1. -From Introduction to Outlook – Ochanomizu University F. Shibata

2. （１） Introduction –Outlook of several theoretical methods （２） Projection operator formalism F. S., T. Arimitsu, M. Ban and S. Kitajima, “Physics of Quanta and Non-equilibrium Systems”, chap. 3, (University of Tokyo Press, 2009, in Japanese) H.-P. Breuer and F. Petruccione, “The Theory of Open Quantum Systems” (2006, Oxford ) （３） Spin relaxation ( i ) F.S., C. Uchiyama, J. Phys. Soc. Jpn., 62 (1993) 381. “Rigorous solution to nonlinear spin relaxation process” （ 4 ） Spin relaxation ( ii ) Y. Hamano, F.S., J. Phys. Soc. Jpn., 51 (1982) 1727. “Theory of spin relaxation for arbitrary time scale”

3. （ 5 ） Low field resonance : exact solution F.S.-I. Sato, Physica A 143 (1987) 468. “Theory of low field resonance and relaxation Ⅰ ” （６） Micro-Laser theory C. Uchiyama, F. S., J. Phys. Soc. Jpn., 69 (2000) 2829. “Self-organized formation of atomic coherence via photon exchange in a coupled atom-photon system”. （ 7 ） Decoherence control S. Kitajima, M. Ban and F.S., J. Phys. B 43 (2010) 135504. “Theory of decoherence control in a fluctuating environment” （ 8 ） Outlook

4. (1) Introduction – Outline of several theoretical methods Damping theory ： (a) Projection operator method (b) Schrodinger picture versus Heisenberg picture (c) Time-convolution (TC) type versus Time-convolutionless (TCL) type Path integral theory ： Feynman – Vernon, Caldeira – Leggett Non-equilibrium Green’s function ： Schwinger - Kerdysh ・

5. (2) Projection operator formalism Basic equations in Interaction picture (Schrodinger's view) ・・ ・・ where Hamiltonian Liouville-von Neumann equation Physics of Quanta and Non-equilibrium Systems”, chap. 3, (University of Tokyo Press, 2009, in Japanese)

6. Time convolutio(TC) type equation

7. where Time convolutionless(TCL) type equation

8. TC type formula Perturbation expansion formulae

9. TCL type formula

10. where

11. Heisenberg picture Mori equation Projection operator where

12. (3) Spin relaxation ( Ⅰ ) ： Rigorous Solution to Nonlinear Spin Relaxation Process J. Phys. Soc. Jpn. 62 (1993) 381 1. Preliminaries Time evolution of the nonlinear spin relaxation process

13. C-number equation for a normally mapped (quasi-)probability density where

14. 2. Method of solution For, where with ・・・ (1)

15. where An exact solution is given by the form of a continued fraction:

16. 3. Averages and Fluctuations The average of the spin operator The second moments of the spin operator

17. J. Phys. Soc. Jpn. 62 (1993) 381

18. (4) Spin relaxation ( Ⅱ ) ： Theory of Spin Relaxation for Arbitrary Time Scale J. Phys. Soc. Jpn. 51 (1982) 1727 1. Reduced Density Operator Hamiltonian

19. Time evolution of a reduced density operator for the relevant system where

20. In the Schrodinger picture ・ The moment equations

21. 2. Longitudinal Relaxation solution

22. J. Phys. Soc. Jpn. 51 (1982) 1727

23. 3. Transverse Relaxation solution

24. J. Phys. Soc. Jpn. 51 (1982) 1727

25. 4. Comparison with Stochastic Theory

26. References 1)F. S., Y. Takahashi and N. Hashitsume, J. Stat. Phys. 17 (1977) 171. 2)S. Chaturvedi and F. S., Z. Phys. B35 (1979) 297. 3)F. S. and T. Arimitsu, J. Phys. Soc. Jpn., 49 (1980) 891.

