We benchmark a selection of semiclassical and perturbative dynamics techniques by investigating the correlated evolution of a cavity-bound atomic system to assess their applicability to study problems involving strong light-matter interactions in quantum cavities. The model system of interest features spontaneous emission, interference, and strong coupling behaviour, and necessitates the consideration of vacuum fluctuations and correlated light-matter dynamics. We compare a selection of approximate dynamics approaches including fewest switches surface hopping, multi-trajectory Ehrenfest dynamics, linearized semiclasical dynamics, and partially linearized semiclassical dynamics. Furthermore, investigating self-consistent perturbative methods, we apply the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy in the second Born approximation. With the exception of fewest switches surface hopping, all methods provide a reasonable level of accuracy for the correlated light-matter dynamics, with most methods lacking the capacity to fully capture interference effects.Here we broaden our scope by investigating the performance of a comprehensive class of approximate quantum dynamics methods for simulating spontaneous emission in an optical cavity, including Ehrenfest meanfield theory 33,34 , Tully's surface hopping algorithm 35 , fully linearized 36 and partially linearized 37,38 semilclassical dynamics techniques, and a selection of approximate closures for the quantum mechanical Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. Through benchmark comparisons with exact numerical results, we assess the accuracy and efficiency of each method and highlight the possibilities and theoretical challenges involved with extending these approaches towards realistic systems.The remainder of this work is divided into four sections: Sec. II gives a short overview of general quantum mechanical arXiv:1909.07177v2 [quant-ph]

We benchmark many-body perturbation theory by studying neutral, as well as non-neutral, excitations of finite lattice systems. The neutral excitation spectra are obtained by time-propagating the Kadanoff-Baym equations in the Hartree-Fock and the second Born approximations. Our method is equivalent to solving the Bethe-Salpeter equation with a high-level kernel while respecting self-consistency, which guarantees the fulfillment of a frequency sum rule. As a result, we find that a time-local method, such as Hartree-Fock, can give incomplete spectra, while already the second Born approximation, which is the simplest time-non-local approximation, reproduces well most of the additional excitations, which are characterized as double excitations.

We investigate the time-resolved quantum transport properties of phonons in arbitrary harmonic systems connected to phonon baths at different temperatures. We obtain a closed analytic expression of the time-dependent one-particle reduced density matrix by explicitly solving the equations of motion for the nonequilibrium Green's function. This is achieved through a well-controlled approximation of the frequency-dependent bath self-energy. Our result allows for exploring transient oscillations and relaxation times of local heat currents, and correctly reduces to an earlier known result in the steady-state limit. We apply the formalism to atomic chains, and benchmark the validity of the approximation against full numerical solutions of the bosonic Kadanoff-Baym equations for the Green's function. We find good agreement between the analytic and numerical solutions for weak contacts and baths with a wide energy dispersion. We further analyze relaxation times from low to high temperature gradients.