Quantum limit to nonequilibrium heat-engine performance imposed by strong system-reservoir coupling

TL;DR

This paper demonstrates that strong system-reservoir coupling in quantum heat engines imposes a fundamental quantum limit on their efficiency and power, due to induced coherence that cannot be described by standard models.

Contribution

It reveals a quantum limit on heat-engine performance caused by finite system-reservoir coupling, highlighting the role of coherence beyond traditional approximations.

Findings

Finite coupling induces coherence that hampers engine performance.

Standard Born-Markov analysis fails to capture these effects.

Quantum limit constrains efficiency and power output.

Abstract

We show that finite system-reservoir coupling imposes a distinct quantum limit on the performance of a non-equilibrium quantum heat engine. Even in the absence of quantum friction along the isentropic strokes, finite system-reservoir coupling induces correlations that result in the generation of coherence between the energy eigenstates of the working system. This coherence acts to hamper the engine's power output, as well as the efficiency with which it can convert heat into useful work, and cannot be captured by a standard Born-Markov analysis of the system-reservoir interactions.

Equations

$$H(t)=H_S(t)+\sum _j(H_{R_j}+H_{I_j}),$$

$$H_{I_j}=-{\sigma }_z\sum _{k_j}{f}_{k_j}(b_{k_j}^{†}+b_{k_j})=-{\lambda }_j{\sigma }_z(a_j^{†}+a_j),$$

$$W=$$

$$L_{dep}[\rho(t)]=\gamma_{dep}\sum_{n}\big{[}\ket{E_{n}}\bra{E_{n}},[\ket{E_{n}}\bra{E_{n}},\rho(t)]\big{]}.$$

Full text

In the same way that fundamental limitations such as the Carnot bound of classical engine cycles impose strict constraints on classical devices, the fundamental limitations of quantum mechanical cycles set the stage for what is achievable with quantum devices. We now have a good understanding of quantum heat engines in the idealized situation of adiabatic dynamics and weak coupling to heat baths Alicki (1979); Kosloff (1984); Rezek and Kosloff (2006); Kosloff and Feldmann (2002); Quan et al. (2007); Linden et al. (2010); Correa et al. (2013); Roßnagel et al. (2014); Kosloff and Levy (2014); Skrzypczyk et al. (2014); Gelbwaser-Klimovsky et al. (2015); Gelbwaser-Klimovsky and Aspuru-Guzik (2015); Hofer and Sothmann (2015); Uzdin et al. (2015, 2016); Hofer et al. (2016); Karimi and Pekola (2016); Brandner et al. (2017); Perarnau-Llobet et al. (2018); Abiuso and Giovannetti (2019); Wiedmann et al. (2019). Recent experimental realisations of nanoscale heat engines Roßnagel et al. (2016); von Lindenfels et al. (2019); Klatzow et al. (2019); de Assis et al. (2019); Peterson et al. (2019), however, operate well outside this regime, and effects resulting from quantum coherence are known to impact a machine’s efficiency. In particular, operating quantum thermodynamic cycles faster than adiabatically typically results in coherence generation during the work extraction strokes Kosloff and Feldmann (2002); Pekola et al. (2019) with correspondingly reduced efficiencies. That this effect, termed quantum friction, can be avoided by dephasing (quantum lubrication Feldmann and Kosloff (2006)) or with control protocols such as shortcuts to adiabaticity Torrontegui et al. (2013); del Campo et al. (2015) is now well established. It thus implies a technical challenge, but not necessarily a fundamental limitation.

In contrast, the impact of quantum coherence entering the cycle via the heat exchange strokes has not so far been considered. Primarily, this is because typical treatments of quantum engine cycles assume idealized weak-coupling, in which system-reservoir correlations are not explicit. In the adiabatic regime where no coherence accumulates during the work extraction strokes, this assumption also results in a reduced system state that remains diagonal in its energy eigenbasis during the heat exchange strokes. Hence, the quantum and classical cycles become largely equivalent.

