
DOI: 10.22661/AAPPSBL.2019.29.6.30
Heat Engines Using Small Quantum Systems
GENTARO WATANABE^{1,2,}*
^{1}DEPARTMENT OF PHYSICS AND ZHEJIANG INSTITUTE OF MODERN PHYSICS,
ZHEJIANG UNIVERSITY, HANGZHOU, ZHEJIANG 310027, CHINA
^{2}ZHEJIANG PROVINCE KEY LABORATORY OF QUANTUM TECHNOLOGY AND DEVICE,
ZHEJIANG UNIVERSITY, HANGZHOU, ZHEJIANG 310027, CHINA
^{* }Email address: gentaro@zju.edu.cn
ABSTRACT
A heat engine with a quantum system as its working substance is called a quantum heat engine (QHE). QHEs are expected to show exotic properties due to their quantum nature. Current experiments have reached the stage to realize such devices using small quantum systems, and active research on QHEs is ongoing. Here we briefly review recent experiments on QHEs. Basic notions of thermodynamic processes in small quantum systems and an example of a quantum version of the Carnot cycle are also explained in depth.
INTRODUCTION
Heat engines are devices that convert thermal energy into mechanical work, and they play a fundamental role in modern civilization. Advances in technology so far have enabled us to downsize heat engines [16] and recent development spurred the fabrication of heat engines at the submicron to nanoscale [712]. Especially in the last decade, great attention has been paid to socalled quantum heat engines (QHEs) [13] (see also Refs. [1418] and references therein), microscopic heat engines whose working substance is a quantum system such as a twolevel system, a single quantum atom, a superconducting device, an ultracold quantum gas, etc. Since the behavior of the working substance is governed by quantum mechanics, QHEs are expected to show exotic properties, which cannot be obtained in conventional macroscopic counterparts governed by classical mechanics (e.g., [1924]).
On the application side, a prominent incentive for the study of QHEs is to identify an enhanced performance due to quantum effects. For example, since the energy gap suppresses excitations, one can easily imagine that the discreteness of the energy levels in quantum systems could allow us to perform reversible quasistatic processes within a finite time scale. Another example concerns the efficiency 畏 of the engine defined as the ratio between the work output W_{out} from the engine and the thermal energy input Q_{i}_{n}^{h} into the engine from a heat bath with a higher temperature (a hot heat bath): 畏 ≡ W_{out }/Q_{i}_{n}^{h}. Using quantum coherence [25] or nonthermal baths [26, 27], one can operate cycles at efficiencies beyond the celebrated Carnot bound. Theoretical studies of QHEs are largely motivated by fundamental questions that address the interplay between thermodynamics and statistical mechanics in the quantum world [18, 28, 3033]. In common wisdom, thermodynamics is regarded as a theory for macroscopic systems. Some interesting and important questions here are: How small can systems go for thermodynamics to be applicable? Is the formulation of thermodynamics possible for quantum systems? How do the laws of thermodynamics emerge from quantum mechanics? These questions form the major agenda of the newly born field called quantum thermodynamics [2830, 3234]. Understanding the property of QHEs would give important insights into these fundamental questions.
In this short review article, we shall overview recent experimental studies on QHEs. First, we will briefly discuss the thermodynamics and thermodynamic processes of small quantum systems. Here, "small" means that the number of degrees of freedom of the system is comparable to unity (or, at least far less than the macroscopic number). Then, the basics of QHEs will be explained taking a simple example of the Carnot cycle made of a twolevel system. Finally, we will overview recent experimental implementations of QHEs using various platforms.
THERMODYNAMICS AND THERMODYNAMIC PROCESSES OF SMALL QUANTUM SYSTEMS
Let us first start with the following question: "What is the meaning of the temperature T of a small quantum system, say a single molecule, and how can it be defined?" To discuss this issue, we introduce heat baths: large systems, in which their temperature is welldefined within conventional macroscopic thermodynamics. Heat baths are assumed to have infinite heat capacity so that their temperature is constant even during and after a thermal contact with other systems to exchange energy.
Suppose our small quantum system described by the Hamiltonian H_{S} is coupled to a heat bath with the temperature T_{B}. After waiting for a sufficiently long time, the whole system is relaxed into a steady state and is in thermal equilibrium at the temperature T_{B}. Provided the coupling between the system S and the bath is weak such that the coupling Hamiltonian H_{C} is negligible compared to H_{S}, the density matrix ρ_{S} of the system S is given by the canonical state:
 (1)

