Quantum Synchronisation in Nanoscale Heat Engines

TL;DR

This paper explores quantum synchronization in nanoscale heat engines, revealing its role in limiting power and efficiency, and establishing it as a key mechanism in quantum thermodynamics.

Contribution

It demonstrates that nanoscale heat engines inherently exhibit quantum synchronization, linking phase coherence with thermodynamic performance and providing insights for quantum technology applications.

Findings

Synchronization bounds the steady-state power of heat engines.

Engine efficiency determines the temperature conditions for synchronization.

Quantum synchronization acts as a mechanism for stable phase coherence.

Abstract

Owing to the ubiquity of synchronization in the classical world, it is interesting to study its behavior in quantum systems. Though quantum synchronisation has been investigated in many systems, a clear connection to quantum technology applications is lacking. We bridge this gap and show that nanoscale heat engines are a natural platform to study quantum synchronization and always possess a stable limit cycle. Furthermore, we demonstrate an intimate relationship between the power of a heat engine and its phase-locking properties by proving that synchronization places an upper bound on the achievable steady-state power of the engine. Finally, we show that the efficiency of the engine sets a point in terms of the bath temperatures where synchronization vanishes. We link the physical phenomenon of synchronization with the emerging field of quantum thermodynamics by establishing quantum synchronization as a mechanism of stable phase coherence.

Figures

Equations

$$\dot{\rho }=-i[H,\rho ]+{\mathcal{L}}_h[\rho ]+{\mathcal{L}}_c[\rho ],$$

$$\mathcal{S}({\phi }_1,{\phi }_2)\equiv \int d\Omega Q(\theta ,\xi ,{\phi }_1,{\phi }_2)-\frac{1}{4{\pi }^2},$$

$$\mathcal{S}({\phi }_1,{\phi }_2)$$

$${\mathcal{S}}^{max}\equiv \underset{{\phi }_1,{\phi }_2}{\max }\mathcal{S}({\phi }_1,{\phi }_2)=\frac{1}{16\pi }{C}_{l_1}({\rho }^{ss}),$$

$$P^{ss}=2\epsilon {\omega }_{32}\text{Im}\left[{\rho }_{23}^{ss}\right].$$

$$\left|P^{ss}\right|\le 16\pi \left|\epsilon \right|{\omega }_{32}{\mathcal{S}}^{max}.$$

$${\rho }_{23}^{ss}=i\epsilon {\Gamma }_{23}^{\ast }{\gamma }_c{\gamma }_h\left({\bar{n}}_c-{\bar{n}}_h\right)/\beta ,$$

$$\text{arrowvert}{n}_3⟩=(\cos \theta ,e^{i{\phi }_1}\cos \xi \sin \theta ,e^{i{\phi }_2}\sin \xi \sin \theta {)}^T,$$

$$dv=d\theta d\xi d{\phi }_{1}d{\phi }_2\cos \theta {\sin }^3\theta \cos \xi \sin \xi ,$$

$${\int }_0^{2\pi }d{\phi }_1{\int }_0^{2\pi }d{\phi }_2{\int }_0^{\pi /2}d\theta \cos \theta {\sin }^3\theta {\int }_0^{\pi /2}d\xi \cos \xi \sin \xi |n_3⟩⟨n_3|=\frac{{\pi }^2}{6}\mathbb{I}.$$

$$Q^{ss}$$

$$H_{0}=\left(\begin{array}[]{ccc}\omega_{1}&0&0\\ 0&\omega_{2}&0\\ 0&0&\omega_{3}\end{array}\right),$$

$$\mathcal{S}({\phi }_1,{\phi }_2)$$

$${\mathcal{S}}^{max}$$

$$\frac{d}{dt}⟨\tilde{H}⟩=\text{tr}\left\{\frac{d\tilde{H}}{dt}\tilde{\rho }\right\}+\text{tr}\left\{\tilde{H}\frac{d\tilde{\rho }}{dt}\right\}.$$

$$\dot{Q}=\text{tr}\left\{H\frac{\partial \rho }{\partial t}\right\}\text{and}P=\text{tr}\left\{\frac{\partial H}{\partial t}\rho \right\}.$$

