Quantum Process Tomography of a Thermal Alkali-Metal Vapor

Abstract

Characterizing the open-system dynamics of multilevel quantum systems (qudits) remains a fundamental challenge due to ensemble inhomogeneities and complex environmental interactions. Here, we introduce a computationally efficient quantum process tomography framework that reconstructs the Liouvillian dynamics of a thermal $^{87}$Rb qutrit ensemble directly in the Bloch-Fano representation. By combining maximum likelihood estimation with post-hoc spectral regularization, our protocol extracts physically admissible, completely positive and trace-preserving maps without repeated numerical integration of the master equation. We rigorously justify selecting the principal branch for the matrix logarithm by demonstrating that experimental eigenvalue phases remain strictly bounded within $[-0.35,0.35]$ radians, avoiding branch-cut ambiguities. The method is validated across relaxation-driven, static-field, and time-dependent regimes, resolving overlapping control signals and subtle dissipative mechanisms such as AC Stark shifts. Our approach establishes a scalable route for generator-level characterization of ambient qudit systems, enabling noise-aware control and precise benchmarking for atomic sensors and simulators.