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arXiv:1209.4402 · Physical Review A

Finite-temperature dipolar ultracold Bose gas with exchange interactions

Sam Cormack, David A. W. Hutchinson

Published Nov 20, 2012• 10 citations
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Abstract

We develop finite-temperature theory for a trapped dipolar Bose gas including thermal-exchange interactions. Previous treatments neglected these difficult-to-compute terms. We present a methodology for numerically evaluating the thermal-exchange contributions, making use of cylindrical symmetry. We then investigate properties of the dipolar gas, including calculating the excitation spectrum over the full range of trap anisotropy. We evaluate the contributions due to thermal exchange noting that, under some regimes, these effects can be at least as significant as the direct interaction. We therefore provide guidance as to when these cumbersome terms can be neglected and when care should be exercised regarding their omission.

DOI
10.1103/physreva.86.053619

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We develop finite-temperature theory for a trapped dipolar Bose gas including thermal-exchange interactions. Previous treatments neglected these difficult-to-compute terms. We present a methodology for numerically evaluating the thermal-exchange contributions, making use of cylindrical symmetry. We then investigate properties of the dipolar gas, including calculating the excitation spectrum over the full range of trap anisotropy. We evaluate the contributions due to thermal exchange noting that, under some regimes, these effects can be at least as significant as the direct interaction. We therefore provide guidance as to when these cumbersome terms can be neglected and when care should be exercised regarding their omission.
We develop a finite-temperature Hartree theory for the trapped dipolar Bose gas. We use this theory to study thermal effects on the mechanical stability of the system and density oscillating condensate states. We present results for the stability phase diagram as a function of temperature and aspect ratio. In oblate traps above the critical temperature for condensation, we find that the Hartree theory predicts significant stability enhancement over the semiclassical result. Below the critical temperature, we find that thermal effects are well described by accounting for the thermal depletion of the condensate. Our results also show that density oscillating condensate states occur over a range of interaction strengths that broadens with increasing temperature.
The work of this thesis concerns the properties of a Bose gas of polarised dipoles that have long-range and anisotropic interactions. Our work is divided into two parts. First, the stability of a dipolar Bose gas at finite temperature (both above and below the critical Bose-Einstein condensation (BEC) temperature Tc). Second, the fluctuations of a dipolar BEC at zero and small finite temperature (T ≪ Tc) in a regime where rotonic excitations emerge. \n \nPart I: Stability of a Trapped Finite Temperature Dipolar Bose Gas: Above Tc we implement a semiclassical Hartree-Fock theory and characterise the dependence of the stability boundary on temperature, trap geometry and the strength of the dipole-dipole interaction and contact interaction. We find that stability is greatly enhanced above Tc and that trap geometry continues to play a key role. Furthermore, we find that for oblate traps a novel double instability feature emerges. \nTo extend our stability analysis to the low temperature regime, T < Tc, we develop a beyond semiclassical Hartree theory. We use this to characterise the stability boundary as a function of geometry. Interestingly, we find large beyond semiclassical effects above Tc for prolate trapping geometries. We characterise thermal effects on biconcave condensate states. \n \nPart II: Rotons and Fluctuations in a Trapped Dipolar Condensate: To study density fluctuations we implement a numerical scheme to solve the Gross-Pitaevskii equation and the Bogoliubov de Gennes equations. We find that the phonon and roton gases spatially separate and we characterise the role of the anomalous density on the density fluctuations of the thermally activated rotons. We develop a numerical scheme that calculates number fluctuations within cells of various shapes and sizes, and find a strong peak in the fluctuations when the cell size is around half the roton wavelength, which should be detectable by current experiments. By tailoring the cell shape we predict that experiments should be able to detect the effects of individual roton modes. \nFor the study of zero temperature fluctuations we deploy the Gross-Pitaevskii and Bogoliubov de Gennes equations to calculate the dynamic and static structure factors for a highly oblate BEC. We find a clear signature of the roton gas dispersion relation within the structure factors. This signature should be detectible in current experiments using Bragg spectroscopy.
We study a Bose-Einstein condensate of a dilute gas with dipolar interactions, at finite temperature, using the Hartree-Fock-Bogoliubov theory within the Popov approximation. An additional approximation involving the dipolar exchange interaction is made to facilitate the computation. We calculate the temperature dependence of the condensate fraction of a condensate confined in a cylindrically symmetric harmonic trap. We show that the biconcave-shaped condensates found in [Ronen et al. Phys. Rev. Lett. 98, 30406 (2007)] in certain pancake traps at zero temperature are also stable at finite temperature. Surprisingly, the dip in the central density of these structured condensates is actually enhanced at low finite temperatures. We explain this effect.
We calculate the stability diagram for a trapped normal Bose gas with dipole-dipole interactions. Our study characterizes the roles of trap geometry, temperature, and short-range interactions on the stability. We predict a robust double instability feature in oblate trapping geometries arising from the interplay of thermal gas saturation and the anisotropy of the interaction. Our results are relevant to current experiments with polar molecules and will be useful in developing strategies to obtain a polar molecule Bose-Einstein condensate.
We develop and apply meanfield methods to two systems of significant interest in cold atom physics. The first system, and the main subject of this thesis, is a gas of polarized dipoles, in which the particles have a long range anisotropic interaction. The second system is a toroidally trapped gas that interacts via short range interactions. \n \nDipolar gases: We develop accurate techniques for dipolar Bose and Fermi gases accounting for both direct and exchange interactions. We detail an efficient numerical implementation of the meanfield treatment. \n \nWe study the thermodynamic and the first and second order correlation properties of the system. We find that exchange interactions cause the correlation functions to be anisotropic in the low temperature regime. We also find that many uniform gas thermodynamic predictions, for which direct interaction effects vanish, are qualitatively unreliable for trapped systems, most notably for oblate traps. \n \nUsing Hartree-Fock theory and analytic approximations, we examine the magnetostrictive position and momentum space distortions that occur in harmonically confined dipolar Bose and Fermi gases. The Bose gas momentum distribution distorts in the opposite sense to that of the Fermi gas. By relating exchange effects to short range correlations between the particles we discuss the energetic origin of this difference. \n \nIntroducing local momentum distortion fields, we develop a practical Hartree-Fock theory in which exchange effects are described by a local effective potential. This theory is applicable at zero and finite temperature, and is in excellent agreement with full Hartree-Fock calculations. \n \nContact gases: We show how considerable simplification of numerical results and analytical derivations can be achieved in the local density approximation using a variety of density of states techniques. \n \nWe examine the process of isentropic loading of a Bose or Fermi gas from a harmonic trap into the scale invariant toroidal regime that exhibits a high degree of system invariance when increasing the radius of the toroid. For bosons a regime of cooling is identified and we find that the Fermi gas is subjected to irreducible heating during loading, caused by the loss of one degree of freedom for thermalization.
We present a finite-temperature analysis of a quasi-two-dimensional (Q2D) dipolar gas. To do this, we use the Hartree-Fock-Bogoliubov method within the Popov approximation. This formalism is a set of nonlocal equations containing the dipole-dipole interaction and the condensate and thermal correlation functions, which are solved self-consistently. We detail the numerical method used to implement the scheme. We present density profiles for a finite-temperature dipolar gas in Q2D and compare these results to those for a gas with zero-range interactions. Additionally, we analyze the excitation spectrum and study the impact of the thermal exchange.