Universal Coherence-Induced Power Losses of Quantum Heat Engines in Linear Response

TL;DR

This paper presents a universal framework for analyzing power losses in quantum heat engines caused by quantum coherence, deriving bounds and demonstrating that coherence can lead to power reduction in the linear-response regime.

Contribution

It introduces a universal scheme to separate classical and quantum coherence contributions to power, deriving bounds and showing coherence-induced power losses in linear response.

Findings

Quantum coherence can cause power losses in heat engines.

Derived bounds on coherent and total power outputs.

Single-qubit engine example illustrates the theory.

Abstract

We introduce a universal scheme to divide the power output of a periodically driven quantum heat engine into a classical contribution and one stemming solely from quantum coherence. Specializing to Lindblad-dynamics and small driving amplitudes, we derive general upper bounds on both, the coherent and the total power. These constraints imply that, in the linear-response regime, coherence inevitably leads to power losses. To illustrate our general analysis, we explicitly work out the experimentally relevant example of a single-qubit engine.

Equations

$$T_t\equiv T+f_t^q,$$

$$H_t=H+f_t^w{G}^w.$$

$$P=-\frac{1}{\mathcal{T}}{\int }_0^{\mathcal{T}}dt\mathrm{tr}\left\{{\dot{H}}_t{\varrho }_t\right\},$$

$$H_t\equiv \sum _n{E}_t^n|n_t⟩⟨n_t|$$

$$P^{\mathrm{d}}\equiv -\frac{1}{\mathcal{T}}{\int }_0^{\mathcal{T}}dt\sum _n{\dot{E}}_t^n⟨n_t|{\varrho }_t|n_t⟩$$

$$P^{\mathrm{c}}\equiv P-P^{\mathrm{d}}=\frac{1}{\mathcal{T}}{\int }_0^{\mathcal{T}}dt\sum _n⟨{\dot{n}}_t|[H_t,{\varrho }_t]|n_t⟩$$

$${\partial }_t{\varrho }_t=-\frac{i}{\hslash }[H,{\varrho }_t]+\mathsf{D}{\varrho }_t,$$

$$\mathsf{D}X\equiv \sum _{\sigma }\frac{{\gamma }_{\sigma }}{2}\left([V_{\sigma }X,V_{\sigma }^{†}]+[V_{\sigma },X{V}_{\sigma }^{†}]\right)$$

$$\mathsf{D}{e}^{-\beta H}=e^{-\beta H}{\mathsf{D}}^{†}.$$

$${\mathsf{D}}^{†}X\equiv \sum _{\sigma }\frac{{\gamma }_{\sigma }}{2}\left(V_{\sigma }^{†}[X,V_{\sigma }]+[V_{\sigma }^{†},X]V_{\sigma }\right).$$

$$P^{\mathrm{d}}$$

$$P^{\mathrm{c}}$$

$$C^{ab}_{t}\equiv\bigl{\llangle}\hat{G}^{a}_{t},\hat{G}^{b}_{0}\bigr{\rrangle}\equiv\int_{0}^{\beta}\!\!\!d\lambda\;\left(\left\langle\hat{G}^{a}_{t}e^{-\lambda H}\hat{G}^{b}_{0}e^{\lambda H}\right\rangle-\left\langle\hat{G}^{a}_{t}\right\rangle\left\langle\hat{G}^{b}_{0}\right\rangle\right),$$

$${\partial }_t{\hat{X}}_t=\frac{i}{\hslash }[H,{\hat{X}}_t]+{\mathsf{D}}^{†}{\hat{X}}_t$$

$$⟨X⟩\equiv \mathrm{tr}\left\{X{e}^{-\beta H}\right\}/\mathrm{tr}\left\{e^{-\beta H}\right\}.$$

$$G^{\mathrm{d}}\equiv \sum _n|n⟩⟨n|G^w|n⟩⟨n|\text{and}{G}^{\mathrm{c}}\equiv G^w-G^{\mathrm{d}},$$

$$\begin{gathered} P^{\mathrm{c}} \\ \le -\frac{L_1^{\mathrm{c}}{\Omega }^2}{{\Omega }^2+L_2^{\mathrm{c}}/L_1^{\mathrm{c}}}{F}^w\le 0, \end{gathered}$$

