Quantum efficiency bound for continuous heat engines coupled to non-canonical reservoirs

TL;DR

This paper establishes a generalized efficiency bound for continuous quantum heat engines operating with squeezed thermal reservoirs, demonstrating potential efficiency amplification in the quantum regime through squeezing effects.

Contribution

It introduces a new efficiency bound for quantum heat engines with non-canonical reservoirs, extending beyond traditional thermodynamics and applicable in the quantum regime.

Findings

Efficiency can be greatly amplified by squeezing in the quantum regime.

Derived a universal maximum power efficiency bound from fluctuation relations.

Validated the theoretical bound with a prototype quantum photoelectric engine.

Abstract

We derive an efficiency bound for continuous quantum heat engines absorbing heat from squeezed thermal reservoirs. Our approach relies on a full-counting statistics description of nonequilibrium transport and it is not limited to the framework of irreversible thermodynamics. Our result, a generalized Carnot efficiency bound, is valid beyond the small squeezing and high temperature limit. Our findings are embodied in a prototype three-terminal quantum photoelectric engine where a qubit converts heat absorbed from a squeezed thermal reservoir into electrical power. We demonstrate that in the quantum regime the efficiency can be greatly amplified by squeezing. From the fluctuation relation we further receive other operational measures in linear response, for example, the universal maximum power efficiency bound.

Equations

$$\ln \left[\frac{P_t(\Delta S)}{P_t(-\Delta S)}\right]=\Delta S.$$

$$\mathcal{Z}(\lambda )\equiv \int d\Delta S{e}^{i\lambda \Delta S}{P}_t(\Delta S),$$

$$\mathcal{Z}({\lambda }_c,{\lambda }_e,{\lambda }_{ph})$$

$$=⟨e^{i{\lambda }_c{\hat{A}}_c+i{\lambda }_e{\hat{A}}_e+i{\lambda }_{ph}{\hat{A}}_{ph}}{e}^{-i{\lambda }_c{\hat{A}}_c(t)-i{\lambda }_e{\hat{A}}_e(t)-i{\lambda }_{ph}{\hat{A}}_{ph}(t)}⟩.$$

$$\frac{P_t(N,E_e,Q_{ph})}{P_t(-N,-E_e,-Q_{ph})}=e^{{\beta }_{el}\Delta \mu N+({\beta }_{el}-{\beta }_{ph})Q_{ph}}.$$

$$\mathcal{Z}({\lambda }_c,{\lambda }_e,{\lambda }_{ph})$$

$$\displaystyle=\mathcal{Z}\big{(}\!-\lambda_{c}\!+i\beta_{el}(\mu_{R}\!-\mu_{L}),-\lambda_{e},\!-\lambda_{ph}\!-i(\beta_{ph}\!-\!\beta_{el})\big{)}$$

$$1=⟨e^{-{\beta }_{el}\Delta \mu N+({\beta }_{ph}-{\beta }_{el})Q_{ph}}⟩.$$

$$⟨\eta ⟩\le \frac{{\beta }_{el}-{\beta }_{ph}}{{\beta }_{el}}=1-\frac{T_{el}}{T_{ph}}.$$

$$\frac{k_d^{ph}}{k_u^{ph}}=\frac{N({\omega }_0)+1}{N({\omega }_0)}.$$

$${\beta }_{eff}({\beta }_{ph},r,{\omega }_0)=\frac{1}{\hslash {\omega }_0}\ln \frac{1+N({\omega }_0)}{N({\omega }_0)}.$$

$${\beta }_{eff}={\beta }_{ph}+\frac{1}{\hslash {\omega }_0}\ln \left[\frac{1+(1+e^{-{\beta }_{ph}\hslash {\omega }_0}){\sinh }^{2}r}{1+(1+e^{{\beta }_{ph}\hslash {\omega }_0}){\sinh }^{2}r}\right].$$

$${\beta }_{eff}\to \frac{{\beta }_{ph}}{1+2{\sinh }^{2}r},$$

$$\mathcal{Z}({\lambda }_c,{\lambda }_e,{\lambda }_{ph})$$

$$=\mathcal{Z}(-{\lambda }_c+i{\beta }_{el}({\mu }_R-{\mu }_L),-{\lambda }_e,-{\lambda }_{ph}-i({\beta }_{eff}-{\beta }_{el})).$$

