Non-thermal quantum engine in transmon qubits
The design and implementation of quantum technologies necessitates the understanding of thermodynamic processes in the quantum domain. In stark contrast to macroscopic thermodynamics, at the quantum scale processes generically operate far from equilibrium and are governed by fluctuations. Thus, experimental insight and empirical findings are indispensable in developing a comprehensive framework. To this end, we theoretically propose an experimentally realistic quantum engine, that utilizes transmon qubits as working substance. We solve the dynamics analytically and calculate its efficiency, that reaches a maximum value of .
Recent advances in nano and quantum technology will necessitate the development of a comprehensive framework for quantum thermodynamics Gemmer2004. In particular, it will be crucial to investigate whether and how the laws of thermodynamics apply to small systems, whose dynamics are governed by fluctuations and which generically operate far from thermal equilibrium. In addition, it has already been recognized that at the nano scale many standard assumptions of classical statistical mechanics and thermodynamics are no longer justified. For instance, the assumption of “ultra-weak coupling” is almost never a good approximation and a consistent thermodynamic description has to account for that Jarzynski2017.
An immediate consequence of the breakdown of ultra-weak coupling is that even in equilibrium quantum systems are generically not well-described by a Maxwell-Boltzmann distribution, or rather a Gibbs state Gelin2009. Thus, the formulation of the statements of quantum thermodynamics have to be carefully re-formulated to account for potential quantum effects in, for instance, the efficiency of heat engines Scully862; PhysRevE.92.042126.
In good thermodynamic tradition, however, this conceptual work needs to be guided by experimental insight and empirical findings. To this end, a cornerstone of quantum thermodynamics has been the description of the working principles of quantum heat engines PhysRevX.7.031044; 0295-5075-88-5-50003; PhysRevE.86.051105; 0295-5075-106-2-20001; PhysRevLett.112.030602; 2015NatSR512953H; 1367-2630-18-8-083012; PhysRevE.93.052120; PhysRevB.96.104304.
However, to date it is not unambiguously clear whether quantum features can always be exploited to outperform classical engines, since to describe the thermodynamics of non-thermal states one needs to consider different perspectives – different than the one established for equilibrium thermodynamics. For instance, it has been shown that the Carnot efficiency cannot be beaten PhysRevE.92.042126 if one accounts for the energy necessary to maintain the non-thermal stationary state PhysRevLett.86.3463; doi:10.1143/PTPS.130.29; Horowitz2014; Yuge2013. However, it has also been argued that Carnot’s limit can be overcome, if one carefully separates the “heat” absorbed from the environment in two different types of energy exchange gers: one is associated with a variation in passive energy Pusz1978; 0295-5075-67-4-565 which would be the part responsible for changes in entropy, and another type that is a variation in ergotropy, a work-like energy that could be extracted by means of a suitable unitary transformation. Having several perspectives to explain the same phenomenon is a clear indication of the subtleties and challenges faced by quantum thermodynamics, and which can only be settled by the execution of well suited experiments. Therefore, theoretical proposals for feasible and relevant experiments appear instrumental.
In this work we propose an experiment to implement a thermodynamic engine with a transmon qubit as the working substance (WS), which interacts with a non-thermal environment composed by two subsystems, a externally excited cavity (a superconducting transmission line) and a classical heat bath 0957-4484-27-36-364003 with temperature . The WS undergoes a non-conventional cycle (different from Otto, Carnot, etc.)callen1985thermodynamics through a succession of non-thermal stationary states obtained by slowly varying its bare energy gap (frequency) and the amplitude of the pumping field applied to the cavity. We calculate the efficiency of this engine, obtain its maximum value (up to ), and compare this with values found in current literature.