$$F^w\equiv \frac{1}{\mathcal{T}}{\int }_0^{\mathcal{T}}dt(f_t^w{)}^2$$

$$L^{{{\rm c}}}_{j}\equiv\int_{0}^{\infty}\!\!\!dt\;\bigl{\llangle}\hat{G}_{t}^{{{\rm c}}(j)},\hat{G}_{0}^{{{\rm c}(j)}}\bigr{\rrangle}\geq 0,$$

$$P\le \frac{L_1^q{F}^q}{4(1+{\psi }_{\Omega })},\text{where}{\psi }_{\Omega }\equiv \frac{(L_1^{\mathrm{c}}/L_1^{\mathrm{d}}){\Omega }^2}{{\Omega }^2+L_2^{\mathrm{c}}/L_1^{\mathrm{c}}}\ge 0$$

$$F^q\equiv \frac{1}{\mathcal{T}}{\int }_0^{\mathcal{T}}dt(f_t^q-{\bar{f}}^q{)}^2\text{with}{\bar{f}}^q\equiv \frac{1}{\mathcal{T}}{\int }_0^{\mathcal{T}}dt{f}_t^q$$

$$P\le L_1^q{F}^q/4.$$

$$G^w=-\mu H/T\text{and}{\mathsf{D}}^{†}H=-\lambda (H-⟨H⟩)$$

$$\bigl{\llangle}\hat{H}_{t},\hat{H}_{0}\bigr{\rrangle}=e^{-\lambda t}\bigl{\llangle}\hat{H}_{0},\hat{H}_{0}\bigr{\rrangle}.$$

$$2{\dot{f}}_t^w=\lambda (f_t^q-{\bar{f}}^q)/\mu -{\dot{f}}_t^q/\mu ,$$

$$\frac{L_1^q{F}^q}{4}=\lambda \frac{⟨H^2⟩-{⟨H⟩}^2}{4k_{\mathrm{B}}{T}^3}{F}^q$$

$$H_t=\frac{\hslash \omega }{2}{\sigma }_z+\frac{\hslash \omega f_t^w}{2}\left(r{\sigma }_z+(1-r){\sigma }_x\right).$$

$$P^{\mathrm{c}}\le -\frac{\hslash \omega \lambda }{2}{r}^{2}g{\psi }_{\Omega }{F}^w\text{and}P\le \frac{\hslash \omega \lambda }{8}\frac{g}{1+{\psi }_{\Omega }}\frac{F^q}{T^2}$$

$$\text{with}{\psi }_{\Omega }=\frac{(1-r{)}^2}{r^2}\frac{\sinh 2\kappa }{4\kappa }\frac{{\Omega }^2}{{\Omega }^2+{\omega }^2+{\lambda }^2/4},$$

$$2{\dot{f}}_t^w=-\Omega (f_t^q-{\bar{f}}^q)/T+{\dot{f}}_t^q/T,$$

Full text

Heat engines are devices that convert thermal energy into useful work. A Stirling motor, for example, uses the varying pressure of a periodically heated gas to produce mechanical motion, Fig. LABEL:FigEnConva. Used by macroscopic engines for two centuries, this elementary operation principle has now been implemented on ever-smaller scales. Over the last decade, a series of experiments has shown that the working fluid of Stirling-type engines can be reduced to tiny objects such as a micrometer-seized silicon spring Steeneken et al. (2010) or a single colloidal particle Blickle and Bechinger (2011); Martínez et al. (2015a, b); Krishnamurthy et al. (2016). These efforts recently culminated in the realization of a single-atom heat engine Roßnagel et al. (2015); Abah et al. (2012). Thus, the dimensions of the working fluid were further decreased by four orders of magnitude within only a few years. In light of this remarkable development, the challenge of even smaller engines operating on time and energy scales comparable to Planck’s constant appears realistic for future experiments.

