Thermodynamic Geometry of Microscopic Heat Engines

TL;DR

This paper introduces a geometric framework for analyzing microscopic heat engines, revealing universal efficiency-power trade-offs and quantum coherence effects, with practical examples in solid-state systems.

Contribution

It develops a unified geometric approach applicable to both classical and quantum microscopic heat engines, highlighting quantum coherence's impact on performance.

Findings

Universal trade-off relation between efficiency and power derived from geometry.

Quantum coherence reduces engine performance regardless of driving strength.

Single-qubit heat engine example demonstrates practical relevance.

Abstract

We develop a geometric framework to describe the thermodynamics of microscopic heat engines driven by slow periodic temperature variations and modulations of a mechanical control parameter. Covering both the classical and the quantum regime, our approach reveals a universal trade-off relation between efficiency and power that follows solely from geometric arguments and holds for any thermodynamically consistent microdynamics. Focusing on Lindblad dynamics, we derive a second bound showing that coherence as a genuine quantum effect inevitably reduces the performance of slow engine cycles regardless of the driving amplitudes. To demonstrate the practical applicability of our results, we work out the example of a single-qubit heat engine, which lies within the range of current solid-state technologies.

Equations

$$\mathbf{\Lambda }\equiv (T,\lambda )\equiv ({\Lambda }^u,{\Lambda }^w)$$

$$W={\int }_0^{\tau }dt{f}_t^w{\dot{\Lambda }}_t^w\text{and}U=-{\int }_0^{\tau }dt{f}_t^u{\dot{\Lambda }}_t^u.$$

$$f^{w}_{t}\equiv-{{{\rm Tr}}}\bigl{[}\rho_{t}\partial_{\lambda}H_{\lambda}\bigr{]}\bigl{|}_{\lambda=\lambda\raisebox{-1.6pt}{{{\scriptsize$t$}}}}\quad\text{and}\quad f^{u}_{t}\equiv-{{{\rm Tr}}}\bigl{[}\rho_{t}\ln\rho_{t}\bigr{]},$$

$$A\equiv U-W=-{\int }_0^{\tau }dt{f}_t^{\mu }{\dot{\Lambda }}_t^{\mu }\ge 0,$$

$$\eta \equiv W/U\le 1$$

$$\rho_{t}=\varrho_{\boldsymbol{\Lambda}\raisebox{-1.6pt}{{{\scriptsize$t$}}}}\quad\text{with}\quad\varrho_{\boldsymbol{\Lambda}}\equiv\exp\bigl{[}-(H_{\lambda}-\mathcal{F}_{\boldsymbol{\Lambda}})/T\bigr{]}$$

$$f^{\mu}_{t}=\mathcal{F}^{\mu}_{\boldsymbol{\Lambda}\raisebox{-1.6pt}{{{\scriptsize$t$}}}}\quad\text{with}\quad\mathcal{F}^{\mu}_{\boldsymbol{\Lambda}}\equiv{{{\rm Tr}}}\bigl{[}\varrho_{\boldsymbol{\Lambda}}F_{\boldsymbol{\Lambda}}^{\mu}\bigr{]}=-\partial_{\mu}\mathcal{F}_{\boldsymbol{\Lambda}},$$

$$F_{\mathbf{\Lambda }}^w\equiv -{\partial }_{\lambda }{H}_{\lambda }\text{and}{F}_{\mathbf{\Lambda }}^u\equiv -\ln {\varrho }_{\mathbf{\Lambda }}$$

$$f_t^{\mu }={\mathcal{F}}_t^{\mu }+R_t^{\mu \nu }{\dot{\Lambda }}_t^{\nu },$$

$$A={\int }_0^{\tau }dt{g}_{{\mathbf{\Lambda }}_t}^{\mu \nu }{\dot{\Lambda }}_t^{\mu }{\dot{\Lambda }}_t^{\nu }\text{with}{g}_{\mathbf{\Lambda }}^{\mu \nu }\equiv -(R_{\mathbf{\Lambda }}^{\mu \nu }+R_{\mathbf{\Lambda }}^{\nu \mu })/2$$

$$A\ge {\mathcal{L}}^2/\tau ,\text{where}\mathcal{L}\equiv {\oint }_{\gamma }\sqrt{g_{\mathbf{\Lambda }}^{\mu \nu }d{\Lambda }^{\mu }d{\Lambda }^{\nu }}$$

