Open quantum systems (OQSs) cannot always be described with the Markov approximation, which requires a large separation of system and environment time scales. An overview is given of some of the most important techniques available to tackle the dynamics of an OQS beyond the Markov approximation. Some of these techniques, such as master equations, Heisenberg equations, and stochastic methods, are based on solving the reduced OQS dynamics, while others, such as path integral Monte Carlo or chain mapping approaches, are based on solving the dynamics of the full system. The physical interpretation and derivation of the various approaches are emphasized, how they are connected is explored, and how different methods may be suitable for solving different problems is examined.

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Dynamics of non-Markovian open quantum systems

Quantum thermodynamics addresses the emergence of thermodynamic laws from quantum mechanics. The viewpoint advocated is based on the intimate connection of quantum thermodynamics with the theory of open quantum systems. Quantum mechanics inserts dynamics into thermodynamics, giving a sound foundation to finite-time-thermodynamics. The emergence of the 0-law, I-law, II-law and III-law of thermodynamics from quantum considerations is presented. The emphasis is on consistency between the two theories, which address the same subject from different foundations. We claim that inconsistency is the result of faulty analysis, pointing to flaws in approximations.

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Quantum Thermodynamics: A Dynamical Viewpoint

Quantum theory of open systems

Quantum thermodynamics supplies a consistent description of quantum heat engines and refrigerators up to a single few-level system coupled to the environment. Once the environment is split into three (a hot, cold, and work reservoir), a heat engine can operate. The device converts the positive gain into power, with the gain obtained from population inversion between the components of the device. Reversing the operation transforms the device into a quantum refrigerator. The quantum tricycle, a device connected by three external leads to three heat reservoirs, is used as a template for engines and refrigerators. The equation of motion for the heat currents and power can be derived from first principles. Only a global description of the coupling of the device to the reservoirs is consistent with the first and second laws of thermodynamics. Optimization of the devices leads to a balanced set of parameters in which the couplings to the three reservoirs are of the same order and the external driving field is in resonance. When analyzing refrigerators, one needs to devote special attention to a dynamical version of the third law of thermodynamics. Bounds on the rate of cooling when T c → 0 are obtained by optimizing the cooling current. All refrigerators as T c → 0 show universal behavior. The dynamical version of the third law imposes restrictions on the scaling as T c → 0 of the relaxation rate γc and heat capacity cV of the cold bath.

Quantum Heat Engines and Refrigerators: Continuous Devices

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Ordinary Differential Equations With Applications

The recently developed technique combining the weak-coupling limit with the Floquet formalism is applied to a model of a two-level atom driven by a strong laser field and weakly coupled to heat baths. First, the case of a single electromagnetic bath at zero temperature is discussed and the formula for resonance fluorescence is derived. The expression describes the well-known Mollow triplet, but its details differ from the standard ones based on additional simplifying assumptions. The second example describes the case of two thermal reservoirs: an electromagnetic one at finite temperature and the second dephasing one, which can be realized as a phononic or buffer gas reservoir. It is shown using the developed thermodynamical approach that the latter system can work in two regimes depending on the detuning sign: a heat pump transporting heat from the dephasing reservoir to an electromagnetic bath or heating both, always at the expense of work supplied by the laser field.

Markovian master equation and thermodynamics of a two-level system in a strong laser field

In this paper, the mathematical framework providing a description of completely positive and trace preserving dynamics of open quantum systems is addressed. Special case of time-dependent Lindbladian governed by periodic Hamiltonian is concerned. It is proven that appropriate trace preserving dynamical map of such periodically driven system may be constructed by application of Floquet theory. Appropriate Lindbladian and Markovian master equation in weak coupling regime and resulting dynamical map are constructed. Some examples of application of developed technique are given.

On the application of Floquet theorem in development of time-dependent Lindbladians

We explore, theoretically and experimentally, a method for cooling a broadband heat reservoir, via its laser-assisted collisions with two-level atoms followed by their fluorescence. This method is shown to be advantageous compared to existing laser-cooling methods in terms of its cooling efficiency, the lowest attainable temperature for broadband baths, and its versatility: it can cool down any heat reservoir, provided the laser is red detuned from the atomic resonance. It is applicable to cooling down both dense gaseous and condensed media.

Laser-induced cooling of broadband heat reservoirs

Solar cells are engines converting energy supplied by the photon flux into work. All known types of macroscopic engines and turbines are also self-oscillating systems which yield a periodic motion at the expense of a usually non-periodic source of energy. The very definition of work in the formalism of quantum open systems suggests the hypothesis that the oscillating 'piston' is a necessary ingredient of the work extraction process. This aspect of solar cell operation is absent in the existing descriptions and the main goal of this paper is to show that plasma oscillations provide the physical implementation of a piston.

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Solar cell as a self-oscillating heat engine

In this paper, the mathematical framework providing a description of completely positive and trace preserving dynamics of open quantum systems is addressed. Special case of time-dependent Lindbladian governed by periodic Hamiltonian is concerned. It is proven, that appropriate trace preserving dynamical map of such periodically driven system may be constructed by application of Floquet theory. Appropriate Lindbladian and Markovian master equation in weak coupling regime and resulting dynamical map are constructed. Some examples of application of developed technique are given.

Krzysztof Szczygielski

Papers

Markovian master equation and thermodynamics of a two-level system in a strong laser field

On the application of Floquet theorem in development of time-dependent Lindbladians

Laser-induced cooling of broadband heat reservoirs

Solar cell as a self-oscillating heat engine

On the application of Floquet theorem in development of time-dependent Lindbladians