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arXiv:0811.2022 · Journal of Physics A Mathematical and Theoretical

Spinor Bose–Einstein condensates in double-well potentials

C Wang, P G Kevrekidis, N Whitaker, Tristram J. Alexander, D J Frantzeskakis, P Schmelcher

Published Dec 9, 2008• 10 citations
arXivHTMLPDF

Abstract

We consider the statics and dynamics of F = 1 spinor Bose–Einstein condensates (BECs) confined in double-well potentials. We use a two-mode Galerkin-type quasi-analytical approximation to describe the stationary states of the system. This way, we are able to obtain not only earlier results based on the single-mode approximation (SMA) frequently used in studies of spinor BECs, but also additional modes that involve either two or all three spinor components of the F = 1 spinor BEC. The results based on this Galerkin-type decomposition are in good agreement with the analysis of the full system. We subsequently analyze the stability of these multi-component states, as well as their dynamics when we find them to be unstable. The instabilities of the symmetric or anti-symmetric states exhibit symmetry-breaking and recurrent asymmetric patterns. Our results yield qualitatively similar bifurcation diagrams both for polar (such as 23 Na) and ferromagnetic (such as 87 Rb) spinor BECs.

DOI
10.1088/1751-8113/42/3/035201

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We consider the statics and dynamics of F = 1 spinor Bose–Einstein condensates (BECs) confined in double-well potentials. We use a two-mode Galerkin-type quasi-analytical approximation to describe the stationary states of the system. This way, we are able to obtain not only earlier results based on the single-mode approximation (SMA) frequently used in studies of spinor BECs, but also additional modes that involve either two or all three spinor components of the F = 1 spinor BEC. The results based on this Galerkin-type decomposition are in good agreement with the analysis of the full system. We subsequently analyze the stability of these multi-component states, as well as their dynamics when we find them to be unstable. The instabilities of the symmetric or anti-symmetric states exhibit symmetry-breaking and recurrent asymmetric patterns. Our results yield qualitatively similar bifurcation diagrams both for polar (such as 23 Na) and ferromagnetic (such as 87 Rb) spinor BECs.
This dissertation concentrates on the existence, stability and dynamical properties of nonlinear waves in Bose-Einstein condensates (BECs) trapped in doublewell potentials (DWPs). The fundamental model of interest will be the nonlinear Schrödinger equation, the so-called Gross-Pitaevskii (GP) equation, contributed to the well-established mean-field description of BECs. In this context of the GP equation with DWP, a Galerkin-type few-mode approach provides us a powerful handle towards studying the stationary states and predicting the bifurcation diagram including the occurrence of spontaneous symmetry breaking (SSB). Such method and the corresponding phenomena are discussed based on a prototypical quasi-1D model in Chapter 2. The systematic analysis progresses by considering various modified models, starting with the ones involving different interatomic interactions, e.g., collisionally inhomogeneous interactions, long-range interactions, and competing of short- and long-range interactions. We observe how the basic SSB bifurcation structure persists or is appropriately modified in the presence of these interactions in Chapter 3. We also extend the study to multi-component systems, including nonlinearly coupled two-component settings and F = 1 spinor BECs (genuinely three-component settings) confined in DWPs, where besides the one-component stationary states, combined states involving two or three components appear as well, and novel SSB phenomena emerge within them. Finally the trapped stationary modes of a twodimensional (2D) GP equation with a symmetric four-well potential are explored, providing the picture of SSB in the fundamental 2D setting. These various systems are studied in Chapter 4 - 6. In all models, our analytical predictions based on the few-mode approximation are in excellent agreement with the numerical results of the full GP equations.
