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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Floquet-Bloch operator for the Bose-Hubbard model with static field

Andrey R. Kolovsky, Andreas Buchleitner

Published Nov 25, 2003• 59 citations
Publisher

Abstract

This paper deals with the spectral properties of the one-dimensional Bose-Hubbard Hamiltonian amended by an external static field-a model for cold spinless atoms loaded in a quasi-one-dimensional optical lattice and subject to an additional static (for example, gravitational) force. The analysis is performed in terms of the Floquet-Bloch operator, defined as the evolution operator of the system over one Bloch period. Depending on the particular choice of parameters, the spectrum is found to be either regular or chaotic. Moreover, in the chaotic case, the matrix of the Floquet-Bloch operator is well characterized as a random matrix of the circular orthogonal ensemble.

DOI
10.1103/physreve.68.056213

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This paper deals with the spectral properties of the one-dimensional Bose-Hubbard Hamiltonian amended by an external static field-a model for cold spinless atoms loaded in a quasi-one-dimensional optical lattice and subject to an additional static (for example, gravitational) force. The analysis is performed in terms of the Floquet-Bloch operator, defined as the evolution operator of the system over one Bloch period. Depending on the particular choice of parameters, the spectrum is found to be either regular or chaotic. Moreover, in the chaotic case, the matrix of the Floquet-Bloch operator is well characterized as a random matrix of the circular orthogonal ensemble.
We study a model for ultracold, spinless atoms in quasi-one-dimensional optical lattices and subjected to a tunable tilting force. Statistical tests are employed to quantitatively characterize the spectrum of the Floquet-Bloch operator of the system along the transition from the regular to the quantum chaotic regime. Moreover, we perturbatively include the coupling of the one-band model to the second energy band. This allows us to study the Landau-Zener interband tunneling within a truly many-body description of ultracold atoms. The distributions of the computed tunneling rates provide an independent and experimentally accessible signature of the regular-chaotic transition.
The extended effective multiorbital Bose–Hubbard-type Hamiltonian which takes into account higher Bloch bands is discussed for boson systems in optical lattices, with emphasis on dynamical properties, in relation to current experiments. It is shown that the renormalization of Hamiltonian parameters depends on the dimension of the problem studied. Therefore, mean-field phase diagrams do not scale with the coordination number of the lattice. The effect of Hamiltonian parameters renormalization on the dynamics in reduced one-dimensional optical lattice potential is analyzed. We study both the quasi-adiabatic quench through the superfluid–Mott insulator transition and the absorption spectroscopy, that is, the energy absorption rate when the lattice depth is periodically modulated.
We study the Bloch dynamics of a quasi-one-dimensional Bose-Einstein condensate of cold atoms in a tilted optical lattice modeled by a Hamiltonian of Bose-Hubbard type. The corresponding mean-field system described by a discrete nonlinear Schr\"odinger equation can exhibit dynamical (or modulation) instability due to chaotic dynamics and equipartition over the quasimomentum modes. It is shown that these phenomena are related to Bogoliubov's depletion of the Bose-Einstein condensate and a decoherence of the condensate in the many-particle description. Three types of dynamics are distinguished: (i) decaying oscillations in the region of dynamical instability and (ii) persisting Bloch oscillations or (iii) periodic decay and revivals in the region of stability.
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.