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Exact Reformulation of the Bosonic Many-Body Problem in Terms of Stochastic Wave Functions: an Elementary Derivation

Iacopo Carusotto, Yvan Castin

Published Jan 1, 2003• 2 citations
Publisher

Abstract

Recently, a new method was developed to reformulate in an exact way the N-body problem of interacting bosons in terms of the stochastic evolution of single particle wavefunctions, see I. Carusotto Y. Castin, and J. Dalibard, Phys. Rev. A 63, 023606 (2001) . The original derivation is however often considered as rather involved. For the specific scheme called `simple Fock scheme’ we present here a very elementary derivation of such an approach.

DOI
10.1007/978-3-0348-7907-1_61

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Recently, a new method was developed to reformulate in an exact way the N-body problem of interacting bosons in terms of the stochastic evolution of single particle wavefunctions, see I. Carusotto Y. Castin, and J. Dalibard, Phys. Rev. A 63, 023606 (2001) . The original derivation is however often considered as rather involved. For the specific scheme called `simple Fock scheme’ we present here a very elementary derivation of such an approach.
We present a numerically tractable method to solve exactly the evolution of a N boson system with binary interactions. The density operator of the system $\ensuremath{\rho}$ is obtained as the stochastic average of particular operators $|{\ensuremath{\Psi}}_{1}〉〈{\ensuremath{\Psi}}_{2}|$ of the system. The states $|{\ensuremath{\Psi}}_{1,2}〉$ are either Fock states $|N:{\ensuremath{\varphi}}_{1,2}〉$ or coherent states $|\mathrm{coh}:{\ensuremath{\varphi}}_{1,2}〉$ with each particle in the state ${\ensuremath{\varphi}}_{1,2}(x).$ We determine the conditions on the evolution of ${\ensuremath{\varphi}}_{1,2},$ which involves a stochastic element, under which we recover the exact evolution of $\ensuremath{\rho}.$ We discuss various possible implementations of these conditions. The well known positive P representation arises as a particular case of the coherent state ansatz. We treat numerically two examples: a two-mode system and a one-dimensional harmonically confined gas. These examples, together with an analytical estimate of the noise, show that the Fock state ansatz is the most promising one in terms of precision and stability of the numerical solution.
There exist methods to reformulate in an exact way the many-body problem of interacting bosons in terms of the stochastic evolution of single particle wave functions. For one such reformulation, the so-called simple Fock scheme, we present an elementary derivation, much simpler than the original one. Furthermore, we show that two other schemes, based on coherent states of the matter field rather than on Fock states, lead to an infinite statistical uncertainty in the continuous time limit. The simple Fock scheme is therefore, up to now, the only one that was proved to lead to a convergent Monte Carlo simulation scheme at all times.
We present a new exact method to numerically compute the thermodynamical properties of an interacting Bose gas in the canonical ensemble. As in our previous paper (Carusotto I, Castin Y and Dalibard J 2001 Phys. Rev. A 63 023606), we write the density operator ρ as an average of Hartree dyadics | N :ϕ 1 ⟩⟨ N :ϕ 2 | and we find stochastic evolution equations for the wavefunctions ϕ 1,2 such that the exact imaginary-time evolution of ρ is recovered after averaging over noise. In this way, the thermal equilibrium density operator can be obtained for any temperature T . The method is then applied to study the thermodynamical properties of a homogeneous one-dimensional N -boson system: although Bose-Einstein condensation cannot occur in the thermodynamical limit, a macroscopic occupation of the lowest mode of a finite system is observed at sufficiently low temperatures. If k B T >>µ, the main effect of interactions is to suppress density fluctuations and to reduce their correlation length. Different effects such as a spatial antibunching of the atoms are predicted for the opposite k B T ⩽µ regime. Our exact stochastic calculations have been compared with existing approximate theories.