$$⟨\eta ⟩\le 1-\frac{{\beta }_{eff}}{{\beta }_{el}}.$$

$$⟨\eta ⟩\le 1-\frac{T_{el}}{T_{ph}}+\frac{1}{{\beta }_{el}\hslash {\omega }_0}\ln \left[\frac{1+(1+e^{{\beta }_{ph}\hslash {\omega }_0}){\sinh }^{2}r}{1+(1+e^{-{\beta }_{ph}\hslash {\omega }_0}){\sinh }^{2}r}\right]$$

$$\ln \left[1+\frac{\left(e^{{\beta }_{ph}\hslash {\omega }_0}-e^{-{\beta }_{ph}\hslash {\omega }_0}\right){\sinh }^{2}r}{1+(1+e^{-{\beta }_{ph}\hslash {\omega }_0}){\sinh }^{2}r}\right]$$

$$\to \frac{{\beta }_{ph}\hslash {\omega }_0\times 2{\sinh }^{2}r}{1+2{\sinh }^{2}r},$$

$$⟨\eta ⟩\le 1-\frac{T_{el}}{T_{ph}(1+2{\sinh }^{2}r)}.$$

$$⟨\eta ⟩\le 1-\frac{T_{el}}{T_{ph}}+\frac{1}{{\beta }_{el}\hslash {\omega }_0}\left[\frac{{\sinh }^{2}r}{1+{\sinh }^{2}r}\times e^{{\beta }_{ph}\hslash {\omega }_0}\right].$$

$$\hat{H}={\hat{H}}_s+{\hat{H}}_{el}+{\hat{H}}_{ph}+{\hat{V}}_{s-el}+{\hat{V}}_{s-ph}.$$

$${\hat{H}}_{el}$$

$${\hat{V}}_{s-el}=g{\hat{\sigma }}_x\left({\hat{c}}_d^{†}{\hat{c}}_a+{\hat{c}}_a^{†}{\hat{c}}_d\right),{\hat{V}}_{s-ph}={\hat{\sigma }}_x\sum _k{g}_k\left({\hat{a}}_k^{†}+{\hat{a}}_k\right).$$

$$\mathcal{G}(\lambda )=-\frac{1}{2}(k_u+k_d)+\frac{1}{2}\sqrt{(k_u-k_d{)}^2+4k_u^{\lambda }{k}_d^{\lambda }}.$$

$$k_d^{\lambda }=[k_d^{el}{]}^{\lambda }+[k_d^{ph}{]}^{\lambda },[k_d^{el}{]}^{\lambda }=[k_d^{\lambda }{]}^{L\to R}+[k_d^{\lambda }{]}^{R\to L}.$$

$$\displaystyle[k_{d}^{\lambda}]^{L\to R}=\int\frac{d\epsilon}{2\pi}\Big{[}f_{L}(\epsilon)(1-f_{R}(\epsilon+\omega_{0}))J_{L}(\epsilon)J_{R}(\epsilon+\omega_{0})$$

$$\displaystyle\times e^{-i(\lambda_{c}+(\epsilon+\omega_{0})\lambda_{e})}\Big{]}.$$

$$[k_d^{\lambda }{]}^{ph}={\Gamma }_{ph}({\omega }_0)[N({\omega }_0)+1]e^{-i{\lambda }_{ph}{\omega }_0},$$

$$[k_u^{\lambda }{]}^{ph}={\Gamma }_{ph}({\omega }_0)N({\omega }_0)e^{i{\lambda }_{ph}{\omega }_0}.$$

$$\displaystyle\langle I_{c}\rangle=\frac{\partial\cal G(\lambda)}{\partial(i\lambda_{c})}\Big{|}_{\lambda=0}=\frac{k_{d}\frac{\partial(k_{u}^{\lambda})}{\partial(i\lambda_{c})}+k_{u}\frac{\partial(k_{d}^{\lambda})}{\partial(i\lambda_{c})}}{k_{u}+k_{d}}.$$

