2 by occupying two adjacent sites of OL. These stable 2D arrays of dipolar superfluid BEC solitons are quite similar to the recently studied dipolar Mott insulator states on 2D lattice in the Bose–Hubbard model by Capogrosso-Sansone et al. [B. Capogrosso-Sansone, C. Trefzger, M. Lewenstein, P. Zoller, G. Pupillo, Phys. Rev. Lett. 104 (2010) 125301].     

Interferencing in Coupled Bose–Einstein Condensates

We consider an exactly soluble model of two Bose–Einstein condensates with a Josephson-type of coupling. Its equilibrium states are explicitly found showing condensation and spontaneously broken gauge symmetry. It is proved that the total number and total phase fluctuation operators, as well as the relative number and relative current fluctuation operators form both a quantum canonical pair. The exact relation between the relative current and phase fluctuation operators is established. Also the dynamics of these operators is solved showing the collapse and revival phenomenon.     

Stability of Attractive Bose–Einstein Condensates

We propose the critical nonlinear Schrödinger equation with a harmonic potential as a model of attractive Bose–Einstein condensates. By an elaborate mathematical analysis we show that a sharp stability threshold exists with respect to the number of condensate particles. The value of the threshold agrees with the existing experimental data. Moreover with this threshold we prove that a ground state of the condensate exists and is orbital stable. We also evaluate the minimum of the condensate energy.     

Bose–Einstein condensation in random potentials

We present a rigorous study of the perfect Bose-gas in the presence of a homogeneous ergodic random potential. It is demonstrated that the Lifshitz tail behaviour of the one-particle spectrum reduces the critical dimensionality of the (generalized) Bose–Einstein Condensation (BEC) to d=1. To tackle the Off-Diagonal Long-Range Order (ODLRO) we introduce the space average one-body reduced density matrix. For a one-dimensional Poisson-type random potential we prove that randomness enhances the exponential decay of this matrix in domain free of the BEC. To cite this article: O. Lenoble et al., C. R. Physique 5 (2004).     

Bose–Einstein condensation of atomic hydrogen

The recent creation of a Bose–Einstein condensate of atomic hydrogen has added a new system to this exciting field. The differences between hydrogen and the alkali metal atoms require other techniques for the initial trapping and cooling of the atoms and the subsequent detection of the condensate. The use of a cryogenic loading technique results in a larger number of trapped atoms. Spectroscopic detection is well suited to measuring the temperature and density of the sample in situ. The transition was observed at a temperature of 50 μK and a density of 2×10¹⁴ cm^-3. The number of condensed atoms is about 10⁹ at a condensate fraction of a few percent. A peak condensate density of 4.8×10¹⁵ cm^-3 has been observed. Received: 22 June 1999 / Published online: 3 November 1999     

Phase fluctuations in Bose–Einstein condensates

We demonstrate the existence of phase fluctuations in elongated Bose–Einstein condensates (BECs) and study the dependence of these fluctuations on the system parameters. A strong dependence on temperature, atom number, and trapping geometry is observed. Phase fluctuations directly affect the coherence properties of BECs. In particular, we observe instances where the phase-coherence length is significantly smaller than the condensate size. Our method of detecting phase fluctuations is based on their transformation into density modulations after ballistic expansion. An analytic theory describing this transformation is developed. Received: 13 July 2001 / Revised version: 28 September 2001 / Published online: 23 November 2001     

Bose–Einstein condensates in magnetic waveguides

In this article, we describe an experimental system for generating Bose–Einstein condensates and controlling the shape and motion of a condensate by using miniaturised magnetic potentials. In particular, we describe the magnetic trap setup, the vacuum system, the use of dispenser sources for loading a high number of atoms into the magneto-optical trap, the magnetic transfer of atoms into the microtrap, and the experimental cycle for generating Bose–Einstein condensates. We present first results on outcoupling of condensates into a magnetic waveguide and discuss influences of the trap surface on the ultra-cold ensembles. Received: 21 August 2002 / Revised version: 10 December 2002 / Published online: 26 February 2003 RID="*" ID="*"Corresponding author. Fax: +49-7071/295-829, E-mail: fortagh@pit.uni-tuebingen.de     

Phase Sensitivity in Bose–Einstein Condensates

We investigate the phase sensitivity of a collection of interacting spins with quantum Fisher information controlled by an external field. By adopting the frozen-spin approximation, we derive the approximate analytical expressions of the maximal quantum Fisher information and the phase sensitivity. It is shown that the maximal quantum Fisher information and the phase sensitivity depend on the strength of the external field. With the increases of the external field, the period of oscillation of quantum Fisher information and phase estimation decrease, while the values of quantum Fisher information and phase sensitivity increase because of the suppress by the external field.     

Time-domain Ramsey interferometry with Bose–Einstein condensates

We report on the observation of time-domain interference with Bose–Einstein condensates, by means of a double separated oscillator technique. We discuss the decay of the Ramsey oscillations amplitude, that in our system occurs on a time scale of tens of microseconds. To elucidate the origin of this fast decay, we compare the behaviour of a condensate with that of a thermal cloud.