Spectral singularities in a non-Hermitian Friedrichs-Fano-Anderson model

Physical Review B - Condensed Matter and Materials Physics

Volumn 80, Issue 16, 2009, Pages

  Longhi, Stefano ^a  

^a CNR   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 72649084319     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.80.165125     Document Type: Article

References (83)

1. C. M. Bender, Rep. Prog. Phys. 70, 947 (2007). 10.1088/0034-4885/70/6/R03
2. P. Dorey, C. Dunning, and R. Tateo, J. Phys. A 40, R205 (2007). 10.1088/1751-8113/40/32/R01
3. A. Mostafazadeh, arXiv:0810.5643 (unpublished).
4. C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998). 10.1103/PhysRevLett.80.5243
5. T. Kato, Perturbation Theory of Linear Operators (Springer, Berlin, 1966).
6. M. V. Berry, Czech. J. Phys. 54, 1039 (2004). 10.1023/B:CJOP.0000044002. 05657.04
7. R. R. D. Kemp, Can. J. Math. 10, 447 (1958)
8. M. A. Naimark, Am. Math. Soc. Transl. 16, 103 (1960)
9. J. Schwartz, Commun. Pure Appl. Math. 13, 609 (1960) 10.1002/cpa. 3160130405
10. V. E. Ljance, Am. Math. Soc. Transl. 60, 185 (1967).
11. B. F. Samsonov, J. Phys. A 38, L397 (2005). 10.1088/0305-4470/38/21/L04
12. A. Mostafazadeh and H. Mehri-Dehnavi, J. Phys. A 42, 125303 (2009). 10.1088/1751-8113/42/12/125303
13. W. D. Heiss, Phys. Rep. 242, 443 (1994) 10.1016/0370-1573(94)90177-5
14. E. Narevicius and N. Moiseyev, Phys. Rev. Lett. 81, 2221 (1998) 10.1103/PhysRevLett.81.2221
15. C. Dembowski, H.-D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, Phys. Rev. Lett. 86, 787 (2001) 10.1103/PhysRevLett.86. 787
16. C. Dembowski, B. Dietz, H.-D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, Phys. Rev. E 69, 056216 (2004) 10.1103/PhysRevE.69.056216
17. T. Stehmann, W. D. Heiss, and F. G. Scholtz, J. Phys. A 37, 7813 (2004) 10.1088/0305-4470/37/31/012
18. J. Rubinstein, P. Sternberg, and Q. Ma, Phys. Rev. Lett. 99, 167003 (2007) 10.1103/PhysRevLett.99.167003
19. U. Günther, I. Rotter, and B. Samsonov, J. Phys. A 40, 8815 (2007) 10.1088/1751-8113/40/30/014
20. P. Cejnar, S. Heinze, and M. Macek, Phys. Rev. Lett. 99, 100601 (2007) 10.1103/PhysRevLett.99.100601
21. S. Klaiman, U. Günther, and N. Moiseyev, Phys. Rev. Lett. 101, 080402 (2008) 10.1103/PhysRevLett.101.080402
22. M. Müller and I. Rotter, J. Phys. A 41, 244018 (2008). 10.1088/1751-8113/41/24/244018
23. A. Mostafazadeh, Phys. Rev. Lett. 102, 220402 (2009). 10.1103/PhysRevLett.102.220402
24. A. Mostafazadeh, J. Phys. A 39, 13495 (2006). 10.1088/0305-4470/39/43/008
25. Z. Ahmedar, arXiv:0908.2876 (unpublished).
26. J. G. Muga, J. P. Palao, B. Navarro, and I. L. Egusquiza, Phys. Rep. 395, 357 (2004). 10.1016/j.physrep.2004.03.002
27. A. Mostafazadeh, Phys. Rev. A 80, 032711 (2009). 10.1103/PhysRevA.80. 032711
28. A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38, L171 (2005). 10.1088/0305-4470/38/9/L03
29. K. O. Friedrichs, Commun. Pure Appl. Math. 1, 361 (1948). 10.1002/cpa.3160010404
30. U. Fano, Phys. Rev. 124, 1866 (1961). 10.1103/PhysRev.124.1866
31. P. W. Anderson, Phys. Rev. 124, 41 (1961). 10.1103/PhysRev.124.41
32. B. Piraux, R. Bhatt, and P. L. Knight, Phys. Rev. A 41, 6296 (1990). 10.1103/PhysRevA.41.6296
33. P. L. Knight, M. A. Lauder, and B. J. Dalton, Phys. Rep. 190, 1 (1990). 10.1016/0370-1573(90)90089-K
34. C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions (Wiley, New York, 1992).
35. A. G. Kofman, G. Kurizki, and B. Sherman, J. Mod. Opt. 41, 353 (1994). 10.1080/09500349414550381
