Autonomous Maxwell's demon in a cavity QED system

We implement an autonomous Maxwell's demon with a single Rydberg atom and a high-quality microwave resonator. The atom encodes both a qubit interacting with the cavity, and a demon carrying information on the qubit state. While the cold qubit crosses the hot cavity, the demon prevents energy absorption from the cavity mode, apparently violating the Second Law of thermodynamics. Taking into account the change of the mutual information between the demon and the qubit-cavity system gives rise to a generalized expression of the Second Law that we establish and measure. Finally, considering the closed qubit-cavity-demon system, we establish and measure that the generalized Second Law can be recast into an entropy conservation law, as expected for a unitary evolution.

Back in the 19 th century Rudolf Clausius has postulated that no spontaneous process exists whose sole result is the transfer of heat Q h from a cold to a hot bath [1]. Introducing the inverse temperatures, β c and β h , of the cold and hot bath, respectively, this empirical result is captured by the formula It was later realized that Eq. (1) is one of the many expressions of the second law of thermodynamics (SLT), Σ being the entropy produced during the heat transfer. The SLT asserts that the total entropy of an isolated system can never decrease and stays constant for a reversible evolution, thus defining a thermodynamical arrow of time.
Later on, Maxwell pointed out a possible limitation of the SLT [2][3][4]. In contemporary terms, he envisioned a feedback mechanism, later dubbed Maxwell's demon, that would exploit the microscopic information on the molecules of a gas in order to establish a gradient of temperature, without any work expenditure. The paradox vanishes by acknowledging that information is a physical resource, that can be defined and quantified as correlations between the gas molecules and the demon's memory [5][6][7][8]. While being consistent with the SLT, consuming these correlations allows performing tasks that would be impossible otherwise. Since these pioneering contributions, the Maxwell's demon has been experimentally exorcized in various setups, whether classical or quantum [9][10][11][12][13][14][15][16][17]. While most devices make use of feedback loops operating between the quantum and the classical world, a few autonomous Maxwell's demons have been demonstrated [10,14].
Here, we report on such a fully self-contained, autonomous Maxwell's demon experiment, realized in a cavity QED system. Information is exploited to transfer heat from a "cold" Rydberg atom (qubit) to a "hot" microwave cavity (harmonic oscillator). In drastic contrast with former experiments, the ensemble of subsys-tems (demon, cold body, and hot body) is closed and, thus, undergoes a unitary, reversible evolution. While the entropy of the total system is constant, the entropy changes of the different subsystems is analyzed (and measured) as entropy production and consumption of correlations [18,19].
Before presenting our experimental setup, we first focus on the general theoretical description of the underlying thermodynamic protocol. We study the transfer of heat Q C from a qubit Q to a cavity C, that are initially prepared in thermal statesĜ β Q andĜ β C with temperatures β Q and β C , respectively. This transfer is controlled by a demon D, the role of which is played by another qubit. We denoteρ X , with X ∈ {QDC, Q, C, QC, D}, the density matrices of the total system and various subsystems, and S X ≡ S[ρ X ] = −Tr[ρ X lnρ X ] their respective von Neumann entropies. The evolution of the joint qubit-demon-cavity system is unitary such that its entropy S QDC remains constant in time. This entropy conservation law is consistent with the SLT as soon as unitary evolutions are considered reversible.
The demon's actions can be schematically split into a readout step and a feedback step. During the readout step, D encodes information on the state of Q. The amount of information owned by D is quantified by the mutual information given by I QC:D = S QC +S D −S QDC . If the qubit is found in its ground state, the demon blocks the Q-C interaction and, thus, the heat exchange between them: This is the feedback step, during which the demon state remains the same. Combined with the total entropy conservation, it yields ∆I QC:D = ∆S QC . Here and in the following, ∆ denotes the change of the corresponding quantity in the feedback step. In the absence of D, there is no correlation and hence ∆S QC = 0, as expected from a unitary evolution. Conversely in the presence of D, consuming correlations (∆I QC:D < 0) allows one to lower the entropy of the QC system at no work cost, which is the key mechanism at play in our experiment.

