Readout of the spectral density of an environment from the dynamics of an open system

Given the initial state of the whole system and the spectral density function of the environment, the dynamics of a non-Markovian open system can be exactly determined. However, it has barely been studied how to obtain, in reverse, the spectral density function from the system dynamics. In this work, focusing on the dissipation dynamics of a single-mode non-Markovian open system, we present a simple formalism to determine both the spectral density function and the system mode frequency from only the system dynamics. As examples, we demonstrate the calculation of spectral densities with arbitrary dynamics. Also, in a one-dimensional open chain of bosonic modes, we show how to determine all of its parameters iteratively from only the dynamics on the first mode. This formalism not only reveals features of non-Markovian open systems, but it is also applicable to single-qubit spectroscopy and the estimation of parameters in a structured system.