$$\dot{\tilde{Q}}=\text{tr}\left\{\tilde{H}\frac{d\tilde{\rho }}{dt}\right\}\text{and}\tilde{P}=\text{tr}\left\{\frac{d\tilde{H}}{dt}\tilde{\rho }\right\},$$

$$\dot{\tilde{Q}}$$

$$\tilde{P}$$

$$\frac{d}{dt}\text{tr}\left\{\rho H_0\right\}$$

$$H=H_0+V(t).$$

$$V(t)=\epsilon \left({\sigma }_{23}{e}^{i{\omega }_{d}t}+{\sigma }_{32}{e}^{-i{\omega }_{d}t}\right),$$

$$U(t)=e^{-i{\omega }_{1}t}{\sigma }_{11}+e^{-i{\omega }_{2}t}{\sigma }_{22}+e^{-i({\omega }_2+{\omega }_d)t}{\sigma }_{33}.$$

$$\dot{\tilde{\rho }}=-i[\tilde{H},\tilde{\rho }]+{\mathcal{L}}_h[\tilde{\rho }]+{\mathcal{L}}_c[\tilde{\rho }],$$

$$\tilde{H}$$

$${\mathcal{L}}_h[\tilde{\rho }]$$

$${\mathcal{L}}_c[\tilde{\rho }]$$

$${\dot{\tilde{\rho }}}_{11}$$

$${\dot{\tilde{\rho }}}_{22}$$

$${\dot{\tilde{\rho }}}_{33}$$

$${\dot{\tilde{\rho }}}_{12}$$

$${\dot{\tilde{\rho }}}_{13}$$

$${\dot{\tilde{\rho }}}_{23}$$

$${\Gamma }_{12}$$

Full text

Furthermore, we demonstrate an intimate relationship between the power of a heat engine and its phase-locking properties by proving that synchronization places an upper bound on the achievable steady-state power of the engine. Finally, we show that the efficiency of the engine sets a point in terms of the bath temperatures where synchronization vanishes. We link the physical phenomenon of synchronization with the emerging field of quantum thermodynamics by establishing quantum synchronization as a mechanism of stable phase coherence.

Introduction.— Synchronization has been observed and studied in a multitude of naturally occurring as well as man-made systems, finding different applications in fields ranging from engineering to medicine Pikovsky et al. (2003); Kuramoto (2012); Balanov et al. (2008); Strogatz (2004). Recently the universality of this phenomenon has also been embraced in the quantum realm. Thus far all the efforts have been aimed at characterising synchronization in quantum systemsHeinrich et al. (2011); Bastidas et al. (2015); Witthaut et al. (2017); Marquardt et al. (2006); Lee and Sadeghpour (2013); Walter et al. (2014); Amitai et al. (2017); Sonar et al. (2018, 2018); Lörch et al. (2017); Nigg (2018); Lörch et al. (2017); Roulet and Bruder (2018a, b); Kwek (2018). It is compelling to ask if/where this phenomenon may play a role in the functionality itself of quantum devices. What is the impact of synchronization on quantum technology platforms? A natural playground to explore this question is quantum thermodynamics Goold et al. (2016); Vinjanampathy and Anders (2016). Nanoscale heat engines can be modelled as multi-level atoms that produce work when cyclically coupled to two or more heat baths Kosloff (2013). In this manuscript we highlight a deep connection between quantum synchronization and the performance of nanoscale heat engines. This relation is particularly evident in the emitted power of the engine displaying an Arnold tongue, a distinct signature of synchronization. We show that synchronization provides an upper bound on the achievable magnitude of the steady-state power of the heat engine. Finally, we determine that the efficiency of the heat engine $\eta$ sets the ratio of bath temperatures where synchronization vanishes. This points at a fundamental relation between the operational regime of a thermal machine and its synchronization properties. An important outcome of this relationship is to bring the study of quantum synchronisation closer to quantum technology applications.

