General formalism of local thermodynamics with an example: Quantum Otto engine with a spin-$1/2$ coupled to an arbitrary spin

We investigate a quantum heat engine with a working substance of two particles, one with a spin-$1/2$ and the other with an arbitrary spin (spin $s$), coupled by Heisenberg exchange interaction, and subject to an external magnetic field. The engine operates in a quantum Otto cycle. Work harvested in the cycle and its efficiency are calculated using quantum thermodynamical definitions. It is found that the engine has higher efficiencies at higher spins and can harvest work at higher exchange interaction strengths. The role of exchange coupling and spin $s$ on the work output and the thermal efficiency is studied in detail. In addition, the engine operation is analyzed from the perspective of local work and efficiency. We develop a general formalism to explore local thermodynamics applicable to any coupled bipartite system. Our general framework allows for examination of local thermodynamics even when global parameters of the system are varied in thermodynamic cycles. The generalized definitions of local and cooperative work are introduced by using mean field Hamiltonians. The general conditions for which the global work is not equal to the sum of the local works are given in terms of the covariance of the subsystems. Our coupled spin quantum Otto engine is used as an example of the general formalism.

Figure 1

Dependence of global work $W$ (a) and efficiency $\eta$ (b) on coupling strength $J$ for temperatures $T_1=1, T_2=0.5$, and magnetic fields $B_1=4, B_2=3$, and spins $s=1/2$ (black line), and $s=1$ (red line), $s=3/2$ (blue line), $s=2$ (green line), $s=5/2$ (yellow line), and $s=3$ (magenta line). The dashed line in (b) indicates the upper bound ${\eta }_b$ of the global efficiency given in Eq. (5) for the case of spin-$1/2$ pair. For the above parameters, we have ${\eta }_c=1-T_2/T_1=0.5$ and ${\eta }_{J=0}=1-B_2/B_1=0.25$. All quantities plotted are dimensionless. In all figures, we use a unit system where $\hslash =1,{\mu }_B=1,k_B=1$ and use $T_1$ as our scaling parameter.

Figure 2

Mutual relation of global work $W$ and efficiency $\eta$ for the same magnetic field and temperature parameters and the coupling range as in Fig. 1. The curves for each spin $s$ nearly coincide.

Figure 3

Work output (a) and efficiency (b) in the broader range of $J$, including the strong coupling region for the same parameters and spin $s$ as in Fig. 1. The direction of arrow in (b) indicates the lines in the order of increasing spin $s$ from $s=1$ to $s=3$. The curves are the same with those in Fig. 1 in the weak coupling regime. Note that after $J\approx 0.5$, the engine cannot produce positive work for the case of coupled spins-$1/2$.

Figure 4

Local work done by the spin $1/2$ (a) and spin $s$ (b) vs coupling strength $J$ for values $T_1=1, T_2=0.5, B_1=4, B_2=3$, and $s=1/2$ (black line), $s=1$ (red line), $s=3/2$ (blue line), $s=2$ (green line), $s=5/2$ (yellow line), and $s=3$ (magenta line).

Figure 5

Local work done by the spin-$1/2$ (a) and spin $s$ (b) in the strong coupling region where $J>0.5$ for the same parameters and spin $s$ as in Fig. 4. The curves are the same with those in Fig. 4 in the weak coupling regime. The direction of arrow in (b) indicates the lines in the order of increasing spin $s$ from $s=1$ to $s=3$. Note that after $J\approx 0.5, w_A=w_B\le 0$ for $s=1/2$ as given by the black line in (a).

Figure 6

Local temperature of spin-$1/2$ vs the parameter $s$ at $J=0.1$ (a) and at $J=4.0$ (b). Here $T_A^H\left(T_A^L\right)$ denotes the local temperature for the hot (cold) heat bath cases with $T_1=1.0(T_2=0.5)$ and $B_1=4(B_2=3)$.