Full text

Introduction.— The efficiency of heat engines, defined by the ratio of the extracted work to the absorbed heat, is fundamentally restricted by the second law of thermodynamics to the Carnot limit. This canonical bound is being challenged nowadays by quantum and classical effects review ; kosloffrev . For example, quantum phenomena such as steady state coherence scully0 ; scully1 ; scully2 and quantum correlations Huber , which persist in multi-level quantum systems, are suggested as a resource for the design of more efficient engines. As well, nonequilibrium, stationary reservoirs that are characterized by additional parameters besides their temperature, are exploited to construct devices with efficiency beyond the Carnot bound Yi ; LutzPRL ; LutzEPL ; Parrondo ; David1 ; Kurizki15 ; Kurizki1 ; imamoglu . In particular, a four-stroke Otto heat engine, operating between two reservoirs, a hot squeezed thermal bath and a cold thermal bath, was examined in Refs. Yi ; LutzPRL ; Parrondo , reaching a unit value in the asymptotic, high squeezing limit.

Beyond the analysis of the averaged efficiency, a quantum mechanical, full counting statistics derivation provides the ultimate, fundamental description of out-of-equilibrium quantum statistical phenomena. Such an approach hands over symmetries, bounds, and noise terms (cumulants) to characterize e.g. particle and energy transport. It is unclear, however, whether the steady state fluctuation symmetry fluct1 ; fluct2 ; fluct3 holds for transport phenomena between non-canonical reservoirs. Another fundamental question is whether quantum principles impose new bounds on energy conversion efficiency in such systems, to extend the second law of thermodynamics.

In this Letter, we fill these gaps by employing a full counting statistics approach to study energy conversion in quantum engines absorbing heat from a non-canonical reservoir. Our device consists a single qubit coupled to hot squeezed photon bath and two cold electronic reservoirs (the source and drain), see Fig. 1. We show that the nonequilibrium fluctuation relation (FR) for entropy production can be recovered once identifying an effective temperature for the squeezed thermal bath. From the fluctuation symmetry, we derive a generalized, quantum efficiency bound for the heat engine, surpassing the Carnot limit. Since the FR encompasses linear-response thermodynamics, we receive immediately other operational measures of heat engines in linear response: the universal maximum power efficiency bound casati ; espmp and properties of fluctuations statistics espfl . Our theory is exemplified with a quantum mechanical, full counting statistics description of a nanoscale photoelectric device.

We begin with a quick review of the fundamentals of the entropy production fluctuation theorem fluct1 ; fluct2 ; fluct3 . Based on the microreversibility of the Hamiltonian dynamics and the canonical form of the initial condition, one can prove a universal relation in steady state,

[TABLE]

Here, $P_{t}(\Delta S)$ is the probability distribution for entropy production $\Delta S$ during a time interval $t$. It is convenient to define the characteristic function

[TABLE]

with $\lambda$ the so-called counting parameter. One can immediately prove the Gallavotti-Cohen fluctuation symmetry from the fluctuation relation (1), $\mathcal{Z}(\lambda)=\mathcal{Z}(-\lambda+i)$ fluct1 ; fluct2 ; fluct3 . Moreover, by using $\lambda=0$ in Eq. (2), it is easy to prove that $1=\langle e^{-\Delta S}\rangle$. This equality immediately leads to the second law of thermodynamics, $\langle\Delta S\rangle\geq 0$, by using Jensen’s inequality for convex functions.

Three-terminal photoelectric devices.— We now apply these considerations onto a quantum heat engine consisting of three terminals. In our construction, see Fig. 1, a qubit is coupled to a photonic heat source ($ph$), which may be canonical (equilibrium) or squeezed (out of equilibrium). As well, the qubit is exchanging energy with an electronic circuit with two metal leads, $L$ and $R$, which can be set out of equilibrium by the application of a finite voltage bias $\Delta\mu=\mu_{R}-\mu_{L}$ and a temperature difference. For simplicity, we assume that the two electrodes are maintained at the same temperature, $\beta_{el}=\beta_{\alpha}$; $\alpha=L,R$, and that the photon bath is hotter than the electronic system, $\beta_{ph}<\beta_{el}$. Our interest here is in the conversion of photon energy into electrical work.

