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arXiv:1108.6188 · Physical Review A

Analytical approach to the two-site Bose-Hubbard model: From Fock states to Schrödinger cat states and entanglement entropy

Luca Dell’Anna

Published May 9, 2012• 15 citations
arXivHTMLPDF

Abstract

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 ${U}_{c}$, 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 ${U}_{c}^{*}$, 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 ${U}_{c}^{*}$ and exceeds its coherent value within the whole range of interaction between $2{U}_{c}$ and zero.

DOI
10.1103/physreva.85.053608

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We consider a bosonic Josephson junction made of NNN ultracold and dilute atoms confined by a quasi-one-dimensional double-well potential within the two-site Bose-Hubbard model framework. The behavior of the system is investigated at zero temperature by varying the interatomic interaction from the strongly attractive regime to the repulsive one. We show that the ground state exhibits a crossover from a macroscopic Schr\"odinger-cat state to a separable Fock state through an atomic coherent regime. By diagonalizing the Bose-Hubbard Hamiltonian we characterize the emergence of the macroscopic cat states by calculating the Fisher information FFF, the coherence by means of the visibility $\ensuremath{\alpha}$ of the interference fringes in the momentum distribution, and the quantum correlations by using the entanglement entropy SSS. Both Fisher information and visibility are shown to be related to the ground-state energy by employing the Hellmann-Feynman theorem. This result, together with a perturbative calculation of the ground-state energy, allows simple analytical formulas for FFF and $\ensuremath{\alpha}$ to be obtained over a range of interactions, in excellent agreement with the exact diagonalization of the Bose-Hubbard Hamiltonian. In the attractive regime the entanglement entropy attains values very close to its upper limit for a specific interaction strength lying in the region where coherence is lost and self-trapping sets in.
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This work contains a detailed analysis of the properties of the ground state of a two-component two-site Bose-Hubbard model, which captures the physics of a binary mixture of Bose-Einstein condensates trapped in a double-well potential. The atom-atom interactions within each species and among the two species are taken as variable parameters, while the hopping terms are kept fixed. To characterize the ground state, we use observables such as the imbalance of population and its quantum uncertainty. The quantum many-body correlations present in the system are further quantified by studying the degree of condensation of each species, the entanglement between the two sites, and the entanglement between the two species. The latter is measured by means of the Schmidt gap, the von Neumann entropy, or the purity obtained after tracing out a part of the system. A number of relevant states are identified, e.g., Schr\"odinger catlike many-body states, in which the outcome of the population imbalance of both components is completely correlated, and other states with even larger von Neumann entropy which have a large spread in Fock space.
Abstract In a recent work, Murmann et al (2015 Phys. Rev. Lett. 114 080402) have experimentally prepared and manipulated a double-well optical potential containing a pair of Fermi atoms as a possible building block of Hubbard model. Here, we carry out a detailed theoretical study on the properties of both fermionic and bosonic two-site Hubbard models with a pair of interacting atoms in a trap with a double-well structure along z -axis and a 2D harmonic confinement along the transverse directions. We consider fermions as of two-component type and bosons as of spinless as well as of two spin components. We first calculate Hubbard parameters using the finite-range interaction potentials of Jost and Kohn. In general, a finite range of interaction leads to on-site, inter-site, exchange and partial-exchange terms We show that, given the same input parameters for both bosonic and fermionic two-site Hubbard models, many of the statistical properties such as the single- and double-occupancy of a site, and the probabilities for the single-particle and pair tunneling are similar in both fermionic and bosonic cases. But, quantum entanglement and quantum fluctuations are found to be markedly different for the two cases. We discuss atom-atom entanglement in two spatial modes corresponding to the two sites of the double-well. Our results show that the entanglement of a pair of spin-half fermions is always greater than that of spinless bosons; and when the fermions are maximally entangled the fluctuation in the two-mode phase difference is largely squeezed. In contrast, spinless bosons never exhibit phase squeezing, but shows squeezing in two-mode population imbalance depending on the system parameters.