[Paper Review] Periodically-driven quantum thermal machines from warming up to limit cycle

Liu, Junjie, Jung, Kenneth A.+1 more|arXiv (Cornell University)|Jun 19, 2021

Advanced Thermodynamics and Statistical Mechanics36 citations

TL;DR

This paper develops a unified thermodynamic framework for periodically-driven quantum thermal machines (PD-QTMs), addressing both transient warming-up and limit-cycle operation phases. By identifying a previously overlooked term A in the first law of thermodynamics—arising from non-periodic ensemble averages of system-bath observables—it shows that this term significantly affects efficiency, especially at strong system-bath coupling, where conventional models fail. The framework is validated via simulations of a driven resonant level quantum Otto engine.

ABSTRACT

Theoretical treatments of periodically-driven quantum thermal machines (PD-QTMs) are largely focused on the limit-cycle stage of operation characterized by a periodic state of the system. Yet, this regime is not immediately accessible for experimental verification. Here, we present a general thermodynamic framework that can handle the performance of PD-QTMs both before and during the limit-cycle stage of operation. It is achieved by observing that periodicity may break down at the ensemble average level, even in the limit-cycle phase. With this observation, and using conventional thermodynamic expressions for work and heat, we find that a complete description of the first law of thermodynamics for PD-QTMs requires a new contribution, which vanishes only in the limit-cycle phase under rather weak system-bath couplings. Significantly, this contribution is substantial at strong couplings even at limit cycle, thus largely affecting the behavior of the thermodynamic efficiency. We demonstrate our framework by simulating a quantum Otto engine building upon a driven resonant level model. Our results provide new insights towards a complete description of PD-QTMs, from turn-on to the limit-cycle stage and, particularly, shed light on the development of quantum thermodynamics at strong coupling.

Motivation & Objective

• To address the lack of a unified thermodynamic description for periodically-driven quantum thermal machines (PD-QTMs) across both transient warming-up and limit-cycle operation phases.
• To resolve the discrepancy between theoretical predictions and experimental measurements by accounting for non-periodic behavior in ensemble averages during limit-cycle operation.
• To identify and quantify a previously overlooked contribution, denoted as A, in the first law of thermodynamics for PD-QTMs, which arises due to non-periodic system-bath observables even in the limit cycle.
• To demonstrate that this A term substantially alters thermodynamic efficiency, particularly at strong system-bath coupling, challenging conventional assumptions in quantum thermodynamics.
• To provide a complete, non-perturbative description of work, heat, and energy balance in PD-QTMs applicable from initial turn-on to steady-state operation.

Proposed method

• Formalizing thermodynamic quantities—work W(m) and heat Qv(m)—over cycle-number-dependent intervals [mT, (m+1)T], enabling analysis of both transient and steady-state regimes.
• Defining work and heat via the full Hamiltonian dynamics, avoiding system-bath partitioning ambiguities, with W(m) = ∫ dt Tr[ ˙H(t)ρtot(t)] and Qv(m) = −∫ dt Tr[HvB ˙ρtot(t)].
• Identifying a new term A(m) = Tr{HSI(mT)[ρtot((m+1)T)−ρtot(mT)]} in the first law, which ensures energy conservation across all times and coupling strengths.
• Demonstrating that A(m) vanishes only in the limit cycle under weak coupling; it remains significant at strong coupling, even in the steady state.
• Using a driven resonant level model as a quantum Otto engine, with a smooth, differentiable protocol for the level energy ϵ(t) over a cycle T.
• Performing numerical simulations based on the Lindblad-von Neumann (DLvN) equation to solve for the time evolution of the total density matrix, with convergence checks on γ, D, ∆ϵ, and δt.

Experimental results

Research questions

• RQ1 How can a unified thermodynamic description be constructed for PD-QTMs that spans both the transient warming-up phase and the limit-cycle phase?
• RQ2 Why do conventional thermodynamic definitions fail at strong system-bath coupling, and what new physical contribution is required to restore energy conservation?
• RQ3 Under what conditions does periodicity break down at the ensemble average level, even in the limit-cycle phase, and how does this affect thermodynamic observables?
• RQ4 What is the quantitative impact of the newly identified term A on the thermodynamic efficiency of PD-QTMs, especially in the strong coupling regime?
• RQ5 Can the proposed framework accurately describe the full evolution of a quantum thermal machine from initial turn-on to steady-state operation?

Key findings

• The term A(m) in the first law of thermodynamics, representing the change in ensemble average of HS(t) + HI(t) over a cycle, is non-zero during both warming-up and limit-cycle phases, particularly at strong coupling.
• The A term vanishes only in the limit cycle under weak system-bath coupling; at strong coupling, it remains substantial even in the steady state, invalidating conventional efficiency calculations.
• Simulations of a driven resonant level quantum Otto engine show that the inclusion of A is essential for energy conservation, especially at strong coupling (Γ = 0.5), where conventional definitions fail.
• Heat exchange Qh(m) and Qc(m) exhibit cycle-number-dependent variations even in the limit cycle, indicating that bath density matrices do not inherit the periodicity of the system, due to non-periodic ensemble averages.
• At strong coupling (Γ = 0.5), the absolute value of cold bath heat |Qc| exceeds that of hot bath heat |Qh|, a seemingly unphysical imbalance that is compensated by a negative A term, preserving the first law.
• The thermodynamic efficiency η(m) converges to a steady value only after several cycles (m ≈ 5 for weak coupling), and its correct value depends critically on including the A term, which is negligible only in the weak coupling limit.