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arXiv:cond-mat/0509592

Diffusive instability of a vortex in a rotating Bose gas

Ashton S. Bradley, Crispin W. Gardiner

Published Sep 23, 2005• 1 citation
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Abstract

The dissipative dynamics of a vortex in a finite temperature trapped Bose-Einstein condensate are shown to be governed by a {\em diffusive instability}. In the weakly interacting regime we find a cross-over from instability to metastability of the vortex, determined only by the relative rotation rate of the vortex and thermal cloud. Even in the thermodynamically stable regime there is a finite lifetime arising from diffusive instability. We find the mean exit time for the vortex from the condensate, and show that near the critical frequency for thermodynamic stability the lifetime has a universal form which is independent of the condensate size and the vortex precession frequency. In the thermodynamically stable regime the steady-state two-time position correlation predicts `thermal bunching', giving an increased probability of positional coincidence for short times.

DOI
10.48550/arxiv.cond-mat/0509592

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The dissipative dynamics of a vortex in a finite temperature trapped Bose-Einstein condensate are shown to be governed by a {\em diffusive instability}. In the weakly interacting regime we find a cross-over from instability to metastability of the vortex, determined only by the relative rotation rate of the vortex and thermal cloud. Even in the thermodynamically stable regime there is a finite lifetime arising from diffusive instability. We find the mean exit time for the vortex from the condensate, and show that near the critical frequency for thermodynamic stability the lifetime has a universal form which is independent of the condensate size and the vortex precession frequency. In the thermodynamically stable regime the steady-state two-time position correlation predicts `thermal bunching', giving an increased probability of positional coincidence for short times.
We study the dynamics of vortices in finite temperature atomic Bose-Einstein condensates, focusing on decay rates, precession frequencies, and core brightness, motivated by a recent experiment [Science 329, 1182 (2010)] in which real-time dynamics of a single vortex was observed. Using the Zaremba, Nikuni, and Griffin (ZNG) formalism based on a dissipative Gross-Pitaevskii equation for the condensate coupled to a semiclassical Boltzmann equation for the thermal cloud, we find a rapid nonlinear increase of both the decay rate and precession frequency with increasing temperatures. The increase, which is dominated by the dynamical condensate-thermal coupling is also dependent on the intrinsic thermal cloud collisional dynamics; the precession frequency also varies with the initial radial coordinate. The integrated thermal cloud density in the vortex core is for the most part independent of the position of the vortex (except when it is near the condensate edge), with its value increasing with temperature. This could potentially be used as a variant to the method of Coddington et al. [Phys. Rev. A 70, 063607 (2004)] for experimentally determining the temperature.
We show that the formation of a vortex lattice in a weakly interacting Bose condensed gas can be modeled with the nonlinear Schrödinger 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.
We discuss dissipative dynamics of a vortex state in a trapped Bose-condensed gas at finite temperature and draw a scenario of decay of this state in a static trap. The interaction of the vortex with the thermal cloud transfers energy from the vortex to the cloud and induces the motion of the vortex core to the border of the condensate. Once the vortex reaches the border, it immediately decays through the creation of excitations. We calculate the characteristic lifetime of a vortex state and address the question of how the dissipative dynamics of vortices can be studied experimentally.
Based on the method of matched asymptotic expansions and on a time-dependent variational analysis, we study the dynamics of a vortex in a three-dimensional disk-shaped nonaxisymmetric condensate in the Thomas-Fermi limit. Both methods show that a vortex in a trapped Bose-Einstein condensate has formally unstable normal mode(s) with positive normalization and negative frequency, corresponding to a precession of the vortex line around the center of the trap. In a rotating trap, the solution becomes stable above an angular velocity Omega(m), characterizing the onset of metastability with respect to small transverse displacements of the vortex from the central axis.
We perform finite-temperature dynamical simulations of the arrest of a rotating Bose-Einstein condensate by a fixed trap anisotropy, using a Hamiltonian classical-field method. We consider a quasi-two-dimensional condensate containing a single vortex in equilibrium with a rotating thermal cloud. Introducing an elliptical deformation of the trapping potential leads to the loss of angular momentum from the system. We identify the condensate and the complementary thermal component of the nonequilibrium field and compare the evolution of their angular momenta and angular velocities. By varying the trap anisotropy we alter the relative efficiencies of the vortex-cloud and cloud-trap coupling. For strong trap anisotropies the angular momentum of the thermal cloud may be entirely depleted before the vortex begins to decay. For weak trap anisotropies, the thermal cloud exhibits a long-lived steady state in which it rotates at an intermediate angular velocity.
A second-order expansion of the Gross-Pitaevskii equation in the interaction parameter determines the thermodynamic critical angular velocity ${\ensuremath{\Omega}}_{c}$ for the creation of a vortex in a small axisymmetric condensate. Similarly, a second-order expansion of the Bogoliubov equations determines the (negative) frequency ${\ensuremath{\omega}}_{a}$ of the anomalous mode. Although ${\ensuremath{\Omega}}_{c}=\ensuremath{-}{\ensuremath{\omega}}_{a}$ through first order, the second-order contributions ensure that the absolute value $|{\ensuremath{\omega}}_{a}|$ is always smaller than the critical angular velocity ${\ensuremath{\Omega}}_{c}.$ With increasing external rotation $\ensuremath{\Omega},$ the dynamical instability of the condensate with a vortex disappears at ${\ensuremath{\Omega}}^{*}=|{\ensuremath{\omega}}_{a}|,$ whereas the vortex state becomes energetically stable at the larger value ${\ensuremath{\Omega}}_{c}.$ Both second-order contributions depend explicitly on the axial anisotropy of the trap. The appearance of a local minimum of the free energy for a vortex at the center determines the metastable angular velocity ${\ensuremath{\Omega}}_{m}.$ A variational calculation yields ${\ensuremath{\Omega}}_{m}=|{\ensuremath{\omega}}_{a}|$ to first order (hence ${\ensuremath{\Omega}}_{m}$ also coincides with the critical angular velocity ${\ensuremath{\Omega}}_{c}$ to this order). Qualitatively, the scenario for the onset of stability in the weak-coupling limit is the same as that found in the strong-coupling (Thomas-Fermi) limit.