Michael Krygier

Probing the Circulation of Ring-shaped Bose-Einstein Condensates

Joint Quantum Institute, National Institute of Standards and Technologyand the University of Maryland, Gaithersburg, MD 20899, USA(Dated: January 10, 2014)This paper reports the results of a theoretical and experimental study of how the initial circulationof ring–shaped Bose–Einstein condensates (BECs) can be probed by time–of–flight (TOF) images.We have studied theoretically the dynamics of a BEC after release from a toroidal trap potential bysolving the 3D Gross–Pitaevskii (GP) equation. The trap and condensate characteristics matchedthose of a recent experiment. The circulation, experimentally imparted to the condensate by stirring,was simulated theoretically by imprinting a linear azimuthal phase on the initial condensate wavefunction. The theoretical TOF images were in good agreement with the experimental data. Wefind that upon release the dynamics of the ring–shaped condensate proceeds in two distinct phases.First, the condensate expands rapidly inward, filling in the initial hole until it reaches a minimumradius that depends on the initial circulation. In the second phase, the density at the inner radiusincreases to a maximum after which the hole radius begins slowly to expand. During this secondphase a series of concentric rings appears due to the interference of ingoing and outgoing matterwaves from the inner radius. The results of the GP equation predict that the hole area is a quadraticfunction of the initial circulation when the condensate is released directly from the trap in whichit was stirred and is a linear function of the circulation if the trap is relaxed before release. Thesescalings matched the data. Thus, hole size after TOF can be used as a reliable probe of initialcondensate circulation. This connection between circulation and hole size after TOF will facilitatefuture studies of atomtronic systems that are implemented in ultracold quantum gases.

Approximate mean-field equations of motion for quasi-two-dimensional Bose-Einstein-condensate systems.

We present a method for approximating the solution of the three-dimensional, time-dependent Gross-Pitaevskii equation (GPE) for Bose-Einstein-condensate systems where the confinement in one dimension is much tighter than in the other two. This method employs a hybrid Lagrangian variational technique whose trial wave function is the product of a completely unspecified function of the coordinates in the plane of weak confinement and a Gaussian in the strongly confined direction having a time-dependent width and quadratic phase. The hybrid Lagrangian variational method produces equations of motion that consist of (1) a two-dimensional (2D) effective GPE whose nonlinear coefficient contains the width of the Gaussian and (2) an equation of motion for the width that depends on the integral of the fourth power of the solution of the 2D effective GPE. We apply this method to the dynamics of Bose-Einstein condensates confined in ring-shaped potentials and compare the approximate solution to the numerical solution of the full 3D GPE.

Approximate mean-field equations of motion for quasi-2D Bose-Einstein condensate systems

We present a method for approximating the solution of the three-dimensional, time-dependent Gross-Pitaevskii equation (GPE) for Bose-Einstein-condensate systems where the confinement in one dimension is much tighter than in the other two. This method employs a hybrid Lagrangian variational technique whose trial wave function is the product of a completely unspecified function of the coordinates in the plane of weak confinement and a Gaussian in the strongly confined direction having a time-dependent width and quadratic phase. The hybrid Lagrangian variational method produces equations of motion that consist of (1) a two-dimensional (2D) effective GPE whose nonlinear coefficient contains the width of the Gaussian and (2) an equation of motion for the width that depends on the integral of the fourth power of the solution of the 2D effective GPE. We apply this method to the dynamics of Bose-Einstein condensates confined in ring-shaped potentials and compare the approximate solution to the numerical solution of the full 3D GPE.

Approximate Mean-field Equations of Motion for Quasi-two-dimensional Bose-Einstein Condensates

We present a method for approximating the solution of the three-dimensional, time-dependent Gross-Pitaevskii equation (GPE) for Bose-Einstein-condensate systems where the confinement in one dimension is much tighter than in the other two. This method employs a hybrid Lagrangian variational technique whose trial wave function is the product of a completely unspecified function of the coordinates in the plane of weak confinement and a Gaussian in the strongly confined direction having a time-dependent width and quadratic phase. The hybrid Lagrangian variational method produces equations of motion that consist of (1) a two-dimensional (2D) effective GPE whose nonlinear coefficient contains the width of the Gaussian and (2) an equation of motion for the width that depends on the integral of the fourth power of the solution of the 2D effective GPE. We apply this method to the dynamics of Bose-Einstein condensates confined in ring-shaped potentials and compare the approximate solution to the numerical solution of the full 3D GPE.