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arXiv:0912.3300 · Physical Review A

Decay of a quantum vortex: Test of nonequilibrium theories for warm Bose-Einstein condensates

S. J. Rooney, Ashton S. Bradley, P. B. Blakie

Published Feb 26, 2010• 78 citations
arXivHTMLPDF

Abstract

The decay of a vortex from a nonrotating high temperature Bose-Einstein condensate is modeled using the stochastic projected Gross-Pitaevskii equation (SPGPE). In order to exploit the tunability of temperature in SPGPE theory while maintaining the total atom number constant, we develop a simple and accurate Hartree-Fock method to estimate the SPGPE parameters for systems close to thermal equilibrium. We then calculate the lifetime of a vortex using three classical field theories that describe vortex decay in different levels of approximation. The SPGPE theory is shown to give the most complete description of the decay process, predicting significantly shorter vortex lifetimes than the alternative theories. Using the SPGPE theory to simulate vortex decay for a trapped gas of $5\ifmmode\times\else\texttimes\fi{}{10}^{5}$ $^{87}\mathrm{Rb}$ atoms, we calculate a vortex lifetime $\mathrm{t\ifmmode \bar{}\else \={}\fi{}}$ that decreases linearly with temperature, falling in the range $20\mathrm{s}>\mathrm{t\ifmmode \bar{}\else \={}\fi{}}>1.5\mathrm{s}$ corresponding to the temperature range $0.78{T}_{c}\ensuremath{\leqslant}T\ensuremath{\leqslant}0.93{T}_{c}$. The vortex lifetimes calculated provide a lower bound for the lifetime of a persistent current with unit winding number in our chosen trap geometry in the limit of vanishing vortex pinning potential.