where Z ≡ Tr[exp(H_{S }/k_{B}T_{B})] is the partition function, Tr[⋯] is the trace over the system S, and k_{B} is the Boltzmann constant. Now the temperature T of the system S can be identified as T = T_{B}. In other words, when the system S is in the canonical state characterized by the temperature T, we can define T as the temperature of the system S, which is welldefined irrespective of the size of the system.
Fig. 1: The pistoncylinder setup. A gas (red region) as a working substance is in a cylinder with a piston. The position of the piston is a timedependent external parameter denoted by 位(t).
Having clarified the meaning of the temperature of small quantum systems, we shall next discuss thermodynamic processes of these systems. Focusing on two kinds of representative thermodynamic processes, adiabatic processes and isothermal processes, we will explain similarities and differences of these processes between quantum thermodynamics and classical thermodynamics for macroscopic systems.
1) Adiabatic processes
Both in classical and quantum thermodynamics, the definition of an adiabatic process is one in which the system exchanges the energy with its external world only in the form of work. Taking a typical schematic setup commonly used in classical thermodynamics, a gas in a cylinder with a piston (see Fig. 1), let us examine the adiabatic processes in classical and quantum thermodynamics in detail. In this setup, the walls of the cylinder and of the piston give the potential energy of gas particles, and the position of the piston can be described by a timedependent external parameter of the system, which is symbolically denoted by 位(t). When we discuss adiabatic processes in classical thermodynamics, the above setup is surrounded by a perfect heat insulator. Consequently, the system does not exchange heat with its external world. Instead, energy exchange with the external world is solely in the form of work through pushing and pulling the piston, i.e., through the timedependence of the external parameter 位(t).
In the case of quantum systems, even if there is no energy exchange with its external world, there are two possibilities: either quantum coherence of the system is preserved or it is destroyed by external disturbances. In usual discussions in quantum thermodynamics, however, it is customary that "adiabatic processes" in quantum systems do not include the latter possibility (see, e.g., Ref. [17]). Namely, the "adiabatic processes" in quantum systems or "quantum adiabatic processes" often mean the processes of closed quantum systems given by the unitary evolution generated by the system Hamiltonian H_{S }[位(t)] with timedependent external parameter(s) 位(t). There, the propagator U (t_{1}, t_{0}) from time t = t_{0} to t_{1} is U (t_{1}, t_{0})= 𝒯 exp{(i/魔) ∫_{t}_{0}^{t1} dt H_{S }[位(t)]}, where 𝒯 is the timeordering operator. It is noted that such a class of processes is narrower compared to those realized by the heat insulated pistoncylinder setup, in which decoherence might occur.
There is also another important difference related to the adiabatic processes in classical and quantum thermodynamics. In classical thermodynamics for macroscopic systems, it is assumed that the adiabatic systems (i.e., systems surrounded by a perfect heat insulator) will relax into a thermal equilibrium state at some temperature after a sufficiently long time. Indeed, this is an empirical fact for macroscopic systems, where a part of the system can effectively work as a heat bath for the remaining part. However, it is not necessarily the case for small systems: adiabatic systems with a small number of degrees of freedom might never relax into an equilibrium state. Therefore, after a quantum adiabatic process, even if the process is quasistatic, the final state is different from the canonical state in general.
In quasistatic adiabatic processes of small systems, because of the absence of diabatic transitions, the probability of the initial and the intermediate (including the final) microstates connected by the processes is conserved. Therefore, if the initial state (at t = t_{0}) of the quasistatic adiabatic process is a canonical state at the temperature T_{0}, ρ_{S}(t_{0})= Z^{1}[位(t_{0})] exp{H_{S }[位(t_{0})]/k_{B}T_{0}}, the state at t in the quasistatic adiabatic process, for example in the quantum case, is given by
 (2)

