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\begin{document}
\title{The world's smallest tunable Mach-Zehnder and efficient optical quantum computing}
\date{\today}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \author{Dimitris \surname{Angelakis}}%
    \author{Marcelo F. \surname{Santos}}%
    \author{Vassilis \surname{Yannopapas}}%
\author{Artur \surname{Ekert}}%
    \affiliation{Centre for Quantum Computation,...\\
   QOLS, ...\\Condensed Matter,....
    }%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 \begin{abstract}
bla bla
  \end{abstract}

  \maketitle

 \preprint{Version 3}

\section{Part I-Tunable MZI in a photonic crystal }
%{\it Guys, we need to cut down a bit this part and add a few lines
%that show clearer that the form of the power transfer function
%obtained indeed suggests that this is a usual MZI. Show that
%similar expressions can be obtained for the amplitude as well}

A typical MZI consists of two arms of equal length, joined in two
beamsplitters. In the simplest case a dielectric medium introduced
in one of the arms induces a phase lag between the two arms which
can be detected in the output port as oscillation of the output
power. This is usually a sinusoidal function of the relative phase
 between the two arms caused by the interference of the
light modes in the beamsplitters.

Our MZI is integrated in a 2-D photonic crystal structure which
consists of an hexagonal array of air holes in a specific
dielectric exhibiting an complete band gap, and utilizes the
lossless propagation of light through line defects created in the
structure. The mechanism is based on the concept of the light
propagation through heavy photon bands. More specifically the
guiding regions defining the arms of the MZI are coupled-cavity
waveguides (CCW) \cite{stef,yar1,oz1}. These are realized by
creating a linear chain of defects. A single defect introduces a
bound state of the EM field within the band gap. This defect can
act as a high-Q cavity since the EM state is highly localized
within the defect area and decays exponentially away from it in
all directions. When many defects are brought together so as to
form a linear chain, the states of neighboring defects interact
with each other giving rise to a (defect) band of modes within the
absolute gap, allowing for light propagation along the direction
of the chain. Due to the small spatial extent of the mode of a
single cavity, the coupling between neighboring modes is weak
enabling us to describe the EM field which corresponds to the
defect band in the context of tight-binding approximation
\cite{stef,yar1,oz1,yanno1}. For a single-mode CCW \cite{soltani}
the resulting dispersion relation of the defect band reads as
\begin{equation}
\omega_{k}=\omega_{0}+W\cos(kR) \label{eq:disp}
\end{equation}
where $\omega_{0}$ is the resonance frequency of an individual
defect cavity, $k$ is the Bloch wavevector ($-\pi/R\leq k <
\pi/R$) and $W$ is a measure of the overlap integral between
neighboring defect states. Due to the weak coupling between the
cavities ($W \ll \omega_{0}$) light propagation along the CCW can
be described  through a hopping mechanism between neighboring
cavities with a very small group velocity. As can been from
Eq.~(\ref{eq:disp}) the group velocity reaches its maximum value
at the band center and vanishes at the band edges. The suppression
of the group velocity is highly desirable in our actual
implementation of the MZI since it facilitates the nonlinear
control of the phase of the guided mode by a group of atoms doped
in a defect. These could be either doped for the case of
dielectric defects in a air, or just be of a gas of cold atoms
cooled down in the void defects for the inverted structure. A
suitable group velocity to provide adequate coupling of the light
wavepacket with the doped atoms inside the defect would be of the
range of $10^{-2}$-$10^{-3}$ of the speed of light. For this range
of values, it's  a good approximation to assume negligible
spreading of the wavepacket along the chain of CCWs.

 The beam-splitter operation is performed by two CCWs placed next to
each other separated by a single layer of the host photonic
crystal, as shown in Fig.~\ref{fig1}. Note that similar couplers
in the near-infrared regime have been fabricated using planar
photonic crystal waveguides \cite{borel}. Assuming that only one
of the two inputs, say, input 1 is active, it has been shown by
Martinez et el.~\cite{marti1} that for the dispersion relation
shown in Eq. ~(\ref{eq:disp}) and for a similar structure, the
power transfer function is exactly the one corresponding to a
usual MZI and is given by \cite{rama}
\begin{eqnarray}
T_{11}=\sin^{2}(\Delta \phi) \nonumber \\
T_{12}=\cos^{2}(\Delta \phi) \label{eq:transfer}
\end{eqnarray}
where $T_{ij}$ represents the power transfer function from input
$i$ to output $j$ and $\Delta \phi$ is the phase lag between the
two output signals. We note that in their case no active control
over the phase difference between the two arms was possible.
Fabricating a new structure with a different ratio of arms length
is needed each time in order to tune the phase shift accordingly.
However this could be avoided if a medium with tunability in the
index of refraction is placed in one of the arms. That could be
 a solution of optically birefringent nematic liquid
crystals filling one or more defects, or some holes of the host
material in one of the arms. An external electric field could be
used to rotate their principal axis and thus change the index of
refraction to the desired one\cite{BJ99}.