DOI
10.1103/physreva.81.023630

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The decay of a vortex from a nonrotating high temperature Bose-Einstein condensate is modeled using the stochastic projected Gross-Pitaevskii equation (SPGPE). In order to exploit the tunability of temperature in SPGPE theory while maintaining the total atom number constant, we develop a simple and accurate Hartree-Fock method to estimate the SPGPE parameters for systems close to thermal equilibrium. We then calculate the lifetime of a vortex using three classical field theories that describe vortex decay in different levels of approximation. The SPGPE theory is shown to give the most complete description of the decay process, predicting significantly shorter vortex lifetimes than the alternative theories. Using the SPGPE theory to simulate vortex decay for a trapped gas of 5\ifmmode×\else\texttimes\fi1055\ifmmode\times\else\texttimes\fi{}{10}^{5}5\ifmmode×\else\texttimes\fi105 87Rb^{87}\mathrm{Rb}87Rb atoms, we calculate a vortex lifetime t\ifmmodeˉ\elseˉ\fi\mathrm{t\ifmmode \bar{}\else \={}\fi{}}t\ifmmodeˉ\elseˉ\fi that decreases linearly with temperature, falling in the range 20s>t\ifmmodeˉ\elseˉ\fi>1.5s20\mathrm{s}>\mathrm{t\ifmmode \bar{}\else \={}\fi{}}>1.5\mathrm{s}20s>t\ifmmodeˉ\elseˉ\fi>1.5s corresponding to the temperature range 0.78Tc\ensuremath⩽T\ensuremath⩽0.93Tc0.78{T}_{c}\ensuremath{\leqslant}T\ensuremath{\leqslant}0.93{T}_{c}0.78Tc​\ensuremath⩽T\ensuremath⩽0.93Tc​. The vortex lifetimes calculated provide a lower bound for the lifetime of a persistent current with unit winding number in our chosen trap geometry in the limit of vanishing vortex pinning potential.
We simulate the dissipative evolution of a vortex in a trapped finite-temperature dilute-gas Bose-Einstein condensate using first-principles open-systems theory. Simulations of the complete stochastic projected Gross-Pitaevskii equation for a partially condensed Bose gas containing a single quantum vortex show that the transfer of condensate energy to the incoherent thermal component without population transfer provides an important channel for vortex decay. For the lower temperatures considered, this effect is significantly larger that the population transfer process underpinning the standard theory of vortex decay, and is the dominant determinant of the vortex lifetime. A comparison with the Zaremba-Nikuni-Griffin kinetic (two-fluid) theory further elucidates the role of the particle transfer interaction, and suggests the need for experimental testing of reservoir interaction theory. The dominance of this particular energetic decay mechanism for this open quantum system should be testable with current experimental setups, and its observation would have broad implications for the dynamics of atomic matter waves and experimental studies of dissipative phenomena.
Providing a complete description of dissipative superfluid dynamics is one of the major challenges of many-body quantum field theory. In this thesis we make a fundamental step towards this goal by implementing the stochastic projected Gross-Pitaevskii equation (SPGPE) in complete form for the first time. The SPGPE is a high-temperature theory of Bose-Einstein condensate dynamics, providing a classical-field description of a low-energy subspace in contact with a thermal reservoir. The reservoir interaction terms account for dissipation and noise from thermal interactions, and arise from two distinct processes described as number-damping and energy-damping. This work advances previous applications of the SPGPE theory, which have only included number-damping processes, by implementing the energy-damping processes. We describe the properties of the deterministic and noise terms corresponding to the energy-damping process, and develop a novel algorithm to accurately and efficiently evaluate the energy-damping terms in the SPGPE. We apply the SPGPE to a range of experimentally accessible systems, considering both non-equilibrium and quasi-equilibrium dynamics. We model the experiment of Neely et al. [Phys. Rev. Lett. 111, 235301 (2013)], where stirring of a toroidally trapped Bose-Einstein condensate generates a disordered array of quantum vortices that decay, via thermal dissipation, to form a macroscopic persistent current. We perform numerical simulations of the experiment using the number-damping SPGPE and ab initio determined reservoir parameters. We quantitatively reproduce both the formation time and size of the persistent current, as measured in the experiment. In the first application of the full SPGPE, we consider the non-equilibrium dynamics of a condensate excited into a large-amplitude breathing mode. We find that in such non-equilibrium regimes, the energy-damping dominates over the number-damping process, leading to qualitatively different system dynamics. In particular, energy damping causes the system to rapidly reach thermal equilibrium without greatly depleting the condensate, showing that energy damping provides a highly coherent dissipation mechanism. Finally, we apply the SPGPE to the quasi-equilibrium dynamics of single-vortex decay. Energy-damping processes have previously been neglected for this system. SPGPE simulations show that in fact energy-damping has a dominant effect on the lifetime of a single vortex, with lifetimes less than half those predicted by the number-damping SPGPE. In contrast to the breathing mode decay, we observe little qualitative difference between the energy-damping and number-damping descriptions of vortex decay. Our findings show that while energy-damping processes are important to quantitatively describe quasi-equilibrium dynamics, the system behavior may be described by the number-damping SPGPE with a suitably modified dissipation rate.
Vortex states of weakly interacting Bose-Einstein condensates confined in three-dimensional rotating harmonic traps are investigated numerically at zero temperature. The ground state in the rotating frame is obtained by propagating the Gross-Pitaevskii equation for the condensate in imaginary time. The total energies between states with and without a vortex are compared, yielding critical rotation frequencies that depend on the anisotropy of the trap and the number of atoms. Vortices displaced from the center of nonrotating traps are found to have long lifetimes for sufficiently large numbers of atoms. The relationship between vortex stability and bound core states is explored.
Recent experiments at the University of Arizona have demonstrated superb control and imaging of flattened superfluids revealing a rich landscape of vortex dynamics. Vortex dipoles have been created and their dynamics observed [Neely et al. Phys. Rev. Lett. 104, 160410 (2010)], and highly charged persistent currents have been formed through a clever combination of stirring and dissipation [T. W. Neely, PhD thesis, (2010)]. Many ultra-cold gas systems, including these, are in a regime that we may characterize as quasi-equilibrium: despite some external forcing, a large fraction of the gas remains in thermal equilibrium. In this thesis we use the stochastic projected Gross-Pitaevskii equation (SPGPE) to make quantitative predictions of the quasi-equilibrium Bose gas. \n\nWe quantitatively compare the SPGPE with experiment by modeling the persistent current formation experiment performed at the University of Arizona [T. W. Neely, PhD thesis, (2010)]. We determine all SPGPE parameters for the toroidal system prior to simulation, enabling quantitative modeling of the experiment with no fitting parameters, giving a true test of the SPGPE theory. We find the SPGPE gives quantitative agreement with experiment, accurately predicting the size of the persistent current as well as the decay time of the vortices. We also observe the crucial role that thermal fluctuations have on enabling this agreement, showing the comprehensive SPGPE treatment is necessary to make quantitatively accurate calculations of quasi-equilibrium systems. This is the first quantitative agreement with experimental observations of vortex dynamics obtained with a theory of dissipation from first principles using no fitting parameters.\n\nWe then systematically study the Kelvin mode excitations on a vortex line in a three dimensional Bose-Einstein condensate using the SPGPE. We give a quantitative measure of the magnitude of vortex bending caused by the activation of Kelvin modes, and find that vortex bending can be suppressed by tightening the confinement along the direction of the vortex line. This leads to a strong suppression of the vortex decay rate as the system enters a regime of two-dimensional vortex dynamics, characterized by a critical oblateness.\n\nIn our final application of the SPGPE we simulate the decay of a vortex dipole. We model a recent experiment of Neely et al. [Neely et al. Phys. Rev. Lett. 104, 160410 (2010)], finding the SPGPE predicts a dipole lifetime consistent with experimental observations. We also show the experiment lies in the two-dimensional regime of vortex dynamics, validating our critical oblateness calculation.
We study the dynamics of a single vortex and a pair of vortices in quasi two-dimensional Bose-Einstein condensates at finite temperatures. To this end, we use the stochastic Gross--Pitaevskii equation, which is the Langevin equation for the Bose--Einstein condensate. For a pair of vortices, we study the dynamics of both the vortex-vortex and vortex-antivortex pairs, which are generated by rotating the trap and moving the Gaussian obstacle potential, respectively. Due to thermal fluctuations, the constituent vortices are not symmetrically generated with respect to each other at finite temperatures. This initial asymmetry coupled with the presence of random thermal fluctuations in the system can lead to different decay rates for the component vortices of the pair, especially in the case of two corotating vortices.
Experimental stirring of a toroidally trapped Bose-Einstein condensate at high temperature generates a disordered array of quantum vortices that decays via thermal dissipation to form a macroscopic persistent current [T. W. Neely et al., Phys. Rev. Lett. 111, 235301 (2013)]. We perform three-dimensional numerical simulations of the experimental sequence within the stochastic projected Gross-Pitaevskii equation using ab initio determined reservoir parameters. We find that both damping and noise are essential for describing the dynamics of the high-temperature Bose field. The theory gives a quantitative account of the formation of a persistent current, with no fitted parameters.