$$\mathcal{W}=-{\oint }_{\gamma }{\mathcal{A}}_{\mathbf{\Lambda }}^{\mu }d{\Lambda }^{\mu }\text{with}{\mathcal{A}}_{\mathbf{\Lambda }}^{\mu }\equiv {\partial }_{\mu }{\mathcal{F}}_{\mathbf{\Lambda }}^w{\Lambda }^w$$

$$(1-\eta )(\mathcal{W}/\mathcal{L}{)}^2\ge \mathcal{W}/\tau =P.$$

$$t=\tau {\int }_0^{{\varphi }_t}ds\sqrt{g_s^{\mu \nu }{\dot{\Lambda }}_s^{\nu }{\dot{\Lambda }}_s^{\mu }}/\mathcal{L}.$$

$${\eta }^{\ast }=1-{\mathcal{L}}^2/\mathcal{W}\tau .$$

$${\partial }_t{\rho }_t={\mathsf{L}}_t{\rho }_t\text{with}$$

$$\displaystyle\mathsf{L}_{\boldsymbol{\Lambda}}X\equiv-\frac{i}{\hbar}[H_{\lambda},X]+\sum\nolimits_{\sigma}\Bigl{(}[V^{\sigma}_{\boldsymbol{\Lambda}}X,V^{\sigma\dagger}_{\boldsymbol{\Lambda}}]+[V^{\sigma}_{\boldsymbol{\Lambda}},XV^{\sigma\dagger}_{\boldsymbol{\Lambda}}]\Bigr{)}.$$

$$R^{\mu\nu}_{\boldsymbol{\Lambda}}=-\frac{1}{T}\int_{0}^{\infty}\!\!\!dt\;\bigl{\llangle}\exp[\mathsf{K}_{\boldsymbol{\Lambda}}t]F^{\mu}_{\boldsymbol{\Lambda}}\bigr{|}F^{\nu}_{\boldsymbol{\Lambda}}\bigr{\rrangle}$$

$$\llangle X|Y\rrangle\equiv\int_{0}^{1}\!\!\!dx\;{{{\rm Tr}}}\bigl{[}\varrho_{\boldsymbol{\Lambda}}^{1-x}X\varrho_{\boldsymbol{\Lambda}}^{x}Y\bigr{]}-{{{\rm Tr}}}\bigl{[}\varrho_{\boldsymbol{\Lambda}}X\bigr{]}{{{\rm Tr}}}\bigl{[}\varrho_{\boldsymbol{\Lambda}}Y\bigr{]}$$

$$\mathsf{K}_{\boldsymbol{\Lambda}}X\equiv\frac{i}{\hbar}[H_{\lambda},X]+\sum\nolimits_{\sigma}\Bigl{(}V^{\sigma\dagger}_{\boldsymbol{\Lambda}}[X,V^{\sigma}_{\boldsymbol{\Lambda}}]+[V^{\sigma\dagger}_{\boldsymbol{\Lambda}},X]V^{\sigma}_{\boldsymbol{\Lambda}}\Bigr{)}.$$

$${\mathsf{K}}_{\mathbf{\Lambda }}[H_{\lambda },X]=[H_{\lambda },{\mathsf{K}}_{\mathbf{\Lambda }}X].$$

$$\mathsf{T}{\varrho }_{\mathbf{\Lambda }}{\mathsf{K}}_{\mathbf{\Lambda }}X={\mathsf{L}}_{\mathbf{\Lambda }}{\varrho }_{\mathbf{\Lambda }}\mathsf{T}X,$$

$$F_{\mathbf{\Lambda }}^w\equiv F_{\mathbf{\Lambda }}^d+i[H_{\lambda },G_{\lambda }]\equiv F_{\mathbf{\Lambda }}^d+F_{\mathbf{\Lambda }}^c.$$

$$R_{\mathbf{\Lambda }}^{\mu \nu }=D_{\mathbf{\Lambda }}^{\mu \nu }+{\delta }_{\mu w}{\delta }_{\nu w}{C}_{\mathbf{\Lambda }}^{ww},$$

