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Quantum dynamics of coupled bosonic wells within the Bose-Hubbard picture

Franzosi, R., Vittorio Penna, Riccardo Zecchina

Published Jan 1, 2000• 21 citations
OpenAlex

Abstract

We relate the quantum dynamics of the Bose–Hubbard model (BHM) to the semiclassical nonlinear equations that describe an array of interacting Bose condensates by implementing a standard variational procedure based on the coherent state method. We investigate the dynamics of the two-site BHM from the purely quantum viewpoint by recasting first the model within a spin picture and using then the related dynamical algebra. The latter allows us to study thoroughly the energy spectrum structure and to interpret quantally the classical symmetries of the two-site dynamics. The energy spectrum is also evaluated through various approximations relying on the coherent state approach.

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•Jan 1, 2016•0 citations
We relate the quantum dynamics of the Bose–Hubbard model (BHM) to the semiclassical nonlinear equations that describe an array of interacting Bose condensates by implementing a standard variational procedure based on the coherent state method. We investigate the dynamics of the two-site BHM from the purely quantum viewpoint by recasting first the model within a spin picture and using then the related dynamical algebra. The latter allows us to study thoroughly the energy spectrum structure and to interpret quantally the classical symmetries of the two-site dynamics. The energy spectrum is also evaluated through various approximations relying on the coherent state approach.
We relate the quantum dynamics of the Bose–Hubbard model (BHM) to the semiclassical nonlinear equations that describe an array of interacting Bose condensates by implementing a standard variational procedure based on the coherent state method. We investigate the dynamics of the two-site BHM from the purely quantum viewpoint by recasting first the model within a spin picture and using then the related dynamical algebra. The latter allows us to study thoroughly the energy spectrum structure and to interpret quantally the classical symmetries of the two-site dynamics. The energy spectrum is also evaluated through various approximations relying on the coherent state approach.
We propose a scheme for investigating the quantum dynamics of interacting electron models by means of a time-dependent variational principle and spin coherent states of space lattice operators. We apply such a scheme to the one-dimensional Hubbard model, and solve the resulting equations in different regimes. In particular, we find that at low densities the dynamics is mapped into two coupled nonlinear Schr\"odinger equations, whereas near half-filling the model is described by two coupled Josephson-junction arrays. Focusing then to the case in which only the phases of the spin variables are dynamically active, we examine a number of different solutions corresponding to the excitations of few macroscopic modes. Based on fixed-point equations of the simpler among them, we show that the standard one-band ground-state phase space is found.
We investigate the energy spectrum structure of a system of two (identical) interacting bosonic wells occupied by N bosons within the Schwinger realization of the angular momentum. This picture enables us to recognize the symmetry properties of the system Hamiltonian H and to use them for characterizing the energy eigenstates. Also, it allows for the derivation of the single-boson picture that is shown to be the background picture naturally involved by the secular equation for H. After deriving the corresponding eigenvalue equation, we recast it in a recursive N-dependent form that suggests a way to generate the level doublets (characterizing the H spectrum) via suitable inner parameters. Finally, we show how the presence of doublets in the spectrum allows us to recover, in the classical limit, the symmetry-breaking effect that characterizes the system classically.
Quantum dynamics of the Bose-Hubbard model is investigated through a semiclassical Hamiltonian picture provided by the time-dependent variational principle method. The system is studied within a factorized slow/fast dynamics. The semiclassical requantization procedure allows one to account for the strong quantum nature of the system when $t/U\ensuremath{\ll}1$. The phase diagram is in good agreement with quantum Monte Carlo results and third order strong coupling perturbative expansion.
We consider a system formed by an array of Bose–Einstein condensates trapped in a harmonic potential with a superimposed periodic optical potential. Starting from the boson field Hamiltonian, appropriate to describe a dilute gas of bosonic atoms, we reformulate the system dynamics within the Bose–Hubbard model scenario. Then we analyse the effective dynamics of the system when the optical-potential depth is suddenly varied according to a procedure applied in many of the recent experiments on superfluid–Mott transition in Bose–Einstein condensates. Initially the condensate array generated in a weak optical potential is assumed to be in the superfluid ground state which is well described in terms of coherent states. At a given time, the optical-potential depth is suddenly increased and, after a waiting time, it is quickly decreased so that the initial depth is restored. We compute the system-state evolution and show that the potential jump brings on an excitation of the system, incorporated in the final condensate wavefunctions, whose effects are analysed in terms of two-site correlation functions and of on-site population oscillations. Also we show how too long a waiting time can destroy completely the coherence of the final state, making it unobservable.
This dissertation discusses a number of theoretical models of coupled bosonic modes, all closely related to the Bose-Hubbard dimer. In studying these models, we will repeatedly return to two unifying themes: the classical structure underlying quantum dynamics and the impact of weakly coupling a system to an environment. Or, more succinctly, semiclassical methods and open quantum systems. \n \nOur primary motivation for studying models such as the Bose-Hubbard is their relevance to ongoing ultracold atom experiments. We review these experiments, derive the Bose-Hubbard model in their context and briefly discuss its limitations in the first half of Chapter 1. In its second half, we review the theory of open quantum systems and the master equation description of the dissipative Bose-Hubbard model. This opening chapter constitutes a survey of existing results, rather than original work. \n \nIn Chapter 2, we turn to the mean-field limit of the Bose-Hubbard model. After reviewing the striking localization phenomena predicted by the mean-field (and confirmed by experiment), we identify the first corrections to this picture for the dimer. The most interesting of these is the dynamical tunneling between the self-trapping points of the mean-field. We derive an accurate analytical expression for the tunneling rate using semiclassical techniques. \n \nWe continue studying the dynamics near the self-trapping fixed points in Chapter 3, focusing on corrections to the mean-field that arise at larger nonlinearities and on shorter time scales than dynamical tunneling. We study the impact of dissipation on coherence and entanglement near the fixed points, and explain it in terms of the structure of the classical phase space. \n \nThe last chapter of the dissertation is also devoted to a dissipative bosonic dimer model, but one arising in a very different physical context. Abandoning optical lattices, we consider the problem of formulating a quantum model of operation of the cylindrical anode magnetron, a vacuum tube crossed-field microwave amplifier. We derive an effective dissipative dimer model and study its relationship to the classical description. Our dimer model is a first step towards the analysis of solid-state analogs of such devices.