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# Experimental realization of a Szilard engine with a single electron

Edited by Peter J. Rossky, Rice University, Houston, TX and approved August 12, 2014 (received for review April 17, 2014)

## Significance

A Maxwell demon makes use of information to convert thermal energy of a reservoir into work. A quantitative example is a thought experiment known as a Szilard engine, which uses one bit of information about the position of a thermalized molecule in a box to extract *k*_{B}*T* ln 2 of work. The second law of thermodynamics remains valid because, according to Landauer principle, erasure of the information dissipates at least the same amount of heat. Here, we present an experimental realization of a Maxwell demon similar to a Szilard engine, in the form of a single electron box. We provide, to our knowledge, the first demonstration of extracting nearly *k*_{B}*T* ln 2 of work for one bit of information.

## Abstract

The most succinct manifestation of the second law of thermodynamics is the limitation imposed by the Landauer principle on the amount of heat a Maxwell demon (MD) can convert into free energy per single bit of information obtained in a measurement. We propose and realize an electronic MD based on a single-electron box operated as a Szilard engine, where *k*_{B}*T* ln 2 of heat is extracted from the reservoir at temperature *T* per one bit of created information. The information is encoded in the position of an extra electron in the box.

The work of Maxwell introduced what is now known as the “Maxwell demon” (MD) (1). This idea was quantified later on by Szilard (2), initiating interest in the relationship between information and thermodynamics; see, e.g., refs. 3⇓⇓–6. MD extracts heat from a thermal reservoir at temperature *T* by observing a thermodynamic system to make a spontaneous, thermally induced, transition into a state with larger-than-average free energy (either because of a larger internal energy or a smaller entropy) and using the feedback to collect this extra free energy as work. Szilard demonstrated that by obtaining a single bit of information as a measurement result of the state of the system, one could collect up to *k*_{B}*T* ln 2 useful work, where *k*_{B} is the Boltzmann constant. Such a direct conversion of heat into work would by itself violate the second law of thermodynamics, because both the measurement and the feedback part of MD operation can in principle be done without generating extra entropy. In particular, a classical measurement can be viewed as a process of copying the state of the system into the memory of the detector. The only fundamentally unavoidable thermodynamic costs of conversion of heat into work by an MD is the creation of information about the state of the measured system. According to the Landauer principle (7⇓–9), erasure of this information generates at least the extracted amount of heat, *k*_{B}*T* ln 2 per bit, restoring the agreement with the second law.

Whereas these general principles of MD operation are well understood in theory [see, e.g., recent discussions (10⇓–12)], only few experimental realizations of an MD exist (13), and thus far none demonstrates a quantitative connection between the MD output and the obtained information. The goal of this work is to realize a system that demonstrates explicitly the extraction of *k*_{B}*T* ln 2 of heat from a thermal reservoir by an MD per one bit of created information. The operating cycle we use is close to the thought experiment suggested by Szilard. It realizes the MD operation using feedback-controlled molecule in a box as the working system. Fig. 1*A*, from left to right, shows the steps of the operation of such a Szilard engine (SE). The molecule is in equilibrium at temperature *T*, and the box is divided initially into two equal sections. After the measurement establishes which section the molecule is in, the container is allowed to expand into the full volume, lifting a weight tied to the dividing wall. In doing so, it extracts work from the thermal molecule. Then a dividing wall is introduced again and the cycle repeats. At the beginning of each cycle, the molecule has equal probabilities to be on the right or on the left, so that the measurement produces precisely one bit of information per cycle. As a result, the maximum of the average extracted work per cycle reaches the fundamental value of *k*_{B}*T* ln 2.

Our experimental realization of the SE cycle is shown in Fig. 1*C*. Its main element is the single-electron box (SEB) (14⇓–16), which consists of two small metallic islands connected by a tunnel junction. The SEB is maintained at the dilution-refrigerator temperatures in the 0.1-K range. Physically, there are two main differences between the SEB and the original single-molecule SE. The electrodes of the box contain electron gas of a large number of electrons, and not just one particle. Consequently, what is being manipulated in the engine operation is not this single particle but the charge configuration of the box, which is determined by the position of one extra electron. Also, this manipulation is achieved not by partitioning and reconnecting the electrodes (which for the SEB would correspond to the modulation of the conductance of the tunnel junction connecting the islands) but by changing the potential difference between the electron gases in the two islands. Apart from these differences, the engine follows the steps (illustrated with the potential profiles in Fig. 1*B*) similar to the operation of the original SE. The difference of the chemical potentials between the islands is controlled by the gate voltage *V*_{g} applied to one of them. Initially, *V*_{g} is such that the extra electron is found equally likely on either of the islands (Fig. 1*C*). This “degeneracy point” is realized when the gate-offset charge *n*_{g} = *C*_{g}*V*_{g}/*e*, where *C*_{g} is the capacitance between the gate and the SEB, is half-integer. A single-electron transistor (SET) electrometer, which can be seen in Fig. 1 *C* and *D*, *Bottom Right*, detects which island the electron is on. Then, *n*_{g} is changed rapidly to capture electron on the corresponding island by increasing the energy required for tunneling out. Finally, *n*_{g} is moved slowly back to the initial degeneracy value, extracting energy from the heat bath in the process, and completing the cycle. An example of four such consecutive experimental cycles is shown in Fig. 1*E*. Dotted vertical lines denote the time when the measurement is performed. We observe that the feedback signal indeed locks the extra electron to the measured state (parts of the trace in the Fig. 1*E*, *Upper* with no jumps), but the charge starts to hop again when *n*_{g} is moved toward the degeneracy point.

