text:
\section{Introduction}{\label{I}}
Second Harmonic Generation (SHG) has nowadays quite a long tradition as a mean of
squeezed light generation \cite{Per88,Siz90,Kur93,Pas94,Ral95,Tsu95,You96}.
The preferred experimental setup has been the doubly resonant configuration
as, at least in principle, permits arbitrarily large squeezing. However, such
scheme has been hampered by the technical difficulties arising from keeping
the resonance in both modes simultaneously. Thus, in spite of the development
of very ingenious stabilizing procedures \cite{Kur93},
for the moment it has been only possible to maintain the double
resonance for a few seconds. Certainly, this kind of experimental delicacy can
hardly surprise when dealing with the generation of non-classical
states of light. In view of such difficulties, some experimental efforts
have been recently redirected to singly resonant configurations
\cite{Pas94,Ral95,Tsu95}. Although, the maximum noise suppression is then
limited to a 90\% \cite{Pas94}, the efforts resulted in very stable
intense squeezed light sources with degrees of squeezing
even surpassing those reported in the doubly resonant counterparts
\cite{Tsu95}. This evolution highlights the importance of reducing
to a minimum the technical demanding of new proposals in a so
experimentally challenging field.

At the same time, singly resonant Optical Parametric Oscillation (OPO), the
most successful method to squeeze the vacuum \cite{Pol92,Bre95}, has been
generalized to singly resonant Optical Parametric Amplification (OPA), i.e.\,
a laser driving in the harmonic mode has been added, again showing an
extraordinary stability at quite high noise suppression values in the
fundamental mode \cite{Sch96a}. Although the squeezed beams are in
this case much less intense than the in SHG counterpart, this setup permits
a control of the phase of the squeezed quadrature, something which
allowed a spectacular demonstration using quantum tomography, of the different
kinds of squeezed states \cite{Bre97}.

In view of this experimental success it seems timely to extend
the quantum mechanical model beyond the pure phase matched cases.
More specifically, we address here quantum noise reduction in an
extension of the conventional singly resonant SHG to include also
a coherent input in the harmonic mode as well as phase mismatch
between the interacting waves.

On the other hand, an increasingly number of papers is being devoted
to study quantum noise in systems combining different
kinds of nonlinearities (see, for instance,
\cite{Sun95,Mar95a,Mar95b,Cab97,Khe98} for some recent
contributions). In particular, the combination of $\chi^{(2)}$
with Kerr-like $\chi^{(3)}$ nonlinearities in c.w.\ cavity systems
has been quite extensively studied
\cite{Cab97,Tom84,Tom86,Gar89,Cab92a,Cab92b,Cab93,Kry96,Khe97}.
Even exact full quantum results have been obtained showing, for instance,
the emergence of tristability not present
in the classical counterpart \cite{Khe97}.
With respect to the squeezing performance the results appear as
very promising  at least in degenerate doubly
resonant configurations \cite{Cab97,Cab93}. The simplest system
from the implementation point of view, combining this two kinds of
nonlinearities that the authors can think of is precisely a singly
resonant second order nonlinear system with phase mismatched
interacting waves as then, by virtue of the cascading effect, an
effective Kerr-like third order nonlinearity appears.

In order to make the search of strong noise reduction through the
parameter space affordable we are bounded to the standard linearization
procedures, the only capable of yielding analytical results. Inside
the linear approximation perfect squeezing is possible at dynamic
instabilities. We make use of this fact to find optimum working
points showing up maximum squeezing. They are, however, an artifact
of the method as the linear approximation breaks down at the
instabilities. What matters for the practical implementation
of new squeezed light sources (our ultimate goal) are the optimum
paths through the parameter space approaching such points. Let us
explain a little more what we mean. Fixed a particular parameter
there will be a set of values of the remaining parameters (including
the frequency as such) tuned up to yield the maximum noise reduction.
When the chosen parameter is varied an optimum path is defined by the
set of parameter values maximizing the noise reduction at any stage
of the variation. The real optimum working point will be somewhere
along these paths before reaching an instability.
Thus, these paths would guide the experimentalist
towards the optimum working point in the real experimental setup.
The essence of our approach to find such paths will be to isolate
the dynamic aspect of the squeezing behavior from the
static contributions to the noise coming from the different inputs.
A simple adequate normalization will disentangle the two aspects of
the quantum noise behavior. This simplifies the analysis
sufficiently to allow a characterization of the optimum paths.

