Power and efficiency of a thermal engine with a coherent bath

We consider a quantum engine driven by repeated weak interactions with a heat bath of identical three-level atoms. This model was first introduced by Scully et al. [Science 299, 862 (2003)], who showed that coherence between the energy-degenerate ground states serves as a thermodynamic resource that allows operation of a thermal cycle with a coherence-dependent thermalization temperature. We consider a similar engine out of the quasistatic limit and find that the ground-state coherence also determines the rate of thermalization, therefore increasing the output power and the engine efficiency only when the thermalization temperature is reduced; revealing a more nuanced perspective of coherence as a resource. This allows us to optimize the output power by adjusting the coherence and relative stroke durations.

Figure 1

The Otto cycle shown for the variables $\overline{n}$ and $\omega$, and as $T\text{−}S$ diagram (inset). Strokes are described in Sec. 3. Isotherms at temperatures $T_h$ (red) and $T_c$ (blue) are shown. The work extracted in a single cycle is the area enclosed by the rectangle, $W=\hslash ({\omega }_h-{\omega }_c)\left({\overline{n}}_h-{\overline{n}}_c\right)$. The heat input to the system is the energy change of the system during the isochoric heating stroke, $Q_h=E_1=\hslash {\omega }_h({\overline{n}}_h-{\overline{n}}_c)$.

Figure 2

The energy cost of the isentropic expansion $V_e$ and compression $V_c$ strokes, using the shortcut-to-adiabaticity protocol. The energy cost is scaled by $1/\hslash \omega$, where $\omega ={10}^{14}\mathrm{Hz}$. Recall that $t_{\text{cycle}}$ is in terms of the dimensionless time parameter (7). The cost is large for short cycle times and approaches zero as they become longer in duration. In this plot we used a hot bath with temperature ${\overline{n}}_{\mathrm{ss}}=2$, and a cold bath with ${\overline{n}}_{\mathrm{ss}}=0.55$, both of which were incoherent. The system interacted with the hot bath for duration $t_h$ and with the cold bath for $t_c$; and we chose $t_h=t_c=t_{\text{cycle}}/4$. Expansion carries a larger cost than compression because it begins with the system at a hotter temperature, and $V\left(\tau \right)\propto \overline{n}\left(0\right)$ (30). The compression stroke begins with the steady cycle initial photon number ${\overline{n}}_{sc}$ (33); the expansion stroke begins with initial photon number ${\overline{n}}_h$, which is found by evolving ${\overline{n}}_{sc}$ under Eq. (13) for time $t_h$. The integration in Eq. (30) was completed numerically.

Figure 3

The average photon number $\overline{n}$ as the system reaches a steady cycle. Regardless the initial average photon number ${\overline{n}}_0$, $\overline{n}$ eventually reaches a closed cycle (36). The photon number is constant during the work strokes, since the shortcut-to-adiabaticity protocol holds the energy populations constant. Each stroke takes a quarter of the total cycle time: $t_{\text{cycle}}/4$.

Figure 4

Diagram and table of three different pairs of baths, all with the same effective temperatures. Temperature according to thermal populations in hard line, effective temperature in broken lines. The coherent hot (CH) baths involve using coherence to create a hot bath from two cold baths; the coherent cold (CC) baths use coherence to create a cold bath from two hot baths. Without the use of coherence, these latter two bath systems would not be usable for running a thermal engine.

Figure 5

The cost of the shortcut-to-adiabaticity protocol $V_e+V_c$ [see Eq. (30) and Fig. 2] for the three pairs of thermal baths. The total energy cost is scaled by $1/\hslash \omega$, where $\omega ={10}^{14}\mathrm{Hz}$. Recall that $t_{\text{cycle}}$ is in terms of the dimensionless time parameter (7). The three pairs of thermal baths are described in Fig. 4. Both the coherent cold (CC) and the coherent hot (CH) systems have a reduced cost derived from their coherence (see Appendix pp2-s2 for details). The cost is large for $t_{\text{cycle}}\to 0$. As $t_{\text{cycle}}\to \infty$, the cost $V_e+V_c\to 0$. Here we used a hot bath with temperature ${\overline{n}}_{\mathrm{ss}}=2$, and a cold bath with ${\overline{n}}_{\mathrm{ss}}=0.55$.

