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arXiv:1910.12489 · AIP conference proceedings

Chaotic and regular dynamics in the three-site Bose-Hubbard model

Anna A. Bychek, Pavel S. Muraev, Dmitrii N. Maksimov, Evgeny N. Bulgakov, Andrey R. Kolovsky

Published Jan 1, 2020• 7 citations
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Abstract

We analyze the energy spectrum of the three-site Bose-Hubbard model. It is shown that this spectrum is a mixture of the regular and irregular spectra associated with the regular and chaotic components of the classical Bose-Hubbard model. We find relative volumes of these components by using the pseudoclassical approach. Substituting these values in the Berry-Robnik distribution for the level spacing statistics we obtain good agreement with the numerical data.

DOI
10.1063/5.0011540

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We analyze the energy spectrum of the three-site Bose-Hubbard model. It is shown that this spectrum is a mixture of the regular and irregular spectra associated with the regular and chaotic components of the classical Bose-Hubbard model. We find relative volumes of these components by using the pseudoclassical approach. Substituting these values in the Berry-Robnik distribution for the level spacing statistics we obtain good agreement with the numerical data.
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 consider a quantum many-body system-the Bose-Hubbard system on three sites-which has a classical limit, and which is neither strongly chaotic nor integrable but rather shows a mixture of the two types of behavior. We compare quantum measures of chaos (eigenvalue statistics and eigenvector structure) in the quantum system, with classical measures of chaos (Lyapunov exponents) in the corresponding classical system. As a function of energy and interaction strength, we demonstrate a strong overall correspondence between the two cases. In contrast to both strongly chaotic and integrable systems, the largest Lyapunov exponent is shown to be a multivalued function of energy.
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.
Motivated by the role that spectral properties play for the dynamical evolution of a quantum many-body system, we investigate the level spacing statistics of the extended Bose–Hubbard model. In particular, we focus on the distribution of the ratio of adjacent level spacings, useful at large interactions, to distinguish between chaotic and non-chaotic regimes. After revisiting the bare Bose–Hubbard model, we study the effect of two different perturbations: next-nearest-neighbor hopping and nearest-neighbor interaction. The system size dependence is investigated together with the effect of the proximity to integrable points or lines. Lastly, we discuss the consequences of a cutoff in the number of onsite bosons on the level statistics.
Systems of interacting bosons in double-well potentials, modeled by two-site Bose-Hubbard models, are of significant theoretical and experimental interest and attracted intensive studies in contexts ranging from many-body physics and quantum dynamics to the onset of quantum chaos. In this work we systematically study a kicked two-site Bose-Hubbard model (Bose-Hubbard dimer) with the on-site potential difference being periodically modulated. Our model can be equivalently represented as a kicked Lipkin-Meshkov-Glick model and thus displays different dynamical behaviors from the kicked-top model. By analyzing spectral statistics of Floquet operator, we unveil that the system undergoes a transition from regularity to chaos with increasing the interaction strength. Then based on semiclassical approximation and the analysis of R\'enyi entropy of coherent states in the basis of Floquet operator eigenstates, we reveal the local chaotic features of our model, which indicate the existence of integrable islands even in the deep chaotic regime. The semiclassical analysis also suggests that the system in chaotic regime may display different dynamical behavior depending on the choice of initial states. Finally, we demonstrate that dynamical signatures of chaos can be manifested by studying dynamical evolution of local operators and out-of-time-order correlation function as well as the entanglement entropy. Our numerical results exhibit the richness of dynamics of the kicked Bose-Hubbard dimer in both regular and chaotic regimes as the initial states are chosen as coherent spin states located in different locations of phase space.