中国物理B ›› 2019, Vol. 28 ›› Issue (10): 100502-100502.doi: 10.1088/1674-1056/ab3f96
• SPECIAL TOPIC—Recent advances in thermoelectric materials and devices • 上一篇 下一篇
Runzu Zhang(张润祖), Weihua Zhang(张为华), Barbara Dietz, Guozhi Chai(柴国志), Liang Huang(黄亮)
收稿日期:
2019-07-13
修回日期:
2019-08-10
出版日期:
2019-10-05
发布日期:
2019-10-05
通讯作者:
Barbara Dietz
E-mail:dietz@lzu.edu.cn
基金资助:
Runzu Zhang(张润祖)1,2, Weihua Zhang(张为华)1,2, Barbara Dietz1,2, Guozhi Chai(柴国志)2, Liang Huang(黄亮)1,2
Received:
2019-07-13
Revised:
2019-08-10
Online:
2019-10-05
Published:
2019-10-05
Contact:
Barbara Dietz
E-mail:dietz@lzu.edu.cn
Supported by:
摘要: We report on the experimental investigation of the properties of the eigenvalues and wavefunctions and the fluctuation properties of the scattering matrix of closed and open billiards, respectively, of which the classical dynamics undergoes a transition from integrable via almost integrable to fully chaotic. To realize such a system, we chose a billiard with a 60° sector shape of which the classical dynamics is integrable, and introduced circular scatterers of varying number, size, and position. The spectral properties of generic quantum systems of which the classical counterpart is either integrable or chaotic are universal and well understood. If, however, the classical dynamics is pseudo-integrable or almost-integrable, they exhibit a non-universal intermediate statistics, for which analytical results are known only in a few cases, e.g., if it corresponds to semi-Poisson statistics. Since the latter is, above all, clearly distinguishable from those of integrable and chaotic systems, our aim was to design a billiard with these features which indeed is achievable by adding just one scatterer of appropriate size and position to the sector billiard. We demonstrated that, while the spectral properties of almost-integrable billiards are sensitive to the classical dynamics, this is not the case for the distribution of the wavefunction components, which was analyzed in terms of the strength distribution, and the fluctuation properties of the scattering matrix which coincide with those of typical, fully chaotic systems.
中图分类号: (Solutions of wave equations: bound states)
张润祖, 张为华, Barbara Dietz, 柴国志, 黄亮. Experimental investigation of the fluctuations in nonchaotic scattering in microwave billiards[J]. 中国物理B, 2019, 28(10): 100502-100502.
Runzu Zhang(张润祖), Weihua Zhang(张为华), Barbara Dietz, Guozhi Chai(柴国志), Liang Huang(黄亮). Experimental investigation of the fluctuations in nonchaotic scattering in microwave billiards[J]. Chin. Phys. B, 2019, 28(10): 100502-100502.
[33] |
Stöckmann H J and Stein J 1990 Phys. Rev. Lett. 64 2215
doi: 10.1103/PhysRevLett.64.2215 |
[1] | Sinai Y G 1990 Russ. Math. Surveys 25 137 |
[34] |
