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Chin. Phys. B, 2019, Vol. 28(10): 100502    DOI: 10.1088/1674-1056/ab3f96
SPECIAL TOPIC—110th Anniversary of Lanzhou University Prev   Next  

Experimental investigation of the fluctuations in nonchaotic scattering in microwave billiards

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
1 School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China;
2 Key Laboratory for Magnetism and Magnetic Materials of MOE, Lanzhou University, Lanzhou 730000, China
Abstract  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.
Keywords:  wave chaos      quantum billiards      microwave billiards      random matrix theory  
Received:  13 July 2019      Revised:  10 August 2019      Published:  05 October 2019
PACS:  03.65.Ge (Solutions of wave equations: bound states)  
  03.65.Sq (Semiclassical theories and applications)  
  05.45.Mt (Quantum chaos; semiclassical methods)  
  24.60.Ky (Fluctuation phenomena)  
Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 11775100, 11775101, and 11961131009).
Corresponding Authors:  Barbara Dietz     E-mail:  dietz@lzu.edu.cn

Cite this article: 

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 2019 Chin. Phys. B 28 100502

[33] Stöckmann H J and Stein J 1990 Phys. Rev. Lett. 64 2215
[1] Sinai Y G 1990 Russ. Math. Surveys 25 137
[34] Exner P 1997 Found. Phys. 27 171
[2] Bunimovich L A 1991 Chaos 1 187
[35] Bogomolny E, Gerland U and Schmit C 2001 Phys. Rev. E 63 036206
[3] Berry M V 1981 Eur. J. Phys. 2 91
[36] Bogomolny E, Giraud O and Schmit C 2002 Phys. Rev. E 65 056214
[4] Bunimovich L A 1979 Commun. Math. Phys. 65 295
[37] Tudorovskiy T, Höhmann R, Kuhl U and Stöckmann H J 2008 J. Phys. A 41 275101
[5] Berry M V and Tabor M 1977 J. Phys. A 10 371
[38] Tudorovskiy T, Kuhl U and Stöckmann H J 2010 New J. Phys. 12 123021
[6] Berry M V 1977 J. Phys. A 10 2083
[39] Tudorovskiy T, Kuhl U and Stöckmann H J 2011 J. Phys. A 44 135101
[7] Casati G, Valz-Gris F and Guarnieri I 1980 Lett. Nuovo Cimento 28 279
[40] Bialous M, Yunko V, Bauch S, Lawniczak M, Dietz B and Sirko L 2016 Phys. Rev. E 94 042211
[8] Bohigas O, Giannoni M J and Schmit C 1984 Phys. Rev. Lett. 52 1
[41] do Carmo R B and de Aguiar F M 2019 Sci. Rep. 9 3634
[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
[10] Rosenzweig N and Porter C 1960 Phys. Rev. 120 1698
[43] Albeverio S, Haake F, Kurasov P, Kuś M and Šeba P 1996 J. Math. Phys. 37 4888
[11] Berry M V 1984 Proc. R. Soc. Lond. A 392 45
[44] Haake F, Kuś M, Šeba P, Stöckmann H J and Stoffregen U 1996 J. Phys. A 29 5745
[12] Lenz G and Haake F 1991 Phys. Rev. Lett. 67 1
[45] Stöckmann H J and Šeba P 1998 J. Phys. A 31 3439
[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
[15] Życzkowski K 1992 Acta Phys. Pol. B 23 245
[16] Życzkowski K 1994 Phys. Rev. E 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
[48] Bogomolny E, Dietz B, Friedrich T, Miski-Oglu M, Richter A, Schäfer F and Schmit C 2006 Phys. Rev. Lett. 97 254102
[18] Shudo A and Shimizu Y 1993 Phys. Rev. E 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
[19] Shudo A, Shimizu Y, Šeba P, Stein J, Stöckmann H J and Życzkowski K 1994 Phys. Rev. E 49 3748
[20] Šeba P 1990 Phys. Rev. Lett. 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
[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
[22] Šeba P and Życzkowski K 1991 Phys. Rev. A 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