36. P. Lambropoulos, G. M. Nikolopoulos, T. R. Nielsen, and S. Bay, Rep. Prog. Phys. 63, 455 (2000). 10.1088/0034-4885/63/4/201
37. G. D. Mahan, Many-Particle Physics (Plenum Press, New York, 1990), pp. 272-285.
38. M. Cini, Topics and Methods in Condensed-Matter Theory (Springer, Heidelberg, 2007), Chap., pp. 81-89.
39. J. W. Gadzuk and M. Plihal, Faraday Discuss. 117, 1 (2000). 10.1039/b008695i
40. N. Stefanou and A. Modinos, Phys. Rev. B 57, 12127 (1998) 10.1103/PhysRevB.57.12127
41. P. A. Orellana, M. L. Ladrón de Guevara, and F. Claro, Phys. Rev. B 70, 233315 (2004) 10.1103/PhysRevB.70.233315
42. L. Zhou, F. M. Hu, J. Lu, and C. P. Sun, Phys. Rev. A 74, 032102 (2006) 10.1103/PhysRevA.74.032102
43. E. Rufeil Fiori and H. M. Pastawski, Chem. Phys. Lett. 420, 35 (2006) 10.1016/j.cplett.2005.12.025
44. G.-B. Zhang, S.-J. Wang, and L. Li, Phys. Rev. B 74, 085106 (2006) 10.1103/PhysRevB.74.085106
45. A. V. Malyshev, P. A. Orellana, and F. Domínguez-Adame, Phys. Rev. B 74, 033308 (2006) 10.1103/PhysRevB.74.033308
46. P. Zedler, G. Schaller, G. Kiesslich, C. Emary, and T. Brandes, Phys. Rev. B 80, 045309 (2009). 10.1103/PhysRevB.80.045309
47. S. Tanaka, S. Garmon, and T. Petrosky, Phys. Rev. B 73, 115340 (2006) 10.1103/PhysRevB.73.115340
48. S. Tanaka, S. Garmon, G. Ordonez, and T. Petrosky, Phys. Rev. B 76, 153308 (2007) 10.1103/PhysRevB.76.153308
49. H. Nakamura, N. Hatano, S. Garmon, and T. Petrosky, Phys. Rev. Lett. 99, 210404 (2007). 10.1103/PhysRevLett.99.210404
50. T. Petrosky, I. Prigogine, and S. Tasaki, Physica A 173, 175 (1991) 10.1016/0378-4371(91)90257-D
51. G. Ordonez, T. Petrosky, and I. Prigogine, Phys. Rev. A 63, 052106 (2001). 10.1103/PhysRevA.63.052106
52. H. Nakazato, M. Namiki, and S. Pascazio, Int. J. Mod. Phys. B 10, 247 (1996) 10.1142/S0217979296000118
53. P. Facchi, H. Nakazato, and S. Pascazio, Phys. Rev. Lett. 86, 2699 (2001) 10.1103/PhysRevLett.86.2699
54. P. Facchi and S. Pascazio, in Quaderni di Fisica Teorica, edited by, S. Boffi, (Bibliopolis, Napoli, 1999).
55. A. G. Kofman and G. Kurizki, Nature (London) 405, 546 (2000) 10.1038/35014537
56. A. G. Kofman and G. Kurizki, Phys. Rev. Lett. 87, 270405 (2001) 10.1103/PhysRevLett.87.270405
57. X.-H. Wang, B.-Y. Gu, R. Wang, and H.-Q. Xu, Phys. Rev. Lett. 91, 113904 (2003) 10.1103/PhysRevLett.91.113904
58. J. Martorell, D. W. L. Sprung, W. van Dijk, and J. G. Muga, Phys. Rev. A 79, 062104 (2009). 10.1103/PhysRevA.79.062104
59. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, Phys. Rev. Lett. 80, 960 (1998) 10.1103/PhysRevLett.80.960
60. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, Phys. Rev. B 59, 15882 (1999) 10.1103/PhysRevB.59.15882
61. P. Chak, S. Pereira, and J. E. Sipe, Phys. Rev. B 73, 035105 (2006) 10.1103/PhysRevB.73.035105
62. L.-L. Lin, Z.-Y. Li, and B. Lin, Phys. Rev. B 72, 165330 (2005). 10.1103/PhysRevB.72.165330
63. Y. Xu, Y. Li, R. K. Lee, and A. Yariv, Phys. Rev. E 62, 7389 (2000). 10.1103/PhysRevE.62.7389
64. S. Longhi, Phys. Rev. A 74, 063826 (2006). 10.1103/PhysRevA.74.063826
65. S. Longhi, Eur. Phys. J. B 57, 45 (2007). 10.1140/epjb/e2007-00143-2
66. S. Longhi, Phys. Rev. Lett. 97, 110402 (2006) 10.1103/PhysRevLett.97. 110402
67. P. Biagioni, G. Della Valle, M. Ornigotti, M. Finazzi, L. Duo, P. Laporta, and S. Longhi, Opt. Express 16, 3762 (2008). 10.1364/OE.16.003762
68. A. Miroshnichenko, S. Flach, and Y. Kivshar, arXiv:0902.3014 (unpublished).
