Ground States of Two-component Bose-Einstein Condensates with an Internal Atomic Josephson Junction

Abstract

Abstract. In this paper, we prove existence and uniqueness results for the ground states of the coupled Gross-Pitaevskii equations for describing two-component Bose-Einstein condensates with an internal atomic Josephson junction, and obtain the limiting behavior of the ground states with large parameters. Efficient and accurate numerical methods based on continuous normalized gradient flow and gradient flow with discrete normalization are presented, for computing the ground states numerically. A modified backward Euler finite difference scheme is proposed to discretize the gradient flows. Numerical results are reported, to demonstrate the efficiency and accuracy of the numerical methods and show the rich phenomena of the ground sates in the problem.

Modeling and Computation of Bose-Einstein Condensates: Stationary States, Nucleation, Dynamics, Stochasticity

“…At the end of the computation, we obtain each component of the stationary state and some interesting physical outputs (see Table 3). We can also simultaneously print out the moduli of the components and conclude that they are the same as the ones reported in [17] (see Figure 12). …”

“…This problem has been studied for example in [16,17]. Let us recall the following result which proves that the energy associated with the solution to (75) is decaying under suitable assumptions.…”

“…In this example, we want to reproduce the numerical simulations obtained in [17] where the following one-dimensional (d = 1) system of GPEs with a Josephson junction is considered…”

“…In addition, the (strong) stopping criterion for computing the stationary states is 10 −6 . Following [17], the values of the physical parameters are: λ = −1, δ = 0, β = 500, β 11 = β , β 12 = 0.94β , β 22 = 0.97β . The initial data is a centered gaussian for each component.…”

Acceleration of the imaginary time method for spectrally computing the stationary states of Gross–Pitaevskii equations

“…adaptive mesh techniques. However, in some situations like for dipolar interactions [11,12] or multi-components BECs [9,10,11], these approaches may be complicate to implement and lead to huge computational costs. The direction that we follow here uses pseudo-spectral (SP) discretization techniques based on the Fast Fourier Transform (FFT) [5,7,11].…”