Quantum Szilard engines with arbitrary spin

The quantum Szilard engine (QSZE) is a conceptual quantum engine for understanding the fundamental physics of quantum thermodynamics and information physics. We generalize the QSZE to an arbitrary spin case, i.e., a spin QSZE (SQSZE), and we systematically study the basic physical properties of both fermion and boson SQSZEs in a low-temperature approximation. We give the analytic formulation of the total work. For the fermion SQSZE, the work might be absorbed from the environment, and the change rate of the work with temperature exhibits periodicity and even-odd oscillation, which is a generalization of a spinless QSZE. It is interesting that the average absorbed work oscillates regularly and periodically in a large-number limit, which implies that the average absorbed work in a fermion SQSZE is neither an intensive quantity nor an extensive quantity. The phase diagrams of both fermion and boson SQSZEs give the SQSZE doing positive or negative work in the parameter space of the temperature and the particle number of the system, but they have different behaviors because the spin degrees of the fermion and the boson play different roles in their configuration states and corresponding statistical properties. The critical temperature of phase transition depends sensitively on the particle number. By using Landauer's erasure principle, we give the erasure work in a thermodynamic cycle, and we define an efficiency (we refer to it as information-work efficiency) to measure the engine's ability of utilizing information to extract work. We also give the conditions under which the maximum extracted work and highest information-work efficiencies for fermion and boson SQSZEs can be achieved.

Figure 1

The working cycle of the classical multiparticle Szilard engine. (a) A wall is isothermally inserted into the box. (b) Then a measurement is performed to ascertain the number of particles on both sides. (c) The system does work to the environment by isothermally moving the wall to the equilibrium position. (d) The wall is removed isothermally and the system is restored to its initial state.

Figure 2

The energy state of the fermion SQSZE when the wall is inserted at position $L/2$ in the low-temperature limit.

Figure 3

The change of $D_F$ and $W_{0F}$ along with $N$, in which $u=5$, $L={10}^{-9}\text{m}$, and $M={10}^{-26}\text{kg}$.

Figure 4

(a) The change of $\overline{W_{0F}}$ along with $N$. (b) The partial enlarged diagram of (a) when $N$ is very big. $u=5$, $L={10}^{-9}\text{m}$, and $M={10}^{-26}\text{kg}$.

Figure 5

(a) The change of the maximum of $D_F$ along with $u(u=s+1/2)$. (b) The change of the maximum of $W_{0F}$ along with $u$ when $n$ has different values. $u=5$, $L={10}^{-9}\text{m}$, and $M={10}^{-26}\text{kg}$.

Figure 6

The change of total work $W_{\text{tot}}$ along with temperature $T$ and particle number $N$ when (a) $u=1$, (b) $u=2$, (c) $u=5$, and (d) $u=10$. $L={10}^{-9}\text{m}$, $M={10}^{-26}\text{kg}$.

Figure 7

The change of $D_B$ and $W_{0B}$ along with $N$ when $s$ gets different values, in which $L={10}^{-9}\text{m}$ and $M={10}^{-26}\text{kg}$.

Figure 8

The change of total work $W_{\text{tot}}$ along with temperature $T$ and particle number $N$ when $s=0$, 1, 2, and 5. $L={10}^{-9}\text{m}$, $M={10}^{-26}\text{kg}$.

Figure 9

The work phase diagram of the fermion QSZE, in which $u=5$, $L={10}^{-9}\text{m}$, and $M={10}^{-26}\text{kg}$.

Figure 10

The work phase diagram of the boson QSZE when $s$ gets different values, in which $L={10}^{-9}\text{m}$ and $M={10}^{-26}\text{kg}$.