Quantum thermodynamic devices: From theoretical proposals to experimental reality

For many students, grasping the concepts of thermodynamics poses significant challenges. Arguably, the reason is that, as a phenomenological theory, thermodynamics relies much more on abstraction of complex problems than many other areas of theoretical physics. The most prominent topic of the theory are heat engine cycles, which are abstract, idealized, and cyclic processes that describe the universal working principles of converting heat to work.

It is often useful to remind oneself that thermodynamics was invented concurrently with the Industrial Revolution,¹ and that its original purpose was to understand and optimize the operation of steam engines. Rather remarkably, we are in a similar situation that occasionally is referred to as the Second Quantum Revolution.² Recent years have witnessed significant domestic and international efforts^3–7 to realize market-ready quantum technologies. However, to fully exploit the capabilities of these new technologies, it appears obvious that we will need to train a “quantum literate workforce.”⁸ However, very similar to the situation the public faced at the invention of steam engines, new quantum technologies are complex⁹ and markedly different from what societies are used to.¹⁰ While many fundamental as well as quite practical questions remain to be addressed,¹¹ it does appear obvious that a thermodynamic approach to quantum technologies may provide an effective and pedagogical entry point for a large audience.

Quantum thermodynamics¹² is a relatively young field, which only rather recently has grown into a vibrant branch of modern research. One of its main objectives is the study and description of converting heat, work, and information in quantum systems. In complete analogy to conventional thermodynamics,¹³ quantum thermodynamics relies on idealized processes that universally describe a wide range of scenarios. Remarkably, the first quantum heat engine was proposed already in the late 1950s.¹⁴ However, only over the last decade (or so) has the literature on quantum thermal devices really grown.¹⁵ This may be explained by the fact that quantum heat engines (cf. Fig. 1) provide simple, pedagogical descriptions of otherwise much more complex quantum scenarios as well as by the fact that the first experimental realizations of genuinely quantum devices have been reported.

The present review paper seeks to provide an overview of the current state-of-the-art and to categorize the plethora of different proposals according to their distinct platforms. To this end, we outline the beginnings of quantum heat engines in Sec. II, before we summarize the main concepts and theoretical tools in Sec. III. The majority of this article, however, is dedicated to the various physical platforms and selected implementations of quantum thermal devices in Sec. IV. The review is concluded with a few remarks in Sec. V.

When writing this review, we strove for a comprehensive, objective, and pedagogically valuable account of the current state-of-the-art. However, given the enormous number of publications on the topic, we had to make some hard choices and not everything could be covered in detail. However, we do hope that we have done justice to the field and that this review may be useful for newcomers as well as seasoned practitioners of quantum thermodynamics.

We begin with a brief historical account of the beginnings of the field. Interestingly, the concept of a quantum heat engine is intimately related to the theoretical description of masers and lasers.

Already more than six decades ago, Scovil and Schultz-DuBois¹⁴ suggested that the performance of a maser (microwave amplification by stimulated emission of radiation)¹⁶ can be assessed akin to a continuous engine. The maser was the direct predecessor of the laser (light amplification by stimulated emission of radiation) and it operates by the same working principle.¹⁷ 

In its simplest form, a maser consists of a three-level system that amplifies a signal with frequency ω_s, by being pumped at frequency ω_p. The excess energy is dumped into an idler mode with frequency $ωi=ωp−ωs$⁠. Scovil and Schultz-DuBois then realized that the maser becomes a continuous heat engine if the idler mode is coupled to a cold heat reservoir with temperature T_c, and the pumping is facilitated by a hot reservoir at T_h, see Fig. 2 for an illustration of the set-up. Amplification is successful if the occupation number of the first excited state is larger than of the ground state, $n2≥n1$⁠. This is commonly known as “population inversion.”

From a thermodynamic perspective, for any quantum of heat $Qh=ℏωp$ that is absorbed from the hot reservoir, the three-level engine exhausts $Qc=ℏωi$ to the cold reservoir, and work $W=ℏωs$ can be extracted. Therefore, the thermodynamic efficiency simply becomes

This can be further elaborated on by considering the thermally driven transitions between the levels 2 and 1. In particular, we have

where n₃ is the occupation number of the second excited state, and $β=1/kBT$ is the inverse temperature.

Equation (2) can be re-written to read

where we used Eq. (1) and where we introduced the Carnot efficiency

It is then interesting to note that for the maser, population inversion is identical to the positive working condition. Namely, work can be extracted if and only if $n2≥n1$⁠. Thus, we also immediately conclude that the Carnot efficiency is the natural upper bound on the maser efficiency (1),

The inequality becomes tight in the limit of vanishing inversion, i.e., $n2≃n1$⁠.

Scovil and Schultz-DuBois¹⁴ concluded their analysis by noting that for $n2≤n1$⁠, the device would operate as a refrigerator. Thus, the maser is a simple and pedagogically valuable system to study the whole range of thermodynamic devices. Remarkably, this three-level system is, however, not only a toy model, but actually a good description of experimentally realistic masers.^16,18,19

However, the original analysis of the maser as a “quantum” heat engine¹⁴ is still somewhat rudimentary. In particular, describing the transitions between the quantum levels by classical, thermal fluctuations does not leave room for genuine quantum effects. Nevertheless, the maser has become the prototypical example and the foundation for the study of continuous heat engines.^20–39 

The next major step toward truly quantum engines was taken by Scully⁴⁰ with the proposal of a “quantum afterburner.” This device consists of a laser–maser system that undergoes a joint Otto cycle. The standard Otto cycle consists of four strokes: (⁠$A→B$⁠) isentropic compression, (⁠$B→C$⁠) isochoric heating, (⁠$C→D$⁠) isentropic expansion, and (⁠$D→A$⁠) isochoric cooling.¹³ A typical sketch of resulting TS-diagram is depicted in the left panel of Fig. 3.

Scully realized that if the working medium is an optical cavity, then a laser on resonance with the cavity mode can extract further work from the low entropy state A. This is possible, since the thermal radiation of the cavity will excite the laser into population inversion, and the total energy of cavity plus laser will decrease through the spontaneous emission of the laser. The correspondingly modified TS-diagram is sketched in the right panel of Fig. 3.

Thus, this quantum afterburner can extract additional energy from an ideal Otto cycle by exploiting the quantum states of the laser. The explicit expressions for extracted work and efficiency depend on the setup and the actual design of the system.^40,41 However, the idea of modifying ideal, classical cycles to harness quantum resources has proven to be versatile and powerful, see, for instance, an engine operating with transmon qubits.⁴² More importantly, we will see in the following that heat engines operating with nonequilibrium or squeezed reservoirs^43–55 also follow essentially the same design principles.

Arguably, the most prominent example among the early analyses of a heat engine that exploits genuine quantum effects is the photo-Carnot engine proposed by Scully et al.⁵⁶ Inspired by the discovery of lasing without inversion,⁵⁷ Scully et al.⁵⁶ considered a scenario in which the hot heat bath supports a small amount of quantum coherence.

More specifically, in the photo-Carnot engine the working medium is simply the radiation pressure, and the actual “engine” is a microlaser cavity.^58,59 Its mechanical equation of state can be written as

where P is the radiation pressure, V is the cavity volume, ω its frequency, and $n¯$ is the average number of photons in the mode in thermal equilibrium.