In order to describe the system quantum mechanically, we use the two-time measurement protocol fluct1 ; fluct2 and define the characteristic function as

[TABLE]

Here, $\lambda_{c,e,ph}$ are counting parameters for charge, electronic energy, and photonic energy, respectively. $\hat{A}_{c}$, $\hat{A}_{e}$ and $\hat{A}_{ph}$ are the respective operators: $\hat{A}_{c}$ is the number operator corresponding to the total charge in e.g. the $R$ lead. $\hat{A}_{e}$ is the Hamiltonian operator for the $R$ electrode and $\hat{A}_{ph}$ is the Hamiltonian operator for the photon bath. Time evolution corresponds to the Heisenberg representation, $\langle...\rangle$ represents an average with respect to the total initial density matrix, which takes a factorized form with respect to the system ($s$) and ($L$, $R$ and $ph$) baths, $\rho_{T}(0)=\rho_{s}(0)\otimes\rho_{L}\otimes\rho_{R}\otimes\rho_{ph}$. The state of the metal leads is described by a grand canonical distribution, $\rho_{\alpha}\!=\!\exp[-\beta_{el}(\hat{H}_{\alpha}-\mu_{\alpha}\hat{N}_{\alpha})]/Z_{\alpha}$, with Z_{\alpha}\!=\!{\rm Tr}\Big{[}\exp[-\beta_{el}(\hat{H}_{\alpha}-\mu_{\alpha}\hat{N}_{\alpha})]\Big{]} as the partition function.

Equilibrium thermal photon bath.— Let us begin by assuming that the state of the photon bath is canonical, $\rho_{ph}\!=\!\exp[-\beta_{ph}\hat{H}_{ph}]/Z_{ph}$, with Z_{ph}={\rm Tr}\big{[}\exp(-\beta_{ph}\hat{H}_{ph})\big{]}. The fluctuation relation (1) translates to

[TABLE]

Here, $N$ denotes the number of electrons transferred from $R$ to $L$ during the time interval $t$. Similarly, $E_{e}$ is the electronic energy and $Q_{ph}$ photonic heat that are exchanged between the baths during the time interval $t$. The characteristics function thus satisfies

[TABLE]

This relation immediately implies that

[TABLE]

Using Jensen’s inequality, we receive $[-\beta_{el}\Delta\mu\langle N\rangle+(\beta_{ph}-\beta_{el})\langle Q_{ph}\rangle]\leq 0$. The efficiency, $\langle\eta\rangle\equiv-\frac{\Delta\mu\langle N\rangle}{\langle Q_{ph}\rangle}$, comment thus obeys the Carnot bound ($T=1/\beta$),

[TABLE]

Non-canonical photon bath.— We now repeat this exercise—with a squeezed, hot thermal reservoir. The electric field of a single-mode wave can be written as a combination of orthogonal (quadrature) components, which oscillate as $\cos\omega t$ and $\sin\omega t$ QObook . Squeezed states have reduced fluctuations in one of the quadratures—but enhanced noise in the other quadrature—so as to satisfy the bosonic commutation relation. Such states are defined by two parameters, the squeezing factor $r$ and phase $\phi$ QObook .

For simplicity, the quantum “working fluid” system includes a single qubit with an energy gap $\hbar\omega_{0}$. The squeezed bath can excite and de-excite the qubit, with rate constants $k_{u}^{ph}$ and $k_{d}^{ph}$, satisfying Yi

[TABLE]

Here breuer , $N(\omega_{0})=N_{th}(\omega_{0})\left(\cosh^{2}r+\sinh^{2}r\right)+\sinh^{2}r$, with the squeezing parameter $r$ reflecting the nonequilibrium nature of the bath. The phase $\phi$ does not appear in this expression, as it only affects transients. In fact, at weak system-photon bath coupling, this effective temperature describes as well harmonic systems. For a canonical thermal bath ($r=0$), the occupation number reduces to the Bose-Einstein distribution function, $N(\omega_{0})\rightarrow N_{th}(\omega_{0})=1/[e^{\beta_{ph}\hbar\omega_{0}}-1]$, and the rate constants satisfy the detailed balance relation with respect to the photon bath, $k_{d}^{ph}(\omega_{0})/k_{u}^{ph}(\omega_{0})=e^{\beta_{ph}\hbar\omega_{0}}$. To restore the detailed balance relation for the $r\neq 0$ case, one can identify an effective temperature, which is unique in the present model Yi ,

[TABLE]

Simple manipulations provide

[TABLE]

It is important to note that: (i) $\beta_{eff}\leq\beta_{ph}$. This observation implies that more work can be extracted from a squeezed bath, than the case with $r=0$. (ii) The effective temperature (9) may depend on system parameters, the energy gap $\omega_{0}$ in the present case. However, in the small $r$ and high temperature limit one recovers a proper, “thermodynamical” temperature

[TABLE]

which is solely described in terms of bath parameters. Therefore, in this limit universal relations of traditional linear irreversible thermodynamics hold.

