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Path-integral approach to bose condensation

Sivinee Sawatdiaree, Virulh Sa-yakanit

Published Jan 1, 1996• 0 citations
Publisher

Abstract

The excitation spectrum of the interacting Bose gas has two branches. The first branch was dominated in the low momentum area and high density of the ground state particles and the second one was dominated in the high momentum area and was existed in the density of the ground state particles which lower than the existed density of the ground state particles of the first branch. The functional integrals method was used. The Hamiltonian was introduced in form of the combination of field operators which was divided into two parts; ground state particles and exicted particles were described respectively the action and partition function were calculated and were introduced partition function poles which were defined as excitation spectrum. According to the definition of excitation energy of ground was equal to zero, parameters were calculated and devided both forms of Bose excitation spectrums in phase space. The interaction between boson particles in the system was replaced by our ansatz interaction. Its behaviour was like the Lennard-Jones potential. The ansatz interaction was mentioned in configuration space was varied with [e -alphar/(betar)2 - e -alphar/(betar)] by assigning α, β as unity with the inversion unit of length.

DOI
10.58837/chula.the.1996.2257

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The excitation spectrum of the interacting Bose gas has two branches. The first branch was dominated in the low momentum area and high density of the ground state particles and the second one was dominated in the high momentum area and was existed in the density of the ground state particles which lower than the existed density of the ground state particles of the first branch. The functional integrals method was used. The Hamiltonian was introduced in form of the combination of field operators which was divided into two parts; ground state particles and exicted particles were described respectively the action and partition function were calculated and were introduced partition function poles which were defined as excitation spectrum. According to the definition of excitation energy of ground was equal to zero, parameters were calculated and devided both forms of Bose excitation spectrums in phase space. The interaction between boson particles in the system was replaced by our ansatz interaction. Its behaviour was like the Lennard-Jones potential. The ansatz interaction was mentioned in configuration space was varied with [e -alphar/(betar)2 - e -alphar/(betar)] by assigning α, β as unity with the inversion unit of length.
Introduction The one-dimensional Bose gas with point-like interaction of the particles (the quantum variant of the nonlinear Schrödinger equation) is one of the main and most important models which can be Solved by means of the Bethe Ansatz. This model has been thoroughly investigated (and. We shall start with the construction of eigenfunctions of the Hamiltonian in a finite volume. Quantities interesting from the physical point of view (in the thermodynamic limit at zero temperature) are considered; the thermodynamics at finite temperatures is investigated in detail as well. A number of essential ideas which will be applied to other models are introduced. The construction of eigenfunctions of the Hamiltonian is explained in section 1. Their explicit form and, in particular, the two-particle reducibility, are common features of the models solvable by means of the Bethe Ansatz method. Periodic boundary conditions are imposed on the wave function in section 2; the Bethe equations for the particle' momenta are introduced and analyzed. Taken in the logarithmic form, these equations realize the extremum condition of a certain functional, the corresponding action being called the Yang-Yang action. The transition to the thermodynamic limit is considered in section 3. In that same section the ground state of the gas is constructed. The density of the particle distribution in momentum space and the energy of the ground state are calculated. The method of the transition to the thermodynamic limit described in this section is rather general and may be applied to any model solvable by means of the Bethe Ansatz. In section 4, excitations above the ground state are constructed and their main characteristics (energy, momentum and scattering matrix) are determined by means of the dressing equations. The ground state of the model is the Dirac sea (also called the Fermi sphere).
We apply Feynman's path integral theory to study Bose-Einstein condensates in a double well potential and in a random potential. The main idea of this thesis is to perform the mean field approximation in two body interaction by the path integral approach. Bose particles are confined in the double well which is taken as a harmonic potential with Gaussian or cosine barrier. Performing the variational calculations we obtain analytical results of the ground state energy, wave functions, and overlap integrals. The overlap integrals enable us to calculate the tunneling rates which are in good agreement with numerical values for the two-mode model of Gross-Pitaevskii's equation. We also consider the model of a Bose system consisting of N particles with finite interacting pair potentials under the influence of random potentials. The porosity or nonhomogeneity of the system can be represented by the density of porosity with amplitude of the fluctuation, and a correlation length. We obtain analytical results of the effective mass and the ground state energy which are in good agreement with that of Bugoliubov approach. The propagator of the system is used to calculate the partition function, specific heat, the critical temperature, condensate density, and the superfluid density of the system.