To exploit quantum effects, the hot reservoir is then assumed to consist of phaseonium, a three-level atom whose lower two states are nearly energetically degenerate. These nearly degenerate states are prepared so that they support a small amount of quantum coherence, see Fig. 4 for a sketch. Scully et al.⁵⁶ showed that the average number of photons in the cavity can be written as

where ϵ measures the magnitude of the coherence, and $ϕ$ is the phase.

Since the radiation field generated by the phaseonium is still thermal, we can further identify

The photo-Carnot engine otherwise operates similarly to a standard Carnot engine. Therefore, it is a simple exercise to show that its efficiency becomes

We immediately observe that in the case of vanishing coherence, ϵ = 0, the standard Carnot efficiency is recovered. However, we also see that there are particular phase values $ϕ$ for which this photo-Carnot engine outperforms classical devices.

In their interpretation of this result, Scully et al.⁵⁶ are very clear. While Eq. (9) is in full agreement with the second law of thermodynamics, it does demonstrate that there are genuine quantum resources that can be exploited through judicious design of heat engines. Thus, it may not be too bold to claim that Scully et al.⁵⁶ put the quantum into quantum heat engines, and it inspired significant work on thermodynamic devices with quantum coherences^60–76 and entanglement.^77–87 

In this review, we will be following a similar logic. Motivated and inspired by experimentally realistic scenarios, we survey existing proposals and realizations of heat engines that exploit quantum resources. However, it is worth emphasizing that none of the following scenarios constitute a violation of the physical principles of thermodynamics.

The principal component of any cyclic heat engine analysis is the identification of the heat exchanged and work done during each stroke. Let us consider the general case of a quantum system, described by density operator ρ, coupled to a thermal environment. We take the Hamiltonian for the system to be $H(λ)$⁠, where λ is an external control parameter. The system's dynamics can then be described by $ρ̇=Lλ(ρ)$⁠, where the superoperator $Lλ$ accounts for the unitary dynamics generated by $H(λ)$ as well as the nonunitary evolution that arises from the interaction of the system with the thermal environment. In the limit of ultraweak coupling, the equilibrium state of the system is the Gibbs state,¹² 

where $Z=tr{exp (−βH)}$ is the partition function. The internal energy of the system can be found from

For the Gibbs state, the thermodynamic entropy is given by the Gibbs entropy, $S=−tr{ρeq ln ρeq}$⁠. For an isothermal, quasistatic process, the change in entropy is then given by¹² 

Using Eq. (11), we can separate the change in internal energy, dE into two contributions, one associated with a change in entropy and the other from a change in the Hamiltonian¹² 

In complete analogy to classical thermodynamics, we can define the first as heat and the second as work.

It is important to note that these definitions of heat and work are valid if and only if $ρeq$ is a Gibbs state.⁸⁸ In the case that the system–environment coupling is not ultraweak, the energetic back-action due to correlations between the system and environment must be accounted for. For a non-Gibbsian equilibrium state $ρss$⁠, the change in entropy can be expressed as⁸⁸ 

where $−dQtot≡tr{dρssH}$ is the total heat and $−dQc≡dF−tr{ρssdH}$ is the energetic price to maintain quantum coherence and correlations. Here, $F$ is the “information free energy,”⁸⁹ which is determined from the Helmholtz free energy and the quantum relative entropy between $ρss$ and $ρeq$⁠.

With heat and work identified, quantum analogues of the classical isothermal, isochoric, adiabatic, and isobaric processes can be found,^90–93 allowing for the study of quantum implementations of heat engine cycles including Carnot, Otto, Stirling, Brayton, and Diesel.

For instance, Ref. 94 provides a detailed examination of the performance optimization of a quantum heat engine consisting of two finite-level quantum systems both coupled to a work source and separately to thermal baths at two different temperatures. Notably this analysis is not limited to the equilibrium regime, with the intermediate states of the engine allowed to be arbitrarily far from equilibrium. For finite-time performance, the Curzon–Ahlborn efficiency⁹⁵ is found to be the lower bound on the efficiency when the work output is maximized, achieved in the macroscopic limit. Furthermore, it is shown that as the efficiency approaches the Carnot efficiency, the work output of the finite-time engine vanishes. However, if the system–bath interaction is optimized, finite work output can be achieved at close to Carnot efficiency, provided that the cycle duration is long.

Outside of the realm of equilibrium systems, the notion of temperature is no longer straightforward to define. Ref. 96 examines this question using a quantum engine consisting of two qubits prepared in different thermal states that then undergo a thermally isolated unitary work extraction process. The final state of the total system is a nonequilibrium one, with a different local temperature for each subsystem. The behavior of three different definitions for the nonequilibrium temperature of the composite system is examined, and it is shown that, while all three definitions agree at mutual equilibrium, in general they show radically different behavior.

A motivating factor in the study of quantum heat engines is the idea that quantum resources can be exploited to enhance the engine performance. This prompts the immediate question of which parameter regimes the engine must operate to maintain its quantum nature. This is the primary question of Ref. 97, which uses violations of the Leggett–Garg inequality to quantify the regimes in which a Otto cycle with a two-level working medium displays nonclassical properties. The cycle is found to operate in three distinct regimes, one in which the dynamics are entirely classical, one in which they are entirely quantum, and a third transition regime in which the dynamics are quantum over certain temperature ranges and classical in others. Furthermore, it is shown that decreasing the cycle duration to avoid decoherence can actually lead to incoherent dynamics arising from thermodynamic constraints.

Finally, it is important to highlight the distinction between cyclic and continuous heat engines. This distinction is the central focus of Ref. 98. In a cyclic engine, the working medium interacts alternately between a hot and cold reservoir in a cyclic process, while for continuous engines, heat is exchanged between the reservoirs via a flow of particles that do work against an external field during this process. For this reason, Ref. 98 refers to continuous thermal machines as “particle-exchange” engines. Common examples of continuous heat engines include thermionic, thermoelectric, and photovoltaic devices. Notably the conditions for reversibility are distinct for each implementation, with cyclic engines achieving reversibility when the heat transfer is isothermal, while continuous engines achieve reversibility when the particle transfer is isentropic.⁹⁸ Reference 98 notes that the prototypical quantum heat engine, the three-level maser (see Sec. II A), should be properly classified as a continuous engine.

In the following, we will elaborate on the various heat engine and refrigerator cycles as well as continuous thermodynamic devices in the context of realistically available physical platforms. Before discussing specific systems, however, it is instructive to outline the main concepts and notions.

In Ref. 99, Eq. (13) was applied to derive bounds on the efficiency of a heat engine consisting of an open quantum system weakly coupled to N independent thermal reservoirs. For slowly varying external conditions, the evolution of the system can be described by a Markovian master equation.⁹⁹ Over the course of a full cycle of period τ, the periodicity conditions,⁹⁹ 

must be followed. The total work performed by the system per cycle is

where Q₁ and Q₂ are the total heat exchanged with the hot and cold reservoirs, respectively, over a full cycle. Taking into account the periodicity condition for the entropy leads to

where $σ(t)$ is the entropy production. Noting that $σ(t)≥0$⁠, Eq. (17) yields⁹⁹ 

By combining Eqs. (16) and (18), it can immediately be seen that the efficiency of this cyclic engine is bounded by the Carnot efficiency⁹⁹ 

Outside of the assumption of weak coupling between the system and the thermal environments, the energetic cost of the system–environment interaction must be carefully accounted for. Such an analysis is carried out in Ref. 100; however, it is important to note that doing so requires specifying the microscopic model of the heat engine.

