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arXiv:1507.06122 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Stability of self-gravitating Bose-Einstein condensates

Kris Schroven, Meike List, Cláus Lämmerzahl

Published Dec 2, 2015• 17 citations
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Abstract

We study the ground state and the first three radially excited states of a self-gravitating Bose-Einstein condensate with respect to the influence of two external parameters, the total mass and the strength of interactions between particles. For this we use the so-called Gross-Pitaevskii-Newton system. In this context we especially determine the case of very high total masses where the ground state solutions of the Gross-Pitaevskii-Newton system can be approximated with the Thomas-Fermi limit. Furthermore, stability properties of the computed radially excited states are examined by applying arguments of the catastrophe theory.

DOI
10.1103/physrevd.92.124008

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We study the ground state and the first three radially excited states of a self-gravitating Bose-Einstein condensate with respect to the influence of two external parameters, the total mass and the strength of interactions between particles. For this we use the so-called Gross-Pitaevskii-Newton system. In this context we especially determine the case of very high total masses where the ground state solutions of the Gross-Pitaevskii-Newton system can be approximated with the Thomas-Fermi limit. Furthermore, stability properties of the computed radially excited states are examined by applying arguments of the catastrophe theory.
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.
Abstract. In the infinite three-dimensional space, we study a homogeneous nonrelativistic Schrödinger Hamiltonian describing a self-gravitating Bose gas with two types of interactions: a repulsive scaled interaction modeled by [Formula: see text] and an attractive Newtonian potential of the form [Formula: see text]. We investigate the mean-field limit as the number of particles [Formula: see text], with coupling parameters [Formula: see text] and [Formula: see text] satisfying [Formula: see text]. In this regime, we rigorously derive the corresponding Gross–Pitaevskii–Poisson theory. Furthermore, we analyze the asymptotic behavior of the condensate profile in two opposite limiting cases: when [Formula: see text], the system is well approximated by a Thomas–Fermi-type model, whereas when [Formula: see text], the condensate approaches a Choquard–Pekar-type profile.
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.
An approximate solution to the Gross-Pitaevskii equation for Bose-Einstein condensate in a spherical harmonic trap is suggested, which is valid in the whole interval of the coupling parameter, correctly interpolating between weak-coupling and strong-coupling limits. This solution is shown to be more accurate than the optimized Gaussian approximation as well as the Thomas-Fermi approximation. The derivation of the solution is based on the self-similar approximation theory. The possibility of obtaining interpolation formulas in the case of nonspherical traps is discussed.
We study the characteristics of Bose-Einstein condensates formed by the action of three different potentials: Newtonian self-gravity, hard-sphere repulsion (the Gross-Pittaevski potential), and a harmonic trap potential as used in laboratory experiments. We show that in a certain regime a condensate can form which is much larger than predicted by the quantum self-gravitational ground state radius. Furthermore, the size is independent of the number of particles in the condensate. We speculate that massive objects of this type may form in extreme isolation through gravitational accretion onto a fixed-size object. We propose that these be called deep-space quantum objects, or DSQOs (discos). Analytical Hartree-Fock solutions for the DSQO ground state are obtained, along with a formula for the limiting DSQO radius and energy. These are then compared with numerical calculations. Prospects for DSQO formation are briefly discussed with emphasis on possible cooling mechanisms and the dynamic stability of structures formed by accretion.