Slow Dynamics and Thermodynamics of Open Quantum Systems

Abstract

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 quasi-static transformations, the state of the system is not able to continuously relax to the equilibrium ensemble. Our approach allows to formally evaluate perturbations up to arbitrary order to the work and heat exchange associated to 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.A central result in the study of open quantum systems [1, 2] is the Markovian Master Equation (MME) approach which, under realistic assumptions, describes the temporal evolution of a system of interest A induced by a weak coupling with a large external environment E. This consists in a first order linear differential equatioṅ ρ(t) = L[ρ(t)] where ρ(t) is the density matrix of A and where the generator of the dynamics is provided by a quantum Liouvillian superoperator L that can be casted in the so called Gorini-Kossakowski-Sudarshan-Lindblad form [3][4][5]. For autonomous systems the latter does not exhibit an explicit time dependence and the dynamics of A exponentially relaxes to a (typically unique) equilibrium steady state ρ 0 identified by the null eigenvector equation L[ρ 0 ] = 0. MMEs can also be employed to describe the temporal evolution of A when it is tampered by the presence of slow varying, external driving forces. Indeed, as long as these operate on a time scale which is much larger than the characteristic bath correlations times and the inverse frequencies of the system of interest, the effective coupling between A and E adapts instantaneously to the driving control, resulting on a MME governed by a time-dependent Liouvillian generator, i.e.An explicit integration of this equation is in general difficult to obtain. Yet, if the control forces are so slow that their associated time scale is also larger with respect to the relaxation time of the system induced by the interaction with E, one expects ρ(t) to approximately follow the instantaneous equilibrium state ρ 0 (t) that nullifies L t . Our aim is to estimate quantitative deviations from this ultra-slow driving regime. For this purpose we develop a perturbation theory valid in the limit of slowly varying Liouvillians L t and derive a formal solution of Eq. (1) which allows one to evaluate non-equilibrium corrections up to arbitrary order. The main motivation of our analysis is to model thermodynamic processes and cycles beyond the usual reversible limit which is strictly valid only for infinitely long quasi-static transformations. Finite-time thermodynamics [7,8]...

Maximum power and corresponding efficiency for two-level heat engines and refrigerators: optimality of fast cycles

“…CA c emerges in various specific models [27,53,57], and it has been derived by general arguments from linear irreversible thermodynamics [56], while the Schmiedl-…”

“…Quantum thermodynamics [6][7][8] has emerged both as a field of fundamental interest, and as a potential candidate to improve the performance of thermal machines [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. The optimal performance of these systems has been discussed within several frameworks and operational assumptions, ranging from low-dissipation and slow driving regimes [24][25][26][27][28], to shortcuts to adiabaticity approaches [29][30][31][32], to endoreversible engines [33,34]. Several techniques have been developed for the optimal control of two-level systems for achieving a variety of goals: from optimizing the speed [35][36][37], to generating efficient quantum gates [38,39], to controlling dissipation [40,41], and to optimizing thermodynamic performances [42][43][44][45][46][47].…”

Maximum efficiency of low-dissipation heat engines at arbitrary power

“…To this end, there is a handful of microscopic models yielding reasonable expressions for σ. For relatively broad class of slowly driven systems (described by generalized Markovian master equation with a symmetric protocol for hot and cold isotherms), the dissipation ratio assumes the form σ = (T h /T c ) 1−ξ , where ξ stands for the exponent in the bath spectral density [33]. The limit σ → 0 thus corresponds to an infinitely super-Ohmic bath (ξ → ∞), while the opposite limit σ → ∞ is obtained for an infinitely sub-Ohmic bath (ξ → −∞).…”

“…Therefore, they should be generally valid for slowly, but not quasi-statically, driven systems. Indeed, the decay of total dissipated heat with the inverse of duration was theoretically predicted for various quantum and classical setups [31][32][33][34] and observed in various experiments [35,36]. The second situation, where the assumption (1) and (2) holds for arbitrary cycle duration, are overdamped Brownian systems driven by special time-dependent protocols (usually minimizing dissipated energy during the isotherms [21,25,27,37,38]).…”

Ultra-cold single-atom quantum heat engines

“…= mK s −1 , obtained maintaining the maximum real efficiency η real . Additional limitations to the maximum real efficiency attainable can come from the finite-time nature of the isothermal transformations, as shown in [36,37].The quantification of the performance of our UAEs can be done using well established techniques. The measurement of the level population of the K atom P n can be inferred by using Raman sideband spectroscopy [38], while the temperature of the Rb bath can be obtained with standard time-of-flight imaging.…”