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arXiv:0806.1181 · Journal of Physics A Mathematical and Theoretical

Some remarks on the coherent-state variational approach to nonlinear boson models

Pierfrancesco Buonsante, Vittorio Penna

Published Apr 15, 2008• 52 citations
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Abstract

The mean-field pictures based on the standard time-dependent variational approach have widely been used in the study of nonlinear many-boson systems such as the Bose-Hubbard model. The mean-field schemes relevant to Gutzwiller-like trial states $|F>$, number-preserving states $|ξ>$ and Glauber-like trial states $|Z>$ are compared to evidence the specific properties of such schemes. After deriving the Hamiltonian picture relevant to $|Z>$ from that based on $|F>$, the latter is shown to exhibit a Poisson algebra equipped with a Weyl-Heisenberg subalgebra which preludes to the $|Z>$-based picture. Then states $|Z>$ are shown to be a superposition of $\cal N$-boson states $|ξ>$ and the similarities/differences of the $|Z>$-based and $|ξ>$-based pictures are discussed. Finally, after proving that the simple, symmetric state $|ξ>$ indeed corresponds to a SU(M) coherent state, a dual version of states $|Z>$ and $|ξ>$ in terms of momentum-mode operators is discussed together with some applications.

DOI
10.1088/1751-8113/41/17/175301

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The mean-field pictures based on the standard time-dependent variational approach have widely been used in the study of nonlinear many-boson systems such as the Bose-Hubbard model. The mean-field schemes relevant to Gutzwiller-like trial states ∣F>|F>∣F>, number-preserving states ∣ξ>|ξ>∣ξ> and Glauber-like trial states ∣Z>|Z>∣Z> are compared to evidence the specific properties of such schemes. After deriving the Hamiltonian picture relevant to ∣Z>|Z>∣Z> from that based on ∣F>|F>∣F>, the latter is shown to exhibit a Poisson algebra equipped with a Weyl-Heisenberg subalgebra which preludes to the ∣Z>|Z>∣Z>-based picture. Then states ∣Z>|Z>∣Z> are shown to be a superposition of N\cal NN-boson states ∣ξ>|ξ>∣ξ> and the similarities/differences of the ∣Z>|Z>∣Z>-based and ∣ξ>|ξ>∣ξ>-based pictures are discussed. Finally, after proving that the simple, symmetric state ∣ξ>|ξ>∣ξ> indeed corresponds to a SU(M) coherent state, a dual version of states ∣Z>|Z>∣Z> and ∣ξ>|ξ>∣ξ> in terms of momentum-mode operators is discussed together with some applications.
We propose a variational approach to the dynamics of the Bose-Hubbard model beyond the mean-field approximation. To develop a numerical scheme, we use a discrete overcomplete set of Glauber coherent states and its connection to the generalized coherent states studied in depth by Perelomov [Perelomov, Generalized Coherent States and Their Applications (Springer-Verlag, Berlin, 1986)]. The variational equations of motion of the generalized coherent state parameters as well as of the coefficients in an expansion of the wave function in terms of those states are derived and solved for many-particle problems with large particle numbers SSS and increasing mode number MMM. For M=6M=6M=6, it is revealed that the number of complex-valued parameters that have to be propagated is more than one order of magnitude less than in an expansion in terms of Fock states.
We summarize some crucial results from mathematical models that help us understand the relevance of gener-alized coherent states (GCS), a fundamental method used for the description of non-linear problems in physics. We mainly concentrate our review in the following models: ferromagnetism, nonlinear Schr¨odinger equation with external potential, nonlinear quantum oscillators, Bose - Einstein condensation (BEC) and the DNA quasi-spin model. Such models and some applications of the wide variety these involve are outlined. As variational trial states, the coherent states (CS) allow the estimation of the ground state energies and properties, yielding results which become exact in a number of nonlinear differential equations for dynamical variable of each model. Mainly we concentrate our review on two types of coherent states, The first one are based on the Heisenberg - Weyl group where it is applied the bosonization effect. The second one is based on the SU(2)/U(1) Lie group. For this case when the Hamiltonian and all physical operators are constructed by the elements of a Lie algebra, the generalized coherent states approach is directly applied without bozonization.
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.
A new method for describing the two-level pairing model in a many-fermion system is presented. The basic idea comes from the Schwinger boson representation for the su(2) ⊗ su(2)-algebra, in which four kinds of boson operators govern the dynamics. It is well known that in the original fermion space, it is almost impossible to introduce any trial state for the variation except the BCS state in the framework of the mean field approximation. However, in the Schwinger boson representation, various trial states can be defined on the condition that the mean field approximation is useful. In these states, the su(2) ⊗ su(1,1)-coherent state is mainly discussed in this paper.
The mean-field approach to two-site Bose-Hubbard systems is well established and leads to nonlinear classical equations of motion for the population imbalance and the phase difference. It can, e.g., be based on the representation of the solution of the time-dependent Schrodinger equation either by a single Glauber state or by a single atomic (SU(2)) coherent state [S. Wimberger et al., Phys. Rev. A 103, 023326 (2021)]. We demonstrate that quantum effects beyond the mean-field approximation are easily uncovered if, instead, a multi-configuration ansatz with a few time-dependent SU(2) basis functions is used in the variational principle. For the case of plasma oscillations, the use of just two basis states, whose time-dependent parameters are determined variationally, already gives good qualitative agreement of the phase space dynamics with numerically exact quantum solutions. In order to correctly account for more non-trivial effects, like macroscopic quantum self trapping, moderately more basis states are needed. If one is interested in the onset of spontaneous symmetry breaking, however, a multiplicity of two gives a big improvement towards the exact result already. In any case, the number of variational trajectories needed for good agreement with full quantum results is orders of magnitude smaller than in the semiclassical case, which is based on multiple mean-field trajectories.
The Bose-Hubbard system has been studied extensively both theoretically and experimentally, in particular in the context of ultracold atomic gases in optical lattices. Even in the two-mode case the many-particle dynamics display complex interference effects resulting in revival and breakdown phenomena as well as tunnelling. The most basic theoretical description is the mean-field approximation, which can be derived from a time-dependent variational principle assuming the many-particle wave function is an SU(2) coherent state. Here we build on this to construct a simple initial-value coherent state propagator, summing over mean-field trajectories and keeping track of their phases, given by the corresponding mean-field actions. This yields an approximation to the full time-dependent many-particle state, and is able to reproduce breakdown and revival dynamics. Applying a time-slicing procedure on top of this, we are able to accurately capture many-particle tunnelling effects. While in this paper we focus our analysis on the Bose-Hubbard dimer, the methods developed can be applied to more general SU(2) Hamiltonians, and can be extended to SU(M) systems.