Avoided level crossings, diabolic points, and branch points in the complex plane in an open double quantum dot

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 71, Issue 3, 2005, Pages

  Rotter, I ^a   Sadreev, A F ^b,c,d  

^a MAX PLANCK INSTITUT FÜR PHYSIK KOMPLEXER SYSTEME   (Germany)

^b KIRENSKY INSTITUTE OF PHYSICS   (Russian Federation)

^c LINKÖPING UNIVERSITY   (Sweden)

^d Astaf'ev Krasnoyarsk State Pedagogical University   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

BRANCH POINTS IN THE COMPLEX PLANE (BPCP); DIABOLIC POINTS (DP); GEOMETRIC PHASES; RIEMANN SHEETS;

EIGENVALUES AND EIGENFUNCTIONS; ELECTRON ENERGY LEVELS; FUNCTIONS; HAMILTONIANS; QUANTUM THEORY; RESONANCE; TOPOLOGY;

SEMICONDUCTOR QUANTUM DOTS;

EID: 41349115222     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.71.036227     Document Type: Article

References (53)

1. T. Kato, Perturbation Theory of Linear Operators, 2nd ed. (Springer, Berlin, 1980).
2. M. V. Berry, Proc. R. Soc. London, Ser. A 392, 45 (1984).
3. M. V. Berry and M. Wilkinson, Proc. R. Soc. London, Ser. A 392, 15 (1984).
4. R. Y. Chiao and Y. S. Wu, Phys. Rev. Lett. 57, 933 (1986); A. Tomita and R. Y. Chiao, ibid. 57, 937 (1986); R. S. Nikam and P. Ring, ibid. 58, 980 (1987); R. Tycko, ibid. 58, 2281 (1987); T. Bitter and D. Dubbers, ibid. 59, 251 (1987); S. Clough, G. J. Barker, K. J. Abed, and A. J. Horsewill, ibid. 60, 136 (1988); R. Bhandari and J. Samuel, ibid. 60, 1211 (1988); R. Y. Chiao, A. Antaramian, K. M. Ganga, H. Jiao, and S. R. Wilkinson, ibid. 60, 1214 (1988); D. Suter, K. T. Mueller, and A. Pines, ibid. 60, 1218 (1988).
5. R. Y. Chiao and Y. S. Wu, Phys. Rev. Lett. 57, 933 (1986); A. Tomita and R. Y. Chiao, ibid. 57, 937 (1986); R. S. Nikam and P. Ring, ibid. 58, 980 (1987); R. Tycko, ibid. 58, 2281 (1987); T. Bitter and D. Dubbers, ibid. 59, 251 (1987); S. Clough, G. J. Barker, K. J. Abed, and A. J. Horsewill, ibid. 60, 136 (1988); R. Bhandari and J. Samuel, ibid. 60, 1211 (1988); R. Y. Chiao, A. Antaramian, K. M. Ganga, H. Jiao, and S. R. Wilkinson, ibid. 60, 1214 (1988); D. Suter, K. T. Mueller, and A. Pines, ibid. 60, 1218 (1988).
6. R. Y. Chiao and Y. S. Wu, Phys. Rev. Lett. 57, 933 (1986); A. Tomita and R. Y. Chiao, ibid. 57, 937 (1986); R. S. Nikam and P. Ring, ibid. 58, 980 (1987); R. Tycko, ibid. 58, 2281 (1987); T. Bitter and D. Dubbers, ibid. 59, 251 (1987); S. Clough, G. J. Barker, K. J. Abed, and A. J. Horsewill, ibid. 60, 136 (1988); R. Bhandari and J. Samuel, ibid. 60, 1211 (1988); R. Y. Chiao, A. Antaramian, K. M. Ganga, H. Jiao, and S. R. Wilkinson, ibid. 60, 1214 (1988); D. Suter, K. T. Mueller, and A. Pines, ibid. 60, 1218 (1988).
7. R. Y. Chiao and Y. S. Wu, Phys. Rev. Lett. 57, 933 (1986); A. Tomita and R. Y. Chiao, ibid. 57, 937 (1986); R. S. Nikam and P. Ring, ibid. 58, 980 (1987); R. Tycko, ibid. 58, 2281 (1987); T. Bitter and D. Dubbers, ibid. 59, 251 (1987); S. Clough, G. J. Barker, K. J. Abed, and A. J. Horsewill, ibid. 60, 136 (1988); R. Bhandari and J. Samuel, ibid. 60, 1211 (1988); R. Y. Chiao, A. Antaramian, K. M. Ganga, H. Jiao, and S. R. Wilkinson, ibid. 60, 1214 (1988); D. Suter, K. T. Mueller, and A. Pines, ibid. 60, 1218 (1988).
