Non-Markovian dynamics in nonstationary Gaussian baths: A hierarchy of pure states approach

Building on the standard hierarchy of pure states (HOPS) approach, we construct a generalized formulation suitable for open quantum systems interacting with nonstationary Gaussian baths, potentially extending its applicability to nonequilibrium baths. This is achieved by extending the conventional exponential decomposition of a bath-correlation function (BCF) for nonstationary cases. Using our formulation of HOPS, we derive the corresponding hierarchy of master equations and, when each term in the BCF expansion can be associated with an independent physical bath, we show how the formalism connects to the well-known pseudomode representation. We demonstrate the method's performance on two examples of nonstationary squeezed reservoirs generated via uniform squeezing and degenerate parametric amplification in a one-sided cavity. Benchmarking against the hierarchy of master equations shows that HOPS is more efficient under hierarchy truncation. The pseudomode representation is shown to be more efficient in the strongly non-Markovian regime. Our results highlight HOPS as a versatile and powerful tool for simulating open quantum systems in nonstationary baths, with potential applications ranging from squeezed light-matter interactions to driven-quantum materials and dissipative phase transitions.