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
IInformation and thermodynamics: fast and precise approach to Landauer’s bound inan underdamped micro-mechanical oscillator
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 k B T ln 2 of energy is required to erase a 1-bit memory,with k B T the thermal energy of the system. We study the effects of inertia on this bound usingas one-bit memory an underdamped micro-mechanical oscillator confined in a double-well potentialcreated by a feedback loop. The potential barrier is precisely tunable in the few k B T range. Wemeasure, within the stochastic thermodynamic framework, the work and the heat of the erasureprotocol. We demonstrate experimentally and theoretically that, in this underdamped system, theLandauer bound is reached with a 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’sprinciple [10]: at least k B T ln 2 of work is required toerase one bit of information from a memory at tem-perature T , with k B the Boltzmann constant. This isa tiny amount of energy, only ∼ × − J at roomtemperature (
300 K ), but it is a general lower bound,independent of the specific type of memory used, andit is related to the generalized Jarzynski equality [11].The Landauer bound (LB) has been measured in severalclassical experiments, using optical tweezers [12, 13], anelectrical circuit [14], a feedback trap [15–17] and nano-magnets [18, 19] as well as in quantum experiments witha 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 longerthan the relaxation time of the above mentioned sys-tems used as one-bit memories. In practice, when theerasure is performed in a short time, the energy neededfor such a process can be minimized using optimal pro-tocols, which have been computed [22–26] and used foroverdamped systems [17]. Another strategy to approachthe asymptotic LB faster is of course to reduce the re-laxation time. However, for very fast protocols, one maywonder whether inertial (inductive) terms in mechanical(electronic) systems play a role in their reliability andenergy cost.The goal of this letter is to analyze this problem andto 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 weakcoupling to the thermostat. Both the relaxation timeand the coupling to the bath of our system are ordersof magnitude smaller than those of the overdamped sys- -8 -6 -4 -2
10 kHz , thecantilever behaves like a resonator at f = 1270 Hz , with aquality factor Q = 10 . We infer from this measurement thevariance σ = h x i = k B T /k , used to normalize all measuredquantities. (c) Measured double-well potential energy (blue)obtained with the feedback on, with x = 0 and V adjustedto have a k B T barrier height. The potential is inferred fromthe measured Probability Density Function (PDF) of x dur-ing a
10 s acquisition and the Boltzmann distribution. The fitusing 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, andthus accumulate much more statistics. Notice that theasymptotic time in our experiment is among the fastest a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b Stage 2Stage 1 U ( z, z , z )
100 V and V (cid:28) V so that in a good approximation, F ∝ ± V up to a static term. This feedback loop results in theapplication of an external force whose sign depends onwhether the cantilever is above or below the threshold z . The reaction time of the feedback loop is very fast(around − f − ), 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 twoparameters z and V . Specifically, the barrier position isset by z and its height is controlled by V which sets thewells centers ± z = ± V ∂ d C ( d ) V / ( kσ ) . The potentialenergy constructed by this feedback is: U ( z, z , z ) = 12 (cid:0) z − S ( z − z ) z (cid:1) , (1)where S ( z ) is the sign function: S ( z ) = − if z < and S ( z ) = 1 if z > . We carefully checked that thefeedback loop creates a proper virtual potential withoutperturbing the response of the system, including hav-ing no effect on the equilibrium velocity distribution [30].The potential in Eq. (1) can be experimentally measuredfrom the Probability Distribution Function (PDF) P ( z )and the Boltzmann