Coherence effects in the performance of the quantum Otto heat engine

The working substance fueling a quantum heat engine may contain coherence in its energy basis, depending on the dynamics of the engine cycle. In some models of quantum Otto heat engines, energy coherence has been associated with entropy production and quantum friction. We considered a quantum Otto heat engine operating at finite time. Coherence is generated and the working substance does not reach thermal equilibrium after interacting with the hot heat reservoir, leaving the working substance in a state with residual energy coherence. We observe an interferencelike effect between the residual coherence (after the incomplete thermalization) and the coherence generated in the subsequent finite-time stroke. We introduce analytical expressions highlighting the role of coherence and examine how this dynamical interference effect influences the engine performance. Additionally, in this scenario in which coherence is present along the cycle, we argue that the careful tuning of the cycle parameters may exploit this interference effect and make coherence acts like a dynamical quantum lubricant. To illustrate this, we numerically consider an experimentally feasible example and compare the engine performance to the performance of a similar engine where the residual coherence is completely erased, ruling out the dynamical interference effect.

Figure 1

Engine cycle. (a) The working substance begins in the cold thermal state ${\rho }_0^{\text{eq,c}}$, with Hamiltonian $H_0$ and inverse temperature ${\beta }_{\text{c}}$. The working substance is driven by an adiabatic expansion which changes the Hamiltonian to $H_{{\tau }_1}$ and leads to the state ${\rho }_{{\tau }_1}$ at time ${\tau }_1$. The second stroke is comprised by hot thermalization, where the working substance interacts with a hot heat reservoir. The interaction time between the working substance and the heat reservoir is sufficiently small such that the thermalization is incomplete. The third stroke is an adiabatic compression changing the Hamiltonian from $H_{{\tau }_1}$ back to $H_0$. The fourth stroke is a complete thermalization with the cold heat reservoir. (b) Representation of the cycle with a dephasing operation in the energy basis after the incomplete thermalization. For this dephased engine, the states at the end of each stroke are ${\rho }_0^{\text{eq,c}},{\rho }_{{\tau }_1},{\rho }_{{\tau }_2}^{\text{deph}}$, and ${\rho }_{{\tau }_3}^{\text{deph}}$. This dephasing operation does not cost energy and erases the residual coherence of the engine.

Figure 2

Dynamics of the coherences. (a) The relative entropy of coherences of the key-working-substance states $\mathcal{C}\left({\rho }_{{\tau }_1}\right)$ (blue dashed line), $\mathcal{C}\left({\rho }_{{\tau }_2}\right)$ (red dash-dotted line), and $\mathcal{C}\left({\rho }_{{\tau }_3}\right)$ (green solid line) for the original and $\mathcal{C}\left({\rho }_{{\tau }_3}^{\text{deph}}\right)$ (purple dotted line) for dephased engine cycles. Additionally, the energy transition probability ${\xi }^{\text{exp}}\left({\tau }_1,0\right)$ (orange long-dashed line) is also shown. All quantities are plotted assuming a fixed thermalization time ${\tau }_{\text{therm}}^{\text{h}}=75.15\text{ms}$ and varying the driving time ${\tau }_1$. The engine works as a heat engine for the parameters on the right of the dashed vertical line. (b) The relative entropy of coherence of the relevant key-working-substance states as a function of the thermalization time ${\tau }_{\text{therm}}^{\text{h}}$ for fixed driving time ${\tau }_1=0.46\text{ms}$. (c) The population component of the thermal divergence of the key-working-substance states, $D\left[\varepsilon \left({\rho }_{{\tau }_1}\right)\left|\right|{\rho }_{{\tau }_1}^{\text{eq,h}}\right]$ (blue dashed line), $D\left[\varepsilon \left({\rho }_{{\tau }_2}\right)\left|\right|{\rho }_{{\tau }_2}^{\text{eq,h}}\right]$ (red dash-dotted line), and $D\left[\varepsilon \left({\rho }_{{\tau }_3}\right)\left|\right|{\rho }_{{\tau }_3}^{\text{eq,c}}\right]$ (green solid line) for the original and $D\left[\varepsilon \left({\rho }_{{\tau }_3}^{\text{deph}}\right)\left|\right|{\rho }_{{\tau }_3}^{\text{eq,c}}\right]$ (purple dotted line) for the dephased engine cycles. All quantities are plotted assuming a fixed thermalization time ${\tau }_{\text{therm}}^{\text{h}}=75.15\text{ms}$ and varying the driving time ${\tau }_1$. (d) The population component of the thermal divergence of the key-working-substance states as a function of the thermalization time ${\tau }_{\text{therm}}^{\text{h}}$ for fixed driving time ${\tau }_1=0.46\text{ms}$. The map $\varepsilon$ is the full dephasing operation as explained in Eq. (5). All entropic quantities are computed in the natural unit of information (nat).

Figure 3

Entropy production and quantum friction as a function of the driving time ${\tau }_1$ (a) and thermalization time ${\tau }_{\text{therm}}^{\text{h}}$ (b). Total entropy production of the engine with and without dynamical interference is $〈{\mathrm{\Sigma }}_{\text{total}}〉$ and $〈{\mathrm{\Sigma }}_{\text{total}}^{\text{deph}}〉$, respectively. Similarly, the quantum friction of the engine with and without dynamical interference is $\mathcal{F}$ and ${\mathcal{F}}^{\text{deph}}$, respectively. All entropic quantities are computed in the natural unit of information (nat).

Figure 4

Efficiency and power output. (a, b) The engine efficiency and power output as a function of the driving time ${\tau }_1$ for the fixed thermalization time ${\tau }_{\text{therm}}^{\text{h}}=75.15\text{ms}$, respectively. (c, d) The engine efficiency and power output as a function of the thermalization time ${\tau }_{\text{therm}}^{\text{h}}$ for the fixed driving time ${\tau }_1=0.46\text{ms}$, respectively. In all figures, the efficiency and power output of the original and dephased engine cycle are depicted in red solid line and blue dashed line, respectively.

Figure 5

Change in the eigenenergies during one cycle. For $n=0$ [red (upper) line] and $n=1$ [blue (lower) line] the curves show the value of the instantaneous eigenenergies during one cycle. From the parameters considered in Sec. 3, the relation between the initial and final frequencies is ${\omega }_{{\tau }_1}=1.8\times {\omega }_0$. With this information, we plotted the renormalized instantaneous eigenenergies $E_n\left(t\right)/\hslash {\omega }_0$ as a function of time. Additionally, we depicted each stroke as well as represented the Hamiltonian at the four key instants of time.