Anisotropy-assisted thermodynamic advantage of a local-spin quantum thermal machine

We study quantum Otto thermal machines with a two-spin working system coupled by anisotropic interaction. Depending on the choice of different parameters, the quantum Otto cycle can function as different thermal machines, including a heat engine, refrigerator, accelerator, and heater. We aim to investigate how the anisotropy plays a fundamental role in the performance of the quantum Otto engine (QOE) operating in different timescales. We find that while the engine's efficiency increases with the increase in anisotropy for the quasistatic operation, quantum internal friction and incomplete thermalization degrade the performance in a finite-time cycle. Further, we study the quantum heat engine (QHE) with one of the spins (local spin) as the working system. We show that the efficiency of such an engine can surpass the standard quantum Otto limit, along with maximum power, thanks to the anisotropy. This can be attributed to quantum interference effects. We demonstrate that the enhanced performance of a local-spin QHE originates from the same interference effects, as in a measurement-based QOE for their finite-time operation.

Figure 1

Schematic diagram of the quantum Otto cycle in the entropy ($S$) vs magnetic field ($B$) plane, when it functions as a heat engine. In other types of thermal machines, the direction of heat flows and work differ.

Figure 2

Variation of the thermodynamic quantities $W, Q_H$, and $Q_L$ as a function of ratio of the temperatures $T_H/T_L$ of the hot and cold baths for different values of the anisotropy parameters (a) $\gamma =1$ and (b) $\gamma =0$. Symbols R, H, A, and E represent refrigerator, heater, accelerator, and engine, respectively. The other parameters are $B_L=1,B_H=4,J=1,T_L=1.$

Figure 3

(a) Variation of efficiency $\eta$ as a function of the ratio of the temperatures $T_H/T_L$ of the hot and cold baths. (b) Parametric plot of efficiency versus work as a function of anisotropy $\gamma$ when $T_H=10$. The $\gamma$ varies from 0 to 1 from the left to the right. The other parameters are the same as in Fig. 2.

Figure 4

Variation of entropy $S_B$ of the global two-spin system at point B of the cycle as a function of the anisotropy parameter $\gamma$, for the quasistatic performance of the engine. We have chosen $T_H=10$ and the other parameters are the same as in Fig. 2.

Figure 5

Variation of (a) transition probability ${\xi }_{\tau }$ between two energy eigenstates, (b) irreversible work $W_{\tau }^{\text{Ir}}$, (c) work in a complete cycle $W_{\tau }$, and (d) efficiency ${\eta }_{\tau }$ as a function of duration $\tau$ of the unitary stages, for different values of $\gamma$. We have chosen $T_H=10$, while the other parameters are the same as in Fig. 2.

Figure 6

Variation of (a) heat absorbed by the system $Q_{Ht}$, (b) work in a complete cycle $W_t$, (c) trace distance $D$, and (d) efficiency of the QHE as a function of $t_h$, for different values of $\gamma$. We have chosen $\mathrm{\Gamma }=0.1, T_H=10$, while the other parameters are the same as in Fig. 2.

Figure 7

Schematic diagram of the quantum Otto cycle on the entropy $S$ versus magnetic field $B$ plane when it functions as a heat engine. We consider a single local spin as a working system when the coupled two-spin global system is operated in the Otto cycle.

Figure 8

On the left axis, variation of the work difference $W_G-2W_L$ as a function of anisotropy parameter $\gamma$. On the right axis, a variation of efficiency for a local system as a function of $\gamma$. The efficiency of a single spin system QOE is 0.75 for $B_L=1,B_H=4$. We have chosen $T_H=10$ and the other parameters are the same as in Fig. 2.

Figure 9

Variation of the transition probability ${\lambda }_{\tau }$ and ${\delta }_{\tau }$ on the left axis, and efficiency of a local spin on the right axis as a function of $\tau$. The solid line on the top represents the quasistatic value of ${\delta }_{\tau }$, at the bottom represents the quasistatic value of ${\lambda }_{\tau }$, and in the middle represents the quasistatic value of the local efficiency, respectively. We have chosen $\gamma =1$ and $T_H=10$, while the other parameters remaining are the same as in Fig. 2.

Figure 10

Variation of efficiency of the local spin heat engine as a function of anisotropy parameter $\gamma$ for different values of the unitary process time $\tau$. $\tau =20$ represents the adiabatic and $\tau =0.3$ represents the nonadiabatic cases of the unitary time evolution. We have chosen $T_H=10$ and the other parameters are the same as in Fig. 2.

Figure 11

Variation of efficiency of the local spin heat engine as a function of power and the time of the unitary processes $\tau$ for $\gamma =1$. The times for the isochoric processes are $t_h=100,t_c=220, \mathrm{\Gamma }=0.1$. We have chosen $T_H=10$, while the other parameters are the same as in Fig. 2. The blue line represents the quasistatic value of the local spin QHE's efficiency [see Eq. (17)] and the black line represents the single spin QOE's efficiency [see Eq. (18)] with zero power for the same values of the parameters as used for the finite time behavior.