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The S-Matrix Formulation Of Quantum Statistical Mechanics, With Applications To Cold Quantum Gas

Pye Ton How

Published Aug 31, 2011• 0 citations
OpenAlex

Abstract

A novel formalism of quantum statistical mechanics, based on the zero-temperature S-matrix of the quantum system, is presented in this thesis. In our new formalism, the lowest order approximation ("two-body approximation") corresponds to the exact resummation of all binary collision terms, and can be expressed as an integral equation reminiscent of the thermodynamic Bethe Ansatz (TBA). Two applications of this formalism are explored: the critical point of a weakly-interacting Bose gas in two dimensions, and the scaling behavior of quantum gases at the unitary limit in two and three spatial dimensions. We found that a weakly-interacting 2D Bose gas undergoes a superfluid transition at Tc [ALMOST EQUAL TO] 2[pi]n/[m log(2[pi]/mg )], where n is the number density, m the mass of a particle, and g the coupling. In the unitary limit where the coupling g diverges, the two-body kernel of our integral equation has simple forms in both two and three spatial dimensions, and we were able to solve the integral equation numerically. Various scaling functions in the unitary limit are defined (as functions of [MICRO SIGN]/T ) and computed from the numerical solutions. For bosons in three spatial dimensions, we present evidence that the gas undergoes 3 a strongly interacting version of Bose-Einstein condensation at n[lamda]T [ALMOST EQUAL TO] 1.3, where n is the number density and [lamda]T is the thermal wavelength. Finally, we look at the ratio of shear viscosity to entropy density [eta]/s of the unitary quantum gas, which has a conjectured lower bound of [eta]/s [GREATER-THAN OR EQUAL TO] /4[pi]kb based on the AdS/CFT correspondence of a strongly coupled Yang-Mills theory.

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A novel formalism of quantum statistical mechanics, based on the zero-temperature S-matrix of the quantum system, is presented in this thesis. In our new formalism, the lowest order approximation ("two-body approximation") corresponds to the exact resummation of all binary collision terms, and can be expressed as an integral equation reminiscent of the thermodynamic Bethe Ansatz (TBA). Two applications of this formalism are explored: the critical point of a weakly-interacting Bose gas in two dimensions, and the scaling behavior of quantum gases at the unitary limit in two and three spatial dimensions. We found that a weakly-interacting 2D Bose gas undergoes a superfluid transition at Tc [ALMOST EQUAL TO] 2[pi]n/[m log(2[pi]/mg )], where n is the number density, m the mass of a particle, and g the coupling. In the unitary limit where the coupling g diverges, the two-body kernel of our integral equation has simple forms in both two and three spatial dimensions, and we were able to solve the integral equation numerically. Various scaling functions in the unitary limit are defined (as functions of [MICRO SIGN]/T ) and computed from the numerical solutions. For bosons in three spatial dimensions, we present evidence that the gas undergoes 3 a strongly interacting version of Bose-Einstein condensation at n[lamda]T [ALMOST EQUAL TO] 1.3, where n is the number density and [lamda]T is the thermal wavelength. Finally, we look at the ratio of shear viscosity to entropy density [eta]/s of the unitary quantum gas, which has a conjectured lower bound of [eta]/s [GREATER-THAN OR EQUAL TO] /4[pi]kb based on the AdS/CFT correspondence of a strongly coupled Yang-Mills theory.
A new treatment of the critical point of the two-dimensional interacting Bose gas is presented. In the lowest order approximation we obtain the critical temperature Tc≈2πn/[mlog(2π/mg)], where n is the density, m the mass, and g the coupling. This result is based on a new formulation of interacting gases at finite density and temperature which is reminiscent of the thermodynamic Bethe ansatz in one dimension. In this formalism, the basic thermodynamic quantities are expressed in terms of a pseudo-energy. Consistent resummation of 2-body scattering leads to an integral equation for the pseudo-energy with a kernel based on the logarithm of the exact 2-body S-matrix.
In three spatial dimensions, in the unitary limit of a non-relativistic quantum Bose or Fermi gas, the scattering length diverges. This occurs at a renormalization group fixed point. Thus these systems present interesting examples of interacting scale-invariant models with dynamical exponent z = 2. We study this problem in two and three spatial dimensions using the S -matrix-based approach to the thermodynamics we recently developed. It is well suited to the unitary limit where the S matrix S = − 1, since it allows an expansion in the inverse coupling. We define a meaningful scale-invariant, unitary limit in two spatial dimensions, where again the scattering length diverges. In the two-dimensional case, the integral equation for the pseudo-energy becomes transcendentally algebraic and we can easily compute the various universal scaling functions as a function of μ/ T , such as the energy per particle. The ratio of the shear viscosity to the entropy density η/ s is above the conjectured lower bound of for all cases except attractive bosons. For attractive two-component fermions, , whereas for attractive bosons .
A new analytic treatment of three-dimensional homogeneous Bose and Fermi gases in the unitary limit of negative infinite scattering length is presented, based on the S -matrix approach to statistical mechanics we recently developed. The unitary limit occurs at a fixed point of the renormalization group with dynamical exponent z = 2 where the S -matrix equals − 1. For fermions we find T c / T F ≈0.1. For bosons we present evidence that the gas does not collapse, but rather has a critical point that is a strongly interacting form of Bose–Einstein condensation. This bosonic critical point occurs at n λ T 3 ≈1.3, where n is the density and λ T the thermal wavelength, which is lower than the ideal gas value of ζ(3/2) = 2.61.
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)
The refinement and the combination of laser cooling and evaporative cooling in dilute systems of alkali atoms lead to the achievement of Bose-Einstein condensation in 1995. Nowadays there is a fruitful interplay between theoretical and experimental results: in this context, the study of uniform Bose gases plays an important role in the comprehension of the static and dynamic properties of a bosonic system. Here we use quantum field theory, within the approach of functional integration, to study an homogeneous system of weakly-interacting bosonic cold atoms. After characterizing their thermodynamical properties, we derive - at a gaussian level - the number density and the superfluid density in three and in two spatial dimensions, expressed as functions of the condensate density and the temperature. Since the equations derived are valid for a generic interaction potential, we implement these formulas for bosons with a zero-range interaction, providing also an extension of these results with the inclusion of a finite-range interaction.
Recent developments in statistical mechanics of quantum integrable systems are reviewed. Those studies are fundamental and have a renewed interest related to newly developing fields such as atomic Bose-Einstein condensations, photonic crystals and quantum computations. After a brief summary of the basic concepts and methods, the following three topics are discussed. First, by the thermal Bethe ansatz (TBA), a hard-core Bose gas is exactly solved. The model includes fully the effect of excluded volume and is identified to be a c=1 conformal field theory. Second, the cluster expansion method based on the periodic boundary condition for the Bethe wave function, which we call the Bethe ansatz cluster expansion (BACE) method, is developed for a δ-function gas and the XXX Heisenberg chain. This directly proves the TBA and reveals intrinsic properties of quantum integrable systems. Third, for a δ-function gas, the integral equations for the distribution functions of the quasi-momentum and the quasi-particle energy are solved in the form of power series. In the weak coupling case, the results reproduce those of Bogoliubov theory.