Abstract
A quantum system coupled to a heat-bath in non-equilibrium environment is considered to study the problem of noise-induced escape rate from a metastable state in the moderate to strong friction limit (Kramers’ regime). It is known that starting from an initial coherent state representation of bath oscillators, one can derive a c-number generalized quantum Langevin equation where the quantum correction terms appear as a coupled infinite set of hierarchy of equations. For practical purpose, one should truncate these equations after a certain order. In our present development, we calculate the quantum correction terms in a closed analytical form based on a systematic perturbation technique and then derive the lowest order quantum correction factor exactly in the case of an Ohmic dissipative bath. Finally, to demonstrate its applicability, the effective equation of motions has been used to study the barrier crossing dynamics which incorporates the quantum correction factors.
Similar content being viewed by others
References
Young, M.R., Singh, S.: Opt. Lett. 13, 21 (1998)
Louisell, W.H.: Quantum Statistical Properties of Radiation. Wiley, New York (1998)
Zhou, X., Gao, W., Zhu, S.: Phys. Lett. A 213, 43 (1996)
Long, Q., Cao, L., Wu, D.-J., Li, Z.-G.: Phys. Lett. A 231, 339 (1997)
Tekalp, A.M., Pavlovic, G.: IEEE Trans. Signal Process. 39, 2136 (1991)
Gerashhchenko, O.V., Ginzburg, S.L., Pustovoit, M.A.: JETP Lett. 67, 887 (1998)
Blanter, Y.M., Buttiker, M.: Phys. Rev. Lett. 81, 4040 (1998)
Li, J.H., Han, Y.X., Chen, S.G.: Physica D 195, 67 (2004)
Banerjee, D., Banik, S.K., Bag, B.C., Ray, D.S.: Phys. Rev. E 66, 051105 (2002)
Barik, D., Banik, S.K., Ray, D.S.: J. Chem. Phys. 119, 680 (2003)
Okamura, K., Tanimura, Y.: Phys. Rev. E 56, 2747 (1997)
Tanimura, Y., Steffen, T.: Phys. Rev. E 69, 4095 (2000)
Kato, T., Tanimura, Y.: J. Chem. Phys. 117, 6221 (2002)
Wendin, G., Shumeiko, V.S.: Low Temp. Phys. 33, 724 (2007)
Tesch, C.M., De Vivie-Riedle, R.: Phys. Rev. Lett. 89, 157901 (2002)
Surek, W.H.: Phys. Today 44, 36 (1971)
Caldeira, A.O., Leggett, A.J.: Physica A 121, 587 (1983)
Weiss, U.: Quantum Dissipative Systems, 3rd edn. World Scientific, Singapore (2008)
Feynman, R.P., Vernon, F.L.: Ann. Phys. (NY) 24, 118 (1963)
Breuer, H.P., Petruccione, F.: The Theory of Open Quantum Systems. Oxford University Press, Oxford (2002)
Ankerhold, J., Pechukas, P., Grabert, H.: Phys. Rev. Lett. 87, 086802 (2001)
Ankerhold, J., Grabert, H., Pechukas, P.: Chaos 15, 026106 (2005)
Koch, W., Grobmann, F., Stockbuger, J.T., Ankerhold, J.: Phys. Rev. Lett. 100, 230402 (2008)
Stockbuger, J.T., Grabert, H.: Chem. Phys. 268, 249 (2001)
Stockbuger, J.T., Grabert, H.: Phys. Rev. Lett. 88, 170407 (2002)
Stockbuger, J.T.: Chem. Phys. 296, 159 (2004)
Yan, Y., Yang, F., Liu, Y., Shao, J.: Chem. Phys. Lett. 395, 216 (2004)
Barik, D., Ray, D.S.: J. Stat. Phys. 120, 339 (2005)
Zwanzig, R.: J. Stat. Phys. 9, 215 (1973)
Ford, G.W., Kac, M., Mazur, P.: J. Math. Phys. 6, 504 (1965)
Gardiner, C.W., Zoller, P.: Quantum Noise. A Handbook of Markovian and Non-Markovian Quantum Stochastic. Springer, Berlin (2004)
Hillery, M., O’Connell, R.F., Scully, M.O., Wigner, E.P.: Phys. Rep. 106, 121 (1984)
Barik, S.K., Ray Chaudhuri, J., Ray, D.S.: J. Chem. Phys. 112, 8330 (2000)
Adelman, S.A.: J. Chem. Phys. 64, 124 (1976)
Mazo, R.M.: In: Garrido, L., Segler, P., Sheppberd, R.J. (eds.) Stochastic Processes in Nonequilibrium Systems. Lecture Notes in Physics, vol. 84. Springer, Berlin (1978)
Ray Chaudhuri, J., Gangopadhyay, G., Ray, D.S.: J. Chem. Phys. 109, 5565 (1998)
Ghosh, P.K., Barik, D., Bag, B.C., Ray, D.S.: J. Chem. Phys. 123, 224104 (2005)
Sundaram, B., Milonni, P.W.: Phys. Rev. E 51, 197 (1995)
Hänggi, P., Talkner, P., Borkovec, M.: Rev. Mod. Phys. 62, 251 (1990)
Wigner, E.P.: Z. Phys. Chem. B 19, 203 (1932)
Larkin, A.I., Ovchinikov, Yu.N.: Sov. Phys. JETP 59, 420 (1984)
Grabert, H., Weiss, U., Hänggi, P.: Phys. Rev. Lett. 52, 2193 (1984)
Langer, J.S.: Ann. Phys. (NY) 41, 108 (1967)
Grabert, H., Olschowski, P., Weiss, U.: Phys. Rev. B 36, 1931 (1987)
Ankerhold, J., Grabert, H., Ingold, G.I.: Phys. Rev. E 51, 4267 (1995)
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
About this article
Cite this article
Bhattacharya, S., Chattopadhyay, S. & Ray Chaudhuri, J. Investigation of Noise-Induced Escape Rate: A Quantum Mechanical Approach. J Stat Phys 136, 733–750 (2009). https://doi.org/10.1007/s10955-009-9802-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-009-9802-5