Vortex Lattice Formation in Bose-Einstein Condensates

Abstract

We show that the formation of a vortex lattice in a weakly interacting Bose condensed gas can be modeled with the nonlinear Schrö dinger equation for both T 0 and finite temperatures without the need for an explicit damping term. Applying a weak rotating anisotropic harmonic potential, we find numerically that the turbulent dynamics of the field produces an effective dissipation of the vortex motion and leads to the formation of a lattice. For T 0, this turbulent dynamics is triggered by a rotational dynamic instability of the condensate. For finite temperatures, noise is present at the start of the simulation and allows the formation of a vortex lattice at a lower rotation frequency, the Landau frequency. These two regimes have different vortex dynamics. We show that the multimode interpretation of the classical field is essential. DOI: 10.1103/PhysRevLett.92.020403 PACS numbers: 03.75.Lm Vortex lattices exist in many domains of physics, from neutron stars to superconductors or liquid helium. In none of these systems has the formation of the lattice been understood at the level of a microscopic theory. Several groups have recently observed the formation of a vortex lattice in weakly interacting Bose gases [1][2][3][4] and are able to monitor this formation in real time. This gives us the chance to understand the problem of lattice formation in a relatively simple system. Indeed there have been theoretical attempts to understand the formation process [5][6][7][8] with simulations of the Gross-Pitaevskii equation for the condensate wave function. All of them stress the need for explicitly including a damping term representing the noncondensed modes to which the vortices have to give away energy to relax to a lattice configuration. In this Letter, we consider this problem in the framework of the classical theory of a complex field [9] whose exact equation of motion is the nonlinear Schrö dinger equation (NLSE). First, we show that lattice formation is predicted within this framework without the addition of damping terms. Second, we provide two distinct scenarios of vortex lattice formation (dynamics, temperature dependence of the formation time, and critical rotation frequency) that can be directly compared with the experiments. We study the formation of the lattice in 3D from an initially nonrotating Bose condensed gas both at T 0 and at finite temperature. Contrary to the common belief, we find that the dynamic instability, which was predicted in [10] to occur above a certain threshold value of the trap rotation frequency, leads to the formation of a vortex lattice. The formation time is in this case only weakly dependent of the temperature and the observed scenario and time scales are comparable to those seen in present experiments. For a lower trap rotation frequency corresponding to the Landau frequency, but only at finite temperature, we identify a new scenario not yet observed experimentally in which the vortices enter a few at a time and gradually spiral towards the center.We start our simulations with...

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