A basic prerequisite for synchronization is the existence of a stable limit cycle Pikovsky et al. (2003). Once a limit cycle has been established in a non-linear dynamical system, we can synchronize the given system to an external frequency standard 111synchronisation is usually reserved for the adjustment of rhythms between two or more physical systems, whereas the adjustment of a systems rhythm to an external frequency is known as entrainment. Following existing literature in quantum synchronisation, we will use the same word for both phenomena.. We show that thermal atoms possess a limit cycle by constructing the quasi-probability distribution for arbitrary three-level atoms. This allows us to perform a synchronization analysis of a thermal three-level system and investigate its phase-locking properties in the context of the engine’s performance.

Thermal three-level atoms.— We consider the three-level maser model introduced by Scovil and Schulz-Dubois as an example of quantum synchronization Scovil and Schulz-DuBois (1959). In view of the connection to heat engines, we will examine its properties when coupled to two baths kept at different temperatures. Though we consider a specific model to develop the results presented here, the techniques are transferrable to generic coherent quantum technologies.

The evolution for the system depicted in Fig. (1) can be written ($\hbar=1$) as

[TABLE]

where $H=H_{0}+V$ represents the sum of the bare Hamiltonian $H_{0}$ and the drive $V=\varepsilon(e^{i\omega_{d}t}\sigma_{23}+e^{-i\omega_{d}t}\sigma_{32})$, with $\sigma_{ij}\equiv|{i}\rangle\langle{j}|$. We model the baths as single-mode thermal fields resonant with the respective atomic transitions that they couple. The bath frequency $\omega$, its temperature $T$ and its average photon number $\bar{n}$ are related by $e^{-\hbar\omega/k_{B}T}=\bar{n}/(\bar{n}+1)$ Breuer et al. (2002). The Lindbladian dissipators are given by $\mathcal{L}_{h}[\rho]\equiv\gamma_{h}\bar{n}_{h}\mathcal{D}[\sigma_{31}]\rho+\gamma_{h}(1+\bar{n}_{h})\mathcal{D}[\sigma_{13}]\rho$, incoherently coupling the $|{1}\rangle\leftrightarrow|{3}\rangle$ transition at temperature $T_{h}$, and $\mathcal{L}_{c}[\rho]\equiv\gamma_{c}\bar{n}_{c}\mathcal{D}[\sigma_{21}]\rho+\gamma_{c}(1+\bar{n}_{c})\mathcal{D}[\sigma_{12}]\rho$ representing a cold bath at temperature $T_{c}$ and incoherently coupling the $|{1}\rangle\leftrightarrow|{2}\rangle$ transition. The Markovian master equation is written in the standard Lindblad form Breuer et al. (2002) as $\mathcal{D}[O]\rho=O\rho O^{{\dagger}}-\frac{1}{2}\{O^{{\dagger}}O,\rho\}$.

The system is weakly driven with small detunings, which ensures that (a) the limit cycles presented below are not deformed by the drive, (b) the populations in the steady states are dominated by the dissipation rates and (c) the adiabatic definitions of heat and work are valid in this regime.

Points (a) and (b) are related and ensure that we are in a regime where it is sensible to study synchronization Roulet and Bruder (2018a). Large driving strength also produces phase locking. However, due to deformation of the limit cycle this regime is known as suppression of natural dynamics Balanov et al. (2008) and is beyond the scope of this manuscript. In general, the power of an engine can be attributed to two parts, one that accounts for the change in the instantaneous eigenvalues and another which accounts for the change in eigenvectors. For perturbative driving the first term does not contribute (indeed our eigenvalues are constant in time).

Synchronization in heat engines.— A stable limit cycle is defined as a phase trajectory that attracts nearby orbits. Such a limit cycle arises as a compromise between dissipation and gain in the system. Another defining property of the limit cycle is that an observable phase is free. It is this freedom of the phase combined with the stability of the limit cycle that allows a weak external signal to influence the phase distribution of the oscillator. To study limit cycles in thermal three-level systems, we define the Husimi-Kano $Q$-representation function Schleich (2011), given for a generic three-level atom as $Q(\theta,\xi,\varphi_{1},\varphi_{2})\equiv(6/\pi^{2})\langle{n_{3}}|\rho|{n_{3}}\rangle$ where $|{n_{3}}\rangle\equiv(\cos\theta,\;e^{i\varphi_{1}}\cos\xi\sin\theta,\;e^{i\varphi_{2}}\sin\xi\sin\theta)^{T}$ is the SU(3) coherent state Nemoto (2000).