$$\mathcal{L}\ge \sqrt{{\mathcal{L}}_d^2+{\mathcal{L}}_c^2}.$$

$${\mathcal{L}}_d\equiv {\oint }_{\gamma }\sqrt{-D_{\mathbf{\Lambda }}^{\mu \nu }d{\Lambda }^{\mu }d{\Lambda }^{\nu }}\text{and}{\mathcal{L}}_c\equiv {\oint }_{\gamma }\sqrt{-C_{\mathbf{\Lambda }}^{ww}d{\Lambda }^{w}d{\Lambda }^w},$$

$${\eta }^{\ast }\le 1-({\mathcal{L}}_d^2+{\mathcal{L}}_c^2)/\mathcal{W}\tau ,$$

$$H_{\lambda}=-\frac{\hbar\Omega}{2}\Bigl{(}\varepsilon\sigma_{x}+\sqrt{\lambda^{2}-\varepsilon^{2}}\sigma_{z}\Bigr{)}.$$

$$[H_{\lambda },V_{\mathbf{\Lambda }}^{\pm }]=\pm \hslash \Omega \lambda V_{\mathbf{\Lambda }}^{\pm },$$

$$\boldsymbol{\Lambda}_{t}=\bigl{(}\hbar\Omega(1+\sin^{2}[\pi\Omega t]),1+\sin^{2}[\pi\Omega t+\pi/4]\bigr{)}.$$

Full text

The laws of thermodynamics put fundamental limits on the performance of thermal machines across all length and energy scales. A prime example is the Carnot bound on efficiency, which applies to James Watt’s steam engine as well as to recent small-scale engines using colloidal particles Blickle and Bechinger (2011); Martínez et al. (2015a, b), single atoms Abah et al. (2012); Roßnagel et al. (2016) or engineered quantum systems Klatzow et al. (2017); Ronzani et al. (2018). Still, despite its universality, this bound is mostly of theoretical value as it can be attained only by infinitely slow cycles producing zero power. Practical devices, however, must operate in finite time and therefore are inevitably subject to frictional energy losses suppressing their efficiency. Hence, we are prompted to ask: how much performance has to be sacrificed for finite speed?

This question, which inspired the development of finite-time thermodynamics in the 1970s Andresen (2011), has recently attracted renewed interest: triggered by the observation that Carnot efficiency at finite power could indeed be possible in systems with broken time-reversal symmetry Benenti et al. (2011), a series of studies discovered quantitative trade-off relations, which limit the finite-time performance of microscopic engines and confirm the conventional expectation that dissipationless machines deliver zero power Brandner et al. (2015); Brandner and Seifert (2015); Proesmans and Van den Broeck (2015); Proesmans et al. (2016); Brandner and Seifert (2016); Shiraishi et al. (2016); Pietzonka and Seifert (2018); Shiraishi and Saito (2019).

These results rely on stochastic models to describe the internal dynamics of small-scale engines. Here, we pursue an alternative strategy that builds on the framework of thermodynamic geometry Ruppeiner (1995). This approach replaces the traditional thermodynamic picture, which mixes control and response variables, with a geometric picture. The properties of the working system are thereby encoded in a vector potential and a Riemannian metric in the space of control parameters, see Fig. 1. The driving protocols define a closed path in this space and can thus be assigned an effective flux and length. In adiabatic response, these quantities provide measures for the two key figures of merit: the work output and the minimal dissipation of the underlying thermodynamic process.

The idea of using geometric concepts to describe the thermodynamics of finite-time operations was originally conceived for macroscopic systems and developed mainly on the basis of phenomenological principles Andresen et al. (1984); Ruppeiner (1995); Andresen (2011). Over the last decades, this approach has been formulated on microscopic grounds Brody and Rivier (1995), linked to information-theoretic quantities Ruppeiner (1995) and extended to classical nano-scale systems Crooks (2007), closed quantum systems far from equilibrium Deffner and Lutz (2013) and, most recently, open quantum systems Scandi and Perarnau-Llobet (2018). Thermodynamic geometry has thus become a powerful tool, which, as its key application, provides an elegant way to determine optimal control protocols minimizing the dissipation of isothermal processes Zulkowski et al. (2012); Sivak and Crooks (2012); Rotskoff and Crooks (2015); Machta (2015); Scandi and Perarnau-Llobet (2018). Yet, this framework has neither been applied systematically to bound the performance of cyclic micro-engines nor to explore the impact of quantum effects on such devices.