Here we move beyond the limitations of the weak-coupling approach to consider the impact of system-reservoir correlations accrued as a result of finite coupling during the heat exchange strokes of a quantum Otto cycle. By remaining adiabatic along the work extraction strokes, we show that coherences resulting from system-reservoir correlations lead to performance losses even where conventional quantum friction plays no role. This constitutes a distinct finite-coupling quantum limit to non-equilibrium engine cycles that cannot be captured by a standard weak-coupling approach. Neither can it be avoided by straightforward generalisation of the techniques applied along the isentropic stokes. As we show, for finite system-reservoir coupling the appropriate basis for employing quantum lubrication comprises both system and reservoir components, and is thus difficult to identify experimentally. Likewise, control protocols are challenging to analyse even theoretically beyond the weak-coupling limit. Alternatively, dynamical decoupling could be employed to prevent the growth of system-reservoir correlations, but as this effectively decouples the system and reservoir, it would also simply stop the machine from working. Our work therefore reveals a crucial restriction on quantum engine performance, which is not yet clearly avoidable even in principle.

We consider a two-level system (TLS) interacting separately with two heat reservoirs, at temperatures $T_{h}$ and $T_{c}$ ($T_{h}>T_{c})$. The Otto cycle consists of four strokes labelled by eight points: $A^{\prime}BB^{\prime}CC^{\prime}DD^{\prime}A$. Hot isochore: at $A^{\prime}$ the TLS is coupled to the hot reservoir with which it interacts for a time $\tau_{i}$ to reach $B$. The interaction is then switched off instantaneously ($B^{\prime}$). Isentropic expansion: the TLS Hamiltonian, $H_{S}$, is tuned over a time $\tau$ to reduce the gap between the two energy eigenvalues, reaching point $C$. Now, the TLS-cold reservoir interaction is switched on suddenly ($C^{\prime}$). Cold isochore: the TLS interacts with the cold reservoir for a time $\tau_{i}$, to reach point $D$. The system is then decoupled from the cold reservoir ($D^{\prime}$). Isentropic compression: $H_{S}$ is tuned for a time $\tau$ to increase the gap between energy eigenvalues back to the level at $A^{\prime}$, reaching point $A$. The cycle is completed by turning on the interaction with the hot reservoir ($A^{\prime}$). These strokes are repeated until a limit cycle is reached Kosloff and Rezek (2017).

The full Hamiltonian reads

[TABLE]

where $H_{S}(t)=\epsilon(t)\sigma_{z}/2+\Delta(t)\sigma_{x}/2$, $H_{R_{j}}=\sum_{k}\omega_{k_{j}}b_{k_{j}}^{\dagger}b_{k_{j}}$, and $H_{I_{j}}=-\sigma_{z}\sum_{k_{j}}f_{k_{j}}(b_{k_{j}}^{\dagger}+b_{k_{j}})$ for $j=h,c$, denoting the hot ($h$) and cold ($c$) reservoir. Here, $\sigma_{z}$ and $\sigma_{x}$ are the usual TLS Pauli operators. The time dependence arises over the isentropic strokes when the splitting $\mu(t)=\sqrt{\epsilon(t)^{2}+\Delta^{2}(t)}$ between the energy eigenlevels of $H_{S}(t)$ is tuned. We label the values of $\mu$ during the hot or cold stage of the cycle as $\mu_{h}$ and $\mu_{c}$, respectively. Bosonic reservoir annihilation operators for excitations at frequencies $\omega_{k_{j}}$ are given by $b_{k_{j}}$. The TLS-reservoir coupling is via $H_{I_{j}}$ with strengths $f_{k_{j}}$, and is present only during the relevant isochore.

