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arXiv:1410.4767 · SIAM Journal on Mathematical Analysis

On Dipolar Quantum Gases in the Unstable Regime

Jacopo Bellazzini, Louis Jeanjean

Published Jan 1, 2016• 133 citations
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Abstract

We study the nonlinear Schrödinger equation arising in dipolar Bose--Einstein condensate in the unstable regime. Two cases are studied: the first when the system is free, the second when gradually a trapping potential is added. In both cases we first focus on the existence and stability/instability properties of standing waves. Our approach leads to the search of critical points of a constrained functional which is unbounded from below on the constraint. In the free case, by showing that the constrained functional has a so-called mountain pass geometry, we prove the existence of standing states with least energy, the ground states, and show that any ground state is orbitally unstable. Moreover, when the system is free, we show that small data in the energy space scatter in all regimes, stable and unstable. In the second case, if the trapping potential is small, we prove that two different kinds of standing waves appear: one corresponds to a topological local minimizer of the constrained energy functional and consists in ground states, and the other is again of mountain pass type but now corresponds to excited states. We also prove that any ground state is a topological local minimizer. Despite the problem being mass supercritical and the functional being unbounded from below, the standing waves associated to the set of ground states turn out to be orbitally stable. Actually, from the physical point of view, the introduction of the trapping potential stabilizes the system that is initially unstable. Related to this we observe that it also creates a gap in the ground state energy level of the system. In addition when the trapping potential is active the presence of standing waves with arbitrary small norm does not permit small data scattering. Eventually some asymptotic results are also given.

DOI
10.1137/15m1015959

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We study the nonlinear Schrödinger equation arising in dipolar Bose--Einstein condensate in the unstable regime. Two cases are studied: the first when the system is free, the second when gradually a trapping potential is added. In both cases we first focus on the existence and stability/instability properties of standing waves. Our approach leads to the search of critical points of a constrained functional which is unbounded from below on the constraint. In the free case, by showing that the constrained functional has a so-called mountain pass geometry, we prove the existence of standing states with least energy, the ground states, and show that any ground state is orbitally unstable. Moreover, when the system is free, we show that small data in the energy space scatter in all regimes, stable and unstable. In the second case, if the trapping potential is small, we prove that two different kinds of standing waves appear: one corresponds to a topological local minimizer of the constrained energy functional and consists in ground states, and the other is again of mountain pass type but now corresponds to excited states. We also prove that any ground state is a topological local minimizer. Despite the problem being mass supercritical and the functional being unbounded from below, the standing waves associated to the set of ground states turn out to be orbitally stable. Actually, from the physical point of view, the introduction of the trapping potential stabilizes the system that is initially unstable. Related to this we observe that it also creates a gap in the ground state energy level of the system. In addition when the trapping potential is active the presence of standing waves with arbitrary small norm does not permit small data scattering. Eventually some asymptotic results are also given.
Abstract. We study the existence and stability / instability properties of standing waves for a nonlinear Schrödinger equation arising in dipolar Bose-Einstein condensate in the unstable regime. Two cases are studied: the first when the system is free, the second when gradually a trapping potential is add. In both cases this leads to the search of critical points of a constrained functional which is unbounded from below on the constraint. In the first case we prove the existence and instability of ground states by showing that the constrained functional has a so-called mountain pass geometry. In the second case we prove that, if the trapping potential is small, two different kind of standing waves appears: one corresponds to a local minima of the constrained energy functional, the other is again of mountain pass type. Despite the problem is mass supercritical and the functional unbounded from below the standing waves associated to the local minima are orbitally stable. Eventually we show that the addition of the trapping potential, however small, create a “gap ” in the ground state energy level of the system. 1.
Starting from the spectrum of the radially symmetric quantum harmonic oscillator in two dimensions, we create a large set of nonlinear solutions. The relevant three principal branches, with nr=0,1{n}_{r}=0,1nr​=0,1, and 2 radial nodes, respectively, are systematically continued as a function of the chemical potential and their linear stability is analyzed in detail, in the absence as well as in the presence of topological charge mmm, i.e., vorticity. It is found that for repulsive interatomic interactions only the ground state is linearly stable throughout the parameter range examined. Furthermore, this is true for topological charges m=0m=0m=0 or 1; solutions with higher topological charge can be unstable even in that case. All higher excited states are found to be unstable in a wide parametric regime. However, for the focusing (attractive) case the ground state with nr=0{n}_{r}=0nr​=0 and m=0m=0m=0 can only be stable for a sufficiently low number of atoms. Once again, excited states are found to be generically unstable. For unstable profiles, the dynamical evolution of the corresponding branches is also followed to monitor the temporal development of the instability.
Density-wave patterns in discrete media with local interactions are known to be unstable. We demonstrate that stable double- and triple-period patterns (DPPs and TPPs), with respect to the period of the underlying lattice, exist in media with nonlocal nonlinearity. This is shown in detail for dipolar Bose-Einstein condensates, loaded into a deep one-dimensional optical lattice. The DPP and TPP emerge via phase transitions of the second and first kind, respectively. The emerging patterns may be stable if the dipole-dipole interactions are repulsive and sufficiently strong, in comparison with the local repulsive nonlinearity. Within the set of the considered states, the TPPs realize a minimum of the free energy. A vast stability region for the TPPs is found in the parameter space, while the DPP stability region is relatively narrow. The same mechanism may create stable density-wave patterns in other physical media featuring nonlocal interactions.
The cubic nonlinear Schrödinger equation with repulsive nonlinearity and an elliptic function potential models a quasi-one-dimensional repulsive dilute gas Bose-Einstein condensate trapped in a standing light wave. New families of stationary solutions are presented. Some of these solutions have neither an analog in the linear Schrödinger equation nor in the integrable nonlinear Schrödinger equation. Their stability is examined using analytical and numerical methods. All trivial-phase stable solutions are deformations of the ground state of the linear Schrödinger equation. Our results show that a large number of condensed atoms is sufficient to form a stable, periodic condensate. Physically, this implies stability of states near the Thomas-Fermi limit.
We analyse systematically, from the viewpoint of the nonlinear physics of solitary waves, the effect of the spatial dimension (D = 1,2,3) on the structure and stability of the Bose-Einstein condensates (BECs) trapped in an external anisotropic parabolic potential. While for the positive scattering length the stationary ground-state solutions of the Gross-Pitaevskii equation are shown to be always stable independently of the spatial dimension, for the negative scattering length the ground-state condensate is stable only in the 1D and 2D cases, whereas in the 3D case it becomes unstable. A direct link between nonlinear modes of BECs and (bright and dark) solitary waves of the nonlinear Schrodinger equation is demonstrated for all the dimensions.