The availability of quantum resources, such as coherence and entanglement, opens up a natural question as to whether these resources can be leveraged to design an engine that can outperform the Carnot efficiency. Reference 88 demonstrates that this is not the case, and that no quantum heat engine operating in a quasistatic Carnot cycle can harness quantum correlations. More specifically, the energetic back action arising from the correlation of system and environment must be accounted for. During any quasistatic process, a portion of the energy exchanged with the environment is paid as an energetic price to maintain the necessarily non-Gibbsian state arising from the presence of coherences and correlations. When this energetic cost is properly accounted for, the Carnot efficiency is maintained as the fundamental bound on cycle efficiency. In Ref. 101, it has even been shown that the Carnot bound holds beyond the typical framework of Hermitian quantum mechanics and can be extended to pseudo-Hermitian systems.

The quantum Carnot cycle has been studied for a range of working mediums including single spins,¹⁰² and two-level systems, where it has been shown that the Carnot efficiency can be achieved in experimentally relevant scenarios by cycling over a range of values for the energy gap.¹⁰³ 

As the basis for the internal combustion engine, the Otto cycle is perhaps the most widely implemented thermodynamic cycle in practice. This lends particular practical importance to understanding how the Otto cycle can be implemented for a quantum working medium. The classical Otto cycle consists of four strokes: (1) adiabatic compression, (2) isochoric heating (3), adiabatic expansion, and (4) isochoric cooling.¹³ 

Let us consider the adiabatic strokes first. By definition, an adiabatic process is one in which no heat transfer between the working medium and thermal environment occurs. Considering Eq. (13) such a process corresponds to one in which the eigenenergies are varied while the populations remain fixed, thus ensuring that any change in the internal energy is associated with work. As the populations of each energy level do not change, we can immediately see that the von Neumann entropy of the system remains constant. In the framework of classical thermodynamics, an adiabatic process is typically accomplished by performing the compression or expansion process rapidly enough that no heat exchange can occur. However, for a quantum working medium, the quantum adiabatic theorem guarantees that any rapid perturbation will generate excitations that will lead to changes in the energy level populations. Thus, to remain in accordance with the adiabatic theorem, the adiabatic strokes of the quantum Otto cycle must be performed quasistatically. This is a crucial difference between the classical and quantum implementations of the Otto cycle.^12,104

The fact that each stroke of the Otto cycle corresponds to an exchange of either work or heat, but never both simultaneously, makes it particularly attractive for implementation in quantum systems. Work and heat can be immediately identified from the change in internal energy during each stroke, without the need to differentiate each contribution, as must be done in the case of, e.g., isothermal processes.

Let us now consider the isochoric strokes. During the heating (cooling) stroke, the working medium is placed in contact with the hot (cold) bath until it achieves thermal equilibrium. The external control parameter of the Hamiltonian is held fixed during this process. Considering Eq. (13), we see that such a process corresponds to one in which the eigenenergies remain fixed while the populations of each energy level change, thus ensuring that any change in internal energy is associated with heat. As the populations of each energy level are varying, we note that this change in internal energy is associated with a change in the von Neumann entropy.

It is important to note that the quantum Otto cycle is inherently irreversible. For most of the cycle, the working medium is not in a state of thermal equilibrium, only achieving such a state at the ends of the heating and cooling strokes. The process of thermalization that takes place during these strokes is fundamentally irreversible and leads to the overall irreversibility of the cycle.

In Ref. 92, the quantum Otto cycle is analyzed for a multilevel working medium with energy eigenvalues E_n (with $n=0,1,2,…$⁠). During the adiabatic processes, it is assumed that the energy gaps between each level change by the same ratio α, such that,

where $Enh$ (⁠$Enl$⁠) is the nth eigenenergy of the system during the isochoric heating (cooling) process. The heat absorbed by the working medium during the isochoric heating process is

where $℘n$ is the occupation probability of the nth energy eigenstate. Similarly, the heat released to the low temperature bath during the isochoric cooling process is

The adiabatic nature of the expansion and compression strokes means that the occupation probabilities must fulfill the conditions

Using Eqs. (21)–(23), we can determine the net work done during the cycle,

The efficiency can now be found as the ratio of the net work to the absorbed heat

It is worth noting that this efficiency is completely analogous to the classical Otto efficiency, with the parameter α playing the role of an inverse “compression ratio.”

The quantum Otto cycle has been examined in a huge variety of contexts including, but not limited to, implementations with working mediums of single spin systems,^37,105,106 coupled spin systems,^82,107,108 harmonic oscillators,^109–111 relativistic oscillators,¹¹² an ideal Bose gas,¹¹³ a Bose–Einstein condensate,¹¹⁴ anyons,^115,116 a two-level atom,¹¹⁷ coupled spin-3/2 biquartits,¹¹⁸ and an $NI2$ dimer.¹¹⁹ Furthermore, it has been shown that the cycle performance can be enhanced with a quantum afterburner¹²⁰ (as elaborated on in Sec. II B), non-Markovian reservoirs,¹²¹ and nonequilibrium effects.¹²² 

While Carnot and Otto are perhaps the most ubiquitously studied, there exist a range of other heat engine cycles including the Stirling, Diesel, and Brayton cycles. As with Carnot and Otto, these cycles can also be generalized to quantum working mediums.

Reference 123 examines the performance of a quantum Stirling engine with a single spin or pair of coupled spins under an external magnetic field as the working medium. Like its classical counterpart, the quantum Stirling cycle consists of four strokes, illustrated in Fig. 5:

1. Isothermal expansion: While the working medium remains in contact with the hot bath, the external magnetic field is decreased slowly from B₂ to B₁ such that the working medium remains in thermal equilibrium with the hot bath at temperature T_h. Work is done by the working medium and heat is absorbed from the hot bath during this process.

2. Isochoric cooling: The working medium is placed in contact with the cold bath and allowed to cool to temperature T_c, while the magnetic field is held constant at B₁. No work is done and heat is absorbed by the cold bath during this process.

3. Isothermal compression: While the working medium remains in contact with the cold bath, the external magnetic field is increased slowly from B₁ to B₂ such that the working medium remains in thermal equilibrium with the cold bath at temperature T_c. Work is done on the working medium and heat is absorbed by the cold bath during this process.

4. Isochoric heating: The working medium is placed in contact with the hot bath and allowed to heat to temperature T_h, while the magnetic field is held constant at B₂. No work is done and heat is absorbed from the hot bath during this process.

The use of a regenerator is often considered when analyzing the performance of the Stirling cycle. The regenerator improves the cycle efficiency by capturing the heat released by the working medium to the hot bath during the isochoric cooling stroke and allowing it to be absorbed back into the working medium during the isochoric heating stroke.

