On the emergence of quantum Boltzmann fluctuation dynamics near a Bose-Einstein Condensate

In this work, we study the quantum fluctuation dynamics in a Bose gas on a torus $\Lambda=(L\mathbb{T})^3$ that exhibits Bose-Einstein condensation, beyond the leading order Hartree-Fock-Bogoliubov (HFB) fluctuations. Given a Bose-Einstein condensate (BEC) with density $N$ surrounded by thermal fluctuations with density $1$, we assume that the system is described by a mean-field Hamiltonian. We extract a quantum Boltzmann type dynamics from a second-order Duhamel expansion upon subtracting both the BEC dynamics and the HFB dynamics. Using a Fock-space approach, we provide explicit error bounds. It is known that the BEC and the HFB fluctuations both evolve at microscopic time scales $t\sim1$. Given a quasifree initial state, we determine the time evolution of the centered correlation functions $\langle a\rangle$, $\langle aa\rangle-\langle a\rangle^2$, $\langle a^+a\rangle-|\langle a\rangle|^2$ at mesoscopic time scales $t\sim\lambda^{-2}$, where $0<\lambda\ll1$ denotes the size of the HFB interaction. For large but finite $N$, we consider both the case of fixed system size $L\sim1$, and the case $L\sim \lambda^{-2-}$. In the case $L\sim1$, we show that the Boltzmann collision operator contains subleading terms that can become dominant, depending on time-dependent coefficients assuming particular values in $\mathbb{Q}$; this phenomenon is reminiscent of the Talbot effect. For the case $L\sim \lambda^{-2-}$, we prove that the collision operator is well approximated by the expression predicted in the literature. In either of those cases, we have $\lambda\sim \Big(\frac{\log \log N}{\log N}\Big)^{\alpha}$, for different values of $\alpha>0$.