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Weakly-Interacting Bosons in a Trap within Approximate Second Quantization Approach

Andrij A. Rovenchak

Published Nov 21, 2012• 3 citations
OpenAlex

Abstract

The theory of Bogoliubov is generalized for the case of a weakly-interacting Bose-gas in harmonic trap. A set of nonlinear matrix equations is obtained to make the diagonalization of Hamiltonian possible. Its perturbative solution is used for the calculation of the energy and the condensate fraction of the model system to show the applicability of the method. PACS numbers: 03.75.Hh, 05.30.−d 1.

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The mean ground-state occupation number and condensate fluctuations of interacting and noninteracting Bose gases confined in a harmonic trap are considered by using a canonical ensemble approach. To obtain the mean ground-state occupation number and the condensate fluctuations, an analytical description for the probability distribution function of the condensate is provided directly starting from the analysis of the partition function of the system. For the ideal Bose gas, the probability distribution function is found to be a Gaussian one for the case of the harmonic trap. For the interacting Bose gas, using a unified approach the condensate fluctuations are calculated based on the lowest-order perturbation method and on Bogoliubov theory. It is found that the condensate fluctuations based on the lowest-order perturbation theory follow the law $〈{\ensuremath{\delta}}^{2}{N}_{0}〉\ensuremath{\sim}N,$ while the fluctuations based on Bogoliubov theory behave as ${N}^{4/3}.$

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The theory of Bogoliubov is generalized for the case of a weakly-interacting Bose-gas in harmonic trap. A set of nonlinear matrix equations is obtained to make the diagonalization of Hamiltonian possible. Its perturbative solution is used for the calculation of the energy and the condensate fraction of the model system to show the applicability of the method. PACS numbers: 03.75.Hh, 05.30.−d 1.
We diagonalize the second-quantized Hamiltonian of a one-dimensional Bose gas with a non-point repulsive interatomic potential and zero boundary conditions. At a weak coupling, the solutions for the ground-state energy E0 and the dispersion law E(k) coincide with the Bogoliubov solutions for a periodic system. In this case, the single-particle density matrix F1(x, x′) at T = 0 is close to the solution for a periodic system and, at T > 0, is significantly different from it. We also obtain that the wave function ⟨w(x, t)⟩ of the effective condensate is close to a constant √︀N0/L inside the system and vanishes on the boundaries (here, N0 is the number of atoms in the effective condensate, and L is the size of the system). We find the criterion of applicability of the method, according to which the method works for a finite system at very low temperature and with a weak coupling (a weak interaction or a large concentration).
We consider a harmonically trapped dilute NNN-boson system described by a low-energy Hamiltonian with pairwise interactions. We determine the condensate fraction, defined in terms of the largest occupation number, of the weakly interacting NNN-boson system ($N\ensuremath{\geqslant}2$) by employing a perturbative treatment within the framework of second quantization. The one-body density matrix and the corresponding occupation numbers are compared with those obtained by solving the two-body problem with zero-range interactions exactly. Our expressions are also compared with high precision ab initio calculations for Bose gases with N=2−−4N=2--4N=2−−4 that interact through finite-range two-body model potentials. Nonuniversal corrections are identified to enter at subleading order, confirming that different low-energy Hamiltonians, constructed to yield the same energy, may yield different occupation numbers. Lastly, we consider the strongly interacting three-boson system under spherically symmetric harmonic confinement and determine its occupation numbers as a function of the three-body ``Efimov parameter.''
We study a system of trapped bosonic particles interacting by model harmonic forces. Our model allows for detailed examination of the notion of an order parameter (a condensate wave function). By decomposing a single particle density matrix into coherent eigenmodes we study an effect of interaction on the condensate. We show that sufficiently strong interactions cause that the condensate disappears even if the whole system is in its lowest energy state. In the second part of our paper we discuss the validity of the Bogoliubov approximation by comparing its predictions with results inferred from the exactl y soluble model. In particular we examine an energy spectrum, occupation, and fluctuations of the condensate. We conclude that Bogoliubov approach gives quite accurate description of the system in the limit of weak interactions.
In Chapter 6, we derived a generalized Gross–Pitaevskii condensate equation which is coupled to a kinetic equation for the distribution function for the thermal atoms. However, the kinetic equation in Chapter 6 is only valid in the semiclassical limit. It involves the assumption that the thermal energy k B T is much greater than the spacing between the harmonic trap energy levels ( k B T ≫ω 0 where ω 0 is the trap frequency) and also much greater than the average interaction energy ( k B T ≫ gn ). The ZNG model, based on HF excitations, is still expected to be adequate down to quite low temperatures in trapped Bose gases, as will be shown by the results in Chapter 12. However, the ZNG model will break down at very low temperatures, where the Hartree–Fock excitations must be replaced by the Bogoliubov spectrum. To deal with this, one has to derive a kinetic equation for the Bogoliubov quasiparticle excitations. This is the goal of the present chapter. In this chapter, we use the second-order Beliaev approximation to discuss the nonequilibrium dynamics of a trapped Bose-condensed gas at finite temperatures. In doing to, we combine the second-order Beliaev self-energies with the lower-order Bogoliubov excitation spectrum, including off-diagonal single-particle propagators but still omitting the anomalous correlation functions. This last condition defines the Bogoliubov–Popov approximation. In this chapter, we consider only the damping which arises from collisions. We will not explicitly calculate corrections that are second-order in g to the quasiparticle energy spectrum or to the condensate chemical potential, both of which are associated with the real parts of the second-order Beliaev self-energies. The present chapter is a natural generalization of work presented in Chapter 6 for the simpler HF excitation spectrum assumed in the ZNG analysis.
The mean ground-state occupation number and condensate fluctuations of interacting and noninteracting Bose gases confined in a harmonic trap are considered by using a canonical ensemble approach. To obtain the mean ground-state occupation number and the condensate fluctuations, an analytical description for the probability distribution function of the condensate is provided directly starting from the analysis of the partition function of the system. For the ideal Bose gas, the probability distribution function is found to be a Gaussian one for the case of the harmonic trap. For the interacting Bose gas, using a unified approach the condensate fluctuations are calculated based on the lowest-order perturbation method and on Bogoliubov theory. It is found that the condensate fluctuations based on the lowest-order perturbation theory follow the law $〈{\ensuremath{\delta}}^{2}{N}_{0}〉\ensuremath{\sim}N,$ while the fluctuations based on Bogoliubov theory behave as N4/3.{N}^{4/3}.N4/3.