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arXiv:0711.4729 · Physical Review A

Bogoliubov theory of quantum correlations in the time-dependent Bose-Hubbard model

Uwe R. Fischer, Ralf Schützhold, Michael Uhlmann

Published Apr 14, 2008• 47 citations
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Abstract

By means of an adapted mean-field expansion for large fillings $n⪢1$, we study the evolution of quantum fluctuations in the time-dependent Bose-Hubbard model, starting in the superfluid state and approaching the Mott phase by decreasing the tunneling rate or increasing the interaction strength in time. For experimentally relevant cases, we derive analytical results for the temporal behavior of the number and phase fluctuations, respectively. This allows us to calculate the growth of the quantum depletion and the decay of off-diagonal long-range order. We estimate the conditions for the observability of the time dependence in the correlation functions in the experimental setups with external trapping present. Finally, we discuss the analogy to quantum effects in the early universe during the inflationary epoch.

DOI
10.1103/physreva.77.043615

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By means of an adapted mean-field expansion for large fillings n⪢1n⪢1n⪢1, we study the evolution of quantum fluctuations in the time-dependent Bose-Hubbard model, starting in the superfluid state and approaching the Mott phase by decreasing the tunneling rate or increasing the interaction strength in time. For experimentally relevant cases, we derive analytical results for the temporal behavior of the number and phase fluctuations, respectively. This allows us to calculate the growth of the quantum depletion and the decay of off-diagonal long-range order. We estimate the conditions for the observability of the time dependence in the correlation functions in the experimental setups with external trapping present. Finally, we discuss the analogy to quantum effects in the early universe during the inflationary epoch.
We study the sweep through the quantum phase transition from the superfluid to the Mott state for the Bose-Hubbard model with a time-dependent tunneling rate J(t). In the experimentally relevant case of exponential decay J(t) proportional variant e -gamma t, an adapted mean-field expansion for large fillings n yields a scaling solution for the fluctuations. This enables us to analytically calculate the evolution of the number and phase variations (on-site) and correlations (off-site) for slow (gamma<<mu), intermediate, and fast (nonadiabatic gamma>>mu) sweeps, where mu is the chemical potential. Finally, we derive the dynamical decay of the off-diagonal long-range order as well as the temporal shrinkage of the superfluid fraction in a persistent ring-current setup.
We study the Bose-Hubbard and Fermi-Hubbard model in the limit of large coordination numbers Z (i.e., many tunnelling partners). Via a controlled expansion into powers of 1/Z, we establish a hierarchy of correlations, which facilitates an approximate analytic solution of the quantum evolution. For the Bose-Hubbard model, we derive the growth of phase coherence after a quench from the Mott to the superfluid phase. For a quench within the Mott phase, we find that various local observables approach a quasi-equilibrium state after a finite period of time. However, this state is not thermal, i.e., real thermalisation -- if it occurs -- requires much longer time scales. For a tilted lattice in the Mott state, we calculate the tunnelling probability and find a remarkable analogy to the Sauter-Schwinger effect (i.e., electron-positron pair creation out of the vacuum due to a strong electric field). These analytical results are compared to numerical simulations for finite lattices in one and two dimensions and we find qualitative agreement. Finally, we generalize these studies to the more involved case of the Fermi-Hubbard model.
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.
We study the quench from the Mott-insulator to the superfluid phase in the Bose-Hubbard model and investigate the spatial-temporal growth of phase coherence, that is, phase locking between initially uncorrelated sites. To this end, we establish a hierarchy of correlations via a controlled expansion into inverse powers of the coordination number 1/Z1/Z1/Z. It turns out that the off-diagonal long-range order spreads with a constant propagation speed, forming local condensate patches, whereas the phase correlator follows a diffusionlike growth rate.
In this paper, we study the dynamics of the Bose-Hubbard model by using time-dependent Gutzwiller methods. In particular, we vary the parameters in the Hamiltonian as a function of time, and investigate the temporal behavior of the system from the Mott insulator to the superfluid (SF) crossing a second-order phase transition. We first solve a time-dependent Schr\"odinger equation for the experimental setup recently done by Braun et al. [Proc. Natl. Acad. Sci. USA 112, 3641 (2015)] and show that the numerical and experimental results are in fairly good agreement. However, these results disagree with the Kibble-Zurek scaling. From our numerical study, we reveal a possible source of the discrepancy. Next, we calculate the critical exponents of the correlation length and vortex density in addition to the SF order parameter for a Kibble-Zurek protocol. We show that beside the ``freeze'' time $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{t}$, there exists another important time, teq{t}_{\mathrm{eq}}teq​, at which an oscillating behavior of the SF amplitude starts. From calculations of the exponents of the correlation length and vortex density with respect to a quench time ${\ensuremath{\tau}}_{\mathrm{Q}}$, we obtain a physical picture of a coarsening process. Finally, we study how the system evolves after the quench. We give a global picture of dynamics of the Bose-Hubbard model.
We study the dynamics of phase transitions in the one-dimensional Bose-Hubbard model. To drive the system from a Mott insulator to a superfluid phase, we change the tunneling frequency at a finite rate. We investigate the buildup of correlations during fast and slow transitions using variational wave functions, dynamical Bogoliubov theory, Kibble-Zurek mechanism, and numerical simulations. We show that time-dependent correlations satisfy characteristic scaling relations that can be measured in optical lattices filled with cold atoms.