PROPOSAL OF A UNIVERSAL QUANTUM COPYING

MACHINE IN CAVITY QED• Joanna Gonzalez• Miguel Orszag• Sergio Dagach• Facultad de Física• Pontificia Universidad Católica de Chile

Quantum Optics IICOZUMEL-MEXICO

The No-Cloning Theorem(Wooters and Zurek,Nature299,802(1982)) showed that it is not possible to construct a device that will produce an exact copy of an arbitrary quantum state.This Theorem is an unexpected quantum effect due to the linearity of Quantum Mechanics, as opposed to Classical Physics, where the copying Process presents no difficulties, and this represents the most significant difference between Classical and Quantum Information.Thus, an operation like:

Thus, an operation like:

0 ´a b x a b x

Q Q

Is not possible, with:

a

0b

xQ

´x

Q

=INPUT QUBIT

=initial state of cloner

=Blank copy=final stateof cloner

Because of this Theorem, scientist ignored the subject up to 1996 when Buzek and Hillery (V.Buzek,M.Hillery,Phys.Rev.A,54,1844(1996) proposed the Universal Quantum Copying Machine(UQCM)-that produced two imperfect copies from an original qubit, the quality of which was independent of the input state.

UNIVERSAL QUANTUM COPYING MACHINE BASIS

2 13 32 13 3

,

12

56

I

I

I

B A A

B A A

B BlankCopy

A A AUXILLIARY QUBITS

F

�

The quality of the copy is measured through the FIDELITY

copy ideal

ideal input

F

In the present work, we propose a protocol that produces 2 copies from an input state, with Fidelity 5

6F

In the context of Cavity QED, in which the information is encoded in the electronic levels of Rb atoms, that interact with two Nb high Q cavities.SOME PREVIOUS BACKGROUND TO THE PROPOSALConsider a two level atom that is prepared in a superposition state , using the Microwave pulses in a Ramsey Zone, with frequency r

Near the e(excited)-g(ground) transition. It generates superpositions

cos sin

sin cos

( )

i

i

r eg

e e e g

g e e g

where

t

Depends on the interaction timeIs prop. detuning

On the other hand, the atom-field interaction is described by the Jaynes Cummings Hamiltonian

0

0

1( ) ( ),.2 2 2

. ,2

egJ C zH a a a a

Coupling constant

The atom-field state evolves like

0 0

0 0

,0 ( ) cos( ) ,0 sin( ) ,12 2

,1 ( ) cos( ) ,1 sin( ) ,02 2

e

g

e t t e t g

g t t g t e

For example, for

0

0

1 1,0 ,0 ,1 , , , , , , ,1 ,0 ,12 2 2

,0 ,1 , , , , , , , , , , ,1 ,0

t e e g g e g

t e g g e

Now, consider an external Classical pulse, interacting with the atom

( )ext gH t

We use the dressed state basis that diagonalizes the J-C Hamiltonian:

2 2

, ) sin , 1 cos ,

, ) cos , 1 sin ,

cos 24 ( 1)

n n

n n

n

n g n e n

n g n e n

n

eg ,

The Energies of the dressed states are

)1(42

)21( 22 nnE

In the limit0

, , ............ .. .. ......... ,1 ,1)

, , 1 .......... .. .. ....... ,0 ,1

n e n in this case e

n g n in this case g

Consider the external field in resonance with the (+,1)-(-,0)Transition, that is

)cos()( ttf sg Where f(t) is some smooth function of time to represent the pulse shape, with(in the dispersive case)

201 3

eg

nns EE

The above Hamiltonian has been studied by several authors (Domokos et al;Giovannetti et al) and arrive to the conclusion thatFor a suitable pulse, a C-NOT gate can be achieved, where the photonNumber (0 or 1) is the control and the atom the target

The mechanism of the above C-NOT gate that forbids, for example the (g,0>--(e,0> transition is the Stark Effect, caused by one photon in the cavity. In order to resolve these two transitions, we have to make sure that

int

1t

g,0or -,-1g,0or -,-1

e,0or ,0e,0or ,0

e,1or ,1g,1or ,0

g,1or -,0e,1or ,1

Where Is the frequency difference between these two transitions.

The exchange

,1 ,1g e

IS POSSIBLE

C-NOT GATE

N=0 ATOMIC STATE IS NOT CHANGEDN=1 ATOMIC STATE IS EXCHANGED

CONTROL TARGET

UQCM

PROPOSED PROTOCOL

ATOM 1

A1, initially at is prepared in a superposition, via a Ramsey Field

1g

1 1 1

2 13 3

1cos3

g g e

ar

A1 interacts with the cavity Ca(initially in )through a Rotation, so

0a

State swapping.The excitation of atom 1 is transferred to the cavity a

1 1 1

2 1 2 1( ) 0 ( 0 1 )3 3 3 3a a a

g e g

0,1,

1,,

eg

goe

ATOM 2

IT CONTAINS THE INFORMATION TO BE CLONED

2 2 2e g

This state can be prepared in the same fashion as the atom 1, for example with a Ramsey Field.Then we apply a Classical pulse, as described before, generating a C-NOT gate ,nothing happens with 0 photons

,1 ,1g e

2 2

2 2 2 2

2 1( )( 0 1 )3 3

2 10 13 3

a a

a a

e g

e g g e

C-not

A3 and A4 are the atoms carrying the two

copies(IDENTICAL)

FINAL STATE

3 4 3,4

3 4 3,4

1 1 3 4

1

1 2

3,4 3 3 4

2 13 3

2 13 3

, ,

2 1( )3 3

0 0

0 1

1 ( )2

I

I

a b

I a b

e e A A

g g A A

wheree g

B g e g g

A g g

A g g

e g g e

DISCUSSION

Experimental numbers(Haroche et al)

KHzKHzKHz 502

,502

,1002

An interaction time of s 50int Marginally satisfies the earlier requirement.

With the flight time of 100 s The whole scheme should

Take about 700 s Which is reasonable in a cavity with a

Relaxation time of 16ms.They achieved a resolution required toDistinguish between 1 or 0 photons

The complete Hamiltonian in the Interaction Picture is:

seg

eg

s

INT

with

tititf

atiatitH

1

112

2

)exp()exp()exp(

)exp()exp()exp(

DISCUSSION OF THE C-NOT GATE

Since the external pulse is resonant with the (+,1)-(-,0)Transition,this imposes a condition on 1

2)(41)(81

222

1

Also, we notice that we have introduced exponential factorsIn both terms of the Hamiltonian just to mimic the passageTime and duration of the pulse, referred to as

and sRespectively.We have done this in order to solve Schrodingers equationWith continuous functions.

Assuming

N

nnn gntbentat

0

,)(,)()(

We have to solve the following set of differential equations

)()exp())(exp(

)(1)exp()(exp

)()exp())(exp(

)()exp()(exp

12

12

12

12

tbtitf

tbntitdt

i

tatitf

tantitdt

i

ns

nnda

ns

nndb

KHZf

KHZ

KHZ

KHZ

ss

s

1.812

25.462

1372

502

5.1327

1

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