To progress in this direction, we consider a general model for a microscopic heat engine consisting of two components: a working medium with tunable Hamiltonian $H_{\lambda}$ and a heat source to control the temperature $T$ of the environment of this system. The device is operated by periodically changing the parameters $\lambda$ and $T$ such that the vector

[TABLE]

passes through a closed path $\gamma:t\mapsto\boldsymbol{\Lambda}_{t}$. Once the system has settled to a periodic state $\rho_{t}$, the average output and input of this process, i.e., the mean generated work and the effective uptake of thermal energy from the heat source, are given by

[TABLE]

Here, dots indicate time derivatives, $\tau$ denotes the cycle time and the thermodynamic forces,

[TABLE]

correspond to the generalized pressure and the entropy of the working system, respectively. The upper panel of Fig. 1 shows a graphical illustration of this scheme. Note that we set Boltzmann’s constant to $1$ throughout.

Using the relations (2), the energy balance of the engine can be formulated as

[TABLE]

where $\mu=w,u$ and summation over identical indices is understood throughout. The quantity $A$ corresponds to the mean energy loss or dissipated availability Salamon and Berry (1983) per cycle; it must be non-negative as a direct consequence of the second law 111To this end, observe that the first law, $\dot{E}_{t}=J_{t}-P_{t}$, implies $-f^{\mu}_{t}\dot{\Lambda}^{\mu}_{t}=T_{t}\Sigma_{t}+\partial_{t}(E_{t}-T_{t}S_{t})$ and that $\Sigma_{t}\equiv\dot{S}_{t}-J_{t}/T_{t}\geq 0$ corresponds to the total rate of entropy production. Here, $E_{t}$ and $S_{t}=f^{u}_{t}$ denote the internal energy and entropy of the system, respectively, $J_{t}$ is the rate of heat uptake from the environment and $P_{t}$ the instantaneous power. . Thus, the dimensionless coefficient

[TABLE]

provides a proper measure for the efficiency of the engine. This figure is well-defined for any control protocols leading to positive work extraction, i.e., $W>0$. In the special case, where the temperature switches between two constant levels, it becomes an upper bound on the traditional thermodynamic efficiency, which involves only the heat uptake during the hot phase of the cycle 222 Specifically, we have $\eta_{{{\rm th}}}/\eta_{{{\rm C}}}\equiv W/Q\eta_{{{\rm C}}}\leq\eta$, where $Q$ denotes the heat uptake during the hot phase, $\eta_{{{\rm C}}}\equiv 1-T_{{{\rm c}}}/T_{{{\rm h}}}$ is the Carnot factor and $T_{{{\rm h}}}$ and $T_{{{\rm c}}}<T_{{{\rm h}}}$ correspond to the hot and the cold temperature..

Under quasi-static driving, the system follows its instantaneous equilibrium state, i.e., we have

[TABLE]

and $\mathcal{F}_{\boldsymbol{\Lambda}}$ denoting the Helmholtz free energy. The generalized forces (3) can then be expressed as

[TABLE]

where we have introduced the force operators

[TABLE]

for later purposes. Inserting (7) into (4) shows that the energy loss $A$ goes to zero in the quasi-static limit; the efficiency (5) thus attains its upper bound $1$. However, since the condition (6) can be met only for infinitely long cycle times, the generated power, $P\equiv W/\tau$, also vanishes and the engine becomes virtually useless.

Increasing the driving speed leads to finite power but inevitably also to energy losses reducing $\eta$. This trade-off can be understood quantitatively in the adiabatic response regime, where the external parameters change slowly compared to the relaxation time of the system. Under this condition, the thermodynamic forces (3) and the control rates $\dot{\boldsymbol{\Lambda}}_{t}$ are connected by the linear relations

[TABLE]

where $\nu=w,u$ and the adiabatic response coefficients R^{\mu\nu}_{\boldsymbol{\Lambda}\raisebox{-1.6pt}{{{\scriptsizet}}}} depend parametrically on the driving protocols $\boldsymbol{\Lambda}_{t}$ D’Alessio and Polkovnikov (2014). The average energy loss (4) thus becomes

[TABLE]

denoting the elements of a, possibly degenerate, metric tensor in the space of control parameters 333Note that the matrix $g^{\mu\nu}_{\boldsymbol{\Lambda}}$ must be positive semi-definite, since the second law requires $A\geq 0$ for any closed path $\gamma$ and any parameterization.. Thus, the Cauchy-Schwarz inequality implies

[TABLE]

corresponds to the thermodynamic length of the path $\gamma$.

