Quantum Otto engine with exchange coupling in the presence of level degeneracy

We consider a quasistatic quantum Otto cycle using two effectively two-level systems with degeneracy in the excited state. The systems are coupled through isotropic exchange interaction of strength $J>0$, in the presence of an external magnetic field $B$ which is varied during the cycle. We prove the positive work condition and show that level degeneracy can act as a thermodynamic resource, so that a larger amount of work can be extracted than in the nondegenerate case, both with and without coupling. We also derive an upper bound for the efficiency of the cycle. This bound is the same as derived for a system of coupled spin-1/2 particles [G. Thomas and R. S. Johal, Phys. Rev. E 83, 031135 (2011)], i.e., without degeneracy, and depends only on the control parameters of the Hamiltonian, being independent of the level degeneracy and the reservoir temperatures.

Figure 1

Schematic of QOC between hot ($T_1$) and cold ($T_2$) heat baths, using as the working substance two effectively two-level systems coupled via an exchange interaction of strength $J$. The degeneracy of excited levels is $n^{\left(1\right)}=2$ and $n^{\left(2\right)}=3$. The different stages of the cycle are marked with circles.

Figure 2

The work extracted from the engine as a function of the coupling strength $J$ for different $(n^{\left(1\right)},n^{\left(2\right)})$ values. The parameter values are $B_1=4, B_2=3, T_1=2$, and $T_2=1$. Inset: Corresponding parametric plots of work versus efficiency as a function of the parameter $J\in [0,J_c]$, traced in a clockwise fashion. The loop-shaped curves show that work and efficiency achieve their maximum values at different values of $J$.

Figure 3

Variation of the efficiency of the engine as a function of the coupling strength $J$, for different $(n^{\left(1\right)},n^{\left(2\right)})$ values. The parameter values are $B_1=4, B_2=3, T_1=2$, and $T_2=1$. $J_0$ marks the value of $J$ where the efficiency for the (1,1) case (without degeneracy) cuts the uncoupled efficiency line, ${\eta }_0=0.25$. The upper bound ${\eta }_{\mathrm{ub}}$ is also shown. Inset: The regimes $P_2>P_2^{\prime }$ and $P_2^{\prime }>P_2$ vs $J$ for the (4,5) case; the probability curves cross each other at the value of $J<J_c=0.5$.

Figure 4

The extracted work ($W$) in a QOC with two interacting $\mathsf{V}$-configuration systems ($n^{\left(1\right)}=n^{\left(2\right)}=2$) as a function of $J$, compared with two noninteracting such systems ($2w_2$), as well as four noninteracting two-level systems ($4w_1$). The lines showing $w_2<2w_1$ illustrate that $w_n$ is bounded by $n{w}_1$ (see Appendix pp2). The parameter values are set at $B_1=4, B_2=3, T_1=2.5$, and $T_2=1.25$.