Observing a quantum Maxwell demon at work

We now discuss the steps of the work extraction cycle. During step 1, we prepare the system in a thermal state at an arbitrary temperature $T_S\ge T_S^0$ (Fig. 1), where $T_S^0=103\pm 9$ mK is the equilibrium system temperature in the dilution refrigerator. This thermalization is realized by driving, in a fraction $p(T_S)$ of all experimental sequences, the qubit with a resonant $\pi$-pulse, which flips the qubit state, thus simulating thermalization with a heat bath (SI Appendix). Conveniently, this technique can also prepare nonthermal quantum states, such as an equal superposition of the qubit, by driving it with a $\pi /2$-pulse.

Step 2 consists in encoding the state of the system into the demon’s memory, which starts in the vacuum ${|0⟩}_D$. Driving port $a$ with a pulse of amplitude ${\alpha }_{in}$ at frequency $f_D$ (Fig. 1) excites the demon’s memory conditioned on the system being in ${|g⟩}_S$. This selectivity requires the pulse duration to be longer than ${\chi }^{-1}$ and shorter than the coherence times of the qubit and cavity (SI Appendix). By design, decoherence of both system and demon’s memory is dominated by spontaneous emission into port $b$ with respective relaxation rates ${\gamma }_S={(2.2\mu \mathrm{s})}^{-1}$ and ${\gamma }_D={(207\mathrm{ns})}^{-1}$. If the system starts in an arbitrary superposition $c_g{|g⟩}_S+c_e{|e⟩}_S$, it becomes entangled with the demon (Fig. 1); ideally, $c_g{|g⟩}_S\otimes {|\alpha ⟩}_D+c_e{|e⟩}_S\otimes {|0⟩}_D$, where ${|\alpha ⟩}_D$ is a coherent state. In practice, the qubit-induced nonlinearity and decoherence of the cavity lead to an impure memory state ${\rho }_{{\alpha }_{in}}$ instead of ${|\alpha ⟩}_D$. The average photon number $\overline{n}=\mathrm{Tr}(d^{†}d{\rho }_{{\alpha }_{in}})$ is determined by fitting the numerical result of the full master equation to match the experimentally obtained system state (SI Appendix).

The work extraction occurs during step 3. A coherent $\pi$-pulse, playing the role of the battery B, is sent through port $b$ at frequency $f_S$ (Fig. 1). Without the demon, the qubit would deterministically absorb (emit) a quantum of energy $h{f}_S$ from (into) the battery, if it is initially in ${|g⟩}_S$ (${|e⟩}_S$). Crucially, the demon prevents this transfer when its memory has $N\ge 1$ photons, because then the pulse is off resonance by $-N\chi$. When the correlation between the demon’s memory having no photons and the system being in ${|e⟩}_S$ is perfect, only stimulated emission is allowed, and work is extracted from system to battery. However, when the correlation is not perfect, in particular when $\overline{n}\ll 1$, the demon sometimes erroneously lets the qubit absorb a quantum of energy from the battery.

The demon thus ends in a state with an entropy $S_D$ of at least the decrease of system entropy, and has to be reset to close the thermodynamic cycle (2). In the final step of this experiment (step 4), we let the demon’s memory thermalize with a second bath that has a low temperature ($72\pm 13$ mK). Physically, this bath differs from the one coupled to the system, because of the various operating frequency ranges of the microwave components on the output line and because of the likely coupling of the transmon qubit to excited vibration modes of the substrate, which couple to the system more than to the demon. So this demon can extract work in a cyclic manner, but it does so using a second bath, thus behaving as a regular heat engine.

Remarkably, the power extracted from the system during step 3 when it is driven at $f_S$ can be directly accessed (SI Appendix) through the difference between the incoming and outgoing photon rates of port $b$,where ${\sigma }_X,{\sigma }_Y$, and ${\sigma }_Z$ are the Pauli matrices for the system; ${\gamma }_b$ is the Purcell coupling rate of the system to the transmission line through port $b$; and $\mathrm{\Omega }\propto |{⟨b_{in}⟩}_B|$ is the frequency of the Rabi oscillations around ${\sigma }_Y$ induced by the drive. The two contributions on the far right side can be identified as spontaneous emission of the system through port $b$ and stimulated emission. The latter is a coherent exchange of energy between the drive and system and, as such, contributes to the work extracted from the system.

In the experiment, at $t=0$, we send a pulse $b_{in}$ with a duration $\pi /\mathrm{\Omega }$ with $\mathrm{\Omega }={(67\mathrm{ns})}^{-1}\gg {\gamma }_S\ge {\gamma }_b$ so that spontaneous emission can be neglected, and ${⟨b_{out}^{†}{b}_{out}⟩}_B-{⟨b_{in}^{†}{b}_{in}⟩}_B$ fully quantifies the work extraction. We measure the field intensity on $b_{out}$ using a near-quantum-limited heterodyne detection setup (SI Appendix) to access directly the average instantaneous power extracted from the system (21). The power is shown in Fig. 2 as a function of time during the pulse in step 3 in units of photons per microsecond for various initial system states (Fig. 2, Inset) and for two values of the average photon number in the demon memory $\overline{n}$. In Fig. 2A, the average photon number $\overline{n}=9$ is large enough for the demon to distinguish the system states well. As expected from the demon’s action, the measured power is positive for all initial states and greater for higher initial system temperature. In contrast, when the demon is unable to distinguish ${|g⟩}_S$ and ${|e⟩}_S$, which happens for ${\alpha }_{in}=\overline{n}=0$, the extracted power is measured to be negative for the system starting in any thermal state (Fig. 2B). This counterproductive action occurs because the demon is ignorant and lets the system drain energy from the battery. This failure uncovers the role of information in the work extraction by the demon.

