Quantum collapses and revivals are fascinating manifestations of interference. Of particular interest in recent years are macroscopic quantum interference effects in Bose-Einstein condensates. In this letter such effects will be studied for the two site Bose-Hubbard model that is a standard model for exploration of Bose-Einstein condensates. An analytic expression that is valid in the weak coupling limit for the difference in the occupation of the two sites is developed and tested numerically. It describes correctly the collapses and revivals. Moreover, it is demonstrated that a calculation to the first order in the interparticle interaction is required for the prediction of the collapse and revival times while the second order is required for the evaluation of the shape of the revival peaks. We believe that the result is relevant for a variety of situations where collapses and revivals are found.
We study the dynamics of ultracold bosons in a double-well potential within the two-mode Bose-Hubbard model by means of semiclassical methods. By applying a Wentzel-Kramers-Brillouin (WKB) quantization we find analytical results for the energy spectrum, which are in excellent agreement with numerically exact results. They are valid in the energy range of plasma oscillations, both in the Rabi and the Josephson regime. Adopting the reflection principle and the Poisson summation formula we derive an analytical expression for the dynamics of the population imbalance depending only on the few relevant parameters of the system. This allows us to discuss its characteristic dynamics, especially the oscillation frequency and the collapse and revival time, as a function of the model parameters, leading to a deeper understanding of Josephson physics. We find that our formulas match previous experimental observations.
The one-dimensional Bose-Hubbard model at a unit filling factor is studied by means of a very high-order symbolic perturbative expansion. Analytical expressions are derived for the ground-state quantities such as energy per site, variance of on-site occupation, and correlation functions: $⟨{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{j}^{\ifmmode\dagger\else\textdagger\fi{}}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{j+r}⟩$ and $⟨{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}}_{j}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}}_{j+r}⟩$. These findings are compared to numerics and good agreement is found in the Mott insulator phase. Our results provide analytical approximations to important observables in the Mott phase, and are also of direct relevance to future experiments with ultracold atomic gases placed in optical lattices. We also discuss the symmetry of the Bose-Hubbard model associated with the sign change of the tunneling coupling.
We study the interpolation from occupation number Fock states to Schr\"odinger cat states on systems modeled by a two-mode Bose-Hubbard Hamiltonian, like, for instance, bosons in a double well or superconducting Cooper pair boxes. In the repulsive interaction regime, by a simplified single particle description, we calculate analytically energy, number fluctuations, stability under coupling to a heat bath, entanglement entropy, and Fisher information, all in terms of hypergeometric polynomials of the single particle overlap parameter. Our approach allows us to find how those quantities scale with the number of bosons. In the attractive interaction regime we calculate the same physical quantities in terms of the imbalance parameter, and find that the spontaneous symmetry breaking, occurring at interaction
Uc, predicted by a semiclassical approximation, is valid only in the limit of infinite number of bosons. For a large but finite number we determine a characteristic strength of interaction
Uc∗, which can be promoted as the crossover point from coherent to incoherent regimes and can be identified as the threshold of fragility of the cat state. Moreover, we find that the Fisher information is always in direct ratio to the variance of on-site number of bosons, for both positive and negative interactions. We finally show that the entanglement entropy is maximum close to
Uc∗ and exceeds its coherent value within the whole range of interaction between
2Uc and zero.
The one-dimensional Bose-Hubbard model in large-
U limit has been studied via reducing and mapping the Hamiltonian to a simpler one. The eigenstates and eigenvalues have been obtained exactly in the subspaces with fixed numbers of single- and double-occupancies but without multiple-occupancies, and the thermodynamic properties of the system have been calculated further. These eigenstates and eigenvalues also enable us to develop a new perturbation treatment of the model, with which the ground-state energy has been calculated exactly to first order in
1/U.
The collapse and revival of localized states in many-body systems is a quintessential example of a quantum interference phenomenon that cannot be explained purely within mean-field theories. Historically, the majority of previous work on quantum revivals has been in a Schrödinger mechanics context, typically for effectively 1D systems with a focus on the autocorrelation function as the key detection observable. The general revival subject acquires a far greater complexity in the bosonic many-body context that offers a larger variety of observables and intrinsically implies high dimensionality. Here, the focus is on the simplest nontrivial (i.e., not 1D) bosonic many-body system, namely a symmetric two-site Bose-Hubbard model. The non-equilibrium situation of a coherent initial state prepared on one of the two sites of this integrable 2D system in the strongly self-trapping regime is considered. The revival behaviors emerging from the resulting dynamics within expectation values, coherences, and the autocorrelation function are theoretically derived, and the entanglement behavior is illustrated. Several new features, unknown so far in the Schrödinger context and identified here in terms of five ``puzzles'', are encountered and discussed, among them, a synchronized resurgence behavior, differing revivals times depending on the specific observable, and a manifestation of a precession that appears in the underlying mean field dynamics. These new phenomena can be entirely understood through a comprehensive semiclassical theory, building upon the underlying classical (mean-field) phase-space structure. A quantitative comparison of the latter with semiclassical predictions and quantum perturbative calculations yields an overall excellent agreement and confirms that a full semiclassical theory is pivotal for the understanding of these genuinely quantum many-body interference phenomena.