Such quantum engines would have access to a non-classical mechanism of energy conversion that relies on the creation of coherent superpositions between the energy levels of the working fluid Uzdin et al. (2015), Fig. LABEL:FigEnConvb. How does this additional freedom affect performance figures like power and efficiency? Having triggered substantial research efforts in recent years, this question constitutes one of the central problems in the emerging field of quantum thermodynamics, see for example Quan et al. (2007); Funo et al. (2013); Roßnagel et al. (2014); Pekola (2015); Zhou et al. (2015); Lostaglio et al. (2015); Ćwikliński et al. (2015); Gardas and Deffner (2015); Tasaki (2016); Jaramillo et al. (2016). However, the available results are so far inconclusive. In fact, current evidence suggests that, depending on the specific setup and benchmark parameters, coherence can, in principle, be both conducive Scully et al. (2011); Abe and Okuyama (2012); Correa et al. (2014); Horowitz and Jacobs (2014); Roßnagel et al. (2014); Brandner et al. (2015a); Mitchison et al. (2015); Gelbwaser-Klimovsky et al. (2015); Uzdin et al. (2015); Uzdin (2016); Watanabe et al. (2017) and detrimental Feldmann and Kosloff (2003, 2004); Brandner and Seifert (2016); Roulet et al. (2016); Karimi and Pekola (2016) to the performance of thermal devices.

In this article, we universally characterize the role of coherence for the power output of cyclic heat engines in linear response. Our analysis builds on the well-established theory of open quantum systems Alicki (1979); Breuer and Petruccione (2006) and a recently developed thermodynamic framework describing periodically driven systems Brandner et al. (2015b); Brandner and Seifert (2016), which has already proven very useful in the classical realm Proesmans and Van den Broeck (2015); Bauer et al. (2016); Proesmans et al. (2016a, b). For a quantum engine, we model the working fluid as an $N$-level system with bare Hamiltonian $H$, which is embedded in a large reservoir with base temperature $T$ Kosloff (2013); Brandner and Seifert (2016). For simplicity, we assume that $N$ is finite. A heat source injects thermal energy into the system by periodically heating its local environment. Hence, the working fluid effectively feels the time-dependent temperature

[TABLE]

where $f^{q}_{t}\geq 0$. For work extraction, a periodic driving field $f^{w}_{t}$ is applied, which couples linearly to the degree of freedom $G_{w}$ of the system. The Hamiltonian thus acquires the time-dependence

[TABLE]

For uniqueness, we assume that the field $f^{w}_{t}$ is dimensionless and that its average over one period $\mathcal{T}$ vanishes.

This engine delivers the mean power output

[TABLE]

where $\varrho_{t}$ denotes the periodic state of the system Alicki (1979). Using the spectral decomposition

[TABLE]

of the time-dependent Hamiltonian, $P$ can be divided into two contributions corresponding to the different mechanisms of work extraction illustrated in Fig. LABEL:FigEnConv. First, the classical power

[TABLE]

is generated by changing the energy levels of the working fluid, i.e., the diagonal elements of its Hamiltonian with respect to the unperturbed energy eigenstates. Second, the coherent power

[TABLE]

arises from creating superpositions between the instantaneous energy eigenstates 111To make these definitions unique, we understand that the time-dependent energy eigenvalues are arranged in increasing order, i.e., $E^{0}_{t}\leq E^{1}_{t}\leq\cdots\leq E^{N}_{t}$ for any $t\in\mathbb{R}$. Furthermore, we note that the expressions (5) and (6) are unaltered if the instantaneous energy eigenstates are multiplied with arbitrary time-dependent phase factors.. Accordingly, $P^{{{\rm c}}}$ vanishes when $\varrho_{t}$ commutes with $H_{t}$ throughout one operation cycle. This condition is met, for example, in the adiabatic limit, where the state of the system follows the instantaneous Boltzmann distribution.

For deriving constraints on the coherent power $P^{{{\rm c}}}$, we have to specify the dissipative dynamics of the working fluid. To this end, we invoke the standard condition of weak coupling between system and reservoir. In equilibrium, i.e., for $f^{q}_{t}=f^{w}_{t}=0$, the state $\varrho_{t}$ evolves according to the Markovian master equation Breuer and Petruccione (2006)

[TABLE]

where the dissipator

[TABLE]

accounts for the influence of the thermal environment Spohn and Lebowitz (1978). Furthermore, $\hbar$ denotes Planck’s constant and $\{\gamma_{\sigma}\}$ is a set of positive rates with corresponding Lindblad operators $\{V_{\sigma}\}$. Due to microreversibility, these quantities are constrained by the quantum detailed balance relation, which can be expressed compactly in terms of the formal identity Alicki (1976); Spohn and Lebowitz (1978)

