Finite-time availability in a quantum system

Classically, availability refers to the work available in any reversible process that brings about equilibrium between the system and its environment. Here we introduce an additional meaning of availability as the maximum work associated with the change of an external parameter in the Hamiltonian of a quantum system. This availability can be gained in a FEAT process and for times larger than or equal to the FEAT time, there exists an optimal control that achieves the available work. For shorter times, quantum friction effects are unavoidable and the available work is thereby lowered. This finite-time availability is quantified here as a function of the time available. Copyright c ⃝ EPLA, 2015 Availability as the quantification of available work from a thermodynamic system has been with us since Gibbs introduced the concept in 1873 [1]. While not being very prominent in physics and chemistry texts on the subject, it was enthusiastically revived in the engineering literature by Joseph Keenan [2] and has remained as an important cornerstone of engineering process analysis. Renamed exergy, it survives to this day with an extensive literature including a dedicated journal, the International Journal of Exergy. In both Gibbs’ and Keenan’s treatments, the concept perforce relates the work available from disequilibrium between the system of interest and an environment that acts as a bath with constant intensities. Classically, availability refers to the work available in any reversible process that brings about equilibrium between the system and such an environment. During the last decades, the concept has been used to bound the work required to bring a system along a specified sequence of states in a given time using a macroscopic [3,4], or a microscopic [5] description of the process. More recently, availability loss and the closely related concept of entropy production have been discussed for applications to heat engines [6], and radiation [7,8], as well as from a more fundamental point of view [9]. And finally, the concept has attained considerable interest in connection with the work available from quantum systems, see, for instance, the work by Deffner and Lutz [10], Allahverdyan and collaborators [11–14], or Sivak and Crooks [15]. In a recent paper [16], we introduced an additional meaning of availability as the maximum work associated with the change of an external parameter in the Hamiltonian of a quantum system. The context of the problem discussed there, the change in the frequency of a harmonic oscillator, made the extension of the term natural since the associated availability in fact equaled the traditional availability of the initial state of the quantum system relative to an environment at the temperature of the final state. As shown in that work, there is a minimum time tmin and for times larger than tmin, there exists a control of the frequency which achieves the available work. For shorter times quantum friction effects [17,18] are unavoidable and the available work is thereby lowered. Similar results have been shown for a system of two interacting spin-12 particles in an external magnetic field [19]. Both examples achieved what traditional adiabatic switching could also achieve. However, they achieve the same final state in a much shorter time than adiabatic switching would require. Accordingly these minimum possible time processes have been dubbed Fastest Effectively Adiabatic Transitions or simply FEATs. The FEAT controls found in [19] extract an amount of work equal to the availability associated with the adiabatic change for any time greater than tFEAT = tmin. What can be achieved for even shorter times is the topic of the present paper. Effectively adiabatic processes have received considerable interest in the context of control of quantum systems [20–26] for instance to bring them close to