Boosting work characteristics and overall heat-engine performance via shortcuts to adiabaticity: Quantum and classical systems

Under a general framework, shortcuts to adiabatic processes are shown to be possible in classical systems. We study the distribution function of the work done on a small system initially prepared at thermal equilibrium. We find that the work fluctuations can be significantly reduced via shortcuts to adiabatic processes. For example, in the classical case, probabilities of having very large or almost zero work values are suppressed. In the quantum case, negative work may be totally removed from the otherwise non-positive-definite work values. We also apply our findings to a micro Otto-cycle-based heat engine. It is shown that the use of shortcuts, which directly enhances the engine output power, can also increase the heat-engine efficiency substantially, in both quantum and classical regimes.

Figure 1

Work function for a parametric oscillator with $\beta =0.2$, ${\omega }_i=10$, and ${\omega }_f=10\sqrt{3}$, with all variables in scaled and hence dimensionless units. The solid (blue) line is for STA, with ${\omega }_i\tau =0.001$ and $\omega \left(t\right)$ chosen to be $\omega \left(t\right)={\omega }_i\sqrt{(a^2+1)/2-[(a^2-1)/2]\cos \left[\pi (t/\tau )\right]}$, with $a={\omega }_f/{\omega }_i=\sqrt{3}$. The dashed (red) line is for a bare process of the same duration. The same results are shown in the inset using a semilogarithmic plot.

Figure 2

Numerical average value of $\langle e^{-\beta W}\rangle$ vs the number of classical trajectories used, with ${\omega }_i=10$, $\beta =0.2$, and ${\omega }_i\tau =0.001$. The upper red (lower blue) line is for a bare nonadiabatic process (STA). The horizontal thin line indicates the theoretical value $1/\sqrt{3}$. The protocol for $\omega \left(t\right)$ is the same as in Fig. 1.

Figure 3

Quantum work function from direct simulations, with $2\pi \hslash =1$, $\beta =0.2$, and $\omega \left(t\right)$ the same as in Fig. 2. The red dots (blue squares) denote $P^q\left(W\right)$ of a bare nonadiabatic process (STA) with $\tau {\omega }_i=0.001$. For clarity only data points with $P^q\left(W\right)\ge 0.0002$ are plotted. The inset displays all data points on a semilogarithmic plot using a wider range of $W$, with the thick line for STA.

Figure 4

Efficiency at maximized work output as a function of ${\beta }_1/{\beta }_2$ for a prototypical quantum heat engine [16, 17], with ${\omega }_i=10$, $2\pi \hslash =1$, and ${\beta }_1=10$ or $0.01$. The two solid lines describe the classical results ${\eta }_{\text{ad}}$ and ${\eta }_{\text{nonad}}$ given in the text. Note the efficiency increase by STA.