Narrow Results

In this paper, we study the stationary and dynamical properties of three-dimensional trapped Bose–Einstein condensates with attractive interactions subjected to a random potential. To this end, a variational method is applied to solve the underlying Gross–Pitaevskii equation. We derive analytical predictions for the energy, the equilibrium width, and evolution laws of the condensate parameter. The breathing mode oscillations’ frequency of the condensate has been also calculated in terms of the gas and disorder parameters. We analyze in addition the dynamics of collapse from the Gaussian approximation. Surprisingly, we find that the intriguing interplay of the attractive interaction and disorder effects leads to prevent collapse of the condensate.

In a previous article with A. Aftalion and X. Blanc, it was shown that the hypercontractivity property of the dilation semigroup in spaces of entire functions was a key ingredient in the study of the Lowest Landau Level model for fast rotating Bose–Einstein condensates. That former work was concerned with the stationary constrained variational problem. This article is about the nonlinear Hamiltonian dynamics and the spectral stability of the constrained minima with motivations arising from the description of Tkatchenko modes of Bose–Einstein condensates. Again the hypercontractivity property provides a very strong control of the nonlinear term in the dynamical analysis.

In this paper, we study the Gross–Pitaevskii energy of a Bose–Einstein condensate in the presence of an optical lattice, modeled by a periodic potential V(x₃) in the third direction. We study a simple case where the wells of the potential V correspond to regions where V vanishes, and are separated by small intervals of size δ where V is large. According to the intensity of V, we determine the limiting energy as δ tends to 0. In the critical case, the periodic potential approaches a sum of delta functions and the limiting energy has a contribution due to the value of the wave function between the wells. The proof relies on Gamma convergence type techniques.

Our aim is to analyze the various energy functionals appearing in the physics literature and describing the behavior of a Bose–Einstein condensate in an optical lattice. We want to justify the use of some reduced models and control the error of approximation. For that purpose, we will use the semi-classical analysis developed for linear problems related to the Schrödinger operator with periodic potential or multiple wells potentials. We justify, in some asymptotic regimes, the reduction to low dimensional problems and analyze the reduced problems.

We consider Bose gases of $N$ particles in a box of volume one, interacting through a repulsive potential with scattering length of order $N^{-1+\kappa }$, for $\kappa >0$. Such regimes interpolate between the Gross–Pitaevskii and thermodynamic limits. Assuming that $\kappa$ is sufficiently small, we determine the ground state energy and the low-energy excitation spectrum, up to errors vanishing in the limit $N\to \infty$.

Following an idea first proposed by Penrose in 1996 to explain the problem of quantum state reduction as a gravitational effect, Moroz, Penrose and Tod¹ have shown that quantum state reduction due to gravitational interactions could take place in about one second for the case of 10¹¹ nucleons. However, keeping 10¹¹ nucleons together in a quantum macroscopic state does not appear to be feasible as yet. The closest physical system to such a situation is provided by Bose–Einstein condensates (BEC) with attractive interactions. We present numerical simulations of the Schrödinger–Newton equations, which show that an attractive BEC with 10³ atoms would yield a decorrelation time of the order of 10^-2 seconds. Hence, a "Penrose-like" reduction, induced by BEC attractive interaction instead of gravity, might be observable and possibly monitored in current BEC experiments with attractive interactions.

Recent developments in simulating fundamental quantum field theoretical effects in the kinematical context of analogue gravity are reviewed. Specifically, it is argued that a curved spacetime generalization of the Unruh–Davies effect — the Gibbons–Hawking effect in the de Sitter spacetime of inflationary cosmological models — can be implemented and verified in an ultracold gas of bosonic atoms.

We review the recent experimental and theoretical advances in the generation of matter wave solitons in Bose–Einstein condensates. In particular, the controlled generation and dynamics of stable bright solitons by mean of Feshbach resonance techniques is discussed in details. Several aspects are taking into account, including the variation of the scattering length due to Feshbach resonance, the safe parameters against the collapse and the experimental implications of our scenario.

We analyze the stability of a Bose–Einstein condensate in a Lorentz violating scenario, which is characterized by a deformation in the dispersion relation. The incorporation of a Lorentz violation within the bosonic statistics has, as a consequence, the emergence of a pseudo-interaction, the one can be associated to a characteristic scattering length. In addition, we calculate the relevant parameters associated to the stability of such condensate incorporating this pseudo-interaction in the nonlinear term of the Gross–Pitaevskii equation. We show that these parameters must be corrected, as a consequence of the quantum structure of spacetime.

We analyze the effects caused by an anomalous single-particle dispersion relation suggested in several quantum-gravity models, upon the thermodynamics of a Bose–Einstein condensate trapped in a generic three-dimensional power-law potential. We prove that the shift in the condensation temperature, caused by a deformed dispersion relation, described as a non-trivial function of the number of particles and the shape associated to the corresponding trap, could provide bounds for the parameters associated to such deformation. In addition, we calculate the fluctuations in the number of particles as a criterium of thermodynamic stability for these systems. We show that the apparent instability caused by the anomalous fluctuations in the thermodynamic limit can be suppressed considering the lowest energy associated to the system in question.

