Abstract
The response of a quantum system in a pure state to an external force is investigated by reconsidering the standard statistical approach to quantum dynamics in the light of the statistical description of equilibrium based on typicality. We prove that the response of the large majority of quantum pure states subjected to the same arbitrary external perturbation tends to be close to the statistical response as the dimension of the Hilbert space increases. This is what we can term dynamical typicality. The theoretical analysis is substantiated by numerical simulations of the response of a spin system to a sudden quench of the external magnetic field. For the considered system we show that not only the system relaxes toward a new equilibrium state after the quench of the Hamiltonian but also that such a new equilibrium is compatible with the description of a thermal equilibrium.
References
M.T.R. Kubo, N. Hashitsume, Statistical Physics II: Nonequilibrium Statistical Mechanics (Springer-Verlag, Berlin, 1985)
H. Spohn, Rev. Mod. Phys. 52, 569 (1980)
D. Kohen, C.C. Marston, D.J. Tannor, J. Chem. Phys. 107, 5236 (1997)
R. Kubo, J. Phys. Soc. Jpn 12, 570 (1957)
S. Goldstein, J.L. Lebowitz, R. Tumulka, N. Zanghì, Phys. Rev. Lett. 96, 050403 (2006)
S. Popescu, A.J. Short, A. Winter, Nat. Phys. 2, 754 (2006)
P. Reimann, Phys. Rev. Lett. 99, 160404 (2007)
P. Reimann, Phys. Rev. Lett. 101, 190403 (2008)
N. Linden, S. Popescu, A.J. Short, A. Winter, Phys. Rev. E 79, 061103 (2009)
B. d’Espagnat, Found. Phys. 20, 1147 (1990)
J. Gemmer, G. Mahler, Eur. Phys. J. D 17, 385 (2001)
J. Gemmer, G. Mahler, Europhys. Lett. 59, 159 (2002)
B. Fresch, G.J. Moro, J. Phys. Chem. A 113, 14502 (2009)
A.Y. Khinchin, Mathematical Foundations of Statistical Mechanics (Dover Publications Inc., New York, 1949)
B. Fresch, G.J. Moro, J. Chem. Phys. 133, 034509 (2010)
B. Fresch, G.J. Moro, J. Chem. Phys. 133, 034510 (2010)
F.B.G. Gallavotti, G. Gentile, Aspects of the Ergodic, Qualitative and Statistical Theory of Motion (Springer-Verlag, Berlin, 2004)
B. Fresch, G.J. Moro, J. Chem. Phys. 134, 054510 (2011)
M.M.J. Gemmer, G. Mahler, Quantum thermodynamics emergence of thermodynamic Behavior Within Composite Quantum Systems (Springer, New York, 2004)
C. Bartsch, J. Gemmer, Phys. Rev. Lett. 102, 110403 (2009)
S. Mukamel, Principles of Nonlinear Optics and Spectroscopy (Oxford University Press, New York, 1995)
R.V. Jensen, R. Shankar, Phys. Rev. Lett. 54, 1879 (1985)
P. Borowski, J. Gemmer, G. Mahler, Eur. Phys. J. B 35, 255 (2003)
S. Yuan, M.I. Katsnelson, H. De Raedt, J. Phys. Soc. Jpn 78, 094003 (2009)
F.G.S.L. Brandão, P. Ćwikliński, M. Horodecki, P. Horodecki, J.K. Korbicz, M. Mozrzymas, Phys. Rev. E 86, 031101 (2012)
Q. Xu, G. Sadiek, S. Kais, Phys. Rev. A 83, 062312 (2011)
T. Monnai, Phys. Rev. E 84, 011126 (2011)
A. Peres, Phys. Rev. Lett. 49, 1118 (1982)
D.G. Cory, A.F. Fahmy, T.F. Havel, Proc. Natl. Acad. Sci. 94, 1634 (1997)
N.A. Gershenfeld, I.L. Chuang, Science 275, 350 (1997)
E. Knill, R. Laflamme, Phys. Rev. Lett. 81, 5672 (1998)
H. De Raedt, K. Michielsen, Comput. Methods Simul. Quantum Comput., in Handbook of Theoretical and Computational Nanotechnology, edited by M. Rieth, W. Schommers (American Scientific Publishers, Los Angeles, 2006), p. 2
W. Zhang, N. Konstantinidis, K.A. Al-Hassanieh, V.V. Dobrovitski, J. Phys.: Condens. Matter 19, 083202 (2007)
J. Schliemann, A.V. Khaetskii, D. Loss, Phys. Rev. B 66, 245303 (2002)
G.A. Álvarez, E.P. Danieli, P.R. Levstein, H.M. Pastawski, Phys. Rev. Lett. 101, 120503 (2008)
N. Van Kampen, Phys. Norv. 5, 279 (1971)
V.M. Kenkre, Phys. Rev. A 4, 2327 (1971)
V.M. Kenkre, M. Dresden, Phys. Rev. Lett. 27, 9 (1971)
V.M. Kenkre, M. Dresden, Phys. Rev. A 6, 769 (1972)
W.M. Visscher, Phys. Rev. A 10, 2461 (1974)
G. Jacucci, Physica A 118, 157 (1983)
J.H. Weare, I. Oppenheim, Physica 72, 1 (1974)
I. Procaccia, D. Ronis, M.A. Collins, J. Ross, I. Oppenheim, Phys. Rev. A 19, 1290 (1979)
A.K. Rajagopal, S. Teitler, Phys. Rev. A 43, 2059 (1991)
M.O. Cáceres, S.A. Grigera, Physica A 291, 317 (2001)
M. Falcioni, A. Vulpiani, Physica A 215, 481 (1995)
M. Bianucci, R. Mannella, B.J. West, P. Grigolini, Phys. Rev. E 50, 2630 (1994)
G.P. Morriss, D.J. Evans, E.G.D. Cohen, H. van Beijeren, Phys. Rev. Lett. 62, 1579 (1989)
R. Kubo, Rep. Prog. Phys. 29, 255 (1966)
C.M. Van Vliet, J. Stat. Phys. 53, 49 (1988)
K.M. van Vliet, J. Math. Phys. 19, 1345 (1978)
K.M. van Vliet, J. Math. Phys. 20, 2573 (1979)
M.V.S. Bonança, Phys. Rev. E 78, 031107 (2008)
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Fresch, B., Moro, G. Typical response of quantum pure states. Eur. Phys. J. B 86, 233 (2013). https://doi.org/10.1140/epjb/e2013-40023-6
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DOI: https://doi.org/10.1140/epjb/e2013-40023-6