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arXiv:1012.1679 · Physical Review A

Dynamical Hartree-Fock-Bogoliubov theory of vortices in Bose-Einstein condensates at finite temperature

B. G. Wild, David A. W. Hutchinson

Published Jun 29, 2011• 0 citations
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Abstract

We present a method utilizing the continuity equation for the condensate density to make predictions of the precessional frequency of single off-axis vortices and of vortex arrays in Bose-Einstein condensates at finite temperature. We also present an orthogonalized Hartree-Fock-Bogoliubov (HFB) formalism. We solve the continuity equation for the condensate density self-consistently with the orthogonalized HFB equations and find stationary solutions in the frame rotating at this frequency. As an example of the utility of this formalism we obtain time-independent solutions for quasi-two-dimensional rotating systems in the corotating frame. We compare these results with time-dependent predictions where we simulate stirring of the condensate.

DOI
10.1103/physreva.83.063635

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We present a method utilizing the continuity equation for the condensate density to make predictions of the precessional frequency of single off-axis vortices and of vortex arrays in Bose-Einstein condensates at finite temperature. We also present an orthogonalized Hartree-Fock-Bogoliubov (HFB) formalism. We solve the continuity equation for the condensate density self-consistently with the orthogonalized HFB equations and find stationary solutions in the frame rotating at this frequency. As an example of the utility of this formalism we obtain time-independent solutions for quasi-two-dimensional rotating systems in the corotating frame. We compare these results with time-dependent predictions where we simulate stirring of the condensate.
In this thesis we develop an orthogonalised Hartree-Fock-Bogoliubov (HFB) formalism that has a zero-energy excitation (in contrast with standard HFB). We demonstrate that this formalism satisfies the number and linear/angular momentum conservation laws (as does standard HFB). This formalism is applied to vortices in 2D Bose-Einstein condensates (BECs) in axially-symmetric harmonic traps, where we initially find solutions for on-axis vortices, determining the energy spectrum and hence the lowest core localised state (LCLS) energy. In the T=0 case we identify this with the anomalous mode which gives a zero excitation energy in the frame rotating at this frequency. For this reason the anomalous mode frequency was identified in the earlier literature with the precessional frequency for an off-axis vortex. However the LCLS energy is positive in the finite temperature case. Hence, associating this LCLS energy with the precessional frequency leads to the erroneous conclusion that the vortex precesses in the opposite direction to the T=0 case, which is clearly physically unreasonable. In order to address this problem, we derive an equation for the prediction of vortex precessional frequencies from the continuity equation, and use this equation solved self-consisently with the orthogonal HFB equations in the frame rotating at this predicted frequency to create off-axis vortices at pre-specified positions. Hence we are able to predict the precessional frequencies and show that these are consistent with the T=0 case, and are entirely unrelated to the LCLS energy. We also consider a two-state model and demonstrate that this model is insufficient for the description of single off-axis precessing vortices. We formulate a generalised multi-state model, using the normalisation conditions for the model to derive an equation predicting the precessional frequency of the vortex, and demonstrate equivalence with the continuity equation prediction. We use the time-dependent HFB equations to simulate creation of vortices by stirring the BEC by means of a Gaussian optical potential, finding very good agreement of the measured precessional frequencies of the vortices in stirred BECs with the predicted values. We find the existence of a critical stirring frequency for the creation of vortices in regions of appreciable superfluid density, in qualitative agreement with experiment. We then investigate the consequences of breaking rotational symmetry and find that breaking the axial symmetry of the harmonic trapping potential leads to loss of angular momentum, and hence to the decay of vortices. Finally we develop a finite temperature treatment of the Bose-Hubbard model based upon the Hartree-Fock-Bogoliubov formalism in the Popov approximation to study the effect of temperature upon the superfluid phase of ultracold, weakly interacting bosons in a one-dimensional optical lattice.
We demonstrate that the precessional frequencies of vortices in Bose-Einstein condensates (BECs) are determined by a conservation law and not by the lowest-lying excitation energy mode. We determine the precessional frequency for a single off-axis vortex and vortex lattices in BECs using the continuity equation and solve this self-consistently with the time-independent Hartree-Fock-Bogoliubov equations in the rotating frame. We find agreement with zero-temperature calculations (Bogoliubov approximation) and a smooth variation in the precession frequency as the temperature is increased. Time-dependent solutions confirm the validity of these predictions.
Abstract We introduce the map of dynamics of quantum Bose gases into dynamics of quasifree states, which we call the “nonlinear quasifree approximation”. We use this map to derive the time-dependent Hartree–Fock–Bogoliubov (HFB) equations describing the dynamics of quantum fluctuations around a Bose–Einstein condensate. We prove global well-posedness of the HFB equations for pair potentials satisfying suitable regularity conditions, and we establish important conservation laws. We show that the space of solutions of the HFB equations has a symplectic structure reminiscent of a Hamiltonian system. This is then used to relate the HFB equations to the HFB eigenvalue equations discussed in the physics literature. We also construct Gibbs equilibrium states at positive temperature associated with the HFB equations, and we establish criteria for the appearance of Bose–Einstein condensation.
We present a detailed finite-temperature Hartree-Fock-Bogoliubov (HFB) treatment of the two-dimensional trapped Bose gas. We highlight the numerical methods required to obtain solutions to the HFB equations within the Popov approximation, the derivation of which we outline. This method has previously been applied successfully to the three-dimensional case and we focus on the unique features of the system which are due to its reduced dimensionality. These can be found in the spectrum of low-lying excitations and in the coherence properties. We calculate the Bragg response and the coherence length within the condensate in analogy with experiments performed in the quasi-one-dimensional regime [Richard et al., Phys. Rev. Lett. 91, 010405 (2003)] and compare to results calculated for the one-dimensional case. We then make predictions for the experimental observation of the quasicondensate phase via Bragg spectroscopy in the quasi-two-dimensional regime.
The dynamics of Bose-Einstein condensate (BEC) is studied at nonzero temperatures using our variational time-dependent-Hartree-Fock-Bogoliubov formalism. We have shown that this approach is an efficient tool to study the expansion and collective excitations of the condensate, the thermal cloud, and the anomalous correlation function at nonzero temperatures. We have found that the condensate and the anomalous density have the same breathing oscillations. We have investigated, on the other hand, the behavior of a single quantized vortex in a harmonically trapped BEC at nonzero temperatures. Generalized expressions for vortex excitations, vortex core size, and Kelvin modes have been derived. An important and somehow surprising result is that the numerical solution of our equations predicts that the vortex core is partially filled by the thermal atoms at nonzero temperatures. We have shown that the effect of thermal fluctuations is important and it may lead to enhancing the size of the vortex core. The behavior of the singly anomalous vortex has also been studied at nonzero temperatures.
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.