Exner P 1997 Found. Phys. 27 171
doi: 10.1007/BF02550448 |
[2] | Bunimovich L A 1991 Chaos 1 187 |
[35] |
Bogomolny E, Gerland U and Schmit C 2001 Phys. Rev. E 63 036206
doi: 10.1103/PhysRevE.63.036206 |
[3] |
Berry M V 1981 Eur. J. Phys. 2 91
doi: 10.1088/0143-0807/2/2/006 |
[36] |
Bogomolny E, Giraud O and Schmit C 2002 Phys. Rev. E 65 056214
doi: 10.1103/PhysRevE.65.056214 |
[4] |
Bunimovich L A 1979 Commun. Math. Phys. 65 295
doi: 10.1007/BF01197884 |
[37] |
Tudorovskiy T, Höhmann R, Kuhl U and Stöckmann H J 2008 J. Phys. A 41 275101
doi: 10.1088/1751-8113/41/27/275101 |
[5] |
Berry M V and Tabor M 1977 J. Phys. A 10 371
doi: 10.1088/0305-4470/10/3/009 |
[38] |
Tudorovskiy T, Kuhl U and Stöckmann H J 2010 New J. Phys. 12 123021
doi: 10.1088/1367-2630/12/12/123021 |
[6] |
Berry M V 1977 J. Phys. A 10 2083
doi: 10.1088/0305-4470/10/12/016 |
[39] |
Tudorovskiy T, Kuhl U and Stöckmann H J 2011 J. Phys. A 44 135101
doi: 10.1088/1751-8113/44/13/135101 |
[7] |
Casati G, Valz-Gris F and Guarnieri I 1980 Lett. Nuovo Cimento 28 279
doi: 10.1007/BF02798790 |
[40] |
Bialous M, Yunko V, Bauch S, Lawniczak M, Dietz B and Sirko L 2016 Phys. Rev. E 94 042211
doi: 10.1103/PhysRevE.94.042211 |
[8] |
Bohigas O, Giannoni M J and Schmit C 1984 Phys. Rev. Lett. 52 1
doi: 10.1103/PhysRevLett.52.1 |
[41] |
do Carmo R B and de Aguiar F M 2019 Sci. Rep. 9 3634
doi: 10.1038/s41598-019-40048-0 |
[9] | Mehta M L 1990 Random Matrices (London: Academic Press) |
[42] |
Dietz B, Eckmann J-P, Pillet C-A, Smilansky U and Ussishkin I 1995 Phys. Rev. E 51 4222
doi: 10.1103/PhysRevE.51.4222 |
[10] |
Rosenzweig N and Porter C 1960 Phys. Rev. 120 1698
doi: 10.1103/PhysRev.120.1698 |
[43] |
Albeverio S, Haake F, Kurasov P, Kuś M and Šeba P 1996 J. Math. Phys. 37 4888
doi: 10.1063/1.531668 |
[11] |
Berry M V 1984 Proc. R. Soc. Lond. A 392 45
doi: 10.1098/rspa.1984.0023 |
[44] |
Haake F, Kuś M, Šeba P, Stöckmann H J and Stoffregen U 1996 J. Phys. A 29 5745
doi: 10.1088/0305-4470/29/18/009 |
[12] |
Lenz G and Haake F 1991 Phys. Rev. Lett. 67 1
doi: 10.1103/PhysRevLett.67.1 |
[45] |
Stöckmann H J and Šeba P 1998 J. Phys. A 31 3439
doi: 10.1088/0305-4470/31/15/009 |
[13] | Kota V K B 2014 Embedded Random Matrix Ensembles in Quantum Physics (Heidelberg: Springer) |
[46] | Stöckmann H J 2000 Quantum Chaos: An Introduction (Cambridge: Cambridge University Press) |
[14] |
Richens P J and Berry M V 1981 Physica D 2 495
doi: 10.1016/0167-2789(81)90024-5 |
[15] | Życzkowski K 1992 Acta Phys. Pol. B 23 245 |
[16] |
Życzkowski K 1994 Phys. Rev. E 49 3748
doi: 10.1103/PhysRevE.49.3748 |
[47] | Richter A 1999 Emerging Applications of Number Theory, The IMA Volumes in Mathematics and its Applications (Hejhal D A, Friedmann J, Gutzwiller M C and Od-lyzko A M, Ed.) (New York: Springer) 109 479 |
