Multivariate Central Limit Theorem in Quantum Dynamics

Abstract

We consider the time evolution of N bosons in the mean field regime for factorized initial data. In the limit of large N, the many body evolution can be approximated by the non-linear Hartree equation. In this paper we are interested in the fluctuations around the Hartree dynamics. We choose k self-adjoint one-particle operators O ₁,…,O _k on $L^{2} ({\mathbb{R}}^{3})$, and we average their action over the N-particles. We show that, for every fixed $t \in{\mathbb{R}}$, expectations of products of functions of the averaged observables approach, as N→∞, expectations with respect to a complex Gaussian measure, whose covariance matrix can be expressed in terms of a Bogoliubov transformation describing the dynamics of quantum fluctuations around the mean field Hartree evolution. If the operators O ₁,…,O _k commute, the Gaussian measure is real and positive, and we recover a “classical” multivariate central limit theorem. All our results give explicit bounds on the rate of the convergence.

Appendix: Properties of ξ N

We collect some properties of the Fock space vector $\xi_{N} = d_{N} W^{*} (\sqrt{N} \varphi) \varphi^{\otimes N}$ which have been used in the proof of Theorem 1.1. The proof of the next lemma can be found in [1].

Lemma A.1

For $\varphi\in L^{2} ({\mathbb{R}}^{3})$, set

$$\xi_N = d_N W^* (\sqrt{N} \varphi) \frac{a^*(\varphi)^N}{\sqrt {N!}} \varOmega $$

with $d_{N} = e^{N/2} \sqrt{N!} N^{-N/2} \simeq N^{1/4}$. Then there exists a constant C>0 such that

$$\| ({\mathcal{N}}+1)^{-1/2} \xi_N \| \leq C $$

uniformly in N. Moreover, we have

$$\xi_N = \sum_{\ell=0}^\infty \xi_N^{(\ell)} a^* (\varphi)^\ell \varOmega $$

with the coefficients

$$\xi_N^{(\ell)} = \sum_{j=0}^\ell(-1)^j N^{j-\ell/2} \frac {N!}{(N-\ell+j)! (\ell-j)! j!} $$

Notice that the coefficients $\xi_{N}^{(\ell)}$ satisfy the recursion

$$\xi_N^{(\ell)} = \frac{1-\ell}{\ell} N^{-1/2} \xi_N^{(\ell-1)} - \frac{1}{\ell} \xi_N^{(\ell-2)} $$

with $\xi_{N}^{(0)} = 1$ and $\xi_{N}^{(1)} = 0$.