General conditions for quantum adiabatic evolution

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 80, Issue 1, 2009, Pages

  Comparat, Daniel ^a  

^a LABORATOIRE AIMÉ COTTON   (France)

Author keywords

[No Author keywords available]

Indexed keywords

ADIABATIC EVOLUTION; ADIABATICITY; EIGEN STATE; PHYSICAL SYSTEMS; QUANTUM ADIABATIC THEOREM; SINUSOIDAL FUNCTIONS; SPECTRAL GAP;

HAMILTONIANS;

EID: 68549092696     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.80.012106     Document Type: Article

References (62)

1. Keith J. Laidler, Can. J. Chem. 72, 936 (1994). 10.1139/v94-121
2. M. Born and V. Fock, Z. Phys. 51, 165 (1928). 10.1007/BF01343193
3. H. Nakamura, Nonadiabatic Transition: Concepts, Basic Theories and Applications (World Scientific, Singapore, 2002).
4. S. Teufel, Adiabatic Perturbation Theory in Quantum Dynamics, Lecture Notes in Mathematics 1821 (Springer-Verlag, Berlin, 2003).
5. J. Du, L. Hu, Y. Wang, J. Wu, M. Zhao, and D. Suter, Phys. Rev. Lett. 101, 060403 (2008). 10.1103/PhysRevLett.101.060403
6. J. Schwinger, Phys. Rev. 51, 648 (1937). 10.1103/PhysRev.51.648
7. A. Messiah, Mécanique Quantique (Dunod, Paris, 1959).
8. m| = m| H |n En - Em comes from the time derivative of m|n = δmn and of m| H |n = En δmn, where δmk is Kronecker's delta.
9. M. S. Sarandy, L.-A. Wu, and D. A. Lidar, Quantum Inf. Process. 3, 331 (2004). 10.1007/s11128-004-7712-7
10. Z. Wu and H. Yang, Phys. Rev. A 72, 012114 (2005). 10.1103/PhysRevA.72. 012114
11. D. M. Tong, K. Singh, L. C. Kwek, and C. H. Oh, Phys. Rev. Lett. 95, 110407 (2005). 10.1103/PhysRevLett.95.110407
12. S. Duki, H. Mathur, and O. Narayan, Phys. Rev. Lett. 97, 128901 (2006). 10.1103/PhysRevLett.97.128901
13. J. Ma, Y. Zhang, E. Wang, and B. Wu, Phys. Rev. Lett. 97, 128902 (2006). 10.1103/PhysRevLett.97.128902
14. K.-P. Marzlin and B. C. Sanders, Phys. Rev. Lett. 97, 128903 (2006). 10.1103/PhysRevLett.97.128903
15. L. I. Schiff, Quantum Mechanics (McGraw-Hill Book Co., New York, 1949).
16. B. H. Bransden and C. J. Joachain, Introduction to Quantum Mechanics (Addison-Wesley, Reading, MA, 1989).
17. A. Joye, G. Mileti, and C.-E. Pfister, Phys. Rev. A 44, 4280 (1991). 10.1103/PhysRevA.44.4280
18. K. Nakamura and S. A. Rice, Phys. Rev. A 49, R2217 (1994). 10.1103/PhysRevA.49.R2217
19. S. Stenholm, Quantum Dynamics of Simple Systems, edited by, G. L. Oppo, S. M. Barnett, E. Riis, and, M. Wilkenson, (Springer, New York, 1996), p. 267.
20. M. Wilkinson and M. A. Morgan, Phys. Rev. A 61, 062104 (2000). 10.1103/PhysRevA.61.062104
21. M.-Y. Ye, X.-F. Zhou, Y.-S. Zhang, and G.-C. Guo (unpublished).
22. R. MacKenzie, E. Marcotte, and H. Paquette, Phys. Rev. A 73, 042104 (2006). 10.1103/PhysRevA.73.042104
23. T. Vértesi and R. Englman, Phys. Lett. A 353, 11 (2006). 10.1016/j.physleta.2005.12.052
24. D. M. Tong, K. Singh, L. C. Kwek, and C. H. Oh, Phys. Rev. Lett. 98, 150402 (2007). 10.1103/PhysRevLett.98.150402
25. M.-Y. Ye, X.-F. Zhou, Y.-S. Zhang, and G.-C. Guo, Phys. Lett. A 368, 18 (2007). 10.1016/j.physleta.2007.03.056
26. D. M. Tong, K. Singh, L. C. Kwek, X. J. Fan, and C. H. Oh, Phys. Lett. A 339, 288 (R) (2005). 10.1016/j.physleta.2005.03.043
27. R. MacKenzie, A. Morin-Duchesne, H. Paquette, and J. Pinel, Phys. Rev. A 76, 044102 (2007). 10.1103/PhysRevA.76.044102
28. Z. Wei and M. Ying, Phys. Rev. A 76, 024304 (2007). 10.1103/PhysRevA.76. 024304
29. Y. Zhao, Phys. Rev. A 77, 032109 (2008). 10.1103/PhysRevA.77.032109
30. J.-D. Wu, M.-S. Zhao, J.-L. Chen, and Y.-D. Zhang, Phys. Rev. A 77, 062114 (2008). 10.1103/PhysRevA.77.062114
31. G. Rigolin, G. Ortiz, and V. H. Ponce, Phys. Rev. A 78, 052508 (2008). 10.1103/PhysRevA.78.052508
32. K.-P. Marzlin and B. C. Sanders, Phys. Rev. Lett. 93, 160408 (2004). 10.1103/PhysRevLett.93.160408