Expanding the efficiency (5) to second order in the driving rates $\dot{\boldsymbol{\Lambda}}_{t}$ yields $\eta=1-A/\mathcal{W}$, where the adiabatic work can be expressed as a line integral,

[TABLE]

being the thermodynamic vector potential. Upon using (11), we thus arrive at our first main result, the power-efficiency trade-off relation

[TABLE]

This bound implies that the power of any cyclic heat engine covered by our model must vanish at least linearly as its efficiency approaches the ideal value $1$. The maximal slope of this decay is determined by the thermodynamic mean force $\mathcal{W}/\mathcal{L}$, where $\mathcal{L}$ and $\mathcal{W}$ are geometric quantities, i.e., they are independent of the parameterization of the control path $\gamma$, see the lower panel of Fig. 1.

Moreover, (13) entails a universal optimization principle, which arises from the observation that the bound (11) becomes an equality if the path $\gamma$ is parameterized in terms of its thermodynamic length. To this end, $t$ has to be replaced with the speed function $\phi_{t}$, which is implicitly defined through the relation

[TABLE]

Since $\mathcal{W}$ is not affected by this transformation, the bound (13) can be saturated for any given control path $\gamma$ by choosing this optimal parameterization. The efficiency then attains its geometric maximum

[TABLE]

Holding for any thermodynamically consistent microdynamics, our general analysis so far applies to classical and quantum heat engines alike. To explore the fundamental differences between these two regimes, we now model the time evolution of the working medium explicitly using the well-established adiabatic Lindblad approach. This scheme rests on the assumption that the modulations of the system Hamiltonian and the rate at which the external heat source provides thermal energy are both slow compared to the relaxation time of the environment. Applying this condition together with the standard weak-coupling approximation and a coarse-graining in time to wipe out memory effects and fast oscillations yields the Markovian master equation

[TABLE]

Here, $\hbar$ denotes Planck’s constant and the Lindblad generator $\mathsf{L}_{\boldsymbol{\Lambda}}$ depends parametrically on the driving protocols $\boldsymbol{\Lambda}_{t}$, for details see Alicki (1979); Albash et al. (2012); Majenz et al. (2013); Brandner and Seifert (2016). Using (16), the periodic state $\rho_{t}$ can be determined by means of an adiabatic perturbation theory Cavina et al. (2017); Weinberg et al. (2017).

This procedure, which we outline in SM , yields the Green-Kubo type expression

[TABLE]

for the adiabatic response coefficients, where the canonical correlation function is defined as

[TABLE]

for arbitrary observables $X$ and $Y$; the force operators $F^{\mu}_{\boldsymbol{\Lambda}}$ were introduced in (8) and $\mathsf{K}_{\boldsymbol{\Lambda}}$ denotes the adjoint Lindblad generator, which is defined by the relation {{{\rm Tr}}}\bigl{[}X\mathsf{K}_{\boldsymbol{\Lambda}}Y\bigr{]}\equiv{{{\rm Tr}}}\bigl{[}Y\mathsf{L}_{\boldsymbol{\Lambda}}X\bigr{]} 444Specifically, the adjoint generator reads

\mathsf{K}_{\boldsymbol{\Lambda}}X\equiv\frac{i}{\hbar}[H_{\lambda},X]+\sum\nolimits_{\sigma}\Bigl{(}V^{\sigma\dagger}_{\boldsymbol{\Lambda}}[X,V^{\sigma}_{\boldsymbol{\Lambda}}]+[V^{\sigma\dagger}_{\boldsymbol{\Lambda}},X]V^{\sigma}_{\boldsymbol{\Lambda}}\Bigr{)}.

.

