Density matrix analysis of systems influenced by periodic Hamiltonians
Soham Sen, Manjari Dutta, Sunandan Gangopadhyay
TL;DR
This work analyzes density-matrix dynamics under time-periodic Hamiltonians using the von Neumann equation and the Lewis-Riesenfeld invariant, demonstrating a direct link between the density matrix and the invariant up to a constant factor. It shows that the invariant framework reproduces the density-matrix evolution for periodic drives, with the Lewis phase growing linearly in time. The study further examines a two-level system under a square-pulse magnetic field, deriving a revival periodicity for pure states and showing sign flips of off-diagonal elements across half-periods, while comparing two coherence measures. It finds that the $l_1$-norm coherence can exhibit time-dependent behavior, whereas the Frobenius-norm coherence remains unity, arguing that the Frobenius measure provides a more faithful estimator of coherence in this regime.
Abstract
In this work, we consider simple systems that are influenced by Hamiltonians with time periodicity. Our analysis is mainly focussed on the density matrix approach and aims to solve the Liouville equation of motion from which one can extract the state of the system when the system is in a pure state. We start our analysis with the standard Rabi-oscillation problem. We consider a density matrix corresponding to the entire model system and solve the Liouville equation of motion. We have then made use of the Lewis-Reisenfeld invariant approach and arrive at the exact same result which implies that the density matrix of the system can indeed be identified with the Lewis invariant. Finally, we consider a two-level system with a constant magnetic field in the $z$-direction and a time dependent magnetic field in the $x$-direction. Finally, we solve the Liouville equation of motion for this system and calculate the various coherence measures and plot them to investigate the time dependence and reliability of different coherence measures.