[TABLE]

Here, $\beta\equiv 1/(k_{{{\rm B}}}T)$, $k_{{{\rm B}}}$ denotes Boltzmann’s constant and the adjoint dissipator is given by

[TABLE]

Provided that the cycle period $\mathcal{T}$ is large compared to the relaxation time of the reservoir, finite driving can be included in this framework by allowing the rates and Lindblad operators to be time-dependent and replacing $H$ and $T$ with $H_{t}$ and $T_{t}$ respectively in (7)-(10) Alicki (1979). Solving the master equation (7) by treating $f^{q}_{t}$ and $f^{w}_{t}$ as first-order perturbations then yields the explicit expressions 222To obtain (Universal Coherence-Induced Power Losses of Quantum Heat Engines in Linear Response) from (5) and (6), standard linear-response theory is applied. This derivation exploits that, due to the detailed balance relation (9), the space of all system operators commuting with the unperturbed Hamiltonian $H$ is invariant under the action of the super operators $\mathsf{D}$ and $\mathsf{D}^{\dagger}$, for details see Brandner and Seifert (2016).

[TABLE]

for the classical and the coherent power, respectively 333The detailed-balance relation (9) implies that the set of Lindblad operators $\{V_{\sigma}\}$ is self-adjoint Alicki (1976); Kossakowski et al. (1977). Additionally, we here assume that this set is irreducible such that $X=1$ is the only solution of $\mathsf{D}^{\dagger}X=0$ Spohn (1977). Under this condition, the improper integrals showing up in (Universal Coherence-Induced Power Losses of Quantum Heat Engines in Linear Response) are well-defined Brandner and Seifert (2016), in the following notation. We abbreviate with $C^{ab}_{t}$ the Kubo correlation function Kubo et al. (1998)

[TABLE]

where $t\geq 0$, $a,b={{\rm d}},{{\rm c}},q$. Hats indicate Heisenberg-picture operators satisfying the adjoint master equation

[TABLE]

with initial condition $\hat{X}_{0}=X$ Breuer and Petruccione (2006). The angular brackets in (12) denote the thermal average, i.e.,

[TABLE]

Finally, we have defined the operator $G^{q}\equiv-H/T$ and split the control variable $G^{w}$ into a diagonal, quasi-classical, and a coherent part,

[TABLE]

where the vectors $\left|n\right\rangle$ correspond to the eigenstates of the unperturbed Hamiltonian $H$.

As a first key-observation, we note that the expression (Universal Coherence-Induced Power Losses of Quantum Heat Engines in Linear Response) for $P^{{{\rm c}}}$ is independent of the temperature profile $f^{q}_{t}$. Thus, under linear-response conditions, it is impossible to convert thermal energy provided by the heat source into positive power output via quantum coherence; rather coherent power can only be injected into the system through mechanical driving. This constraint is captured quantitatively by the bound

[TABLE]

which is saturated in the two limits $\Omega\rightarrow 0$ and $\Omega\rightarrow\infty$, for the proof see SM . Besides the cycle frequency $\Omega\equiv 2\pi/\mathcal{T}$, the bound (16) involves the mean square amplitude

[TABLE]

of the driving field, and the Green-Kubo type coefficients

[TABLE]

where the index $j$ in brackets means a time-derivative of respective order.

The bound (16) can be understood intuitively by identifying the parameter $L^{{{\rm c}}}_{2}/L^{{{\rm c}}}_{1}$ as an estimator for the decoherence strength of the reservoir, i.e., the square of the mean rate, at which its influence destroys coherent superpositions between the energy levels of the working fluid. In the incoherent limit $L^{{{\rm c}}}_{2}/L^{{{\rm c}}}_{1}\gg\Omega^{2}$, the coherent power can approach zero due to frequent interactions with the environment constantly forcing the system into a state that is diagonal in the instantaneous energy eigenbasis. This behavior resembles the quantum Zeno effect with the role of the observer played by the thermal reservoir Breuer and Petruccione (2006). If $L^{{{\rm c}}}_{2}/L^{{{\rm c}}}_{1}\ll\Omega^{2}$, the bath-induced decoherence is slow compared to the external driving. In this limit, coherences can be fully established such that maximal coherent power is injected into the system. Accordingly, the upper bound (16) reduces to $P^{{{\rm c}}}\leq-L^{{{\rm c}}}_{1}F^{w}$, its minimum with respect to $\Omega$.

