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arXiv:cond-mat/0609340 · Physical Review A

Stationary solutions of Gross-Pitaevskii equations in a double square well

Weidong Li

Published Dec 11, 2006• 20 citations
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Abstract

We present analytical stationary solutions for the Gross-Pitaevskii equation (GPE) of a Bose-Einstein condensate (BECs) trapped in a double-well potential. These solutions are compared to those described by [Mahmud et al., Phys. Rev. A 66, 063607 (2002)]. In particular, we provide further evidence that symmetry preserving stationary solutions can be reduced to the eigenstates of the corresponding linear Schr\"odinger equation. Moreover, we have found that the symmetry-breaking solutions can emerge not only from bifurcations, but also from isolated points in the chemical potential--nonlinear interaction diagram. We also have found that there are some moving nodes in the symmetry-breaking solutions.

DOI
10.1103/physreva.74.063612

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We present analytical stationary solutions for the Gross-Pitaevskii equation (GPE) of a Bose-Einstein condensate (BECs) trapped in a double-well potential. These solutions are compared to those described by [Mahmud et al., Phys. Rev. A 66, 063607 (2002)]. In particular, we provide further evidence that symmetry preserving stationary solutions can be reduced to the eigenstates of the corresponding linear Schr"odinger equation. Moreover, we have found that the symmetry-breaking solutions can emerge not only from bifurcations, but also from isolated points in the chemical potential--nonlinear interaction diagram. We also have found that there are some moving nodes in the symmetry-breaking solutions.
An analytical approximation method to clarify the symmetry-breaking stationary solutions to the Gross–Pitaevskii equation with symmetric double-well external potential is presented. By this method, we can understand that each symmetry-breaking solution bifurcates from a symmetry-preserving solution as one varies the coupling constant of the nonlinear interaction, and predict the bifurcation point. Also, the method gives a good approximation to the energy eigenvalues of the lower states, symmetry-breaking as well as symmetry-preserving, as long as the nonlinear interaction is not too strong.
We treat the behavior of Bose-Einstein condensates in double square well potentials of both equal and different depths. For even depth, symmetry preserving solutions to the relevant nonlinear Schrödinger equation are known, just as in the linear limit. When the nonlinearity is strong enough, symmetry breaking solutions also exist, side by side with the symmetric one. Interestingly, solutions almost entirely localized in one of the wells are known as an extreme case. Here we outline a method for obtaining all these solutions for repulsive interactions. The bifurcation point at which, for critical nonlinearity, the asymmetric solutions branch off from the symmetry preserving ones is found analytically. We also find this bifurcation point and treat the solutions generally via a Josephson junction model. When the confining potential is in the form of two wells of different depth, interesting phenomena appear. This is true of both the occurrence of the bifurcation point for the static solutions and also of the dynamics of phase and amplitude varying solutions. Again a generalization of the Josephson model proves useful. The stability of solutions is treated briefly.
We investigate the Gross-Pitaevskii equation for a Bose-Einstein condensate in a PT symmetric double-well potential by means of the time-dependent variational principle and numerically exact solutions. A one-dimensional and a fully three-dimensional setup are used. Stationary states are determined and the propagation of wave function is investigated using the time-dependent Gross-Pitaevskii equation. Due to the nonlinearity of the Gross-Pitaevskii equation the potential dependson the wave function and its solutions decide whether or not the Hamiltonian itself is PT symmetric. Stationary solutions with real energy eigenvalues fulfilling exact PT symmetry are found as well as PT broken eigenstates with complex energies. The latter describe decaying or growing probability amplitudes and are not true stationary solutions of the time-dependent Gross-Pitaevskii equation. However, they still provide qualitative information about the time evolution of the wave functions.
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 a representative set of analytic stationary-state solutions of the nonlinear Schr\"odinger equation for a symmetric double-square-well potential for both attractive and repulsive nonlinearity. In addition to the usual symmetry-preserving even and odd states, nonlinearity introduces quite exotic symmetry-breaking solutions---among them are trains of solitons with different number and sizes of density lumps in the two wells. We use the symmetry-breaking localized solutions to form macroscopic quantum superposition states and explore a simple model for the exponentially small tunneling splitting.
The stationary solutions of the Gross-Pitaevskii equation can be divided in two classes: those which reduce, in the limit of vanishing nonlinearity, to the eigenfunctions of the associated Schrödinger equation and those which do not have linear counterpart. Analytical and numerical results support an existence condition for the solutions of the first class in terms of the ratio between their proper frequency and the corresponding linear eigenvalue. For one-dimensional confined systems, we show that solutions without linear counterpart do exist in presence of a multi-well external potential. These solutions, which in the limit of strong nonlinearity have the form of chains of dark or bright solitons located near the extrema of the potential, represent macroscopically excited states of a Bose-Einstein condensate and are in principle experimentally observable.