Studies of quantum Otto cycles generally invoke the weak coupling assumption, i.e. that the interaction terms are (negligibly) small. This leads to a tractable analysis in terms of a quantum state of the TLS and reservoirs approximated to remain in tensor product form at all times. In the finite coupling regime of interest, where the interaction terms cannot be neglected, it is a more involved task to compute the state around the cycle Perarnau-Llobet et al. (2018); Abiuso and Giovannetti (2019); Wiedmann et al. (2019). In order to access this regime, we extend the reaction-coordinate (RC) formalism  Nazir and Schaller (2019); Iles-Smith et al. (2016, 2014); Newman et al. (2017); Strasberg et al. (2016); Schaller et al. (2018) from the infinite time Otto cycle considered in Ref. Newman et al. (2017) to inherently non-equilibrium, finite time cycles. In this approach, the interaction terms in Eq. (1) are unitarily mapped to collective modes (the RCs) such that

[TABLE]

where $a_{j}$ annihilates an excitation in the RC mode for reservoir $j$ with natural frequency $\Omega_{j}$, and $\lambda_{j}=({\sum_{k}f_{k_{j}}^{2}})^{1/2}$. The reservoir Hamiltonians become $H_{R_{j}}=\Omega_{j}a_{j}^{\dagger}a_{j}+\sum_{k_{j}}g_{k_{j}}(a_{j}^{\dagger}+a_{j})(r_{k_{j}}^{\dagger}+r_{k_{j}})+\sum_{k_{j}}\nu_{k_{j}}r_{k_{j}}^{{\dagger}}r_{k_{j}}$, while the system Hamiltonian remains unchanged. Here, $r_{k_{j}}$ annihilates an excitation at frequency $\nu_{k_{j}}$ in a redefined residual reservoir which interacts weakly with the corresponding RC mode via couplings $g_{k_{j}}$. The full Hamiltonian becomes $H(t)=H_{S^{\prime}}(t)+\sum_{j}H_{R^{\prime}_{j}}$, where $H_{S^{\prime}}(t)={\epsilon(t)}\sigma_{z}/2+{\Delta(t)}\sigma_{x}/2-\sum_{j}\lambda_{j}\sigma_{z}(a_{j}^{\dagger}+a_{j})+\Omega_{j}a_{j}^{\dagger}a_{j}$, and $H_{R^{\prime}_{j}}=\sum_{k_{j}}g_{k_{j}}(a_{j}^{\dagger}+a_{j})(r_{k_{j}}^{\dagger}+r_{k_{j}})+\sum_{k_{j}}\nu_{k_{j}}r_{k_{j}}^{\dagger}r_{k_{j}}$. The RC mapping involves an enlarged view of a redefined system $S^{\prime}$ whose self-energy $H_{S^{\prime}}(t)$ now additionally incorporates the self-energies of the reservoir RCs as well as the TLS-RC coupling terms. The remaining terms represent residual environments $R^{\prime}_{j}$ and their interactions with the corresponding RC, which may be treated as Markovian Iles-Smith et al. (2016, 2014). Nevertheless, through the RCs, correlations between the TLS and the reservoirs will form and indeed persist even in the limit cycle.

In the original representation we characterise the TLS-reservoir interactions via spectral densities $J_{j}(\omega)\equiv\sum_{k}{f^{2}_{k_{j}}}\delta(\omega-\omega_{k_{j}})={\alpha\omega\omega_{c}}/{({\omega}^{2}+{\omega}_{c}^{2})}$, taken to be the same for each reservoir, with coupling strength $\alpha$ and cutoff frequency $\omega_{c}$. As the residual reservoirs are traced out when deriving a master equation for the enlarged system $S^{\prime}$, to determine the RC mapping one simply needs to find the spectral density $\tilde{J_{j}}(\nu)\equiv\sum_{k}{g^{2}_{k_{j}}}\delta(\nu-\nu_{k_{j}})$ that characterizes the coupling between $S^{\prime}$ and the residual reservoirs, as well as $\Omega_{j}$ and $\lambda_{j}$, such that the Heisenberg equations of motion for operators in the TLS subspace are equivalent in both pictures. Imposing this results in $\Omega_{j}=2\pi\gamma\omega_{c}$, $\lambda_{j}=\sqrt{{\pi\alpha\Omega_{j}}/{2}}$, and $\tilde{J}_{j}(\nu)=\nu{\sqrt{\epsilon^{2}+\Delta^{2}}}/{2\pi\omega_{c}}$ Iles-Smith et al. (2014).