[17] |
Biswas D and Jain S R 1990 Phys. Rev. A 42 3170
doi: 10.1103/PhysRevA.42.3170 |
[48] |
Bogomolny E, Dietz B, Friedrich T, Miski-Oglu M, Richter A, Schäfer F and Schmit C 2006 Phys. Rev. Lett. 97 254102
doi: 10.1103/PhysRevLett.97.254102 |
[18] |
Shudo A and Shimizu Y 1993 Phys. Rev. E 47 54
doi: 10.1103/PhysRevE.47.54 |
[49] |
Dietz B, Friedrich T, Metz J, Miski-Oglu M, Richter A, Schäfer F and Stafford C A 2007 Phys. Rev. E 75 027201
doi: 10.1103/PhysRevE.75.027201 |
[19] |
Shudo A, Shimizu Y, Šeba P, Stein J, Stöckmann H J and Życzkowski K 1994 Phys. Rev. E 49 3748
doi: 10.1103/PhysRevE.49.3748 |
[20] |
Šeba P 1990 Phys. Rev. Lett. 64 1855
doi: 10.1103/PhysRevLett.64.1855 |
[50] |
Dietz B, Friedrich T, Harney H L, Miski-Oglu M, Richter A, Schäfer F and Weidenmüller H A 2008 Phys. Rev. E 78 055204
doi: 10.1103/PhysRevE.78.055204 |
[21] | Haake F, Lenz G, Šeba P, Stein J, Stöckmann H J and Życzkowski K 1991 Phys. Rev. A 44 R6161 |
[51] |
Dietz B, Friedrich T, Miski-Oglu M, Richter A, Schäfer F and Seligmann T H 2009 Phys. Rev. E 80 036212
doi: 10.1103/PhysRevE.80.036212 |
[22] |
Šeba P and Życzkowski K 1991 Phys. Rev. A 44 3457
doi: 10.1103/PhysRevA.44.3457 |
[52] |
Dietz B, Friedrich T, Harney H L, Miski-Oglu M, Richter A, Schäfer F and Weidenmüller H A 2010 Phys. Rev. E 81 036205
doi: 10.1103/PhysRevE.81.036205 |
[23] | Shigehara T, Yoshinaga N, Cheon T and Mizusaki T 1993 Phys. Rev. E 47 R3822 |
[53] |
Dietz B and Richter A 2015 Chaos 25 097601
doi: 10.1063/1.4915527 |
[24] |
Shigehara T 1994 Phys. Rev. E 50 4357
doi: 10.1103/PhysRevE.50.4357 |
[54] |
Dörr U, Stöckmann H J, Barth M and Kuhl U 1998 Phys. Rev. Lett. 80 1030
doi: 10.1103/PhysRevLett.80.1030 |
[25] |
Shigehara T and Cheon T 1996 Phys. Rev. E 54 1321
doi: 10.1103/PhysRevE.54.1321 |
[26] | Cheon T and Shigehara T 1996 Phys. Rev. E 54 3300 |
[55] |
Dembowski C, Gräf H D, Hofferbert R, Rehfeld H, Richter A and Weiland T 1999 Phys. Rev. E 60 3942
doi: 10.1103/PhysRevE.60.3942 |
[27] | Weaver R L and Sornette D 1995 Phys. Rev. E 52 3341 |
[56] |
Maier L and Slater J 1952 J. Appl. Phys. 23 68
doi: 10.1063/1.1701980 |
[28] |
Legrand O, Mortessagne F and Weaver R L 1997 Phys. Rev. E 55 7741
doi: 10.1103/PhysRevE.55.7741 |
[57] | Kuhl U 2007 Eur. Phys. J. 145 103 |
[29] |
Rahav S and Fishman S 2002 Nonlinearity 15 1541
doi: 10.1088/0951-7715/15/5/311 |
[58] |
Dembowski C, Dietz B, Friedrich T, Gräf H D, Harney H L, Heine A, Miski-Oglu M and Richter A 2005 Phys. Rev. E 71 046202
doi: 10.1103/PhysRevE.71.046202 |
[30] | Rahav S, Richman O and Fishman S 2003 J. Phys. A 36 L529 |
[59] | Porter C E 1965 Statistical Theories of Spectra: Fluctuations (New York: Academic) |
[31] | Bogomolny E, Gerland U and Schmit C 1999 Phys. Rev. E 59 R1315 |