This super operator is subject to three general consistency requirements. First, since we now work on a coarse-grained time scale, where coherent oscillations have been averaged out, the operators $V^{\sigma}_{\boldsymbol{\Lambda}}$ can only induce jumps between the energy levels of the working system Majenz et al. (2013). Hence, the eigenstates of $H_{\lambda}$ form the preferred basis of the dynamics and $\mathsf{K}_{{{\boldsymbol{\Lambda}}}}$ obeys the invariance condition

[TABLE]

Second, owing to microreversibility, the generators $\mathsf{K}_{\boldsymbol{\Lambda}}$ and $\mathsf{L}_{\boldsymbol{\Lambda}}$ are connected by symmetry relation Brandner and Seifert (2016)

[TABLE]

where the super operator $\mathsf{T}$ induces time reversal Brandner and Seifert (2016) and we assume that no magnetic field is applied to the system, i.e., $\mathsf{T}H_{\lambda}=H_{\lambda}$ and $\mathsf{T}V^{\sigma}_{\boldsymbol{\Lambda}}=V^{\sigma}_{\boldsymbol{\Lambda}}$. Together with (19), this property implies the adiabatic reciprocity relation $R^{\mu\nu}_{\boldsymbol{\Lambda}}=R^{\nu\mu}_{\boldsymbol{\Lambda}}$, which resembles the familiar Onsager symmetry of linear irreversible thermodynamics Onsager (1931a, b). Third, as a technical requirement, we understand that the jump operators $V^{\sigma}_{\boldsymbol{\Lambda}}$ form a self-adjoint and irreducible set; this condition ensures that, for $\boldsymbol{\Lambda}$ fixed, the mean of any observable relaxes to its unique equilibrium value under the dynamics generated by $\mathsf{K}_{\boldsymbol{\Lambda}}$ in the Heisenberg picture Spohn (1977). The expression (17) is then well-defined over the entire space of control parameters.

We are now ready to analyze the impact of quantum effects on slowly driven heat engines from a geometric perspective. To this end, we first divide the mechanical force operator into a diagonal and a coherent part,

[TABLE]

Here, $F^{d}_{\boldsymbol{\Lambda}}$ commutes with $H_{\lambda}$ and $G_{\lambda}$ corresponds to an adiabatic gauge potential Weinberg et al. (2017). Upon inserting this decomposition into (17), the adiabatic response coefficients decay into two components,

[TABLE]

where $D^{\mu\nu}_{\boldsymbol{\Lambda}}$ and $C_{\boldsymbol{\Lambda}}^{ww}$ are given by the formula (17) with $F^{w}_{\boldsymbol{\Lambda}}$ replaced by $F^{d}_{\boldsymbol{\Lambda}}$ and $F^{c}_{\boldsymbol{\Lambda}}$, respectively; the cross-terms between $F^{c}_{\boldsymbol{\Lambda}}$ and the diagonal operators $F^{d}_{\boldsymbol{\Lambda}}$ and $F^{u}_{\boldsymbol{\Lambda}}$ vanish due to the property (19) of the adjoint generator. Next, by plugging (22) into the definition (11) of the thermodynamic length and using the concavity of the square-root function, we arrive at the bound 555To derive (23), observe that $\mathcal{L}\geq\sqrt{\alpha}\mathcal{L}_{d}+\sqrt{1-\alpha}\mathcal{L}_{c}$ for $0\leq\alpha\leq 1$ by the concavity of the square root function. Maximizing the right-hand side of this inequality with respect to $\alpha$ yields the desired result.

[TABLE]

The two quantities on the right, which are defined as

[TABLE]

thereby describe two genuinely different types of energy losses: the reduced thermodynamic length $\mathcal{L}_{d}$ accounts for the dissipation of heat in the environment and the quantum correction $\mathcal{L}_{c}$ arises from the decay of superpositions between the energy levels of the working system, a mechanism known as quantum friction Kosloff and Feldmann (2002); Feldmann and Kosloff (2003, 2004, 2006, 2012).

The constraint (23) puts an upper limit on the optimal finite-time efficiency (15). This bound,

[TABLE]

which is our second main result, is saturated in the quasi-classical limit, where $F^{c}_{\boldsymbol{\Lambda}}=0$; the energy eigenstates of the system are then time-independent and the periodic state $\rho_{t}$ is diagonal in this basis throughout the cycle. In fact, since the adiabatic work $\mathcal{W}$ is independent of $F^{c}_{\boldsymbol{\Lambda}}$, the bound (24) shows that injecting coherence into the working system can only reduce the maximum efficiency of the engine at given power. These coherence-induced performance losses are a universal feature of the slow-driving regime, where superpositions between different energy levels are irreversibly destroyed by the environment before their work content can be extracted through mechanical operations. While similar conclusions were drawn before for specific models Kosloff and Feldmann (2002); Feldmann and Kosloff (2003, 2004, 2006, 2012) and small driving amplitudes Brandner and Seifert (2016); Brandner et al. (2017), our new bound (24) applies to any heat engine that is covered by Lindblad dynamics and operated in adiabatic response. Thus, it further corroborates the emerging picture that quantum effects can enhance the performance of thermal machines only far from equilibrium Uzdin et al. (2015); Uzdin (2016); Brandner et al. (2017).