The coefficients (18) vanish if and only if $G^{c}=0$, which means that the control variable $G^{w}$ commutes with the unperturbed Hamiltonian $H$. Thus, according to (16), any non-classical driving will inevitably reduce the net output $P=P^{{{\rm d}}}+P^{{{\rm c}}}$ of the engine. In fact, $P$ is subject to the upper bound

[TABLE]

provides a measure for the relative strength of coherent and classical driving and

[TABLE]

corresponds to the mean square magnitude of the local temperature variation induced by the heat source. This bound is proven in SM . As the bound (16), it involves a set of protocol-independent parameters $L^{a}_{j}$, which are reminiscent of linear transport coefficients. For $a={{{\rm d}}}$ and $a=q$, these quantities are defined analogously to (18) with $G^{{{\rm c}}}$ replaced by $G^{{{\rm d}}}$ and $G^{q}$, respectively.

In the special case of purely coherent driving, $G^{{{\rm d}}}=0$, the coefficient $L^{{{\rm d}}}_{1}$ vanishes. The coherence parameter $\psi_{\Omega}$ then diverges and (19) reduces to $P\leq 0$. Consequently in line with our analysis above, no cyclic engine relying only on coherent work extraction can properly operate in the linear-response regime. For $G^{{{\rm c}}}=0$, i.e., quasi-classical driving, $\psi_{\Omega}$ vanishes and the constraint (19) assumes its weakest form

[TABLE]

This bound can be saturated if and only if

[TABLE]

for some real scalars $\mu$ and $\lambda>0$, see SM . Thus, the control field $f^{w}_{t}$ has to couple directly to the free Hamiltonian $H$, and the energy correlation function must decay exponentially with rate $\lambda$, i.e.,

[TABLE]

If these two requirements are fulfilled, as we show in SM , the protocol for optimal power extraction is determined by the condition

[TABLE]

which leads to $P=L^{q}_{1}F_{q}/4$ for any temperature profile $f^{q}_{t}$ and sufficiently short operation cycles 444For long cycles, the reduced temperature profile $f^{q}_{t}-\bar{f}^{q}$ would oscillate slowly and thus be either positive or negative over substantial time ranges. Consequently, integrating (24) would yield a driving protocol $f^{w}_{t}$ with large amplitude, which violates the linear-response condition underlying the derivations leading to (21) and (24).. Furthermore, using relation (23), the upper bound (21) can be expressed in a physically transparent way. Specifically, we obtain

[TABLE]

by evaluating (18). Hence, the strength and the decay rate of the energy fluctuations in equilibrium essentially determine the maximum power output of a cyclic $N$-level engine in the linear-response regime. A similar result was obtained only recently for classical machines obeying Fokker-Planck type dynamics Bauer et al. (2016); Brandner et al. (2015b).

We will now explore the quality of our general bounds under practical conditions. To this end, we consider a two-level engine with time-dependent Hamiltonian

[TABLE]

Here, $\sigma_{x,y,z}$ are the usual Pauli matrices and the dimensionless parameter $0\leq r\leq 1$ determines the relative weight of the classical and the coherent parts, $G^{{{\rm d}}}=r(\hbar\omega/2)\sigma_{z}$ and $G^{{{\rm c}}}=(1-r)(\hbar\omega/2)\sigma_{x}$, of the control variable $G^{w}$. The corresponding equilibrium dissipator (8) involves two Lindblad operators, $V_{\pm}=(\sigma_{x}\pm i\sigma_{y})/2$, acting at the rates $\gamma_{\pm}\equiv\gamma e^{\mp\kappa}$, respectively, where $\kappa\equiv\hbar\omega\beta/2$. This setup lies within the range of forthcoming experiments using a superconducting qubit to realize the system and ultra fast electron thermometers for calorimetric work measurements Gasparinetti et al. (2015); Viisanen et al. (2015); Pekola et al. (2013). Its coherent and total power are subject to the bounds

[TABLE]

$g\equiv\kappa/\cosh^{2}\kappa$ and $\lambda\equiv 2\gamma\cosh\kappa$, which follow from (16) and (19) upon evaluation of the coefficients (18), see SM .

