Transcript Projection Operator Method and Related Problems

Ochanomizu Univ.
F. Shibata
（１） Brief historical survey
Environment
System
（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
Damping theory
General formalism
D and N
Transport, diffusion
W. Heitler (～1936)
..
Schrodinger picture (SP)
R. Kubo
Explicitly
cited in:
Time-Convolution (TC)
S. Nakajima (1958)
R. Zwanzig (1960)
“Micro-Langevin” H. Mori (1965)
Several work on
Time-Convolutionless(TCL)
Relaxation and
decoherence
Cumulant expansion
Heisenberg picture (HP)
S-Takahashi
-Hashitsume (1977)
Expansion formulae
SP & HP, TC & TCL
R. Kubo (1963)
van Kampen (1974)
M. Tokuyama
“Micro-Langevin” -H. Mori (1976)
Chaturvedi-S (1979)
S- Arimitsu (1980)
Uchiyama-S (1999)
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)
Phys. Lett. A 374 (2010) 2324
1. Reduced dynamics of an open quantum system
Liouville-von Neumann equation
Formal solution
super operator
Initial density operator of the total system ：
Unitary superoperator ：
The reduced density operator of the relevant system ：
The reduced density operator of the relevant system ：
・・・ (1)
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.
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 .
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.
3. Reduced dynamics of the boson-detector model
The reduced density operator of the propagating particle ：
・・・ (4)
(4)
The reduced density operator in the interaction picture ：
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.
Phys. Rev. A 82, (2010) 022111
1. Stochastic Liouville equation ：
(A) Time-evolution by stochastic Hamiltonian
((Time-evolution equation))
Formal solution
Density operator averaged over the stochastic process
Joint probability
When there is no initial correlation between the quantum system and
stochastic process, we obtain the time-convolutionless (TCL)
quantum master equation
(B) Time-evolution of joint density operator
Time-evolution equation of the transition probability
condition
： Probability vector
Time-evolution of the probability vector
Time-evolution of the joint density operator
・・・ (5)
Matrix form ：
The interaction picture
The initial joint state
The formal solution
The differential operator
The stochastic Liouville equation
The reduced density operator
The probability density function
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
The time evolution of the density operator
with the Lindblad operator,
Taking
(B) Discrete stochastic variable
The quantum master equation ：
(C) Continuous stochastic variable
The quantum master equation
The density operator
The differential equation for the system operator
The differential equation for the system operator
The time evolution equation of the joint density operator
3. Reduced dynamics with initial correlation ：
(A) General formulation
A qubit state is represented by
The characteristic function of the stochastic variable
The coherence of a qubit is characterized by
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：
(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
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
The time evolution of the coherence (a) and the concurrence (b)
for the two-state-jump Markov process.
Phys. Rev. A 82, (2010) 022111
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 nonstationary and is essential for the creation of the coherence and the
entanglement.
Appendix ： Perturbative expansion for master equation
The projection operator
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.