 We suggest the use of
quantum optical means for the phase shift through the
near-resonant interaction of the hopping photon with a collection
of atoms doped in the defects. Let us assume that all the defects
in both arms contain identical atoms and let us pick two atomic
states namely $g$ and $h$. Initially the atomic transition is far
off resonant with the hopping photon and the defects. The defects
on the other hand are supporting the mode of the ``hopper". In
other words $\omega_{gh}>>\omega_p=\omega_d$.

 Assume now that through say a fiber, a photon is
inserted in the structure from the left,  and starts hopping its
way to the right-Fig. 1. As it reaches the arms area a DC field is
applied in upper part of our MZI, through the full extend of the
structure bringing the atoms in those defects close but still of
resonance to the hopping photon. Assuming the photon was at defect
number N at that time and the coupling with the field is $\Omega$,
the detuning $\Delta$ and $\Omega^2/\Delta<<1$, the un-coupled
atom-defect photon state undergoes the well known AC-stark shift
and acquires a phase proportional to $\Omega^2/\Delta \times T$.
This can be tuned to any desired value by changing the time the
atoms interact with the field. We stress the fact that knowledge
for the exact location of the photon is not essential, as it will
interact only with the atoms in the specific defect that it
happened to reside in at the time of the field shift. Due to the
physics of the hopping mechanism and the strong localization of
the photon within only one defect each time, the photon spends
most of its life in defects rather than between them.

 Once the desired phase rotation is acquired,
the electric field shifts the atoms residing in defects $M\leq N$
(defects on the right of M- back to far off resonant making them
``transparent" again). A rough estimate of the time since the
photon entered the structure suffices in this case to estimate M.
We note that in this way we also control backwards hopping in the
following way: Defects on the left of M contain atoms still close
to resonance with their defect mode frequencies, phenomenon known
as frequency pulling, and thus these are not supporting the mode
of the hopping photon anymore. We note that this technique of
global switching also increases the quality factor of defect N
during the phase shift operation as the photon cannot leak out.
This idea will also be the main tool for the implementation of
quantum gates between the input photon states as we discuss in the
quantum information processing part of this letter.

%As shown by Yelena et el. ~\cite{scherer}, defects of this form
%can act a high-Q cavity with quality factor of the order of
%$10^4-10^5$. Which is as we show later in the range of parameters
% section is more than sufficient.

%For the one atom in defect case this could  problem as This could
%be tackled by using a 3D structure where quality factors of
%defects modes can be two or three orders of magnitudes higher.
%Also by scaling the current structure to infrared or longer
%wavelength higher quality factors are possible.

\section{Part II-Efficient quantum computing using number states}
In our system, the qubits are the $|0 \rangle$ and $|1 \rangle$
Fock states of input field modes in the MZI. In order to achieve
universal quantum computation, we need to operate two quantum
gates in those qubits: a single qubit phase shift, and a
controlled (two qubit gate) phase shift. In this section we show
that three level-atoms doped in the defects in the arms of the
interferometer allows us to implement both gates efficiently.