33. S. Jansen, M.-B. Ruskai, and R. Seiler, J. Math. Phys. 48, 102111 (2007). 10.1063/1.2798382
34. T. Kato, J. Phys. Soc. Jpn. 5, 435 (1950). 10.1143/JPSJ.5.435
35. M. V. Berry, Proc. R. Soc. London, Ser. A 392, 45 (1984). 10.1098/rspa.1984.0023
36. J. E. Avron and A. Elgart, Commun. Math. Phys. 203, 445 (1999). 10.1007/s002200050620
37. M. S. Sarandy and D. A. Lidar, Phys. Rev. A 71, 012331 (2005). 10.1103/PhysRevA.71.012331
38. The path parametrization is here s (t) =t/T=εt[0,1] and Hε (t) =H [s (t)], where T is the evolution time. The Schrödinger equation is then i
39. E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, and D. Preda, Science 292, 472 (2001). 10.1126/science.1057726
40. J. Liu and L.-B. Fu, Phys. Lett. A 370, 17 (2007). 10.1016/j.physleta. 2007.05.021
41. E. E. Nikitin, Handbooks of Atomic, Molecular, and Optical Physics (Springer, New York, 2006), Chap..
42. In this paper, we use x = x† x = ∑ i=i N | xi | 2 and x 1 = ∑ i=1 N | xi |. They verify x 2 ≤ x 1 ≤N x 2. For the matrix norm M = maxx≠0 Mx x.
43. G. Schaller, S. Mostame, and R. Schützhold, Phys. Rev. A 73, 062307 (2006). 10.1103/PhysRevA.73.062307
44. For real Hamiltonian H, |m is real, so the (Pancharatnam's phase) arg m| is zero and d dt m|m =2 m| m =0.
45. Interestingly enough, very similar considerations have been used by Born and Fock in their seminal paper.
46. J. Paldus, Handbooks of Atomic, Molecular, and Optical Physics (Springer, New York, 2006), Chap..
47. Simple condition (similar to Weyl's theorem) for the smallness of δ′-1 off-diagonal terms exists. For instance, by choosing H0 as the diagonal part of H′ in the Brillouin-Wigner equation, we can replace conditions 13 essentially by H (t) 1 ΔE (t) ≈ Δ En (t)
48. Y. Kayanuma, Phys. Rev. A 50, 843 (1994). 10.1103/PhysRevA.50.843
49. Interestingly enough, the reverse case, namely, the diabatic limit (p1 ≈1) can lead (for instance when α1)
50. S. Giller, Acta Phys. Pol. B 35, 551 (2004).
51. M. H. S. Amin, Phys. Rev. Lett. 102, 220401 (2009). 10.1103/PhysRevLett. 102.220401
52. More practical definition of a nonoscillating Hamiltonian than based on the small number of monotonicity change of the adiabatic Hamiltonian H′ would be useful. We conjecture (and hope that someone could demonstrate it) that real Hamiltonians containing sum, product, multiplication, or division of composition of (real) exponential or polynomial functions are of this type. A possible clue for this proof may be based on iterative use of Eq. 6 with the iterative U″, P″, U (3), P (3),... matrices converging toward the wanted diagonalization, as done in the Jacobi algorithm which iterates the N=2 case. During each Jacobi step, the P (m) elements are still of the sum, product, multiplication, or division of composition of (real) exponential or polynomial types. The exponent of the polynomial functions grows, as well as the number of monoticity changes, but slowly enough to be always bounded (they never reach infinity due to the isolated zero theorem).
53. V. M. Akulin, Coherent Dynamics of Complex Quantum Systems (Springer, New York, 2006).
54. M. V. Berry, J. Phys. A 17, 1225 (1984). 10.1088/0305-4470/17/6/018
55. A. E. Allahverdyan and T. M. Nieuwenhuizen, Phys. Rev. E 71, 046107 (2005). 10.1103/PhysRevE.71.046107
56. Y. Rezek and R. Kosloff, New J. Phys. 8, 83 (2006). 10.1088/1367-2630/8/ 5/083
57. I. C. F. Ipsen, Linear Algebr. Appl. 358, 239 (2003). 10.1016/S0024-3795(01)00571-7
58. X. Chen and W. Li, J. Comput. Appl. Math. 196, 338 (2006). 10.1016/j.cam.2005.08.019
59. W. Li and W. Sun, SIAM J. Matrix Anal. Appl. 29, 643 (2007). 10.1137/060648969
60. We wrote En′ 1 simply as En′.
61. J. Roland and N. J. Cerf, Phys. Rev. A 65, 042308 (2002). 10.1103/PhysRevA.65.042308
62. M. Grifoni and P. Hänggi, Phys. Rep. 304, 229 (1998). 10.1016/S0370-1573(98)00022-2