Ii System description
We consider a multipartite system, comprised of a transmon qubit of tunable frequency , which interacts with a transmission line (cavity) of natural frequency with coupling strengh . The cavity is pumped by an external field of amplitude and single frequency . Both systems are in contact with a classical heat bath at temperature . Such a set-up is experimentally realistic and several implementations have already been reported in different contexts Majer2007; 0957-4484-27-36-364003. Here and in the following, the transmon is used as a working substance (WS) and the (non-standard) “bath” is represented by the other two systems: the cavity and the cryogenic environment (classical bath). There are two subtleties that must be noted here: (i) the bath “seen” by the qubit does not only consist of a classical reservoir at some fixed temperature, but it has an additional system, represented by the pumped cavity. By changing the pumping, several cavity states can be addressed. Such a feature gives the possibility of making this composed bath non-thermal on demand. And (ii), the proposed engine is devised as containing only one bath (cavity + environment), which does not pose any problem considering that it is an out-of-equilibrium bath.
The engine’s Hamiltonian describes a tunable qubit, interacting with a single mode pumped cavity through a Jaynes-Cummings interaction
where is the diagonal Pauli matrix, and are the canonical bosonic creation and annihilation operators associated with the cavity excitations, is the qubit-cavity coupling strength. The last term represents a monochromatic pumping of amplitude and frequency applied to the cavity. In the rotating wave approximation (RWA) scully_zubairy_1997, Eq. (1) becomes
Thus, the qubit-cavity dissipative dynamics can be written as 0957-4484-27-36-364003
with being the cavity damping (excitation) rate and the qubit relaxation (excitation) rate.
However, we are interested in the observed dynamics of the WS and hence it is necessary to find the qubit reduced density matrix , where represents the partial trace on the cavity’s degrees of freedom. In order to proceed, that the system state is in a qubit-cavity product state, i.e., , which shall emerge due to the decoherence process imposed by their common environment. In addition, the cavity stationary state is assumed to be mainly determined by the external pumping, which can be easily found for situations of a strong pumping and/or weak strength coupling . This closely resembles a situation, in which the cavity acts as a work source of effectively infinite inertia PhysRevX.3.041003. Hence changing the state of qubit does nor affect the state of the cavity, but it is still susceptible to the applied field and the cryogenic bath, and we have
Hence, the reduced master equation (3) can be written as 111The cavity stationary state condition together with the approximation of effectively infinite inertia implies that
Note that the effective qubit Hamiltonian carries information about the interaction with the cavity through and , which are dependent on the cavity state.
Iii Non-equilibrium thermodynamics
iii.1 Non-thermal equilibrium states
The only processes that are fully describable by means of conventional thermodynamics are infinitely slow successions of equilibrium states. For the operating principles of heat engines, the second law states that the maximum attainable efficiency of a thermal engine operating between two heat baths is limited by Carnot’s efficiency.
An extension of this standard description is considering infinitely slow successions on non-Gibbsian, but stationary states doi:10.1143/PTPS.130.29; PhysRevLett.86.3463; Sasa2006. In the present case, namely, a heat engine with transmon qubit as working substance, non-Gibbsianity is induced by the external excitation applied as a driving field to the cavity. We will see in the following that, therefore, caution should be taken when determining the work- and heat-like character of the energy exchange, which is crucial when computing the entropy variation during the engine operation.
The stationary state can be found by solving the master equation Eq. (5), which we write as
where the matrix elements can be computed explicitly and are summarized in Appendix A.
We observe that for the cases of effective qubit-cavity weak coupling, i.e., , the obtained non-thermal state asymptotically approaches a thermal one, namely, and . In addition, as expected, the high temperature limit gives as the qubit stationary state a thermal maximally mixed state, when the cavity is not strongly pumped.
iii.2 Thermodynamic cycle
In equilibrium thermodynamics the cycles are constructed by following a closed path on a surface obtained by the equation of state callen1985thermodynamics, which characterizes the possible equilibrium states for a given set of macroscopic variables. This procedure can be generalized in the context of steady state thermodynamics, where a equation of state is also constructed.
