Matter–wave interference in Bose–Einstein condensates: A dispersive hydrodynamic perspective

Abstract

The interference pattern generated by the merging interaction of two Bose–Einstein condensates reveals the coherent, quantum wave nature of matter. An asymptotic analysis of the nonlinear Schrödinger equation in the small dispersion (semiclassical) limit, experimental results, and three-dimensional numerical simulations show that this interference pattern can be interpreted as a modulated soliton train generated by the interaction of two rarefaction waves propagating through the vacuum. The soliton train is shown to emerge from a linear, trigonometric interference pattern and is found by use of the Whitham modulation theory for nonlinear waves. This dispersive hydrodynamic perspective offers a new viewpoint on the mechanism driving matter–wave interference.

Introduction

Rich, nonlinear, dispersive wave dynamics abound in the study of Bose–Einstein condensates (BECs). In particular, the interaction of initially separated BECs, the BEC merging problem, has yielded interesting experimental results, including interference patterns enabling atomic interferometry and illustrative of the fundamental wave nature of matter [1], [2]. Coherent nonlinear wave structures such as dispersive shock waves [3], [4], solitons, vortices, and vortex rings [4], [5], [6], [7], [8] have also been observed in the context of BEC merging experiments. Motivated by these intriguing experimental results, we study the BEC merging problem from a hydrodynamic perspective. An asymptotic analysis of the governing Gross–Pitaevskii (GP) equation in the small dispersion limit, experiments, and three-dimensional (3D) simulations are used to demonstrate the dispersive hydrodynamic origins of matter–wave interference.

The merging problem we consider here has been studied theoretically in various contexts. The interference of BECs was proposed and studied in the linear regime in [9]. Numerical studies in 1D showed the existence of interference fringes when two BECs, initially residing in a double well potential, interact [10], [11]. In [12], 1D simulations of the GP equation with two initially trapped, separated condensates were performed. It was argued that solitons are generated during the merging interaction that ensues. Indeed, in this work, we present 3D numerical as well as experimental data observing such soliton formation. Furthermore, the approximate, asymptotic (small dispersion) solution we calculate in this work demonstrates how these solitons naturally arise from a linear interference pattern when the BEC density is increased. The solution is a modulated elliptic function that reduces to linear, trigonometric waves in the small density limit. Thus, our results bridge the linear theory of matter–wave interference for sufficiently small densities [9], [10], [11] to a nonlinear theory that describes the interference pattern as a modulated train of solitons. Soliton trains in BECs have been studied previously in 1D with the aid of the Inverse Scattering Transform (IST) [13], [14]. The IST was also used to gain information about the long time behavior and interference fringe spacing of two interacting Gaussian wave packets [15].

In this work, we consider, both theoretically and experimentally, the case of an elongated, anisotropically confined BEC, often referred to as cigar shaped, following the experiments of [4]. This configuration is studied analytically in the quasi-one-dimensional (1D) regime. A dispersive hydrodynamic viewpoint is useful in this regard. Using Whitham averaging theory [16], [17] for the small-dispersion limit of the Nonlinear Schrödinger (NLS) equation, we asymptotically solve the BEC merging problem for piecewise-constant initial data that correspond to the merging interaction of two semi-infinite BECs. We demonstrate the emergence of a soliton train (interference pattern) due to the interaction of two rarefaction waves propagating through a region of zero density (vacuum). Via multiple scale arguments, this asymptotic result is extended to the 3D GP equation. New experimental results are presented and compared favorably with our asymptotic calculations and 3D numerical simulations of the GP equation.

The results we present here are closely related to dispersive shock waves (DSWs). DSWs evolve after a dispersive hydrodynamic system develops large gradients and “breaks”. They are slowly varying, expanding, oscillatory wave solutions and were studied in the context of the small dispersion limit of the NLS equation in Refs. [18], [19] with the use of Whitham averaging theory. These and classical results of viscous shock wave theory were applied to study BEC DSWs in Refs. [20], [21]. Further studies of BEC DSWs were performed in [22], [23], with the latter including experimental observation of DSW structures. In Ref. [24], Whitham averaging theory was used to describe the interactions of DSWs. The merging problem considered here (two semi-infinite, steady BECs separated by the vacuum) is related to the so-called collision problem of Ref. [24] where two DSWs counter-propagate on a nonzero background density toward one another and interact leading to modulated quasi-periodic or 2-phase behavior. In contrast, the merging problem involves the interaction of rarefaction waves propagating on the vacuum leading to a modulated periodic or 1-phase region, the BEC interference pattern.

The organization of this article is as follows. Section 2 is concerned with the calculation of the asymptotic solution to the 1D merging problem for the case of piecewise-constant initial data. We then show how this result can be used to study the merging problem for a 3D BEC in strong anisotropic confinement in Section 3. Finally, we present new experimental results and 3D numerical simulations that qualitatively agree with the aforementioned asymptotic results in Section 4.