[60] |
Guhr T, Müller-Groeling G A and Weidenmüller H A 1998 Phys. Rep. 299 189
doi: 10.1016/S0370-1573(97)00088-4 |
[32] |
Bogomolny E, Gerland U and Schmit C 2001 Eur. Phys. J. B 19 121
doi: 10.1007/s100510170357 |
[61] |
Dittes F 2000 Phys. Rep. 339 215
doi: 10.1016/S0370-1573(00)00065-X |
[33] |
Stöckmann H J and Stein J 1990 Phys. Rev. Lett. 64 2215
doi: 10.1103/PhysRevLett.64.2215 |
[62] | Mahaux C and Weidenmüller H A 1969 Shell Model Approach to Nuclear Reactions (Amsterdam: North Holland) |
[34] |
Exner P 1997 Found. Phys. 27 171
doi: 10.1007/BF02550448 |
[63] |
Dietz B, Harney H L, Richter A, Schäfer F and Weidenmüller H A 2010 Phys. Lett. B 685 263
doi: 10.1016/j.physletb.2010.01.074 |
[35] |
Bogomolny E, Gerland U and Schmit C 2001 Phys. Rev. E 63 036206
doi: 10.1103/PhysRevE.63.036206 |
[64] |
Kumar S, Nock A, Sommers H J, Guhr T, Dietz B, Miski-Oglu M, Richter A and Schäfer F 2013 Phys. Rev. Lett. 111 030403
doi: 10.1103/PhysRevLett.111.030403 |
[36] |
Bogomolny E, Giraud O and Schmit C 2002 Phys. Rev. E 65 056214
doi: 10.1103/PhysRevE.65.056214 |
[65] |
Dietz B, Heusler A, Maier K H, Richter A and Brown B A 2017 Phys. Rev. Lett. 118 012501
doi: 10.1103/PhysRevLett.118.012501 |
[37] |
Tudorovskiy T, Höhmann R, Kuhl U and Stöckmann H J 2008 J. Phys. A 41 275101
doi: 10.1088/1751-8113/41/27/275101 |
[66] |
Kumar S, Dietz B, Guhr T and Richter A 2017 Phys. Rev. Lett. 119 244102
doi: 10.1103/PhysRevLett.119.244102 |
[38] |
Tudorovskiy T, Kuhl U and Stöckmann H J 2010 New J. Phys. 12 123021
doi: 10.1088/1367-2630/12/12/123021 |
[67] |
Verbaarschot J J M, Weidenmüller H A and Zirnbauer M R 1985 Phys. Rep. 129 367
doi: 10.1016/0370-1573(85)90070-5 |
[68] |
Fyodorov Y V, Savin D V and Sommers H J 2005 J. Phys. A 38 10731
doi: 10.1088/0305-4470/38/49/017 |
[39] |
Tudorovskiy T, Kuhl U and Stöckmann H J 2011 J. Phys. A 44 135101
doi: 10.1088/1751-8113/44/13/135101 |
[40] |
Bialous M, Yunko V, Bauch S, Lawniczak M, Dietz B and Sirko L 2016 Phys. Rev. E 94 042211
doi: 10.1103/PhysRevE.94.042211 |
[41] |
do Carmo R B and de Aguiar F M 2019 Sci. Rep. 9 3634
doi: 10.1038/s41598-019-40048-0 |
[42] |
Dietz B, Eckmann J-P, Pillet C-A, Smilansky U and Ussishkin I 1995 Phys. Rev. E 51 4222
doi: 10.1103/PhysRevE.51.4222 |
[43] |
Albeverio S, Haake F, Kurasov P, Kuś M and Šeba P 1996 J. Math. Phys. 37 4888
doi: 10.1063/1.531668 |
[44] |
Haake F, Kuś M, Šeba P, Stöckmann H J and Stoffregen U 1996 J. Phys. A 29 5745
doi: 10.1088/0305-4470/29/18/009 |
[45] |
Stöckmann H J and Šeba P 1998 J. Phys. A 31 3439
doi: 10.1088/0305-4470/31/15/009 |
[46] | Stöckmann H J 2000 Quantum Chaos: An Introduction (Cambridge: Cambridge University Press) |
