Slow Dynamics and Thermodynamics of Open Quantum Systems

We develop a perturbation theory of quantum (and classical) master equations with slowly varying parameters, applicable to systems which are externally controlled on a time scale much longer than their characteristic relaxation time. We apply this technique to the analysis of finite-time isothermal processes in which, differently from quasistatic transformations, the state of the system is not able to continuously relax to the equilibrium ensemble. Our approach allows one to formally evaluate perturbations up to arbitrary order to the work and heat exchange associated with an arbitrary process. Within first order in the perturbation expansion, we identify a general formula for the efficiency at maximum power of a finite-time Carnot engine. We also clarify under which assumptions and in which limit one can recover previous phenomenological results as, for example, the Curzon-Ahlborn efficiency.

Figure 1

Efficiency at maximum power ${\eta }^{*}$ with respect to the temperature ratio $T_C/T_H$ for different values of the spectral density exponent $\alpha$; see Eq. (24). Light gray line represents the Carnot bound; blue circles and green triangles are exact numerical results based on a two-level system heat engine coupled to a flat ($\alpha =0$) and Ohmic bath ($\alpha =1$), respectively. The details of the simulation are reported in the Supplemental Material (Appendix C) [40].