For a single spin, the Hamiltonian of the working medium is simply, $H=Bσz/2$⁠, where B is the strength of the external magnetic field and σ^z is the Pauli spin operator. The heat exchanged during the isothermal processes can be found by using $−dQ=TdS$ where S is, as before, the von Neumann entropy. Thus,

Similarly,

During the isochoric processes, as no work is done, the change in internal energy of the working medium can be entirely attributed to heat. As such,

Similarly,

The net work per cycle can then be found using the first law, $W=∑Q$⁠. Thus,

The heat transfer between the system and the regenerator is given by the sum of the heats exchanged during the isochoric strokes

For a classical ideal gas working medium, the Stirling cycle is perfectly regenerative, with $ΔQ=0$ always. Reference 123 highlights that this is not always the case for the quantum Stirling cycle, allowing for three possibilities. If $ΔQ=0$⁠, perfect regeneration is achieved; however, this only occurs for a restrictive set of values of the bath temperatures and magnetic field strengths. If $ΔQ<0$⁠, the regenerator absorbs more heat than it releases. This redundant heat must be dissipated to the cold bath to keep it from building up in the regenerator. If $ΔQ>0$⁠, the regenerator absorbs less heat than it releases. In this case, additional heat from the hot bath must be absorbed by the working medium in compensation.

The efficiency can be found in the typical manner, by taking the ratio of the net work and the heat absorbed from the hot bath. For the classical Stirling cycle, the efficiency is identical to the Carnot efficiency. This is no longer true for the quantum Stirling cycle, with the efficiency depending not just on the bath temperatures but also on the initial and final magnetic field strengths. The efficiency is maximized for initial and final magnetic field strengths that meet the perfect regeneration of $ΔQ=0$⁠. For some parameter regimes, the maximum efficiency can even exceed the Carnot efficiency; however, Ref. 123 notes that this is not in violation of the second law as in these regimes the regenerator consumes additional energy.

The authors extend the analysis to a working medium of two coupled spins, resulting in an additional tunable parameter in the coupling strength. Notably for coupled spins, the condition of perfect regeneration can more easily be met for arbitrary bath temperatures and magnetic field strengths by adjusting the coupling strength. The analysis of the quantum Stirling cycle has been extended to the finite-time regime, using a two-level system as the working medium.¹²⁴ 

The Diesel cycle, consisting of one isobaric, one isochoric, and two isentropic strokes, has also been examined for ideal Bose and Fermi gas working mediums.¹²⁵ Similarly, the quantum Brayton cycle, consisting of two isobaric and two isentropic strokes, has been analyzed for a variety of working mediums, including a harmonic oscillator,¹²⁶ coupled spins,¹²⁷ noninteracting spins,¹²⁸ and an ideal Bose gas.¹²⁹ Quantum working mediums also allow for the implementation of nonconventional cycles. This includes cycles that extract work from a single heat bath whose energy input arises from nonselective measurement of the working medium,¹³⁰ cycles that incorporate isoenergetic strokes during which the expectation value of the Hamiltonian is held constant,^131–137 and cycles that utilize nonthermal baths.⁴² 

In equilibrium thermodynamics, the optimal efficiency of any heat engine cycle is bounded by the Carnot efficiency, regardless of the properties of the working medium.¹³ However, this efficiency is obtained in the limit of infinitely slow, quasistatic strokes, resulting in zero power output. A figure of merit of more practical use, the efficiency at maximum power (EMP), was introduced by Curzon and Ahlborn using the framework of endoreversible thermodynamics.^95,138,139 In endoreversible thermodynamics, the system is assumed to be in a state of local equilibrium at all times, but with dynamics that occur quickly enough that global equilibrium with the environment is not achieved. This results in a process that is locally reversible, but globally irreversible.¹³⁹ In the context of heat engines, this means that while the working medium may be in a local equilibrium state at temperature T, there is a thermal gradient at the boundaries where the working medium comes into contact with the bath at temperature $Tbath$⁠.

Curzon and Ahlborn⁹⁵ found the EMP of a endoreversible Carnot engine to be

where T_c (T_h) is the cold (hot) reservoir temperature. Remarkably, this result, now known as the Curzon–Ahlborn (CA) efficiency, has emerged in the analysis of heat engines in a wide variety of contexts outside the original case of a classical working medium undergoing a finite-time Carnot cycle. The EMP of endoreversible Otto,¹⁴⁰ Brayton,¹⁴⁰ and Stirling¹⁴¹ cycles have all been shown to be equivalent to the CA efficiency. Even quantum working mediums, including an open system of harmonic oscillators undergoing a quasistatic Otto cycle,¹⁴² a single harmonically trapped particle undergoing an Otto cycle,^46,110 and an endoreversible Otto cycle for a relativistic quantum particle in a Dirac oscillator potential,¹¹² have been found to assume the CA efficiency.

This wide applicability seems to suggest that the CA efficiency may be a universal characterization of finite-time performance, akin to the bound on pure efficiency given by the Carnot efficiency. However, despite its ubiquitous appearances, the CA efficiency is not universal and can be exceeded under the right circumstances. For the case of the finite-time Carnot cycle, whether or not CA efficiency is achieved at maximum power is determined by the symmetry of the dissipation during the cold and hot isothermal strokes, with the CA efficiency being achieved in the limit of symmetric dissipation and exceeded when the dissipation is significantly larger during the hot isotherm.¹⁴³ Furthermore, within the regime of linear response, it has been shown that an Otto cycle with a working medium of a classical harmonic oscillator can come arbitrarily close to the Carnot efficiency at finite power with a specific choice for the parameterization of the potential.¹⁴⁴ 

In Ref. 145, it was shown that the EMP of an endoreversible Otto engine with a single harmonically trapped quantum particle as the working medium can exceed the Curzon–Ahlborn efficiency. For an equilibrium thermal state of such a particle, $ρ∝exp(−βH)$⁠, the corresponding internal energy reads

where, as always, $β=1/kBT$⁠. The four strokes of the Otto cycle are as follows:

1. Isentropic compression: During this stroke, the working medium remains in a state of constant entropy, exchanging no heat with the environment. Using the first law $ΔE=Q+W$⁠, the change in internal energy can be identified completely with work

   $Wcomp=E(TB,ω2)−E(TA,ω1).$

   (34)
2. Isochoric heating: During this stroke, the externally controlled work parameter (the trap frequency in the case of the harmonic engine) is held constant, resulting in zero work. By the first law, the change in internal energy can be completely identified with heat

   $Qh=E(TC,ω2)−E(TB,ω2).$

   (35)
   In the endoreversible regime, the working medium does not fully thermalize with the hot reservoir during this stroke. This yields the condition $TB≤TC≤Th$⁠. The change in temperature during the stroke depends on the properties of the working medium and can be determined using Fourier's law,¹³ 

   $dTdt=−αh(T(t)−Th),$

   (36)

   where α_h is a constant determined by the heat capacity and thermal conductivity of the working medium.

3. Isentropic expansion: In exactly the same manner as the compression stroke, the change in internal energy during the expansion stroke can be identified with work

   $W exp =E(TD,ω1)−E(TC,ω2).$

   (37)
4. Isochoric cooling: As in the heating stroke, the change in internal energy during the cooling stroke is identified with heat

   $Qc=E(TA,ω1)−E(TD,ω1).$

   (38)

The temperature change can again be determined from Fourier's law

where $TD>TA≥Tc$⁠.

The efficiency of the engine is given by the ratio of the total work and the heat exchanged with the hot reservoir

and the power output by the ratio of the total work to the cycle duration, $τcyc$⁠,

Note that only the durations of the heating and cooling strokes are accounted for explicitly, with γ serving as a multiplicative factor that implicitly incorporates the duration of the isentropic strokes, $τcyc≡γ(τh+τc)$⁠.