Identifying the entropy production associated with the photon energy flow by $\langle\Delta S\rangle=(\beta_{el}-\beta_{eff})\langle Q_{ph}\rangle$, we perform a quantum mechanical, counting statistics analysis, similarly to the canonical case, and confirm the symmetry Eq. (4), only replacing $\beta_{ph}$ by $\beta_{eff}$,

[TABLE]

The FR implies that $1=\langle e^{-\beta_{el}\Delta\mu N+(\beta_{eff}-\beta_{el})Q_{ph}}\rangle$, thus the averaged efficiency, $\langle\eta\rangle\equiv-\Delta\mu\langle N\rangle/\langle Q_{ph}\rangle$, is bounded by

[TABLE]

This bound is universal, holding beyond the squeezed-bath case. It is valid for any nonequilibrium thermal bath that can be characterized by a unique, stationary, effective temperature, see Ref. David1 for some examples. Explicitly, the efficiency bound for our photoelectric engine is given by

[TABLE]

which is the main result of our work. It was derived from the fluctuation theorem, and it is valid to describe continuous quantum heat engines, unlike earlier studies, which were focused on four-stroke engines, see e.g. Ref. David1 . Since the third term in this expression is positive for nonzero $r$, squeezing of a thermal bath always increases the heat-to-work efficiency bound.

We now discuss several interesting limits of Eq. (13). First, we expand it close to thermal equilibrium assuming $\sinh^{2}r$ is a small parameter. As well, we assume that the temperature of the photon bath is high, $\beta_{ph}\hbar\omega_{0}\ll 1$. The expression in the square brackets reduces to

[TABLE]

and Eq. (13) becomes

[TABLE]

Remarkably, this agrees with Ref. LutzPRL ; Parrondo . Recall that our derivation concerns continuous heat engines; Refs. LutzPRL ; Parrondo , in contrast, received this limit by constructing a four-stroke cycle. This agreement can be rationalized by noting that Eq. (15) should be regarded as a linear response limit for $r$, which is a resource to drive energy current between equal-temperature baths Yi .

Another interesting case is the deep quantum regime, $\beta_{ph}\hbar\omega_{0}\gg 1$. Assuming small $r$, we receive from Eq. (13) an exponential quantum enhancement in comparison to the classical case,

[TABLE]

Note that the expansion assumes that the term inside the square bracket is kept below 1. Finally, at large $r$, the natural logarithm term in (13) cancels out the second contribution for both high and low $T_{ph}$. The efficiency bound then saturates to a unit value, $\langle\eta\rangle\rightarrow 1$, realizing a complete conversion of heat to work. We display these results in Fig. 2: Squeezing enhances the efficiency beyond the Carnot limit. In the quantum regime, $\beta_{ph}\omega_{0}>1$, the bound is greatly reinforced beyond the “thermodynamical” value, Eq. (15).

A squeezed bath coupled to a qubit can be described by a single-unique effective temperature in the thermodynamical limit of high $T_{ph}$ and small $r$. Since the fluctuation theorem embodies linear irreversible thermodynamics, all linear response operational results immediately follow. In particular, the averaged maximum power efficiency (MPE) satisfies the universal linear response result casati $\langle\eta^{*}\rangle=\langle\eta_{M}\rangle/2$, with $\langle\eta_{M}\rangle$ the upper bound in Eq. (15). For a four-stroke Otto engine, the MPE is given by the Curzon-Ahlborn bound (beyond linear response), $\langle\eta^{*}\rangle=1-\sqrt{\frac{T_{cold}}{T_{eff}}}$, with the identification of the thermodynamic temperature (10). This agrees with Ref. LutzPRL .

