×

zbMATH — the first resource for mathematics

Statistics of time delay and scattering correlation functions in chaotic systems. II: Semiclassical approximation. (English) Zbl 1322.81049
The scattering properties of a quantum mechanical, chaotic cavity with \(M\) open channels can be described by its scattering matrix \(S(E)\) which is an energy dependent, complex \(M\times M\) matrix. In this article, the correlation function
\[ C_n(\epsilon,M) := \frac{1}{M}\left\langle\text{Tr}\left[S^\dagger\left(E-\frac{\epsilon\hbar}{2\tau_D}\right)S\left(E+\frac{\epsilon\hbar}{2\tau_D}\right)\right]^n\right\rangle, \] where \(\tau_D\) is the classical dwell time and \(\langle\bullet \rangle\) denotes the average over \(E\), is studied for a system without time reversibility and a semiclassical approximation is calculated. This calculation is based on a semiclassical approximation of the scattering matrix which takes into account a systematic pairing of classical trajectories and which has been established in physics literature [R. A. Jalabert et al., Phys. Rev. Lett. 65, 2442 (1990); K. Richter and M. Sieber, Phys. Rev. Lett. 89, 206801 (2002); S. Müller et al. New J. Phys. 9, 12 (2007)].
From the semiclassical expression of \(C_n(\epsilon,M)\), the author derives an expression for the energy averages of the time delay moments \(\langle\mathcal M_m\rangle\) where \(\mathcal M_m := \frac{1}{M}\text{Tr}(Q^m)\) and \(Q\) being the Wigner time delay matrix \(Q:= -i\hbar S^\dagger\frac{dS}{dE}\). The explicit expression of \(\langle \mathcal M_m\rangle\) is of complicated nature. It is nevertheless possible to compare the expression for the first 8 moments (\(m\leq 8\)) to expressions which have been obtained by the author via random matrix theory [J. Math. Phys. 56, No. 6, 062110, 6 p. (2015; Zbl 1322.81050)] and show that they coincide.

MSC:
81Q50 Quantum chaos
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
81U35 Inelastic and multichannel quantum scattering
15B52 Random matrices (algebraic aspects)
60B20 Random matrices (probabilistic aspects)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Wigner, E. P., Phys. Rev., 98, 145, (1955) · Zbl 0064.21804
[2] Smith, F. T., Phys. Rev., 118, 349, (1960) · Zbl 0092.19001
[3] Berkolaiko, G.; Kuipers, J., J. Phys. A, 43, 035101, (2010) · Zbl 1183.81069
[4] Jalabert, R. A.; Baranger, H. U.; Stone, A. D., Phys. Rev. Lett., 65, 2442, (1990)
[5] Berry, M. V., Proc. R. Soc. A, 400, 229, (1985) · Zbl 0875.35061
[6] Cvitanović, P.; Eckhardt, B., J. Phys. A, 24, L237, (1991)
[7] Hannay, J. H.; Ozorio de Almeida, A. M., J. Phys. A, 17, 3429, (1984)
[8] Richter, K.; Sieber, M., Phys. Rev. Lett., 89, 206801, (2002)
[9] Müller, S.; Heusler, S.; Braun, P.; Haake, F., New J. Phys., 9, 12, (2007)
[10] Berkolaiko, G.; Kuipers, J., Phys. Rev. E, 85, 045201, (2012)
[11] Novaes, M., Europhys. Lett., 98, 20006, (2012)
[12] Balian, R.; Bloch, C., Ann. Phys., 85, 514, (1974) · Zbl 0281.35029
[13] Altmann, E. G.; Portela, J. S. E.; Tél, T., Rev. Mod. Phys., 85, 869, (2013)
[14] Eckhardt, B., Chaos, 3, 613, (1993)
[15] Vallejos, R. O.; Ozorio de Almeida, A. M.; Lewenkopf, C. H., J. Phys. A, 31, 4885, (1998) · Zbl 0953.81018
[16] Kuipers, J.; Sieber, M., Nonlinearity, 20, 909, (2007) · Zbl 1124.81023
[17] Kuipers, J.; Sieber, M., Phys. Rev. E, 77, 046219, (2008)
[18] Lewenkopf, C. H.; Vallejos, R. O., J. Phys. A, 37, 131, (2004) · Zbl 1045.81051
[19] Berkolaiko, G.; Kuipers, J., New J. Phys., 13, 063020, (2011) · Zbl 1275.82007
[20] Kuipers, J.; Savin, D. V.; Sieber, M., New J. Phys., 16, 123018, (2014)
[21] Novaes, M., J. Phys. A, 46, 502002, (2013) · Zbl 1298.82062
[22] Verbaarschot, J. J. M.; Weidenmüller, H. A.; Zirnbauer, M. R., Phys. Rep., 129, 367, (1985)
[23] Lehmann, N.; Savin, D. V.; Sokolov, V. V.; Sommers, H.-J., Physica D, 86, 572, (1995) · Zbl 0878.60087
[24] Fyodorov, Y. V.; Sommers, H.-J., J. Math. Phys., 38, 1918, (1997) · Zbl 0872.58072
[25] Kumar, S.; Nock, A.; Sommers, H.-J.; Guhr, T.; Dietz, B.; Miski-Oglu, M.; Richter, A.; Schäfer, F., Phys. Rev. Lett., 111, 030403, (2013)
[26] Novaes, M., J. Math. Phys., 56, 062110, (2015) · Zbl 1322.81050
[27] Kuipers, J.; Waltner, D.; Petitjean, C.; Berkolaiko, G.; Richter, K., Phys. Rev. Lett., 104, 027001, (2010)
[28] Kuipers, J.; Engl, T.; Berkolaiko, G.; Petitjean, C.; Waltner, D.; Richter, K., Phys. Rev. B, 83, 195315, (2011)
[29] Adagideli, I., Phys. Rev. B, 68, 233308, (2003)
[30] Rahav, S.; Brouwer, P. W., Phys. Rev. Lett., 95, 056806, (2005)
[31] Whitney, R. S.; Jacquod, Ph., Phys. Rev. Lett., 96, 206804, (2006)
[32] Waltner, D.; Kuipers, J.; Richter, K., Phys. Rev. B, 83, 195315, (2011)
[33] Müller, S.; Heusler, S.; Braun, P.; Haake, F.; Altland, A., Phys. Rev. E, 72, 046207, (2005)
[34] Berkolaiko, G.; Kuipers, J., J. Math. Phys., 54, 112103, (2013) · Zbl 1288.81048
[35] Berkolaiko, G.; Kuipers, J., J. Math. Phys., 54, 123505, (2013) · Zbl 1288.82053
[36] Di Francesco, P.; Brezin, É.; Kazakov, V., Applications of Random Matrices in Physics, (2006), Springer
[37] Bouttier, J.; Akemann, G.; Baik, J.; Di Francesco, P., The Oxford Handbook of Random Matrix Theory, Chap. 26, (2011), Oxford University Press · Zbl 1225.15004
[38] Morris, T. R., Nucl. Phys. B, 356, 703, (1991)
[39] Samuel, S., J. Math. Phys., 21, 2695, (1980)
[40] Esposti, M. D.; Knauf, A., J. Math. Phys., 45, 4957, (2004) · Zbl 1064.81036
[41] Collins, B., Int. Math. Res. Not., 17, 953, (2003) · Zbl 1049.60091
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.