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Physica Scripta

Nonlocal energy functionals and long range interactions in collapsing condensates in a box

Stavros Theodorakis, Nikolas Charalambous

Published Jun 25, 2019• 0 citations
Publisher

Abstract

Abstract The collapse of attractive Bose–Einstein condensates in a box with tunable interatomic interactions was studied experimentally recently. Not only were remarkably stable remnant condensates observed, but furthermore they often seem to involve two stable plateaus. We suggest that these plateaus correspond in fact to two minima of the energy, the attractive atomic interactions being long-range. We show in detail that all the experimental data for these remnant condensates can be accounted for by minimising a nonlocal variational energy involving a long-range interatomic potential that is attractive everywhere and whose range is of the order of the magnetic length.

DOI
10.1088/1402-4896/ab2c71

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Abstract The collapse of attractive Bose–Einstein condensates in a box with tunable interatomic interactions was studied experimentally recently. Not only were remarkably stable remnant condensates observed, but furthermore they often seem to involve two stable plateaus. We suggest that these plateaus correspond in fact to two minima of the energy, the attractive atomic interactions being long-range. We show in detail that all the experimental data for these remnant condensates can be accounted for by minimising a nonlocal variational energy involving a long-range interatomic potential that is attractive everywhere and whose range is of the order of the magnetic length.
The collapse of attractive Bose-Einstein condensates in a box with tunable interatomic interactions was studied experimentally recently. Not only were remarkably stable remnant condensates observed, but furthermore they often seem to involve two stable plateaus. We suggest that these plateaus correspond in fact to two minima of the energy, the attractive atomic interactions being nonlocal. We show in detail that all the experimental data for these remnant condensates can be accounted for by minimising a variational energy involving a nonlocal interatomic potential that is attractive everywhere and that has a range of the order of the magnetic length.
Abstract Bose–Einstein condensates with tunable interatomic interactions have been studied intensely in recent experiments. The investigation of the collapse of a condensate following a sudden change in the nature of the interaction from repulsive to attractive has led to the observation of a remnant condensate that did not undergo further collapse. We suggest that this high-density remnant is in fact the absolute minimum of the energy, if the attractive atomic interactions are nonlocal, and is therefore inherently stable. We show that a variational trial function consisting of a superposition of two distinct gaussians is an accurate representation of the wavefunction of the ground state of the conventional local Gross–Pitaevskii field equation for an attractive condensate and gives correctly the points of emergence of instability. We then use such a superposition of two gaussians as a variational trial function in order to calculate the minima of the energy when it includes a nonlocal interaction term. We use experimental data in order to study the long range of the nonlocal interaction, showing that they agree very well with a dimensionally derived expression for this range.
We present an action that can be used to study variationally the collapse of Bose-Einstein condensates. This action is real, even though it includes dissipative terms. It adopts long-range interactions between the atoms, so that there is always a stable minimum of the energy, even if the remaining number of atoms is above the number that in the case of local interactions is the critical one. The proposed action incorporates the time needed for the abrupt and delayed onset of collapse, yielding in fact its dependence on the scattering length. We show that the evolution of the condensate is equivalent to the motion of a particle in an effective potential. The particle begins its motion far from the point of stable equilibrium and it then proceeds to oscillate about that point. We prove that the resulting large oscillations in the shape of the wave function after the collapse have frequencies equal to twice the frequencies of the traps. Our results agree with the experimental observations.
We consider dynamics of Bose-Einstein condensates with long-range attractive interaction proportional to 1/rb1/{r}^{b}1/rb and arbitrary angular dependence. It is shown exactly that collapse of a Bose-Einstein condensate without contact interactions is possible only for $b\ensuremath{\geqslant}2$. The case b=2b=2b=2 is critical and requires the number of particles to exceed a critical value to allow collapse. The critical collapse in that case is a strong one, trapping into the collapsing region a finite number of particles. The case b>2b>2b>2 is supercritical with an expected weak collapse which traps a rapidly decreasing number of particles during approach to collapse. For b<2b<2b<2 a singularity at r=0r=0r=0 is not strong enough to allow collapse but an attractive 1/rb1/{r}^{b}1/rb interaction admits stable self-trapping (a stable three-dimensional soliton solution) even in the absence of an external trapping potential.
The time-dependent extended Gross-Pitaevskii equation for Bose-Einstein condensates with attractive 1/r1/r1/r interaction is investigated with both a variational approach and numerically exact calculations. We show that these condensates exhibit signatures known from the nonlinear dynamics of autonomous Hamiltonian systems. The two stationary solutions created in a tangent bifurcation at a critical value of the scattering length are identified as elliptical and hyperbolical fixed points, corresponding to stable and unstable stationary states of the condensate. The stable stationary state is surrounded by elliptical islands, corresponding to condensates periodically oscillating in time, whereas condensate wave functions in the unstable region undergo a collapse within finite time. For negative scattering lengths below the tangent bifurcation no stationary solutions exist, i.e., the condensate is always unstable and collapses.
Summary form only given. Bose-Einstein condensation (BEC) of trapped atomic vapor has been realized in several atomic species. The static and dynamical properties of BEC crucially depend on the sign of the interatomic interaction. When the interaction is attractive, BEC in a spatially uniform 3D system is unstable to collapse into a denser phase. In a spatially confined system, however, the zero-point energy serves as a kinetic obstacle against collapse, allowing metastable BEC to be formed if the number of BEC atoms is below a certain critical number. Above the critical number of atoms, BEC collapses. The BEC with attractive interactions, therefore, has been restricted to a small number of atoms (/spl sim/1000) and the number has not been controllable.