More quantitatively, the working space of the engine is spanned by the number *n* of excess electrons on one of the box islands, whereas equilibrium electron gas in the box islands plays the role of the thermal reservoir at temperature *T*. Because there is only capacitive coupling between the box and the rest of the circuitry, electron tunneling takes place only between the two box islands. Therefore, the total electric charge on the two islands is conserved, and the state with *n* excess electrons on one island has −*n* excess electrons on the other island, as in a regular capacitor made of two electrodes. The internal energy of the engine is given then by the charging energy of these states, *E*_{n} = *E*_{c}(*n* − *n*_{g})^{2}, averaged over their occupation probabilities *p*_{n}. Here *E*_{c} = *e*^{2}/2*C*_{tot} is the usual charging energy of the total capacitance *C*_{tot} between the box islands. In the low-temperature regime relevant for this work, the charge dynamics is reduced to the two states, *n* = 0, 1. Thermodynamics of the engine cycle described above qualitatively is characterized quantitatively (17) by (*i*) the work done by the gate voltage source, *ii*) the heat *Q* transferred to the electron gas of the box islands, i.e., to the thermal reservoir, by electron tunneling events. Note that electron tunneling events which change the charge state *n* make the integral in the expression for work *W* dependent on the specific realization of the history of the tunneling transitions. Each tunneling event produces the heat*n*: 0 → 1 transitions, and the minus sign describes the *n*: 1 → 0 transitions. These relations enable us to measure *Q* directly, as was done previously in refs. 18, 19, by detecting the electron tunneling events in real time and evaluating the corresponding energy difference *E*_{0} − *E*_{1} at the moments of these events.

In the closed cycle of our experiment, energy conservation makes the total heat −*Q* extracted from the reservoir equal to the work −*W* extracted from the engine; see *SI Text* for a proof. The cycle starts with the SEB at degeneracy, and at this point the charge state is measured by the external SET detector. One bit of information represented by the (equally probable) position of the extra electron on one or the other island of the box is copied into the detector and stored for the subsequent feedback process, where it is used to determine the polarity of the rapid gate-voltage drive. If the box is found in the state *n* = 0, the gate voltage is changed rapidly so that the offset charge *n*_{g} changes from the degeneracy value *n*_{g} = 1/2 to *n*_{g} = 0; if the measured state is *n* = 1, *n*_{g} changes from *n*_{g} = 1/2 to *n*_{g} = 1. Such a rapid feedback drive traps the electron to the measured state. Ideally, this drive is so fast that no electron transitions have a chance to occur during it and, as a result, no heat is transferred to the reservoir.

The final part of the engine cycle is the quasistatic ramp which returns the box to the degeneracy. In the fully quasistatic limit, the heat *Q* dissipated in the reservoir can be found by considering the change of the total entropy *S* of the box. This change, Δ*S* = Δ*S*_{r} + Δ*S*_{ch}, consists of the standard entropy change of the thermal reservoir in equilibrium at temperature *T* due to heat flow into it, Δ*S*_{r} = 〈*Q*〉/*T*, where 〈*Q*〉 is the average dissipated heat, and the change of the Boltzmann entropy of the charge states*p*_{n} (14), one can find the rate of change of entropy *S* due to electron tunneling in a general evolution process as_{mn} is the rate of electron tunneling from state *n* to *m*. The tunneling rates satisfy the detailed-balance condition, Γ_{mn} = Γ_{nm} exp{(*E*_{n} − *E*_{m})/*k*_{B}*T*} (see *SI Text* for details). Eq. **3** shows that *S* never decreases, and remains constant in the fully adiabatic evolution, when the probabilities *p*_{n} maintain local equilibrium, *p*_{n} ∝ exp{−*E*_{n}(*t*)/*k*_{B}*T*} and the detailed-balance condition ensures that the probability fluxes vanish: *p*_{n}Γ_{mn} = *p*_{m}Γ_{nm}. In this case, the total entropy is conserved, Δ*S* = 0, and the two components of *S* change in the opposite directions Δ*S*_{r} = 〈*Q*〉/*T* = −Δ*S*_{ch}. The average heat 〈*Q*〉 is determined by the change of the entropy of the charge states,**4** gives *Q* = −*k*_{B}*T* ln 2 for every ramp. Qualitatively, this means that we are extracting *k*_{B}*T* ln 2 of heat from the reservoir by creating a bit of information determined by the electron position on one or the other island of the SEB. In terms of work *W*, it is first extracted from the box by rapid lowering of the potential, as sketched in Fig. 1*B*. Work is then applied to drive the box back to the degeneracy; however, the required work is lowered by the amount of heat *k*_{B}*T* ln 2 absorbed from the thermal bath. The ability to reach this fundamental maximum distinguishes the SEB setup in this work from other proposed electronic MDs (20⇓–22) and is important for establishing the link between the extracted heat and information.