Another crucial issue when dealing with the squeezing performance of a
system is precisely the election of the most relevant parameter to compare the
different configurations with each other or with the reported related
experiments. There is no universal criterion to determine the
squeezing efficiency of a given device. An efficient setup
regarding power consumption, i.e., when compared for fixed input power
could well be deceptive when compared for the same output power and
perhaps inadequate to some spectroscopic applications.
However, within the state of the art of the present squeezed light
generators, the main concern is to improve the squeezing
figures themselves having other considerations such as the power
consumption a relative importance.  Under this perspective probably the
parameter of utmost importance as far as c.w.\ resonant systems are concerned is
the energy load inside the cavity. Indeed, the usual causes of squeezing
degradation such as blue-light-induced red absorption come from an
excessive mean photon number inside the cavity capable of
significantly degrade the material optical response at the relevant
frequencies. These considerations will lead us to define another normalization
this time useful for the evaluation of the squeezing efficiency with
respect to the intra-cavity photon number.

The sketch of the article is as follows. In section \ref{QMM} the
quantum mechanical model is presented. In section \ref{LEE} the
evolution equations are linearized, the fixed points of the system
obtained and their stability studied. Section \ref{SS} gives all the
formulae regarding quantum noise spectra in the system. In section
\ref{SP} a general approach to one-mode systems is developed which
allows the definition of general criteria to characterize the optimum
paths and applied to the specific case here addressed.
Finally the limits of the model and possible implementations
are thoroughly discussed in section \ref{DC}, concluding the article
with a summary of the most relevant results obtained.

\section{Quantum mechanical model}{\label{QMM}}
The system we want to address consists in a second order nonlinear
medium coupling two modes of frequency $\omega$ (fundamental) and $2
\omega$ (harmonic) respectively and placed inside a ring cavity resonant
only with the fundamental mode. We will also assume just one input-output
mirror of finite reflectivity.
The effect of phase mismatch when only the fundamental mode is driven
has been experimentally studied in \cite{Whi96a} where bistability
induced by cascading was demonstrated. The classical
evolution equation of the fundamental mode, $\alpha$, as given in
\cite{Whi96a}, reads
\begin{equation}
\label{eq:claeqa}
\frac{d\alpha}{dt} = -\left [ \gamma + i\delta + \nu K(\Delta k)
|\alpha|^{2} \right ]\alpha + \sqrt{2 \gamma_{c}}\, \alpha_{in} \,.
\end{equation}

\noindent
The nonlinear coupling depends on the wave vector mismatch
$\Delta k = k(2\omega)-k(\omega)$ as $K(\Delta k) =
2 \int_{0}^{L_{m}} \int_{0}^{z} u^{*}(\Delta k,z) u(\Delta k, z^{\prime})
d z^{\prime} d z / L_{m}^{2}$ being $L_{m}$ the length of the nonlinear
medium, $u(k,z)$ the spatial dependence of the resonator mode and $\nu$ is
proportional to the second order nonlinear susceptibility (see below).
Splitting $K$ in its real and imaginary parts, Eq.\ (\ref{eq:claeqa}) can
be recast as
\begin{equation}
\label{eq:claeqa2}
\frac{d\alpha}{dt} =
-\left [ \gamma + \mu |\alpha|^{2} +
i(\delta+\Gamma |\alpha|^{2}) \right ]\alpha +
\sqrt{2 \gamma_{c}} \alpha_{in} \,.
\end{equation}

\noindent
For a plane wave geometry
\begin{mathletters}
\label{eq:mu&Ga}
\begin{eqnarray}
\label{eq:mu}
\mu\;\equiv\;\nu K_{r} & = &\nu
\left ( {\rm sinc}\frac{\Delta k L_{m}} {2} \right )^{2} \\
\label{eq:Ga}
\Gamma\;\equiv\;\nu K_{i} & = &\frac{2\nu} {\Delta k L_{m}}
\left[{\rm sinc} \frac{\Delta k L_{m}} {2}
\cos \frac{\Delta k L_{m}} {2} -1 \right]\,,
\end{eqnarray}
\end{mathletters}

\noindent
where $K_{r}$ and $K_{i}$ denote the real and imaginary part of $K(\Delta k)$.
In this way the nonlinear dynamics is divided in a nonlinear
absorption (the up conversion of photons) and a nonlinear dispersion
(the cascading effect). The behavior of both parameters with
$\Delta k$ is plotted in Fig.\ \ref{fg:fg}. Notice that for any
finite $\Delta k$ the nonlinear dispersion is also finite, periodically
completely dominating (null frequency doubling).
\begin{figure}
\centerline{\includegraphics*[scale=0.5]{fkgk.eps}}
\caption{The dependence of $K_{i}$ and $K_{r}$ with respect to the phase
mismatch.}
\label{fg:fg}
\end{figure}