For the present purposes, we use the steady state (7) to devise a cycle for our heat engine. The equation of state in our case is represented by the stationary state’s von Neumann entropy , which is fully determined by the pair of controllable variables , the transmon’s frequency, and , amplitude field of the pumping applied to the cavity. In order to do that, the stationary state is slowly varied 222The timescale for which the changes made can be considered slow is such that the conditions imposed to the system state are satisfied, namely, the state is a product state and the cavity steady state is a coherent state with Eq. (4). by changing the “knobs” ,. The cycle is composed of four strokes where we keep one of the two controllable variables constant and vary the other one, for example, at the first stroke we keep and vary from to . The complete cycle is sketched in Fig. (2).
For experimentally relevant parameters the corresponding cycle lies on the von Neumann entropy surface depicted in Fig. 3.
In the present analysis all parameters are chosen from the experimentally relevant regime 0957-4484-27-36-364003, which we collect in Tab. 1.
Iv Thermodynamic quantities and efficiency
The first law of thermodynamics, , states that a variation of internal energy along a thermodynamic process can be divided into two different parts, work and heat , where for Lindblad dynamics we have 0305-4470-12-5-007,
Typically, work is understood as a controllable energy exchange, which can be used for something useful, while heat cannot be controlled, emerging from the unavoidable interaction of the engine with its environment. For non-thermal situations, one important feature is that part of the heat can be shown not to be responsible for changing the entropy gers. Nevertheless, to compute for the efficiency , no distinction between different “flavors” of heat becomes necessary.
The present cycle is such that in each stroke one of the knobs is kept fixed, while the other one is changed. Recall that the cavity is assumed to be a subpart of the bath seen by the WS, and that its state is modified by . Since the WS is always in contact with the environment, heat and work are exchanged in each stroke. Here, such a calculation is done by using Eqs. (8), considering the stationary state Eq. (7) and the effective WS Hamiltonian Eq. (6). Then, for the th stroke, the corresponding and integrals can be parametrized in terms of the respective knob variation.
Once these quantities are determined, we calculate the efficiency of this engine, given by
with being the heat that was given to the WS in a complete cycle. It is worth mentioning that this efficiency is obtained in a quasi-static manner as a succession of steady states, so it actually represents the efficiency upper bound. Consequently, any experimental implementation of this protocol will end up with a smaller efficiency. In Fig. 4 is shown our engine efficiency , by always starting from the initial values , as a function of the values attained in the execution of the strokes, as depicted in Fig. 2.
V Conclusion and final remarks
Theoretical research of small heat engines in the quantum domain is a common place in quantum thermodynamics 0305-4470-12-5-007; article; PhysRevLett.93.140403; PhysRevE.76.031105; PhysRevLett.105.130401; 2014NatSR...4E3949C; PhysRevX.5.031044; PhysRevLett.109.203006; PhysRevLett.112.150602. In the present work, we have devised a transmon-based heat engine using an experimentally realistic regime of parameters reaching a maximum efficiency of , which turns out to be a reasonable value when compared with the state of the art in quantum heat engines. One of the most recent experiments in quantum heat engine was implemented by Peterson et al. 2018arXiv180306021P using a spin system and nuclear resonance techniques, performing an Otto cycle with efficiency in excess of at maximum power. It is important to stress that implementing small heat engines constitutes a hard task, even when dealing with classical systems. Indeed, a representative example is the single ion confined in a linear Paul trap, which was used to implement a Stirling engine Rob with efficiency of only . By devising this theoretical protocol for the implementation of a quantum heat engine, we hope to help the community, and in particular the experimentalists, in the formidable task to design and implement quantum thermodynamic systems and to consolidate the concepts of this new exiting field of research.
Acknowledgements.We thank F. Rouxinol and V. F. Teizen for valuable discussions. CC would like to thank the hospitality of UMBC, where most of this research was conducted. CC and FB acknowledge financial support in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. During his stay at UMBC, CC was supported by the CAPES scholarship PDSE/process No. 88881.132982/2016-01. FB is also supported by the Brazilian National Institute for Science and Technology of Quantum Information (INCT-IQ). SD acknowledges support from the U.S. National Science Foundation under Grant No. CHE-1648973.
Appendix A Non-thermal equilibrium states
In this appendix we summarize the explicit expressions of the matrix elements of (7). We have,