Unraveling a Brownian particle's memory with effective mode chains

Memory effects in quantum dynamical processes involving structured environments are presently difficult, if not impossible, to investigate using standard approaches. Progress can be made by transforming the environmental variables to a suitable chain representation which effectively performs a Markovian embedding of the dynamics. Here, we show that this effective-mode chain representation provides a unique way of unraveling the memory kernel κ(t) as a function of time. Truncated or Markov-closed chains with n effective modes exactly reproduce κ(t) to the 4nth order in time, up to an irrelevant constant of order κ(0)/n. These favorable convergence properties pave the way for efficient quantum simulations of fast (non-Markovian) processes by reduced dynamical models.