We will now show how our general results can be applied to practical devices. To this end, we consider a simple model for a solid-state quantum heat engine that is inspired by a recent experiment Ronzani et al. (2018). The working system consists of a superconducting qubit with Hamiltonian

[TABLE]

Here, $\sigma_{x}$ and $\sigma_{z}$ are the usual Pauli matrices, $\hbar\Omega$ denotes the overall energy scale and the dimensionless parameters $\varepsilon\geq 0$ and $\lambda\geq\varepsilon$ correspond to the tunneling energy and the flux-tunable level-splitting of the qubit Niskanen et al. (2007); Karimi and Pekola (2016). The role the environment is played by a normal-metal island, whose temperature can be accurately controlled with established techniques Giazotto et al. (2006) and monitored by means of sensitive electron thermometers, a technology that could soon enable calorimetric work measurements Campisi et al. (2015); Viisanen et al. (2015); Gasparinetti et al. (2015); Kupiainen et al. (2016); Donvil et al. (2018); Wang et al. (2018). This reservoir can be described in terms of two jump operators, $V^{+}_{\boldsymbol{\Lambda}}$ and $V^{-}_{\boldsymbol{\Lambda}}$, defined by the conditions

[TABLE]

where $\Gamma$ determines the average jump frequency.

We proceed in three steps. First, we evaluate the adiabatic response coefficients for the single-qubit engine using the formula (17). Second, we calculate the geometric quantities entering the bounds (13) and (24) and the optimal speed function $\phi_{t}$ defined in (14). For simplicity, we thereby assume that the device is driven by harmonic temperature and energy modulations, i.e., we set

[TABLE]

Hence, the control path $\gamma$ is a circle in the $\Lambda^{u}-\Lambda^{w}$ plane. Third, in order to assess the quality of our bounds, we determine the periodic state $\rho_{t}$ of the system exactly by solving the time-inhomogeneous master equation (16) for both constant and optimal driving speed. Using the expressions (2) and (3), the power and the efficiency of the engine can thus be obtained for any cycle time $\tau$.

The results of this analysis are summarized in Fig. 2, for details see SM . We find that, for optimal driving speed, our bound (13) is practically attained in the range $\eta\gtrsim 0.8$, which corresponds to $\tau\gtrsim 2/\Omega$. The optimal protocols \boldsymbol{\Lambda}^{\ast}_{t}\equiv\boldsymbol{\Lambda}_{\phi\raisebox{-1.6pt}{{{\scriptsizet}}}} thereby outperform the harmonic profiles (26) by roughly a factor $1.2$ in power at given efficiency. Remarkably, this increase in performance persists even for $\eta<0.8$, i.e., for short cycle times $\tau<2/\Omega$, which are not covered by the slow-driving approximation (9). This phenomenon, whose degree of universality is yet to be established, raises the appealing perspective that it might be possible to extend our geometric description of microscopic heat engines beyond the limits of adiabatic response.

The lower panel of Fig. 2 shows that the single-qubit engine operates most efficiently in the quasi-classical configuration $\varepsilon=0$. For this setting, the eigenstates of the Hamiltonian (25) are independent of $\lambda$ and our bounds (23) and (24) are saturated. Raising the value of $\varepsilon$ leads to increasing quantum friction. Hence, the thermodynamic length grows and the optimal efficiency drops, whereby both figures closely follow their upper and lower bound, respectively. This behavior underlines our general result that coherence only reduces the efficiency of thermodynamic cycles in adiabatic response. Although this conclusion does not extend to the fast-driving regime, it still provides a valuable guideline for future theoretical and experimental studies seeking new strategies to gain a quantum advantage in the design of thermal machines.