arXiv:2001.07445v1 [quant-ph] 21 Jan 2020
In a similar way, the QC entropy change can be written as ∆S QC = ∆S Q + ∆S C − ∆I Q:C . Since the initial states of Q and C are thermal, their individual entropy changes are ∆S X = Q X β X − D ρ X ||Ĝ β X , with X ∈ {Q, C},ρ X the respective density matrices at the end of the feedback step and Q X the heat absorbed by a system X [20]. They depend on the relative entropy which is a strictly non-negative quantity measuring how much the system X in stateρ X is far from a Gibbs stateĜ β X . From the two expressions for ∆S QC we get eventually (2) δβ = β C −β Q is the difference between the inverse temperatures of the C and Q respectively. Q C δβ is reminiscent of the quantity appearing in Clausius inequality and in Eq. 1. From now on we shall dub it a physical entropy production. The relative entropy of the QC system is defined [20]. In the demon absence, Eq. (2) simply reads Q C δβ = D QC and the physical entropy production is positive as expected for a spontaneous process. Conversely, in the demon presence, the physical entropy production can reach negative values, revealing the role of information as a resource that allows for otherwise impossible dynamics. We refer to the inequality Q C δβ − ∆I QC:D ≥ 0 as a generalized SLT [22]. Our experimental setup, used to verify the equality (2), is depicted in Fig. 1(a) [23,24]. The high-finesse Fabry-Perot cavity C, made of two superconducting mirrors, has a resonance frequency ω/2π = 51 GHz and an energy relaxation time of 25 ms at a 1.5-K temperature. The source S C is used to inject a microwave field in C by diffraction on the mirrors' edges. Individual circular Rydberg atoms with principal quantum number 50 (level |g ) are prepared out of a thermal beam of rubidium atoms in B out of a velocity selected (250 m/s) beam. The mean atom number per atomic sample is about 0.04, keeping the probability to have several atoms per sample negligible. The atomic transition frequency between |g and the next higher circular level |e is close to ω. All circular states have a relaxation lifetime of about 30 ms. The two levels |e , |g are involved in our qubit Q, see below. The source S Q drives the atoms between |g and |e in the zone R Q . The atom-cavity interaction is controlled by means of the quadratic Stark shift in an electric field applied with potential V across C. The Rabi frequency of the resonant atom-cavity interaction is about 49 kHz. The atomic levels are finally measured in the detector M by state-selective field-ionisation. The detection efficiency is 0.5 and the detection error, i.e., the probability to erroneously attribute the atomic state, is 0.05. The time of flight between B and M is 1.2 ms, much shorter than all relaxation times. The Ramsey interferometer, The cavity C is prepared in a thermal state at temperature β C using microwave source S C . The three-level atom, initially in state |g , is prepared in a thermal state (temperature β Q ) in the {|g , |e } subspace using source S Q . The demon D is realized by source S D resonant with |f −|g transition and applied before Q enters C. The Q-C energy exchange is induced through adiabatic passage technique (Q-C detuning is controlled by electric field applied by voltage potential V). The atomic state is measured by detector M. The cavity state is reconstructed by the Ramsey interferometer (zones R1 and R2 fed by S R ) with a sequence of QND atoms.
composed of low-quality cavities R 1 and R 2 fed by the source S R , allows us to perform a quantum nondemolition (QND) measurement of the cavity photon number with a long sequence of probe atoms [24,25]. The dispersive atom-cavity interaction for this measurement is tuned to realize a π/2 phase-shift per photon between atomic states |g and |e .
The demon D involves an auxiliary level of the Rydberg atom, namely a lower circular state |f with principal quantum number 49. The |f −|g transition has a frequency of ω D /2π = 54 GHz and is driven by the source S D . The read-out is realized by transferring the atomic population from |g to |f before the atom-cavity interaction. Since level |f does not couple to C, the demon eliminates the possibility for the atom to extract energy from C, while the probability to transfer the energy into C when the atom is in |e stays unchanged. The thermodynamic description of the experiment leading to (2) is recovered by using the following mapping between the three atomic levels and the logical states of Q and D: The forth state of the QD system, |1 Q ⊗ |1 D , is never populated in the considered demon scheme and, thus, does not need to be taken into account [20]. Figure 1(b) presents schematically the experimental sequence. The cavity C is initially prepared in a thermal stateĜ β C by means of S C . We perform 10 microwave injections of 0.1-ms duration each with the same amplitude, but random phase [20]. The thermal field preparation is calibrated and verified by reconstructing the photonnumber distribution P (n) inĜ β C with a sequence of 800 QND atoms. The estimated P (n) is in excellent agreement with the Boltzmann thermal distribution providing us a thermal photon number n th and defining β C = ( ω) −1 ln[(1 + n th )/n th ]. We set n th = 0.63 ± 0.04 photons, corresponding to a temperature of 2.6 ± 0.1 K.
