HAL Id: ensl-03240746
https://hal-ens-lyon.archives-ouvertes.fr/ensl-03240746
Submitted on 28 May 2021
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Information and thermodynamics: fast and precise
approach to Landauer’s bound in an underdamped
micro-mechanical oscillator
Salambô Dago, Jorge Pereda, Nicolas Barros, Sergio Ciliberto, Ludovic Bellon
To cite this version:
Salambô Dago, Jorge Pereda, Nicolas Barros, Sergio Ciliberto, Ludovic Bellon. Information and
ther-modynamics: fast and precise approach to Landauer’s bound in an underdamped micro-mechanical
oscillator.
Physical Review Letters, American Physical Society, 2021, 126 (17),
Salambô Dago, Jorge Pereda, Nicolas Barros, Sergio Ciliberto, and Ludovic Bellon∗
Univ Lyon, ENS de Lyon, Univ Claude Bernard Lyon 1, CNRS, Laboratoire de Physique, F-69342 Lyon, France
The Landauer principle states that at least kBT ln 2of energy is required to erase a 1-bit memory,
with kBT the thermal energy of the system. We study the effects of inertia on this bound using
as one-bit memory an underdamped micro-mechanical oscillator confined in a double-well potential created by a feedback loop. The potential barrier is precisely tunable in the few kBT range. We
measure, within the stochastic thermodynamic framework, the work and the heat of the erasure protocol. We demonstrate experimentally and theoretically that, in this underdamped system, the Landauer bound is reached with a 1 % uncertainty, with protocols as short as 100 ms.
The thermodynamic energy cost of information pro-cessing is a widely studied subject both for its
funda-mental aspects and for its potential applications [1–9].
This energy cost has a lower bound, fixed by Landauer’s
principle [10]: at least kBT ln 2 of work is required to
erase one bit of information from a memory at
tem-perature T , with kB the Boltzmann constant. This is
a tiny amount of energy, only ∼ 3 × 10−21J at room
temperature (300 K), but it is a general lower bound, independent of the specific type of memory used, and
it is related to the generalized Jarzynski equality [11].
The Landauer bound (LB) has been measured in several
classical experiments, using optical tweezers [12,13], an
electrical circuit [14], a feedback trap [15–17] and
nano-magnets [18,19] as well as in quantum experiments with
a trapped ultracold ion [20] and a molecular
nanomag-net [21]. The LB can be reached asymptotically in
quasi-static erasure protocols whose duration is much longer than the relaxation time of the above mentioned sys-tems used as one-bit memories. In practice, when the erasure is performed in a short time, the energy needed for such a process can be minimized using optimal
pro-tocols, which have been computed [22–27] and used for
overdamped systems [17]. Another strategy to approach
the asymptotic LB faster is of course to reduce the re-laxation time. However, for very fast protocols, one may wonder whether inertial (inductive) terms in mechanical (electronic) systems play a role in their reliability and energy cost.
