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Control of instability of one dimensional stationary state Bose-Einstein condensates

Wenhua Hai, Guishu Chong

Published Apr 30, 2007• 0 citations
OpenAlex

Abstract

We explore how to control the dynamical instability of a quasi-one dimensional (1D) stationary state Bose-Einstein condensate (BEC) consisting of fixed $N$ atoms with arbitrary time-independent external potential and atomic scattering length. The dependence of the perturbed solutions on the velocity field, external potential, condensed atom number, and the initial perturbations is demonstrated analytically, that shows the stability, instability and undetermined stability of the system. Some sufficient conditions on the stability and controlling criteria to the instability are established. We take a BEC held in an optical lattice potential as an example establishing the stability regions on parameter space and illustrating the scheme for controlling instability. The results contain the known stability assertions for the lattice potential case and agree with the recent experimental findings qualitatively.

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We explore how to control the dynamical instability of a quasi-one dimensional (1D) stationary state Bose-Einstein condensate (BEC) consisting of fixed NNN atoms with arbitrary time-independent external potential and atomic scattering length. The dependence of the perturbed solutions on the velocity field, external potential, condensed atom number, and the initial perturbations is demonstrated analytically, that shows the stability, instability and undetermined stability of the system. Some sufficient conditions on the stability and controlling criteria to the instability are established. We take a BEC held in an optical lattice potential as an example establishing the stability regions on parameter space and illustrating the scheme for controlling instability. The results contain the known stability assertions for the lattice potential case and agree with the recent experimental findings qualitatively.
Using the direct perturbation technique, this paper obtains a general perturbed solution of the Bose—Einstein condensates trapped in one-dimensional tilted optical lattice potential. We also gave out two necessary and sufficient conditions for boundedness of the perturbed solution. Theoretical analytical results and the corresponding numerical results show that the perturbed solution of the Bose-Einstein condensate system is unbounded in general and indicate that the Bose—Einstein condensates are Lyapunov-unstable. However, when the conditions for boundedness of the perturbed solution are satisfied, then the Bose-Einstein condensates are Lyapunov-stable.
Energy band structure and stability of Bose-Einstein condensate with both two- and three-body interactions in one-dimensional periodic optical lattices are discussed. It is demonstrated that the lowest energy band can show a loop structure at the boundary of the first Brillouin zone only when the number of atoms is larger than a critical value. When the loop structure is excited, an analytical expression of the loop width is obtained. The energetical and dynamical stabilities of the system are determined by considering the behavior of the Gross-Pitaevskii energy functional when the condensate wave function diverge away from a stationary state upon small perturbation, and its dependence on the number of atoms, the magnitude of the periodic potential, especially, the strength of the three-body force are studied. In addition, a criteria of the stability for excitations with long wavelengths is given by applying the hydrodynamic approach. The influence of the three-body interaction on the band properties and the stabilities are discussed in detail. It is shown that the critical density for developing the loop structure, the loop width, and the instability criteria are modified dramatically by the three-body interaction.
It is shown that the quasi-one-dimensional Bose-Einstein condensate is experimentally accessible and rich in intriguing phenomena. We demonstrate numerically and analytically the existence, stability and perturbation-induced dynamics of all types of stationary states of the quasi-one-dimensional nonlinear Schrödinger equation for both repulsive and attractive cases. Among our results are: the connection between stationary states and solitons; creation of vortices from such states; manipulation of such states with simple phase profiles; demonstration of the fragility of the condensate phase in response to shock; and a robust stabilization of the attractive Bose-Einstein condensate.
Motivated by recent experimental observations [Parker, Hua, and Chin, Nat. Phys. 9, 769 (2013)], we analyze the stability of a Bose-Einstein condensate (BEC) in a one-dimensional lattice subjected to periodic shaking. In such a system there is no thermodynamic ground state, but there may be a long-lived steady state, described as an eigenstate of a ``Floquet Hamiltonian.'' We calculate how scattering processes lead to a decay of the Floquet state. We map out the phase diagram of the system and find regions where the BEC is stable and regions where the BEC is unstable against atomic collisions. We show that Parker et al. performed their experiment in the stable region, which accounts for the long lifetime of the condensate $(\ensuremath{\sim}1\mathrm{s})$. We also estimate the scattering rate of the bosons in the region where the BEC is unstable.
We investigated the stability properties of breather soliton trains in a three-dimensional Bose–Einstein condensate (BEC) with Feshbach-resonance management of the scattering length. This is done so as to generate both attractive and repulsive interaction. The condensate is confined only by a one-dimensional optical lattice and we consider strong, moderate and weak confinement. By strong confinement we mean a situation in which a quasi two-dimensional soliton is created. Moderate confinement admits a fully three-dimensional soliton. Weak confinement allows individual solitons to interact. Stability properties are investigated by several theoretical methods such as a variational analysis, treatment of motion in effective potential wells, and collapse dynamics. Armed with all the information forthcoming from these methods, we then undertake a numerical calculation. Our theoretical predictions are fully confirmed, perhaps to a higher degree than expected. We compare regions of stability in parameter space obtained from a fully three-dimensional analysis with those from a quasi two-dimensional treatment, when the dynamics in one direction are frozen. We find that in the three-dimensional case the stability region splits into two parts. However, as we tighten the confinement, one of the islands of stability moves toward higher frequencies and the lower frequency region becomes more and more like that for the quasi two-dimensional case. We demonstrate these solutions in direct numerical simulations and, importantly, suggest a way of creating robust three-dimensional solitons in experiments in a BEC in a one-dimensional lattice.