Quantization of Eq.\ (\ref{eq:claeqa2}) is then
accomplished independently for each effect. Regarding the nonlinear
absorption we use the two-photon model proposed in \cite{Col91}, while
nonlinear dispersion is accounted for by a fourth order
Hamiltonian, $H = (\hbar \Gamma/2)\, a^{\dagger \,2} a^{2}$ as in the
standard theory of optical Kerr effect. It represents a
Hamiltonian modification of the two-photon absorption model so that the
quantum mechanical equation reads
\begin{equation}
\label{eq:quant}
\frac{d a}{dt} =
-\left [ \gamma + i \delta + (\mu + i \Gamma)a^{\dagger}a \right ]a +
2\sqrt{\mu}\,a^{\dagger}b_{in}+\sqrt{2 \gamma_{c}}\, a_{in} +
\sqrt{2\gamma_{s}}\,w_{in}\,,
\end{equation}

\noindent
where Latin characters denote the annihilation operators for the
corresponding classical (Greek characters) modes.
Two extra terms not present in the classical analog appear, namely, a
white noise input, $w_{in}$, accounting for the fluctuations induced
by the scattering and the absorption in the crystal
($\gamma_{s} = \gamma-\gamma_{c}$) and a parametric ``gain''
term coming from the, classically empty, incoming harmonic mode, $b_{in}$.
Eq.\ (\ref{eq:quant}) is complemented with the boundary conditions
\cite{Pas94}
\begin{mathletters}
\label{eq:boundary}
\begin{eqnarray}
\label{eq:boundarya}
a_{out} & = & \sqrt{2\gamma_c}\, a - a_{in} \,,\\
\label{eq:boundaryb}
b_{out} & = & \sqrt{\mu}\,a^{2} - b_{in} \,,
\end{eqnarray}
\end{mathletters}

\noindent
from which the output spectra can be computed. Input fields are assumed
to be in coherent states. In particular, allowing a coherent state
different from the vacuum for the incoming harmonic mode we generalize
the system to the case of driving both modes. In the case
$\Gamma = \delta = 0$, the squeezing properties as well as the applicability
to quantum nondemoliton measurements of this system have been studied in
detail in \cite{Sch95}.

The used definitions for the creation
operators give the following relations with the usual experimental
parameters (see appendix in \cite{Sch95}): the input and output powers
are $P_{\omega,in/out} = \hbar \omega \langle a^{\dagger}_{in/out}
a_{in/out} \rangle$ and $P_{2\omega,in/out}= \hbar 2 \omega \langle
b^{\dagger}_{in/out} b_{in/out}\rangle$; the circulating power is $\hbar
\omega \langle a^{\dagger} a \rangle / \tau$ being $\tau$ the
round-trip time and the single-pass power-conversion efficiency (in
W$^{-1}$) is $2 \tau^{2} \nu/ \hbar \omega$.

\section{Linearized evolution equations and linear stability
analysis}{\label{LEE}}
 Defining fluctuation operators as
\begin{mathletters}
\label{eq:fluctope}
\begin{eqnarray}
a & = &\alpha +\delta a \,,
\label{eq:fluctopea}\\
a_{in,out} & = & \alpha_{in,out} +\delta a_{in,out} \,,
\label{eq:fluctopeain}\\
b_{in,out} & = & \beta_{in,out}+\delta b_{in,out} \,,
\label{eq:fluctopebin}
\end{eqnarray}
\end{mathletters}

\noindent
a linearization of Eqs.\ (\ref{eq:quant}) and (\ref{eq:boundary}) yields
\begin{eqnarray}
\label{eq:linquant}
\frac{d\,\delta a}{dt} & = &
-\left [ \gamma + i \delta + 2(\mu + i \Gamma)|\alpha|^{2}
\right]\delta a + \left [ 2\sqrt{\mu}\beta_{in} -
(\mu+i \Gamma)\alpha^{2}\right ] \delta a^{\dagger} \nonumber \\
& + & 2\sqrt{\mu}\,\alpha^{*}\delta b_{in}+
\sqrt{2 \gamma_{c}}\,\delta a_{in} + \sqrt{2\gamma_{s}}\,w_{in}\,,
\end{eqnarray}

\noindent
and
\begin{mathletters}
\label{eq:linboundary}
\begin{eqnarray}
\delta a_{out} & = & \sqrt{2 \gamma_{c}}\,\delta a - \delta a_{in}\,,
\label{eq:linboundarya} \\
\delta b_{out} & = & 2 \alpha \sqrt{\mu}\, \delta a - \delta b_{in}\,,
\label{eq:linboundaryb}
\end{eqnarray}
\end{mathletters}