In order to ensure that the compression and expansion strokes are isentropic, we have the conditions

Applying these conditions, along with the solutions to Eqs. (36) and (39), the efficiency is determined to be

where $κ=ω1/ω2$⁠. Similarly, the expression for power can be found in terms of the hot and cold bath temperatures, the thermal conductivities, the initial and final frequencies, and the stroke times.

The EMP can then be determined by maximizing the power with respect to the compression ratio κ. The EMP for the engine operating in the classical regime, corresponding to $ℏω2/kBTc=0.1$⁠, and in the quantum regime, corresponding to $ℏω2/kBTc=10$⁠, is shown in Fig. 6, with the Carnot and CA efficiencies given for comparison. It is clear that the engine significantly exceeds the CA efficiency when operating in the quantum regime.

As demonstrated by these results, the EMP of the Otto cycle is not universally given by the CA efficiency but determined by the nature of the working medium. Further work has investigated the power and EMP of Otto cycles using working mediums with polynomial fundamental relations (e.g., photonic gases),¹⁴⁶ two-level working mediums,^37,147 degenerate quantum gas working mediums,¹⁴⁸ quantum statistics,^111,115,116 and Bose–Einstein condensate working mediums.¹¹⁴ The power and EMP of other cycle types has also been examined, including Carnot cycles with two-level quantum systems as a working medium,¹⁴⁹ and a uniquely quantum cycle consisting of isoenergetic, isothermal, and adiabatic strokes.^133,134

As a field, thermodynamics has always been aware of the principle that “more is different.”¹⁵⁰ Many-body systems display collective behavior beyond the sum of the individual constituent behaviors, with phase transitions serving as a prime example. Quantum many-body systems introduce a range of new collective behaviors such as wave function symmetrization, super-radiance, quantum phase transitions, and many-body localization. For quantum thermal machines with multi-particle working mediums, many-body effects can have significant impacts on performance.^151,152 Thus, we continue with a particularly instructive situation, for which single versus many-particle performance has been thoroughly investigated.

We begin with a single particle trapped in a harmonic oscillator, which has probably become the most studied setup for finite-time quantum Otto engines, see for instance Refs. 30, 109, 110, and 153. Its Hamiltonian reads

where x and p are the position and momentum operators of an oscillator of mass m. The angular frequency plays the role of “inverse volume,” and we consider situations in which $ω(t)$ can be varied between ω₁ and ω₂. In addition, the particle is alternatingly coupled to two heat baths at inverse temperatures, β₁ and β₂, see Fig. 7 for an illustration of the scenario.

The quantum Otto cycle is then implemented through the following four strokes:

1. Isentropic compression $A(ω1,β1)→B(ω2,β1)$⁠: the frequency is varied during time τ₁ while the system is isolated. The evolution is unitary and the von Neumann entropy of the oscillator is, thus, constant. Note that state B is nonthermal even for slow (adiabatic) processes. In this step, work $W1=⟨H⟩B−⟨H⟩A$ is performed on the medium, and the average thermal energy is again given by Eq. (33). At the end of the unitary stroke, we have^154–157 

   $⟨H⟩B=ℏω22Qcomp∗coth(β1ℏω12).$

   (45)

   Here $Qcomp*$ is a dimensionless parameter that measures the degree of adiabaticity of the isentropic compression and expansion strokes, respectively.^154–157 Its exact form is determined by the protocol under which the trapping frequency is modulated, but in general $Q*≥1$ with $Q*=1$ corresponding to a completely adiabatic stroke.

2. Hot isochore $B(ω2,β1)→C(ω2,β2)$⁠: the oscillator is weakly coupled to a reservoir at inverse temperature β₂ at fixed frequency and allowed to relax during time τ₂ to the thermal state C. This equilibration is much shorter than the expansion/compression strokes and only an amount of heat $Q2=⟨H⟩C−⟨H⟩B$ is transferred. Note that at C the system is again in equilibrium, and its energy is accordingly given by Eq. (33).

3. Isentropic expansion $C(ω2,β2)→D(ω1,β2)$⁠: the frequency is changed back to its initial value during time τ₃. The isolated oscillator evolves unitarily into the nonthermal state D at constant entropy. An amount of work $W3=⟨H⟩D−⟨H⟩C$ is extracted from the medium during this stroke, which we can compute with^154–157 

   $⟨H⟩D=ℏω12Q exp ∗coth(β2ℏω22).$

   (46)

4. Cold isochore $D(ω1,β2)→A(ω1,β1)$⁠: the system is weakly coupled to a reservoir at inverse temperature $β1>β2$ and quickly relaxes to the initial thermal state A during τ₄. The frequency is again kept constant and an amount of heat $Q4=⟨H⟩A−⟨H⟩D$ is transferred from the working medium.

During the first and third strokes (compression and expansion), the quantum oscillator is isolated, and the corresponding work values are

During the thermalization strokes (isochoric processes), heat is exchanged with the reservoirs, and we have

The efficiency of this quantum engine, defined as the ratio of the total work per cycle and the heat received from the hot reservoir, is then as follows:¹¹⁰ 

and the power output per cycle, $P=−(W1+W3)/τcyc$ becomes

It is easy to see that for slow driving (adiabatic limit), during the isentropic processes $Qi∗=1$⁠, the thermal machine efficiency is $ηOAD=1−ω1/ω2$⁠, whereas the power vanishes. As we will see shortly, this single particle engine was realized in ion traps based on a theoretical proposal.¹¹⁰ 

The natural question then is how the engine performance changes if not one, but two quantum particles are trapped in the harmonic oscillator. To this end, Ref. 111 examined how “exchange forces,” a collective phenomenon that arises from the fundamentally indistinguishable nature of quantum particles, affects the performance of this finite-time Otto cycle with a working medium of two particles, either bosons or fermions. Due to the underlying symmetry of the collective wave function, bosons are more likely to be found bunched together while fermions, with an underlying antisymmetric wave function, are more likely to be found spread apart. This can be clearly seen in the thermal state position distributions, as illustrated in Fig. 8. Physically, these exchange forces manifest as an effective attraction between bosons and repulsion between fermions.

As the differences between bosons and fermions manifests entirely in terms of the symmetry of the total wave function, the Hamiltonian is identical for both working mediums

By solving the full time-dependent dynamics for the unitary strokes, and assuming the heating and cooling strokes are of sufficient duration for the working medium to fully thermalize, the internal energy at each corner of the cycle is determined as follows:

where the top sign corresponds to bosons and the bottom to fermions.

As in the single particle case, these internal energies are sufficient to characterize the finite-time performance of the cycle. Examining multiple figures of merit, including efficiency, power output, and efficiency at maximum power, Ref. 111 showed that a bosonic working medium displays enhanced performance, in comparison to a working medium of distinguishable particles, while a fermionic working medium displays reduced performance. This “bosonic enhancement” even extends to the inherent trade-off between efficiency and power quantified by the efficient power, given by the product of efficiency and power. Furthermore, Ref. 111 demonstrated that the bosonic working medium functions as an engine or refrigerator over a wider range of the total parameter space in comparison to the fermionic medium. Notably, the differences in bosonic and fermionic performance are fundamentally an effect of the nonequilibrium nature of the cycle and vanish in the limit of long stroke times.

