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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
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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:
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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.
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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
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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
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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
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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
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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
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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
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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
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Where Is the frequency difference between these two transitions.
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The exchange
,1 ,1g e
IS POSSIBLE
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C-NOT GATE
N=0 ATOMIC STATE IS NOT CHANGEDN=1 ATOMIC STATE IS EXCHANGED
CONTROL TARGET
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UQCM
PROPOSED PROTOCOL
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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
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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
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A3 and A4 are the atoms carrying the two
copies(IDENTICAL)
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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
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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
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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
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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.
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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
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KHZf
KHZ
KHZ
KHZ
ss
s
1.812
25.462
1372
502
5.1327
1
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BIBLIOGRAPHY1.-W.K.Wooters and W.H.Zurek,Nature,London,299,802(1982)2.-V.Buzek,M.Hillery,Phys.Rev.A 54,1844(1996)3.-D.Bruss et al, Phys.Rev.A 57,2368(1998)4.-N.Gisin,S.Massar, Phys.Rev.Lett,794,153(1997)5.-D.Bruss et al, Phys.Rev.Lett,81,2598(1998)6.-V.Buzek,S.L.Braunstein,M.Hillery,D.Bruss, Phys.Rev.A,56,3446(1998)7.-C.Simon,G.Weihs,A.Zeilinger, Phys.Rev.Lett,84,2993(2000)9.-P.Milman,H.Olivier,J.M.Raimond, Phys.Rev.A,67,012314(20003)10.-M.Paternostro,M.S.Kim,G.M.Palma,J.of Mod.Opt,50,2075(2003)11.-M.Brune et alPhys.Rev.A,78,1800(1995)12.-V.Giovannetti,D.Vitali,P.Tombesi,A.Eckert,Phys.Rev.A,52,3554(1995)13.-M.Orszag,J.Gonzalez,S.Dagach,sub Phys.Rev.A14.- M.Orszag,J.Gonzalez,Open Sys and Info Dyn,11,1(2004)