\noindent
being $\alpha_{in,out},\beta_{in.out}$ the mean values of the
corresponding input and output modes and $\alpha$ a
stable fixed point of the classical counterpart of
Eq.\ (\ref{eq:quant}), i.e.\,
\begin{equation}
\label{eq:class}
\frac{d \alpha}{dt} =
-\left [ \gamma + i \delta + (\mu + i \Gamma) |\alpha|^{2} \right
]\alpha +
2\sqrt{\mu}\,\alpha^{*}\beta_{in}+\sqrt{2 \gamma_{c}}\,\alpha_{in}
\,.
\end{equation}

\noindent
Equating to zero the l.h.s.\ of Eq.\ (\ref{eq:class}) a ``state equation''
for the fixed points is obtained, namely,
\begin{equation}
\label{eq:state}
\alpha = \frac{\sqrt{2 \gamma_{c}}\left \{ \left[ \gamma+\mu n -i
(\delta + \Gamma n) \right ] \alpha_{in}+ 2 \sqrt{\mu} \,\beta_{in}\,
\alpha_{in}^{*} \right \}}
{(\gamma + \mu n)^{2}+(\delta+\Gamma n)^{2}-4 \mu |\beta_{in}|^{2}}
\,,
\end{equation}

\noindent
with $n = |\alpha|^{2}$. Let be $\theta,\, \phi$ and $\varphi$ the
phases of $\alpha,\, \alpha_{in}$ and $\beta_{in}$ respectively.
Then, dividing both sides of Eq.\ (\ref{eq:state})
by $e^{i\varphi/2}$
\begin{eqnarray}
\label{eq:statea}
|\alpha|e^{i(\theta-\varphi/2)} \left [ (\gamma + \mu n)^{2}
+ (\delta+\Gamma n)^{2} - 4 \mu |\beta_{in}|^{2} \right ] \: = \: & &
\nonumber \\
|\alpha_{in}| \sqrt{ 2 \gamma_{c}} \left [ \left ( \gamma + \mu n- i
(\delta+\Gamma n) \right ) e^{i(\phi-\varphi/2)}+
2 \sqrt{\mu}\,|\beta_{in}|e^{-i(\phi-\varphi/2)} \right ] \,. & &
\end{eqnarray}

\noindent
Taking the squared modulus in both sides a quintic equation for $n$ is
obtained
\begin{eqnarray}
\label{eq:neqn}
0 & = & n \left [ (\gamma + \mu n)^{2}
+ (\delta+\Gamma n)^{2} - 4 \mu |\beta_{in}|^{2} \right ]^{2} -
2 \gamma_{c}|\alpha_{in}|^{2} \left\{ (\gamma + \mu n)^{2}
+ (\delta+\Gamma n)^{2} + 4 \mu |\beta_{in}|^{2} + \right . \nonumber \\
& & \left .
4 \sqrt{\mu}\,|\beta_{in}| \left[ (\gamma+\mu n) \cos(2 \phi-\varphi)+
(\delta+\Gamma n) \sin(2 \phi -\varphi) \right] \right\} \,.
\end{eqnarray}

\noindent
The real and the imaginary part of Eq.\ ($\ref{eq:statea}$) determine the
$\sin(\theta-\varphi/2)$ and $\cos(\theta-\varphi/2)$ as functions of
the solutions of Eq.\ (\ref{eq:neqn})
\begin{mathletters}
\label{eq:phases}
\begin{eqnarray}
\cos(\theta-\varphi/2) & = & \frac{|\alpha_{in}|}{|\alpha|}
\sqrt{2\gamma_{c}}\,
\frac{(\gamma + \mu n+2\sqrt{\mu}\,|\beta_{in}|)\cos
(\phi-\varphi/2)+(\delta+\Gamma n) \sin (\phi-\varphi/2)}
{(\gamma+\mu n)^{2}+(\delta+\Gamma n)^{2}-4 \mu |\beta_{in}|^{2}}
\label{eq:cos}
\\
\sin(\theta-\varphi/2) & = & \frac{|\alpha_{in}|}{|\alpha|}
\sqrt{2 \gamma_{c}}\,
\frac{(\gamma + \mu n-2\sqrt{\mu}\,|\beta_{in}|)\sin
(\phi-\varphi/2)-(\delta+\Gamma n) \cos (\phi-\varphi/2)}
{(\gamma+\mu n)^{2}+(\delta+\Gamma n)^{2}-4 \mu |\beta_{in}|^{2}} \,.
\end{eqnarray}
\end{mathletters}
\label{eq.sin}

