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arXiv:1909.08828

Excited states on Bose-Einstein condensates with attractive interactions

Peng Luo, Shuangjie Peng, Shusen Yan

Published Sep 19, 2019• 4 citations
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Abstract

We study the Bose-Einstein condensates (BEC) in two or three dimensions with attractive interactions, described by $L^{2}$ constraint Gross-Pitaevskii energy functional. First, we give the precise description of the chemical potential of the condensate $μ$ and the attractive interaction $a$. Next, for a class of degenerated trapping potential with non-isolated critical points, we obtain the existence and the local uniqueness of excited states by precise analysis of the concentrated points and the Lagrange multiplier. To our best knowledge, this is the first result concerning on excited states of BEC in Mathematics. Also, our results show that $ka_*$ are critical values in two dimension when the concentration occurs for any positive integer $k$ with some positive constant $a_*$. And we point out that our results on degenerated trapping potential with non-isolated critical points are also new even for the classical Schrödinger equations. Here our main tools are finite-dimensional reduction and various Pohozave identities. The main difficulties come from the estimates on Lagrange multiplier and the different degenerate rate along different directions at the critical points of $V(x)$.

DOI
10.48550/arxiv.1909.08828

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We study the Bose-Einstein condensates (BEC) in two or three dimensions with attractive interactions, described by L2L^{2}L2 constraint Gross-Pitaevskii energy functional. First, we give the precise description of the chemical potential of the condensate μμμ and the attractive interaction aaa. Next, for a class of degenerated trapping potential with non-isolated critical points, we obtain the existence and the local uniqueness of excited states by precise analysis of the concentrated points and the Lagrange multiplier. To our best knowledge, this is the first result concerning on excited states of BEC in Mathematics. Also, our results show that ka∗ka_*ka∗​ are critical values in two dimension when the concentration occurs for any positive integer kkk with some positive constant a∗a_*a∗​. And we point out that our results on degenerated trapping potential with non-isolated critical points are also new even for the classical Schrödinger equations. Here our main tools are finite-dimensional reduction and various Pohozave identities. The main difficulties come from the estimates on Lagrange multiplier and the different degenerate rate along different directions at the critical points of V(x)V(x)V(x).
This article is devoted to studying the model of two-dimensional attractive Bose-Einstein condensates in a trap V(x)V(x)V(x) rotating at the velocity $\Omega $. This model can be described by the complex-valued Gross-Pitaevskii energy functional. It is shown that there exists a critical rotational velocity 000 000 denotes the absolute product for the number of particles times the scattering length, and w>0w>0w>0 is the unique positive solution of Δw−w+w3=0\Delta w-w+w^3=0Δw−w+w3=0 in R2\mathbb{R}^2R2. If V(x)=∣x∣2V(x)=|x|^2V(x)=∣x∣2 and $ 0<\Omega <\Omega^*(=2)$ is fixed, we prove that, up to a constant phase, all minimizers must be real-valued, unique and free of vortices as a↗a∗a \nearrow a^*a↗a∗, by analyzing the refined limit behavior of minimizers and employing the non-degenerancy of www.
In this paper, we mainly investigate complex-valued vortex states of trapped attractive Bose-Einstein condensates (BECs), which can be described by an $ L^2- $critical constraint minimization problem. We prove that both the existence and nonexistence of vortex states may occur at the threshold $ a^*(m) $ depending on the value of $ V(0) $. When there is no vortex states at the threshold $ a^*(m) $, the limit behavior of vortex states as $ a \nearrow a^*(m) $ is also investigated if the trapping potential $ V(x) $ vanishes on the unit circle $ B(0,1) $ in $ {{\mathbb{R}}}^2 $.
Amplifying on a proposal by O'Dell et al. for the realization of Bose-Einstein condensates of neutral atoms with attractive 1∕r1∕r1∕r interaction, we point out that the instance of self-trapping of the condensate, without an external trap potential, is physically best understood by introducing appropriate ``atomic'' units. This reveals a remarkable scaling property: the physics of the condensate depends only on the two parameters N2a∕au{N}^{2}a∕{a}_{u}N2a∕au​ and $\ensuremath{\gamma}∕{N}^{2}$, where NNN is the particle number, aaa the scattering length, au{a}_{u}au​ the ``Bohr'' radius, and $\ensuremath{\gamma}$ the trap frequency in atomic units. We calculate accurate numerical results for self-trapping wave functions and potentials, and for energies, sizes, and peak densities, and compare with previous variational results. We point out the existence of a second solution of the extended Gross-Pitaevskii equation for negative scattering lengths, with and without trapping potential, which is born together with the ground state in a tangent bifurcation. This indicates the existence of an unstable collectively excited state of the condensate for negative scattering lengths.
We consider ground states of two-dimensional Bose–Einstein condensates in box-shaped trapping potentials Vext(x) with inhomogeneous attractive interactions am(x), which can be described equivalently by minimizers of Gross–Pitaevskii energy functional in bounded domains. In this paper, we prove that there is a threshold a* > 0 such that minimizers exist for 0 < a < a* and the minimizer does not exist for any a > a*. However, if a = a*, it is shown that whether minimizers exist depends sensitively on the asymptotic behaviors of m(x) near its maximum points. Moreover, based on a detailed analysis on the limit behavior of minimizers as a ↗ a*, we prove local uniqueness of minimizers under some suitable assumptions on m(x).
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.