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AUTHOR(S): 

Alexander Iomin

 

TITLE

Continuous Time Quantum Walks and Recurrences in the Hilbert Space

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ABSTRACT

An analytical consideration of quantum walks in the Hilbert space is suggested for dynamical systems. It is shown that in the semiclassical limit, statistics of the quantum recurrences relates to statistics of the Poincaré recurrences of the classical counterpart. It is shown that the statistics of the quantum recurrences is sensitive to the statistics of the corresponding quantum spectrum. The difference in the statistics of quantum recurrences in the Hilbert space for the chaotic and integrable systems follows from the essential difference between the level statistics of integrable and chaotic systems. In particular, when the integrable part of the phase space emerges due to bifurcation, and the exponential distribution of the Poincar´e recurrences of chaotic trajectories is changed into the power law, the statistics of the quantum walks in the Hilbert space follows exactly its classical counterpart.

KEYWORDS

Chaos, Random Walk, Poincar?e recurrences, Statistics of quantum spectrum, Almost periodic functions

REFERENCES

[1] G.M. Zaslavsky, Phys. Rep. 371, 461 (2002). [1] G.M. Zaslavsky, Phys. Rep. 371, 461 (2002). 

[2] G.A. Margulis, Functs. Ann. i Prilozh. 3, 80 (1969); 4, 62 (1970). 

[3] M.Kac, Probability and Related Topics in Physical Sciences. (Interscience, New York, 1958). 

[4] J.D. Meiss, Chaos 7, 39 (1997). 

[5] G.M. Zaslavsky and M.K. Tippett, Phys. Rev. Lett. 67, 325 (1991). 

[6] V. Afraimovich and G.M. Zaslavsky, Chaos 13, 519 (2003). 

[7] J.H. Eberly N.B. Narozhny, and J.J. Sanches- ´ Mondragon, Phys. Rev. Lett. ´ 44, 1323 (1980); J. Parker and C.R. Stroud Jr., Phys. Rev. Lett. 56, 716 (1986);B. Yurke and D. Stoler, Phys. Rev. Lett. 57, 13 (1986); G. Alber, H. Ritsch, and P. Zoller, Phys. Rev. A 34, 1058 (1986). 

[8] C.F.F. Karney, Physica D 8, 360 (1983). 

[9] V.K. Melnikov, in Transport, Chaos and Plasma Physics, II, Proceedings, Marseille, edited by F. Doveil, S. Benkadda, and Y. Elskens (World Scientific, Singapore, 1996), p. 142. 

[10] V. Rom-Kedar and G.M. Zaslavsky, Chaos, 9, 697 (1999). 

[11] G.M. Zaslavsky, M. Edelman, and B.A. Niyazov, Chaos, 7, 159 (1997). 

[12] J.D. Hanson, E. Ott, and T.M. Antonsen, Phys. Rev. A 29, 819 (1984). 

[13] I. Dana, Phys. Rev. E 69, 016212 (2004); O. Barash and I. Dana, Phys. Rev. E 71, 036222 (2005). 

[14] A. Iomin and G.M. Zaslavsky, Phys. Rev. E 60, 7580 (1999); G. M. Zaslavsky and M. Edelman, Chaos 10, 135 (2000). 

[15] S. Benkadda, S. Kassibrakis, R. White, and G. Zaslavsky, Phys. Rev. E 55, 4909 (1997); S. Benkadda, S. Kassibrakis, R. White, and G. Zaslavsky, Phys. Rev. E 59, 3761 (1999). 

[16] B. Sundaram and G.M. Zaslavsky, Phys. Rev. E 59, 7231 (1999). 

[17] A. Iomin and G.M. Zaslavsky, Phys. Rev. E 63, 047203 (2001); Chem. Phys. 284, 3 (2002). 

[18] A. Iomin, S. Fishman, and G.M. Zaslavsky, Phys. Rev. E 65, 036215 (2002). 

[19] M.V. Berry and M. Tabor, Prc. R. Soc. A 356, 375 (1977). 

[20] E.P. Wigner, Math. Proc. Cambridge Philos. Soc. 47, 790 (1951). 

[21] F.J. Dyson, J. Math. Phys. 3, 140 (1962). 

[22] M.V. Berry and M. Robnik, J. Phys. A: Math. Gen. 17, 2413 (1984). 

[23] A.S. Besicovitch, Almost periodic functions (Cambridge University Press, London, 1932) 

[24] L.S. Schulman, Phys. Rev. A 18, 2379 (1978). 

[25] T. Hogg and B.A. Huberman, Phys. Rev. Lett. 48, 711 (1982). 

[26] F. Haake, Quantum Signature of Chaos (Springer, Berlin, 2001). 

[27] L.E. Reichl, The Transition to Chaos (Springer, New York, 2004). 

[28] E.W. Montroll and M.F. Shlesinger, in J. Lebowitz and E.W. Montroll (eds) Studies in Statistical Mechanics, v. 11 (Noth Holland, Amsterdam, 1984)

Cite this paper

Alexander Iomin. (2016) Continuous Time Quantum Walks and Recurrences in the Hilbert Space. Mathematical and Computational Methods, 1, 287-293

 

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