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arXiv:cond-mat/0003399 · Physical Review A

N -boson time-dependent problem: A reformulation with stochastic wave functions

Iacopo Carusotto, Yvan Castin, Jean Dalibard

Published Jan 11, 2001• 80 citations
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Abstract

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.

DOI
10.1103/physreva.63.023606

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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 \ensuremathρ\ensuremath{\rho}\ensuremathρ is obtained as the stochastic average of particular operators ∣\ensuremathΨ1〉〈\ensuremathΨ2∣|{\ensuremath{\Psi}}_{1}〉〈{\ensuremath{\Psi}}_{2}|∣\ensuremathΨ1​〉〈\ensuremathΨ2​∣ of the system. The states ∣\ensuremathΨ1,2〉|{\ensuremath{\Psi}}_{1,2}〉∣\ensuremathΨ1,2​〉 are either Fock states ∣N:\ensuremathφ1,2〉|N:{\ensuremath{\varphi}}_{1,2}〉∣N:\ensuremathφ1,2​〉 or coherent states ∣coh:\ensuremathφ1,2〉|\mathrm{coh}:{\ensuremath{\varphi}}_{1,2}〉∣coh:\ensuremathφ1,2​〉 with each particle in the state \ensuremathφ1,2(x).{\ensuremath{\varphi}}_{1,2}(x).\ensuremathφ1,2​(x). We determine the conditions on the evolution of \ensuremathφ1,2,{\ensuremath{\varphi}}_{1,2},\ensuremathφ1,2​, which involves a stochastic element, under which we recover the exact evolution of \ensuremathρ.\ensuremath{\rho}.\ensuremathρ. 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 investigate a reformulation of the dynamics of interacting fermion systems in terms of a stochastic extension of time-dependent Hartree-Fock equations. From a path-integral representation of the evolution operator, we show that the exact N-body state can be interpreted as a coherent average over Slater determinants evolving in a random mean-field. The imaginary time propagation is also presented and gives a similar scheme which converges to the exact ground state. In addition, the growth of statistical errors is examined to show the stability of this stochastic formulation.
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.
A common approximate way to study the time evolution of a superfluid Fermi gas is to consider the mean-field time-dependent BCS equations. We show that, by inclusion of an appropriate fluctuating term in these mean-field equations, the exact quantum evolution of the Fermi gas is recovered, after an average over all possible realizations of the noise term: this leads to an exact quantum Monte Carlo technique for the real-time evolution of an interacting fermionic system. In practice, we derive the general conditions on the noise term, ensuring that the exact dynamics is reproduced, and we explore several explicit realizations of this noise term, trying to minimize the statistical spreading of the quantum Monte Carlo simulations. The convergence issue is discussed in detail, with upper bounds obtained on the statistical spreading, and the relation between the stochastic and the mean-field equations is analyzed. Finally, we apply the method to a simple two-site model, to exemplify that the simulations can display effects out of reach of the mean-field approximation.
The time-dependent fermionic Hartree-Fock equations can be stochastically extended in such a way as to become the exact representation of quantum dynamics. This fact was first observed in the work of Juillet and Chomaz [Phys. Rev. Lett. 88, 142503 (2002)]. During the past decade, this observation has led to the emergence of a whole family of stochastic wave-function methods for fermions. The common feature of all these methods is that they are based on the expansion of the density operator over the dyadic product of the two fermionic Slater determinant states. In this work, we develop a unified and rigorous foundation for this family of methods. We find a general form of stochastic equations and describe the sufficient conditions under which these methods converge towards exact quantum dynamics. To achieve these goals, we employ the representation of quantum dynamics in generalized phase space. In particular, we consider the quasiprobability distributions which emerge in these stochastic methods and their master equations. It is shown that the convergence towards exact quantum dynamics is controlled by the problem of boundary terms. We provide an example of stochastic Hartree-Fock method which is well-defined and free from this problem.