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We study the classical and quantum correspondence for a two-dimensional Sinai billiard system. By using the Stationary state expansion method and Gutzwiller's periodic orbit theory, we analyze the quantum length spectrum obtained through the Fourier transformation of the quantum density of state for the Sinai billiard system, and by comparing the peak position with the length of the classical periodic orbit we find their excellent correspondence. We observe that some quantum states are localized near some short period orbits, forming the quantum scarred states or superscarred states. In this paper we also investigate the nearest-neighbor spacing distribution of levels for both concentric and nonconcentric Sinai billiard systems, and find that the concentric Sinai billiard system is nearintegrable, and for the nonconcentric Sinai billiard system with =3/8 its nearest-neighbor spacing distribution of levels transits from nearintegrable to the Wigner distribution as the distance between the two centers increases.
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Keywords:
- quantum billiard /
- periodic orbit theory /
- quantum length spectrum /
- level nearest neighbor spacing distribution
[1] Gutzwiller M C 1990 Chaos in Classical and Quantum Mechanics (New York: Springer-Verlag) pp254-281, 282-321
[2] Robinett R W 1996 Am. J. Phys. 64 440
[3] Lu J, Du M L 2004 Acta Phys. Sin. 53 2450 (in Chinese) [陆军, 杜孟利 2004 物理学报 53 2450]
[4] Berry M V 1981 Eur. J. Phys. 2 91
[5] Heller E J, O'Connor P W, Gehlen J 1989 Physica Scripta 40 354
[6] Cheon T, Cohen T D 1989 Phys. Rev. Lett. 62 2769
[7] Tuan P H, Yu Y T, Chiang P Y, Liang H C, Huang K F, Chen Y F 2012 Phys. Rev. E 85 026202
[8] Shigehara T 1994 Phys. Rev. E 50 4357
[9] Nakamura K, Thomas H 1988 Phys. Rev. Lett. 61 247
[10] Wilkinson P B, Fromhold T M, Eaves L, Sheard F W, Miura N, Takamasu T 1996 Nature 380 606
[11] Marcus C M, Rimberg A J, Westervelt R M, Hopkins P F, Gossard A C 1992 Phys. Rev. Lett. 69 506
[12] Shudo A, Shimizu Y 1993 Phys. Rev. E 47 54
[13] Šeba P, Zyczkowski 1991 Phys. Rev. A 44 3457
[14] Kudroli A, Kidambi V, Sridhar S 1995 Phys. Rev. Lett. 75 822
[15] Stöckmann H J 1999 Quantum Chaos (Cambridge: Cambridge Vniversity Press) pp86-92
[16] Stöckmann H J, Stein J 1990 Phys. Rev. Lett. 64 2215
[17] Kaufman D L, Kosztin I, Schulten K 1999 Am. J. Phys. 67 133
[18] Heller E J 1984 Phys. Rev. Lett. 53 1515
[19] Bogomolny E, Schmit C 2004 Phys. Rev. Lett. 92 244102
[20] Bogomolny E, Dietz B, Friedrich T 2006 Phys. Rev. Lett. 97 254102
[21] Du M L, Delos J B 1988 Phys. Rev. A 38 1896
[22] Zhang Y H, Xu X Y, Lin S L 2009 Chin. Phys. B 18 35
[23] Du M L, Zhang Y H, Xu X Y 2005 Acta Phys. Sin. 54 4538 (in Chinese) [杜孟利, 张延惠, 徐学友 2005 物理学报 54 4538]
[24] Balian R, Bloch C 1974 Ann. Phys. 85 514
[25] Haak F 1991 Quantum Signature of Chaos (Berlin, Heidelberg: Springer) pp52-54
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[1] Gutzwiller M C 1990 Chaos in Classical and Quantum Mechanics (New York: Springer-Verlag) pp254-281, 282-321
[2] Robinett R W 1996 Am. J. Phys. 64 440
[3] Lu J, Du M L 2004 Acta Phys. Sin. 53 2450 (in Chinese) [陆军, 杜孟利 2004 物理学报 53 2450]
[4] Berry M V 1981 Eur. J. Phys. 2 91
[5] Heller E J, O'Connor P W, Gehlen J 1989 Physica Scripta 40 354
[6] Cheon T, Cohen T D 1989 Phys. Rev. Lett. 62 2769
[7] Tuan P H, Yu Y T, Chiang P Y, Liang H C, Huang K F, Chen Y F 2012 Phys. Rev. E 85 026202
[8] Shigehara T 1994 Phys. Rev. E 50 4357
[9] Nakamura K, Thomas H 1988 Phys. Rev. Lett. 61 247
[10] Wilkinson P B, Fromhold T M, Eaves L, Sheard F W, Miura N, Takamasu T 1996 Nature 380 606
[11] Marcus C M, Rimberg A J, Westervelt R M, Hopkins P F, Gossard A C 1992 Phys. Rev. Lett. 69 506
[12] Shudo A, Shimizu Y 1993 Phys. Rev. E 47 54
[13] Šeba P, Zyczkowski 1991 Phys. Rev. A 44 3457
[14] Kudroli A, Kidambi V, Sridhar S 1995 Phys. Rev. Lett. 75 822
[15] Stöckmann H J 1999 Quantum Chaos (Cambridge: Cambridge Vniversity Press) pp86-92
[16] Stöckmann H J, Stein J 1990 Phys. Rev. Lett. 64 2215
[17] Kaufman D L, Kosztin I, Schulten K 1999 Am. J. Phys. 67 133
[18] Heller E J 1984 Phys. Rev. Lett. 53 1515
[19] Bogomolny E, Schmit C 2004 Phys. Rev. Lett. 92 244102
[20] Bogomolny E, Dietz B, Friedrich T 2006 Phys. Rev. Lett. 97 254102
[21] Du M L, Delos J B 1988 Phys. Rev. A 38 1896
[22] Zhang Y H, Xu X Y, Lin S L 2009 Chin. Phys. B 18 35
[23] Du M L, Zhang Y H, Xu X Y 2005 Acta Phys. Sin. 54 4538 (in Chinese) [杜孟利, 张延惠, 徐学友 2005 物理学报 54 4538]
[24] Balian R, Bloch C 1974 Ann. Phys. 85 514
[25] Haak F 1991 Quantum Signature of Chaos (Berlin, Heidelberg: Springer) pp52-54
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