distribution P ( z ) ∝ e − U ( z ) . Figure1(c) presents an example of an experimental symmet-ric double-well potential generated by the feedback loop,tuned to have a barrier of z = 5 ( z = 0 , z = √ ).The dashed red line is the best fit with Eq. (1), demon-strating that the feedback-generated potential behaves asan 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, thesystem is at equilibrium either in the state 0 (left-handwell centered in − Z ) or in the state 1 (right-hand wellcentered in Z ) with a probability p i = . The processmust result in the final state 0 with probability p f = 1 , toperform a full erasure process. To do so, during the firststage we drive the wells closer until they merge into onesingle harmonic well centered in . After a short equili-bration time, the barrier position (dashed green line) ispushed away to prevent the cantilever from visiting theright hand well (black line). It thus remains in the lefthand well (red line), which is driven back to its initialposition in − Z . The so-called stage 2 ends when thethreshold is brought back in , to reach the final state 0in the bi-stable potential, regardless of the initial state. ⌧
100 ms (with τ = 30 ms and τ = 3 τ r = 7 . ), the energy costof erasure is only 10% larger than the LB. We didn’t ex-plore cycles faster than τ = 6 f − per ramp, which arealready only twice the relaxation time τ r = π Qf − .One may wonder if the extra cost at high speed is dueeither to the erasure protocol itself, or to the damping FIG. 5. Energy cost from erasure protocols at different speeds ˙ z = Z /τ , with Z ∼ . Experimental data (blue) are fittedby L ∞ + B Z /τ (dashed red), with B = (437 ± µ s and L ∞ = 0 . ± . . (Inset) Experimental mean work andheat during stage 2 (green) at different speeds. The theoreti-cal prediction B Z /τ (dashed red) with B = Z /Qω worksperfectly (no adjustable parameters). losses during the ramps. In the inset of Fig. 5, we plotthe contribution W or Q of the ramp of stage 2. In afirst approximation (neglecting transients), the cantileverwill follow the well center z at speed ˙ z , while experienc-ing a viscous drag force γσ ˙ z . We thus expect the rampcost to be hW i = hQ i ∼ B Z /τ , with B = Z /Qω .This approximation with no adjustable parameters is re-ported in the inset of Fig. 5, and matches perfectly theexperimental data. We compute B ∼ µ s for Z ∼ :the ramp contribution is not enough to explain the extracost of fast erasures, since B (one contribution for eachstage) would explain only
30 % of the overhead to ln 2 .As a conclusion, let us summarize the highlights ofthis work. We demonstrate in this letter the benefitsof exploring stochastic thermodynamics with an under-damped micro-oscillator: owing to its fast dynamics andsmall relaxation time, we can accumulate large statisticsand lower uncertainties for the measured quantities. Theuse of a high-precision interferometer coupled with a sim-ple feedback loop allows for a well-defined and tunabledouble-well potential energy. Combining these two fea-tures, we demonstrate experimentally that the Landauerbound can be reached with a very high accuracy in ashort time. Furthermore, we can check every step withanalytical predictions. Inertia and weak coupling to thebath have no effect on the limit, which is reached for a fewnatural relaxation times of the oscillator. Notice that, byusing oscillators with a higher resonance frequency, therelaxation time can be considerably reduced. Thus, thepresent approach is promising to further explore stochas-tic thermodynamics, with unprecedented statistics andmore complex behaviors brought on by the inertia.
Acknowledgments
This work has been financially sup-ported by the Agence Nationale de la Recherche throughgrant 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 availability
