Computational studies of a BEC trapped in multiple connected wells requiring efficient sparse matrix routines in a massively parallel processing environment

Mary Ann Leung, University of Washington

BECs are a new state of matter where dilute but strongly interacting gases behave like a single macroscopic quantum object. They provide a unique opportunity to study the peculiarities of quantum mechanics (such as entangled or Schrödinger Cat states), superfluidity, properties and interactions of cold atoms, as well as nonlinear matter wave optics. Originally predicted by Einstein and Bose in 1924-5, the first gaseous atomic BEC was made in the laboratory in 1995.

We are currently studying the dynamics of two coupled double well BECs in either a linear or a ring configuration, i.e. a mesoscopic quantum fluid in 4 neighboring wells, with tunable couplings. We work with a time dependent control Hamiltonian, H(t), an N by N matrix, and numerically solve the dynamics by time propagating N-coupled ordinary differential equations, where N=(n+w-1)!/(n!(w-1)!) with n=total number of particles, w=number of wells=4. In typical BEC experiments, n~50-1,000, and so N may be as large as 23,426 to 167,668,501; and thus the problem size necessitates deployment on massively parallel processor systems. We will focus on moderate size systems for exact solutions and to explore the physics such that appropriate approximation techniques can be developed to examine the larger system sizes.

The key computationally intensive portion of the problem is linear time propagation of the wave-function expansion coefficients vector, c(t), of dimension N, subject to initial and time dependent phase and gate control, and thus involves a time-dependent Hamiltonian. This implies solution of the coupled complex linear system i dc(t)/dt = H(t)c(t). Additionally, as the Hamiltonian matrix is highly, although irregularly, sparse an efficient sparse matrix-vector multiplication routine is essential.

We are currently exploring the physics, and possible quantum information applications, of a BEC in the four well system on a single fast processor, as well as generalizing our algorithms for N particles in M wells, and working on implementating codes running on massively parallel systems.

Abstract Author(s): Mary Ann Leung* and William P. Reinhardt**<br />Department of Chemistry, University of Washington, Seattle, WA 98195-1700 <br />&nbsp;<br /> * Department of Energy Computational Science Graduate Fellow <br />** work supported, in part by NSF Physics