Before detailing the dynamics of the atom-photon interaction, let
us just recap a bit the mechanism of a controlled-phase gate. The
transformation we want to implement for a controlled phase gate is
$|00 \rangle \rightarrow |00 \rangle$, $|01 \rangle \rightarrow
|01 \rangle$, $|10 \rangle \rightarrow |10 \rangle$, $|11 \rangle
\rightarrow -|11 \rangle$. Assume now that the states before the
arrows are the entries of a Mach-Zender interferometer, and the
states after
the arrows are its outcomes. %The intermediate step of the gate
%transformation which is the state
%inside the interferometer.% Our MZI obeys to the following modes
%relations (where, a and b are the modes before the first
%beam-splitter, c and d are the modes inside the interferometer and
%e and f are the modes after the second beam-splitter):$a =
%(c+d)/\sqrt{2}$,
%$b=(c-d)/\sqrt{2}$, and $e=a$, $f=b$. \\
We also note that the action on the entry states of an MZI as
follows:$|00 \rangle \rightarrow |00 \rangle \rightarrow |00
\rangle$, $|01 \rangle \rightarrow \frac{|01 \rangle - |10
\rangle}{\sqrt{2}} \rightarrow |01 \rangle$, $|10 \rangle
\rightarrow \frac{|01 \rangle + |10 \rangle}{\sqrt{2}} \rightarrow
|10 \rangle$, $|11 \rangle \rightarrow \frac{|20 \rangle - |02
\rangle}{\sqrt{2}} \rightarrow |11 \rangle$.

Assume we have designed the interferometer with one non-linear
medium in each arm that does not affect the vacuum $|00 \rangle$
and 1-photon states $|10\rangle$, $|01\rangle$ but rotate state
$|20 \rangle$ to $-|02 \rangle$, then, states $|00 \rangle$, $|01
\rangle$ and $|10 \rangle$ do not change, while states $|20
\rangle$ and $|02 \rangle$ both acquire the same $\pi$ phase. It's
easy to see that in this case, after the second beam-splitter of
the interferometer, we achieve exactly the truth table of the
controlled-phase gate. Next we show how to implement this rotation
in our specific scheme.

Initially note that groups of atoms are placed inside the defects
micro-cavities, similarly to the case of the general MZI phase
shift(see previous section) in the arms of the Mach-Zender. The
atoms and the photons are coupled through typical linear,
field-dipole interactions as shown in Fig. 1. In the rotating wave
approximation, the dynamics are described by the following
Hamiltonian.  ($\hbar=1$).

\begin{eqnarray}
H_0=\omega_{0}a^\dagger a + \omega_{g} \sigma_{gg} + \omega_{e}
\sigma_{ee} + \omega_{h} \sigma_{hh} +\nonumber\\
 g_1(\sigma_{gh} a^\dagger
+ \sigma_{hg} a) + g_2(\sigma_{he} a^\dagger + \sigma_{eh} a)
\end{eqnarray}
which, written in the interaction picture, reads:
\begin{eqnarray}
H_{int}=g_1(\sigma_{gh} a^\dagger e^{i\delta t} +
 \sigma_{hg} a e^{-i \delta t}) \nonumber \\
 + g_2(\sigma_{he} a^\dagger e^{-i
\delta t} + \sigma_{eh} a e^{i \delta t})
\end{eqnarray}
where, $\delta=\omega_0-(\omega_h-\omega_g) =
(\omega_e-\omega_h)-\omega_0$. $a^\dagger$, $a$ are the creation
and annihilation operators for a photon in the defect microcavity
mode, $\sigma_{ij},$ are the corresponding atom operators with
$i,j=g,h,e$. $g_{1},g_{2}$ are the corresponding coupling
constants for the two transitions. The value of $\delta$ is chosen
by applying an external static field over the atom.
%These could either be through coupling each one of them
%to extra atomic levels like Zeeman level - radio frequency,
%excited levels - optical frequency, or they can be shifted
%together by means of static electric fields.

The atom will always start in the ground level $|g \rangle$ and we
will show that this interaction can be manipulated in order to
give us any kind of gate we want. First, we show how to implement
the usually more difficult one, the control phase shift.

%In one arm of the interferometer, we displace atomic levels $|e \rangle$ and
%$|g \rangle$ so that the respective atom is very far from resonance and does
%not interact with the field. In the other arm, we only displace atomic level
%$|e \rangle$, so that $\delta_2 >> \delta_1 > g_1, g_2$, but the atom and the
%field still interact in the dispersive regime. In this case, since the atom
%starts in the ground state, the corresponding effective hamiltonian is given
%by:
%\begin{equation}
%H_{eff}=-\frac{g_1^2}{\delta_1}(\sigma_{gg} a^\dagger a)
%\end{equation}
%which implements the phase gate, leaving state $|g \rangle |0\rangle$ untouched
%and shifting state $|g \rangle |1 \rangle$ to $e^{i\frac{g_1^2}{\delta_1}t} |g
%\rangle |1 \rangle$. This phase shift works well if we feed only one arm of the
%interferometer with the qubit.

