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Complex environments, such as molecular matrices and biological material,play a fundamental role in many important dynamic processes in condensedphases. Because it is extremely difficult to conduct full quantum dynamicssimulations on such environments due to their many degrees of freedom, here wetreat in detail the environment only around the main system of interest (thesubenvironment), while the other degrees of freedom needed to maintain theequilibrium temperature are described by a simple harmonic bath, which we calla quantum thermostat. The noise generated by the subenvironment is spatiallynon-local and non-Gaussian and cannot be characterized by thefluctuation-dissipation theorem. We describe this model by simulating thedynamics of a two-level system (TLS) that interacts with a subenvironmentconsisting of a one-dimensional $XXZ$ spin chain. The hierarchicalSchr\"odinger equations of motion are employed to describe the quantumthermostat, allowing time-irreversible simulations of the dynamics at arbitrarytemperature. To see the effects of a quantum phase transition of thesubenvironment, we investigate the decoherence and relaxation processes of theTLS at zero and finite temperatures for various values of the spin anisotropy.We observed the decoherence of the TLS at finite temperature, even when theanisotropy of the $XXZ$ model is enormous. We also found that the populationrelaxation dynamics of the TLS changed in a complex manner with the change ofthe anisotropy and the ferromagnetic or antiferromagnetic orders of the spins.

Open quantum dynamics theory for a complex subenvironment system with a quantum thermostat: Application to a spin heat bath