Direct measurement of Kramers turnover with a levitated nanoparticle


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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Figure 1: Double-trap set-up.
Figure 2: Transition kinetics.
Figure 3: Experimentally measured jumping rate R as a function of gas pressure Pgas (black dots) compared with the analytical model of equation (3).


  1. 1

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

    Article  Google Scholar 

  2. 2

    Chung, H. S., Piana-Agostinetti, S., Shaw, D. E. & Eaton, W. A. Structural origin of slow diffusion in protein folding. Science 349, 1504–1510 (2015).

    CAS  Article  Google Scholar 

  3. 3

    Garcìa-Müller, P. L., Borondo, F., Hernandez, R. & Benito, R. M. Solvent-induced acceleration of the rate of activation of a molecular reaction. Phys. Rev. Lett. 101, 178302 (2008).

    Article  Google Scholar 

  4. 4

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

    CAS  Article  Google Scholar 

  5. 5

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

    CAS  Article  Google Scholar 

  6. 6

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

    Article  Google Scholar 

  7. 7

    Silvestrini, P., Pagano, S., Cristiano, R., Liengme, O. & Gray, K. E. 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).

    CAS  Article  Google Scholar 

  8. 8

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

    CAS  Article  Google Scholar 

  9. 9

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

    CAS  Article  Google Scholar 

  10. 10

    McCann, L. I., Dykman, M. & Golding, B. Thermally activated transitions in a bistable three-dimensional optical trap. Nature 402, 785–787 (1999).

    CAS  Article  Google Scholar 

  11. 11

    Li, T., Kheifets, S., Medellin, D. & Raizen, M. G. Measurement of the instantaneous velocity of a Brownian particle. Science 328, 1673–1675 (2010).

    CAS  Article  Google Scholar 

  12. 12

    Gieseler, J., Deutsch, B., Quidant, R. & Novotny, L. Subkelvin parametric feedback cooling of a laser-trapped nanoparticle. Phys. Rev. Lett. 109, 103603 (2012).

    Article  Google Scholar 

  13. 13

    Gieseler, J., Quidant, R., Dellago, C. & Novotny, L. Dynamic relaxation of a levitated nanoparticle from a non-equilibrium steady state. Nat. Nanotech. 9, 358–364 (2014).

    CAS  Article  Google Scholar 

  14. 14

    Millen, J., Deesuwan, T., Barker, P. & Anders, J. Nanoscale temperature measurements using non-equilibrium Brownian dynamics of a levitated nanosphere. Nat. Nanotech. 9, 425–429 (2014).

    CAS  Article  Google Scholar 

  15. 15

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

    CAS  Article  Google Scholar 

  16. 16

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

  17. 17

    Mel'nikov, V. I. The Kramers problem: fifty years of development. Phys. Rep. 209, 1–71 (1991).

    Article  Google Scholar 

  18. 18

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

    Article  Google Scholar 

  19. 19

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

    CAS  Article  Google Scholar 

  20. 20

    Han, S., Lapointe, J. & Lukens, J. E. Thermal activation in a two-dimensional potential. Phys. Rev. Lett. 63, 1712–1715 (1989).

    CAS  Article  Google Scholar 

  21. 21

    Bowman, R. W. & Padgett, M. J. Optical trapping and binding. Rep. Prog. Phys. 76, 026401 (2013).

    Article  Google Scholar 

  22. 22

    Dykman, M. I. & Ryvkine, D. Activated escape of periodically modulated systems. Phys. Rev. Lett. 94, 070602 (2005).

    CAS  Article  Google Scholar 

  23. 23

    Gammaitoni, L., Hänggi, P., Jung, P. & Marchesoni, F. Stochastic resonance. Rev. Mod. Phys. 70, 223–287 (1998).

    CAS  Article  Google Scholar 

  24. 24

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

    CAS  Article  Google Scholar 

  25. 25

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

    CAS  Article  Google Scholar 

  26. 26

    Fonseca, P. Z. G., Aranas, E. B., Millen, J., Monteiro, T. S. & Barker, P. F. Nonlinear dynamics and strong cavity cooling of levitated nanoparticles. Phys. Rev. Lett. 117, 173602 (2016).

    CAS  Article  Google Scholar 

  27. 27

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

    Article  Google Scholar 

  28. 28

    Guantes, R., Vega, J. L., Miret-Artes, S. & Pollak, E. Kramers turnover theory for diffusion of Na atoms on a Cu(001) surface measured by He scattering. J. Chem. Phys. 119, 2780–2791 (2003).

    CAS  Article  Google Scholar 

  29. 29

    Toyabe, S., Sagawa, T., Ueda, M., Muneyuki, E. & Sano, M. Experimental demonstration of information-to-energy conversion and validation of the generalized Jarzynski equality. Nat. Phys. 6, 988–992 (2010).

    CAS  Article  Google Scholar 

  30. 30

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

    Article  Google Scholar 

  31. 31

    Dechant, A., Kiesel, N. & Lutz, E. All-optical nanomechanical heat engine. Phys. Rev. Lett. 114, 183602 (2015).

    Article  Google Scholar 

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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.

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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.

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Correspondence to Lukas Novotny.

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The authors declare no competing financial interests.

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Rondin, L., Gieseler, J., Ricci, F. et al. Direct measurement of Kramers turnover with a levitated nanoparticle. Nature Nanotech 12, 1130–1133 (2017).

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