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A gapless theory of Bose-Einstein condensation in dilute gases at finite temperature

S. A. Morgan

Published Jan 1, 1999• 73 citations
OpenAlex

Abstract

In this thesis we develop a gapless theory of BEC which can be applied to both trapped and homogeneous gases at zero and finite temperature. The starting point for the theory is the second quantized, many-body Hamiltonian for a system of structureless bosons with pairwise interactions. A number conserving approach is used to rewrite this Hamiltonian in a form which is approximately quadratic with higher order cubic and quartic terms. The quadratic part of the Hamiltonian can be diagonalized exactly by transforming to a quasiparticle basis, while requiring that the condensate satisfy the Gross-Pitaevskii equation. The non-quadratic terms are assumed to have a small effect and are dealt with using first and second order perturbation theory. The conventional treatment of these terms, based on factorization approximations, is shown to be inconsistent. Infra-red divergences can appear in individual terms of the perturbation expansion, but we show analytically that the total contribution beyond quadratic order is finite. The resulting excitation spectrum is gapless and the energy shifts are small for a dilute gas away from the critical region, justifying the use of perturbation theory. Ultra-violet divergences can appear if a contact potential is used to describe particle interactions. We show that the use of this potential as an approximation to the two-body T-matrix leads naturally to a high-energy renormalization. The theory developed in this thesis is therefore well-defined at both low and high energy and provides a systematic description of Bose-Einstein condensation in dilute gases. It can be used to calculate the energies and decay rates of the excitations of the system at temperatures approaching the phase transition. We present analytical results for these quantities in the homogeneous limit and describe how they can be calculated numerically for trapped gases. (author)

Related papers

A gapless theory of Bose-Einstein condensation in dilute gases at finite temperature

S. A. Morgan

In this paper we develop a gapless theory of Bose-Einstein condensation (BEC) which can be applied to both trapped and homogeneous gases at zero and finite temperature. The starting point for the theory is the second quantized, many-body Hamiltonian for a system of structureless bosons with pairwise interactions. A number-conserving approach is used to rewrite this Hamiltonian in a form which is approximately quadratic with higher-order cubic and quartic terms. The quadratic part of the Hamiltonian can be diagonalized exactly by transforming to a quasiparticle basis, while requiring that the condensate satisfy the Gross-Pitaevskii equation. The non-quadratic terms are assumed to have a small effect and are dealt with using first- and second-order perturbation theory. The conventional treatment of these terms, based on factorization approximations, is shown to be inconsistent. Infrared divergences can appear in individual terms of the perturbation expansion, but we show analytically that the total contribution beyond quadratic order is finite. The resulting excitation spectrum is gapless and the energy shifts are small for a dilute gas away from the critical region, justifying the use of perturbation theory. Ultraviolet divergences can appear if a contact potential is used to describe particle interactions. We show that the use of this potential as an approximation to the two-body T -matrix leads naturally to a high-energy renormalization. The theory developed in this paper is therefore well defined at both low and high energy and provides a systematic description of BEC in dilute gases. It can therefore be used to calculate the energies and decay rates of the excitations of the system at temperatures approaching the phase transition.

arXiv:cond-mat/9911278•Journal of Physics B Atomic Molecular and Optical Physics•Sep 14, 2000•116 citations