The master equation governing the dynamics of $S^{\prime}$ during the $j$ ($=h,c$) isochore is $\dot{\rho}(t)=L_{j}[\rho(t)]$, with $L_{j}[\rho(t)]\equiv-i[H_{S^{\prime}_{j}},\rho(t)]-[A_{j},[\chi_{j},\rho(t)]]+[A_{j},\{\Xi_{j},\rho(t)\}]$ and $\rho(t)$ the state of the TLS plus both RCs. Here, $A_{j}=a_{j}+a_{j}^{\dagger}$, the self-Hamiltonian $H_{S^{\prime}_{j}}$ only includes interaction terms for the $j$ RC, $\chi_{j}=\gamma\int^{\infty}_{0}d\tau\int^{\infty}_{0}d\omega\omega\cos(\omega\tau)\coth\left(\frac{\beta_{h}\omega}{2}\right)A_{j}(-\tau)$, and $\Xi_{j}=\gamma\int^{\infty}_{0}d\tau\int^{\infty}_{0}d\omega\cos(\omega\tau)\left[H_{S_{j}^{\prime}},A_{j}(-\tau)\right]$, for $A_{j}(\tau)\equiv e^{iH_{S^{\prime}_{j}}\tau}A_{j}e^{-iH_{S^{\prime}_{j}}\tau}$ Iles-Smith et al. (2014).

Thermodynamic treatments of the Otto cycle usually consider thermal reservoir resources. We wish to isolate strong-coupling effects from any due to coupling to heat reservoirs that are out of equilbrium at the start of each isochore. To this end, we include a mechanism in our finite time cycle to ensure that any uncoupled reservoir returns to thermal equilibrium by the time it is coupled to the TLS once more. For the uncoupled reservoir $j$, which has been driven out of equilibrium during the previous isochore, to return to thermal equilibrium at temperature $T_{j}$ while the other reservoir and TLS interact, we can add terms to the master equation that act only on the uncoupled RC and hence do not depend on the full system plus RC eigenstructure. The uncoupled RC is a harmonic oscillator so we add standard dissipators $L_{d_{j}}[\rho(t)]=\gamma_{d}(N_{j}+1){\mathcal{L}}_{a_{j}}[\rho(t)]+\gamma_{d}N_{j}{\mathcal{L}}_{a^{\dagger}_{j}}[\rho(t)]$, where ${\mathcal{L}}_{O}[\rho(t)]=O\rho(t)O^{\dagger}-\frac{1}{2}\{O^{\dagger}O,\rho(t)\}$ and $N_{j}=(e^{\beta_{j}\Omega_{j}}-1)^{-1}$, with $\beta_{j}=1/T_{j}$ Breuer and Petruccione (2002); Carmichael (1999) . We choose $\gamma_{d}$ to ensure thermalisation occurs over a timescale such that the TLS re-couples to a thermal reservoir at the start of the subsequent isochore.

The cycle work output is given by the net energy change of the system across each of the isentropic strokes. For strong coupling, this also involves accounting for the energetic costs associated with turning off the interaction term at the end of the isochores; there are no coupling costs due to rethermalisation of the RCs when uncoupled. This leads to work output Newman et al. (2017)

[TABLE]

where $h$ and $c$ superscripts indicate that the TLS splitting is set to $\mu_{h}$ and $\mu_{c}$, respectively. The density operator $\rho$ is labelled with superscripts $A-D$ indicating the various points around the cycle and as before represents the state of the enlarged $S^{\prime}$. The energy transferred into the system during the hot isochore is given by $Q=\textrm{tr}[H_{S^{\prime}_{h}}^{h}\rho^{B}]-\textrm{tr}[H_{S^{\prime}_{h}}^{h}\rho^{A}]$, and we shall term this heat. Through $H_{S^{\prime}_{h}}$ this expression contains contributions from the TLS-hot RC interaction energy as well as from the hot RC being pulled out of equilibrium, both of which are neglected in weak-coupling treatments.