The goal of this letter is to analyze this problem and to provide a new experimental and theoretical explo-ration of Landauer’s principle on an underdamped micro-mechanical oscillator confined in a very specific double-well potential. Erasure procedures in underdamped sys-tems have never been studied before, and this new ap-plication field opens significant possibilities, since oscil-lators are fundamental building blocks to many systems. Moreover, it is interesting to verify the LB for a weak coupling to the thermostat. Both the relaxation time and the coupling to the bath of our system are orders of magnitude smaller than those of the overdamped
sys-102 103 104 10-8 10-6 10-4 10-2 <latexit sha1_base64="OLxJUKVVpzWEAGoh76cATqQKZPA=">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</latexit> x <latexit sha1_base64="sUwDnQGPmgaIcgHXzdVoW5BgaF0=">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</latexit> F <latexit sha1_base64="rhKj0vED8j075GbWNhtpDr9YPbM=">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</latexit> x Differential interferometer <latexit sha1_base64="Zf1c1+jsBUHg5KevVk70MjLFf8o=">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</latexit> V1 <latexit sha1_base64="f6bq24mZpMmH233bUJfflpX7M9M=">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</latexit> x0 <latexit sha1_base64="tX1THjs9krjxwPuMbbbkVlBOS4I=">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</latexit> x > x0 Feedback loop a) b) c) V =<latexit sha1_base64="+uWOu207zS7O72h4JDjLsjIj0NM=">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</latexit> ±V1 V<latexit sha1_base64="Cojfb1TAz/2V915LaOfDpskoqfs=">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</latexit> 0
FIG. 1. (a) Schematic diagram of the experiment. The con-ductive cantilever is sketched in yellow. Its deflection x is measured with a differential interferometer [28], through two laser beams focused respectively on the cantilever and on its base. The cantilever at voltage V = ±V1is facing an electrode
at V0. The voltage difference V − V0 between them creates
an attractive electrostatic force F ∝ (V − V0)2. The dashed
box encloses the feedback controller made of a comparator and a multiplier, which create the double-well potential. (b) Measured Power Spectrum Density (PSD) of the cantilever deflection thermal noise with no feedback (V1= 0, blue), and
best fit by the theoretical thermal noise spectrum of a Simple Harmonic Oscillator (SHO, dashed red). Up to 10 kHz, the cantilever behaves like a resonator at f0 = 1270 Hz, with a
quality factor Q = 10. We infer from this measurement the variance σ2
= hx2i = kBT /k, used to normalize all measured
quantities. (c) Measured double-well potential energy (blue) obtained with the feedback on, with x0 = 0and V1 adjusted
to have a 5 kBT barrier height. The potential is inferred from
the measured Probability Density Function (PDF) of x dur-ing a 10 s acquisition and the Boltzmann distribution. The fit using Eq. (1) is excellent (dashed red).
tems of previous demonstrations. We approach the LB in
100 ms, compared to the 30 s [13] previously needed, and
thus accumulate much more statistics. Notice that the asymptotic time in our experiment is among the fastest
2 0 1 0 1 Stage 2 Stage 1 U (z, z<latexit sha1_base64="GTfDUZVorDFfYh3WaeXdxHgXH1Q=">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</latexit> 0, z1) P (z, z0, z1) <latexit sha1_base64="aSi/XvhKNG2U22dM+xA+4vXuGJc=">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</latexit> a) b)
FIG. 2. Description of the erasure protocol. (a) Evolution of the potential U(z, z0, z1). (b) Experimental static PDF P (z, z0, z1)
(blue), best fit to the Boltzmann distribution (red), and position of the threshold z0 (dashed green).
erasure times of the systems mentioned above. Further-more, our choice of confining potential allows both a pre-cise experimental control and an analytical computation of the work and heat on a trajectory. These predictions can be compared quantitatively with the experimental results because they include only quantities that can be measured with high precision in our experimental set-up,
schematically described in Fig.1(a).
The underdamped oscillator is a conductive
can-tilever [29] which is weakly damped by the surrounding
air. Its deflection x is measured with very high accu-racy and signal-to-noise ratio by a differential
interfer-ometer [28]. The Power Spectrum Density (PSD) of the
thermal fluctuations of x is plotted in Fig. 1(b). The
fit of this PSD with the thermal noise spectrum of a Simple Harmonic Oscillator (SHO) gives its resonance
frequency f0 = ω0/2π = 1270 Hz and quality factor
Q = mω0/γ = 10, where m, k = mω20 and γ are
re-spectively: the mass, stiffness and damping coefficient of the SHO. From the PSD we compute the variance
σ2=hx2i = k
BT /k∼ 1 nm2, which is used to normalize
all measured quantities: deflections are from here on
ex-pressed as z = x/σ, and energies are in units of kBT. The
SHO potential energy is for example 1
2kx2/kBT = 12z2,
and its kinetic energy Ek =12m ˙x2/kBT =12˙z2/ω02.