From the definition of $|{n_{3}}\rangle$ we can see that $(\varphi_{1},\varphi_{2})$ and $(\theta,\xi)$ carry information about the coherences and the populations of the system, respectively. Under the influence of a bare Hamiltonian written in the energy eigenbasis, the SU(3) coherent state evolves as $|{n_{3}}\rangle=|{\theta,\xi,\varphi_{1},\varphi_{2}}\rangle\rightarrow|{\theta,\xi,\varphi_{1}-\omega_{21}t,\varphi_{2}-\omega_{31}t}\rangle$, where $\omega_{ij}\equiv\omega_{i}-\omega_{j}$. This implies that the angles $(\varphi_{1},\varphi_{2})$ are the relevant dynamical phases to the study of synchronization.

Similarly to a spin-1 atom considered in Roulet and Bruder (2018a) the system becomes synchronized when the distribution of $\varphi_{1}$ and $\varphi_{2}$ becomes localized. The difference between the localized and uniform distributions serves as a measure of synchronization,

[TABLE]

where $\int d\Omega\equiv\int_{0}^{\frac{\pi}{2}}d\theta\cos\theta\sin^{3}\theta\int_{0}^{\frac{\pi}{2}}d\xi\cos\xi\sin\xi$.

This is a generalisation to the full SU(3) group of a previous definition presented for SO(3) subgroup of SU(3) Roulet and Bruder (2018a).

Under the influence of the bare Hamiltonian and the dissipators, the system settles into a stable limit cycle given by no phase preference in the angles $(\varphi_{1},\varphi_{2})$. When the system is driven, it develops a phase preference characterised by localization of $Q(\theta,\xi,\varphi_{1},\varphi_{2})$ in the relevant angles $(\varphi_{1},\varphi_{2})$. The synchronization measure $\mathcal{S}(\varphi_{1},\varphi_{2})$ in the steady state is evaluated to be

[TABLE]

Here $\rho^{ss}$ is the steady-state density matrix. For the evolution given in Eq. (1), this synchronization measure simplifies to $\mathcal{S}(\varphi_{1},\varphi_{2})=\text{Re}\left[e^{i(\varphi_{2}-\varphi_{1})}\rho_{23}\right]/8\pi$. We are primarily concerned with how much the phases are localised in the steady state, given by the maximum of $\mathcal{S}(\varphi_{1},\varphi_{2})$ which can be readily computed,

[TABLE]

where $C_{l_{1}}(\rho)\equiv\sum_{i\neq j}|\rho_{ij}|$ is the $l_{1}$-norm of coherence Baumgratz et al. (2014).

The synchronization measure $\mathcal{S}^{max}$ is shown in Fig. 2(a) as a function of the drive’s detuning $\Delta=\omega_{32}-\omega_{d}$ and its strength $\varepsilon$. In analogy to classical synchronization, note the Arnold tongue-like behaviour of the figure indicating the phase locking of the atom to the external drive. The range of detuning for which the atom displays phase locking to the external drive increases with the strength of the drive.

Engine Performance & Power Arnold tongue.— Efficiency and power are the characteristics that most often define the operation of an engine Millen and Xuereb (2016); Goold et al. (2016); Vinjanampathy and Anders (2016). Scovil and Schulz-Dubois Scovil and Schulz-DuBois (1959) showed that a maser can be operated as a heat engine whose efficiency is bounded by Carnot efficiency, namely $\eta={\omega_{32}}/{\omega_{31}}\leq\eta_{c}\equiv 1-{T_{c}}/{T_{h}}$. This bound justifies our choice of driving a generic three-level atom, whose Husimi-Kano $Q$ function is defined in terms of SU(3) coherent states as opposed to considering an equally spaced three-level atom of Roulet and Bruder (2018a). Our generalisation reveals the complex dynamics inherent to the full SU(3) group while simultaneously allowing the efficiency to be varied by the choice of $\omega_{ij}$.

