Beyond mean-field dynamics of small Bose-Hubbard systems based on the number-conserving phase-space approach

Abstract

The number-conserving quantum phase space description of the Bose-Hubbard model is discussed for the illustrative case of two and three modes, as well as the generalization of the two-mode case to an open quantum system. The phase-space description based on generalized SU͑M͒ coherent states yields a Liouvillian flow in the macroscopic limit, which can be efficiently simulated using Monte Carlo methods even for large systems. We show that this description clearly goes beyond the common mean-field limit. In particular it resolves well-known problems where the common mean-field approach fails, such as the description of dynamical instabilities and chaotic dynamics. Moreover, it provides a valuable tool for a semiclassical approximation of many interesting quantities, which depend on higher moments of the quantum state and are therefore not accessible within the common approach. As a prominent example, we analyze the depletion and heating of the condensate. A comparison to methods ignoring the fixed particle number shows that in this case artificial number fluctuations lead to ambiguities and large deviations even for quite simple examples.

Dynamics of the central-depleted-well regime in the open Bose-Hubbard trimer

“…This issue has been discussed in a series of papers examining the spectral properties of quantum trimer [20]- [22], its description within the phase-variable [23] and the Husimidistribution [24] pictures, and the inclusion of higherorder quantum correlations within the multiconfigurational Hartree method [25]. Quantum trimer has been also used to model coherent transport with weak interaction [26] and thermalization effects within the FokkerPlanck theory [35].…”

Quantum signatures of an oscillatory instability in the Bose-Hubbard trimer

“…On the other hand, it is well known (see, e.g., [12][13][14]16,17]) that for each finite N , one may construct SU(f ) coherent states (f = 3 for a trimer), for which the time evolution obtained from the Bose-Hubbard model exactly reproduces the classical DNLS dynamics in the limit N → ∞. [The SU(f ) coherent states are equivalent to the Hartree states of [18], but differ from the Glauber coherent states used, e.g., in [19,20] essentially in that the former conserve total particle number for any N , while the latter conserve the rescaled particle number, the DNLS norm, only in the classical limit N → ∞.]…”

“…An SU(3) coherent state, converging to a classical SDW stationary solution in the classical limit, can be constructed as described, e.g., in [12][13][14]16,17]. In particular, we may use the explicit expansion in Fock eigenstates given in Eq.…”

“…Following [12][13][14][16][17][18] and making an ansatz of the wave function as an SU(3) coherent state with a given N = 3 i=1 n i , we obtain a dynamical equation for the coherentstate parameters a i , which is the DNLS equation corresponding to the following effective classical Hamiltonian:…”

Spin squeezing in dipolar spinor condensates

“…In order to analyze phase portraits we have to reduce the number of parameters through slicing [61,62]. The subspace of interest is the mean-field counterpart of the Bloch sphere spanned by the {Ĵ x ,Ĵ y ,Ĵ z } operators.…”