Transfer of entanglement from a Gaussian field to

remote qubits

Myungshik KimQueen’s University,

Belfast

UniMilano 14 December 2004

In collaboration with

• Mauro Paternostro

Wonmin Son

Helen McAneney, Ebyung Park, Pamela Wilson, Mark Tame, Jingak Jang

Why entanglement in two remote qubits?

Quantum teleportation to send

Quantum repeater

Distributed quantum computation

NON LOCAL CNOT Eisert,Jacobs, Papadopoulos,Plenio,

PRA 62, 52317 (2000); Collins,Linden,Popescu, PRA64, 32302 (2000)

Motivation 1

Why Gaussian?Gaussians are as natural as orange juice and sunshine

Motivation 2

Contents

• Generation of entanglement between two quibits in a cavity

• Entanglement transfer through local reservoirs

• Remarks

Generation of entanglement

• Two atoms in a cavity of a single-mode field

egtcggtbgeta

t

eg

,,0)(,,1)(,,0)(

is system total the,at then

0cavity and in are atomsInitially

++

• When the cavity is in a thermal field

state ground itsin other the

state excited itsin initially atoms one

numberphoton average :10=n

MSK,Lee, Ahn, Knight., PRA 65, 040101(R) (2002)

Generation of entanglement

• Two atoms in a cavity of a single-mode field

Interaction time

entangleme

nt

eg Initial

• Even when a many-mode thermal field is concerned, entanglement is generated. Braun, PRL 89, 277901 (2002)

Generation of entanglement

• Two atoms in a cavity of a single-mode field

ggin atoms &

state thermalain cavity

Initially 1=n

1.0=n

Generation of entanglement

• Two atoms in a cavity of a single-mode field

egtcgetbggta

t

gg

,,0)(,,0)(,,1)(

in is system total the,at then,,1cavity -

happens. nothingthen ,0cavity -

and in are atomsInitially

Erasure of “which-path information” effectively entangles two qubits (Bose al,(‘99), Browne et al, (‘03)).

Duan et al. Nature 414, 413 (2001)

√ egge

2

1

• via an entangled system

entangler

eggegg 2

1001001

2

1

Entanglement transfer through Gaussian fields

local environment a

local environment b

Q1 Q2

driving field: broadband squeezed field

qubit-bosonic mode interaction: 1111 ˆˆˆˆˆ aaH aa

Paternostro, Son, Kim , PRL 92, 197901 (2004)Kraus & Cirac, PRL92, 013602 (2004)

(02). 1739 49, JMO qubits”, to variablescontinuous

fromer ent transf“Entanglem MSK, Son, W.

,tanhcosh

vacuumsqueezed mode- twoPure

0,

1

,

n

ba

n

bannrr

Rabi oscillation depends on the photon number

t

p

Questions• Qubits are located in respective cavities• When the channel is mixed• Entanglement in the steady state

– related to minimum control• To include spontaneous emission

Evolution of the qubitsConditions

• driving-field carrier frequency resonant with local modes and its bandwidth is larger than the cavity decay rate

Then we use

• second-order perturbation theory

• Markov approximation

To get the master equation for the field inside the cavities

Adiabatic elimination of the local field modes: 21, ba

4

1,12121212 ,ˆˆ

2

1ˆˆˆ

OOOOL Dt eff

Find the evolution of the qubits

214213

212211

ˆˆ ˆˆ

ˆˆ ˆˆ

yx

yx

11

11

OO

OO

BC

CAD

4x4 Kossakowski matrix

Benatti et al.,PRL(03), Ficek&Tanas, Phys.Rep.(02)

Entanglement condition

Consider the Gaussian channel

with its variance matrix

mc

mc

cn

cn

00

00

00

00

+ qubits prepared in the ground state THEN 2)1)(1( cmn

0cm,n ),(

..still a bit obscure...BUT: consider the uncertainty principle for the driving field. This can be written as

then it is

0)1)(1( 22 Tcmn

Entanglement condition

0)1)(1( 22 Tcmn

Two remote qubits can be entangled , at some instant of time of their interaction with a correlated Gaussian channel, if and only if the channel itself is entangled.

Example: 1= 2 , n=m and solve the ME. Then

c=1.58

n=2.4

c=1.804

c=2.18

n=2.4

c=2.18

c=1.804

c=1.58

initial state , 12

gg

c=1.58

n=2.4

c=1.804

c=2.18

n=2.4

c=2.18

c=1.804

c=1.58

initial state 12

ee

11)1(1

)( 222 nnnn

ncc ss

Channel entanglement condition

Steady state entanglement

The two qubits are entangled at their steady-state if and only if c > css

For 1= 2 , n=m it is

een

ngg

n

n

12

1

12

statesteady in the are qubits the

entangled,maximally and pure is channel When the

Kraus & Cirac, PRL92, 013602 (2004)Paternostro, Son, Kim , PRL 92, 197901 (2004)

Entanglement transfer through Gaussian fields

local environment a

local environment b

Q1 Q2

driving field: broadband squeezed field

qubit-bosonic mode interaction: 1111 ˆˆˆˆˆ aaH aa

Paternostro, Son, Kim , PRL 92, 197901 (2004)Kraus & Cirac, PRL92, 013602 (2004)

0

,

1

,,tanhcosh

vacuumsqueezed mode- twoPure

nba

n

bannrr

Rabi oscillation depends on the photon number

t

p

Questions• Qubits are located in respective cavities • When the channel is mixed • Entanglement in the steady state

– related to minimum control • To include spontaneous emission

Summary

• Entanglement transfer from a Gaussian field to a qubit system

• Entangled remote local reservoirs suggested

• Entanglement condition obtained