Optimal efficiency of a noisy quantum heat engine

In this article we use optimal control to maximize the efficiency of a quantum heat engine executing the Otto cycle in the presence of external noise. We optimize the engine performance for both amplitude and phase noise. In the case of phase damping we additionally show that the ideal performance of a noiseless engine can be retrieved in the adiabatic (long time) limit. The results obtained here are useful in the quest for absolute zero, the design of quantum refrigerators that can cool a physical system to the lowest possible temperature. They can also be applied to the optimal control of a collection of classical harmonic oscillators sharing the same time-dependent frequency and subjected to similar noise mechanisms. Finally, our methodology can be used for the optimization of other interesting thermodynamic processes.

Figure 1

(a) For ${\gamma }_a=0,{\omega }_h{\gamma }_p=0.01$ (phase damping), and ${\omega }_c/{\omega }_h=1/3$ we plot the optimal parameter $\delta$ (blue solid line), as well as the same quantity for the frequency profile (15) used in Ref. [21] and for durations $T_n,n=1,2,3,4,5$ from (16) (red circles). Observe that the optimized $\delta$ is lower, corresponding to a higher efficiency. Note that there is a minimum necessary time for a feasible solution of the optimization problem, ${\omega }_{h}T=1.85$. For large $T$ parameter $\delta$ approaches zero, the ideal value corresponding to the noiseless case. The reason is that for phase damping and long-enough durations, the evolution of the system can take place close to a noise-free path. (b) For both cases we display a measure of the undesirable remaining energy in the $L,C$ modes. Observe that this parasitic energy is lower in the optimized case (blue solid line).

Figure 2

For ${\gamma }_a=0,{\omega }_h{\gamma }_p=0.01$, and ${\omega }_c/{\omega }_h=1/3$ we plot the numerically obtained optimal frequency profile for various values of the normalized time. Observe that for shorter available times the optimal frequency takes values on the boundary of the allowed region (23), while for larger times it is smoother.

Figure 3

(a) For ${\omega }_h{\gamma }_a=0.02,{\gamma }_p=0$ (amplitude noise), and ${\omega }_c/{\omega }_h=1/3$ we plot the optimal parameter $\delta$ (blue solid line), as well as the same quantity for the frequency profile (15) used in Ref. [21] and for durations $T_n,n=1,2,3,4,5$ from (16) (red circles). Again, the optimized $\delta$ is lower, corresponding to a higher efficiency, and there is a minimum necessary time for a feasible solution of the optimization problem, ${\omega }_{h}T=1.89$. For small $T$ the quantity $\delta$ increases due to the limited available time, while for large $T$ increases since for amplitude noise the evolution does not take place along a noise-free path. As a result, there is an intermediate time $T$ where $\delta$ is minimized. (b) The parasitic energy is again lower in the optimized case (blue solid line).

Figure 4

For ${\omega }_h{\gamma }_a=0.02,{\gamma }_p=0$, and ${\omega }_c/{\omega }_h=1/3$ we plot the numerically obtained optimal frequency profile for various values of the normalized time. Observe that for shorter available times the optimal shape is similar to the corresponding cases of Fig. 2, while for larger times where relaxation dominates it differs.