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Virtual potential created by a feedback loop: taming the feedback demon to explore stochastic thermodynamics of underdamped systems

Salambô Dago, Nicolas Barros, Jorge Pereda, Sergio Ciliberto, Ludovic Bellon
in Bouju, X., Joachim, C. (eds) Crossroad of Maxwell Demon. CMD 2023. Advances in Atom and Single Molecule Machines. Springer, Cham.

doi: 10.1007/978-3-031-57904-2_6
arXiv: 2311.12687

Virtual potentials are an elegant, precise and flexible tool to manipulate small systems and explore fundamental questions in stochastic thermodynamics. In particular double-well potentials have applications in information processing, such as the demonstration of Landauer’s principle. In this chapter, we detail the implementation of a feedback loop for an underdamped system, in order to build a tunable virtual double-well potential. This feedback behaves as a demon acting on the system depending on the outcome of a continuously running measurement. It can thus modify the energy exchanges with the thermostat and create an out-of-equilibrium state. To create a bi-stable potential, the feedback consists only in switching an external force between two steady values when the measured position crosses a threshold. We show that a small delay of the feedback loop in the switches between the two wells results in a modified velocity distribution. The latter can be interpreted as a cooling of the kinetic temperature of the system. Using a fast digital feedback, we successfully address all experimental issues to create a virtual potential that is statistically indistinguishable from a physical one, with a tunable barrier height and energy step between the two wells.

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Reliability and operation cost of underdamped memories during cyclic erasures

Salambô Dago, Sergio Ciliberto, Ludovic Bellon, Adv. Phys. Res. 2023 2300074 (2023)

[Article] doi: 10.1002/apxr.202300074
[Dataset] doi: 10.5281/zenodo.8307734

The reliability of fast repeated erasures is studied experimentally and theoretically in a 1-bit underdamped memory. The bit is encoded by the position of a micro-mechanical oscillator whose motion is confined in a double well potential. To contain the energetic cost of fast erasures, we use a resonator with high quality factor Q: the erasure work W is close to Landauer’s bound, even at high speed. The drawback is the rise of the system’s temperature T due to a weak coupling to the environment. Repeated erasures without letting the memory thermalize between operations result in a continuous warming, potentially leading to a thermal noise overcoming the barrier between the potential wells. In such case, the reset operation can fail to reach the targeted logical state. The reliability is characterized by the success rate Rsi after i successive operations. W, T and Rsi are studied experimentally as a function of the erasure speed. Above a velocity threshold, T soars while Rsi collapses: the reliability of too fast erasures is low. These experimental results are fully justified by two complementary models. We demonstrate that Q≃10 is optimal to contain energetic costs and maintain high reliability standards for repeated erasures at any speed.

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Adiabatic computing for optimal thermodynamic efficiency of information processing

Salambô Dago, Sergio Ciliberto, Ludovic Bellon, PNAS 120 e2301742120 (2023)

[Article] doi: 10.1073/pnas.2301742120
[Dataset] doi: 10.5281/zenodo.6572643

Handling information in the physical world requires energy: Landauer’s principle makes a Landauer’s principle makes a strong connection between information theory and thermodynamics by stating that erasing a one-bit memory at temperature T0 requires an average energy larger than WLB = kBT0 ln2, with kB Boltzmann’s constant. This tiny limit has been saturated in model experiments using quasistatic processes. For faster operations, an overhead proportional to the processing speed and to the memory damping appears. In this article, we show that underdamped systems are a winning strategy to reduce this extra energetic cost. We prove both experimentally and theoretically that, in the limit of vanishing dissipation mechanisms in the memory, the physical system is thermally insulated from its environment during fast erasures, i.e., fast protocols are adiabatic as no heat is exchanged with the bath. Using a fast optimal erasure protocol, we also show that these adiabatic processes produce a maximum adiabatic temperature Ta = 2T0, and that Landauer’s bound for fast erasures in underdamped systems becomes the adiabatic bound: Wa = kBT0.

