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Physical Review E

Semiclassical analysis of the Bogoliubov spectrum in the Bose-Hubbard model

Andrey R. Kolovsky

Published Aug 13, 2007• 8 citations
Publisher

Abstract

We analyze the Bogoliubov spectrum of the Bose-Hubbard model with a finite number of sites and Bose particles by using a semiclassical approach. This approach allows us to take into account the finite-size effects responsible for evolution of the Bogoliubov spectrum into an irregular (chaotic) spectrum at higher energies. A manifestation of this transition for the excitation dynamics of the Bose-Hubbard system is discussed as well.

DOI
10.1103/physreve.76.026207

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We analyze the Bogoliubov spectrum of the Bose-Hubbard model with a finite number of sites and Bose particles by using a semiclassical approach. This approach allows us to take into account the finite-size effects responsible for evolution of the Bogoliubov spectrum into an irregular (chaotic) spectrum at higher energies. A manifestation of this transition for the excitation dynamics of the Bose-Hubbard system is discussed as well.
We analyze the Bogoliubov spectrum of the three-site Bose-Hubbard model with a finite number of Bose particles by using a semiclassical approach. The Bogoliubov spectrum is shown to be associated with the low-energy regular component of the classical Hubbard model. We identify the full set of the integrals of motion of this regular component and, quantizing them, obtain the energy levels of the quantum system. The critical values of the energy, above which the regular Bogoliubov spectrum evolves into a chaotic spectrum, is indicated as well.
We analyze the spectrum of the three-site Bose-Hubbard model with periodic boundary conditions using a semiclassical method. The Bohr-Sommerfeld quantization is applied to an effective classical Hamiltonian which we derive using resonance normal form theory. The derivation takes into account the 1:1 resonance between frequencies of a linearized classical system and brings nonlinear terms into a corresponding normal form. The obtained expressions reproduce the exact low-energy spectrum of the system remarkably well even for a small number of particles NNN corresponding to fillings of just two particles per site. Such small fillings are often used in current experiments, and it is inspiring to get insight into this quantum regime using essentially classical calculations.
A semiclassical theory is provided for the metastability regime diagram of atomtronic superfluid circuits. Such circuits typically exhibit high-dimensional chaos, and nonlinear resonances that couple the Bogoliubov excitations manifest. Contrary to the expectation, these resonances do not originate from the familiar Beliaev and Landau damping terms. Rather, they are described by a variant of the Cherry Hamiltonian of celestial mechanics. Consequently, we study the induced decay process and its dependence on the number of sites and condensed particles.
We present a strong-coupling expansion of the Bose-Hubbard model which describes both the superfluid and the Mott phases of ultracold bosonic atoms in an optical lattice. By performing two successive Hubbard-Stratonovich transformations of the intersite hopping term, we derive an effective action which provides a suitable starting point to study the strong-coupling limit of the Bose-Hubbard model. This action can be analyzed by taking into account Gaussian fluctuations about the mean-field approximation as in the Bogoliubov theory of the weakly interacting Bose gas. In the Mott phase, we reproduce results of previous mean-field theories and also calculate the momentum distribution function. In the superfluid phase, we find a gapless spectrum and compare our results with the Bogoliubov theory.
One-dimensional Bose-Hubbard models are well known to obey a transition from regular to quantum-chaotic spectral statistics. We are extending this concept to relatively simple two-dimensional many-body models. Also in two dimensions a transition from regular to chaotic spectral statistics is found and discussed. In particular, we analyze the dependence of the spectral properties on the bond number of the two-dimensional lattices and the applied boundary conditions. For maximal connectivity, the systems behave most regularly in agreement with the applicability of mean-field approaches in the limit of many nearest-neighbor couplings at each site.