Path integral formalism of open quantum systems with non-diagonal system-bath coupling

Most path integral expressions for quantum open systems are formulated with diagonal system-bath coupling, where the influence functional is a functional of scalar-valued trajectories. This formalism is enough if only a single bath is under consideration. However, when multiple baths are present, non-diagonal system-bath couplings need to be taken into consideration. In such a situation, using an abstract Liouvillian method, the influence functional can be obtained as a functional of operator-valued trajectories. The value of the influence functional itself also becomes a superoperator rather than an ordinary scalar, whose meaning is ambiguous at first glance and its connection to the conventional understanding of the influence functional needs extra careful consideration. In this article, we present another concrete derivation of the superoperator-valued influence functional based on the straightforward Trotter–Suzuki splitting, which can provide a clear picture to interpret the superoperator-valued influence functional.