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?