Figure 6

Efficiency curves (43) of the different baths. Recall that $t_{\text{cycle}}$ is in terms of the dimensionless time parameter (7). The three pairs of thermal baths are described in Fig. 4. The coherent cold CC always has the highest efficiency [Eq. (45)]. The coherent hot CH has the worst [Eq. (47)]. Before $t^{\prime }$, the work extracted is less than the energy cost of the shortcut-to-adiabaticity protocol, $W<V_e+V_c$. With longer cycle times, the cost of implementing the shortcut-to-adiabaticity protocol becomes negligible, and the efficiency approaches its maximum. Here we used a hot bath with temperature ${\overline{n}}_{\mathrm{ss}}=2$, and a cold bath with ${\overline{n}}_{\mathrm{ss}}=0.55$.

Figure 7

The power curves (49) for the three pairs of heat baths. The power is scaled by $1/\hslash \omega$, where $\omega ={10}^{14}\mathrm{Hz}$. Recall that $t_{\text{cycle}}$ is in terms of the dimensionless time parameter (7). The three pairs of thermal baths are described in Fig. 4. The CC baths always have a higher output power than the incoherent baths (51). The CH baths do not necessarily have a lesser output, but they do for these temperatures (see Appendix pp2-s4 for details). As $t_h,t_c\to \infty$, the difference ${\overline{n}}_h-{\overline{n}}_c\to \frac{E_h}{{\mathrm{\Delta }}_h}-\frac{E_c}{{\mathrm{\Delta }}_c}$, $W$ approaches a constant value and $P\to 0$. Before $t^{\prime }$, the engine is too costly to run as the cost of using shortcut-to-adiabaticity techniques is larger than the work extracted $W<V_e+V_c$. Here we used a hot bath with temperature ${\overline{n}}_{\mathrm{ss}}=2$, and a cold bath with ${\overline{n}}_{\mathrm{ss}}=0.55$.

Figure 8

The power curves of the engines can be increased by changing the relative times of the four strokes of the Otto cycle. This will change the amount of heat transfer to the system in each cycle, and alter the energy cost of using shortcut-to-adiabaticity techniques. The power is scaled by $1/\hslash \omega$, where $\omega ={10}^{14}\mathrm{Hz}$. The three pairs of thermal baths are described in Fig. 4. The cycle time $t_{\text{cycle}}$ is parametrized in Eqs. (54) and (55). Here we numerically find the optimal values of $p,q,r$ for the CC baths. These are three cross sections of the function Max Power $(p,q,r)$, one for each variable (56). In panel (a), $q=r=0.5$, and the peak occurs at $p\approx 0.58$ for the CC baths. In panel (b), $p=0.58$, $r=0.5$ and the peak occurs at $q\approx 0.69$. In panel (c), $p=0.58,q=0.69$ and the peak occurs at $r\approx 0.59$. See Fig. 9 for the increase in the output power of the heat engine using these values. Here we used a hot bath with temperature ${\overline{n}}_{\mathrm{ss}}=2$, and a cold bath with ${\overline{n}}_{\mathrm{ss}}=0.55$.

Figure 9

The output power can be improved by altering the relative times of each stroke of the Otto cycle. See Fig. 8 for the numerical search for the optimal values of $p,q,r$ for the CC baths. Power with $p=0.58$, $q=0.69$, and $r=0.59$ in thick lines, showing noticeable enhancement. Dotted lines show $p=q=r=0.5$ (where the duration of all strokes is $t_{\text{cycle}}/4$, i.e., Fig. 7). The power is scaled by $1/\hslash \omega$, where $\omega ={10}^{14}\mathrm{Hz}$. Recall that $t_{\text{cycle}}$ is in terms of the dimensionless time parameter (7). The three pairs of thermal baths are described in Fig. 4. Here we used a hot bath with temperature ${\overline{n}}_{\mathrm{ss}}=2$, and a cold bath with ${\overline{n}}_{\mathrm{ss}}=0.55$.