Excitations of Bose-Einstein condensates at finite temperatures

Martin Rusch, Rusch, Martin

Recent experimental observations of collective excitations of Bose condensed atomic vapours have stimulated interest in the microscopic description of the dynamics of a Bose-Einstein condensate confined in an external potential. We present a finite temperature field theory for collective excitations of trapped Bose-Einstein condensates and use a finite-temperature linear response formalism, which goes beyond the simple mean-field approximation of the Gross-Pitaevskii equation. The effect of the non-condensed thermal atoms we include using perturbation theory in a quasiparticle basis. This presents a simple scheme to understand the interaction between condensate and non-condensed atoms and enables us to include the effect the condensate has on collision dynamics. At first we limit our treatment to the case of a spatially homogeneous Bose gas. We include the effect of pair and triplet anomalous averages and thus obtain a gapless theory for the excitations of a weakly interacting system, which we can link to well known results for Landau and Beliaev damping rates. A gapless theory for trapped systems with a static thermal component follows straightforwardly. We then investigate finite temperature excitations of a condensate in a spherically symmetric harmonic trap. We avoid approximations to the density of states and thus emphasise finite size aspects of the problem. We show that excitations couple strongly to a restricted number of modes, giving rise to resonance structure in their frequency spectra. Where possible we derive energy shifts and lifetimes of excitations. For one particular mode, the breathing mode, the effects of the discreteness of the system are sufficiently pronounced that the simple picture of an energy shift and width fails. Experiments in spherical traps have recently become feasible and should be able to test our detailed quantitative predictions.

•Jan 1, 2000•0 citations

Mean-field theory for excitations of trapped Bose condensates at finite temperatures

M. Rusch, Keith R. Burnett

We describe a method for studying the dynamics of a condensate in a weakly interacting, dilute Bose gas which is based on a finite-temperature extension of the Gross-Pitaevskii equation. We present a general finite-temperature linear response formalism for nonuniform systems and give explicit results for homogeneous ones, where we can link our treatment to well-known results. This treatment leads to a consistent ${V}^{2}$ calculation of the excitations for a weakly interacting Bose gas, including Landau and Beliaev damping as well as collisional lifetimes. We do not make use of the Popov approximation, but include pair and triplet anomalous averages to second order in the interaction potential and thus obtain a gapless theory for the excitations of a weakly interacting homogeneous gas. We also show how the formalism can be used to obtain self-consistent theories. We explicitly derive gapless equations for a weakly coupled finite-temperature condensate with a static thermal component and give a clear criterion of their validity in trapped systems.

Physical Review A•May 1, 1999•25 citations

Resonant and nonresonant interactions in cold quantum gases

J. Wächter

In the first part of this thesis, we present a unified kinetic theory that describes the finite-temperature, non-equilibrium dynamics of a Bose-Einstein condensed gas interacting with a thermal cloud in a trap. This theory includes binary interactions to second order in the interaction potential and reduces to a diagonal quantum Boltzmann equation for Bogoliubov quasiparticles. The Hartree-Fock-Bogoliubov interactions include the pairing field and are expressed as many-body T matrices to second order. The interactions thus include the correct renormalized scattering physics. This renormalized theory is automatically gapless. Thus, the excited Bogoliubov modes are naturally orthogonal to the condensate ground state. This kinetic theory is a complete second-order theory that reduces to the Gross-Pitaevskii equation and the quantum Boltzmann equation in the respective limits and thus is capable of describing the system over a wide temperature range. In the second part, we consider a many-body theory of a dilute Fermi gas near a Feshbach resonance. Experiments explore the crossover physics between the Bardeen-Cooper-Schrieffer (BCS) superfluidity of a two-spin Fermi gas, and the Bose-Einstein condensation (BEC) of composite bosons. We consider correlations between a composite boson and a fermion pair and show that such correlations are the minimal ingredients needed in a many-body theory to generate the correct boson-boson scattering length in the Bose-Einstein limit of the crossover. We also use imaginary-time propagation to find zero-temperature ground states in the BCS/BEC crossover. A cumulant expansion allows us to systematically include higher-order interactions between bosons and fermions. In particular, we calculate the Hartree term across the resonance. We further apply the cumulant-expansion method to thermal fermions and composite bosons interacting above the transition temperature in the normal phase. We numerically calculate the full time dependence in ramps across the resonance in this regime and find different two-body and many-body time scales in the system. We calculate molecular conversion efficiencies as a function of temperature and phase-space density, and find good agreement with results from JILA potassium experiments.

