Projection Operator Method and Related Problems

1. Projection Operator Methodand Related Problems Ochanomizu Univ. F. Shibata

2. Environment System （１） Brief historical survey （２） Reduced dynamics and the master equation of open quantum systems:M. Ban, S. Kitajima and F. S., Phys. Lett. A 374(2010) 2324. （３） Relaxation process of quantum system: B-K-S, Phys. Rev. A 82, (2010) 022111

3. (1) Brief historical survey of the method Damping theory W. Heitler (～1936) General formalism D and N .. Schrodinger picture (SP) R. Kubo Explicitly cited in: Time-Convolution (TC) S. Nakajima (1958) Transport, diffusion Heisenberg picture (HP) R. Zwanzig (1960) “Micro-Langevin” H. Mori (1965) Several work on Time-Convolutionless(TCL) M. Tokuyama -H. Mori (1976) Relaxation and decoherence S-Takahashi -Hashitsume (1977) “Micro-Langevin” Chaturvedi-S (1979) S- Arimitsu (1980) Uchiyama-S (1999) Cumulant expansion Expansion formulae SP & HP, TC & TCL R. Kubo (1963) van Kampen (1974) Relevant Books: 1) H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (2006, Oxford ) 2) F. S., T.Arimitsu, M.Ban and S.Kitajima, Physics of Quanta and Non-equilibrium Systems (2009, Univ. of Tokyo press, in Japanese)

4. (2) Reduced dynamics and the master equation of open quantum systems Phys. Lett. A 374 (2010) 2324 1. Reduced dynamics of an open quantum system Liouville-von Neumann equation super operator Formal solution

5. Initial density operator of the total system ： Unitary superoperator ： The reduced density operator of the relevant system ：

6. The reduced density operator of the relevant system ： ・・・ (1)

7. 2. The master equation for the reduced density operator We can obtain up to the second order with respect to the interaction Formal solution ・・・ (2) where the time ordering is to be done as indicated by the subscript quantity G which is different from the time ordering With respect to S.

8. The condition of the second order master equation to be exact is found by differentiating (1), It should be noted that the quantity G(t) can not be placed across the time-ordering symbol because of its time integral up to t .

9. The condition of the second order equation becomes exact is given by ・・・ (3) which can be cast into the statement: The final necessary and sufficient condition for the second order master equation becomes exact is that the system operators S(t)’s are commutable each other at different times.

10. 3. Reduced dynamics of the boson-detector model

11. The reduced density operator of the propagating particle ： ・・・ (4) (4)

12. The reduced density operator in the interaction picture ：

13. References 1) R.P. Feynman, F.L. Vernon Jr., Ann. Phys. 24 (1963) 118. 2) A.O. Caldeira, A.J. Leggett, Physica A 121 (1983) 587. 3) H.-P. Breuer, F. Petruccione, Phys. Rev. A 63 (2001) 032102. 4) H.-P. Breuer, A. Ma, F. Petruccione, LANL, quant-ph/0209153, 2002; in: A. Leggett, B. Ruggiero, P. Silvestrini(Eds.), Quantum Computing and Quantum Bits in Mesoscopic Systems, Kluwer, New York, 2004, pp. 263-271. 5) A. Ishizaki, Y. Tanimura, Chem. Phys. 347 (2008) 185.

14. (3) Relaxation process of quantum system: stochastic Liouville equation and initial correlation Phys. Rev. A 82, (2010) 022111 • Stochastic Liouville equation ： • (A) Time-evolution by stochastic Hamiltonian ((Time-evolution equation)) Formal solution Density operator averaged over the stochastic process

15. Joint probability When there is no initial correlation between the quantum system and stochastic process, we obtain the time-convolutionless (TCL) quantum master equation

16. (B) Time-evolution of joint density operator Time-evolution equation of the transition probability condition ：　Probability vector Time-evolution of the probability vector

17. Time-evolution of the joint density operator ・・・ (5) Matrix form ： The interaction picture

18. The initial joint state The formal solution

19. The differential operator The stochastic Liouville equation The reduced density operator The probability density function

20. 2.Derivation from the quantum master equation ： (A) General consideration The whole system is composed of the relevant quantum system and an interaction mode under the influence of a narrowing limit environment. Phys. Rev. A 82, (2010) 022111

21. The time evolution of the density operator with the Lindblad operator, Taking

22. (B) Discrete stochastic variable The quantum master equation ：

23. (C) Continuous stochastic variable The quantum master equation The density operator

24. The differential equation for the system operator

25. The differential equation for the system operator The time evolution equation of the joint density operator

26. 3.Reduced dynamics with initial correlation ： (A) General formulation

27. A qubit state is represented by The characteristic function of the stochastic variable The coherence of a qubit is characterized by

28. For a two-qubit system A and B, The two-qubit Hamiltonian in the interaction picture： The reduced density operator of the two-qubit system：

30. (B) Gauss-Markov process The time evolution of coherence for the Gauss-Markov fluctuation for the slow (a) and the fast (b) modulation. Phys. Rev. A 82, (2010) 022111

31. The time evolution of concurrence for the Gauss-Markov process for the slow (a) and the fast (b) modulation. Phys. Rev. A 82, (2010) 022111

32. The time evolution of the coherence (a) and the concurrence (b) for the two-state-jump Markov process. Phys. Rev. A 82, (2010) 022111

33. 4.Concluding remarks We have systematically developed a theory of stochastic Liouville equation and the phenomenological feature of the theory is examined on the basis of the microscopic ground. The coherence and the entanglement of the quantum system are induced by the initial correlation between the relevant system and the environment. In the presence of the initial correlation, the process becomes non-stationary and is essential for the creation of the coherence and the entanglement.

34. Appendix ： Perturbative expansion for master equation The projection operator

35. References 1) R.Kubo, J. Math. Phys. 4 (1962) 174. 2) R. Kubo, Adv. Chem. Phys. 15 (1969) 101. 3) Y. Tanimura, J. Phys. Soc. Jpn 75 (2006) 082001 and references therein. Initial correlation by TCL equation ： 4) H.-P. Breuer, B. Kappler and F. Petryccione, Ann. of Phys. 291 (2001) 36. Initial correlation by other view point ： 5) P. Stelmachovic and V. Buzek, Phys. Rev. A 64 (2001) 062106. 6) N. Boulant, J. Emerson, T. F. Havel and D. G. Cory, J. Chem. Phys. 121 (2004) 2955. 7) T. F. Jordan, A. Shaji and E. C. G. Sudarshan, Phys. Rev. A 70 (2004) 052110. Quantum mechanical two-state-jump model ： 8) T. Arimitsu, M. Ban and F. S., Physica A 123 (1984) 131. 9) M. Ban, S. Kitajima, K. Maruyama and F.S., Phys. Lett. A 372 (2008) 351. Quantum mechanical Gaussian model ： 10) Y. Hamano and F. S., J. Phys. Soc. Jpn., 51 (1982) 1727.