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arXiv:2503.17207

Nonadiabatic master equation for a linearly driven harmonic oscillator

Sinan Altinisik, Eric Lutz

Published Mar 21, 2025• 1 citation
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Abstract

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.

DOI
10.48550/arxiv.2503.17207

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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 study a generic quantum Markovian master equation for a linearly displaced or driven harmonic oscillator. It was known that the displacement dynamics of Gaussian mixed states depends on the unitary part of the Liouvillian, the decay rate of the system but not on the bath temperature. Here we further show that the fast-rotating modes do not affect the system's displacement dynamics under linear driving forces. Analytical solutions of the quantum master equation are obtained for displaced Gaussian mixed states. Because the non-driven and driven Liouvillians are related by a unitary displacement operator, they are expected to share the same exceptional points structure. At the exceptional points, the displacement of critically damped oscillator displays a characteristics polynomial-in-time prefactor multiplied by an exponential decay. We discuss how external time-dependent forces affect the displacement dynamics using impulsive force and harmonic force as examples. The results obtained for constant driving remain valid in the presence of time-dependent driving.
We develop an exact analytical framework for studying the nonequilibrium dynamics of interacting quantum harmonic oscillators coupled to a modelled thermal reservoir and driven by external classical fields. The total Hamiltonian is diagonalized via successive Bogoliubov transformations, enabling a nonperturbative treatment of dissipation and driving. For two coupled oscillators with an external source applied to the first, we derive explicit energy expressions and analyze their dependence on coupling and bath parameters. The phase-space structure is characterized through Husimi QQQ-functions for initial separable coherent and number states. We obtain the reduced density matrix elements in the number-state basis and demonstrate that, in the absence of driving and at zero temperature, the reduced density matrix of the main system satisfies the Lindblad master equation. A generating function for the Husimi function is introduced to facilitate computation of higher-order correlations. The analysis is then generalized to nnn interacting oscillators, with analytical energy expressions provided and the case n=3n=3n=3 examined in detail. Our results establish a versatile and exact framework for driven-dissipative quantum systems, with potential applications in quantum thermodynamics, open quantum systems, and many-body physics.
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.
Nonequilibrium energy transport serves as one of fundamental problems in quantum thermodynamics and quantum technologies. Driven quantum master equation in the dressed picture provides an efficient way of investigating nonequilibrium energy flow in general driven-dissipative quantum systems, where the systems are simultaneously driven by the finite thermodynamic bias and coherent driving field. The validity and general applicability of driven quantum master equation is confirmed by comparing with Floquet master equation, by analyzing energy currents in generic spin and boson models. The additional driving phase reserved in system-reservoir interactions, will apparently modify microscopic energy exchange processes. The steady-state energy currents are dramatically enhanced, in particular near the resonant regimes. In contrast, the traditional dressed master equation yields distinct behaviors of the energy currents. We hope that the driven quantum master equation may provide an efficient utility for the control of quantum transport and thermodynamic performances in driven-dissipative nanodevices.
Quantum master equations are an invaluable and highly successful tool for the approximate, yet accurate, description of undriven open quantum systems weakly coupled to their environment. In contrast, results on the description of the equally important case of driven open quantum systems using quantum master equations remain scarce. Existing approaches typically impose restrictions on the driving protocol, including, for instance, Floquet master equations for periodic driving and adiabatic master equations for slow driving. However, no generally accepted formulation for arbitrary driving has yet been established. A promising candidate from which all existing approaches can be derived by approximations based on protocol constraints is the driven Redfield master equation. Strikingly, it has received very little attention so far. A main reason is that, in analogy to its undriven counterpart, the driven Redfield master equation notoriously violates complete positivity. In the undriven case, this problem can be successfully addressed using the rotating wave approximation (RWA). For approaches derived from the driven Redfield master equation for specific protocols, such as Floquet master equations and adiabatic master equations, a generalized form of the RWA can still be applied. However, a full generalization directly applicable to the driven Redfield master equation under arbitrary driving has not yet been developed. To address this issue, we propose a straightforward and natural generalization of the RWA based on the version of the RWA used for adiabatic master equations. We then apply the driven Redfield master equation to the paradigmatic, exactly solvable model of a driven quantum harmonic oscillator coupled to a bosonic bath and benchmark it against the exact solution. In particular, we compare the original driven Redfield master equation without an RWA to its generalized RWA version. We find that the driven Redfield master equation without an RWA exhibits excellent agreement with the exact solution across all driving speeds, while the violation of complete positivity shows no notable effect. By contrast, the RWA-based version of the driven Redfield master equation fails to reproduce the dynamics accurately beyond slow driving at short times. Additionally, we analyze known adiabatic master equations, both without and with an RWA. Remarkably, the non-RWA versions agree with the exact dynamics far beyond the strictly adiabatic regime, while the RWA-based versions already show noticeable deviations even for slow driving, contrasting previous observations in the literature. Taken together, these results indicate that the driven Redfield master equation and adiabatic master equations without RWA have been significantly underestimated in their practical usefulness, whereas RWA-based approaches in driven settings may have been overestimated. This strongly suggests that future work should focus on alternative approximation methods applied to the driven Redfield master equation that preserve its high numerical accuracy while restoring complete positivity.