The data that support the findings ofthis study are available in the figures of this article, andavailable from the corresponding author upon request.They will be uploaded in an open repository and prop-erly referenced as soon as the article is accepted. ∗ [email protected][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 , 131 (2015).[3] E. Lutz and S. Ciliberto, Physics Today , 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. , 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, andM. Sano, Nature Phys. , 988 (2010).[8] É. Roldán, I. A. Martínez, J. M. R. Parrondo, andD. Petrov, Nature Physics , 457 (2014).[9] J. V. Koski, V. F. Maisi, J. P. Pekola, and D. V. Averin,Proceedings of the National Academy of Sciences ,13786 (2014).[10] R. Landauer, IBM Journal of Research and Development , 183 (1961).[11] A. Bérut, A. Petrosyan, and S. Ciliberto, EPL (Euro-physics Letters) , 60002 (2013).[12] A. Bérut, A. Arakelyan, A. Petrosyan, S. Ciliberto,R. Dillenschneider, and E. Lutz, Nature , 187 (2012).[13] A. Bérut, A. Petrosyan, and S. Ciliberto, Journal ofStatistical Mechanics: Theory and Experiment ,P06015 (2015).[14] A. O. Orlov, C. S. Lent, C. C. Thorpe, G. P. Boechler,and G. L. Snider, Japanese Journal of Applied Physics , 06FE10 (2012).[15] Y. Jun, M. Gavrilov, and J. Bechhoefer, Phys. Rev. Lett. , 190601 (2014).[16] M. Gavrilov and J. Bechhoefer, EPL (Europhysics Let- ters) , 50002 (2016).[17] K. Proesmans, J. Ehrich, and J. Bechhoefer, Phys. Rev.Lett. , 100602 (2020).[18] J. Hong, B. Lambson, S. Dhuey, and J. Bokor, Sci. Adv. , 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 , 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, andM. Feng, Phys. Rev. Lett. , 210601 (2018).[21] R. Gaudenzi, E. Burzuri, S. Maegawa, H. S. J. van derZant, and F. Luis, Nucl. Phys. , 565 (2018).[22] E. Aurell, K. Gaw¸edzki, C. Mejía-Monasterio, R. Mo-hayaee, and P. Muratore-Ginanneschi, Journal of Statis-tical Physics , 487 (2012).[23] G. Diana, G. B. Bagci, and M. Esposito, Phys. Rev. E , 012111 (2013).[24] T. Schmiedl and U. Seifert, Phys. Rev. Lett. , 108301(2007).[25] A. B. Boyd, D. Mandal, and J. P. Crutchfield, Phys. Rev.X , 031036 (2018).[26] A. B. Boyd, A. Patra, C. Jarzynski, and J. P. Crutchfield,Shortcuts to thermodynamic computing: The cost of fastand faithful erasure (2018), arXiv:1812.11241.[27] P. Paolino, F. Aguilar Sandoval, and L. Bellon, Rev. Sci.Instrum. , 095001 (2013).[28] Doped silicon cantilever OCTO1000S from MicromotiveMikrotechnik, nominal stiffness × − N / m .[29] H.-J. Butt, B. Cappella, and M. Kappl, Surface ScienceReports , 1 (2005).[30] See Supplemental Material at http://[link provided bythe editorial office] for details on the feedback loop, thederivation of the theoretical predictions on work andheat, an example of high speed protocol, and a PDF of Q .[31] To set τ , we use the rule of thumb that a perturb systemreturns to equilibrium after τ r , with τ r the characteristicrelaxation time in the exponential decay.[32] K. Sekimoto, Stochastic Energetics , Lecture Notes inPhysics, Vol. 799 (Springer, 2010).[33] K. Sekimoto and S. Sasa, Journal of the Physical Societyof Japan , 3326 (1997).[34] U. Seifert, Reports on Progress in Physics , 126001(2012).[35] C. Jarzynski, Annual Review of Condensed MatterPhysics , 329 (2011).[36] S. Ciliberto, Phys. Rev. X , 021051 (2017).[37] hWi does not depend on τ f but on τ only. Indeed as ˙ z = 0 outside the ramps period then in Eq. (2) theintegration limits reduces to τ [30].[38] C. H. Bennett, Scientific American , 108 (1987). upplemental material - Information and thermodynamics: fast and preciseapproach to Landauer’s bound in an underdamped micro-mechanical oscillator 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
FEEDBACK AND DOUBLE WELL POTENTIAL
To create the virtual double well potential energy U ( z, z , z ) , we use a feedback loop as sketched in Fig. 1of the letter, which must meet two criteria: no hystere-sis ( z is independent of z ), and negligible delay τ FB .The latter necessitates a loop response time much shorterthan the natural response time of the system. This isachieved with a high-speed comparator ( τ FB ∼