In order to use the cantilever as a one-bit memory, we need to confine its motion in an energy potential con-sisting of two wells separated by a barrier, whose shape can be tuned at will. This potential U is created by a feedback loop, which compares the cantilever deflection
z to an adjustable threshold z0. After having multiplied
the output of the comparator by an adjustable voltage
V1, the result is a feedback signal V which is +V1 if
z > z0 and −V1 if z < z0. The voltage V is applied
to the cantilever which is at a distance d from an
elec-trode kept at a voltage V0. The cantilever-electrode
volt-age difference V0± V1 creates an electrostatic attractive
force F = 1
2∂dC(d)(V0 ± V1)2 [30], where C(d) is the
cantilever-electrode capacitance. Since d σ, ∂dC(d)
can be assumed constant. We apply V0 ∼ 100 V and
V1 V0 so that in a good approximation, F ∝ ±V1
up to a static term. This feedback loop results in the application of an external force whose sign depends on whether the cantilever is above or below the threshold
z0. The reaction time of the feedback loop is very fast
(around 10−3f−1
0 ), thus the switching transient is
trans-parent thanks to the inertia of the cantilever. As a con-sequence, the oscillator evolves in a virtual static double-well potential, whose features are controlled by the two
parameters z0and V1. Specifically, the barrier position is
set by z0and its height is controlled by V1which sets the
wells centers ±z1 = ±V1∂dC(d)V0/(kσ). The potential
energy constructed by this feedback is:
U (z, z0, z1) = 1
2 z− S(z − z0)z1
2
, (1)
where S(z) is the sign function: S(z) = −1 if z < 0 and S(z) = 1 if z > 0. We carefully checked that the feedback loop creates a proper virtual potential without perturbing the response of the system, including
hav-ing no effect on the equilibrium velocity distribution [31].
The potential in Eq. (1) can be experimentally measured
from the Probability Distribution Function (PDF) P (z)
and the Boltzmann distribution P (z) ∝ e−U (z). Figure
1(c) presents an example of an experimental
symmet-ric double-well potential generated by the feedback loop,
tuned to have a barrier of 1
2z12 = 5 (z0 = 0, z1 =√10).
The dashed red line is the best fit with Eq. (1),
demon-strating that the feedback-generated potential behaves as an equilibrium one.
The erasure protocol corresponds to the potential
evo-lution described in Fig. 2, with the associated
experi-mental static position distribution P (z). Initially, the system is at equilibrium either in the state 0 (left-hand
centered in Z1) with a probability pi = 12. The process
must result in the final state 0 with probability pf = 1, to
perform a full erasure process. To do so, during the first stage we drive the wells closer until they merge into one single harmonic well centered in 0. After a short equili-bration time, the barrier position (dashed green line) is pushed away to prevent the cantilever from visiting the right hand well (black line). It thus remains in the left hand well (red line), which is driven back to its initial
position in −Z1. The so-called stage 2 ends when the
threshold is brought back in 0, to reach the final state 0 in the bi-stable potential, regardless of the initial state.
3⌧<latexit sha1_base64="RACK4XLJJL+QTsyE+3G8/ox1o1g=">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</latexit> 0
⌧0
