Maxwell’s Lesser Demon: A Quantum Engine Driven by Pointer Measurements

We discuss a self-contained spin-boson model for a measurement-driven engine, in which a demon generates work from thermal excitations of a quantum spin via measurement and feedback control. Instead of granting it full direct access to the spin state and to Landauer’s erasure strokes for optimal performance, we restrict this demon’s action to pointer measurements, i.e., random or continuous interrogations of a damped mechanical oscillator that assumes macroscopically distinct positions depending on the spin state. The engine can reach simultaneously the power and efficiency benchmarks and operate in temperature regimes where quantum Otto engines would fail.

Figure 1

Sketch of the demon system consisting of a qubit (working medium) and a harmonic oscillator (pointer). The qubit can be thermally excited by a hot bath at the rate ${\kappa }_h$ and temperature $T_h$, and it displaces the equilibrium position of the pointer to $\pm x_0$ depending on its state. A cold bath of temperature $T_c$ thermalizes the pointer around its equilibrium point at the rate ${\kappa }_c$. Work can be extracted coherently or incoherently from the excited spin by the demon’s interrogation of the pointer position.

Figure 2

Steady-state output power (a) and efficiency (b) as a function of $x_0/x_{\mathrm{th}}=x_0\sqrt{\tanh (\hslash \omega /2k_B{T}_c)}$ ($T_c=0$ on the right). The blue (thick), red, purple, and green (thin) curves correspond to measurement rates $\gamma /\omega ={10}^{-1}$, ${10}^{-2}$, ${10}^{-3}$, and ${10}^{-4}$, respectively. The horizontal dotted lines represent the approximations $\gamma W_{\max }$ and (8), the shaded regions mark ${\overline{n}}_c\ge {\overline{n}}_h$, while the black solid line in (b) shows the Carnot efficiency ${\eta }_{\text{Carnot}}=1-T_c/T_h$. The Otto efficiency ${\eta }_{\text{Otto}}=1-\omega /\mathrm{\Omega }=0.99$ is constant. We fix $\mathrm{\Omega }=100\omega$, $x_0=2.5$, ${\kappa }_h={10}^{-3}\omega$, ${\kappa }_c=0.1\omega$, ${\overline{n}}_h=1$, i.e., $T_h=\hslash \mathrm{\Omega }/k_B\ln 2$.

Figure 3

Output power (a) and efficiency (b) against the detuning $\mathrm{\Delta }$ of the driving field for $f(x)=\mathrm{\Theta }(-x)$ (solid) and $f(x)=1$ (dotted). We fix $\mathrm{\Omega }=100\omega$, $x_0=2.5$, ${\kappa }_h={10}^{-3}\omega$, ${\kappa }_c=0.1\omega$, $\zeta =0.1\omega$ and the same hot and cold bath occupancy ${\overline{n}}_c={\overline{n}}_h=1$. The Carnot and Otto efficiencies are both 0.99.

Figure 4

Output power (a) and efficiency (b) against the rate $\gamma$ or $\zeta$, comparing the active measurement-feedback scheme (black solid) with the passive scheme (blue dotted) at optimal detuning $\mathrm{\Delta }=2\omega x_0^2$ and $f(x)=\mathrm{\Theta }(-x)$. The other parameters are taken from Fig. 3. Note that underlying master equation model may no longer be reliable for $\zeta \sim \omega$.