8. R. Y. Chiao and Y. S. Wu, Phys. Rev. Lett. 57, 933 (1986); A. Tomita and R. Y. Chiao, ibid. 57, 937 (1986); R. S. Nikam and P. Ring, ibid. 58, 980 (1987); R. Tycko, ibid. 58, 2281 (1987); T. Bitter and D. Dubbers, ibid. 59, 251 (1987); S. Clough, G. J. Barker, K. J. Abed, and A. J. Horsewill, ibid. 60, 136 (1988); R. Bhandari and J. Samuel, ibid. 60, 1211 (1988); R. Y. Chiao, A. Antaramian, K. M. Ganga, H. Jiao, and S. R. Wilkinson, ibid. 60, 1214 (1988); D. Suter, K. T. Mueller, and A. Pines, ibid. 60, 1218 (1988).
9. R. Y. Chiao and Y. S. Wu, Phys. Rev. Lett. 57, 933 (1986); A. Tomita and R. Y. Chiao, ibid. 57, 937 (1986); R. S. Nikam and P. Ring, ibid. 58, 980 (1987); R. Tycko, ibid. 58, 2281 (1987); T. Bitter and D. Dubbers, ibid. 59, 251 (1987); S. Clough, G. J. Barker, K. J. Abed, and A. J. Horsewill, ibid. 60, 136 (1988); R. Bhandari and J. Samuel, ibid. 60, 1211 (1988); R. Y. Chiao, A. Antaramian, K. M. Ganga, H. Jiao, and S. R. Wilkinson, ibid. 60, 1214 (1988); D. Suter, K. T. Mueller, and A. Pines, ibid. 60, 1218 (1988).
10. R. Y. Chiao and Y. S. Wu, Phys. Rev. Lett. 57, 933 (1986); A. Tomita and R. Y. Chiao, ibid. 57, 937 (1986); R. S. Nikam and P. Ring, ibid. 58, 980 (1987); R. Tycko, ibid. 58, 2281 (1987); T. Bitter and D. Dubbers, ibid. 59, 251 (1987); S. Clough, G. J. Barker, K. J. Abed, and A. J. Horsewill, ibid. 60, 136 (1988); R. Bhandari and J. Samuel, ibid. 60, 1211 (1988); R. Y. Chiao, A. Antaramian, K. M. Ganga, H. Jiao, and S. R. Wilkinson, ibid. 60, 1214 (1988); D. Suter, K. T. Mueller, and A. Pines, ibid. 60, 1218 (1988).
11. R. Y. Chiao and Y. S. Wu, Phys. Rev. Lett. 57, 933 (1986); A. Tomita and R. Y. Chiao, ibid. 57, 937 (1986); R. S. Nikam and P. Ring, ibid. 58, 980 (1987); R. Tycko, ibid. 58, 2281 (1987); T. Bitter and D. Dubbers, ibid. 59, 251 (1987); S. Clough, G. J. Barker, K. J. Abed, and A. J. Horsewill, ibid. 60, 136 (1988); R. Bhandari and J. Samuel, ibid. 60, 1211 (1988); R. Y. Chiao, A. Antaramian, K. M. Ganga, H. Jiao, and S. R. Wilkinson, ibid. 60, 1214 (1988); D. Suter, K. T. Mueller, and A. Pines, ibid. 60, 1218 (1988).
12. R. Y. Chiao and Y. S. Wu, Phys. Rev. Lett. 57, 933 (1986); A. Tomita and R. Y. Chiao, ibid. 57, 937 (1986); R. S. Nikam and P. Ring, ibid. 58, 980 (1987); R. Tycko, ibid. 58, 2281 (1987); T. Bitter and D. Dubbers, ibid. 59, 251 (1987); S. Clough, G. J. Barker, K. J. Abed, and A. J. Horsewill, ibid. 60, 136 (1988); R. Bhandari and J. Samuel, ibid. 60, 1211 (1988); R. Y. Chiao, A. Antaramian, K. M. Ganga, H. Jiao, and S. R. Wilkinson, ibid. 60, 1214 (1988); D. Suter, K. T. Mueller, and A. Pines, ibid. 60, 1218 (1988).
13. H. M. Lauber, P. Weidenhammer, and D. Dubbers, Phys. Rev. Lett. 72, 1004 (1994).
14. J. Okołowicz, M. Płoszajczak, and I. Rotter, Phys. Rep. 374, 271 (2003).
15. N. Moiseyev, Phys. Rep. 302, 211 (1998).
