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Laser Physics Letters

Finite-time isothermal and isochoric control via mixed state inverse engineering

B Liu, W Ma

Published Dec 1, 2025• 0 citations
Publisher

Abstract

Abstract We derive the transport dynamics of a driven harmonic oscillator coupled to a thermal reservoir by formulating the driven Markovian master equation within the Lewis–Riesenfeld invariant framework. Using the mixed-state inverse engineering method, we design accurate control protocols for finite-time isothermal and isochoric processes. Analytical expressions for work and heat are obtained, providing clear thermodynamic characterization of the quantum transport.

DOI
10.1088/1612-202x/ae18a6

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Abstract We derive the transport dynamics of a driven harmonic oscillator coupled to a thermal reservoir by formulating the driven Markovian master equation within the Lewis–Riesenfeld invariant framework. Using the mixed-state inverse engineering method, we design accurate control protocols for finite-time isothermal and isochoric processes. Analytical expressions for work and heat are obtained, providing clear thermodynamic characterization of the quantum transport.
We investigate the dynamics of the driven open double two-level system by deriving a driven Markovian master equation based on the Lewis-Riesenfeld invariant theory. The transitions induced by coupling to the heat reservoir occur between the instantaneous eigenstates of the Lewis-Riesenfeld invariant. Therefore, different driving protocols associated with corresponding Lewis-Riesenfeld invariants result in different open system dynamics and symmetries. In particular, we show that since the instantaneous steady state of the driven double two-level system is one of eigenstates of the Lewis-Riesenfeld invariant at ultralow reservoir temperature, the inverse engineering method based on the Lewis-Riesenfeld invariants has a good performance in rapidly preparing the quantum state of open quantum systems. As an example, a perfect entangled state is generated by means of the inverse engineering method.
We derive a Markovian master equation for driven open quantum systems based on the Lewis-Riesenfeld-invariants theory, which is available for arbitrary driving protocols. The role of the Lewis-Riesenfeld invariants is to help us bypass the time-ordering obstacle in expanding the propagator of the free dynamics, such that the Lindblad operators in our driven Markovian master equation can be determined easily. We also illustrate that, for the driven open quantum systems, the spontaneous emission and the thermal excitation induce the transitions between eigenstates of the Lewis-Riesenfeld invariant, but not the system Hamiltonian's. As an example, we present the driven Markovian master equation for a driven two-level system coupled to a heat reservoir. By comparing to the exactly solvable models, the availability of the driven Markovian master equation is verified. Meanwhile, the adiabatic limit and inertial limit of the driven Markovian master equation are also discussed, which result in the same Markovian master equations as those presented before in the corresponding limits.
The mixed-state inverse engineering scheme is a control scheme used for engineering the quantum state of a driven open quantum system from an initial steady state to a final steady state. In this paper, we present an analytical study of this scheme applied to the driven two-level model coupled to a heat reservoir. Typically, when the purity of the quantum state varies, incoherent control techniques are required for mixed-state engineering. However, we show that for both Markovian and non-Markovian dynamics, coherent control protocols can transfer the quantum state into the target state. This simplification comes at a cost, as the evolution of the quantum state must be limited to restricted conditions, resulting in special trajectories in its Hilbert space that connect the initial and target states.
Abstract We propose using the dynamical invariants, also known as the Lewis–Riesenfeld invariants, to speed-up the equilibration of a driven open quantum system. This allows us to reverse engineer the time-dependent master equation that describes the dynamics of the open quantum system and systematically derive a protocol that realizes a shortcut to equilibration. The method does not require additional constraints on the timescale of the dynamics beside the Born–Markov approximation and can be generically applied to boost single particle quantum engines significantly. We demonstrate this with the damped harmonic oscillator, and show that our protocol can achieve high-fidelity control on shorter timescales than simple non-optimized protocols. We find that the system is heated during the dynamics to speed-up the equilibration, which can be considered as an analogue of the Mpemba effect in quantum control.
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.
Abstract We study the thermodynamic performance of a finite-time non-regenerative quantum Stirling-like cycle used as a heat engine. We consider specifically the case in which the working substance (WS) is a two-level system (TLS). The Stirling cycle is made of two isochoric transformations separated by a compression and an expansion stroke during which the WS is in contact with a thermal reservoir. To describe these two strokes we derive a non-Markovian master equation which allows to study the real-time dynamics of a driven open quantum system with arbitrary fast driving. Following the real-time dynamics of the WS using this master equation, the endpoints of the isotherms can deviate from the equilibrium thermal states. The role of this deviation in the performance of the heat engine is addressed. We found that the finite-time dynamics and thermodynamics of the cycle depend non-trivially on the different time scales at play. In particular, driving the WS at a time scale comparable to the resonance time of the bath enhances the performance of the cycle and allows for an efficiency higher than the efficiency of the quasistatic cycle, but still below the Carnot bound. However, by adding thermalization of the WS with the baths at the end of compression/expansion processes one recovers the conventional scenario in which efficiency decreases by speeding up the processes. In addition, the performance of the cycle is dependent on the compression/expansion speeds asymmetrically, which suggests new freedom in optimizing quantum heat engines. The maximum output power and the maximum efficiency are obtained almost simultaneously when the real-time endpoints of the compression/expansion processes are considered instead of the equilibrium thermal endpoint states. However, the net extractable work always declines by speeding up the drive.