Liouvillian Geometry of Multidimensional Spectra: Pathway Transport and Observational Holonomy in Open Quantum Systems

Abstract

Liouville pathways provide the conceptual foundation for interpreting multidimensional spectroscopies and are commonly represented as independent contributions to the nonlinear response. In open quantum systems, however, the effective reduced Liouvillian need not preserve this observational pathway decomposition. Environmental interactions can transport amplitude between pathway sectors during the free-evolution intervals, generating measurable signatures in the nonlinear spectroscopic response.} We develop a geometric framework in which pathway transport is governed by a Liouvillian connection, its associated curvature, and the resulting observational holonomy. The framework applies to open quantum systems in which the environment selects a pointer basis distinct from the observational basis used to construct the spectroscopic response. This basis incompatibility induces transport among Liouville pathways, generating characteristic spectral distortions and a nontrivial Liouvillian curvature. Using a Duhamel expansion of the Liouvillian propagator, we derive a reconstruction procedure that identifies the transport operators responsible for the observed redistribution of pathway weight, accurate throughout the full range of basis misalignment. This perspective reframes spectral features as determined not only by which pathways exist but by how amplitude is transported among them. Spectral distortions, peak shifts, and otherwise-forbidden pathway contributions become geometric signatures of a curved Liouville-space manifold rather than phenomenological broadening corrections, identifying pathway geometry as a complementary layer of organization in nonlinear spectroscopy.