<latexit sha1_base64="Y1Z4OoaIJAYelkylwOqfFXkaOuY=">AAAC9XicjVLLSsNAFD3GV31XXboJFsFVSVTQZcGNywq2ClokidM6miYhmail+A9uFdyJW7/HP9CF/+CZMYJaRCckOXPuOffOnRk/CWWmHOd5yBoeGR0bL01MTk3PzM6V5xeaWZyngWgEcRinB76XiVBGoqGkCsVBkgqv64di3z/f1vH9C5FmMo72VC8Rra7XiWRbBp4i1TxSXn7sHJcrTtUxwx4EbgEqKEY9Lr/hCCeIESBHFwIRFHEIDxmfQ7hwkJBroU8uJZImLnCNSXpzqgQVHtlzfjucHRZsxLnOmRl3wCoh35ROGyv0xNSlxLqabeK5yazZ33L3TU69th7/fpGrS1bhlOxfvk/lf326F4U2tkwPkj0lhtHdBUWW3OyKXrn9pSvFDAk5jU8YT4kD4/zcZ9t4MtO73lvPxF+MUrN6HhTaHK9mlb/3l7FGTq2k+kqvjJfB/Xn0g6C5VnXXq2u7G5XacnEtSljCMlZ59puoYQd1NFjpDDe4xZ11ad1bD9bjh9QaKjyL+Dasp3e9BZrt</latexit> ⌧
<latexit sha1_base64="WFrcS5YIEmudbe6Kqc7jsh3Anac=">AAAC83icjVLLSsNAFD2Nr1pfVZdugkVwVZIq6LLgxmUF+4C2SJJO26F5kcyIpfgLbhXciVs/yD/Qhf/gnTEFtRSdkOTMuefcO3dm3NjnqbCs15yxsLi0vJJfLaytb2xuFbd3GmkkE4/VvciPkpbrpMznIasLLnzWihPmBK7Pmu7oTMWb1yxJeRReinHMuoEzCHmfe45QVEc48qpYssqWHuYssDNQQjZqUfEDHfQQwYNEAIYQgrAPByk9bdiwEBPXxYS4hBDXcYZbFMgrScVI4RA7ou+AZu2MDWmucqba7VEVn96EnCYOyBORLiGsqpk6LnVmxc7LPdE51drG9HezXAGxAkNi//JNlf/1qV4E+jjVPXDqKdaM6s7Lski9K2rl5reuBGWIiVO4R/GEsKed0302tSfVvau9dXT8TSsVq+ZeppV416uc319KNSRpOalv1MroMti/j34WNCpl+6hcuTguVfeza5HHHvZxSGd/girOUUOdKg1xh3s8GNJ4NJ6M5y+pkcs8u/gxjJdPATGaSg==</latexit> <latexit sha1_base64="WFrcS5YIEmudbe6Kqc7jsh3Anac=">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</latexit>⌧ ⌧0
<latexit sha1_base64="Y1Z4OoaIJAYelkylwOqfFXkaOuY=">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</latexit>
FIG. 3. Time recording of the cantilever deflection z on a single trajectory (blue, starting in state 1 in this example), superposed with the two wells’ centers +z1 (black) and −z1
(red), and the threshold z0(dashed green). Stage 1 (red
back-ground) et 2 (green backback-ground) both last τ = 200 ms. τ is very long compared to the natural relaxation time τr of the
cantilever: f0τ ∼ 250 f0τr = Q/π ∼ 3. The equilibrium
periods around stages 1 and 2 are chosen freely as long as they allow the cantilever to relax, in this example τ0= 50 ms τr.
To implement the erasure procedure, the position of
the center of the wells ±z1 (black and red lines), and
the barrier position z0(green dashed line) are driven
ac-cordingly to the profiles in Fig. 3. Initial equilibrium
is ensured by a steady potential for a duration of τ0,
chosen longer than the natural relaxation time of the
system τr = 2Q/ω0 = 2.5 ms. For clarity purposes,
τ0= 50 msin Fig.3. During stage 1 (red background) the
center of the wells moves linearly from the initial state
z1(0) = Z1= 4.55to z1(τ ) = 0in a time τ = 200 ms. Z1
sets the height of the barrier in the initial state 1
2Z12& 10,
chosen high enough to secure the initial and final states. When the two wells are close enough, the cantilever starts switching from one well to the other: the information is progressively lost. The cantilever is then allowed to relax
at equilibrium in this single well for 3τ0. During stage
2 (green background), the threshold z0 is set at a large
value (of order Z1or higher): the cantilever can no longer
switch to the right hand side well. The cantilever is then
brought back to the state 0, as z1 follows a linear ramp
from 0 to Z1, in the same time τ. The protocol ends at
τf = 2τ + 5τ0& 2τ + 15τr [32], after letting the system
relax during τ0 and bringing back z0 to 0.
The data plotted in Fig.3contains all we need to
com-pute the stochastic work and heat along a single trajec-tory. Indeed, if we apply to the underdamped regime the
generic computations of these stochastic quantities [33–
37], we obtain the following expressions [31]:
W = Z τf 0 X i=0,1 ∂U ∂zi ˙zidt = Z τf 0 S(z−z0)z− z1 ˙z1dt (2) Q = − Z τf 0 ∂U ∂z ˙zdt− h Ek iτf 0 = Z τf 0 S(z− z0)z1− z ˙zdt − 1 2ω2 0 h ˙z2i τf 0 . (3) From the above definitions, we deduce the energy balance of the system:
∆U + ∆Ek =W − Q. (4)
Since at τf the system has relaxed to equilibrium, and the
initial and the final states of the protocol are the same,
h∆Ui = h∆Eki = 0, thus in average hWi = hQi [38].
