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arXiv:cond-mat/0507711 · Physical Review A

Local density approximation for a perturbative equation of state

G. E. Astrakharchik

Published Dec 27, 2005• 37 citations
arXivHTMLPDF

Abstract

Knowledge of a series expansion of the equation of state provides a deep insight into the physical nature of a quantum system. Starting from a generic ``perturbative'' equation of state of a homogeneous ultracold gas we make predictions for the properties of the gas in the presence of harmonic confinement. The local density approximation is used to obtain the chemical potential, total and release energies, Thomas-Fermi size, and density profile of a trapped system in three-, two-, and one-dimensional geometries. The frequencies of the lowest breathing modes are calculated using scaling and sum-rule approaches and could be used in an experiment as a high-precision tool for obtaining the expansion terms of the equation of state. The derived formalism is applied to dilute Bose and Fermi gases in different dimensions and to integrable one-dimensional models. The physical meaning of the expansion terms in a number of systems is discussed.

DOI
10.1103/physreva.72.063620

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Knowledge of a series expansion of the equation of state provides a deep insight into the physical nature of a quantum system. Starting from a generic ``perturbative'' equation of state of a homogeneous ultracold gas we make predictions for the properties of the gas in the presence of harmonic confinement. The local density approximation is used to obtain the chemical potential, total and release energies, Thomas-Fermi size, and density profile of a trapped system in three-, two-, and one-dimensional geometries. The frequencies of the lowest breathing modes are calculated using scaling and sum-rule approaches and could be used in an experiment as a high-precision tool for obtaining the expansion terms of the equation of state. The derived formalism is applied to dilute Bose and Fermi gases in different dimensions and to integrable one-dimensional models. The physical meaning of the expansion terms in a number of systems is discussed.
We study normal state properties of an interacting Fermi gas in an isotropic harmonic trap of arbitrary dimensions. We exactly calculate the first-order perturbation terms in the ground-state energy and chemical potential, and obtain simple analytic expressions of the total energy and chemical potential. At zero temperature, we find that the Thomas–Fermi approximation agrees well with the exact results for any dimension even though the system is small. In the high-temperature (classical) region, we find the interaction energy decreases in proportion to T − d /2 , where T is the temperature and d is the dimension of the system.
We consider a Fermi gas with short-range attractive interactions that is confined along one direction by a tight harmonic potential. For this quasi-two-dimensional (quasi-2D) Fermi gas, we compute the pressure equation of state, radiofrequency spectrum, and the superfluid critical temperature Tc{T}_{c}Tc​ using a mean-field theory that accounts for all the energy levels of the harmonic confinement. Our calculation for Tc{T}_{c}Tc​ provides a natural generalization of the Thouless criterion to the quasi-2D geometry, and it correctly reduces to the 3D expression derived from the local density approximation in the limit where the confinement frequency ${\ensuremath{\omega}}_{z}\ensuremath{\rightarrow}0$. Furthermore, our results suggest that Tc{T}_{c}Tc​ can be enhanced by relaxing the confinement and perturbing away from the 2D limit.
In this Thesis we will theoretically address some issues concerning the Physics of ultracold Fermi gases, all of them with experimental relevance. The field of ultracold gases, and more recently of ultracold Fermi gases, is gathering a lot of experimental and theoretical interest for two main reasons: the great control on the relevant parameters of the problem and the relative simplicity of the minimal theory which is able to correctly describe the system. Building upon this solid background, ultracold gases provided an ideal laboratory for testing more refined theories and also for addressing fundamental issues of quantum mechanics as well as for simulating more complex physical systems such as those encountered in condensed matter. Even if the effective hamiltonian of ultracold gases can be simple, due to the diluteness of the system and the low temperature, which imply low energy physics, the solution of the quantum mechanic equations governing the state of these systems is not always simple. For the static properties usually mean-field solutions exist or perturbative expansions can be produced in some regimes. However Quantum Monte Carlo (QMC) techniques provide more accurate results especially in