INTRODUCTION
Quantum gates and entanglement are important part of quantum communication
and computation (Prevedel et al., 2007). Many researchers work
on theories of quantum computation, such as entanglement, quantum decoherence,
quantum cryptography and quantum error correction. Now Quantum computation
and communication move closer to being a reality (Benenti et al.,
2004; Benenti and Strini, 2007; Nielsen and Chuang, 2000). For transition
from theory to applications, we need to realize quantum computer by using
the implementation of quantum gates. Standard versions of Planar Lightwave
Circuit (PLC) technology are well documented and have been practiced since
late 1980`s. They combine significant features of optical fiber and integrated
circuit technologies. Basically, lightguiding channels, similar in function
to optical fibers, are defined on a silicon platform. These are fabricated
by depositing sequential glass layers onto silicon wafers. Typically,
an intermediate core layer with an elevated refractive index is patterned
using photolithography and dryetching. This patterned structure becomes
the lightguiding channel. This baseline technology is well suited to
the fabrication of passive devices such as couplers which depend on the
proximity, controlled spacing and path lengths of parallel waveguides.
The silicon chip provides a robust support for the waveguides. Such PLC
components are interfaced to fiber optic networks via edge attached fibers.
This technology supports complex and versatile photonic integration. Implementation
of quantum gates by using all optical circuits is very important in all
optical communication networks. Since the trend for photonics integration
is toward smaller and more denselypacked components, it is necessary
to be able to model these small components perfectly (Okamoto, 1999; Doerr,
2006; Suzuki and Sugita, 2005).
Coupled mode theory: Coupled mode theory is used in optical filters
and switches. In these devices, two coupled waveguides that the coupling
between them is controlled by applied voltage are used. The waveguides
are made from electrooptics material, that its permittivity is controlled
by voltage. By controlling the permittivity, we can transfer power from
one limb to another.
The set of coupled differential equations that we have just derived can
be used to analyze transferred energy from one optical waveguide to another,
when the guiding structures are brought into proximity. Consider two parallel
waveguides, 1 and 2 in Fig. 1, for which the total field
solutions can be written as linear combinations of individual waveguide
modes:

Fig. 1: 
Integrated optics coupler on a chip 

Fig. 2: 
Transferred power between guide 1 and 2 
The amplitudes of a(z) and b(z) satisfy
Where:
K_{ab} and K_{ba} 
= 
The coupling coefficients 
β_{a} and β_{b} 
= 
The propagation constants in waveguides 1 and 2, respectively 
If at z = 0, the optical power is incident only in waveguide 1, (a(0)
= 1, b(0) = 0), we find:
Where:
Therefore, atthe
power transfer from guide 1 to guide 2 is maximum. If β_{a}
= β_{b}, we have Ψ = K_{ab} and the solutions
are
Where, K = K_{ab} = K_{ba}, β = β_{a}
= β_{b}, have been used. Complete power transfer occurs for
synchronous coupling. For Kl = (2n+1)π/2, complete power transfer occurs;
this is called a cross state. For Kl = nπ, there is no power transfer
from guide 1 to guide 2, which called a parallel state.
Power transferring between guide 1 and 2 is shown in Fig.
2. As can be seen in this figure, if the length of coupler is L =
π/2K, then the power of guide 1 transfer to guide 2 completely.
INTERFEROMETRIC ELECTROOPTIC MODULATOR
The basic electrooptic interferometer is shown in Fig.
3. In this device, a single input is split between two waveguides.
The space between the waveguides is designed suitably so that the evanescent
coupling does not take place. The two outputs are added together in a
single output guide where the two waves interfere, with amount of interference
depending upon the difference in propagation time along the two limbs.
The propagation constant of the two guides can be changed differentially
by using the electrooptic effect to change the guide refractive indices.
A DC bias can be applied to equalize the phase difference of the two limbs
in the absence of a modulation voltage. Interferometric modulator using
LiNbO_{3 }with bandwidths up to 40 GHz have been reported and
they have a considerable future in optical communication and computation
(Tamir, 1988). The transfer function matrix of this element is:
Where:
Φ 
= 
Phase difference between the two limbs of modulator
that adjusted by applied voltage 