16. I. Rotter, Phys. Rev. E 64, 036213 (2001).
17. I. Rotter, Phys. Rev. E 68, 016211 (2003).
18. I. Rotter and A. F. Sadreev, Phys. Rev. E 69, 066201 (2004).
19. H. Feshbach, Ann. Phys. (N.Y.) 5, 357 (1958); 19, 287 (1962).
20. H. Feshbach, Ann. Phys. (N.Y.) 5, 357 (1958); 19, 287 (1962).
21. Sometimes, the widths of the resonance states vanish. This may happen due to selection rules or due to the resonance trapping phenomenon at certain values of the tuning parameter [6,15,27]. The selection rules do never hold with 100% accuracy because of higher-order contributions that become important as soon as the main contribution is forbidden. The resonance trapping phenomenon may cause a vanishing (or nearly vanishing minimum) value of the width of a certain resonance state at a certain value of the tuning parameter. The width of this state for neighboring values of the tuning parameter is however nonvanishing, in any case. Concrete examples are considered in atoms [15] and in double quantum dots [27]. It is resonable therefore to generalize the term resonance state and to call all states that are embedded in the continuum, "resonance states" independently of the value of their widths.
22. M. Müller, F. M. Dittes, W. Iskra, and I. Rotter, Phys. Rev. E 52, 5961 (1995).
23. E. Persson, T. Gorin, and I. Rotter, Phys. Rev. E 54, 3339 (1996).
24. A. I. Magunov, I. Rotter, and S. I. Strakhova, J. Phys. B 32, 1669 (1999); 34, 29 (2001).
25. A. I. Magunov, I. Rotter, and S. I. Strakhova, J. Phys. B 32, 1669 (1999); 34, 29 (2001).
26. I. Rotter, Rep. Prog. Phys. 54, 635 (1991).
27. W. D. Heiss, M. Müller, and I. Rotter, Phys. Rev. E 58, 2894 (1998); W. D. Heiss, Eur. Phys. J. D 7, 1 (1999); Phys. Rev. E 61, 929 (2000).
28. W. D. Heiss, M. Müller, and I. Rotter, Phys. Rev. E 58, 2894 (1998); W. D. Heiss, Eur. Phys. J. D 7, 1 (1999); Phys. Rev. E 61, 929 (2000).
29. W. D. Heiss, M. Müller, and I. Rotter, Phys. Rev. E 58, 2894 (1998); W. D. Heiss, Eur. Phys. J. D 7, 1 (1999); Phys. Rev. E 61, 929 (2000).
30. C. Dembowski, H. D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, Phys. Rev. Lett. 86, 787 (2001).
31. C. Dembowski, B. Dietz, H. D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, Phys. Rev. Lett. 90, 034101 (2003).
32. C. Dembowski, B. Dietz, H. D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, Phys. Rev. E 69, 056216 (2004).
33. F. Keck, H. J. Korsch, and S. Mossmann, J. Phys. A 36, 2125 (2003).
34. In Refs. [17-21,23], the BPCP are called "exceptional points" although the function space at these singular points is not incomplete. Instead, the phases of the wave functions of the two crossing states are undetermined, see Figs. 4, 5, and 6, the discussion on Gamow states in Sec. VII, and Refs. [6,10]. The incompleteness of the Hilbert space is characteristic of exceptional points [1]. The chirality of the states at the singular points is determined by the sign in Eq. (21).
35. W. D. Heiss and H. L. Harney, Eur. Phys. J. D 17, 149 (2001).