The impacts of wave function symmetry on heat engine performance were further explored in Ref. 115, which extended the results of Ref. 111 to anyonic statistics. Numerous other many-body phenomena have been explored in the context of thermal devices including (quantum) phase transitions,^158–162 many-body localization,¹⁶³ super-radiance and collective enhancement of energy exchange,^164–167 inter-particle interactions,^116,168,169 many-body quantum interference,¹⁷⁰ and the non-Markovian anti-Zeno effect.¹⁷¹ 

Equation (50) highlights that quantum engines also generally fail to produce finite work in infinitely slow processes. A possible way to achieve finite power in fast processes has been sought in so-called shortcuts to adiabaticity (STA).^172–176 In particular, it has been found that STA methods enhance the performance of thermal devices by reducing irreversible losses that suppress efficiency and power.^{109,168,177–179} Note, however, that this is typically only true if the energy balance of the external controller can be disregarded.¹⁸⁰ 

A particularly powerful technique for STA is called counterdiabatic driving.^181–184 Within this paradigm, all transitions away from the adiabatic manifold of a time-dependent Hamiltonian, H(t), are suppressed by applying the counterdiabatic field. Thus, a quantum system evolves under a new Hamiltonian $Heff(t)=H(t)+HCD(t)$⁠, which reads explicitly

where $|n⟩≡|n(t)⟩$ denotes the nth eigenstate of the original Hamiltonian H(t).

For instance, for a time-dependent harmonic oscillator, the counterdiabatic term becomes¹⁸⁴ 

Note that for implementation in heat engine cycles, boundary conditions are chosen to ensure $⟨HCD(0)⟩=⟨HCD(τ)⟩$ for all strokes.

There has been some debate in the literature over how to appropriately assess the thermodynamic cost of STA, see for instance Refs. 104, 179, 180, and 185–187. Reference 179 took the arguably simplest approach by considering a modified efficiency of the corresponding engine, which can be expressed as

where $Wi=⟨HCD(τ)⟩−⟨H(0)⟩$ is corresponding mean work of the STA protocol and $⟨Hi⟩τ=(1/τ)∫0τdt ⟨HCDi(t)⟩$ is the time-average of the mean STA driving term. Note that Eq. (55) is nothing but the ratio of output and input average energies. Accordingly, the power output becomes¹⁸⁷ 

It has been noted that the such defined efficiency and power represent the “true” performance of the Otto cycle.¹⁸⁸ 

Similar studies looked at thermodynamic cycles with STA in multiferroic material,¹⁸⁹ harmonic oscillators,^190,191 anyonic working mediums,¹¹⁵ Bose-Einstein condensates (BECs) with nonlinear interactions,¹⁸⁷ spin-1/2 systems,^192–194 superconducting qubits,¹⁹⁵ and multi-spin systems^196,197 as working mediums. More recently, techniques from machine learning have been employed to find optimal STA.^198,199 Moreover, it is worth noting that complementary approaches based on more “traditional” optimal control strategies^{144,200–205} have also been developed.

We conclude this section by briefly commenting on fluctuations in the performance of quantum thermodynamic devices. For a comprehensive discussion of notions such as quantum work^206–208 and their corresponding fluctuation theorems, we refer to the literature.^12,209 For our present purposes, it is interesting to note that similar consideration were published in the context of engine cycles.^210–218 

Within the two-time energy measurement paradigm for quantum work, the stochastic performance for the finite-time Otto cycle quantum heat engine is evaluated with the joint probability distribution $P(W1,Q2,W3)$⁠.²¹¹ To this end, projective measurements are taken at the end points of all strokes, and for instance, the probability distribution of the compression work W₁ reads²⁰⁹ 

Here, $Ej0$ and $Ekτ$ are the respective initial and final energy eigenvalues, $℘j0= exp (−β1Ej0)/Z0$ is the initial thermal occupation probability with partition function Z⁰, and $℘jkcom=|⟨j|Ucom(τ)|k⟩|2$ denotes the transition probability between the instantaneous eigenstates $|j⟩$ and $|k⟩$ in time τ with the corresponding unitary $Ucom$⁠. The probability distribution of the heat Q₂ after the second stroke, given the compression work W₁, is equal to the conditional distribution

where the occupation probability $℘iτ=δki$ at time τ after the second projective energy measurement. The quantum work distribution for the expansion stroke, given the compression work W₁ and the heat Q, is²¹⁹ 

where the probability for finding the system in eigenstate $|l⟩$ after the third projective measurement simply reads $℘rτ=δrl$⁠, and the transition probability $℘rmcom$ is determined by the unitary time evolution for expansion $U exp$⁠.

Based on the chain rule for conditional probabilities, the joint probability of a cycle of the quantum engine reads²²⁰ 

By defining the stochastic total extracted work $W=−(W1+W3)$⁠, the joint distribution for work and heat is given by

Then, defining the stochastic efficiency $η=W/Q$⁠, its probability distribution is obtained by integrating over W and Q as²²⁰ 

This probability distribution can be exploited to analyze the effects of thermal and quantum fluctuations on the efficiency of a quantum Otto heat engine.

The study of fluctuations for finite-time quantum heat engines has been attracting some attention mainly focused on two-level systems or harmonic oscillators.^{215,216,220–222} However, its stands to reason that interesting physics is encoded in the fluctuations of any working medium.

The remainder of this review is dedicated to highlighting a selection of particularly important studies in a variety of physical platforms. When collecting the various publication and scenarios, special emphasis was put on giving a broad overview of the many different systems at the expense of being rather brief with regard to technical, physical, and mathematical details.

We begin with recent implementations of quantum thermodynamic devices in ion traps. It is interesting to note that while laser-cooled ions in linear Paul traps were originally invented for the experimental study of quantum computation and information processing applications, they have become a prominent testbed for quantum thermodynamic notions.^{46,110,223–227}

A particularly important breakthrough for the field was achieved with the realization of a heat engine that operates with a single ion as a working medium.²²⁵ To achieve this, a single $40Ca+$ ion was trapped in a linear Paul trap with a funnel-shaped electrode geometry, as illustrated in Fig. 9. The authors of Ref. 225 engineered cold and hot reservoirs by using laser cooling and electric field noise, respectively, and the temperature of the ion was determined by fast thermometry methods.²²⁸ The realized thermodynamic cycle resembles a Stirling engine and its thermodynamic properties were analyzed in this context.

The tapered trap realized a harmonic pseudopotential of the form

where m is the atomic mass and $x,y$ denote the trap axes. The electrodes were driven symmetrically at a radio frequency voltage of 830 V_pp at 21 MHz. Applying constant voltages on the two end-cap electrodes resulted in the axial confinement with a trap frequency of $ωz/2π= 81$ kHz. The resulting radial trap frequencies $ωx,y$ reads

where $ω0x,0y$ are the eigenfrequencies in the radial directions at the trap minimum $z = 0$ and satisfy the cylindrical symmetry with $r2= x2+y2$ a mean radial trap frequency ω_r. The electrode angle is denoted by θ while the radial distance of the electrode is r₀ at axial position z = 0.

To compensate for stray fields, a second set of outer electrodes was used. Then the trapped ion was cooled by a laser beam and the resulting fluorescence was recorded by a rapidly gated intensified charge-coupled device camera. An equilibrium cold bath was realized by exposing the ion to a laser cooling beam at temperature $TC=3.4$ mK (Ref. 225), while a hot reservoir with finite temperature T_H was mimicked by exposing the ion to additional white noise from the electric field.