•Jan 1, 2007•0 citations

Interactions in low-dimensional Bose-Einstein condensates

Lee, Mark David

Recent experiments have been moving ever closer to the goal of creating cold quantum-degenerate Bose gases in one and two dimensional systems. In this thesis we investigate such systems by adapting the mean-field theories used so successfully to describe Bose-Einstein condensates in the three dimensional case. We find that the key to describing the low dimensional systems lies in an accurate description of the interactions within those systems. In three dimensions the interactions may be well approximated by a simple form in which atoms are assumed to interact in a vacuum. Many-body effects on the scattering then lead to only a small perturbation. However, in one and two dimensions we find that the inclusion of many-body effects on the scattering is crucial, even for a first order calculation. We use a many-body T-matrix to describe the interactions, and develop a simple approximation in the zero-temperature limit that involves the off-shell two-body T-matrix evaluated at a shifted interaction energy. We therefore derive expressions for the off-shell two-body T-matrix in one, two, and three dimensions, and so find expressions for the coupling parameters which describe interactions in one and two dimensional Bose gases. These lead to interaction terms in the one and two dimensional Gross-Pitaevskii equations which differ significantly from the three dimensional case. We present numerical solutions of the Gross-Pitaevskii equation in both one and two dimensions, and in one dimension we find that the results are in agreement with exact results in the extreme limits of the interaction strength. We also investigate higher order theories in the one-dimensional case, with a view to describing the cross-over from a weakly-interacting Bose-Einstein condensate to a quasicondensate as the interactions become stronger. Finally, we study the Bogoliubov-de Gennes excitation spectrum for a one dimensional condensate containing solitons. We show that it has an anomalous mode with a negative energy, and that this represents the collective motion of the soliton relative to the background condensate. (author)

•Jan 1, 2002•0 citations

Collisionless dynamics of dilute Bose gases: Role of quantum and thermal fluctuations

S. Giorgini

We study the low-energy collective oscillations of a dilute Bose gas at finite temperature in the collisionless regime. By using a time-dependent mean-field scheme we derive for the dynamics of the condensate and noncondensate components a set of coupled equations, which we solve perturbatively to second order in the interaction coupling constant. This approach is equivalent to the finite-temperature extension of the Beliaev approximation and includes corrections to the Gross-Pitaevskii theory due to both quantum and thermal fluctuations. For a homogeneous system we explicitly calculate the temperature dependence of the velocity of propagation and damping rate of zero sound. In the case of harmonically trapped systems in the thermodynamic limit, we calculate, as a function of temperature, the frequency shift of the low-energy compressional and surface modes.

