Nonlinear waves in Bose–Einstein condensates: physical relevance and mathematical techniques

Abstract

Abstract. The aim of the present review is to introduce the reader to some of the physical notions and of the mathematical methods that are relevant to the study of nonlinear waves in Bose-Einstein Condensates (BECs). Upon introducing the general framework, we discuss the prototypical models that are relevant to this setting for different dimensions and different potentials confining the atoms. We analyze some of the model properties and explore their typical wave solutions (plane wave solutions, bright, dark, gap solitons, as well as vortices). We then offer a collection of mathematical methods that can be used to understand the existence, stability and dynamics of nonlinear waves in such BECs, either directly or starting from different types of limits (e.g., the linear or the nonlinear limit, or the discrete limit of the corresponding equation). Finally, we consider some special topics involving more recent developments, and experimental setups in which there is still considerable need for developing mathematical as well as computational tools.PACS numbers: 03.75. Kk, 03.75.Lm, 05.45 • EP: Ermakov-Pinney (Equation)• GP: Gross-Pitaevskii (Equation)• KdV: Korteweg-de Vries (Equation)• LS: Lyapunov-Schmidt (Technique)• MT: Magnetic Trap• NLS: Nonlinear Schrödinger (Equation)• NPSE: Non-polynomial Schrödinger Equation IntroductionThe phenomenon of Bose-Einstein condensation is a quantum phase transition originally predicted by Bose [1] and Einstein [2,3] in 1924. In particular, it was shown that below a critical transition temperature T c , a finite fraction of particles of a boson gas (i.e., whose particles obey the Bose statistics) condenses into the same quantum state, known as the Bose-Einstein condensate (BEC). Although BoseEinstein condensation is known to be a fundamental phenomenon, connected, e.g., to superfluidity in liquid helium and superconductivity in metals (see, e.g., Ref.[4]), BECs were experimentally realized 70 years after their theoretical prediction: this major achievement took place in 1995, when different species of dilute alkali vapors confined in a magnetic trap (MT) were cooled down to extremely low temperatures [5][6][7], and has already been recognized through the 2001 Nobel prize in Physics [8,9]. This first unambiguous manifestation of a macroscopic quantum state in a manybody system sparked an explosion of activity, as reflected by the publication of several thousand papers related to BECs since then. Nowadays there exist more than fifty experimental BEC groups around the world, while an enormous amount of theoretical work has followed and driven the experimental efforts, with an impressive impact on many branches of Physics.From a theoretical standpoint, and for experimentally relevant conditions, the static and dynamical properties of a BEC can be described by means of an effective mean-field equation known as the Gross-Pitaevskii (GP) equation [10,11]. This is a variant of the famous nonlinear Schrödinger (NLS) equation [12] (incorporating an external potential used to confine th...

Riemann surface and quantization

“…It also works in reverse, that is, the use of classical hydro-dynamic solutions in quantum mechanics. From here we can see the possibility of the existence of quantum shock wave systems, to which many studies are devoted to [34][35][36][37][38][39][40][41].…”

Statics and dynamics of atomic dark-bright solitons in the presence of impurities

“…Assuming that the trap is highly anisotropic, with the longitudinal and transverse trapping frequencies being such that ω x ω ⊥ , we may describe this system by the following two coupled GPEs [3,4]:…”

“…In particular, there have been a vast amount of research efforts devoted to the study of macroscopic nonlinear excitations of BECs (see, e.g., Refs. [3][4][5][6][7] for reviews on this topic). In that regard, of particular interest are the so-called matter-wave solitons, of either the bright [6] or the dark [7] type, that can be supported in BECs with attractive or repulsive interactions, respectively.…”

“…[17][18][19])-a system that can be approximated by two coupled 1D Gross-Pitaevskii equations (GPEs) (see, e.g., Refs. [3,4,7]). We assume that both components are confined by the usual harmonic trap, while an additional small-amplitude localized (δ-like) impurity potential is also incorporated in both components.…”

Nonlinear modes and symmetry breaking in rotating double-well potentials

“…Refs. [27,28]), differs by replacing the propagation distance, z, with time t [29]. Stationary modes with real propagation constant −µ (in terms of the BEC, µ is the chemical potential) are sought for as ψ (θ ′ , z) = exp (−iµz) φ(θ ′ ), with complex function φ (θ ′ ) obeying equation…”