The qubit Q is prepared in a thermal stateĜ β Q by means of the source S Q . Since there is no phase in the state of C and since the atomic state detection is phase-insensitive, we ignore the phase between |g and |e . The qubit temperature in the {|g , |e } subspace is defined by the population p e in the excited state as β Q = ( ω) −1 ln[(1 − p e )/p e ] and is controlled by the duration of the S Q pulse. We vary β Q from large positive values (states close to |g ) to large negative values (states close to |e ) passing through β Q = β C (mutual equilibrium state) and β Q = 0 (infinite temperature with equal state population).
The demon transfers the atomic population from |g to |f . The duration of the S D pulse, applied before Q enters C, is pre-adjusted to maximize the population transfer, realizing a Rabi π-pulse. The maximal population transfer, 0.95, defines the demon read-out efficiency. When Q enters C, their energy exchange is made independent of the photon number of C by using an adiabatic passage technique. The Q-C detuning is swept from 100 to −60 kHz in 60 µs by means of the electric field applied across C. The method ensures that an atom in |e always injects a photon into C, while an atom in |g always absorbs a photon from C if there was one (population transfer efficiency is better than 0.99). As expected, the overall demon's action has thus two effects: read-out (mapping of the state of Q into the state of D) and feedback (conditional protection of Q from the following energy exchange with C). The demon's action is autonomous since it does not involve any information processing at the classical level.
The experimental sequence is repeated 25 000 times and the final state of C is reconstructed separately for different detected states of Q and D [25]. In this way we obtain the probability P (s Q , s D , n) for each joint QDC state |s Q ⊗ |s D ⊗ |n , with s ∈ {0, 1} and n photons in C. This probability represents the diagonal elements (populations) of the stateρ QDC in the {|s Q ⊗ |s D ⊗ |n } basis. The stateρ QDC is reconstructed for different qubit temperatures β Q [26]. All measurements and state reconstructions are repeated with a reference experimental protocol which has no demon read-out.
The Q-C heat exchange Q is shown in Fig. 2 versus the relative qubit effective temperature δβ = 1 − T C /T Q . The heat received by C is defined by Q C = ω ∆n, where the change of the mean photon number, ∆n, is calculated from the photon-number distributions reconstructed with and without D. Similarly, the heat received by Q is defined by the change of the atomic population in |e as Q Q = ω ∆p e . The lines in Fig. 2 are theoretically computed taking into account the main experimental imperfections: the relaxation of Q and C, the limited demon efficiency, and the nonideal detection resolution [20]. Figure 2 clearly shows that in the demon absence (open symbols and dashed lines) the hotter system always gives off heat to the colder one and that the physical entropy production is positive, Q C δβ > 0, as postulated by Clausius and expressed by Eq. (1). At δβ = 0, i.e., when Q and C have the same temperature, the net heat exchange is zero. In the demon presence (filled symbols and solid lines), however, the natural direction of heat flow can be reversed. Since D is designed to block the energy transfer from the cavity to the qubit, Q always gives off heat while C always absorbs heat, irrespective of their temperatures. The sign change in Q for δβ < −2 is mainly due to the limited demon efficiency providing the atom with a small probability to enter the cavity in level |g and, thus, to absorb energy from C.
We characterize the performance of the demon by the heat gain, that we define as ε = (Q d C −Q ∅ C )/ ω. Here, Q d C is the heat transferred from Q to C in the demon presence (shown by the red solid line in Fig. 2). The quantity Q ∅ C is the largest heat that can be transferred from Q to C in a protocol with no microscopic demon involved. The best classical strategy to maximize Q ∅ C , i.e., knowing only the individual temperatures T Q and T C , is to bring Q and C in contact if T Q > T C (Q ∅ C is then given by the red dashed line for positive δβ in Fig. 2) and to inhibit their interaction if T Q ≤ T C (Q ∅ C = 0 for δβ ≤ 0). The inset in Fig. 2 presents the temperature dependence of the heat gain ε. For large positive δβ, the transferred heat saturates close to its fundamental limit given by the energy of a single photon, ω. Therefore, there is no more energy left for D to further increase Q d C . On the other side, for large negative δβ, Q has no energy at all to transfer to C, and thus neither method is able to extract the heat from Q. In the intermediate regime, however, the demon read-out (i.e., the information on the microscopic state of the system) allows the extraction of more energy than in the classical case where only macroscopic information (i.e., temperatures) is available.