In this work we continue our previous studies concerning the possibility of the existence of a Bose–Einstein condensate in the interior of a static black hole, a possibility first advocated by Dvali and Gómez. We find that the phenomenon seems to be rather generic and it is associated to the presence of a horizon, acting as a confining potential. We extend the previous considerations to a Reissner–Nordström black hole and to the de Sitter cosmological horizon. In the latter case the use of static coordinates is essential to understand the physical picture. In order to see whether a BEC is preferred, we use the Brown–York quasilocal energy, finding that a condensate is energetically favorable in all cases in the classically forbidden region. The Brown–York quasilocal energy also allows us to derive a quasilocal potential, whose consequences we explore. Assuming the validity of this quasilocal potential allows us to suggest a possible mechanism to generate a graviton condensate in black holes. However, this mechanism appears not to be feasible in order to generate a quantum condensate behind the cosmological de Sitter horizon.

We study the exact solution for a two-mode model describing coherent coupling between atomic and molecular Bose–Einstein condensates (BEC), in the context of the Bethe ansatz. By combining an asymptotic and numerical analysis, we identify the scaling behaviour of the model and determine the zero temperature expectation value for the coherence and average atomic occupation. The threshold coupling for production of the molecular BEC is identified as the point at which the energy gap is minimum. Our numerical results indicate a parity effect for the energy gap between ground and first excited state depending on whether the total atomic number is odd or even. The numerical calculations for the quantum dynamics reveals a smooth transition from the atomic to the molecular BEC.

We examine in terms of exact solutions of the time-dependent Schrödinger equation, the quantum tunnelling process in Bose–Einstein condensates of two interacting species trapped in a double well configuration. Based on the two series of time-dependent SU(2) gauge transformations, we diagonalize the Hamilton operator and obtain analytic time-evolution formulas of the population imbalance and the berry phase. The particle population imbalance of species A between the two wells is studied analytically.

The Rabi regime for a Bose-Einstein condensate (BEC) in double-well potential occurring for sufficiently strong cross-collision strengths is analyzed. It is shown that in this regime the potential barrier acts as a temporal atomic beam splitter. An ideal 50:50 atomic beam splitter reached at specific intervals of time is employed for a balanced homodyne detection of the condensate relative phase.

We present a formal derivation of the mean-field expansion for dilute Bose–Einstein condensates with two-particle interaction potentials which are weak and finite-range, but otherwise arbitrary. The expansion allows for a controlled investigation of the impact of microscopic interaction details (e.g. the scaling behavior) on the mean-field approach and the induced higher-order corrections beyond the s-wave scattering approximation.

We present in this paper an analytical model for a cold bosonic gas on an optical lattice (with densities of the order of 1 particle per site) targeting the critical regime of the Bose - Einstein Condensate superfluid - Mott insulator transition.

We sketch the major steps in a functional integral derivation of a new set of stochastic Gross-Pitaevsky equations (GPE) for a Bose-Einstein condensate (BEC) confined to a trap at zero temperature with the averaged effects of non-condensate modes incorporated as stochastic sources. The closed-time-path (CTP) coarse-grained effective action (CGEA) or the equivalent influence functional method is particularly suitable because it can account for the full back-reaction of the noncondensate modes on the condensate dynamics self-consistently. The Langevin equations derived here containing nonlocal dissipation together with colored and multiplicative noises are useful for a stochastic (as distinguished from a kinetic) description of the nonequilibrium dynamics of a BEC. This short paper contains original research results not yet published anywhere.

By developing a three-mode approximation, we derive dynamical equations governing a condensate in a symmetric three-well potential. Based on the dynamical equations, we numerically simulate the dynamical properties of a Bose–Einstein condensate in a symmetric three-well potential. It is shown that, for the zero-phase mode, atomic population in each well oscillates periodically with the amplitude dependent on the initial conditions, and there may exist a critical initial distribution. However, for the π-phase mode, it is possible for the system to exhibit a set of behaviors directly dependent on the ratio between the nonlinearity induced by the atom-atom interactions and the coupling of neighboring wells.

By developing a multiple-scale method, we study analytically the dynamics of the soliton inside the semi-infinite band gap (SIBG) of quasi-one-dimensional Bose–Einstein condensates trapped in an optical lattice. In the linear case, a stable condition of soliton formation is obtained. For a weak nonlinearity, whether there occurs a spatially propagating or localized gap soliton is determined by the lattice depth. Meanwhile, we predict the existence of the dark (bright) gap solitons for the repulsive (attractive) interactions in the SIBG, different from that of the gap solitons in other energy gaps. And the collision of two dark (or bright) solitons is nearly elastic under a safe range of atomic numbers. An experimental protocol is further designed for observing these phenomena.

Due to their relevance to physics and technology, the Bose–Einstein condensates (BECs) are of current interest. Certain dynamics of the BECs, such as the cigar-shaped condensate confined in a cylindrically symmetric parabolic trap, can be described by the Gross–Pitaevskii (GP) equation with a time-dependent trap. In this paper, by virtue of the Painlevé analysis and symbolic computation, we derive the integrable condition for the GP equation with the time-dependent scattering length in the presence of a confining or expulsive time-dependent trap. Lax pair for this equation is directly obtained via the Ablowitz–Kaup–Newell–Segur scheme under the integrable condition. Bright one-soliton-like solution of the GP equation is presented via the Bäcklund transformation and some analytic solutions with variable amplitudes are obtained by the ansatz method. In addition, an infinite number of conservation laws are also derived. Those results could be of some value for the studies on the lower-dimensional condensates.