[47] | Richter A 1999 Emerging Applications of Number Theory, The IMA Volumes in Mathematics and its Applications (Hejhal D A, Friedmann J, Gutzwiller M C and Od-lyzko A M, Ed.) (New York: Springer) 109 479 |
[48] |
Bogomolny E, Dietz B, Friedrich T, Miski-Oglu M, Richter A, Schäfer F and Schmit C 2006 Phys. Rev. Lett. 97 254102
doi: 10.1103/PhysRevLett.97.254102 |
[49] |
Dietz B, Friedrich T, Metz J, Miski-Oglu M, Richter A, Schäfer F and Stafford C A 2007 Phys. Rev. E 75 027201
doi: 10.1103/PhysRevE.75.027201 |
[50] |
Dietz B, Friedrich T, Harney H L, Miski-Oglu M, Richter A, Schäfer F and Weidenmüller H A 2008 Phys. Rev. E 78 055204
doi: 10.1103/PhysRevE.78.055204 |
[51] |
Dietz B, Friedrich T, Miski-Oglu M, Richter A, Schäfer F and Seligmann T H 2009 Phys. Rev. E 80 036212
doi: 10.1103/PhysRevE.80.036212 |
[52] |
Dietz B, Friedrich T, Harney H L, Miski-Oglu M, Richter A, Schäfer F and Weidenmüller H A 2010 Phys. Rev. E 81 036205
doi: 10.1103/PhysRevE.81.036205 |
[53] |
Dietz B and Richter A 2015 Chaos 25 097601
doi: 10.1063/1.4915527 |
[54] |
Dörr U, Stöckmann H J, Barth M and Kuhl U 1998 Phys. Rev. Lett. 80 1030
doi: 10.1103/PhysRevLett.80.1030 |
[55] |
Dembowski C, Gräf H D, Hofferbert R, Rehfeld H, Richter A and Weiland T 1999 Phys. Rev. E 60 3942
doi: 10.1103/PhysRevE.60.3942 |
[56] |
Maier L and Slater J 1952 J. Appl. Phys. 23 68
doi: 10.1063/1.1701980 |
[57] | Kuhl U 2007 Eur. Phys. J. 145 103 |
[58] |
Dembowski C, Dietz B, Friedrich T, Gräf H D, Harney H L, Heine A, Miski-Oglu M and Richter A 2005 Phys. Rev. E 71 046202
doi: 10.1103/PhysRevE.71.046202 |
[59] | Porter C E 1965 Statistical Theories of Spectra: Fluctuations (New York: Academic) |
[60] |
Guhr T, Müller-Groeling G A and Weidenmüller H A 1998 Phys. Rep. 299 189
doi: 10.1016/S0370-1573(97)00088-4 |
[61] |
Dittes F 2000 Phys. Rep. 339 215
doi: 10.1016/S0370-1573(00)00065-X |
[62] | Mahaux C and Weidenmüller H A 1969 Shell Model Approach to Nuclear Reactions (Amsterdam: North Holland) |
[63] |
Dietz B, Harney H L, Richter A, Schäfer F and Weidenmüller H A 2010 Phys. Lett. B 685 263
doi: 10.1016/j.physletb.2010.01.074 |
[64] |
Kumar S, Nock A, Sommers H J, Guhr T, Dietz B, Miski-Oglu M, Richter A and Schäfer F 2013 Phys. Rev. Lett. 111 030403
doi: 10.1103/PhysRevLett.111.030403 |
[65] |
Dietz B, Heusler A, Maier K H, Richter A and Brown B A 2017 Phys. Rev. Lett. 118 012501
doi: 10.1103/PhysRevLett.118.012501 |
[66] |
Kumar S, Dietz B, Guhr T and Richter A 2017 Phys. Rev. Lett. 119 244102
doi: 10.1103/PhysRevLett.119.244102 |
[67] |
Verbaarschot J J M, Weidenmüller H A and Zirnbauer M R 1985 Phys. Rep. 129 367
doi: 10.1016/0370-1573(85)90070-5 |
[68] |
Fyodorov Y V, Savin D V and Sommers H J 2005 J. Phys. A 38 10731
doi: 10.1088/0305-4470/38/49/017 |
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