Decoherence and Classical dynamics in Markovian Quantum Open Systems

Decoherence and Classical dynamics in Markovian

Quantum Open Systems

O. Brodier(1) A. M. Ozorio de Almeida(2)

(1)M.P.I.P.K.S. Dresden, ALLEMAGNE

(2)C.B.P.F. Rio de Janeiro, BRASIL

Phase Space Representation

x

p

Classical propagation

Quantum propagation

dQQ

qQ

qpQi

qp tt 2ˆ

2exp,W

,P,P 0 ttt qpqp Liouville propagation:

Wigner function:

S. Habib, K. Shizume, W. H. ZurekPhys.Rev.Lett. 80 (1998) 4361-4365

Separation time

Z. P. Karkuszewski, J. Zakrzewski, W. H. ZurekPhys. Rev. A 65, 042113 (2002)

texpp

p

tH ln1

Decoherence due to coupling to environment:

effect on separation time?

Characteristic function

xxx di

tqpt Wexp,

xxξxξξ dSi

tqpt 0W,exp,

01q̂p̂

ξξξ nq

mp

tmnmn

nmn

i

Momenta:

Semiclassical analysis:

LLLLLLH

i

tˆˆˆ

2

1ˆˆˆ

2

1ˆˆˆˆ,ˆˆ

Markovian Quantum Open System:

ξξlξξξ St

SH

t

S 2,

Hamilton Jacobi:

Result

xxxxxx dtbdtq 00

22 W,W,q̂

texp, tb x

qvm

pqpH

2,

2

m

v with

Conclusion

• analysis of decoherence of a Markovian Quantum Open System

• Dynamical dependence of decoherence

A. M. Ozorio de Almeida, O. Brodier in press Ann. Phys. (2006)

W.K.B.

qSq

i

exp

qdq

dSp

q

qqSqq ,

iexp

q

q

qqqq

Spp

,,2

Entanglement dynamics

A

B

A

B

tc k

Measuring environment → Pure entanglement

Entanglement of a mixture

tc ̂

k

kk ttN

t 1ˆ

tctcN

tck

k ˆ1

Setup 1

01 cttc

A

B

0

011

10

A

0

0

11

01

B

11

10

01

00

:1 21 tpp

:1 tp

:2 tp t1

Setup 2

22

BAi

A

B

i

22

BAi

22

BA

22

BA

11

10

01

00

tctctttc ˆ2021 211

22

tt2

21

Conclusion

• For this example there exists an optimal experimental setup which gives the exact entanglement measure.

• General Prescription for Measuring Environment?