arXiv:cond-mat/9911377•Physical Review A•May 17, 2000•106 citations
In this thesis we develop a gapless theory of BEC which can be applied to both trapped and homogeneous gases at zero and finite temperature. The starting point for the theory is the second quantized, many-body Hamiltonian for a system of structureless bosons with pairwise interactions. A number conserving approach is used to rewrite this Hamiltonian in a form which is approximately quadratic with higher order cubic and quartic terms. The quadratic part of the Hamiltonian can be diagonalized exactly by transforming to a quasiparticle basis, while requiring that the condensate satisfy the Gross-Pitaevskii equation. The non-quadratic terms are assumed to have a small effect and are dealt with using first and second order perturbation theory. The conventional treatment of these terms, based on factorization approximations, is shown to be inconsistent. Infra-red divergences can appear in individual terms of the perturbation expansion, but we show analytically that the total contribution beyond quadratic order is finite. The resulting excitation spectrum is gapless and the energy shifts are small for a dilute gas away from the critical region, justifying the use of perturbation theory. Ultra-violet divergences can appear if a contact potential is used to describe particle interactions. We show that the use of this potential as an approximation to the two-body T-matrix leads naturally to a high-energy renormalization. The theory developed in this thesis is therefore well-defined at both low and high energy and provides a systematic description of Bose-Einstein condensation in dilute gases. It can be used to calculate the energies and decay rates of the excitations of the system at temperatures approaching the phase transition. We present analytical results for these quantities in the homogeneous limit and describe how they can be calculated numerically for trapped gases. (author)
In this paper we develop a gapless theory of Bose-Einstein condensation (BEC) which can be applied to both trapped and homogeneous gases at zero and finite temperature. The starting point for the theory is the second quantized, many-body Hamiltonian for a system of structureless bosons with pairwise interactions. A number-conserving approach is used to rewrite this Hamiltonian in a form which is approximately quadratic with higher-order cubic and quartic terms. The quadratic part of the Hamiltonian can be diagonalized exactly by transforming to a quasiparticle basis, while requiring that the condensate satisfy the Gross-Pitaevskii equation. The non-quadratic terms are assumed to have a small effect and are dealt with using first- and second-order perturbation theory. The conventional treatment of these terms, based on factorization approximations, is shown to be inconsistent. Infrared divergences can appear in individual terms of the perturbation expansion, but we show analytically that the total contribution beyond quadratic order is finite. The resulting excitation spectrum is gapless and the energy shifts are small for a dilute gas away from the critical region, justifying the use of perturbation theory. Ultraviolet divergences can appear if a contact potential is used to describe particle interactions. We show that the use of this potential as an approximation to the two-body T -matrix leads naturally to a high-energy renormalization. The theory developed in this paper is therefore well defined at both low and high energy and provides a systematic description of BEC in dilute gases. It can therefore be used to calculate the energies and decay rates of the excitations of the system at temperatures approaching the phase transition.
Recent experimental observations of collective excitations of Bose condensed atomic vapours have stimulated interest in the microscopic description of the dynamics of a Bose-Einstein condensate confined in an external potential. We present a finite temperature field theory for collective excitations of trapped Bose-Einstein condensates and use a finite-temperature linear response formalism, which goes beyond the simple mean-field approximation of the Gross-Pitaevskii equation. The effect of the non-condensed thermal atoms we include using perturbation theory in a quasiparticle basis. This presents a simple scheme to understand the interaction between condensate and non-condensed atoms and enables us to include the effect the condensate has on collision dynamics. At first we limit our treatment to the case of a spatially homogeneous Bose gas. We include the effect of pair and triplet anomalous averages and thus obtain a gapless theory for the excitations of a weakly interacting system, which we can link to well known results for Landau and Beliaev damping rates. A gapless theory for trapped systems with a static thermal component follows straightforwardly. We then investigate finite temperature excitations of a condensate in a spherically symmetric harmonic trap. We avoid approximations to the density of states and thus emphasise finite size aspects of the problem. We show that excitations couple strongly to a restricted number of modes, giving rise to resonance structure in their frequency spectra. Where possible we derive energy shifts and lifetimes of excitations. For one particular mode, the breathing mode, the effects of the discreteness of the system are sufficiently pronounced that the simple picture of an energy shift and width fails. Experiments in spherical traps have recently become feasible and should be able to test our detailed quantitative predictions.
