Abstract.
We present the general solutions for the classical and quantum dynamics of the anharmonic oscillator coupled to a purely diffusive environment. In both cases, these solutions are obtained by the application of the Baker-Campbell-Hausdorff (BCH) formulas to expand the evolution operator in an ordered product of exponentials. Moreover, we obtain an expression for the Wigner function in the quantum version of the problem. We observe that the role played by diffusion is to reduce or to attenuate the the characteristic quantum effects yielded by the nonlinearity, as the appearance of coherent superpositions of quantum states (Schrödinger cat states) and revivals.
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References
M. Brune et al., Phys. Rev. Lett. 77, 4887 (1996)
M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000)
N. Gisin, G. Ribordy, W. Titel, H. Zbinden, Rev. Mod. Phys. 77, 145 (2002)
D. Giulini et al., Decoherence and the Appearence of a Classical World in Quantum Theory (Springer, Berlin, 1996)
S. Habib, K. Shizume, W.H. Zurek, Phys. Rev. Lett. 80, 4361 (1998)
N. Wiebe, L.E. Ballentine, e-print arXiv:quant-ph/0503170v1
B. Yurke, D. Stoler, Phys. Rev. Lett. 57, 13 (1986)
G.J. Milburn, Phys. Rev. A 33, 674 (1986)
G.J. Milburn, C.A. Holmes, Phys. Rev. Lett. 56, 2237 (1986)
D.J. Daniel, G.J. Milburn, Phys. Rev. A 39, 4628 (1989)
V. Pe\(\check{\mathrm{r}}\)inová, A. Luk\(\check{\mathrm{s}}\), Phys. Rev. A 41, 414 (1990)
S. Chaturvedi, V. Srinivasan, Phys. Rev. A 43, 4054 (1991)
K.V. Kheruntsyan, J. Opt. B: Quantum Semiclass. Opt. 1, 225 (1999)
G.P. Berman, A.R. Bishop, F. Borgonovi, D.A.R. Dalvit, Phys. Rev. A 69, 062110 (2004)
M. Greiner, O. Mandel, T.W. Hänsch, I. Bloch, Nature 419, 51 (2002)
K. Kumar, J. Math. Phys. 6, 1928 (1965)
R.M. Wilcox, J. Math. Phys. 8, 962 (1967)
R. Gilmore, J. Math. Phys. 15, 2090 (1974)
W. Witschel, Int. J. Quantum Chem. 20, 1233 (1981)
A.C. Oliveira, J.G. Peixoto de Faria, M.C. Nemes, Phys. Rev. E 73, 046207 (2006)
F. Toscano, R.L. de Matos Filho, L. Davidovich, Phys. Rev. A 71, 010101 (2005)
S.J. Wang, M.C. Nemes, A.N. Salgueiro, H.A. Weidenmüller, Phys. Rev. A 66, 033608 (2002)
A. Royer, Phys. Rev. A 43, 44 (1991)
K. Wódkiewicz, J.H. Eberly, J. Opt. Soc. Am. B 2, 458 (1985)
G. Dattoli, M. Richetta, A. Torre, Phys. Rev. A 37, 2007 (1988)
E.P. Wigner, Phys. Rev. 40, 749 (1932)
M. Hillery, R.F. O'Connel, M.O. Scully, E.P. Wigner, Phys. Rep. 106, 121 (1984)
H.-W. Lee, Phys. Rep. 259, 147 (1995)
S. Steinberg, in Lie Methods in Physics, edited by J.S. Mondragón, K.B. Wolf, Lecture Notes in Physics 250 (Springer, Berlin, 1985)
A.M. Perelomov, Generalized Coherent States and their Applications (Springer, Berlin, 1986)
J.J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, Reading, MA, 1994)
M.A. Marchiolli, Rev. Bras. Ens. Fís. 24, 421 (2002)
I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series and Products (Academic Press, San Diego, 1980)
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Peixoto de Faria, J. Time evolution of the classical and quantum mechanical versions of diffusive anharmonic oscillator: an example of Lie algebraic techniques. Eur. Phys. J. D 42, 153–162 (2007). https://doi.org/10.1140/epjd/e2006-00278-8
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DOI: https://doi.org/10.1140/epjd/e2006-00278-8