A natural question arises, namely if the aforementioned thermodynamical characteristics depend on the underlying dynamical properties of the system, undergoing a transition from synchronized to unsynchronized regime. The efficiency only depends on the transition frequencies of the system and is independent of any dynamical properties. This is because the efficiency tracks the energy transactions from the bath to the piston via the working medium, and is unconcerned about the dynamical processes involved in the perturbative driving regime.

On the other hand, the power of an engine is intimately related with dynamics Kosloff (2013). To operate an engine at the Carnot efficiency, one must perform adiabatic strokes which will not generate any heat. Since adiabatic strokes take infinite time, such an engine does not generate power. Likewise in the quantum regime, adiabatic strokes are transitionless and do not generate any excitations, allowing the engine to follow the instantaneous eigenstate of the total Hamiltonian. Adiabatic driving has to thermalize with the bath at each instant in time, also implying the slowing down of dynamics Suri et al. (2018); Dann et al. (2018). Furthermore, perturbative driving does not change eigenvalues, and hence the contribution to power cannot arise from such terms. Such a scenario hence precludes generating power from diagonal density matrices. To generate power, the density matrix hence has to be off-diagonal in the Hamiltonian eigenbasis Feldmann and Kosloff (2003). Any definition of power has to satisfy all these constraints and has to be frame independent, saturate in the steady state and satisfy the second law of thermodynamics for all values of detuning $\Delta$ unlike the definition presented in Alicki (1979). This intuition is captured by the steady-state power $P^{ss}=-i\;\text{tr}\{[H,\rho^{ss}]H_{0}\}$ due to Tannor and Boukobza Boukobza and Tannor (2006, 2007) and satisfies all the conditions stated above. A detailed analysis of the definitions of heat and work, alongside the engine performance is presented in the supplemental material for completeness.

The magnitude of steady-state power $|P^{ss}|$ is presented in Fig. 2(b) as a function of $\Delta$ and $\varepsilon$, and displays an Arnold tongue-like behaviour. To understand this, we note that the power is non-zero only when the steady state density matrix is off-diagonal with respect to the bare Hamiltonian. This means that to observe a powerful quantum engine, the system has to be driven externally by a Hamiltonian that does not commute with the bare Hamiltonian. Power is only produced in the region where we observe phase locking to the external drive. If the system is sufficiently detuned at a given value of the driving strength $\varepsilon$, negligible power is emitted by the maser. The underlying mechanism for this Arnold tongue-like behaviour is synchronization which produces coherence in the Hamiltonian eigenbasis.

To make this connection between synchronization and power quantitative we begin by calculating the steady-state power explicitly,

[TABLE]

Using $\left|\text{Im}\left[\rho^{ss}_{23}\right]\right|\leq\left|\rho^{ss}_{23}\right|$ along with Eq. (4) leads to a synchronization bound on the magnitude of the steady-state power of the engine,

[TABLE]

The bound becomes an equality when the engine is driven resonantly, $\Delta=0$, because in this case $\rho^{ss}_{23}$ becomes pure imaginary. This can be seen by computing $\rho_{23}^{ss}$,

[TABLE]

where $\Gamma_{23}=[\gamma_{h}(\bar{n}_{h}+1)+\gamma_{c}(\bar{n}_{c}+1)]/2-i\Delta$ and $\beta$ is a real number whose exact form is given in the supplemental material. Inequality (6) is our main result. It directly links the power of a heat engine with the amount of synchronization in the system. Synchronization sets an upper bound on the achievable power of an engine for a fixed driving strength $\varepsilon$.

