Complementary study of the standing wave solutions of the Gross-Pitaevskii equation in dipolar quantum gases

Abstract

ABSTRACT. We study the stability of the standing wave solutions of a Gross-Pitaevskii equation describing Bose-Einstein condensation of dipolar quantum gases and characterize their orbit. As an intermediate step, we consider the corresponding constrained minimization problem and establish existence, symmetry and uniqueness of the ground state solutions. INTRODUCTIONSince the experimental realization of the first Bose-Einstein condensate (BEC) by Eric Cornell and Carl Wieman in 1995, tremendous efforts have been undertaken by mathematicians to exploit this achievement especially in atomic physics and optics. In the last years, a new kind of quantum gases with dipolar interaction, which acts between particles as a permanent magnetic or electric dipole moment has attracted the attention of a lot of scientists. The interactions between particles are both long-range and non-isotropic. Describing the corresponding BEC via Gross Pitaevskii approximation, one gets the following nonlinear Schrödinger equation (1.1) i ∂ t ψ = − 2 2m ∆ψ + g|ψ| 2 ψ + d 2 (K * |ψ| 2 )ψ + V (x)ψ, t ∈ R, x ∈ R 3 , where |g| = 4π 2 N |a| m , N ∈ N is the number of particles, m denotes the mass of individual particles and a its corresponding scattering length. The external potential V (x) describes the electromagnetic trap and has the following harmonic confinementThe factor d 2 denotes the strength of the dipole moment in Gaussian units andwhere θ = θ(x) is the angle between x ∈ R 3 and the dipole axis n ∈ R 3 . The local term g|ψ| 2 ψ describes the short-range interaction forces between particles, while the non-local potential K * |ψ| 2 describes their long-range dipolar interactions. For the mathematical analysis, it is more convenient to rescale (1.1) into the following dimensionless formThis work was supported by the French ANR projects SchEq (ANR-12-JS01-0005-01) and BoND (ANR-13-BS01-0009-01). , and a 0 = m .In the following, we assume that λ 1 and λ 2 are two given real-valued parameters.In [5], the authors have studied the existence and uniqueness of the equation (1.3) with initial condition ψ 0 ∈ H 1 (R 3 ),They have established that (1.4) has a unique, global solution if λ 1 4 3 πλ 2 0. They called this situation stable regime, referring to the fact that no singularity in formed in finite time. In this paper, we study another notion of stability, that is, the stability of standing waves. They have also showed that in the unstable regime (λ 1 < 4 3 πλ 2 ), finite time blow up may occur, hence the denomination.The evidence of blow-up relies on a function for which the corresponding energy is strictly negative ([5, Lemma 5.1]). They concluded using the virial approach of Zakharov and Glassey. Some refinements of the above result have been discussed in [5, Proposition 5.4].The most important issue in view of the applications of (1.4) in atomic physics and quantum optics seems to be the study of ground state solutions of (1.4). These solutions are the "only" observable states in experiments. A standing wave solution of (1.4) is a wave func...

Derivation of the Dipolar Gross--Pitaevskii Energy

“…First, since V ∈ L 1 loc (R 3 ) and A ∈ L 2 loc (R 3 ) the quadratic form associated with the first two terms of the energy E a,b GP is closed on a domain included in H 1 (R 3 ) and provides a self-adjoint realization of H 0 = −(∇ + iA) 2 + V by the method of Friedrichs [19]. Then, to prove i), it suffices to follow the same proof as in [5] and to use, when necessary, that H 0 = −(∇ + iA) 2 + V has compact resolvent [17]. More precisely, let us take a minimizing sequence (u n ) for E GP .…”

“…Notice that the value of the constant R > 0 in (5) has no importance because of the cancellation property of K, see Lemma 10 below.…”

“…This variational problem has been extensively studied, both theoretically [6,5,2] and numerically [3,4] in the dipolar case (2). Our aim is to justify the validity of the Gross-Pitaevskii minimization (5), starting with the exact many-body problem based on the Hamiltonian…”

Three-dimensional vortex structures in a rotating dipolar Bose–Einstein condensate

“…However, this can be circumvented by evaluating the integral in momentum space [27,31]. For the parameters we used in this work for the dipolar GP with (a = 0), there is no minimizer [32,33], which means there is no actual ground state. So the states we refer here are in fact lowest local minimum states.…”

On the Stability of 2D Dipolar Bose--Einstein Condensates

“…It has been proved in [4,9] (see also [1,10,7,6,29] for other mathematical studies of 3D dipolar BECs, in particular for the existence of non energy minimizing stationary states) that stability holds if and only if β 0 and − β 2 λ β.…”