Abstract 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 influence functional
itself also becomes a superoperator rather than an ordinary scalar,
whose meaning is ambiguous at first glance and its connection to
conventional understanding of the influence functional needs extra
careful considerations. 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.