69. G. Sudarshan, in Field Theory, Quantization and Statistical Physics, edited by, E. Tirapegui, (Reidel, Dordrecht, 1981), pp. 237-245.
70. M. Miyamoto, Phys. Rev. A 72, 063405 (2005). 10.1103/PhysRevA.72.063405
71. For Im (Ea) =0, we recover the Hermitian limit of the FFA Hamiltonian. In this case a divergence of the resolvent for z→ E0, obtained from Eq. with V (E0) =0, corresponds to a bound state embedded into the continuum (see, e.g., Refs.) rather than to a spectral singularity of H.
72. In fact, if z were a real-valued root of Eq. outside the interval (E1, E2), according to Eq. one would have Σ (z) =Δ (z) because V (z) =0. Hence, Σ (z) turns out to be real valued. To satisfy Eq., the condition Im (Ea) =0 must be thus satisfied. This means that a real-valued root z of Eq. necessarily requires H to be Hermitian.
73. Since the divergence of G (z) at z= E0 on the continuous spectrum occurs when z approaches E0 solely from one side (either from the top or from the bottom) of the complex plane, E0 does not belong to the point spectrum of H; rather it is a spectral singularity. In fact, if E0 were an eigenvalue of H corresponding to a square-integrable eigenfunction | E0, denoting by | E0† the eigenfunction of the adjoint H† corresponding to the same eigenvalue and taking in Eq. |χ = | E0† and |φ = | E0, one would obtain Gχ,φ (z) = E0† | E0 / (z- E0). Since E0† | E0 is finite, it would then follow that Gχ,φ (z) should diverge when both z= E0 +i 0+ and z= E0 -i 0+.
74. For a featureless continuum, for which Δ (E) and V (E) are smooth functions of E, the resonance curve has a Lorentzian shape and the quantum system, initially prepared in state |a, would decay into the continuum following a (nearly) exponential decay law with lifetime 1/ [πV (Ea)] (Weisskopf-Wigner or Breit-Wigner approximation). See also the discussion in Sec..
75. We assume here that the spectrum of H is purely continuous, i.e., we assume that there are not bound states (outside or embedded in the continuum). In this case, for the Hermitian FFA model the decay of the survival probability P (t) is always complete.
76. The proper determination of the square root on the right-hand side of Eq. must be chosen such that Im [Σ (z=E±i 0+)] =πV (E), according to Eq..
77. F. Dǒgan, W. Kim, C. M. Blois, and F. Marsiglio, Phys. Rev. B 77, 195107 (2008). 10.1103/PhysRevB.77.195107
78. O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, Phys. Rev. Lett. 103, 030402 (2009). 10.1103/PhysRevLett.103.030402
79. More precisely, far from the lattice boundary a wave packet of the form cn (t) = □dkQ (k) exp [-ikn-iE (k) t], with spectrum Q (k) narrow at around k= k0 (0< k0 <π), propagates along the lattice with a group velocity vg =dn/dt=-2 κ0 sin k0 <0, i.e., the wave packet is incident onto the lattice boundary. Conversely, a wave packet formed by the superposition cn (t) = □dkQ (k) exp [ikn-iE (k) t], with spectrum Q (k) narrow at around k= k0, propagates along the lattice with a group velocity vg =2 κ0 sin k0 >0, i.e. it is reflected from the lattice edge.
80. At the coalescent point, the resolvent G (z) has a second-order pole at z=0.
81. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, Opt. Lett. 32, 2632 (2007) 10.1364/OL.32.002632
82. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, 103904 (2008) 10.1103/PhysRevLett.100.103904
83. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009). 10.1103/PhysRevLett.103.093902