  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 (SHEM)
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.