the strongly interacting regimes. For confined systems it is possible to use QMC only for a few particles, so that, for large number of particles, a fruitful combined use of Density Functional Theory (DFT) and QMC is necessary. The study of the dynamics of ultracold gases has received little attention with QMC techniques, due to the intrinsic computational difficulty of the many-body problem, so that general hydrodynamic equations are often used for studying the propagation of smoothly varying perturbations. In this Thesis we use QMC techniques for studying the problem of ferromagnetism in repulsive or effectively repulsive ultracold gases without a lattice and the problem of the Bardeen-Cooper-Schrieffer to Bose-Einstein-Condensation (BCS-BEC) crossover in two dimensions. We use DFT in the Local Density Approximation (LDA) for calculating the density profiles of ultracold Fermi gases in harmonic magneto-optical traps, starting from QMC equations of state. We study the propagation of first and second sound in ultracold Fermi gases in cylindrical geometry, using the hydrodynamic equations of superfluids.
We study the effect of interatomic correlations beyond the mean-field approximation on the ground-state properties of a trapped Bose gas interacting through a repulsive potential. To this end we employ a variational procedure that is based on the energy functional derived, in the local-density approximation, using the low-density expansion for the ground-state energy of the homogeneous gas. We also make analytic estimates of the corrections due to the higher-order terms in the expansion. We find that corrections are of the order of 11%.1 However, there is a qualitative change in the density profile for large N (number of atoms) due to presence of a logarithmic term in the expansion.
Abstract The critical properties of the phase transition from a normal gas to a Bose–Einstein condensate (superfluid) of a harmonically confined Bose gas are addressed with the knowledge of an equation of state of the underlying homogeneous Bose fluid. It is shown that while the presence of the confinement trap arrests the usual divergences of the isothermal compressibility and heat capacities, the critical behavior manifests itself now in the divergence of derivatives of the mentioned susceptibilities. This result is illustrated with a mean-field like model of an equation of state for the homogeneous particle density as a function of the chemical potential and temperature of the gas. The model assumes the form of an ideal Bose gas in the normal fluid while in the superfluid state a function is proposed such that, both, asymptotically reaches the Thomas–Fermi solution of a weakly interacting Bose gas at large densities and low temperatures and, at the transition, matches the critical properties of the ideal Bose gas. With this model we obtain the global thermodynamics of the harmonically confined gas, from which we analyze its critical properties. We discuss how these properties can be experimentally tested.
The local-density approximation (LDA) is the central technical tool in the modeling of quantum gases in trapping potentials. It consists in treating the gas as an assembly of independent mesoscopic fluid cells at equilibrium with a local chemical potential, and it is justified when the correlation length is larger than the size of the cells. The LDA is often regarded as a crude approximation, particularly in the ground state of the one-dimensional (1D) Bose gas, where the correlation length is ``therefore said to be'' infinite (in the sense that correlation functions decay as a power law). Here we take another look at the LDA. The local density $\ensuremath{\rho}(x)$ is viewed as a functional of the trapping potential V(x)V(x)V(x), to which one applies a gradient expansion. The zeroth order in that expansion is the LDA. The first-order correction in the gradient expansion vanishes due to reflection symmetry. At second order, there are two corrections proportional to d2V/dx2{d}^{2}V/d{x}^{2}d2V/dx2 and (dV/dx)2{(dV/dx)}^{2}(dV/dx)2, and we propose a method to determine the corresponding coefficients by a perturbative calculation in the Lieb-Liniger model. This leads to an expression for the coefficients in terms of matrix elements of the density operator, which can in principle be evaluated numerically for an arbitrary coupling constant; here we show how to efficiently evaluate the coefficient associated to the curvature of the potential d2V/dx2{d}^{2}V/d{x}^{2}d2V/dx2, which dominates the deviation to LDA near local minima or maxima of the trapping potential. Both coefficients are evaluated analytically in the limits of infinite repulsion (hard-core bosons) and small repulsion (quasicondensate). The corrected LDA density profiles are compared to density-matrix renormalization group calculations, with significant improvement compared to zeroth-order LDA.