In order to reach the LB, we should proceed in a
quasi-static fashion. From Eqs. (2) and (3), using the
Boltz-mann equilibrium distribution and the potential energy
of Eq. (1), we demonstrate that in the quasi-static regime
stage 1 requires hWi = hQi = ln 2, and that stage 2
doesn’t contribute [31]. Actually, in a quasi-static
evo-lution, any intermediate state can be theoretically
de-scribed with our specific choice of U(z, z0, z1), and the
mean power computed all along the cycle. Long
proto-cols satisfying τ τr can be considered as quasi-static
erasures, hence we present in Fig.4the results obtained
from 2000 trajectories, equivalent to the one of Fig.3. It
should be emphasized that the protocol is perfectly effi-cient from an information processing point of view: all 2000 trajectories ended in the prescribed well, regardless of the initial condition.
As expected for a slow translation of a single well, stage
2 requires nearly no power: hW2i = 0 ± 0.008.
Dur-ing stage 1 however, the power increases when the can-tilever starts switching between the wells. Of course, the equilibrium stage never contributes to the erasure cost. Summing the dissipated power along the process gives:
hW1i = 0.702 ± 0.006 and hW2i = 0 ± 0.008.
Further-more, we also compute the mean dissipated heat during the whole procedure: hQi = 0.68 ± 0.03. These results are in perfect agreement with the Landauer principle: in a quasi-static process, the mean work or heat required to erase 1 bit of information is ln 2 = 0.693.
After providing experimental evidence that our system matches the LB in the quasi-static limit, we study how fast the procedure can be performed before paying an ex-tra energetic cost for the erasure. We therefore repeat the
4 -1 0 1 2 0 0.5 1 1.5 2
FIG. 4. Time evolution of the mean power over 2000 trajec-tories following the slow protocol of Fig. 3. The work takes off when the cantilever starts switching between the wells, during stage 1 (red background). Since the process is quasi-static, stage 2 (green background) doesn’t contribute on av-erage. The red dashed line is the analytical prediction in the quasi-static regime [31]. (Inset) Work distributions for the 2000 trajectories during stage 1 (W1, red) and 2 (W2, green)
fitted by gaussians (plain lines). As theoretically predicted for a slow erasure, stage 1 reaches the LB of 0.693 (dashed red) while stage 2 requires negligible work (dashed green). PDF of Q are available in the Suppl. Mat. [31]
report the results in Fig.5. The initial distance between
the wells may vary slightly from one set of experiments to
the other (Z1= 4.5to 6, but Z1 is constant for the 2000
experiments of any given τ), so we use the speed Z1/τ to
represent how far from a quasi-static protocol we stand.
As expected, the slow procedures meet the LB (Figs. 3
and4present the slowest one), and quick ones require an
extra cost. For finite τ, it has been reported in earlier
demonstrations of the LB [12,15] that hWi ∼ ln 2+B/τ,
with B a constant depending on the system and applied protocol. More generally, this 1/τ asymptotical behavior is expected for the mean stochastic work or heat for
fi-nite time transformations both in overdamped [22,27,34]
and underdamped [39] systems. This suggests a fit of
our results with L∞+ B0˙z1= L∞+ B0Z1/τ. It leads to
B0 = (437± 27) µs and L
∞ = 0.695± 0.012 which
vali-dates, again, with great accuracy, the Landauer principle.
It is noteworthy that using protocols of only τf ∼ 100 ms
(with τ = 30 ms and τ0= 3τr= 7.5 ms), the energy cost
of erasure is only 10% larger than the LB. We didn’t
ex-plore cycles faster than τ = 6f−1
0 per ramp, which are
already only twice the relaxation time τr= π1Qf0−1.
