Faraday Patterns in Bose-Einstein Condensates

Parametric excitation of Bose-Einstein condensates (BECs) has been shown numerically to create similar patterns as those produced when a liquid is vibrated up and down sinusoidally. In this project, we reproduce the results of single-frequency forcing in BECs and conduct a prelimi-nary investigation of double-frequency forcing. We analyze the stability of the spatially homogeneous, temporally periodic solution of the Gross-Pitaevskii (GP) equation, which describes the macroscopic dynamics of BECs, by perturbing it and investigating the linear Mathieu equation governing the dynamics of the perturbation numerically and analytically. Patterns in different stability regimes can be visualized by numerically integrating the GP equation. The manifestation of similar patterns in disparate systems sug-gests that there is a universality in appropriate coarse-grained de-scriptions of their underlying dynamics. The Faraday experiment provides one important pattern-forming system that serves as an ex-ample of such ubiquity. By forcing waves on the surface of a fluid parametrically one can produce ‘Faraday patterns ’ of various forms [1]. If two different sinusoidal forcings are applied to the same fluid, patterns can appear on different scales—these are called superlat-tice patterns [3], [4]. It has been shown numerically that parametric excitation of Bose-Einstein condensates yields Faraday-like patterns, exemplifying the universality discussed above. This paper describes the duplication of known results about the Faraday patterns formed in Bose-Einstein condensates with one sinusoidal forcing term [5], and a preliminary investigation of the patterns formed when two sinusoidal forcing terms are present.