Equivalence between Redfield and master equation approaches for a time-dependent quantum system and coherence control

TL;DR

This paper demonstrates the equivalence of Redfield and master equation approaches for a time-dependent quantum system, showing they produce identical relaxation times and proposing a protocol to mitigate decoherence via frequency modulation.

Contribution

It derives the Redfield formalism for time-dependent systems, compares it with the quantum master equation approach, and introduces a decoherence mitigation protocol applicable in NMR.

Findings

Redfield and master equation approaches yield identical Bloch equations and relaxation times.

The characteristic times T1 and T2 are related to operator-sum and phenomenological-operator representations.

A protocol using frequency modulation can circumvent decoherence related to energy loss.

Abstract

We present a derivation of the Redfield formalism for treating the dissipative dynamics of a time-dependent quantum system coupled to a classical environment. We compare such a formalism with the master equation approach where the environments are treated quantum mechanically. Focusing on a time-dependent spin-1/2 system we demonstrate the equivalence between both approaches by showing that they lead to the same Bloch equations and, as a consequence, to the same characteristic times $T_{1}$ and $T_{2}$ (associated with the longitudinal and transverse relaxations, respectively). These characteristic times are shown to be related to the operator-sum representation and the equivalent phenomenological-operator approach. Finally, we present a protocol to circumvent the decoherence processes due to the loss of energy (and thus, associated with $T_{1}$). To this end, we simply associate the time-dependence of the quantum system to an easily achieved modulated frequency. A possible implementation of the protocol is also proposed in the context of nuclear magnetic resonance.