The well-known results concerning a dilute Bose gas with the short-range repulsive interaction should be reconsidered due to a thermodynamic inconsistency of the method being basic to much of the present understanding of this subject. The aim of our paper is to propose another way of treating the dilute Bose gas with an arbitrary strong interaction. Using the reduced density matrix of the second order and a variational procedure, this way allows us to escape the inconsistency mentioned and operate with singular potentials of the Lennard-Jones type. The derived expansion of the condensate depletion in powers of the boson density n=N/V reproduces the familiar result, while the expansion for the mean energy per particle is of the form epsilon=2 pi planck(2)an/m[1+128/(15 square root of [pi]) square root of [na(3)](1-5b/8a)+...], where a is the scattering length and b> or =0 stands for one more characteristic length depending on the shape of the interaction potential (in particular, for the hard spheres a=b). All the consideration concerns the zero temperature.
Bose-Einstein condensates have been experimentally realized in cold Bose gases for the first time in 1995.This discovery has renewed interest in the theory of interacting Bose gases; however, the mathematical understanding of Bose-Einstein condensation is still incomplete.In this thesis we derive effective theories for the dynamical and spectral properties of Bose gases.Our theoretical description is based on scaling regimes where the number of particles N becomes large and the two-body interaction potential scales as N 3β-1 V (N β •), for a fixed β ∈ (0, 1].The case β = 1 is known as Gross-Pitaevskii regime.Our first result concerns the time evolution of Bose-Einstein condensates after the trapping potential is removed.In previous works, it was shown that the many-body dynamics can be approximated by a one-body nonlinear Schrödinger equation at the level of reduced density matrices.We provide a more precise approximation of the time evolution: in a second-quantized setting, we construct a unitary effective dynamics, which, in the limit of large N , approximates the fully evolved state in Fock space norm.Our result comes with an explicit bound on the rate of convergence.It is valid for β < 1.Our second result addresses the properties of the ground state wave function.It has been shown before that the ground state wave function exhibits condensation in the limit N → ∞, i.e. the one-particle density matrix associated to the ground state approaches a projection.We prove the optimal bound for the convergence rate, for a translation invariant system.To do this, we estimate uniformly in N the number of excited particles.This result is valid for β = 1.Our last result concerns the ground state energy and the excitation spectrum of the translation invariant system.We prove the validity of Bogoliubov theory; this allows us to compute the ground state energy and the low-lying excitation spectrum up to an error that vanishes for large N .This result holds for β < 1.This thesis is based on the articles [11], [9] and [10].i
By making a systematic analysis of correlation functions in terms of irreducible and reducible parts, Ma and others have developed a dielectric formulation of the dynamics of a condensed Bose system which fully includes the background (or excited) atoms. We show that this formulation quite generally leads to the density and single-particle-correlation functions both having two resonances ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{1}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{2}$ although with different weights. If the condensate n0=0{n}_{0}=0n0​=0, the ${\ensuremath{\omega}}_{2}$ mode only appears in the single-particle spectrum while ${\ensuremath{\omega}}_{1}$ only appears in the density-fluctuation spectrum. For finite values of n0{n}_{0}n0​, these modes are coupled and renormalized to ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{1}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{2}$. As a specific illustration, we use the shielded-potential approximation (SPA) for the reducible self-energies. In this model, we find the free-particle excitations (${\ensuremath{\omega}}_{2}$) are coupled to the zero-sound density fluctuations (${\ensuremath{\omega}}_{1}$) through the action of the condensate. In the SPA, the reducible self-energies have a pole at ${\ensuremath{\omega}}_{1}$. The usual Bogoliubov, Hartree-Fock, and ordinary ttt-matrix approximations involve nonsingular self-energies and hence do not exhibit the ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{1}$ mode. Experimentally, it appears that the high-frequency ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{2}$ mode has never been detected, but this is probably owing to the fact that it is strongly damped as a result of its coupling to the zero-sound mode. Using the two-fluid hydrodynamic equations, one can argue that at low frequencies the ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{1}$ excitation corresponds to first sound and the ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{2}$ excitation corresponds to second sound. Finally, we briefly discuss the possibility of singularities in the irreducible self-energies at high frequencies and the resulting two-roton bound states.