At $\overline{n}=0$, a distinctive feature appears in Fig. 2B when the system starts in a quantum superposition of ${|g⟩}_S$ and ${|e⟩}_S$ (green). Even though the total work is zero, just like for the equally mixed state (red), the instantaneous power now oscillates, illustrating the work potential of coherences. In contrast, for an efficient demon ($\overline{n}=9$ in Fig. 2A), there is no quantum signature in the extracted work. Note that the peak in the green curve arises due to overlapping of steps 2 and 3 to avoid transients (SI Appendix).

Integrating the extracted power over the duration of step 3 gives the work $W={\int }_0^{\pi /\mathrm{\Omega }}P\mathrm{d}t$, whose magnitude is, at most, $h{f}_S$. As ${\alpha }_{in}$ increases, the demon’s encoding improves, and the work increases from negative to positive values (Fig. 3A). This extracted work is given by the change in the system internal energy $U_S=h{f}_S⟨e|{\rho }_S|e⟩$ during step 3, $W=-\mathrm{\Delta }{U}_S-Q\approx -\mathrm{\Delta }{U}_S$, where $Q$ is the heat arising from spontaneous emission, which is negligible. Although the work was measured on the battery, we independently measure $U_S$ as a function of ${\alpha }_{in}$ (Fig. 3B) at the end of step 4 using the cavity as a dispersive detector (20) (SI Appendix). The variations of work (Fig. 3A) indeed mirror the change of internal system energy (Fig. 3B) between steps 1 (dashed lines) and 4 (symbols). As ${\alpha }_{in}$ increases, the demon extracts more energy from the system, making it end up close to the ground state (residual excitation of $2.7\pm 1\%$) whatever the initial state (Fig. 3B). Indeed, the thermodynamic cycle can be used to cool down superconducting qubits in practice, as previously demonstrated in its continuous version (22). The full decay of $U_S$ and the increase of $W$ as a function of ${\alpha }_{in}$ are well reproduced numerically (solid lines in Fig. 3). It is natural to compare the extracted work with Landauer’s work cost of erasure, $k_{B}T\ln 2$ (2, 10). Because the system is connected to a thermal bath only during step 1, the work extraction is not optimal. Indeed, W is limited by the initial internal energy $U_S=h{f}_S/[1+\exp (h{f}_S/k_{B}T)]$, which is at most 40% of the Landauer bound.

A key signature of Maxwell’s demon is the transfer of entropy from the system to the memory (2). In contrast to previous realizations of Maxwell demons (3–6), our experiment not only allows a direct measurement of work but also gives full access to the density matrix ${\rho }_D$ of the demon’s memory, including its von Neumann entropy $S_D=-\mathrm{Tr}({\rho }_D\ln {\rho }_D)$. We perform a full quantum tomography of ${\rho }_D$ using the qubit as a measurement apparatus right after step 3 (SI Appendix) (23). When the qubit starts close to ${|g⟩}_S$, the maximum likelihood reconstruction of the demon’s state gives ${\rho }_{{\alpha }_{in}}$ (SI Appendix) (24). When we set ${\alpha }_{in}=0.25$, the measured state ${\rho }_{{\alpha }_{in}}$ is found to be entropic and far from a coherent state, as expected (Fig. 4A). In contrast, when the system starts in ${|e⟩}_S$, the memory stays close to ${|0⟩}_D$, with a small residual entropy (Fig. 4B). Most interesting is the comparison of the effect on the demon when the system starts in a quantum superposition (Fig. 4C) versus in a thermal state at large temperature (Fig. 4D). In the first case, $S_D=1.0\pm 0.05$, and ${\rho }_D$ exhibits coherences between $|0⟩$ and higher Fock states, whereas coherences are missing in the second case, leading to a larger entropy $S_D=1.2\pm 0.1$. This transfer of nonclassicality from the system to the memory is a signature of the quantum Maxwell demon. Although the entropies of these two states are ordered as expected, their values are much larger than a bit of entropy, $\ln 2\approx 0.7$. This excess entropy is quantitatively reproduced by simulations (SI Appendix) and arises because dissipation and nonlinearity of the memory results in encoding in a large number of energy levels rather than in just two dimensions. Using a full tomography of the system (SI Appendix), we have checked that the memory entropy $S_D$ is always higher than the system entropy decrease $S_S(\text{step}1)-S_S(\text{step}3)$.

Future developments of this experiment could involve superconducting circuits with a widely tunable frequency, which would allow the implementation of optimal quasistatic processes, where the system stays in equilibrium. A test of Landauer’s principle could then be realized in the quantum regime. The encoding fidelity of the demon is quantified by the mutual information between system and demon. By adding an extra qubit and readout cavity, one could demonstrate the expected proportionality between extracted work and mutual information (25–27). Finally, with the level of control shown in the experiment, superconducting circuits provide an exciting platform to explore single-shot quantum thermodynamics (28) and quantum heat engines (29).

We thank M. Devoret, P. Degiovanni, E. Flurin, Z. Leghtas, F. Mallet, V. Manucharyan, J. Pekola, J. M. Raimond, M. Ueda, and the late M. Clusel for fruitful discussions and feedback. Nanofabrication has been made within the consortium Salle Blanche Paris Center. This work was supported by the Agence Nationale de la Recherche under Grants 12-JCJC-TIQS and 13-JCJC-INCAL, by Ville de Paris through Grant Qumotel of the Emergence program, and by the COST network MP1209, “Thermodynamics in the quantum regime.” J.A. acknowledges support from Engineering and Physical Sciences Research Council, Grant EP/M009165/1, and the Royal Society.