Example.— So far, we derived an efficiency bound for continuous quantum heat engines based on the fluctuation symmetry. We now proceed and describe a device where a closed-form expression for the cumulant generating function (CGF) ${\cal G}(\lambda)=\lim_{t\to\infty}\frac{1}{t}\ln{\cal Z}(\lambda)$ is achieved. Here, $\lambda$ collectively refers to the three counting fields. From the CGF, all cumulants of the charge current, electronic energy current and photonic current are available. The closed-form expression for the efficiency of the engine allows us to examine its actual performance under different conditions. Our model photoelectric heat engine is described by the Hamiltonian

[TABLE]

It comprises a single qubit $\hat{H}_{s}=\frac{\hbar\omega_{0}}{2}\hat{\sigma}_{z}$ of energy gap $\hbar\omega_{0}$. The photon bath is written in terms of bosonic creation $\hat{a}_{k}^{\dagger}$ and annihilation $\hat{a}_{k}$ operators, $\hat{H}_{ph}=\sum_{k}\omega_{k}\hat{a}_{k}^{\dagger}\hat{a}_{k}$. The electronic circuit includes two sites (quantum dots) denoted by ’d’ and ’a’, each coupled to their respective metal leads, $L$ and $R$. The corresponding Hamiltonian is

[TABLE]

Here $\hat{c}$ ($\hat{c}^{\dagger}$) are fermionic annihilation (creation) operators. Energy is exchanged between the qubit and the reservoirs via the interaction terms

[TABLE]

In words, the excitation or relaxation of the qubit couples to exchange of electrons between the two sites and the displacement of harmonic modes. The CGF is derived using a quantum master equation that is correct to second order in the electron-qubit and the photon-qubit couplings PCCP ; Bijay15 ,

[TABLE]

Here, $k_{d,u}^{\lambda}$ are the relaxation $(d)$ and excitation $(u)$ rate constants of the qubit, with transitions induced by the reservoirs, e.g.,

[TABLE]

Specifically, $[k_{d}^{\lambda}]^{L\to R}$ describes a de-excitation process of the qubit, induced by an electron moving from the $L$ to the $R$ metal. It involves the release of energy at the right metal (where counting is performed),

[TABLE]

Here, e.g., $J_{L}(\epsilon)=g\frac{\Gamma_{L}(\epsilon)}{(\epsilon-\epsilon_{d})^{2}+\Gamma_{L}(\epsilon)^{2}/4}$ is the spectral function of the $L$ metal, determined by the dot-metal hybridization energy $\Gamma_{\alpha}(\epsilon)=2\pi\sum_{j}|v_{\alpha,j}|^{2}\delta(\epsilon-\epsilon_{\alpha,j})$. Transitions induced by the squeezed photon bath satisfy PCCP

[TABLE]

Here, $\Gamma_{ph}(\omega)=2\pi\sum_{k}|g_{k}|^{2}\delta(\omega-\omega_{k})$, $N(\omega_{0})$ was defined below Eq. (7). It can be shown that the CGF (20) obeys the FR. The electron charge current is given by

[TABLE]

An analogous expression is written for $\langle I_{ph}\rangle$. In Fig. 3 we display the averaged efficiency of the engine $\langle\eta\rangle=-\Delta\mu\langle I_{c}\rangle/\langle I_{ph}\rangle$ for certain parameters, once we set $T_{ph}>T_{el}$ and $\mu_{R}>\mu_{L}$. The device operates as a photoelectric engine when heat is absorbed from the photon bath and charge current is flowing against the potential bias. We operate it in the quantum regime, $\omega_{0}\beta_{ph}\sim 10$, and reveal a significant enhancement of efficiency, largely exceeding the Carnot bound for small squeezing, $r=0.1$.

Summary.— We investigated the operation of heat engines coupled to a squeezed thermal bath. Based on the fluctuation symmetry, we derived a generalized quantum Carnot efficiency bound, as well as other thermodynamical linear response operational bounds. We exemplified our approach with a quantum-mechanical full counting statistics description of a photoelectric device. In multi-level systems it may be necessary to define multiple effective temperatures for a non-canonical bath, corresponding to different transitions in the system. The identification of an effective temperature here and in other studies Yi ; David1 was achieved in the limit of weak coupling between the qubit and the environment. Quantum systems that are strongly coupled to equilibrium thermal reservoirs are expected to bring in new design rules for energy conversion devices DavidS ; Kosloff ; Cao ; Gernot ; Jarzynski ; esposito17 ; Nazir . Describing heat engines that are strongly coupled to non-canonical reservoirs remain a challenge for future work.