Work and efficiency of quantum Otto cycles in power-law trapping potentials

We study the performance of a quantum Otto cycle operating in trapping potentials of different shapes. We show that, while both the mean work output and the efficiency of two Otto cycles in different trapping potentials can be made equal, the work probability distribution will still be strongly affected by the difference in structure of the energy levels. To exemplify our results, we study the family of potentials of the form $V_t\left(x\right)\sim x^{2q}$. This family of potentials possesses a simple scaling property that allows for analytical insights into the efficiency and work output of the cycle. We perform a comparison of quantum Otto cycles in various physically relevant scenarios and find that in certain instances, the efficiency of the cycle is greater when using potentials with larger values of $q$, while in other cases, the efficiency is greater with harmonic traps.

Figure 1

(a),(b) Mean energy $\langle E\rangle$ vs trapping parameter ${\omega }_q$ for a quantum Otto cycle. The continuous blue lines represent the cycle in a harmonic trap ($q=1$), while the dashed red lines represent a cycle in an anharmonic trap with $q=2$. (c),(d) Standard deviation of the work distribution ${\sigma }_{W_{\text{cycle}}}$ vs the anharmonic parameter $q$. In all the plots, the mean energy at each vertex of the cycle is the same (“matched energies condition”). In (a) and (c) the initial temperature $1/{\beta }_{A_q}$ is the same for the two different confining potentials, while in (b) and (d) the initial volume, $V_{A_q}$, is matched. The extremal temperatures for the cycle in a harmonic trap are $1/{\beta }_{A_1}=1/2{\left(\hslash {\omega }_1^{\prime }\right)}^{-1}$ and $1/{\beta }_{C_1}=5/4{\left(\hslash {\omega }_1^{\prime }\right)}^{-1}$.

Figure 2

(a),(b) Probability distribution of work, $P\left(W_p\right)$, for the adiabatic compression and expansion processes of the cycle in the harmonic trap ($q=1$). (c),(d) $P\left(W_p\right)$ for the compression and expansion processes in the anharmonic trap with $q=2$. (e),(f) Probability distribution of work for the full quantum Otto cycle for the harmonic (e) and anharmonic, $q=2$, trap (f). Cycle parameters (corresponding to “matched energies condition” and also matching of the initial temperature $1/{\beta }_{A_q}$) are ${\omega }_1^{\prime \prime }/{\omega }_1^{\prime }=2$, ${\omega }_2^{\prime }/{\omega }_1^{\prime }=1.392$, ${\omega }_2^{\prime \prime }/{\omega }_1^{\prime }=2.341$, ${\beta }_{A_1}=\hslash {\omega }_1^{\prime }$, and ${\beta }_{C_1}=1/4\hslash {\omega }_1^{\prime }$.

Figure 3

(a),(b) Efficiency of Otto cycles for different values of $q$ vs ${\omega }_1^{\prime \prime }$. (c),(d) Comparison of average work done in a cycle $\langle W_{\text{cycle}}\rangle$ vs ${\omega }_1^{\prime \prime }$. In all the plots, the continuous blue line is used for $q=1$, the dashed pink line for $q=2$, and the dot-dashed red line for $q=3$. The compared cycles have the same extremal temperatures (${\beta }_{A_i}={\beta }_{A_j}=10\hslash {\omega }_1^{\prime }$ and ${\beta }_{C_i}={\beta }_{C_j}=\hslash {\omega }_1^{\prime }$ with $i,j=1,2,3$) and, for (a) and (c), matched extremal volumes ($V_{A_i}=V_{A_j}$ and $V_{C_i}=V_{C_j}$ with $i,j=1,2,3$), while in (b) and (d) matched extremal energies ($\langle E_{A_i}\rangle =\langle E_{A_j}\rangle$ and $\langle E_{C_i}\rangle =\langle E_{C_j}\rangle$ with $i,j=1,2,3$).

Figure 4

(a) Efficiency of the cycle vs ${\omega }_1^{\prime \prime }$ for different anharmonic parameters $q$. The cycles compared have the same extremal temperatures (${\beta }_{A_i}={\beta }_{A_j}=10\hslash {\omega }_1^{\prime }$ and ${\beta }_{C_i}={\beta }_{C_j}=\hslash {\omega }_1^{\prime }$ with $i,j=1,2,3,4$) and perform the same average work $\langle W_{\text{cycle}}\rangle$. Moreover, the initial mean energy $\langle E_{A_q}\rangle$ is the same. (b) Average work produced $\langle W_{\text{cycle}}\rangle$ vs ${\omega }_q^{\prime \prime }$ for the same initial temperature ${\beta }_{A_q}$ and mean energy $\langle E_{A_q}\rangle$. In (a) and (b) the continuous blue and black lines are used for $q=1$, the dashed pink line for $q=2$, the dot-dashed red line for $q=3$, and the dotted green line for $q=4$. ${\omega }_2^{\prime }/{\omega }_1^{\prime }=0.9567,{\omega }_3^{\prime }/{\omega }_1^{\prime }=0.9137,{\omega }_4^{\prime }/{\omega }_1^{\prime }=0.8802$. The pink arrow indicates the maximum work attainable for $q>1$ (for this matching condition), while the gray, dashed double arrows highlight the boundaries of the region for which engine cycles within a trap with $q>1$ cannot match the work of the harmonic case.

Figure 5

(a) Efficiency and (b) average work of cycles with Hamiltonian (9) $(q=2)$ vs $\varphi$. The parameters of the anharmonic cycle are computed by matching to a harmonic cycle with ${\omega }_1^{\prime \prime }=2{\omega }_1^{\prime }$. Moreover, the temperatures ${\beta }_{A_i}={\beta }_{A_j}=10\hslash {\omega }_1^{\prime }$, ${\beta }_{C_i}={\beta }_{C_j}=\hslash {\omega }_1^{\prime }$, and either the extremal volumes (continuous blue line) or extremal average energies (dashed red line) are matched to the harmonic case ($\varphi =0$).