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arXiv:cond-mat/0006446 · Physical Review A

Spectral properties of coupled Bose-Einstein condensates

Roberto Franzosi, Vittorio Penna

Published Mar 15, 2001• 45 citations
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Abstract

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.

DOI
10.1103/physreva.63.043609

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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.
We study the energy spectrum structure of a system of two interacting bosonic wells (dimer model) occupied by N bosons and compare it with the phase-space topology of the same model within the mean-field approach. To this end, we characterize the structure of energy eigenstates by using the symmetry properties of the Hamiltonian, and show that the energy levels are nondegenerate. The presence of doublets leads to recover in the classical limit the self-trapping effect exhibited by the mean-field dimer model. Finally, we do some comments on trimer dynamics to show how the interaction with a third well can cause both dimer-like regimes and strongly chaotic behaviors.
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.
The ground-state properties of attractive bosons trapped in a ring lattice including a single attractive potential well with an adjustable depth are investigated. The energy spectrum is reconstructed both in the strong-interaction limit and in the superfluid regime within the Bogoliubov picture. The analytical results thus obtained are compared with those found numerically from the exact Hamiltonian, in order to identify the regions in the parameter space where this picture is effective. The single potential introduced is the simplest way to break the translational symmetry and to observe, through a completely analytical approach, how the absence of symmetry affects the properties of the low-excited eigenstates of the system. This model gives a first insight into the properties of systems including more complex potentials.
We determine the low-energy spectrum and the eigenstates for a two-bosonic mode nonlinear model by applying the Inönü-Wigner contraction method to the Hamiltonian algebra. This model is known to well represent a Bose-Einstein condensate rotating in a thin torus endowed with two angular-momentum modes as well as a condensate in a double-well potential characterized by two space modes. We consider such a model in the presence of both an attractive and a repulsive boson interaction and investigate regimes corresponding to different values of the inter-mode tunneling parameter. We show that the results ensuing from our approach are in many cases extremely satisfactory. To this end we compare our results with the ground state obtained both numerically and within a standard semiclassical approximation based on su(2) coherent states.
We determine the low-energy spectrum and the eigenstates for a two-bosonic mode nonlinear model by applying the In\"{o}n\"{u}-Wigner contraction method to the Hamiltonian algebra. This model is known to well represent a Bose-Einstein condensate rotating in a thin torus endowed with two angular-momentum modes as well as a condensate in a double-well potential characterized by two space modes. We consider such a model in the presence of both an attractive and a repulsive boson interaction and investigate regimes corresponding to different values of the inter-mode tunneling parameter. We show that the results ensuing from our approach are in many cases extremely satisfactory. To this end we compare our results with the ground state obtained both numerically and within a standard semiclassical approximation based on su(2) coherent states.