Fig. 3: 
Interferometric electrooptic modulator 
Another way for producing phase difference between two limbs of modulator
is using of kerrlike nonlinear waveguide. In the Kerrlike medium, intensity
of electromagnetic fields changes refractive index of the waveguide and
provides phase shift.
QUANTUM CIRCUIT FOR QUANTUM ENTANGLEMENT
We can model entangled photons by using quantum gates (Cerf et al.,
1998). This model includes Hadamard and CNOT gates (Fig.
4). If the input of this circuit is one of states 00〉,
01〉,
10〉,
11〉,
then four different entangled states, called Bell states, are produced
at the output. In the first state, suppose that the input is 00〉.
Hadamard gate converts state 0〉
to .
Therefore, the state at the input of CNOT gate is:
At the output of the CNOT gate, this state becomes,
If we consider the other three states 01〉,
10〉,
11〉
as input, then the output of circuit is one of the following Bell states,
These states are entangled states and are the basis of the quantum computation
and communication. From Fig. 4, we see that quantum
entanglement implementation consist of realization of quantum gates. We
propose the new method for integrated optics implementation of quantum
entanglement.

Fig. 4: 
Quantum circuit for producing entangled photon 
ALL OPTICAL IMPLEMENTATION OF QUANTUM NOT GATE
Qubits can be realized by the two normal modes of the dualmode waveguides,
such as the zero logical state 0〉
encoded into one normal mode, TM_{0} and the logical one 1〉
given by other orthogonal normal mode, TM_{1} (Fig.
5). A qubit`s state space consists of all superpositions of the basic
normal modes 0〉
and 1〉.
As mentioned above, in this study, an all optical method is proposed
for implementation of quantum gates. Realization of quantum NOT gate using
interferometric electrooptic modulator is shown in Fig.
6. By applying the suitable V_{mod} to the electrodes, we
can adjust phase difference between the two limbs of the modulator.
With due attention to Eq. 5, the relation between input
and output of interferometric electrooptic modulator is:
If Φ = 0, then all inputs are unchanged at the gate output, but if
Φ is adjusted to the value of π, the modulator acts as a quantum
NOT gate. All inputs to 0〉
appear as the 1〉
output and vice versa, extra an additional phase. Superposition states
are generated by adjusting the phase difference. For example by choosing
Φ = π/2, we have the following states,
The Beam Propagation Method (BPM) simulation results for quantum NOT
gate have been shown in Fig. 79.

Fig. 5: 
Electricfield profiles for TM_{0} and TM_{1
}modes 

Fig. 6: 
Realization of quantum NOT gate using interferometric
electro optic modulator. The width of waveguides is 12 μm and the
length of gate is 2.8 cm 

Fig. 7: 
Quantum NOT: 0〉→
1〉
. (a) Optical field amplitude and (b) output electric field profile 

Fig. 8: 
Quantum NOT: 1〉→
0〉. (a) Optical field amplitude and (b) output electric field profile 

Fig. 9: 
Superposition state generation: .
(a) optical field amplitude and (b) output electric field profile 
ALL OPTICAL IMPLEMENTATION OF HADAMARD GATE
The quantum Hadamard gate operates on a single qubit. It is represented
by the following equation (Andrecut and Ali, 2004),
An integrated optics beam splitter will be proposed for realization of
this gate. Beam splitter is a basic element of many optical fiber communication
systems, often providing a Yjunction by which signals from separate sources
can be combined, or the received power can be divided between two or more
channels. A passive Yjunction beam splitter is shown in Fig.
10. Unfortunately, the power transmission through such a splitter
decreases sharply with increasing half angle θ, the power is radiated
into the substrate. The passive Yjunction beam splitter finds application
where equal power division of incident beam is required.
We consider waveguide branch that supports the two lowest order normal
modes namely TM_{0} and TM_{1} (0〉and
1〉).
Transferred powers between these modes in the branches are described by
the coupled mode equations. Because of linearity of these equations and
superposition properties of Maxwell`s equations, the solution can be obtained
for each local normal mode independently. The achieved solutions are superimposed
at the output branches. Normal TM_{0} and TM_{1 }modes
(0〉
and 1〉)
with equal powers and phases are applied on an ideal power dividing branch.
Considering each normal mode independently, the output mode amplitudes
in branch 1 are same and have values equal to ,
but in branch 2, the output amplitudes of TM_{0} and TM_{1}
modes are and,
respectively. Therefore, the output of branch 1 is the same as the output
of Hadamard gate when its input is 0〉.
Also, the output of branch 2 is the same as the output of Hadamard gate,
when its input is 1〉.
The relation between the output and input of this element is:
By comparing the Eq. 11 and 12, we can
see that the output of this element is exactly same as the Hadamard gate.
The Beam Propagation Method (BPM) simulation results for this gate are
shown in Fig. 1113.