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Logical and thermodynamical reversibility: optimized experimental implementation of the NOT operation

Salambô Dago and Ludovic Bellon, Phys. Rev. E 108, L022101

[Article] doi: 10.1103/PhysRevE.108.L022101
[Data set] doi: 10.5281/zenodo.8099300

The NOT operation is a reversible transformation acting on a 1-bit logical state, and should be achievable in a physically reversible manner at no energetic cost. We experimentally demonstrate a bit-flip protocol based on the momentum of an underdamped oscillator confined in a double well potential. The protocol is designed to be reversible in the ideal dissipationless case, and the thermodynamic work required is inversely proportional to the quality factor of the system. Our implementation demonstrates an energy dissipation significantly lower than the minimal cost of information processing in logically irreversible operations. It is moreover performed at high speed: a fully equilibrated final state is reached in only half a period of the oscillator. The results are supported by an analytical model that takes into account the presence of irreversibility. The Letter concludes with a discussion of optimization strategies.

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Accelerating the heat diffusion: Fast thermal relaxation of a microcantilever

Basile Pottier, Carlos Plata, Emmanuel Trizac, David Guéry-Odelin and Ludovic Bellon, Phys. Rev. Applied  19, 034072 (2023)

[Article] doi: 10.1103/PhysRevApplied.19.034072
[Data set] doi: 10.5281/zenodo.7669037

In most systems, thermal diffusion is intrinsically slow with respect to mechanical relaxation. We devise here a generic approach to accelerate the relaxation of the temperature field of a 1D object, in order to beat the mechanical time scales. This approach is applied to a micro-meter sized silicon cantilever, locally heated by a laser beam. A tailored driving protocol for the laser power is derived to reach arbitrarily fast the thermal stationary state. The model is implemented experimentally yielding a significant acceleration of the thermal relaxation, up to a factor 30. An excellent agreement with the theoretical predictions is reported. This strategy allows to reach a thermal steady state significantly faster than the natural mechanical relaxation.

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Linking fluctuation and dissipation in spatially extended out-of-equilibrium systems

Alex Fontana and Ludovic Bellon, Physics. Rev. E 107, 034118 (2023)

[Article] doi: 10.1103/PhysRevE.107.034118
[Data set] doi: 10.5281/zenodo.7640707

For systems in equilibrium at a temperature T, thermal noise and energy damping are related to T through the fluctuation-dissipation theorem (FDT). We study here an extension of the FDT to an out-of-equilibrium steady state: a microcantilever subject to a constant heat flux. The resulting thermal profile in this spatially extended system interplays with the local energy dissipation field to prescribe the amplitude of mechanical fluctuations. Using three samples with different damping profiles (localized or distributed), we probe this approach and experimentally demonstrate the link between fluctuations and dissipation. The thermal noise can therefore be predicted a priori from the measurement of the dissipation as a function of the maximum temperature of the micro-oscillator.

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Instrumentation pour la mesure diélectrique locale de films polymériques chargés de nanoparticules

Baptiste Ferrero, PhD Thesis, Université de Lyon (2023)

hal: tel-04056084

La dispersion d’oxydes métalliques au sein d’une matrice polymère permet de décupler la permittivité diélectrique de cette dernière tout en conservant ses propriétés mécaniques. Une compréhension fine des interactions entre les constituants est néanmoins nécessaire pour maîtriser la formulation et l’optimisation du matériau composite final.
Cette thèse porte sur la création d’un instrument permettant la caractérisation à l’échelle des nanoparticules de la réponse du matériau : un microscope à force atomique (AFM) pour réaliser une spectroscopie diélectrique locale. Cette méthode donnera accès à la dynamique microscopique des chaînes polymères en interaction avec les particules. L’instrument développé se distingue par sa grande précision, liée à une détection interférométrique de la déflexion de la sonde AFM et à un environnement très faible bruit, combinant cage de Faraday, chambre sourde et isolation des vibrations mécaniques. Une chambre à vide et un contrôle de température parachèvent la maîtrise des conditions de mesure.
Pour caractériser la précision et la fiabilité de la mesure interférométrique, une étude poussée des imperfections optiques est réalisée. Une méthode de calibration innovante permettant de passer outres ces imperfections est proposée. Elle améliore notablement la linéarité de la mesure par rapport aux méthodes traditionnelles des interféromètres à quadrature de phase. Pour finir, une preuve de principe de la mesure diélectrique locale est réalisée sur un échantillon de PVdF-HFP chargé de particules de BaTiO3. Elle démontre le potentiel de la méthode pour la cartographie des propriétés diélectriques à l’échelle de particules de 50 nm de diamètre.

