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New Journal of Physics

Thermodynamic cost of a periodically driven quantum system beyond the adiabatic regime

Xiaoyun Wang, Haowei Wang, Dazhi Xu

Published Sep 30, 2026• 0 citations
Publisher

Abstract

Abstract Finite-time periodic control of an open quantum system is governed by the competition between nonadiabatic driving and reservoir-induced relaxation. We study this competition in a driven two-level system weakly coupled to a single thermal reservoir. Using a nonadiabatic time-dependent quantum master equation, we construct the periodic steady state and separate the dissipated work per cycle into population and nonadiabatic contributions in the instantaneous energy basis. The cycle work is nonmonotonic in the driving frequencies studied and is dominated near its maximum by the nonadiabatic contribution. Nonmonotonicity persists in the examined coupling, temperature, and transverse-amplitude scans. At low frequencies, the work recovers the inverse-period scaling of slow-driving thermodynamics. At higher frequencies, the nonadiabatic work rate ceases to grow in proportion to the driving frequency, and the accumulated work per cycle decreases. We also find that, at finite speed, the ratio of the upward and downward transition coefficients is not determined by the instantaneous energy gap alone and obtain the leading correction to instantaneous detailed balance through a short-memory expansion. These results identify how nonadiabatic transitions and relaxation jointly set the thermodynamic cost of sustaining a periodic quantum process.

DOI
10.1088/1367-2630/aeae66

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Abstract Finite-time periodic control of an open quantum system is governed by the competition between nonadiabatic driving and reservoir-induced relaxation. We study this competition in a driven two-level system weakly coupled to a single thermal reservoir. Using a nonadiabatic time-dependent quantum master equation, we construct the periodic steady state and separate the dissipated work per cycle into population and nonadiabatic contributions in the instantaneous energy basis. The cycle work is nonmonotonic in the driving frequencies studied and is dominated near its maximum by the nonadiabatic contribution. Nonmonotonicity persists in the examined coupling, temperature, and transverse-amplitude scans. At low frequencies, the work recovers the inverse-period scaling of slow-driving thermodynamics. At higher frequencies, the nonadiabatic work rate ceases to grow in proportion to the driving frequency, and the accumulated work per cycle decreases. We also find that, at finite speed, the ratio of the upward and downward transition coefficients is not determined by the instantaneous energy gap alone and obtain the leading correction to instantaneous detailed balance through a short-memory expansion. These results identify how nonadiabatic transitions and relaxation jointly set the thermodynamic cost of sustaining a periodic quantum process.
We investigate the dynamic evolution and thermodynamic process of a driven quantum system immersed in a finite-temperature heat bath. A Born-Markovian quantum master equation is formally derived for the time-dependent system with discrete energy levels. This quantum master equation can be applied to situations with a broad range of driving speeds and bath temperatures and thus be used to study the finite-time quantum thermodynamics even when nonadiabatic transition and dissipation coexist. The dissipative Landau-Zener model is analyzed as an example. The population evolution and transition probability of the model reveal the importance of the competition between driving and dissipation beyond the adiabatic regime. Moreover, local maxima of irreversible entropy production occur at intermediate sweep velocity and finite temperature, which the low-dissipation model cannot describe.
We derive a Markovian master equation for a linearly driven dissipative quantum harmonic oscillator, valid for generic driving beyond the adiabatic limit. We solve this quantum master equation for arbitrary Gaussian initial states and investigate its departure from the adiabatic master equation in the regime of fast driving. We concretely examine the behavior of dynamical variables, such as position and momentum, as well as of thermodynamic quantities, such as energy and entropy. We additionally study the influence of the nonequilibrium driving on the quantum coherence of the oscillator in the instantaneous energy eigenbasis. We further analyze the approach to the adiabatic limit and the relaxation to the instantaneous steady state as a function of the driving speed.
We construct a quantum Markovian master equation for a driven system coupled to a thermal bath. The derivation utilizes an explicit solution of the propagator of the driven system. This enables the validity of the master equation to be extended beyond the adiabatic limit. The nonadiabatic master equation (NAME) is derived employing the weak system-bath coupling limit. The NAME is valid when a separation of timescales exists between the bath dynamics and the external driving. In contrast to the adiabatic master equation, the NAME leads to coupled equations of motion for the population and coherence. We employ the NAME to solve the example of an open driven time-dependent harmonic oscillator. For the harmonic oscillator the NAME predicts the emergence of coherence associated with the dissipation term. As a result of the nonadiabatic driving the thermalization rate is suppressed. The solution is compared with both numerical calculations and the adiabatic master equation.
We present the stochastic thermodynamics analysis of an open quantum system weakly coupled to multiple reservoirs and driven by a rapidly oscillating external field. The analysis is built on a modified stochastic master equation in the Floquet basis. Transition rates are shown to satisfy the local detailed balance involving the entropy flowing out of the reservoirs. The first and second law of thermodynamics are also identified at the trajectory level. Mechanical work is identified by means of initial and final projections on energy eigenstates of the system. We explicitly show that this two step measurement becomes unnecessary in the long time limit. A steady-state fluctuation theorem for the currents and rate of mechanical work is also established. This relation does not require the introduction of a time reversed external driving which is usually needed when considering systems subjected to time asymmetric external fields. This is understood as a consequence of the secular approximation applied in consistency with the large time scale separation between the fast driving oscillations and the slower relaxation dynamics induced by the environment. Our results are finally illustrated on a model describing a thermodynamic engine.
The dynamics of open quantum systems is commonly described by quantum master equations derived under the assumption of weak system-bath coupling and a separation of timescales between system and bath. When the system is additionally subjected to a periodic driving, the validity of the resulting Floquet master equations is further restricted to regimes of weak or high-frequency driving. Here, we benchmark a set of commonly used Floquet master equations for a model of two locally driven spins coupled to a shared Ohmic reservoir at finite temperature. We systematically probe the accuracy of the equations as a function of the driving parameters, thus identifying limits of their applicability. Dynamical maps predicted by each master equation are compared against numerically exact non-Markovian simulations, tracking the full relaxation dynamics. We find that the accuracy of each master equation closely reflects the assumptions underlying its derivation. For the Floquet-Lindblad equation, errors can be strongly amplified near resonances where the secular approximation breaks down, while approaches that avoid the secular approximation perform better and exhibit a more systematic dependence of the error on driving frequency and amplitude.
We present an approach based on a density matrix expansion to study thermodynamic properties of a quantum system strongly coupled to two or more baths. For slow external driving of the system, we identify the adiabatic and nonadiabatic contributions to thermodynamic quantities, and we show how the first and second laws of thermodynamics are manifested in the strong coupling regime. Particularly, we show that the entropy production is positive up to second order in the driving speed. The formulation can be applied both for bosonic and fermionic systems, and recovers previous results for the equilibrium case [Phys. Rev. B 98, 134306 (2018)]. The approach is then demonstrated for the driven resonant level model as well as the driven Anderson impurity model, where the hierarchical quantum master equation method is used to accurately simulate the nonequilibrium quantum dynamics.