where the Boltzmann factor is characterized by the parameter 位(t_{0}) and the temperature T_{0} of the initial state. Here, n;位(t)> and 蔚_{n }[位(t)] are the nth instantaneous eigenstate and eigenvalue of the Hamiltonian H_{S }[位(t)] at t, respectively. Note that Eq. (2) is not a canonical state unless there exists some appropriate T satisfying
 (3)

up to a common constant for any n.
2) Isothermal processes
Isothermal processes are ones during which the system is in thermal contact with a heat bath at a constant temperature. Again, taking the pistoncylinder setup, an isothermal process can be performed by making a contact between the cylinder and a heat bath, or by putting the whole setup in an environment at a constant temperature, while pushing and pulling the piston. Note that, unless the pushing and pulling of the pistion is slow enough, it is possible that the temperature of the system can deviate from that of the bath, or the system is even driven out of an equilibrium state in the middle of the process.
Fig. 2: The pressure versus volume (PV) diagram of the Carnot cycle working with a hot heat bath at temperature T_{h} and a cold one at T_{c}. The strokes 1 → 2 and 3 → 4 are the quasistatic isothermal expansion at the temperature T_{h} and compression at T_{c}, respectively. The strokes 2 → 3 and 4 → 1 are the quasistatic adiabatic expansion and compression, respectively.
As long as the coupling between the system and the bath is weak and the change of the external parameter is slow enough, there would be no essential difference in the isothermal processes between classical and quantum thermodynamics except for the system being classical or quantum. More specifically, in quasistatic isothermal processes, either in the classical or quantum case, the system is in thermal equilibrium at the temperature T_{B} of the bath throughout the process, and the system is in the canonical state corresponding to the instantaneous parameter value at each time: ρ_{S}(t) = Z^{1}[位(t)] exp{H_{S}[位(t)]/k_{B}T_{B}}.
AN EXAMPLE OF A QHE: QUANTUM CARNOT CYCLE
Taking an example of the Carnot cycle, let us illustrate QHEs using the thermodynamic processes in small quantum systems discussed in the previous section. The Carnot cycle is a reversible cycle operating with two (hot and cold) heat baths which consists of two quasistatic isothermal strokes and two quasistatic adiabatic strokes as shown in Fig. 2. Since there is no irreversibility which causes extra dissipation, classical thermodynamics tells that the efficiency of the Carnot cycle is the maximum possible value at the socalled Carnot efficiency:
 (4)

where T_{h} and T_{c} are the temperatures of the hot and cold heat bath, respectively.
In the following, we shall see that the Carnot engine using a small quantum system (quantum Carnot engine) is also consistent with the above prediction by classical thermodynamics yielding the same efficiency. Here, as an example, we take a twolevel system (TLS) for the working substance.
The strokes 1 → 2 and 3 → 4 in Fig. 2 are the quasistatic isothermal expansion at the temperature T_{h} and compression at T_{c}, respectively. Since these isothermal processes are reversible, the average amount of heat Q_{i}_{n}^{h} and Q_{i}_{n}^{c} absorbed by the working substance in each of these processes is given by the entropy change as
 (5)

 (6)

respectively. Here, S(i) is the entropy at point i given by
 (7)

with P_{n}(i)= Z^{1}(i) exp[蔚_{n}(i)/k_{B}T_{i}] being the probability of the nth energy eigenstate at point i following the canonical distribution, where T_{i}, 蔚_{n}(i), and Z(i) are the temperature, energy eigenvalue of the nth state, and the partition function at point i, respectively.
The strokes 2 → 3 and 4 → 1 in Fig. 2 are quasistatic adiabatic processes. For TLSs, the condition (3) gives two equations for two eigenvalues while there are two unknowns (T and an arbitrary constant common for n = 1 and 2). Therefore, in the case of TLS, there always exists some value of T satisfying the condition (3). This means that, for TLSs, the final state of any quasistatic adiabatic process starting from an equilibrium state can be identified as the canonical state at some appropriate temperature. Consequently, by properly choosing the parameter 位 at the final points of the adiabatic processes, the final state of the adiabatic strokes 2 → 3 and 4 → 1 can be set to the canonical state at T_{c} and T_{h}, respectively. Only with such a proper choice, there is no heat exchange between the engine and the heat bath at the connection from the adiabatic process to the isothermal process at points 3 and 1 [i.e., Q_{i}_{n}^{h} and Q_{i}_{n}^{c} given by Eqs. (5) and (6) are the only heat transfers throughout the cycle], and the cycle becomes reversible.
Since the difference ΔE in the mean internal energy between the initial and the final states of any cycle is zero, the first law of thermodynamics reads ΔE = W_{in} + Q_{in} = 0 with W_{in} and Q_{in} ≡ Q_{i}_{n}^{h} + Q_{i}_{n}^{c} being the mean net amount of work and heat input into the working substance during a cycle, respectively. Therefore, the mean work output W_{out} = W_{in} from the engine is
 (8)