We present an analytical model for the theoretical analysis of spin dynamics and spontaneous symmetry breaking in a spinor Bose-Einstein condensate (BEC). This allows for an excellent intuitive understanding of the processes and provides good quantitative agreement with the experimental results of Scherer et al. [Phys. Rev. Lett. 105, 135302 (2010)]. It is shown that the dynamics of a spinor BEC initially prepared in an unstable Zeeman state mF=0{m}_{F}=0mF​=0 ($\left|0\right\ensuremath{\rangle}$) can be understood by approximating the effective trapping potential for the state $\left|\ifmmode\pm\else\textpm\fi{}1\right\ensuremath{\rangle}$ with a cylindrical box potential. The resonances in the creation efficiency of these atom pairs can be traced back to excitation modes of this confinement. The understanding of these excitation modes allows for a detailed characterization of the symmetry-breaking mechanism, showing how a twofold spontaneous breaking of spatial and spin symmetry can occur. In addition, a detailed account of the experimental methods for the preparation and analysis of spinor quantum gases is given.
We study spin-polarized states and their stability in the antiferromagnetic phase of spinor (F=1)(F=1)(F=1) quasi-one-dimensional Bose-Einstein condensates. Using analytical approximations and numerical methods, we find various types of polarized states, including patterns of the Thomas-Fermi type, structures featuring a pulse in one component inducing a hole in the other components, states with holes in all three components, and domain walls (DWs). The stability analysis based on the Bogoliubov--de Gennes equations reveals intervals of weak oscillatory instability in families of these states, except for the DWs, which are always stable. The development of the instabilities is examined by means of direct simulations.
We theoretically study the stationary-state vortex lattice configurations of rotating spin-orbit (SO)- and coherently-coupled spin-1 Bose-Einstein condensates (BECs) trapped in quasi-two-dimensional harmonic potentials. The combined effects of rotation, SO and coherent couplings are analyzed systematically from the single-particle perspective. Through the single-particle Hamiltonian, which is exactly solvable for one-dimensional coupling, we illustrate that a boson in these rotating SO- and coherently-coupled condensates are subjected to effective toroidal, symmetric double-well, or asymmetric double-well potentials under specific coupling and rotation strengths. In the presence of mean-field interactions, using the coupled Gross-Pitaevskii formalism at moderate to high rotation frequencies, the analytically obtained effective potential minima and the numerically obtained coarse-grained density maxima position are in excellent agreement. On rapid rotation, we further find that the spin-expectation per particle of an antiferromagnetic spin-1 BEC approaches unity indicating a similarity in the response with ferromagnetic SO-coupled condensates.
In this paper we study the solutions to a nonlinear Schrödinger equation with harmonic and periodic potentials, motivated from the recent interest in these models as mean-field descriptions of Bose–Einstein condensates. We use a two-mode Galerkin approximation to study the dynamics of the full model. The phase plane and stability analysis of the reduced model yield very good agreement with the findings of the full partial differential equation. A particularly interesting finding of the stability analysis is a spontaneous symmetry breaking through a branching bifurcation, resulting in the stabilization of asymmetric states and the destabilization of symmetric or anti-symmetric ones. We also highlight the important differences between the cases of symmetric potentials and those of weakly asymmetric potentials.
We examine the nonlinear dynamical behavior of a spinor Bose-Einstein condensate in a double-well potential. Considering a condensate with large number of atoms, such that it can be described using the mean field theory, we separate the spinor dynamics from the spatial dynamics under the single-mode approximation. We limit ourselves to certain initial conditions under which the spatial mode is frozen so that we can focus on the spinor dynamics only. Identifying collective spin variables of our system, we derive the corresponding nonlinear equations of motion for them. Employing standard stability analysis, we find and characterize fixed points of the system. For a wide range of physical parameters such as tunneling strength and non-linear interactions, as well as for various initial preparations of the system, we identify qualitatively different dynamical regimes possible in the system. In particular, complete and incomplete oscillations of spin variables between quantum wells are found. We also show that by bringing some fixed points close to each other in the phase space of the system, it is possible to induce amplitude modulation to those otherwise regular tunneling oscillations.