Continuous measurement of a non-Markovian open quantum system

Continuous quantum measurement is the backbone of various methods in quantum control, quantum metrology, and quantum information. Here, we present a generalized formulation of dispersive measurement of a complex quantum systems. We describe the complex system as an open quantum system that is strongly coupled to a non-Markovian environment, enabling the treatment of a broad variety of natural or engineered complex systems. The system is monitored via a probe resonator coupled to a broadband (Markovian) reservoir. Based on this model, we derive a formalism of stochastic hierarchy equations of motion describing the decoherence dynamics of the system conditioned on the measurement record. Furthermore, we demonstrate a spectroscopy method based on weak quantum measurement to reveal the non-Markovian nature of the environment, which we term weak spectroscopy.