Finally, from Eqs. (5) and (8), the efficiency of the quantum Carnot cycle is obtained as
 (9)

(9)
This agrees with the efficiency of the Carnot cycle obtained in classical thermodynamics.
Fig. 3: The schematic picture of the setup of the experiment by Ro脽nagel et al. [8]. A ^{40}Ca^{+} ion (green circle in panel A and green wave packet in panel B) is trapped in a linear Paul trap with funnelshaped electrodes (red brown bars in panels A and B). Reddetuned laser beams for cooling and imaging (blue arrows in panel A) are constantly shining on the ion. White electricfield noise is generated by the outer electrodes (horizontal gray bars in panel A). This figure is taken from Ref. [8].
RECENT EXPERIMENTAL IMPLEMENTATIONS OF A QHE
In this section, we shall see recent experimental implementations of heat engines using small quantum systems. To perform various thermodynamic processes, it is required that the system allows us to control its external parameters at high precision in space and time. In addition, protocols in many QHE experiments so far need to prepare the canonical state externally to mimick the effect of heat baths. In such cases, the quantum state of the system should also be controllable. Furthermore, since it is often necessary to assess the performance of the engine, measurability is also a key factor. Among quantum systems with high controllability and measurability, heat engines (or refrigerators as their inverse process) have been implemented with trapped ion(s) [8, 9, 11], liquidstate NMR [12], diamond NV^{} centers (negatively charged nitrogen vacancy centers) [10], and a superconducting circuit [7].
1) Singleatom heat engine experiment using a trapped ion
First, taking one of the early experiments by Ro脽nagel et al. [8] as an example, which realized a heat engine using a single trapped ion, we shall see the actual implementation of a heat engine using a small quantum system in detail.
The setup of this experiment is shown in Fig. 3: a single ^{40}Ca^{+} ion (green circle in panel A) in a linear Paul trap with funnelshaped electrodes (red brown bars). Such a funnelshaped geometry of the electrodes causes the trap frequency ω_{xy} in the radial (xy) directions to be dependent on the zcoordinate: ω_{xy} = ω_{xy}(z). This position dependence of the trap frequency enables one to transfer the thermal energy of the ion in the radial components into a mechanical motion of the ion in the axial (z) direction. The motion of the ion in the axial direction is subjected to friction generated by the laser beam. If the ion is heated, the width of its wave packet expands. As a consequence, the ion gets a net force in the positive z direction in which the radial trap frequency ω_{xy} decreases. Conversely, if the ion is cooled, it moves toward the negative z direction. Therefore, by heating and cooling the ion alternately, the thermal energy can be transferred into the kinetic energy of an oscillatory motion of the ion against the friction, which is identified as work.
In this experiment, effects of heat baths are mimicked by the laser cooling for a cold bath, and by white noise of an electric field for a hot bath. Reddetuned cooling laser beams (blue lines in Fig. 3A) are constantly shining on the ion; heating and cooling the ion is done by alternately switching the electricfield noise on and off. Suppose the ion is now at a larger z, and only the cooling laser is on. After being cooled down, the ion moves in the negative z direction. Then the electric field noise is turned on for some period (around a few 碌sec) to heat up the ion. Finally the heated ion moves to the original position in the positive z direction, and the cycle is closed.
During this cycle, the working substance (^{40}Ca^{+} ion) is always coupled to either of the heat baths so that the dynamics of the engine is similar to the Stirling cycle, which consists of two isothermal strokes and two isochoric strokes (An isochoric process means a "constantvolume" process. There, the working substance is in contact with the hot or cold bath during which the parameter 位 is fixed.). The highest efficiency obtained in this experiment is 畏 ≃ 2.8 脳 10^{3}.
2) Various platforms
There are three basic components to experimentally implement QHEs: 1) a working substance, 2) heat baths, and 3) a work extraction scheme or a work repository on which the engine does work. To construct a QHE for practical use, we need to employ a platform which allows us to implement all of these components. Various platforms such as trapped ion(s), liquidstate NMR, diamond NV^{} centers, etc. have been used in QHE experiments, and each of them has different pros and cons.