%Alternatively, we can use the same strategy for the two qubit gate, i.e. we
%prepare both atoms (one in each arm) in the same interaction conditions, and we
%feed the interferometer with the qubit in one port and the vacuum in the other.
%In this way, the state pre-beam splitter $\frac{|0 \rangle + |1
%\rangle}{\sqrt{2}}|0 \rangle$, splits into $\frac{|00 \rangle}{\sqrt{2}} +
%\frac{|01 \rangle + |10 \rangle}{2}$. Now, the atom in each arm won't
%interferer with the first state (00), but will input the same phase
%$e^{i\frac{g_1^2}{\delta_1}t}$ to states 01 and 10. After the second arm of the
%balanced Mach-zender, the output state is the vacuum in one port and state
%$\frac{|0 \rangle + e^{i\frac{g_1^2}{\delta_1}t}|1 \rangle}{\sqrt{2}}|0
%\rangle$ on the other.

We feed the photon in from the left and then we apply an electric
field to the whole extent of the two arms so as to shift the level
$|e \rangle$ of the atoms in all defects so that
$|\delta_1|=|\delta_2|>>g_1,g_2$. The dynamics are described by
the following effective Hamiltonian
\begin{equation}
H_{eff}=-\frac{g_1^2}{\delta}\sigma_{gg} (a^\dagger a) +
\frac{g_2^2}{\delta}(\sigma_{ee} a a^\dagger) + \frac{g_1
g_2}{\delta}(\sigma_{ge} {a^\dagger} ^2 + \sigma_{eg} a^2)
\end{equation}

Due to the off-resonant interaction there will be a shift of the
atomic levels from the initial positions which overall will amount
to $2\frac{g_1^2}{\delta} - \frac{g_2^2}{\delta}$ between the
ground and the most excited state. Let us now assume, that we can
displace the energy of level $|e \rangle$ to tune to resonance the
transition $|g2 \rangle \rightarrow |e0 \rangle$ (by means of an
external electric field). That means, adding an extra amount
$2\frac{g_1^2}{\delta} - \frac{g_2^2}{\delta}$ of energy to the
most excited level. Again, this can be achieved by addressing this
level with external classical fields.

Assume the atom is initially in level $|g \rangle$. The two
photons drive the atom in Jaynes Cummings manner-we note here that
this system has been extensively studied in two-photon micromaser
literature as well. When the two photon are on resonance with the
$|g \rangle  \rightarrow |e \rangle$, is easy to see that the
following table for the evolution of the joint atom-field state
after a time T is true.
\begin{eqnarray}
&&|g \rangle |00 \rangle \rightarrow |g\rangle |00 \rangle, \\
&&|g \rangle |01 \rangle \rightarrow e^{(-i\frac{(g_1)^2
T}{\delta})}|g \rangle |01 \rangle, \\
&&|g \rangle |10 \rangle \rightarrow e^{(-i\frac{(g_1)^2
T}{\delta})}|g \rangle |10 \rangle,\\
&&|g \rangle |20 \rangle \rightarrow \cos k T|g \rangle |2 \rangle
+
\sin{kT} |e \rangle |0 \rangle,\\
&&|g \rangle |02 \rangle \rightarrow \cos k T|g \rangle |02
\rangle + \sin{kT} |e \rangle |0 \rangle
\end{eqnarray}
 where $k=\frac{(g_1 g_2
\sqrt{2})}{\delta}$.

Notice that this effective Rabi frequency differs from the normal
one by a $\sqrt{2}$ factor, because this is a two-photon, second
order process. This extra factor is of much use, cause now, if
$g_1/g_2 = 2(\sqrt{2})$, the time it takes for states $|g01
\rangle$ and $|g10 \rangle$ to gain a $2 \pi$ phase ($\frac{2 \pi
\delta}{(g_1)^2}$), is exactly the same time it takes for states
$|g20 \rangle$ and $|g02 \rangle$ to perform a $\pi$ Rabi
oscillation, which means acquiring a minus sign, which means,
implementing the quantum phase gate we were interested in. Indeed
\begin{equation}
kT = \frac{(g_1 g_2 \sqrt{2} T)}{\delta} = \frac{(g_2 \sqrt{2} 2 \pi)}{g_1} =
\pi
\end{equation}

All we would need to find is atoms exhibiting two very close
transitions in frequency and which differ by that amount in
coupling constant. Of course there are other alternatives to this
scheme, all of them playing with the ratios of the coupling
constants, versus multiples of $\pi$ but this is the simplest one.

