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.
Subscribe to Journal
Get full journal access for 1 year
only $14.08 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Rent or Buy article
Get time limited or full article access on ReadCube.
All prices are NET prices.
Best, R. & Hummer, G. Diffusive model of protein folding dynamics with Kramers turnover in rate. Phys. Rev. Lett. 96, 228104 (2006).
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).
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).
Badzey, R. L. & Mohanty, P. Coherent signal amplification in bistable nanomechanical oscillators by stochastic resonance. Nature 437, 995–998 (2005).
Kramers, H. Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7, 284–304 (1940).
Hänggi, P., Talkner, P. & Borkovec, M. Reaction-rate theory: fifty years after Kramers. Rev. Mod. Phys. 62, 251–341 (1990).
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).
Turlot, E. et al. Escape oscillations of a Josephson junction switching out of the zero-voltage state. Phys. Rev. Lett. 62, 1788–1791 (1989).
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).
McCann, L. I., Dykman, M. & Golding, B. Thermally activated transitions in a bistable three-dimensional optical trap. Nature 402, 785–787 (1999).
Li, T., Kheifets, S., Medellin, D. & Raizen, M. G. Measurement of the instantaneous velocity of a Brownian particle. Science 328, 1673–1675 (2010).
Gieseler, J., Deutsch, B., Quidant, R. & Novotny, L. Subkelvin parametric feedback cooling of a laser-trapped nanoparticle. Phys. Rev. Lett. 109, 103603 (2012).
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).
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).
Chandler, D. Statistical mechanics of isomerization dynamics in liquids and the transition state approximation. J. Chem. Phys. 68, 2959–2970 (1978).
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).
Mel'nikov, V. I. The Kramers problem: fifty years of development. Phys. Rep. 209, 1–71 (1991).
Pollak, E. & Ankerhold, J. Improvements to Kramers turnover theory. J. Chem. Phys. 138, 164116 (2013).
Hershkovitz, E. & Pollak, E. Multidimensional generalization of the Pollak–Grabert–Haenggi turnover theory for activated rate processes. J. Chem. Phys. 106, 7678–7699 (1997).
Han, S., Lapointe, J. & Lukens, J. E. Thermal activation in a two-dimensional potential. Phys. Rev. Lett. 63, 1712–1715 (1989).
Bowman, R. W. & Padgett, M. J. Optical trapping and binding. Rep. Prog. Phys. 76, 026401 (2013).
Dykman, M. I. & Ryvkine, D. Activated escape of periodically modulated systems. Phys. Rev. Lett. 94, 070602 (2005).
Gammaitoni, L., Hänggi, P., Jung, P. & Marchesoni, F. Stochastic resonance. Rev. Mod. Phys. 70, 223–287 (1998).
Ricci, F. et al. Optically levitated nanoparticle as a model system for stochastic bistable dynamics. Nature Commun. 8, 15141 (2017).
Kiesel, N. et al. Cavity cooling of an optically levitated submicron particle. Proc. Natl Acad. Sci. USA 110, 14180–14185 (2013).
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).
Jain, V. et al. Direct measurement of photon recoil from a levitated nanoparticle. Phys. Rev. Lett. 116, 243601 (2016).
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).
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).
Bérut, A. et al. Experimental verification of Landauer's principle linking information and thermodynamics. Nature 483, 187–189 (2012).
Dechant, A., Kiesel, N. & Lutz, E. All-optical nanomechanical heat engine. Phys. Rev. Lett. 114, 183602 (2015).
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.
The authors declare no competing financial interests.
About this article
Cite this article
Rondin, L., Gieseler, J., Ricci, F. et al. Direct measurement of Kramers turnover with a levitated nanoparticle. Nature Nanotech 12, 1130–1133 (2017). https://doi.org/10.1038/nnano.2017.198
Nature Communications (2020)
Advanced Quantum Technologies (2020)
Optics Express (2020)
Proceedings of the National Academy of Sciences (2020)