[en] We address the time decay of the Loschmidt echo, measuring the sensitivity of quantum dynamics to small Hamiltonian perturbations, in one-dimensional integrable systems. Using a semiclassical analysis, we show that the Loschmidt echo may exhibit a well-pronounced regime of exponential decay, similar to the one typically observed in quantum systems whose dynamics is chaotic in the classical limit. We derive an explicit formula for the exponential decay rate in terms of the spectral properties of the unperturbed and perturbed Hamilton operators and the initial state. In particular, we show that the decay rate, unlike in the case of the chaotic dynamics, is directly proportional to the strength of the Hamiltonian perturbation. Finally, we compare our analytical predictions against the results of a numerical computation of the Loschmidt echo for a quantum particle moving inside a one-dimensional box with Dirichlet-Robin boundary conditions, and find the two in good agreement.
Disciplines :
Physics
Author, co-author :
Dubertrand, Rémy ; Université de Liège - ULiège > Département de physique > Optique quantique
Goussev, A
Language :
English
Title :
Origin of the exponential decay of the Loschmidt echo in integrable systems
Publication date :
February 2014
Journal title :
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
ISSN :
1539-3755
eISSN :
1550-2376
Publisher :
American Physical Society, College Park, United States - Maryland
T. Gorin, T. Prosen, T. H. Seligman, and M. Znidaric, Phys. Rep. 435, 33 (2006). PRPLCM 0370-1573 10.1016/j.physrep.2006.09.003 (Pubitemid 44696158)
Ph. Jacquod and C. Petitjean, Adv. Phys. 58, 67 (2009). ADPHAH 0001-8732 10.1080/00018730902831009
A. Goussev, R. A. Jalabert, H. M. Pastawski, and D. A. Wisniacki, Scholarpedia 7, 11687 (2012). 1941-6016 10.4249/scholarpedia.11687
R. A. Jalabert and H. M. Pastawski, Phys. Rev. Lett. 86, 2490 (2001). PRLTAO 0031-9007 10.1103/PhysRevLett.86.2490 (Pubitemid 32286687)
Ph. Jacquod, P. G. Silvestrov, and C. W. J. Beenakker, Phys. Rev. E 64, 055203 (R) (2001). 1063-651X 10.1103/PhysRevE.64.055203
F. M. Cucchietti, H. M. Pastawski, and D. A. Wisniacki, Phys. Rev. E 65, 045206 (R) (2002). 1063-651X 10.1103/PhysRevE.65.045206 (Pubitemid 40618897)
F. M. Cucchietti, H. M. Pastawski, and R. A. Jalabert, Phys. Rev. B 70, 035311 (2004). PRBMDO 1098-0121 10.1103/PhysRevB.70.035311
T. Prosen, Phys. Rev. E 65, 036208 (2002). 1063-651X 10.1103/PhysRevE.65. 036208 (Pubitemid 40629646)
Ph. Jacquod, I. Adagideli, and C. W. J. Beenakker, Europhys. Lett. 61, 729 (2003). EULEEJ 0295-5075 10.1209/epl/i2003-00289-y (Pubitemid 36357298)
R. Sankaranarayanan and A. Lakshminarayan, Phys. Rev. E 68, 036216 (2003). 1063-651X 10.1103/PhysRevE.68.036216
T. Prosen and M. Žnidarič, New J. Phys. 5, 109 (2003). NJOPFM 1367-2630 10.1088/1367-2630/5/1/109
A. Goussev, Phys. Rev. E 83, 056210 (2011). PLEEE8 1539-3755 10.1103/PhysRevE.83.056210
Y. S. Weinstein and C. S. Hellberg, Phys. Rev. E 71, 016209 (2005). PLEEE8 1539-3755 10.1103/PhysRevE.71.016209 (Pubitemid 40629778)
W.-ge Wang, G. Casati, and B. Li, Phys. Rev. E 75, 016201 (2007). PLEEE8 1539-3755 10.1103/PhysRevE.75.016201 (Pubitemid 46134196)
A. Peres, Phys. Rev. A 30, 1610 (1984). 0556-2791 10.1103/PhysRevA.30. 1610
M. Nauenberg, J. Phys. B: At. Mol. Opt. Phys. 23, L385 (1990). JPAPEH 0953-4075 10.1088/0953-4075/23/15/001
R. W. Robinett, Phys. Rep. 392, 1 (2004). 10.1016/j.physrep.2003.11.002 (Pubitemid 38187529)
See, e.g., C. Leichtle, I. Sh. Averbukh, and W. P. Schleich, Phys. Rev. A 54, 5299 (1996). PLRAAN 1050-2947 10.1103/PhysRevA.54.5299
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, edited by A. Jeffrey and D. Zwillinger (Academic, New York, 2007), 7th ed.
D. A. Wisniacki, E. G. Vergini, H. M. Pastawski, and F. M. Cucchietti, Phys. Rev. E 65, 055206 (R) (2002). 1063-651X 10.1103/PhysRevE.65.055206 (Pubitemid 40606832)