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Dipoles and Disorder: Applications of Hartree-Fock-Bogoliubov Theory

Sam Cormack

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

We apply the Hartree-Fock-Bogoliubov-Popov (HFBP) method to two problems. The first is the phenomenon of Anderson localisation of a BEC in a two-dimensional bichromatic lattice. The second is the case of a harmonically trapped BEC with an-isotropic, long-range, dipolar interactions between particles. The HFBP method allows us to calculate the condensate properties and excitation spectrum of the sys-tem at finite temperature. For the bichromatic lattice, we examine the effects of in-teractions and finite temperature on the the localisation of the condensate. We also include a synthetic magnetic field and determine its effect. For the dipolar BEC, we examine the effect of the dipolar interactions on the condensate and thermal cloud as well as the excitation spectrum of the system. At finite temperature, we calculate the effects of both direct and exchange dipolar interactions of the thermal cloud on the condensate. The exchange interactions are numerically intensive to calculate and have been excluded from most previous studies. i

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We apply the Hartree-Fock-Bogoliubov-Popov (HFBP) method to two problems. The first is the phenomenon of Anderson localisation of a BEC in a two-dimensional bichromatic lattice. The second is the case of a harmonically trapped BEC with an-isotropic, long-range, dipolar interactions between particles. The HFBP method allows us to calculate the condensate properties and excitation spectrum of the sys-tem at finite temperature. For the bichromatic lattice, we examine the effects of in-teractions and finite temperature on the the localisation of the condensate. We also include a synthetic magnetic field and determine its effect. For the dipolar BEC, we examine the effect of the dipolar interactions on the condensate and thermal cloud as well as the excitation spectrum of the system. At finite temperature, we calculate the effects of both direct and exchange dipolar interactions of the thermal cloud on the condensate. The exchange interactions are numerically intensive to calculate and have been excluded from most previous studies. i
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.
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 present two gapless mean-field theories for trapped Bose-Einstein condensates, and compare them to the better known Hartree-Fock-Bogoliubov (HFB)-Popov approach. The proposed theories are based on suitable inclusion of the simplest anomalous average of the Bose field operator. This leads to an effective interaction between two atoms which is both temperature and density dependent, as opposed to the HFB-Popov approach, for which it is constant. These theories are compared to the HFB-Popov approach for a trapped one-dimensional condensate of 2000 atoms. The density profiles and excitation frequencies of the proposed theories differ from the corresponding HFB-Popov ones by at most a few percent.
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 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.