27. (5)Low field resonance : exact results Theory of low field resonance and relaxation Ⅰ Physica 143A (1987) 468 1. Basic formulation Hamiltonian Liouville-von Neumann equation in the interaction picture The projection operator

28. The average of an arbitrary operator The time-convolution (TC) equation

29. A basic equation The “self-energy”

30. A power spectrum The longitudinal spectrum The transverse spectrum

31. 2. Two-state jump Markoff process Longitudinal relaxation Transverse relaxation Non-adiabatic case

32. The time evolution of the longitudinal function for the low field; (a) for overall and (b) for the short time. The process is the two-state-jump. Physica 143A (1987) 468

33. The time evolution of the transverse function for the low field; (a) for overall and (b) for the short time. The process is the two-state-jump. Physica 143A (1987) 468

34. 3. Gaussian-Markoffian process 3.1 The longitudinal relation : Non-adiabatic case

36. 3.2 The transverse relation : Non-adiabatic case

37. Physica 143A (1987) 468

39. References 1)R. Kubo and T. Toyabe, in: Proc. of the XIVth Colloque Ampere Ljubljana 1966, Magnetic Resonance and Relaxation, R. Blinc, ed. (North-Holland, Amsterdam, 1967) p.810. 2)R. S. Hayano, Y. J. Uemura, J. Imazato, N. Nishida, T. Yamazaki and R. Kubo, Phys. Rev. B20 (1979) 850. 3)Y. J. Uemura and T. Yamazaki, Physica 109 & 110B (1982) 1915; and references cited therein. Y. J. Uemura, Doctor Thesis, University of Tokyo (1982). 4) E. Torikai, A. Ito, Y. Takeda, K. Nagamine, K. Nishiyama, Y. Syono and H. Takei, Solid State Commun. 58 (1986) 839.

40. (6) Micro-Laser theory : Self-organized formation of atomic coherence via photon exchange in a coupled atom-photon system via photon exchange in a coupled atom-photon system J. Phys. Soc. Jpn., 69 (2000) 2829 1. Basic equations Hamiltonian Liouville-von Neumann equation for a reduced density operator

41. 2. Method of solution The normally mapped (quasi-)probability density

42. J. Phys. Soc. Jpn., 69 (2000) 2829

50. (7) Decoherence control : Theory of decoherence control in a fluctuating environment Theory of decoherence control in a fluctuating environment J. Phys. B 43 (2010) 135504 1. Preliminaries The characteristic function

51. Initial row vector The conditional probability where The characteristic function

52. Stochastic relaxation under control pulses Extending to include the effect of control pi pulses The Liouville-von Neumann equation in the interaction picture

53. The density operator for a relevant system where with

54. where

55. 2. Two-state jump Markov process The conditional probability

56. Time evolution of the characteristic function for the two- state-jump process for various values of the parameters. J. Phys. B 43 (2010) 135504

57. The time evolution of the characteristic function for the two-state-jump process; (a) natural decay and (b) with pulses. J. Phys. B 43 (2010) 135504

58. The time evolution of the characteristic function for the two-state-jump process: (a) natural decay and (b) with pulses. J. Phys. B 43 (2010) 135504

59. 3. Gauss-Markov process The characteristic function Fokker-Planck-type equation The initial condition

60. 3.1 Stochastic characteristic function under control pulses ---stationary process--- In the limit of equal separation time of pulses, the result coincides with the previous one ： M. Ban, F. S. and S. Kitajima, JPB 40 (2007) S229.

61. 3.2 Non-stationary Gauss-Markov process under control pulses the initial condition.

63. The time evolution of the characteristic function for the Gauss- Markov process with pulses. J. Phys. B 43 (2010) 135504

64. The time evolution of the characteristic function for the stationary and non-stationary Gauss-Markov processes. J. Phys. B 43 (2010) 135504

65. The time evolution of the characteristic function for the Gauss-Markov process; (a) natural decay and (b) with pulses. J. Phys. B 43 (2010) 135504

66. (8) Outlook Decoherence of qubits ～ Spin relaxation Quantum communication Quantum processing Decoherence of photons ～ Laser dissipation Decoherence control Solvable model －－－ Lower excitations New type of expansion theory