Dynamical BEC formation and vanishing entropy for trapped bosons in non-relativistic QED via nonlinear resonance cascades

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, $$\underset{T\to \mathrm{\infty }}{\lim }\mathrm{\parallel }{\kappa }_T-\mathrm{\mid }{\chi }_0⟩⟨{\chi }_0\mathrm{\mid }{\mathrm{\parallel }}_{{\mathfrak{S}}_1(\mathfrak{h})}=0\mathrm{.}$$ Furthermore, the von Neumann entropy of the macroscopic one-particle density matrix is shown to vanish asymptotically, $$\underset{T\to \mathrm{\infty }}{\lim }S[{\kappa }_T]=0,$$ without any claim of monotonicity in $T$. Thus, any initially mixed, positive-entropy state ($0<S[{\kappa }_0]<\mathrm{\infty }$) is purified to the ground state $\mathrm{\mid }{\chi }_0⟩⟨{\chi }_0\mathrm{\mid }$. 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 $N\to \mathrm{\infty }$. The limits are taken successively; first $N\to \mathrm{\infty }$, then the weak-coupling limit, and finally $T\to \mathrm{\infty }$. 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.