One may wonder if the extra cost at high speed is due either to the erasure protocol itself, or to the damping
losses during the ramps. In the inset of Fig.5, we plot
the contribution W2 or Q2 of the ramp of stage 2. In a
first approximation (neglecting transients), the cantilever
will follow the well center z1at speed ˙z1, while
experienc-FIG. 5. Energy cost from erasure protocols at different speeds ˙
z1= Z1/τ, with Z1∼ 5. Experimental data (blue) are fitted
by L∞+ B0Z1/τ (dashed red), with B0 = (437 ± 27)µs and
L∞ = 0.695 ± 0.012. (Inset) Experimental mean work and
heat during stage 2 (green) at different speeds. The theoreti-cal prediction B0
2Z1/τ(dashed red) with B02= Z1/Qω0works
perfectly (no adjustable parameters).
ing a viscous drag force γσ ˙z1. We thus expect the ramp
cost to be hW2i = hQ2i ∼ B20Z1/τ, with B02= Z1/Qω0.
This approximation with no adjustable parameters is
re-ported in the inset of Fig.5, and matches perfectly the
experimental data. We compute B0
2∼ 63 µs for Z1 ∼ 5:
the ramp contribution is not enough to explain the extra
cost of fast erasures, since 2B2(one contribution for each
stage) would explain only 30 % of the overhead to ln 2. As a conclusion, let us summarize the highlights of this work. We demonstrate in this letter the benefits of exploring stochastic thermodynamics with an under-damped micro-oscillator: owing to its fast dynamics and small relaxation time, we can accumulate large statistics and lower uncertainties for the measured quantities. The use of a high-precision interferometer coupled with a sim-ple feedback loop allows for a well-defined and tunable double-well potential energy. Combining these two fea-tures, we demonstrate experimentally that the Landauer bound can be reached with a very high accuracy in a short time. Furthermore, we can check every step with analytical predictions. Inertia has no effect on the limit, which is reached for just a few natural relaxation times of the oscillator. Notice that, by using oscillators with a higher resonance frequency, the relaxation time could be considerably reduced. Thus, the present approach is promising to further explore stochastic thermodynamics, with unprecedented statistics and more complex behav-iors brought on by the inertia.
AcknowledgmentsThis work has been financially
sup-ported by the Agence Nationale de la Recherche through grant ANR-18-CE30-0013 and by the FQXi Foundation, Grant No. FQXi-IAF19-05, “Information as a fuel in
col-loids and superconducting quantum circuits.”
Data availabilityThe data that support the findings
of this study are openly available in Zenodo at https:
//doi.org/10.5281/zenodo.4626559[40]
∗ ludovic.bellon@ens-lyon.fr
[1] H. Leff and e. Rex, A.F., Maxwell’s Demon: Entropy, Classical and Quantum Information, Computing (Insti-tute of Physics, Philadelphia, 2003).
[2] J. M. R. Parrondo, J. M. Horowitz, and T. Sagawa, Na-ture Physics 11, 131 (2015).
[3] E. Lutz and S. Ciliberto,Physics Today 68, 30 (2015). [4] C. Lent, A. Orlov, W. Porod, and G. Snider, Energy
Lim-its in Computation (Springer, 2018).
[5] M. López-Suárez, I. Neri, and L. Gammaitoni,Nat Com-mun. 7, 12068 (2016).
[6] M. Konopik, T. Korten, E. Lutz, and H. Linke, Fun-damental energy cost of finite-time computing (2021),
arXiv:2101.07075.
[7] S. Toyabe, T. Sagawa, M. Ueda, M. Muneyuki, and M. Sano,Nature Phys. 6, 988 (2010).
[8] É. Roldán, I. A. Martínez, J. M. R. Parrondo, and D. Petrov,Nature Physics 10, 457 (2014).
[9] J. V. Koski, V. F. Maisi, J. P. Pekola, and D. V. Averin,
Proceedings of the National Academy of Sciences 111, 13786 (2014).
[10] R. Landauer,IBM Journal of Research and Development 5, 183 (1961).
[11] A. Bérut, A. Petrosyan, and S. Ciliberto, EPL (Euro-physics Letters) 103, 60002 (2013).
[12] A. Bérut, A. Arakelyan, A. Petrosyan, S. Ciliberto, R. Dillenschneider, and E. Lutz,Nature 483, 187 (2012). [13] A. Bérut, A. Petrosyan, and S. Ciliberto, Journal of Statistical Mechanics: Theory and Experiment 2015, P06015 (2015).