Usingρ QDC reconstructed at each experimental step, i.e., after initial state preparation, demon read-out, and feedback, we compute the quantities ∆I QC:D and D QC entering Eq. (2). The results with and without the demon are shown in Fig. 3. There is no correlation between D and QC without read-out (open symbols in Fig. 3(a)).
With the read-out activated, D gets correlated with Q (in blue). Part of this correlation is consumed during the feedback, but D still keeps some non-consumed information on C after the feedback (in red). For an equal mixture of |0 Q and |1 Q , i.e., for δβ = 1, the mutual information between D and Q is the largest. Here, it is smaller than its absolute maximum value of ln(2) because of experimental imperfections mixing the atomic levels.
In the limit |δβ| 0, the qubit's state tends to one of the pure states, |0 Q or |1 Q , having zero entropy. Thus, the mutual information I QC:D being limited by the entropies of individual partitions also tends to zero. Note also, that for large negative δβ, the atom is mostly in level |g , which can relax to |f or can be erroneously detected as |f . This effect creates small spurious correlations between logical states of Q and D, resulting in non-zero values of I fb QC:D for δβ ≤ −2, see the red curve in Fig. 3(a). Figure 3(b) shows the generalized SLT. As expected, the consumption of the correlation I QC:D can compensate the possible negative physical entropy production such that the quantity Qδβ − ∆I QC:D stays strictly nonnegative. The relative entropy D QC is shown in black in Fig. 3(c). Its dominant term comes from the qubit, which starts in a thermal state and ends in an almost pure state |0 Q . This also implies that the QC correlations are nearly zero and, thus, ∆I Q:C ≈ 0. The relative entropy of C is rather small and is maximal for β Q → −∞ when the heat transfer is maximal (one photon). Figure 3(d) shows the entropy conservation of the total system as described by (2). As expected, this equality is nearly valid both in the demon absence and presence. The residual deviation from zero comes from the non-negligible relaxations of Q and C and from the finite atomic state discrimination creating spurious correlations.
Any comparison between the experimental data and theoretical predictions requires the knowledge of the precision with which the thermodynamic quantities have been calculated from the experimental state reconstruction. The variance of any observable in the reconstructed state can be computed directly from the measured data used for the state tomography [25,27]. This approach is directly applied in Fig. 2 to the errors on the heat Q, since the internal energy is a quantum observable. However, the same method cannot be applied to the entropic entities in Fig. 3, which are not expectation values of a linear operator. To estimate the error of any non-linear function ofρ reconstructed with the maximum likelihood algorithm based on individual trajectories [25], we apply a mathematical recipe developed in Ref. [28].
In conclusion, we have quantitatively evidenced that information is a resource to implement thermodynamically forbidden tasks, like the transfer of heat from cold to hot bodies. Originally, our experiment can be analyzed as a global, reversible unitary evolution coupling a qubit, a demon and a cavity. Entropy production and information consumption appear as entropy changes of subparts of the joint systems, the total entropy remaining constant -offering a striking experimental evidence that irreversibility and information are local physical concepts. The extension of the current protocol should allow us for the cyclic heat exchange between two cavities [25] controlled by the demon and to study the performance of an information-driven quantum engine. The very same platform and mechanisms lead to tripartite entangled states, that can be further used to investigate the energetic value of quantum entanglement [29], the impact of quantum coherence on work extraction [30], and new quantum fluctuation theorems.
We thank J. P. Paz for useful discussions. We acknowledge support by European Community (SIQS project) and by the Agence Nationale de la Recherche (QuDICE project). P. A. C. acknowledges Templeton World Charity Foundation, Inc. This publication was made possible through the support of the grant TWCF0338 from Templeton World Charity Foundation, Inc. The opinions expressed in this publication are those of the author(s) and do not necessarily reflect the views of Templeton World Charity Foundation, Inc. B.-L. N.-S. and P. A. C. contributed equally to this work.