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We extend a standard stochastic theory to study open quantum systems coupledto generic quantum environments including the three fundamental classes ofnoninteracting particles: bosons, fermions and spins. In this unifiedstochastic approach, the generalized stochastic Liouville equation (SLE)formally captures the exact quantum dissipations when noise variables withappropriate statistics for different bath models are applied. Anharmoniceffects of a non-Gaussian bath are precisely encoded in the bath multi-timecorrelation functions that noise variables have to satisfy. Staring from theSLE, we devise a family of generalized hierarchical equations by averaging outthe noise variables and expand bath multi-time correlation functions in acomplete basis of orthonormal functions. The general hiearchical equationsconstitute systems of linear equations that provide numerically exactsimulations of quantum dynamics. For bosonic bath models, our generalhierarchical equation of motion reduces exactly to an extended version ofhierarchical equation of motion which allows efficient simulation for arbitraryspectral densities and temperature regimes. Similar efficiency and exibilitycan be achieved for the fermionic bath models within our formalism. The spinbath models can be simulated with two complementary approaches in the presetnformalism. (I) They can be viewed as an example of non-Gaussian bath models andbe directly handled with the general hierarchical equation approach given theirmulti-time correlation functions. (II) Alterantively, each bath spin can befirst mapped onto a pair of fermions and be treated as fermionic environmentswithin the present formalism.

A Unified Stochastic Formulation of Dissipative Quantum Dynamics. I. Generalized Hierarchical Equations