Decoupling of External and Internal Dynamics in Driven Two-level Systems

TL;DR

This paper develops a framework to decouple and analyze driven two-level quantum systems with external degrees of freedom, providing analytic solutions and insights into control and noise effects.

Contribution

It introduces a representation-free method to decouple external and internal dynamics, enabling analytic solutions and perturbative analysis of driven two-level systems.

Findings

Analytic expressions for Rabi oscillations with external degrees of freedom

Decoupling into classical oscillator equations when detuning operator is diagonalizable

Perturbative approach for weak external potentials

Abstract

We show how a laser driven two-level system including quantized external degrees of freedom for each state can be decoupled into a set of oscillator equations acting only on the external degrees of freedom with operator valued damping representing the detuning. We give a way of characterizing the solvability of this family of problems by appealing to a classical oscillator with time-dependent damping. As a consequence of this classification we (a) obtain analytic and representation-free expressions for Rabi oscillations including external degrees of freedom with and without an external linear potential, (b) show that whenever the detuning operator can be diagonalized (analytically or numerically) the problem decomposes into a set of classical equations and (c) we can use the oscillator equations as a perturbative basis to describe Rabi oscillations in weak but otherwise arbitrary external potentials. Moreover, chirping of the driving fields phase emerges naturally as a means of compensating the Ehrenfest/mean-value part of the detuning operator's dynamics while the presence of driving phase noise leads to a stochastic evolution equation of Langevin type. Lastly, our approach is representation free with respect to the external degrees of freedom and as consequence a suitable representation or basis expansion can be chosen a posteriori depending on the desired application at hand.