Fig. 2 shows the results of the measurements that illustrate such an extraction of heat from the reservoir. We drive our SEB starting from *n*_{g} = 0 toward *n*_{g} = 1 at various rates *n* continuously to measure the total dissipated heat *Q* with Eq. **1**. We see that as the rate of the drive decreases, the average dissipated heat approaches the prediction of Eq. **4**: 〈*Q*〉 tends to −*k*_{B}*T* ln 2 for *n*_{g} = 0.5. This process can also be viewed as the reversal of the Landauer erasure of one bit of information, in which the system is driven from the degeneracy with two equally occupied states to one certain configuration. Such an erasure produces at least *k*_{B}*T* ln 2 of heat as demonstrated explicitly by recent experiments on a colloidal bead controlled with optical tweezers (9). Because the drive in Fig. 2 starts with *n*_{g} = 0, such that the SEB is in a definite state *n* = 0 and thus initially *S*_{ch} = 0, the lowest curve in this plot approaching Eq. **4** can be viewed as direct measurement of the equilibrium entropy *S*_{ch} of the system of the two charge states *n* = 0, 1. When the quasistatic ramp to *n*_{g} = 0.5, as illustrated in Fig. 2, is complemented with an ideal measurement and immediate feedback that follows our SE protocol, the SEB operates as a reversible MD, abstract models of which have been discussed theoretically recently (10⇓–12).

Fig. 3 demonstrates the experimental performance of our SE; see *SI Text* for details about the measurement protocol. The distribution of *W* is obtained from an ensemble of measured trajectories of *n*. The applied work *W* coincides with *Q* determined by Eq. **1** for each individual realization. Because the slow part of the cycle is not fully quasistatic, there are cycle-to-cycle fluctuations in *W*, such that *W* forms a continuous distribution. The cycles with correct gate-voltage feedback (Fig. 3, *Left Inset*) trap the electron on the SEB island, on which it actually sits at degeneracy after the measurement. Then no electron tunneling occurs in the feedback process, and *W* is close to the ideal limit −*k*_{B}*T* ln 2. Such successful cycles produce the large peak at negative values of *W* in Fig. 3, around the ideal value that is indicated by the vertical dashed line. An error in the measurement or feedback drives the SEB to the excited charge state with excess energy Δ*E* = 2*E*_{C}|Δ*n*_{g}|, where Δ*n*_{g} is the total change in *n*_{g} during the fast drive. Subsequent tunneling to the low-energy state (Fig. 3, *Lower Inset*) dissipates energy Δ*E* ≫ *k*_{B}*T* ln 2 extracted in the quasistatic part. Such cycles produce the small peak at positive values of *W* in Fig. 3. For this measurement, we have chosen the optimized |Δ*n*_{g}| = 0.125 to keep the contribution of the positive *W* as small as possible, without significantly reducing the heat extracted from the thermal bath during the quasistatic drive. With this choice, we obtain an average extracted work per cycle of 〈−*W*〉 ∼ 0.75 × *k*_{B}*T* ln 2. For comparison, if no measurement were performed, only 50% of the cycles would be successful, and one would do positive work 〈*W*〉 ∼ 1.55 × *k*_{B}*T* ln 2 on the average.

To summarize, our experiment is a realization of an MD, similar to an SE, with an SEB. We demonstrate quantitatively the extraction of *k*_{B}*T* ln 2 of heat by creating a bit of information encoded in the position of the extra electron on one of the two islands of the box. Under a practical feedback cycle, our engine achieves a fidelity of about 75%. The heat transfer measurements performed as a part of MD demonstration provide also a direct measurement of the equilibrium entropy of a two-state system.

## Acknowledgments

This work has been supported in part by the European Union Seventh Framework Programme Information, Fluctuations and Energy Control in Small Systems (INFERNOS) (FP7/2007-2013) under Grant Agreement 308850, Academy of Finland (Projects 139172, 250280, and 272218), Väisälä Foundation, the National Doctoral Programme in Nanoscience (NGS-NANO) (to V.F.M.), and the National Science Foundation Grant PHY-1314758 (D.V.A.). We acknowledge The Otaniemi Research Infrastructure for Micro- and Nanotechnologies for providing the processing facilities and technical support.

## Footnotes

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^{1}To whom correspondence should be addressed. Email: jonne.koski{at}aalto.fi.

Author contributions: J.V.K., V.F.M., J.P.P., and D.V.A. designed research; J.V.K., V.F.M., J.P.P., and D.V.A. performed research; J.V.K., V.F.M., J.P.P., and D.V.A. contributed new reagents/analytic tools; J.V.K. analyzed data; and J.V.K., V.F.M., J.P.P., and D.V.A. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1406966111/-/DCSupplemental.

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