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Harmonic calibration of quadrature phase interferometry

Baptiste Ferrero and Ludovic Bellon, EPL 139, 55002 (2022)

[Article] doi: 10.1209/0295-5075/ac8761
[Data set] doi:10.5281/zenodo.6801118

The two output signals of quadrature phase interferometers allow to benefit both from the high sensitivity of interferometry (working inside a fringe) and from an extended input range (counting fringes). Their calibration to reach a linear output is traditionally performed using Heydemann’s correction, which involves fitting one output versus the other by an ellipse. Here we present two alternative methods based on the linear response of the measurement to a sinusoidal input in time, which enables a direct calibration with an excellent linearity. A ten fold improvement with respect to the usual technique is demonstrated on an optical interferometer measuring the deflection of scanning force microscopy cantilevers.

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Stochastic thermodynamics: driving of micro-oscillators applied to the study and the optimisation of information processing

Salambô Dago, PhD Thesis, Université de Lyon (2022)

hal: tel-03771837

This thesis extends by theoretical and experimental studies our understanding of the dynamics of systems ruled by thermal fluctuations in order to better control them and, in particular, use them as 1-bit logic gates. This work falls within the framework of out-of-equilibrium statistical physics and of thermodynamics of information based on stochastic thermodynamics. In this respect, we study the minimal work required to perform irreversible operations on 1-bit of information ([RESET] to 0 or 1), or reversible ones ([NOT] operation), and we aim to optimise the energetic cost and the speed of these processes. Our strategy to enhance the processing efficiency and speed consists in using as 1-bit memory a low dissipation micro-mechanical oscillator, therefore evolving at much smaller time-scales than the over-damped test systems used to date (colloidal particles in solution). The feedback control designed to create a virtual energy potential in which evolves the micro-resonator is a major step forward in coding and handling the 1-bit information: it represents the fastest and most energy-efficient device among those which perform logic operations at the thermal energy scale. We furthermore provide a solid theoretical basis, validated by experimental and numerical simulation results, to model energy exchanges. Taken as a whole, this work results in the theoretical prediction of the energetic cost of any logical operation and opens perspectives for information processing optimisation in term of reliability, speed and energy saving.

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Virtual double-well potential for an underdamped oscillator created by a feedback loop

Salambô Dago, Jorge Pereda, Sergio Ciliberto and Ludovic Bellon, J. Stat. Mech. 2022, 053209 (2022)

[Article] doi: 10.1088/1742-5468/ac6d62
[Erratum] doi: 10.1088/1742-5468/acd697
[Dataset] doi: 10.5281/zenodo.6497247

Virtual potentials are a very elegant, precise and flexible tool to manipulate small systems and explore fundamental questions in stochastic thermodynamics. In particular double-well potentials have applications in information processing, such as the demonstration of Landauer’s principle. Nevertheless, virtual double-well potentials had never been implemented in underdamped systems. In this article, we detail how to face the experimental challenge of creating a feedback loop for an underdamped system (evolving at much smaller time scale than its overdamped counterpart), in order to build a tunable virtual double-well potential. To properly describe the system behavior in the feedback trap, we express the commutation time in the double-well for all barrier heights, combining for the first time  Kramer’s description, valid at high barriers, with an adjusted model for lower ones. We show that a small hysteresis or delay of the feedback loop in the commutation between the two wells results in a modified velocity distribution, interpreted as a cooling of the kinetic temperature of the system. We successfully address all issues to create experimentally a virtual potential that is statistically indistinguishable from a physical one, with a tunable barrier height and energy step between the two wells.

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