Optimizing quantum refrigerators with a qubit on the hot side

Eur. Phys. J. Plus (2024) 139:619 https://doi.org/10.1140/epjp/s13360-024-05437-z Regular Article Optimizing quantum refrigerators with a qubit on the hot side J. J. Fernándeza Departamento de Física Fundamental, Universidad Nacional de Educación a Distancia (UNED), 28040 Madrid, Spain Received: 11 January 2024 / Accepted: 8 July 2024 © The Author(s) 2024 Abstract We study the performance of one-qubit quantum and semiclassic refrigerators with the qubit on the hot side. We obtain the cooling power and the  function. We discover that the quantum refrigerators perform differently to classic ones. We also prove that even in the high-temperature limit, where the semiclassic version of the refrigerator is valid, the results of the optimization are different to those obtained by optimizing classic refrigerators. Quantum Thermodynamics (QTD) is the branch of Quantum Mechanics (QM) dealing with the study of engines that produce work on the nanoscale [1]. A general definition of a quantum thermodynamic machine says quantum machines are thermal engines that work under the time evolution rules that are dictated by the QM. This definition is very suitable since it extends the current definitions of quantum heat engines (QHEs) in such a way that it allows quantum refrigerators (QRs) to be included. In the last 70 years, some attention to QHEs and QRs has been devoted. In recent years, furthermore, research in the field of QTD has become quite specialized. This means that we can define three types of jobs within this field. In a first group, we find jobs on the basis of QTD. In them, the authors study the basis of the QTD by proving how the basic principles of thermodynamics arise from the laws of QM. In a second group, we find works that explore the general properties of thermodynamic quantum systems, trying to understand them and to obtain some properties such as energy fluxes Q h and Q c , the system entropy change S, and other thermodynamic properties [1–8]. In them, the authors also use the results to asses or to discard conjectures about the similarity between QTDs and classic thermodynamics. Finally, we find another set of works that are, somehow, more applied: those studying how a concrete quantum thermodynamic system performs in the best possible way. Among the works belonging to the last group, we find works studying how much work is produced by a quantum system or how a classic thermodynamic cycle can be reproduced using a concrete quantum system. In this group, we also find other works that are even more applied. In them, the authors choose a concrete version of a quantum machine and, using the Finite-time Thermodynamics (FTTD) [9–30], obtain its relevant thermodynamic functions using, to calculate the relevant fluxes in the machine, the theory of open-quantum systems [31–35].The authors also optimize the performance of the engines [36–39]. As it is seen looking at the references [36–39], the strategy mentioned in the previous paragraph has only been used to study qubit-based QHEs, but it has not been used to study qubit-based QRs. From a theoretical standpoint, there is no reason not to extend the works on qubit-based QHEs to qubit-based QRs. In this work, we start with this extension proposing the study of a one-qubit QR (OQR) with the qubit on the hot side, see Fig. 1. Doing this, we prove that the methods used in [36–39] to study QHEs are also applicable to the study of QRs and, by extension, to other thermodynamic engines. Moreover, and from a more practical standpoint, the study presented in this work also serves to understand whether QRs perform as classic refrigerators or not, thus also opening the way to carry out studies on quantum systems that could work as refrigerators on the nanoscale. To achieve this last target, we study here the cooling power Q c and the  function () [27] of OQRs. When doing it, we also comment on the effects of the strength of the coupling between the qubit and the thermal baths, thus following the research lines scoped in [40–42]. Finally, it is worthy to mention that our OQR is, somehow, similar to other devices that are studied in the literature [43, 44], which makes our study (i) comparable and (ii) demonstrates that the study of this type of machines is essential to improve our understanding of the fundamental principles of QTD. The work is organized as follows: In Sect. 1, we present the OQR under study in this work. We start calculating (by means of the solutions of the Lindbland equation [45]) the relevant heat fluxes of the machine. Then we deduce, using the techniques of the FTTD [9–30], Q c and . In Sect. 2, we optimize the performance of different OQRs. First, we optimize them considering that the temperatures Th and Tc of the hot and cold baths are fixed to understand the performance of OQRs that work coupled to fixed environments. Then, we carry a more general investigation considering OQRs of different Carnot COPs (εC ). This second kind of studies allow us to optimize the performance of OQRs working in different environments. In Sect. 3, we explore the behavior of the OQR in the limit in which E and Tc satisfy E/Tc  1. Our intention is to understand whether, in this limit, OQRs reduce to their classic counterpart or not. It is also our objective to study how semiclassic refrigerators can be optimized, which are their optimal a e-mail: (corresponding author) 0123456789().: V,-vol 123 619 Page 2 of 10 Eur. Phys. J. Plus (2024) 139:619 working properties and how do they compare to those of classic refrigerators. We end this work with a summary of our findings and a perspective of future work. 1 One-qubit quantum refrigerator with the qubit on the hot side Figure 1 depicts the OQR that is studied in this work. It has two baths filled with bosons on its extremes. One of them is called hot bath and is characterized by the temperature Th . The other one is called cold bath, and it is characterized by the temperature Tc . Both baths are filled with bosons and have the spectrum of an oscillator. The bath of temperature Tc is in perfect contact with a Carnot Refrigerator (CR) that extracts power from it and rejects power to the bath of temperature T1 . This bath is connected with that of temperature Th via a qubit of energy E. Thus, Tc , Th and T1 in our engine satisfy T1 > Th > Tc . In our engine, the amount of energy that passes from the bath of temperature T1 to that of temperature Th is controlled by the qubit of energy E. As it has been proven in previous works [36–39], controlling the energy E of the qubit, the amount of energy that ends in the bath of temperature Th is increased (if E gets bigger) or decreased (if E gets smaller). In order to calculate the energy fluxes circulating through the OQR, we assume that the Hamiltonian of the qubit is   E 1 0 H . (1) 2 0 −1 The energy levels of the qubit are ±E/2, and its state is characterized by the 2 × 2 time-dependent density matrix ρ(t). The time evolut (...truncated)