[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
[24] Shigehara T 1994 Phys. Rev. E 50 4357
[54] Dörr U, Stöckmann H J, Barth M and Kuhl U 1998 Phys. Rev. Lett. 80 1030
[25] Shigehara T and Cheon T 1996 Phys. Rev. E 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
[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
[28] Legrand O, Mortessagne F and Weaver R L 1997 Phys. Rev. E 55 7741
[57] Kuhl U 2007 Eur. Phys. J. 145 103
[29] Rahav S and Fishman S 2002 Nonlinearity 15 1541
[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
[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
[32] Bogomolny E, Gerland U and Schmit C 2001 Eur. Phys. J. B 19 121
[61] Dittes F 2000 Phys. Rep. 339 215
[33] Stöckmann H J and Stein J 1990 Phys. Rev. Lett. 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
[63] Dietz B, Harney H L, Richter A, Schäfer F and Weidenmüller H A 2010 Phys. Lett. B 685 263
[35] Bogomolny E, Gerland U and Schmit C 2001 Phys. Rev. E 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
[36] Bogomolny E, Giraud O and Schmit C 2002 Phys. Rev. E 65 056214
[65] Dietz B, Heusler A, Maier K H, Richter A and Brown B A 2017 Phys. Rev. Lett. 118 012501
[37] Tudorovskiy T, Höhmann R, Kuhl U and Stöckmann H J 2008 J. Phys. A 41 275101
[66] Kumar S, Dietz B, Guhr T and Richter A 2017 Phys. Rev. Lett. 119 244102
[38] Tudorovskiy T, Kuhl U and Stöckmann H J 2010 New J. Phys. 12 123021
[67] Verbaarschot J J M, Weidenmüller H A and Zirnbauer M R 1985 Phys. Rep. 129 367
[68] Fyodorov Y V, Savin D V and Sommers H J 2005 J. Phys. A 38 10731
[39] Tudorovskiy T, Kuhl U and Stöckmann H J 2011 J. Phys. A 44 135101
[40] Bialous M, Yunko V, Bauch S, Lawniczak M, Dietz B and Sirko L 2016 Phys. Rev. E 94 042211
[41] do Carmo R B and de Aguiar F M 2019 Sci. Rep. 9 3634
[42] Dietz B, Eckmann J-P, Pillet C-A, Smilansky U and Ussishkin I 1995 Phys. Rev. E 51 4222
[43] Albeverio S, Haake F, Kurasov P, Kuś M and Šeba P 1996 J. Math. Phys. 37 4888
[44] Haake F, Kuś M, Šeba P, Stöckmann H J and Stoffregen U 1996 J. Phys. A 29 5745
[45] Stöckmann H J and Šeba P 1998 J. Phys. A 31 3439
[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
[49] Dietz B, Friedrich T, Metz J, Miski-Oglu M, Richter A, Schäfer F and Stafford C A 2007 Phys. Rev. E 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
[51] Dietz B, Friedrich T, Miski-Oglu M, Richter A, Schäfer F and Seligmann T H 2009 Phys. Rev. E 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
[53] Dietz B and Richter A 2015 Chaos 25 097601
[54] Dörr U, Stöckmann H J, Barth M and Kuhl U 1998 Phys. Rev. Lett. 80 1030
[55] Dembowski C, Gräf H D, Hofferbert R, Rehfeld H, Richter A and Weiland T 1999 Phys. Rev. E 60 3942
[56] Maier L and Slater J 1952 J. Appl. Phys. 23 68
[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
[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
[61] Dittes F 2000 Phys. Rep. 339 215
[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
[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
[65] Dietz B, Heusler A, Maier K H, Richter A and Brown B A 2017 Phys. Rev. Lett. 118 012501
[66] Kumar S, Dietz B, Guhr T and Richter A 2017 Phys. Rev. Lett. 119 244102
[67] Verbaarschot J J M, Weidenmüller H A and Zirnbauer M R 1985 Phys. Rep. 129 367
[68] Fyodorov Y V, Savin D V and Sommers H J 2005 J. Phys. A 38 10731
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