[14] A. O. Orlov, C. S. Lent, C. C. Thorpe, G. P. Boechler, and G. L. Snider, Japanese Journal of Applied Physics 51, 06FE10 (2012).
[15] Y. Jun, M. Gavrilov, and J. Bechhoefer,Phys. Rev. Lett. 113, 190601 (2014).
[16] M. Gavrilov and J. Bechhoefer,EPL (Europhysics Let-ters) 114, 50002 (2016).
[17] K. Proesmans, J. Ehrich, and J. Bechhoefer,Phys. Rev. Lett. 125, 100602 (2020).
[18] J. Hong, B. Lambson, S. Dhuey, and J. Bokor,Sci. Adv. 2, e1501492 (2016).
[19] L. Martini, M. Pancaldi, M. Madami, P. Vavassori,
G. Gubbiotti, S. Tacchi, F. Hartmann, M. Emmerling, S. Höfling, L. Worschech, and G. Carlotti,Nano Energy 19, 108 (2016).
[20] L. L. Yan, T. P. Xiong, K. Rehan, F. Zhou, D. F. Liang, L. Chen, J. Q. Zhang, W. L. Yang, Z. H. Ma, and M. Feng,Phys. Rev. Lett. 120, 210601 (2018).
[21] R. Gaudenzi, E. Burzuri, S. Maegawa, H. S. J. van der Zant, and F. Luis,Nucl. Phys. 14, 565 (2018).
[22] E. Aurell, K. Gaw¸edzki, C. Mejía-Monasterio, R. Mo-hayaee, and P. Muratore-Ginanneschi,Journal of Statis-tical Physics 147, 487 (2012).
[23] G. Diana, G. B. Bagci, and M. Esposito,Phys. Rev. E 87, 012111 (2013).
[24] T. Schmiedl and U. Seifert,Phys. Rev. Lett. 98, 108301 (2007).
[25] A. B. Boyd, D. Mandal, and J. P. Crutchfield,Phys. Rev. X 8, 031036 (2018).
[26] A. B. Boyd, A. Patra, C. Jarzynski, and J. P. Crutchfield, Shortcuts to thermodynamic computing: The cost of fast and faithful erasure (2018),arXiv:1812.11241.
[27] P. Muratore-Ginanneschi and K. Schwieger,Entropy 19, 379 (2017).
[28] P. Paolino, F. Aguilar Sandoval, and L. Bellon,Rev. Sci. Instrum. 84, 095001 (2013).
[29] Doped silicon cantilever OCTO1000S from Micromotive Mikrotechnik, nominal stiffness 3 × 10−3
N/m.
[30] H.-J. Butt, B. Cappella, and M. Kappl,Surface Science Reports 59, 1 (2005).
[31] See Supplemental Material at http://[link provided by the editorial office] for details on the feedback loop, the derivation of the theoretical predictions on work and heat, an example of high speed protocol, and a PDF of Q.
[32] To set τ0, we use the rule of thumb that a perturb system
returns to equilibrium after 3τr, with τrthe characteristic
relaxation time in the exponential decay.
[33] K. Sekimoto, Stochastic Energetics, Lecture Notes in Physics, Vol. 799 (Springer, 2010).
[34] K. Sekimoto and S. Sasa,Journal of the Physical Society of Japan 66, 3326 (1997).
[35] U. Seifert, Reports on Progress in Physics 75, 126001 (2012).
[36] C. Jarzynski, Annual Review of Condensed Matter Physics 2, 329 (2011).
[37] S. Ciliberto,Phys. Rev. X 7, 021051 (2017).
[38] hWi does not depend on τf but on τ only. Indeed as
˙
z1 = 0 outside the ramps period then in Eq. (2) the
integration limits reduces to 2τ [31].
[39] A. Gomez-Marin, T. Schmiedl, and U. Seifert,The Jour-nal of Chemical Physics 129, 024114 (2008).
[40] S. Dago, J. Pereda, N. Barros, S. Ciliberto, and L. Bellon,
