Quantum Process Tomography of a Thermal Alkali-Metal Vapor

TL;DR

This paper presents a scalable quantum process tomography method for characterizing the dynamics of thermal alkali-metal vapor qudits, enabling precise noise modeling and control without extensive numerical simulations.

Contribution

It introduces a computationally efficient framework combining maximum likelihood estimation and spectral regularization for direct Liouvillian reconstruction in the Bloch-Fano basis.

Findings

Successfully reconstructs Liouvillian dynamics of a thermal Rb-87 qutrit ensemble.

Validates the method across various regimes, resolving subtle dissipative effects.

Avoids branch-cut ambiguities by bounding eigenvalue phases within a specific range.

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.