%Therefore, once we put one atom in each arm of the Mach-zender, we can choose,
%by tuning this interaction on and off in each one of them, if we want a
%dephaser (only one atom is tuned and one photon gains a $\phi = kT$ phase), or
%a controlled-phase gate in which both atoms are tuned and we choose $kT=\pi$.


In order to implement a one-qubit gate, one way is to absorb the
incoming photon, mapping the photon qubit onto the atomic state,
then operate in the atom releasing the photon afterwards in the
new state. The first operation is a complete resonant $\pi$ Rabi
pulse, taking state $|g \rangle (\alpha |0 \rangle + \beta |1
\rangle)$ into state $(\alpha |g \rangle + \beta |e \rangle) |0
\rangle$. Then, a classical field rotate the atom to the new state
$(\alpha' |g \rangle + \beta' |e \rangle) |0 \rangle$ and another
complete $\pi$ Rabi pulse bring it back to $|g \rangle (\alpha' |0
\rangle + \beta' |1 \rangle)$. The shifting

 {\it Two questions arise. The first one
concerns the timescale for this operations to happen. Since both
resonant pulses are one-photon transitions, they are much faster
than the two-photon transitions necessary for the non-linear gate,
which means that if the second is possible, this one-qubit gates
should also be (the atom rotation is even faster since it's
implemented by an external field - the same that shifts the atom
energy level).

The second question is more delicate. Would the photon keep hoping in the
proper direction? A possible solution for this problem would be to close one of
the directions. How? By changing the resonance frequency of the previous
microcavity (ies). The system would calculate the time of flight of the photon.
When the photon hops from cavity $n-1$ to cavity $n$ (where the gate is
implemented), the shifting matrix (the one put on top of the crystal and used
to operate on the atoms) shifts the energy level of the impurity of cavity
$n-1$, changing the cavity's frequency and making it much more probable for the
photon to hop to cavity $n+1$ which is in resonance to its frequency, then to
come backwards. This would be the control mechanism of the direction of flow of
the qubits.

\subsection{order of magnitude}

Based on ref.~\cite{scherer}, the order of magnitude of the interactions inside
those cavities would be very close to what we need (maybe pushing a little too
far right now, but close enough). In this paper, they talk about the following
quantities: mode volume $V_{mode}$, critical atom $N_0= \frac{2 \kappa
\gamma_t}{g^2}$ and photon $m_0=(\frac{\gamma_t}{2g})^2$ numbers, where $\kappa
= \frac{\omega_0}{4 \pi Q}$ is the cavity decay rate (Q is the quality factor
and $\omega_0$ the resonance frequency), $\gamma_t$ is the atomic dipole decay
rate (2,6 Mhz for Cesium) and g is the coupling constant ($g_0 = \gamma_t \sqrt
{\frac{V_0}{V_{mode}}}$ is the Rabi coupling constant).

Now, they give examples of some of their cavities. As an example,
the cavity described in section IV has a quality factor of
$Q=10^4$, and $N_0=6 * 10^{-3}$ and $m_0= 1.4 * 10^{-7}$, which
means $\kappa = 10^{10}$. On the other hand, the coupling constant
g is equal to $g=3 * 10^9$, which means, we are still $1/3$ of
order of magnitude shy of what we want. However, since we don't
worry about the leaking of the photon (as long as we've done with
our rotation - in fact we do want the photon to leak!), it means
that an improvement of one order of magnitude in the quality
factor of the cavity, reducing the decay constant in one order of
magnitude, puts us already within reach!}

\section{Part III}

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\end{thebibliography}
%\begin{figure}
%\includegraphics[width=0.45\textwidth]{machzend.eps}
%\caption{Single particle interference with a Mach-Zender
%Interferometer.} \label{fig:machzend}
%\end{figure}
\begin{figure}
\centering \epsfxsize=5 cm \epsfbox{atom1.eps} \caption{The atmic
level structure} \label{fig1}
\end{figure}

\begin{figure}
%\includegraphics[width=0.45\textwidth]{MZI_1.eps}
\centering \epsfxsize=18 cm \epsfbox{MZI_1b.eps} \caption{The
proposed MZI} \label{fig1}
\end{figure}


\end{document}