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Journal of Sichuan Normal University

Critical Value and Stability for the Two-dimensional Attractive Bose-Einstein Condensates

Jian Zhang

Published Jan 1, 2002• 0 citations
OpenAlex

Abstract

We study the two-dimensional Gross-pitaevskii(GP) equation as a model of attractive Bose-Einstein condensation. By the rigorous theory of partial differential equations and variational arguements, we derive the critical value of condensate particles. And the value is well consistent with the existing experimental data. Moreover with this value we prove that a ground state of the condensate exists and is orbital stable. And the result agrees with the predict of Einstein.

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We study the two-dimensional Gross-pitaevskii(GP) equation as a model of attractive Bose-Einstein condensation. By the rigorous theory of partial differential equations and variational arguements, we derive the critical value of condensate particles. And the value is well consistent with the existing experimental data. Moreover with this value we prove that a ground state of the condensate exists and is orbital stable. And the result agrees with the predict of Einstein.
The characteristic features of the collapse of the ground state in trapped one-component attractive Bose-Einstein condensates are studied by applying the catastrophe theory. From numerically obtained stable and unstable solutions of the Gross-Pitaevskii equation, we derive the catastrophe function defining the stability of the stationary points on the Gross-Pitaevskii energy functional. The bifurcation diagram and the universal scaling laws stemming from the catastrophe function show quantitative agreement with the numerical results.
We study the ground state energy of trapped two-dimensional Bose gases with mean-field type interactions that can be attractive. We prove the stability of second kind of the many-body system and the convergence of the ground state energy per particle to that of a non-linear Schrödinger (NLS) energy functional. Notably, we can take any polynomial scaling of the interaction, and even exponential scalings arbitrarily close to the Gross–Pitaevskii regime, which is a drastic improvement on the best-known result for systems with attractive interactions. As a consequence of the stability of second kind we also obtain Bose–Einstein condensation for the many-body ground states for a much improved range of the diluteness parameter.
An attractive Bose-Einstein condensate in two spatial dimensions is expected to collapse for supercritical values of the interaction strength. Moreover, it is known that for nonzero quanta of angular momentum and infinitesimal attraction the gas prefers to fragment and distribute its angular momentum over different orbitals. In this work we examine the two-dimensional trapped Bose gas for finite values of attraction and describe the ground state in connection to its angular momentum by theoretical methods that go beyond the standard Gross-Pitaevskii theory. By applying the best-mean-field approach over a variational ansatz whose accuracy has been checked numerically, we derive analytical relations for the energy, the fragmentation of the ground states, and the critical (for collapse) value of the attraction strength as a function of the total angular momentum LLL.