[Paper Review] Periodically driven quantum open systems: Tutorial

Robert Alicki, David Gelbwaser-Klimovsky+1 more|arXiv (Cornell University)|May 21, 2012

Advanced Thermodynamics and Statistical Mechanics3 references12 citations

TL;DR

This tutorial derives a Markovian master equation for periodically driven quantum open systems using Floquet theory and weak-coupling approximations, generalizing the Davies generator to time-periodic Hamiltonians. It establishes a thermodynamically consistent definition of heat currents and applies the formalism to a modulated qubit, showing that effective temperatures and non-zero heat currents emerge due to parametric driving, enabling engine-like behavior even with a single bath.

ABSTRACT

We present a short derivation and discussion of the master equation for an open quantum system weakly coupled to a heat bath and then its generalization to the case of with periodic external driving based on the Floquet theory. Further, a single heat bath is replaced by several ones. We present also the definition of heat currents which satisfies the second law of thermodynamics and apply the general results to a simple model of periodically modulated qubit.

Motivation & Objective

• To generalize the standard Davies master equation for open quantum systems to the case of periodic driving using Floquet theory.
• To ensure thermodynamic consistency by defining heat currents that satisfy the second law of thermodynamics in periodically driven systems.
• To analyze the interplay between system dynamics, bath correlations, and control Hamiltonians in non-equilibrium quantum thermodynamics.
• To demonstrate the emergence of effective temperatures and non-zero heat currents in a periodically driven qubit, even with a single heat bath.
• To clarify misconceptions about Markovianity and bath correlation times, showing they are not independent of system dynamics.

Proposed method

• Derives the reduced dynamics of a weakly coupled open quantum system in the interaction picture using the cumulant expansion and Born-Markov approximations.
• Applies Floquet theory to time-periodic Hamiltonians, decomposing the system operator into Fourier components with frequencies corresponding to Bohr transitions.
• Uses the Markov approximation to replace time-convolution integrals with time-local generators, leading to a time-independent Lindblad-Gorini-Kossakowski-Sudarshan (LGKS) generator.
• Introduces a thermodynamically consistent definition of heat currents via the entropy production rate, ensuring compliance with the second law.
• Applies the formalism to a two-level system (qubit) with diagonal periodic modulation, using a time-dependent Rabi frequency and coupling to multiple heat baths.
• Derives the effective generator as a sum over harmonic components, with transition rates modulated by the Fourier coefficients of the driving field.

Experimental results

Research questions

• RQ1 How can the standard Markovian master equation for open quantum systems be generalized to include periodic driving via Floquet theory?
• RQ2 What is the correct thermodynamic definition of heat currents in periodically driven open quantum systems that satisfies the second law?
• RQ3 Can a single heat bath induce a net heat current in a periodically driven system, and if so, under what conditions?
• RQ4 How do effective temperatures emerge in periodically driven systems, and what is their physical interpretation?
• RQ5 To what extent can arbitrary control Hamiltonians be applied to Markovian systems without violating thermodynamic laws?

Key findings

• The master equation for periodically driven systems is derived as a time-local, Markovian generator by combining weak-coupling and Markov approximations, with the generator expressed in terms of Fourier components of the system operators.
• The heat current definition satisfies the second law, with total entropy production non-negative, and the steady-state heat current being time-independent and consistent with energy conservation.
• In the driven qubit model, a single bath can induce a net heat current due to parametric modulation, violating the expectation that a single bath cannot support a heat engine.
• Effective temperatures emerge for each harmonic component of the driving, given by $ T_a(q) = rac{ ho_0}{ ho_0 - q ilde{ u}} T_a $, which can be negative for certain $ q $, indicating non-equilibrium behavior.
• The steady-state density matrix is diagonal and equivalent to a Gibbs state with an effective temperature $ T_{ ext{eff}} $, determined by the ratio of averaged decay and excitation rates.
• The total heat current (equal to negative power) is non-zero due to the $ T_a / T_a(q) $ factor in the current expression, demonstrating engine-like operation even with a single bath.