PROPOSAL OF A UNIVERSAL QUANTUM COPYING MACHINE IN CAVITY QED

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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 II COZUMEL-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
Is not possible, with
INPUT QUBIT
initial state of cloner
final state of cloner
Blank copy
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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The quality of the copy is measured through the
FIDELITY
UNIVERSAL QUANTUM COPYING MACHINE
BASIS
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In the present work, we propose a protocol that
produces 2 copies from an input state, with
Fidelity
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
PROPOSAL Consider a two level atom that is
prepared in a superposition state , using the
Microwave pulses in a Ramsey Zone, with frequency
Near the e(excited)-g(ground) transition. It
generates superpositions
Depends on the interaction time
Is prop. detuning
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On the other hand, the atom-field interaction is
described by the Jaynes Cummings Hamiltonian
Coupling constant
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The atom-field state evolves like
For example, for
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Now, consider an external Classical pulse,
interacting with the atom
We use the dressed state basis that diagonalizes
the J-C Hamiltonian
,
The Energies of the dressed states are

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In the limit
Consider the external field in resonance with the
(,1)-?(-,0) Transition, that is
Where f(t) is some smooth function of time to
represent the pulse shape, with(in the
dispersive case)
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The above Hamiltonian has been studied by several
authors (Domokos et alGiovannetti et al) and
arrive to the conclusion that For a suitable
pulse, a C-NOT gate can be achieved, where the
photon Number (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,0gt--?(e,0gt 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
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Where
Is the frequency difference between these two
transitions.
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The exchange
IS POSSIBLE
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C-NOT GATE
N0 ATOMIC STATE IS NOT CHANGED N1 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
A1 interacts with the cavity Ca(initially in
)through a Rotation, so
State swapping.The excitation of atom 1 is
transferred to the cavity a
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ATOM 2 IT CONTAINS THE INFORMATION TO BE CLONED
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
C-not
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A3 and A4 are the atoms carrying the two
copies(IDENTICAL)
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FINAL STATE
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DISCUSSION
Experimental numbers(Haroche et al)
An interaction time of
Marginally satisfies the earlier requirement.
The whole scheme should
With the flight time of 100
Take about 700
Which is reasonable in a cavity with a
Relaxation time of 16ms.They achieved a
resolution required to Distinguish between 1 or 0
photons
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DISCUSSION OF THE C-NOT GATE
The complete Hamiltonian in the Interaction
Picture is
Since the external pulse is resonant with the
(,1)-?(-,0) Transition,this imposes a condition
on
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Also, we notice that we have introduced
exponential factors In both terms of the
Hamiltonian just to mimic the passage Time and
duration of the pulse, referred to as
and
Respectively.
We have done this in order to solve Schrodingers
equation With continuous functions.
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Assuming
We have to solve the following set of
differential equations
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