To assess the quality of these constraints, we choose a temperature profile $f^{q}_{t}$ that mimics the Stirling cycle illustrated in Fig. LABEL:FigEnConv and a work protocol satisfying

[TABLE]

both shown in Fig. LABEL:FigProtPowa. This choice renders the amplitude and shape of $f^{w}_{t}$ independent of the cycle frequency $\Omega$. In Fig. LABEL:FigProtPowb, the resulting coherent power is plotted as a function of $\Omega/\lambda$ for $r=1/2$. If the level splitting $\omega$ is significantly smaller than the dissipation rate $\lambda$, it decays monotonically while closely following its upper bound (27). With increasing $\omega$, a resonant dip emerges close to $\Omega=\omega$. This feature is not reproduced by our bound, which is, however, still saturated in the limits $\Omega/\lambda\rightarrow 0$ and $\Omega/\lambda\rightarrow\infty$. For $r=1$, the coherent power vanishes and the two conditions (22) are fulfilled with $\mu=-T$. The total power $P$ plotted in Fig. LABEL:FigProtPowc then reaches its upper bound (27) at $\Omega=\lambda$, i.e., when the work protocol (28) satisfies the maximum-power condition (24). As $r$ varies from $1$ to [math], the total power decreases more and more due to coherence-induced losses and the bound (27) lies well above the actual value of $P$ for any cycle frequency. This result underlines our general conclusion that coherence has a purely detrimental effect on power in the linear-response regime.

For a perspective beyond linear response, we stress that our key expressions (5) and (6) are valid for arbitrarily strong driving and any thermodynamically consistent time-evolution of the working fluid. The coherent power (6) thus constitutes a universal indicator for the impact of quantum effects on thermal power generation. It can therefore be used as a unifying performance benchmark across various different types of cyclic quantum machines. In particular, it would be applicable to rapidly driven Cuetara et al. (2015); Gelbwaser-Klimovsky et al. (2015); Alicki and Gelbwaser-Klimovsky (2015); Gelbwaser-Klimovsky and Kurizki (2014); Gelbwaser-Klimovsky et al. (2013); Kolář et al. (2012) and strongly coupled Esposito et al. (2015); Uzdin et al. (2016); Strasberg et al. (2016); Carrega et al. (2016); Newman et al. (2016) engines, which are currently subject to active investigations. Furthermore, the general framework introduced in this article could lead to a new perspective on a phenomenon earlier interpreted as a quantum analogue of classical friction, which was observed in models describing the working fluid as an interacting spin system Feldmann and Kosloff (2012, 2006, 2004, 2003); Kosloff and Feldmann (2002).

As one of the earliest quantum heat engines, the three-level maser relies solely on non-classical work extraction Geva and Kosloff (1994); Scovil and Schulz-DuBois (1959). This example, which does not admit a linear-response description Brandner and Seifert (2016), shows that the coherent power can indeed become positive if the driving is strong. Coupled to two reservoirs with time-independent temperature, the three-level maser works in a steady state with respect to a rotating basis of its Hilbert space. This operation principle is similar to the one used by thermoelectric nano devices, where a spatial temperature gradient drives an electric current Humphrey and Linke (2005). Extending the concept of coherent power to this second class of quantum engines, which has recently attracted remarkable interest Brandner et al. (2013); Brandner and Seifert (2013); Venturelli et al. (2013); Whitney (2014); Matthews et al. (2014); Bergenfeldt et al. (2014); Hofer et al. (2016); Sánchez et al. (2015); Brandner and Seifert (2015); Marchegiani et al. (2016); Samuelsson et al. (2016); Zheng et al. (2016) represents a challenge promising to reveal rich and interesting physics. Eventually, our approach could lead to a comprehensive understanding of the role of quantum effects for one of the most fundamental thermodynamic operations: the conversion of heat into power.