We explore this connection further and study how the synchronization bound in (6) varies with bath temperatures. Fig. 2(c) shows the magnitude of the steady-state power $|P^{ss}|$ (solid lines) as a function of the ratio of bath temperatures $T_{c}/T_{h}$, controlled by varying $\bar{n}_{c}$ and keeping $\bar{n}_{h}$ fixed. Strong power is achieved for vanishing $T_{c}/T_{h}$ which is expected and then decreases as $T_{c}/T_{h}$ increases. We see that $|P^{ss}|$ saturates the synchronization bound (dashed lines) when the engine is driven resonantly while in the case of finite detuning $|P^{ss}|$ is lower than the synchronization bound. As $T_{c}/T_{h}$ increases both $|P^{ss}|$ and the synchronization bound decrease until we reach an interesting point where $\bar{n}_{c}=\bar{n}_{h}$ and the system stops behaving like an engine. It can be shown that this point corresponds to $T_{c}/T_{h}=\omega_{21}/\omega_{31}=1-\eta$. This shows that the engine efficiency $\eta$ determines the point where the synchronization measure $\mathcal{S}^{max}$ and therefore $|P^{ss}|$ vanish as can be seen from Eq. (7) and is also shown in the inset of Fig. 2(c). As the temperature ratio increases past $T_{c}/T_{h}=1-\eta$ the synchronization bound becomes finite again and so does $|P^{ss}|$, and the system enters a new regime where it behaves like a fridge, where heat currents and power reverse sign and the energy starts flowing from the cold bath to the hot one as detailed in the supplemental material. Note that the synchronization bound in (6) holds true regardless of whether the system is operated as an engine or a fridge.

Discussion.— We have demonstrated an intimate connection between synchronization and nanoscale heat engines and showed how the properties of the output power can be understood as a consequence of synchronization developing in the working fluid of the engine. An explicit example of this connection is the case of three-level atoms connected to two thermal reservoirs. We have derived an upper bound on the engine power in terms of the measure of synchronization, $\mathcal{S}^{max}$. The maximum amount of synchronization as measured by $\mathcal{S}^{max}$ can be operationally understood as the maximum magnitude of steady-state power of a three-level engine when it is driven resonantly. Finally, we have showed that the engine’s efficiency $\eta$ determines the ratio of the bath temperatures where synchronization vanishes.

This connection offers, in addition, a very important route towards the experimental observation of quantum synchronization since masers are mature quantum platforms. Three-level atoms have been experimentally coupled to thermal baths using magneto-optical traps Zou et al. (2017) pumped by incoherent laser beams. We envisage however that the connection is far more general and it applies to any generic multi-level quantum engine. We hence anticipate that signatures of quantum synchronization can be observed experimentally. We note that the underlying principle of synchronization of multi-level systems with linear baths is universal and can be applied wherever a stable phase relationship between atomic levels is needed. This will pave the way for future quantum technologies based on quantum synchronization.

acknowledgements

Centre for Quantum Technologies is a Research Centre of Excellence funded by the Ministry of Education and the National Research Foundation of Singapore. This research is supported by the National Research Foundation, Prime Minister’s Office, Singapore under its Competitive Research Programme (CRP Award No. NRF-CRP14-2014-02). SV also acknowledges support from an IITB-IRCC grant number 16IRCCSG019 and a DST-SERB Early Career Research Award (ECR/2018/000957).

Supplementary Material

Measure of synchronization.— In this section we extend the synchronization measure to a general three-level system using SU(3) coherent states. We follow the SU(3) coherent state analysis due to Nemoto Nemoto (2000) where the SU(3) coherent state is

[TABLE]

where $0\leq\theta,\xi\leq\pi/2$, and $0\leq\varphi_{1},\varphi_{2}\leq 2\pi$. The SU(3) group measure is given by Nemoto (2000)

[TABLE]

which leads to the completeness relation for the coherent state $|{n_{3}}\rangle$,

[TABLE]

The steady state Husimi-Kano $Q$ function can be written as

[TABLE]

One can see that in the absence of off-diagonal elements, $Q^{ss}$ is independent of the phases $\varphi_{1}$ and $\varphi_{2}$, and hence the Eq. (11) establishes the limit cycle for a system evolving under thermal Lindbladians.

