Density matrix analysis of systems influenced by periodic Hamiltonians

TL;DR

This paper analyzes quantum systems with periodic Hamiltonians using the density matrix approach, solving the Liouville equation, and compares results with the Lewis-Reisenfeld invariant method, focusing on two-level systems and coherence measures.

Contribution

It demonstrates that the density matrix can be identified with the Lewis invariant in periodically driven quantum systems and applies this to analyze coherence dynamics.

Findings

Density matrix aligns with Lewis invariant in periodic systems

Exact solutions for Rabi oscillations using density matrix and invariants

Time-dependent coherence measures are computed and analyzed

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.