Extracting Work from Quantum Measurement in Maxwell’s Demon Engines

The essence of both classical and quantum engines is to extract useful energy (work) from stochastic energy sources, e.g., thermal baths. In Maxwell’s demon engines, work extraction is assisted by a feedback control based on measurements performed by a demon, whose memory is erased at some nonzero energy cost. Here we propose a new type of quantum Maxwell’s demon engine where work is directly extracted from the measurement channel, such that no heat bath is required. We show that in the Zeno regime of frequent measurements, memory erasure costs eventually vanish. Our findings provide a new paradigm to analyze quantum heat engines and work extraction in the quantum world.

Figure 1

Maxwell’s demon assisted engines. (a) Thermally driven engine. The working agent is a qubit of transition frequency ${\omega }_0$. A demon measuring in the qubit energy basis $\{|0⟩;|1⟩\}$ allows us to convert the heat $Q_{\mathrm{hot}}$ extracted from a bath into work $W_{\mathrm{ext}}$. The demon’s memory is erased by some extra-work source at the minimal work cost $W_{\mathrm{er}}$, while the heat $Q_{\mathrm{cold}}$ is evacuated in a hidden cold bath. (b) Measurement powered engine. In case the demon measures in the $\{|{+}_{\mathbf{x}}⟩;|{-}_{\mathbf{x}}⟩\}$ basis, work can be extracted from the measurement channel and no hot bath is required. (c) Evolution of the qubit state in the Bloch sphere. Between two measurements delayed by ${\tau }_w$, the evolution induced by the driving field is a rotation around the $Y$ axis at Rabi frequency $\mathrm{\Omega }$, green thin arrow). The measurement projects the qubit on states $|{\pm }_{\mathbf{x}}⟩$ (orange thick arrow). (d) Evolution of the qubit internal energy $U(t)=\mathrm{Tr}[\rho (t)H(t)]$ as a function of the Rabi angle $\theta =\mathrm{\Omega }{\tau }_w$. During the Hamiltonian evolution the work $W_{\mathrm{ext}}=-\mathrm{\Delta }U$ is extracted into the driving field. The measurement provides the energy $E_{\mathrm{meas}}=W_{\mathrm{ext}}$ back to the qubit.

Figure 2

Performance of the measurement powered engine. (a) Orange dotted-dashed line: normalized extracted power $P_{{+}_{\mathbf{x}}}/\mathrm{\Omega }\hslash {\omega }_0$; Solid blue line: MPE yield $\eta (\theta )$ as a function of the Rabi angle $\theta$; Dashed blue line: thermally driven engine’s yield ${\eta }_{\mathrm{cl}}$. We chose $\hslash {\omega }_0/[2k_B{T}_{\mathcal{D}}\log (2)]=2$. (b) Normalized extracted power $P_{\mathbf{n}}(\theta )/\mathrm{\Omega }\hslash {\omega }_0$ in the Zeno limit, where $\theta \to 0$, as a function of the chosen operating point $|{+}_{\mathbf{n}}⟩$ for $\theta =\pi /2$, $\theta =\pi /4$ and in the Zeno limit $\theta \to 0$.

Figure 3

Implementation of the MPE in circuit QED. (a) Scheme of the experiment. The qubit is a transmon in a 3D cavity. Work is extracted as microwave propagating photons. (b) Sequence of pulses corresponding to one engine cycle. Green: Drive. Orange: $\pi /2$ pulses. Purple: readout tone. Bottom: Work extracted (green dashed) and measurement energy (orange) as a function of time. (c) Extracted power $P_{\gamma }(t)$ in units of $\mathrm{\Omega }\hslash {\omega }_0$ as a function of time for a single realization $\gamma$ of the engine. (d) Mean power $P(t)$ as a function on time. The blue points are averaged over ${\mathcal{N}}_r=10^4$ stochastic realizations. The size of the point corresponds to the error bar ${\sigma }_P(t)/\sqrt{{\mathcal{N}}_r}$, where ${\sigma }_P(t)$ is the variance of $P_{\gamma }(t)$. Red-solid and purple-dashed curves are obtained from the analytical formula provided in the Supplemental Material [43], for two different values of $\mathrm{\Omega }$. Parameters: ${\tau }_{\mathrm{me}\text{a}s}=70\mathrm{ns}$, ${\tau }_{\mathrm{w}}=70\mathrm{ns}$, $\mathrm{\Omega }=0.2\mathrm{MHz}$ (red solid) and $\mathrm{\Omega }=0.6\mathrm{MHz}$ (purple dashed).