36. A. Mondragon and E. Hernanández, J. Phys. A 26, 5595 (1993); 29, 2567 (1996); E. Hernández, A. Jáuregui, and A. Mondragon, ibid. 33, 4507 (2000); Phys. Rev. A 67, 022721 (2003).
37. A. Mondragon and E. Hernanández, J. Phys. A 26, 5595 (1993); 29, 2567 (1996); E. Hernández, A. Jáuregui, and A. Mondragon, ibid. 33, 4507 (2000); Phys. Rev. A 67, 022721 (2003).
38. A. Mondragon and E. Hernanández, J. Phys. A 26, 5595 (1993); 29, 2567 (1996); E. Hernández, A. Jáuregui, and A. Mondragon, ibid. 33, 4507 (2000); Phys. Rev. A 67, 022721 (2003).
39. A. Mondragon and E. Hernanández, J. Phys. A 26, 5595 (1993); 29, 2567 (1996); E. Hernández, A. Jáuregui, and A. Mondragon, ibid. 33, 4507 (2000); Phys. Rev. A 67, 022721 (2003).
40. I. Rotter, Phys. Rev. E 65, 026217 (2002); 67, 026204 (2003).
41. I. Rotter, Phys. Rev. E 65, 026217 (2002); 67, 026204 (2003).
42. A. F. Sadreev and I. Rotter, J. Phys. A 36, 11413 (2003).
43. I. Rotter and A. F. Sadreev, Phys. Rev. E (to be published), e-print cond-mat/0403184.
44. C. Mahaux and H. A. Weidenmüller, Shell Model Approach in Nuclear Reactions (Amsterdam, North Holland, 1969).
45. F. M. Dittes, Phys. Rep. 339, 215 (2000).
46. eff coalesce are therefore of first order. For narrow resonances, the fixed-point solutions coincide with a pole of the S matrix.
47. E. Persson, I. Rotter, H. J. Stöckmann, and M. Barth, Phys. Rev. Lett. 85, 2478 (2000); H. J. Stöckmann, E. Persson, Y. H. Kim, M. Barth, U. Kuhl, and I. Rotter, Phys. Rev. E 65, 066211 (2002).
48. E. Persson, I. Rotter, H. J. Stöckmann, and M. Barth, Phys. Rev. Lett. 85, 2478 (2000); H. J. Stöckmann, E. Persson, Y. H. Kim, M. Barth, U. Kuhl, and I. Rotter, Phys. Rev. E 65, 066211 (2002).
49. S. Rotter, F. Libisch, J. Burgdörfer, U. Kuhl, and H. J. Stöckmann, Phys. Rev. E 69, 046208 (2004).
50. F. Pistolesi and N. Manini, Phys. Rev. Lett. 85, 1585 (2000); N. Manini and F. Pistolesi, ibid. 85, 3067 (2000).
51. F. Pistolesi and N. Manini, Phys. Rev. Lett. 85, 1585 (2000); N. Manini and F. Pistolesi, ibid. 85, 3067 (2000).
52. A. I. Magunov, I. Rotter, and S. I. Strakhova, Phys. Rev. B 68, 245305 (2003).
53. A method alternative to the continuum shell model for the description of unstable nuclear states is the Gamow shell model ("shell model in the Berggren basis"), see the recent review [6] and references therein. In the continuum shell model, the shell model states are constructed by means of bound single-particle states. The discrete shell model states are embedded in the continuum, i. e., the completeness relation of the wave functions contains both the discrete shell model states (Q subspace) as well as the nonresonant continuum (P subspace). Although the Gamow shell model is based on the unbound Gamow states (together with the bound single-particle states), the Berggren completeness relation of the wave functions contains additionally the nonresonant continuum. That means, in spite of taking into account the finite lifetime of the unbound single-particle states, the states of the Gamow shell model are also embedded in the common continuum.