We describe a method for studying the dynamics of a condensate in a weakly interacting, dilute Bose gas which is based on a finite-temperature extension of the Gross-Pitaevskii equation. We present a general finite-temperature linear response formalism for nonuniform systems and give explicit results for homogeneous ones, where we can link our treatment to well-known results. This treatment leads to a consistent V2{V}^{2}V2 calculation of the excitations for a weakly interacting Bose gas, including Landau and Beliaev damping as well as collisional lifetimes. We do not make use of the Popov approximation, but include pair and triplet anomalous averages to second order in the interaction potential and thus obtain a gapless theory for the excitations of a weakly interacting homogeneous gas. We also show how the formalism can be used to obtain self-consistent theories. We explicitly derive gapless equations for a weakly coupled finite-temperature condensate with a static thermal component and give a clear criterion of their validity in trapped systems.
In the first part of this thesis, we present a unified kinetic theory that describes the finite-temperature, non-equilibrium dynamics of a Bose-Einstein condensed gas interacting with a thermal cloud in a trap. This theory includes binary interactions to second order in the interaction potential and reduces to a diagonal quantum Boltzmann equation for Bogoliubov quasiparticles. The Hartree-Fock-Bogoliubov interactions include the pairing field and are expressed as many-body T matrices to second order. The interactions thus include the correct renormalized scattering physics. This renormalized theory is automatically gapless. Thus, the excited Bogoliubov modes are naturally orthogonal to the condensate ground state. This kinetic theory is a complete second-order theory that reduces to the Gross-Pitaevskii equation and the quantum Boltzmann equation in the respective limits and thus is capable of describing the system over a wide temperature range. In the second part, we consider a many-body theory of a dilute Fermi gas near a Feshbach resonance. Experiments explore the crossover physics between the Bardeen-Cooper-Schrieffer (BCS) superfluidity of a two-spin Fermi gas, and the Bose-Einstein condensation (BEC) of composite bosons. We consider correlations between a composite boson and a fermion pair and show that such correlations are the minimal ingredients needed in a many-body theory to generate the correct boson-boson scattering length in the Bose-Einstein limit of the crossover. We also use imaginary-time propagation to find zero-temperature ground states in the BCS/BEC crossover. A cumulant expansion allows us to systematically include higher-order interactions between bosons and fermions. In particular, we calculate the Hartree term across the resonance. We further apply the cumulant-expansion method to thermal fermions and composite bosons interacting above the transition temperature in the normal phase. We numerically calculate the full time dependence in ramps across the resonance in this regime and find different two-body and many-body time scales in the system. We calculate molecular conversion efficiencies as a function of temperature and phase-space density, and find good agreement with results from JILA potassium experiments.
Recent experiments have been moving ever closer to the goal of creating cold quantum-degenerate Bose gases in one and two dimensional systems. In this thesis we investigate such systems by adapting the mean-field theories used so successfully to describe Bose-Einstein condensates in the three dimensional case. We find that the key to describing the low dimensional systems lies in an accurate description of the interactions within those systems. In three dimensions the interactions may be well approximated by a simple form in which atoms are assumed to interact in a vacuum. Many-body effects on the scattering then lead to only a small perturbation. However, in one and two dimensions we find that the inclusion of many-body effects on the scattering is crucial, even for a first order calculation. We use a many-body T-matrix to describe the interactions, and develop a simple approximation in the zero-temperature limit that involves the off-shell two-body T-matrix evaluated at a shifted interaction energy. We therefore derive expressions for the off-shell two-body T-matrix in one, two, and three dimensions, and so find expressions for the coupling parameters which describe interactions in one and two dimensional Bose gases. These lead to interaction terms in the one and two dimensional Gross-Pitaevskii equations which differ significantly from the three dimensional case. We present numerical solutions of the Gross-Pitaevskii equation in both one and two dimensions, and in one dimension we find that the results are in agreement with exact results in the extreme limits of the interaction strength. We also investigate higher order theories in the one-dimensional case, with a view to describing the cross-over from a weakly-interacting Bose-Einstein condensate to a quasicondensate as the interactions become stronger. Finally, we study the Bogoliubov-de Gennes excitation spectrum for a one dimensional condensate containing solitons. We show that it has an anomalous mode with a negative energy, and that this represents the collective motion of the soliton relative to the background condensate. (author)
We study the low-energy collective oscillations of a dilute Bose gas at finite temperature in the collisionless regime. By using a time-dependent mean-field scheme we derive for the dynamics of the condensate and noncondensate components a set of coupled equations, which we solve perturbatively to second order in the interaction coupling constant. This approach is equivalent to the finite-temperature extension of the Beliaev approximation and includes corrections to the Gross-Pitaevskii theory due to both quantum and thermal fluctuations. For a homogeneous system we explicitly calculate the temperature dependence of the velocity of propagation and damping rate of zero sound. In the case of harmonically trapped systems in the thermodynamic limit, we calculate, as a function of temperature, the frequency shift of the low-energy compressional and surface modes.