To evaluate the heat and net work output we need to compute the states $\rho^{A-D}$ of the enlarged system $S^{\prime}$ when the engine is operating in the limit cycle. In the infinite time (equilibrium) version of the cycle, this involves taking the steady state solution of the RC master equation along each isochore, followed by unitary evolution along the isentropic strokes. For the non-equilibrium case considered here, the calculation is more involved and we must numerically solve the dynamical equation of motion for the full state of $S^{\prime}$ for a particular isochore time $\tau_{i}$.

During the isentropic strokes we tune $\epsilon(t)$ and $\Delta(t)$ such that $(\epsilon_{h},\Delta_{h})\leftrightarrow(\epsilon_{c},\Delta_{c})$, with [$H_{S}(t),H_{S}(t^{\prime})]=0$. This ensures that the TLS Hamiltonians at the start of the stroke, $H_{S}(0)$, and at the end, $H_{S}(\tau)$, share a common energy eigenbasis. The TLS quantum state then adiabatically follows the change in splitting and no coherence develops even when the stroke is carried out in a finite time $\tau$. We thus avoid any quantum friction along the isentropic strokes, and any variations in performance due to the generation of coherence can instead be attributed to the isochores. For the purposes of maximising power output, it is then preferable to complete this stroke quickly and we consider the limit $\tau\rightarrow 0$.

In Fig. 1 (a) we plot the engine’s power output and efficiency at finite coupling. The power output initially increases with coupling strength until a turnover is reached as reservoir decoupling costs begin to dominate over the increase in work output. Note that we are considering here a finite isochore time, $\tau_{i}$, shorter than that necessary for a stationary state to be reached along the isochores. The energy absorbed from the hot reservoir increases with coupling strength too and this leads to an engine efficiency which decreases monotonically with coupling strength. A similar finding is also reflected in the analysis of the strong coupling regime for a heat engine in Ref. Perarnau-Llobet et al. (2018) but there the isochores are long enough that the engine is considered to have reached arbitrarily close to equilibrium. In the present case, we show that in fact power output can be maximised before this equilibrium has been reached in Fig. 1 (b). Here we plot the work and power outputs as a function of the isochore time $\tau_{i}$ for an intermediate coupling strength. As the system $S^{\prime}$ approaches equilibrium, the work output saturates. Power output, however, is maximised before this equilibrium is reached. If the desired engine metric is how much power can be produced, it is thus preferable to operate out of equilibrium, by choosing shorter isochore times and intermediate coupling strengths.

We now wish to explore the quantum nature of the heat engine, and specifically isolate the effects on engine performance of system-reservoir correlations accrued during the (heat exchange) isochoric strokes as a result of finite coupling with the reservoirs. These correlations manifest themselves in finite coherences in the working system state. We shall therefore make a distinction between a fully quantum version of the cycle, where system-reservoir correlations lead to quantum coherence being generated along the isochores, and one where coherence is prevented from accumulating. In this latter version of the cycle, we introduce into the master equation terms that induce pure dephasing Breuer and Petruccione (2002); Schlosshauer (2007); Tempel and Aspuru-Guzik (2011); Rebentrost et al. (2009); Feldmann and Kosloff (2006) in the energy eigenbasis of the working system, while taking care that these have no energetic contribution to the overall evaluation of work output or energy exchange with the reservoirs. To meet with these criteria, the pure dephasing terms must commute with the mapped Hamiltonian $H_{S^{\prime}}$, which we diagonalise and write as $H_{S^{\prime}}=\sum_{n}E_{n}\ket{E_{n}}\bra{E_{n}}$. We then construct a pure dephasing Liouvillian

[TABLE]

We are free to choose the value of $\gamma_{dep}$ to ensure dephasing occurs on an appropriate timescale, such that coherence is prevented from developing during the isochoric strokes of the engine. We can then compute the states of the working system at various points around the cycle as previously described, but with the addition of these terms to both isochores. We define this as the incoherent engine. We stress that these terms achieve pure dephasing in the energy eigenbasis of the enlarged TLS plus RCs (i.e. $H_{S^{\prime}}$) rather than just the TLS energy eigenbasis. This is the natural choice for a system interacting strongly with an environment. Introducing pure dephasing terms that only act on the TLS would be inappropriate since they do not commute with the unitary part of the master equation, and thus have a non-zero energetic contribution.

