Quantum Averaging Theory: effective dynamics from a multi-timescale and symmetry-preserving perturbation theory and its applications

Time-dependent perturbations in quantum-mechanical systems often lead to dynamics lacking a known analytic solution. In particular, when the characteristic frequencies of perturbations lead to dynamics on both fast and slow timescales, perturbation theory and numerical methods both become inefficient at effectively modeling the system behavior for long evolution times. We introduce a quantum averaging theory approach that provides a unitary, analytic, approximate propagator for multi-timescale problems. The solution yields both an effective Hamiltonian that describes the time-averaged behavior and a dynamical phase that reproduces fast-timescale dynamics. We provide example applications to some familiar, touchstone problems in quantum optics that showcase the technique's ability to capture multi-timescale detail.