Fig. 10: 
Yjunction beam splitter as a Hadamard gate 

Fig. 11: 
The Hadamard gate implementation. The width of waveguides
is 12 μm and the length of gate is 1.5 cm 


Fig. 12: 
(a) The amplitude of optical filed propagation for state
0〉,
(b) the electrical field profile amplitude of input state 0〉,
(c) the amplitude of optical filed propagation for state 1〉
and (d) the electrical field profile amplitude of input state 1〉 

Fig. 13: 
(a) The amplitude of optical filed propagation for superimpose
state and (b) the electrical field profile amplitude of input state 
ALL OPTICAL IMPLEMENTATION OF CONTROLLED NOT GATE
In the quantum controlled NOT gate, if input control bit is 0〉,
then the control bit and the target bit do not change. In other case,
for input control bit 0〉,
the target bit changes as below:
Both the coupled mode theory and the interferometric electrooptic modulator
are used for implementation of this gate. The planar lightwave integrated
optics of this gate is shown in Fig. 14. There is a
coupling region in this scheme. The length of this coupler is designed
for power transferring between the two couplers for TM_{1 }mode.
The electrooptic material such as LiNbO_{3} is used in limbs
of modulator. Two electrodes are placed on two limbs. When the optical
power is detected at the coupler output, the applied voltages on the electrodes
are adjusted as the phase difference π is created between the two limbs
of the modulator. If there is not any optical power on the output branch
of the coupler, then the applied voltages on electrodes are adjusted as
the phase difference 0 is created between the two limbs of the modulator.
When 0〉
is present at the control bit, the intensity of the qubit is never coupled
into the output of coupler. Therefore the control and target qubits are
left unchanged. When 1〉
is present at the control bit, the intensity of the qubit is coupled into
the output of coupler. Thus a phase shift of π is created between the
two limbs of the modulator and the states of the target bit will be flipped,
namely 0〉→1〉
and 1〉→0〉.
The Beam Propagation Method (BPM) simulation results for this gate are
shown in Fig. 1518.

Fig. 14: 
The CNOT gate realization. The width of waveguides is
12 μm. The length of coupler is 200 μm, separation between
coupler waveguides is 11 μm and gate length is 2.8 cm 


Fig. 15: 
The CNOT quantum gate: 00〉→00〉.
(a) optical field amplitude, (b) input electric field profiles and
(c) output electric field profiles 


Fig. 16: 
The CNOT quantum gate: 01〉→
〉®
01〉
. (a) optical field amplitude, (b) input electric field profiles and
(c) output electric field profiles 


Fig. 17: 
The CNOT quantum gate: 10〉→
〉®
11〉
. (a) optical field amplitude, (b) input electric field profiles and
(c) output electric field profiles 


Fig. 18: 
The CNOT quantum gate: 11〉→10〉.
(a) optical field amplitude, (b) input electric field profiles and
(c) output electric field profiles 

Fig. 19: 
All optical method to perform quantum entanglement for
integrated optics circuit 
ALL OPTICAL IMPLEMENTATION OF QUANTUM ENTANGLEMENT
We proposed the schemes for all optical implementation of quantum gates
in the previous sections. We can achieve quantum entanglement by combination
of these quantum gates. We propose a fully optical method to perform quantum
entanglement. This proposal is shown in Fig. 19. The
Yjunction beam splitter is used as a Hadamard gate. The upper branch
state of the beam splitter is equal to the output of the Hadamard gate,
when its input is 0〉
and state of the other branch is the same as the output of Hadamard gate
when its input is 1〉.
The output of each branch of the Yjunction beam splitter is used for
the input control bit of each CNOT gate. This scheme is consist of two
parts. Bell states Φ^{+}〉
and Ψ^{+}〉
are generated by the upper part and Φ^{}〉
and Ψ^{}〉
are generated by the other part.
In conclusion, we proposed an all optical method for implementation of
the quantum gates and entanglement. By using planar lightwave technology,
all single qubit and double qubit quantum logic gates are feasible. By
using a dual mode waveguide interferometric electrooptic modulator, directional
couplers and Yjunction beam splitter, we propose a fully optical method
to perform quantum gates and entanglement as the basis of quantum communication
and computation.