On the rate of convergence for the mean field approximation of Bosonic many-body quantum dynamics

Abstract

We consider the time evolution of quantum states by many-body Schrödinger dynamics and study the rate of convergence of their reduced density matrices in the mean field limit. If the prepared state at initial time is of coherent or factorized type and the number of particles n is large enough then it is known that 1/n is the correct rate of convergence at any time. We show in the simple case of bounded pair potentials that the previous rate of convergence holds in more general situations with possibly correlated prepared states. In particular, it turns out that the coherent structure at initial time is unessential and the important fact is rather the speed of convergence of all reduced density matrices of the prepared states. We illustrate our result with several numerical simulations and examples of multi-partite entangled quantum states borrowed from quantum information.Mathematics subject classification: 81S30, 81S05, 81T10, 35Q55, 81P40

On the Mean Field and Classical Limits of Quantum Mechanics

“…for ǫ fixed, has been proved by other methods in this case (see [29,4]). Moreover, quantitative estimates for that limit for ǫ fixed and N → ∞ have been obtained in [28,26,2]. Therefore, only the case where both N → ∞ and ǫ → 0 remains to be treated, and the present work answers precisely this question.…”

“…The mean field limit is the asymptotic regime where N → ∞, with ǫ > 0 fixed. Set the initial data in (2) and let Ψ ǫ,N be the solution of the Cauchy problem (2) -which exists for all times provided that V is such that…”

Fluctuations of $N$-particle quantum dynamics around the nonlinear Schrödinger equation

“…In the simpler case β = 0, i.e. in the mean-field regime, the convergence of the one-particle reduced density towards the orthogonal projection onto the solution of the nonlinear Hartree equation i∂ t ϕ t = −∆ϕ t + (V * |ϕ t | 2 )ϕ t (5) has been proved in several situations; see, e.g., [47,6,20,1,16,4,21,22,31,30,3,13,2]. In the present paper, we are interested in the norm approximation to the manybody evolution, which is more precise than the convergence of the one-particle reduced density.…”

“…is the Fock space vacuum and, for any f ∈ L 2 (R 3 ), W (f ) = exp(a * (f ) − a(f )) is a Weyl operator. The normalization of ϕ guarantees that W ( √ Nϕ)Ω, N W ( √ N ϕ)Ω = N. The time-evolution of initial coherent states of the form (8), generated by the natural extension of the Hamiltonian (2) to the Fock space F…”

Central limit theorem for Bose gases interacting through singular potentials

“…For β = 0 (mean-field regime), the solution of (1.2) can be approximated by products of the Hartree equation i∂ t ϕ t = −∆ϕ t + V * |ϕ t | 2 ϕ t with initial data ϕ 0 ∈ L 2 (R 3 ). See for example [1,2,3,4,5,9,15,19,20,21,23,24,31]. For 0 < β ≤ 1, on the other hand, the solution ψ N,t of (1.2) can be approximated by the non-linear Schrödinger equation…”