Emulating a non-Markovian localization transition with a parametrically driven cavity

Every system occurring in nature interacts with its environment. Often, the environment is modeled in an effective ``memoryless'' way via a Markov approximation. Going beyond the Markovian regime can often prove difficult, but may also give rise to interesting phenomena such as partial decays, revivals, and engineered steady-states. We propose a straightforward experimental method to emulate the non-Markovian dynamics of an arbitrary quantum system coupled to a harmonic-oscillator bath. The bath is synthesized from the modes of a parametrically driven cavity. The spectral density of this bath is in principle arbitrary and can be directly controlled by the drive signal without any alteration of the physical setup. Furthermore, we provide analytic bounds on the deviation between system observables that would be measured in this setup, relative to their ideal realization in the presence of a bath described by a power-law spectral density. Finally, we theoretically consider a simple application of this proposal using classical light in optical-fiber loops, illustrating a potential experimental realization of the localization transition in the Caldeira-Leggett model.