Under the action of the bare Hamiltonian,

[TABLE]

the coherent state evolves as $e^{-iH_{0}t}\arrowvert n_{3}(\theta,\xi,\varphi_{1},\varphi_{2})\rangle=\arrowvert n_{3}(\theta,\xi,\varphi_{1}-\omega_{21}t,\varphi_{2}-\omega_{31}t)\rangle$, where $\omega_{ij}\equiv\omega_{i}-\omega_{j}$. There are two parameters, phases $\varphi_{1}$ and $\varphi_{2}$, that are evolving with time, thus one has to look at the system’s preference for both of these phases. We can define the synchronization measure for a general three-level system as

[TABLE]

We can now maximize $\mathcal{S}(\varphi_{1},\varphi_{2})$ over the two phases,

[TABLE]

where we expressed the coherences in polar coordinates $\rho^{ss}_{ij}=e^{i\vartheta_{ij}}|\rho^{ss}_{ij}|$. $\mathcal{S}^{max}$ saturates this bound only when $\vartheta_{23}=\vartheta_{13}-\vartheta_{12}$ or when two of the three coherences vanish. In the main text, $\mathcal{S}(\varphi_{1},\varphi_{2})=\text{Re}[e^{i(\varphi_{2}-\varphi_{1})}\rho^{ss}_{23}]/8\pi$ and $\mathcal{S}^{max}=C_{l_{1}}(\rho^{ss})/16\pi$ because two of the coherences vanish in the steady state, namely $\rho^{ss}_{12}=\rho^{ss}_{13}=0$.

Heat and Work for Quantum Engines.— Let $\tilde{H}$ be the Hamiltonian in an arbitrary frame obtained by the unitary $U=e^{-ixt}$, ie. $\tilde{H}={U}^{\dagger}HU$, where $H$ is the Hamiltonian in the Schrödinger picture. As the expectation values of an operator is independent of the frame, we have $\langle\tilde{H}\rangle=\langle H\rangle$. By taking the derivative of $\langle\tilde{H}\rangle$ with respect to time, we get

[TABLE]

In the Schrödinger picture this becomes $\frac{d}{dt}\langle H\rangle=P+\dot{Q}$, where $\dot{Q}$ is the heat current and $P$ is the power which are respectively given by Alicki (1979)

[TABLE]

Following the above definitions, if we define heat currents and power in an arbitrary frame as

[TABLE]

then we have

[TABLE]

From Eq. (16) and Eq. (18), we can see that the heat current and power are frame dependent. Authors of reference Boukobza and Tannor (2007) pointed out a further complication and showed that at certain values of detuning Eq.(16) violates the second law of thermodynamics. To circumvent these problems, Boukobza and Tannor considered the system energy alone and redefined the heat currents and power in terms of the energy currents of the bare Hamiltonian Boukobza and Tannor (2006). The energy flux for the system alone is given by

[TABLE]

where the first term is due to the Hamiltonian part of the evolution which is defined as the power and the second term is due to the dissipation part of the evolution which is defined as the heat current.

Steady States, Heat and Work for a 3-level Quantum Engines.— The Hamiltonian describing our system in the Schrödinger picture can be written as a sum of the bare Hamiltonian and a time-dependent drive,

[TABLE]

The drive $V(t)$ has the following form,

[TABLE]

where $\sigma_{ij}\equiv|i\rangle\langle j|$. The Hamiltonian may be written in time-independent form by transforming into a rotating frame via unitary,

[TABLE]

In the rotating frame the Lindblad master equation becomes

[TABLE]

with

[TABLE]

where $\Delta\equiv\omega_{32}-\omega_{d}$. The equations of motion for the density matrix elements are

[TABLE]

where the dissipation rates are defined by

[TABLE]

Solving for the steady state by setting $\dot{\tilde{\rho}}_{ij}=0$, we get

[TABLE]

with the remaining coherences vanishing, $\tilde{\rho}^{ss}_{12}=\tilde{\rho}^{ss}_{13}=0$, and where

[TABLE]

The power and heat currents in the steady state are

[TABLE]

Using Eq. (28) the heat currents can be rewritten,

[TABLE]

We see that for $\bar{n}_{h}>\bar{n}_{c}$ we have $\dot{Q}^{ss}_{h}>0$ and $\dot{Q}^{ss}_{c}<0$ meaning the energy flows from the hot bath to the cold bath and the system behaves like an engine. On the other hand when $\bar{n}_{h}<\bar{n}_{c}$ the signs of the heat currents and power reverse and the system behaves like a fridge.