We compare these two versions of the Otto cycle and analyse the effect of quantum coherence on engine performance in Fig. 2 as a function of isochore time $\tau_{i}$. At large $\tau_{i}$, the incoherent and coherent engines converge. Here, the state of $S^{\prime}$ approaches thermal equilbrium with the relevant residual reservoir which is maintained at the hot or cold temperature. This state is then diagonal in the energy eigenbasis of $S^{\prime}$, and so no coherence is present in either type of engine at points $B$ or $D$ if the isochore is long enough. For shorter times, however, this is not the case, and pure dephasing does have an appreciable effect on the engine metrics. The dephased engine absorbs more heat along the hot isochore (not shown) but outputs a net work large enough to compensate, yielding a higher efficiency at short times than the fully coherent engine. Decoupling costs remain comparable. These are dominated by the RC being driven out of equilibrium along the isochores, which is not prevented by dephasing. In Fig. 3 we show this improved engine performance in parametric plots of power versus efficiency for a selection of coupling strengths. Even at weaker but finite coupling, where coherence generation is less severe, an improvement over the coherent engine can be achieved by dephasing.

In a weak coupling treatment of the Otto cycle the TLS equilibrates to a Gibbs state for long isochore times, which is diagonal in the energy eigenbasis. With the isentropic strokes carried out adiabatically, populations in these energy eigenstates are kept constant and no coherence accumulates in the working system state. In fact, if the isentropic strokes remain adiabatic when the cycle is treated at finite time (as we consider), then no coherence is present during the weak-coupling cycle at any time. At non-negligible system-reservoir interaction strengths, however, correlations that are generated between the TLS and reservoirs lead to quantum coherence in the energy eigenbasis at the end of the isochores. This coherence will, in general, persist during the isentropic strokes even if they are performed adiabatically. In this way, the strongly coupled TLS Otto cycle is inherently quantum in nature.

We further illustrate how dephasing in the TLS plus RC basis accelerates the process of equilibration in Fig. 4. Of crucial importance for work calculations are the TLS population differences in the eigenbasis of $H_{S}$ [see Eq. (3)]. In Fig. 4 we show how these are affected by dephasing at point B, just before the hot adiabatic stroke begins, and at point D, just before the cold adiabatic stroke begins. In a weak-coupling analysis the TLS populations would be completely insensitive to pure dephasing (which for weak coupling would be defined relative to $H_{S}$ to posses no energetic contribution). In stark contrast, for the finite coupling theory presented here, TLS populations are significantly affected by pure dephasing in the correct strong-coupling basis of $H_{S^{\prime}}$. Specifically, the dephased engine displays a faster approach to the steady state than the fully coherent engine. At point B there is therefore a smaller population difference (a) at shorter $\tau_{i}$ and at point D a larger population difference (b). This results in greater work extracted during the hot adiabat (c) and smaller work invested during the cold adiabat (d) for the dephased engine. The reason is that at point B we wish to have as much population as possible in the excited state as this implies a higher (effective) TLS temperature. This entails that more heat has been absorbed from the hot reservoir during the hot isochore and that the subsequent hot adiabatic stroke can extract a larger amount of work. At point D, on the other hand, it is desirable to have as little population in the excited state as possible. This corresponds to a lower temperature, which means that as much heat as possible has been dissipated into the cold reservoir and that on the subsequent adiabatic stroke, the work performed on the system can be kept to a minimum.

To summarise, we have shown that finite system-reservoir coupling imposes a bound on quantum heat engine performance that is not captured by standard weak-coupling approaches. Our results highlight that even when work extraction is adiabatic, quantum coherence enters the cycle through finite system-reservoir interactions. Protocols designed to avoid quantum friction along the work strokes are not straightforwardly extendable to suppress the generation of coherence along the isochores. We thus expect our findings to be of broad relevance to practical realisations of quantum engine cycles, and to stimulate a concerted effort to devise schemes to mitigate this quantum disadvantage.