Dynamical BEC formation and vanishing entropy for trapped bosons in non-relativistic QED via nonlinear resonance cascades
Authors: Thomas Chen and Ali Mezher
Published in arXiv, 2026
Abstract
The dynamical formation of a complete Bose-Einstein condensate and the asymptotic purification of trapped bosons coupled to a quantized radiation field are rigorously established. By tracking the macroscopic one-particle density matrix $\kappa_T$ through a nonlinear resonance cascade, it is demonstrated that radiative relaxation irreversibly depletes all excited trap modes, even when the initial macroscopic state $\kappa_0$ is genuinely mixed. Here $\kappa_T$ is an average, with respect to a de Finetti measure $\mu_0$, over mean-field particle-field orbits, so that the mixedness of $\kappa_0$ reflects a statistical ensemble of orbits. Under the assumptions of a non-vanishing initial ground state component and strictly positive downward Fermi-Golden-Rule rates, strong trace-norm convergence to a pure condensate is proven,
\[\lim_{T\to\infty} \left\lVert \kappa_T-\lvert\chi_0\rangle\langle\chi_0\rvert \right\rVert_{\mathfrak{S}_1(\mathfrak{h})} =0.\]Furthermore, the von Neumann entropy of the macroscopic one-particle density matrix is shown to vanish asymptotically,
\[\lim_{T\to\infty} S[\kappa_T]=0,\]without any claim of monotonicity in $T$. Thus, any initially mixed, positive-entropy state ($0 \lt S[\kappa_0] \lt \infty$) is purified to the ground state $\lvert\chi_0\rangle\langle\chi_0\rvert$. For the proof, we derive the Hartree hierarchy of the coupled system of bosons and photons from the finite-$N$ BBGKY hierarchy of the microscopic quantum many-body system, establishing trace-norm convergence in the mean-field limit as $N\to\infty$. The limits are taken successively: first $N\to\infty$, then the weak-coupling limit, and finally $T\to\infty$. Although the global particle-field evolution remains unitary, the macroscopic one-particle state converges to a pure state as energy and correlations are continuously radiated into the dispersive field.