A minimisation approach for computing the ground state of Gross–Pitaevskii systems
Introduction
The field of low-temperature physics has been fascinating and inspiring many scientists, in particular in the last decade, see [16] and references given therein. Memorable achievements were the first experimental realisations [1], [19], [22] of a single Bose–Einstein Condensate (BEC) in 1995, and of BECs for a mixture of two and three different interacting atomic species, respectively. Mathematically, BECs are modelled by nonlinear time-dependent Schrödinger equations; more precisely, the order parameters of the condensates are solutions of a system of coupled Gross–Pitaevskii Equations (GPEs, [17], [20]).
In the present paper, we are concerned with computing the ground state of a system of GPEs, a special solution of minimal energy. To this purpose, as suggested in [5], we directly minimise the energy functional. We mention that an alternative approach for the ground state computation is provided by the imaginary time method which can also be considered as a steepest descent method, see [2], [3], [25] e.g. resulting in a parabolic evolution equation. Besides, an optimal damping algorithm based on the inverse power method is used in [11]; then, the conjugate gradient method is applied for the solution of the arising linear systems. We do not exploit these and other ([12], [23]) approaches here. Our objectives are twofold. In a general context, we present a Newton-like method for the numerical solution of a constrained minimisation problem and study its convergence. Moreover, we apply the minimisation approach specifically to a system of GPEs to simulate the ground state solution. Numerical results for two- and three-component two-dimensional condensate illustrating the effect of phase segregation [6], [7], [24] are provided.
The structure of the paper is as follows. In Section 2, we first introduce the system of GPEs in a normalised form and then state the minimisation problem. To avoid technicalities, we give a detailed description for the case of a single equation and then sketch the extension to systems. Our approach relies on a spectral decomposition of the ground state solution into Hermite basis functions (see also [4], [10]); we point out that similar ideas could be used in combination with Fourier techniques or other spectral methods. Inserting the resulting representation into the energy functional leads to a constrained nonlinear minimisation problem for the spectral coefficients. Section 3 is devoted to the description and analysis of a numerical method for the minimisation problem. We use a Newton-like method involving an approximate line-search strategy and continuation techniques. Finally, in Section 4, we illustrate the capability of our method by three numerical examples for systems of two or three coupled Gross–Pitaevskii equations in two space dimensions.
Section snippets
Ground state of Gross–Pitaevskii systems
In this section, we present a constrained minimisation approach for computing the ground state of systems of coupled Gross–Pitaevskii equations. For our purposes, it is useful to employ a normalised form of the problem which we introduce in Section 2.1. As the discussion of the general case would involve additional technicalities, we first restrict ourselves to the case of a single Gross–Pitaevskii equation; the extension to systems of coupled Gross–Pitaevskii equations is then sketched in
Constrained minimisation
For the numerical solution of the constrained minimisation problem (13), we apply a Newton-like method with line-search; the algorithm is described and analysed in the following sections. Note that in several space dimensions, a full Newton iteration is computationally expensive due to the large number of unknowns. Our approach is based on a simplified iteration where the costs of the solution of the arising linear system grow only linearly with the number of unknowns. It turns out that our new
Numerical implementation and illustrations
We have implemented the above algorithm in Matlab and it can be used without any restriction also in GNU Octave, version greater or equal to 3.0.0. In the present version, our code can treat coupled systems of GPEs in one or two space dimensions. Gauss–Hermite quadrature points are computed using the stable and efficient routines provided in [13], [14]. Hermite functions at quadrature points are computed once and for all using the stable recurrence relation (4). The program is freely available
Conclusions
In this paper, we were concerned with the numerical computation of the ground state of Gross–Pitaevskii systems. By means of a spectral discretisation, we transformed the problem into a constrained minimisation problem and employed a Newton-like method with approximate line-search for its numerical solution. The algorithm was implemented in Matlab (and successfully tested in GNU Octave); the code is available from the authors on request. The enclosed numerical examples clearly demonstrate the
Acknowledgment
The authors wish to thank Marco Squassina for providing the settings used for the experiment illustrated in Fig. 5.
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