In this setup, heating and cooling act on the radial degrees of freedom. Based on the geometry of the tapered potential, the ion experiences a temperature-dependent average force in axial direction is given by

where the time-averaged spatial distribution of the ion thermal state is described by

The time-averaged width $σ(T)$ of this two-dimensional Gaussian probability distribution depends on the temperature T as $σr=kBT/mωr2$⁠, where k_B is the Boltzmann constant. The heat engine is driven by alternately heating and cooling the ion in radial direction.

The dynamics of the ion when driven in a four-stroke thermodynamic cycle is depicted in Fig. 9. In the first part of the cycle, the ion is heated, which results in the width $σr(T)$ increasing. During the second step, the ion moves along the z-axis to a weaker radial confinement, which leads to the increase in total potential energy of the ion and thereby produces work. During the third step, the ion is cooled to the initial temperature as the radial width of its probability distribution is reduced. For the final step, the ion moves back to its initial position due to the restoring force of the axial potential. However, the resulting cycle deviates from an ideal Stirling cycle due to the fact that full thermalization with the reservoirs is not reached. For each radial cycle, work produced is transferred to the axial degree of freedom and stored in the amplitude of oscillation.

Roßnagel et al.²²⁵ computed the power output, $P=W/tcyc$⁠, during steady-state operation in three independent ways. First, the power is determined from the area enclosed by the engine cycle, where $W=∫dωr 2ℏn¯r$ is the work computed from the area of the cycles from the two radial directions and $tcyc=2π/ωz$ is the cycle time.

In the second approach, the power was directly deduced from the measurement of the axial oscillation amplitudes A_z. The dissipated power of a driven damped harmonic oscillator at steady state is²²⁹ 

with the damping parameter γ determined from the oscillation decay.

The third approach involved the analytical calculation of the engine power output using the expression for work performed over one cycle period $tcyc$⁠,

The analytical power output $Pana=Wana/tcyc$ depends only on the temperature variation and the trap geometry. The analysis of Roßnagel et al.²²⁵ showed good agreement between the measured and the analytical power output as a function of temperature difference, see Fig. 9.

Furthermore, the engine efficiency from the measured data was evaluated as $η=W/QH$⁠, by deducing the heat absorbed from the hot reservoir, $QH=∫H dS T$⁠. The entropy S of the harmonic oscillator engine is calculated using $S=kB[1+ln (kBT/ℏωr)]$⁠. This basically involves transforming the heat engine cycle from the {$n,ω$}-diagram to a temperature-entropy representation. Likewise, an analytical expression of the engine efficiency was derived, which depends solely on temperature variation and trap geometry by calculating the heat absorbed from the reservoir as $QH=∫Hdt T(t) Ṡ(t)$⁠. The analysis of Ref. 225 showed that the engine operates at efficiency of 0.28% and could be increased by varying either the angle of the taper or the absolute radial trap frequencies.

This experimental realization has inspired investigations of quantum thermodynamic devices such as, e.g., quantum absorption refrigerators,²²⁷ spin heat engines with quantum flywheels,^226,230 and potential applications as sensitive thermal probes.²³¹ 

An attempt to unravel the role of thermodynamic fluctuations and quantum effects in the performance of quantum thermal devices led to the experimental study of a spin heat engine coupled to a harmonic-oscillator flywheel (load),²²⁶ see also related theoretical analyses.^232,233 von Lindenfels et al.²²⁶ experimentally realized a heat engine using the valence electron spin of a trapped $40Ca+$ ion as a working medium, and the heat reservoirs were mimicked by controlling the spin polarization via optical pumping. The role of the flywheel coupled to the working medium was to absorb the output energy of the engine. The four-stroke Otto cycle operation is illustrated in Fig. 10.

The authors²²⁶ trapped an ion in a Paul trap at a secular trap frequency $ωt∼2π×1.4$ MHz along the x-axis. The ion was placed in an optical standing wave (SW), which mediates the coupling between the engine and flywheel via a spin-dependent optical dipole force. The spin-oscillator system is described by

where ω_z represents the Zeeman splitting of the spin, Δ_S is the amplitude of the SW depending on the ac-Stark shift variation, $kSW$ is the effective wavenumber, and σ_z is the Pauli z-operator. The bare flywheel Hamiltonian is simply $HHO=ℏω(t) (n+1/2)$⁠, where $ω(t)$ is the time-dependent trapping frequency and n is the number operator. Optically pumping at the trap frequency $ω(t)$ plays the role of reservoirs, with the spin population corresponding to the temperature. The cold and hot reservoirs' temperatures are deduced from the population of the Zeeman sublevel that can be expressed as

where $ωz′=ωz+ΔSkSW⟨x⟩, ⟨σz⟩≳−1$⁠, and $⟨σz⟩≲0$ corresponds to the cold (hot) reservoir temperature T_C (T_H). The engine's cyclic operation results in an increasing amplitude of the harmonic oscillation, stored as energy in the flywheel.

von Lindenfels et al.²²⁶ characterized the state and energetics of the flywheel both theoretically and experimentally. For this purpose, they evaluated the ergotropy of the flywheel, i.e., the maximum work that can be extracted via a cyclic unitary transformation.^234–237 This can be described as

where ρ is the state of the flywheel and ρ_p is the passive state unitarily related to ρ.²³⁴ They found that the flywheel's ergotropy $E$ is always less than the mean energy $E=⟨HHO⟩$ due intrinsic fluctuations in machines operating with single atomic degrees of freedom.

The experiment of a spin-1/2 heat engine coupled to a harmonic oscillator flywheel agrees well with the theoretical analysis²²⁶ and opens the door for more investigations of quantum thermodynamic devices with a load attached.²³⁰ In addition, a recent experimental study of Van Horne et al.²³⁰ considered a quantum refrigerator coupled to a load within the same framework of trapped ions technology. Finally, it has been recognized that “quantum rotors” can simultaneously act as both flywheel and quantum clock,^238,239 which however is more easily implemented in superconducting systems.²⁴⁰ 

Another experimentally feasible single-ion quantum heat engine was proposed by Chand and Biswas.²⁴¹ Their design aims at a quantum Otto engine that mimics continuous heat engine cycles.³⁶ To this end, they considered an Otto model of a single-ion quantum heat engine with continuous interaction between the working medium and a thermal environment. The proposal relies on controlling the magnetic field adiabatically and performing a projective measurement of the electronic states.

Chand and Biswas²⁴¹ imagined the internal state of a single-ion as a the engine's working medium. In the Lamb–Dicke limit, the Hamiltonian describing the interaction between the internal and motional states of the ion is written as

where $HS=gσx+B σz$ describes the internal states of the ion with 2g being the Rabi frequency and B is the strength of the magnetic field. $Hph=ωa†a$ represents the vibrational degree of freedom, and $Hint=k(a†σ−+σ+a)$ is the interaction between the internal and the vibrational degrees of freedom of the ion. Thus, the eigenvalues of the working medium Hamiltonian are given by $E1,2=±g2+B2$⁠.

For this heat engine design, the quantum Otto engine cycle is the implemented as follows:

1. Ignition stroke: During this isochoric process, the working medium thermalizes with the hot bath at temperature T_H. In this stroke, the magnetic field is fixed at B = B_H and the amount of heat transferred can be written as $QH=∑n=12EnH(℘nH−℘nL)$ where, as before, $EnH$ is the nth energy eigenvalue and $℘nL$ is the initial probability of occupying the nth energy eigenstates and $℘nH$ describes the final probability after the thermalization.