10 ns )and a slow cantilever (natural free oscillation period f − ∼ . ). Implementing a comparator without anyhysteresis is less straightforward, since we must preventthe detection noise of the system from causing the com-parator to switch back-and-forth when the cantilever isnear the threshold. Indeed, this fast switching could re-sult in a mean voltage V actually stabilizing the can-tilever around z ! We solve this issue by implementinga temporal lock-up that prevents the comparator fromswitching back for f − / : if the oscillator crosses thethreshold, it will evolve in the new well for at least aquarter of its natural period. Consequently, the oscilla-tor has time to reach the well center, far enough from z that detection noise cannot cause subsequent switching.By the time the dynamics brings the cantilever back tothe threshold, the comparator will be functional again.The distance between the wells z is driven by thevoltage V . When V (cid:29) V we can consider that theelectrostatic force F , and as a consequence z , is propor-tional to V . To confirm this linear behavior, we plot inFig. S1 the half distance between the wells z returnedby the fit of the static distribution function of positions P ( z, z , z ) ∝ e − U ( z,z ,z ) , as a function of the voltage V .The expected linearity is checked, and z can be preciselyset during the experiments.One may wonder whether the feedback loop perturbsthe system and impacts its exchange with the thermalbath, and therefore its kinetic energy E k = ˙ z /ω . Fig-ure S2 aims at showing that the system behaves as ina real potential, whatever the strength of the feedback.A good indicator that this is the case is whether h E k i remains equal to / according to the energy equiparti-tion. The experimental results confirm that the velocitydistribution remains gaussian and that h E k i = 1 / forany z . FIG. S1. Half distance between the wells z returned by the fitof the static probability density function of the cantilever, asa function of the adjustable voltage V . The linear fit (dashedred) is in good agreement with the experimental points (blue). FIG. S2. (Top) PDF of the dimensionless velocity of the can-tilever ˙ z/ω for increasing values of z (from in blue to . in red) imposed by the feedback loop. The blue curve cor-responds to z = 0 (no feedback). The red line correspondsto very separated double wells: z = 5 . . For each distancebetween the wells, the experimental PDF fits to the stan-dard normal distribution expected (dashed black). (Bottom)Mean kinetic energy h E k i = h ˙ z i /ω for increasing distancebetween the wells. E k is not influenced by the feedback loop,and it matches the value one would expect from the equipar-tition theorem. a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b STOCHASTIC WORK AND HEATEXPRESSIONS
The behavior of an underdamped system of mass m ina potential U ( x, x i ( t )) , and with a damping coefficient γ ,is described by the Langevin equation: m ¨ x = − ∂U∂x ( x, x i ) − γ ˙ x + F th , (S1)where x i ( t ) are the external control parameters of thepotential, and F th is the random external force due tothe coupling with the thermal bath. Noting that ˙ U ( x, x i ) = ∂U∂x ( x, x i ) ˙ x + X i ∂U∂x i ( x, x i ) ˙ x i , (S2)we follow Refs. 1 and 2 and multiply Eq.(S1) by ˙ x : ˙ E k = X i ∂U∂x i ˙ x i − ˙ U + ( − γ ˙ x + F th ) ˙ x. (S3)Integrating this equation leads to the energy balance ofthe system (with the positive exit heat flux convention) ∆ U + ∆ E k = W − Q , (S4)where the heat and the work have been defined as: W = Z X i ∂U∂x i ˙ x i d t (S5) Q = − Z ( − γ ˙ x + F th ) ˙ xdt = − Z ∂U∂x ˙ x d t − ∆ E k (S6)In our experiment, the potential energy U is given by U ( z, z , z ) = 12 (cid:0) z − S ( z − z ) z (cid:1) . (S7)The contribution of ( ∂U/∂z ) ˙ z ∝ δ ( z − z ) z ˙ z disap-pears in the work expression: in our protocol ˙ z = 0 ,except at two particular times corresponding to the be-ginning and end of stage 2. The former corresponds to z = 0 , and the latter corresponds to the situation where z and z cannot cross, which constrains this term to 0. Inthe heat expression, the contribution of the ∂S/∂z termin ( ∂U/∂z ) ˙ z is proportional to δ ( z − z ) z z ˙ z and van-ishing as well: assuming that z ( t ) and z ( t ) intersect at t = t , this contribution is Z ∼ t δ ( z − z ) z z ˙ z d t = z ( t ) z ( t ) ˙ z ( t )˙ z ( t ) − ˙ z ( t ) = 0 , (S8)since during our protocol z and z only intersect duringstage 1 where ˙ z = 0 at all times.Eventually W and Q write: W = − Z τ f (cid:0) S ( z − z ) z − z (cid:1) ˙ z d t (S9) Q = Z τ f (cid:0) S ( z − z ) z − z (cid:1) ˙ z d t − ω h ˙ z i τ f . (S10) It should be noted that in the computation of the meandissipated heat, we did not include the kinetic term whichvanishes on average. These general expressions can beused to deduce work and heat during stage 1 or dur-ing stage 2, by adapting the integration bounds. It isstraightforward to compute W , since ˙ z = 0 outside theramps. For Q , we add at least after reaching thefinal state, i.e. at least 2 relaxation times so that thesystem is very close to equilibrium. During stage 1 we have z = 0 and z = Z (1 − t/τ ) ,the potential energy is therefore simply U ( z, z ) = 12 (cid:0) | z | − z ) . (S11)The work is then expressed: W = − Z τ (cid:0) | z | − z (cid:1) ˙ z d t (S12) = Z Z ( | z | − z )d z . (S13)For slow protocols ( τ f (cid:29) Q ), we can assume that thePDF P ( z ) satisfies at all times the static expression givenby the Boltzmann distribution: P ( z ) = 1 Z e − U ( z,z ) = 1 Z e − ( | z |− z ) , (S14)where Z = √ π (cid:16) z / √ (cid:17) . (S15) P ( z ) is even, so we can deduce that: h| z |i = Z ∞−∞ | z | P ( z )d z = 2 Z ∞ zP ( z )d z (S16) = 2 Z Z ∞ ze − ( z − z ) d z (S17) = 2 Z (cid:18) z Z ∞ P ( z )d z − h e − ( z − z ) i ∞ (cid:19) (S18) = z + r π e − z z / √ (S19)All in all, we have the mean work in stage 1: hW i = h Z Z ( | z | − z )d z i (S20) = r π Z Z e − z z / √
2) d z (S21) = ln (cid:16) Z / √ (cid:17) −→ Z (cid:29) ln 2 . (S22)For a robust 1-bit of information, and equivalently, fora complete and reliable erasure, we should take Z (cid:29) in the initial state. As a result, for quasi-static erasureprocesses of 1 bit, hW i = k B T ln 2 .3During the quasi-static merging of the wells, we canalso compute the average instantaneous power: h ∂U∂z ˙ z i = −h (cid:0) | z | − z (cid:1) ˙ z i (S23) = r π e − z z / √ Z τ . (S24)This expression is plotted in Fig. 4 of the letter, and itclosely follows the experimental data. During stage 2 , the cantilever no longer sees the sec-ond well and S ( z − z ) = − , so that the potential sim-plifies into: U ( z, z ) = 12 ( z + z ) . (S25)The mean work becomes hW i = h− Z τ (cid:0) z + z (cid:1) ˙ z d t i (S26) = − Z τ (cid:0) h z i + z (cid:1) ˙ z d t. (S27)During the quasi-static translation, h z i = − z , and so hW i = 0 . FAST ERASURE PROTOCOLS
In this section, we reproduce the results presented inFig. 3 and Fig. 4 of the article, but for protocols that canno longer be considered as quasi-static. For fast protocolscorresponding to a few oscillations of the cantilever suchas the one in Fig. S3, the system can switch only onceor twice between the wells. The work required is abovethe Landauer limit as pointed out in Fig. S4. Fig. S5details the mean power required during the process: stage1 power exceeds the quasi-static curve and the stage 2contribution no longer vanishes. ⌧
We present in Fig. S6 the distribution of the workand the heat of iterations of the erasure with theslow protocol detailed in the letter Fig. 3. The workdistribution is gaussian. Meanwhile, the heat distribu-tion presents exponential tails. This observation is inagreement with the study of Ref. 3: Q is the convolutionproduct of a gaussian and an exponential distribution.Besides, both distributions are centered on the LB be-cause the protocol under study is close to the quasi-staticregime.4 FIG. S6. Work (dark blue) and heat (light blue) distributionsof erasure procedures of Fig. 3 of the letter. The gaussianshape of the work PDF is highlighted by the best fit to agaussian distribution (dashed red line). The heat is far moredispersed than the work and presents exponential tails. Sinceit is a slow protocol, both are centered on the LB (dottedblack vertical line). ∗ [email protected][1] K. Sekimoto, Stochastic Energetics , Lecture Notes inPhysics, Vol. 799 (Springer, 2010).[2] K. Sekimoto and S. Sasa, Journal of the Physical Societyof Japan , 3326 (1997).[3] S. Ciliberto, Phys. Rev. X7