Among them, the system of trapped ion(s) has been used in most experiments so far such as the one on a singleatom heat engine [8] discussed in the previous section, a quantum absorption refrigerator [9], and a quantum Otto engine coupled to a harmonic oscillator as a work repository [11]. Advantages of this platform for the implementation of QHEs are: 1) the preparation and the detection of the quantum state of the ion are possible, and 2) it is easy to couple with other degrees of freedom through the electric charge and the motion of the ion, which facilitates, e.g., the extraction of work. On the other hand, disadvantages of this platform are as follows: 1) it is not easy to prepare heat baths with large heat capacity which can be coupled to the ion. Therefore, instead of using heat baths, they are often mimicked by the preparation of a canonical state using, e.g., optical pumping or laser cooling. 2) The coherence time might not be sufficiently large compared to the period of the engine cycle [8]. Recent developments of experimental techniques to suppress the effects of external disturbances (see, e.g., Ref. [35] and references therein) should allow us to overcome this issue in the near future.
Regarding other platforms, liquidstate NMR was used to study the statistical properties of the performance of a quantum Otto engine operating in a finite time [12]; an ensemble of diamond NV^{} centers was used to experimentally demonstrate dynamical properties of QHEs under weak and periodic operation [10]; and a superconducting circuit was used to realize a Szilard engine with a single electron [7]. As in trapped ion systems, preparation and detection of the quantum state are possible while there is also a similar difficulty to install heat baths in these platforms as well. In superconducting circuits, although it is possible to introduce a heat bath whose temperature is set by that of the substrate, it is still difficult to install two heat baths with different temperatures.
FUTURE PROSPECTS
So far QHE experiments are at the stage of proofofprinciple, and have not reached the realization of QHEs in practical use that generate work using heat instead of, e.g., artificially prepared canonical states. Toward this goal, one of the practical difficulties is to set two heat baths with different temperatures in a small (typically nanoscale) region separately, and couple them to the engine alternately. In addition, another crucial difficulty is to introduce heat baths without causing unexpected decoherence in the working substance and the work repository. Since the working substance, work repository, and heat baths are close together in a microscopic scale due to the smallness of the engine, isolation and heat insulation between them are not trivial tasks.
After overcoming these obstacles, potential novel properties of QHEs emerging from quantum mechanics would have an impact on their practical use, and QHEs and quantum thermodynamics is expected to become one of the key cornerstones of quantum technology. In the author's own perspective, the quantum nature of QHEs would be more apparent when they are coupled to other quantum systems on which the engines do work. Enhanced work output for the operation of a QHE over multiple cycles [36] and that for indistinguishable multiple QHEs [37] are interesting examples which could soon be realized in microscopic engines operating in the quantum world in the future.
Acknowledgement: The author thanks B. Prasanna Venkatesh and MewBing Wan for helpful comments. This work is supported by the Zhejiang Provincial Natural Science Foundation Key Project (Grant No. LZ19A050001), by NSF of China (Grant No. 11674283), by the Zhejiang University 100 Plan, and by the Junior 1000 Talents Plan of China.
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Gentaro Watanabe is a ZJU Young Professor in the Department of Physics, Zhejiang University. After getting his PhD in the University of Tokyo, he worked at Nordic Institute for Theoretical Physics (NORDITA), University of Trento, RIKEN, Asia Pacific Center for Theoretical Physics (APCTP), and Institute for Basic Science (IBS) before joining Zhejiang University. His current research interest lies in quantum control and quantum thermodynamics, theory of ultracold atomic gases and its related topics. 