2. Expansion stroke: Here, the magnetic field is adiabatically changed from B_H to B_L. No heat exchange occurs with the heat bath or the phonon modes, whereas the work performed by the system is $W1=∑n=12℘nH(EnL−EnH)$⁠, where $E1,2H=±g2+BH2$ are the eigenvalues of H_S at the initial stage of the stroke.

3. Exhaust stroke: This isochoric process results in transferring the amount of heat Q_L out of the system as the local driving fields are kept fixed. Chand and Biswas²⁴¹ proposed to employ a projective measurement of the state of the system S to enable the release of heat. This purification process is equivalent to cooling down the system. Assuming that the system is prepared in the ground state $|g⟩$⁠, the heat removed from the system reads $QL=∑n=12EnL(℘nL−℘nH)$⁠.

4. Compression stroke: In this final stroke, the magnetic field strength is adiabatically changed from B_L to B_H. The occupation probabilities remain unchanged, i.e., the system remains in the ground state, and the work done during this stroke is $W2=∑n=12℘nL(EnH−EnL)$⁠.

Accounting for the measurement back action on the working medium, the effective efficiency of the engine is written as

where the energy cost associated with the projective measurement of the qubit is $M≤kBTH ln 2$⁠. The equality sign holds for a maximally mixed state and corresponds to maximal change in entropy. Chand and Biswas²⁴¹ further outlined how to prevent heating of the system when averaged over many cycles using feedback control.

This theoretical proposal motivated measurement-based engines^130,242 that offer the possibility of enhanced engine performance, which we will elaborate on next.

From the point of view of quantum information theory, thermal environments do nothing but perform measurements on a system of interest.^243–247 Thus, it becomes easy to recognize that quantum heat engine cycles can be designed in which “thermodynamic strokes” are replaced by quantum measurements, which is relevant, e.g., in the context of quantum error correcting algorithms²⁴⁸ or in the diagnostics of thermodynamic performance.²⁴⁹ Beyond the work of Chand and Biswas²⁴¹ and Elouard and Jordan,²⁵⁰ the following studies are particularly noteworthy.

Ding et al.¹³⁰ proposed to construct a thermal device that operates between a measurement apparatus and a single heat bath. The working medium is again the parametric harmonic oscillator (44). The engine operates in a four-stroke cycle.

1. Adiabatic compression: The engine is initialized in thermal equilibrium at the inverse temperature β, before the working substance is compressed by changing the frequency from initial ω₁ to final ω₂. During this stroke, no heat is added to the working medium and the work simply reads $WI=E2−E1$⁠, where E₁ and E₂ are the initial and final eigenvalues of the Hamiltonian, respectively. The energies at the initial and final state are deduced using projective measurements.

2. Isochoric heating: In the second stroke, a measurement of the oscillator position is performed at a constant frequency ω₂. This stroke provides an input energy into the system by ensuring that the measured observable does not commute with the working substance Hamiltonian. Accordingly, the energy change $EM$ induced by the measurement is $EM=E3−E2$⁠, where E₃ is the energy eigenvalue at the end of the stroke.

3. Adiabatic expansion: The third stroke involves expanding the working substance back to $H(ω1)$⁠. The amount of work performed on the working substance during this stroke is $WII=E4−E3$⁠, where E₄ is the corresponding eigenvalue at the stroke completion.

4. Isochoric cooling: In the final stroke, the working substance is brought back to its initial state by weakly coupling with the thermal reservoir at the initial inverse temperature β. The heat exchange is the thermal reservoir is $Q=E1′−E4$⁠, where $|1′⟩$ is the initial state of the next cycle.

Using the positive sign convention, the averages of the total work $⟨W⟩=⟨WI⟩+⟨WII⟩$ and the supplied energy $⟨EM⟩$ are calculated for the large class of minimally disturbing generalized measurements.²⁵¹ As usual, the efficiency is $η=−⟨W⟩/⟨EM⟩$⁠. Interestingly, the efficiency takes the universal form $η=1−1/κ$⁠, where κ is again the compression ratio, for a uniform adiabatic compression and expansion of the working substance.

Ding et al.¹³⁰ analyzed the averages and fluctuations of the total work, supplied energy, and performance of this measurement-driven engine. They showed that the power of a measurement engine is considerably larger than that of the Otto engine running with the same average work output because of the shorter measurement engine cycle time. This theoretical proposal has spurred interest to potential devices using qubits as working substance.^242,252,253

A more versatile device was proposed by Buffoni et al.²⁵² They considered a prototype two-qubit device that exploits quantum measurements as a fuel. The model consists of two qubits governed by the Hamiltonian $H=H1+H2$⁠, and each prepared by thermalization with a thermal bath at positive inverse temperatures β₁ and β₂, respectively. Thus, the initial state reads

where $Hi=ℏωiσzi/2$ denotes the Hamiltonian of ith qubit in terms of its Pauli matrix $σzi$ and its resonance frequency ω_i. Finally, $Zi=tr{ exp (−βiHi)}$ is the canonical partition function.

The two-stroke cycle is schematically depicted in Fig. 11. In the first stroke, the two-qubit system interacts with a measurement apparatus. This stroke erases all coherences of the two qubit compound state in the measurement basis. Hence, the postmeasurement state reads $ρ′=Φ[ρ]=∑kπkρπk$⁠, where π_k is a projector²⁴⁵ into the eigenstates of the measured observable, sometimes also called “pointer states.”²⁵⁴ The change in the expectation energy of the ith qubit is $⟨ΔEi⟩=tr{Hi (Φ[ρ]−ρ)}$⁠. Based on the unital property of $Φ$⁠, the second law of thermodynamics can then be expressed as

In the second stroke, each qubit is restored to its initial Gibbs state by putting it back in contact with its thermal bath. During this stroke, each qubit releases an average energy $⟨ΔEi⟩$⁠, gained during the first stroke. Combining the energy conservation $⟨ΔE⟩=⟨ΔE1⟩+⟨ΔE2⟩$ and the second law relation [Eq. (75)] as well as assuming the condition $0<β1<β2$⁠, Buffoni et al.²⁵² identified the operation regime where the device can be function as a refrigerator (R), heat engine (E), thermal accelerator (A), or heater (H).

Buffoni et al.²⁵² further elaborated on how to experimentally realize the two-stroke quantum cooler using solid-state superconducting circuitry by a combination of circuit QED and circuit quantum thermodynamics. Specifically, they suggested a device that comprises two superconducting qubits coupled to an on-chip microwave line resonator.²⁵⁵ The first stroke can be implemented by the combination of two-state manipulation and standard measurement while the second stroke can be implemented by inductively coupling each qubit to an on-chip resistor kept at inverse temperature β_i. This study has attracted a follow-up proposal on the design of quantum magnetometry using a two-stroke engine.²⁵⁶ Similar conceptual tools also proved useful in characterizing the D-Wave machine as a thermodynamic device,²⁵⁷ and a corresponding heat engine cycle has already been realized on IBM's quantum computer.²⁵⁸ 

Even more recently, Bresque et al.²⁵³ proposed a thermal device exploiting quantum entanglement to deepen the understanding of measurement as fuel. The operation of the quantum engine demonstrates that both local measurements and entanglement are crucial for work extraction. The working medium involves two qubits A and B, whose Hamiltonian, $H2qb=Hloc+V$⁠, reads explicitly