Letter

Direct measurement of Kramers turnover with a levitated nanoparticle

  • Nature Nanotechnology volume 12, pages 11301133 (2017)
  • doi:10.1038/nnano.2017.198
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Abstract

Understanding the thermally activated escape from a metastable state is at the heart of important phenomena such as the folding dynamics of proteins1,2, the kinetics of chemical reactions3 or the stability of mechanical systems4. In 1940, Kramers calculated escape rates both in the high damping and low damping regimes, and suggested that the rate must have a maximum for intermediate damping5. This phenomenon, today known as the Kramers turnover, has triggered important theoretical and numerical studies6. However, as yet, there is no direct and quantitative experimental verification of this turnover. Using a nanoparticle trapped in a bistable optical potential, we experimentally measure the nanoparticle's transition rates for variable damping and directly resolve the Kramers turnover. Our measurements are in agreement with an analytical model that is free of adjustable parameters. The levitated nanoparticle presented here is a versatile experimental platform for studying and simulating a wide range of stochastic processes and testing theoretical models and predictions.

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References

  1. 1.

    & Diffusive model of protein folding dynamics with Kramers turnover in rate. Phys. Rev. Lett. 96, 228104 (2006).

  2. 2.

    , , & Structural origin of slow diffusion in protein folding. Science 349, 1504–1510 (2015).

  3. 3.

    , , & Solvent-induced acceleration of the rate of activation of a molecular reaction. Phys. Rev. Lett. 101, 178302 (2008).

  4. 4.

    & Coherent signal amplification in bistable nanomechanical oscillators by stochastic resonance. Nature 437, 995–998 (2005).

  5. 5.

    Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7, 284–304 (1940).

  6. 6.

    , & Reaction-rate theory: fifty years after Kramers. Rev. Mod. Phys. 62, 251–341 (1990).

  7. 7.

    , , , & Effect of dissipation on thermal activation in an underdamped Josephson junction: first evidence of a transition between different damping regimes. Phys. Rev. Lett. 60, 844–847 (1988).

  8. 8.

    et al. Escape oscillations of a Josephson junction switching out of the zero-voltage state. Phys. Rev. Lett. 62, 1788–1791 (1989).

  9. 9.

    , & Photoisomerization of trans-stilbene in compressed solvents: Kramers-turnover and solvent induced barrier shift. Z. Phys. Chem. 188, 287–306 (1995).

  10. 10.

    , & Thermally activated transitions in a bistable three-dimensional optical trap. Nature 402, 785–787 (1999).

  11. 11.

    , , & Measurement of the instantaneous velocity of a Brownian particle. Science 328, 1673–1675 (2010).

  12. 12.

    , , & Subkelvin parametric feedback cooling of a laser-trapped nanoparticle. Phys. Rev. Lett. 109, 103603 (2012).

  13. 13.

    , , & Dynamic relaxation of a levitated nanoparticle from a non-equilibrium steady state. Nat. Nanotech. 9, 358–364 (2014).

  14. 14.

    , , & Nanoscale temperature measurements using non-equilibrium Brownian dynamics of a levitated nanosphere. Nat. Nanotech. 9, 425–429 (2014).

  15. 15.

    Statistical mechanics of isomerization dynamics in liquids and the transition state approximation. J. Chem. Phys. 68, 2959–2970 (1978).

  16. 16.

    & in Advanced Computer Simulation Approaches for Soft Matter Sciences III (eds Holm, P. C. & Kremer, P. K.) 167–233 (Springer, 2009).

  17. 17.

    The Kramers problem: fifty years of development. Phys. Rep. 209, 1–71 (1991).

  18. 18.

    & Improvements to Kramers turnover theory. J. Chem. Phys. 138, 164116 (2013).

  19. 19.

    & Multidimensional generalization of the Pollak–Grabert–Haenggi turnover theory for activated rate processes. J. Chem. Phys. 106, 7678–7699 (1997).

  20. 20.

    , & Thermal activation in a two-dimensional potential. Phys. Rev. Lett. 63, 1712–1715 (1989).

  21. 21.

    & Optical trapping and binding. Rep. Prog. Phys. 76, 026401 (2013).

  22. 22.

    & Activated escape of periodically modulated systems. Phys. Rev. Lett. 94, 070602 (2005).

  23. 23.

    , , & Stochastic resonance. Rev. Mod. Phys. 70, 223–287 (1998).

  24. 24.

    et al. Optically levitated nanoparticle as a model system for stochastic bistable dynamics. Nature Commun. 8, 15141 (2017).

  25. 25.

    et al. Cavity cooling of an optically levitated submicron particle. Proc. Natl Acad. Sci. USA 110, 14180–14185 (2013).

  26. 26.

    , , , & Nonlinear dynamics and strong cavity cooling of levitated nanoparticles. Phys. Rev. Lett. 117, 173602 (2016).

  27. 27.

    et al. Direct measurement of photon recoil from a levitated nanoparticle. Phys. Rev. Lett. 116, 243601 (2016).

  28. 28.

    , , & Kramers turnover theory for diffusion of Na atoms on a Cu(001) surface measured by He scattering. J. Chem. Phys. 119, 2780–2791 (2003).

  29. 29.

    , , , & Experimental demonstration of information-to-energy conversion and validation of the generalized Jarzynski equality. Nat. Phys. 6, 988–992 (2010).

  30. 30.

    et al. Experimental verification of Landauer's principle linking information and thermodynamics. Nature 483, 187–189 (2012).

  31. 31.

    , & All-optical nanomechanical heat engine. Phys. Rev. Lett. 114, 183602 (2015).

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Acknowledgements

This research was supported by the Swiss National Science Foundation (no. 200021L_169319) in cooperation with the Austrian Science Fund (no. I 3163), ERC-QMES (no. 338763), CoG ERC-QnanoMECA (no. 64790), Fundació Privada CELLEX and the severo Ochoa programme. L.R. acknowledges support from an ETH – Marie Curie Cofund Fellowship. The authors thank M. Frimmer, V. Jain, E. Hebestreit, C. Moritz, P. Mestres, E. Pollak and P. Bharadwaj for discussions and experimental support.

Author information

Author notes

    • Loïc Rondin
    •  & Jan Gieseler

    Present address: Laboratoire Aimé Cotton, CNRS, Université Paris-Sud, ENS Cachan, Université Paris-Saclay, 91405 Orsay Cedex, France (L.R.); Physics Department, Harvard University, Cambridge, Massachusetts 02318, USA (J.G.).

Affiliations

  1. ETH Zürich, Photonics Laboratory, 8093 Zürich, Switzerland

    • Loïc Rondin
    • , Jan Gieseler
    •  & Lukas Novotny
  2. ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain

    • Francesco Ricci
    •  & Romain Quidant
  3. ICREA-Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Spain

    • Romain Quidant
  4. Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Wien, Austria

    • Christoph Dellago
  5. Erwin Schrödinger International Institute for Mathematics and Physics, University of Vienna, Boltzmanngasse 9, 1090 Wien, Austria

    • Christoph Dellago

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Contributions

L.R. and L.N. designed and conceived the experiment. L.R. performed the experiment and analysed the data, with input from J.G., C.D. and L.N. All authors discussed the results and contributed to writing the manuscript.

Competing interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to Lukas Novotny.

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