\noindent
Eq.\ (\ref{eq:neqn}) allows for numerical calculation of the fixed
points given the input fields. But it can be interpreted also as a
linear equation for $|\alpha_{in}|^{2}$, i.e.\,

\begin{equation}
\label{eq:ain}
2 \gamma_{c}|\alpha_{in}|^{2} = \frac{n \left [ (\gamma + \mu n)^{2}
+ (\delta+\Gamma n)^{2} - 4 \mu |\beta_{in}|^{2} \right ]^{2}}
{|\gamma + \mu n + 2 \sqrt{\mu} |\beta_{in}|
e^{i (2 \phi-\varphi)}|^{2} +
4 \sqrt{\mu}\,|\beta_{in}| (\delta+\Gamma n) \sin(2 \phi -\varphi)} \,.
\end{equation}

\noindent
The positive character of the r.h.s.\ is not always guaranteed and therefore
not for every value of the parameters a real positive $n$ is
possible. Notice, however, that in the cases in which this happens a
simultaneous change of the sign of $\delta$ and $\Gamma$ yields a
consistent set of parameter values. As we shall see, this fact will have
useful consequences in regarding the analysis of the quantum noise
behavior in the system.

The stability of the fixed points is governed by the real part of the
eigenvalues of the drift matrix associated with the linearized
evolution equation (\ref{eq:linquant}). Very simple algebra yields
\begin{equation}
\label{eq:eigen}
\lambda_{\pm}=-(\gamma+2 \mu \,n) \pm \sqrt{|(\mu+
i \Gamma)\alpha^{2}-2 \sqrt{\mu}\,\beta_{in}|^{2}-
(\delta+2\Gamma\,n)^{2}}\,.
\end{equation}

\noindent
Provided that the real part of both eigenvalues are negative the fixed
point will be stable. With respect to the phase-matched SHG case ($\Gamma =0$ and
$\beta_{in}=0$, always stable), although both $\Gamma$ and $\delta$ alone
tend to stabilize the dynamics, in combination are able of destabilize
the system.  A finite $\beta_{in}$, on the other hand, can promote
instability depending on its relative phase with respect to $\alpha$, the case
$\theta - \varphi/2 =\pm \pi/2$ maximizing the effect. All of these new
instabilities, however, correspond to zero eigenvalues without a finite
imaginary part. In other words, contrary to the double resonant SHG
there is no Hopf bifurcation and, consequently, no selfpulsing solution.

\section{Squeezing spectra}{\label{SS}}

For a given quadrature of the electric field,
$X_{\theta_{s}}^{out}(t) \equiv a_{out}(t) e ^{-i \theta_{s}}+
a_{out}^{\dagger}(t) e^{i\theta_{s}}$, the
squeezing spectrum is simply the noise spectrum of such
a quantity, i.e.\,
\begin{eqnarray}
\label{eq:Sw}
S(\omega) & = & C \int_{-\infty}^{\infty} \langle \delta
X_{\theta_{s}}^{out}(t) \delta X_{\theta_{s}}^{out}(t+\tau) \rangle \,
e^{-i\omega \tau} d \tau \nonumber \\
& = & C \int_{-\infty}^{\infty}  \langle \delta
X_{\theta_{s}}^{out}(\omega) \delta X_{\theta_{s}}^{out}(-\omega^{\prime})
\rangle d \omega^{\prime} \,,
\end{eqnarray}

\noindent
being $C$ some normalization constant and the averages are assumed
stationary. As a function of the annihilation and creation operators
Eq.\ (\ref{eq:Sw}) is rewritten as
\begin{equation}
\label{eq:Sw2}
S(\omega) = C \left [
\langle \delta a^{\dagger}_{out}(\omega) \delta a_{out}(-\omega)\rangle +
{\rm Re}\{\exp(-i2\theta_{s})
\langle \delta a_{out}(\omega) \delta a_{out}(-\omega) \rangle \}
\right ]\,,
\end{equation}

\noindent
where use has been made of the stationarity of the average and Re
denotes real part. From this expression it is evident that the noise is
minimized and therefore the squeezing effect maximized for a quadrature
phase such as
\begin{equation}
\label{eq:Sw3}
S(\omega) = C \left [
\langle \delta a^{\dagger}_{out}(\omega) \delta a_{out}(-\omega)\rangle -
\left |
\langle \delta a_{out}(\omega) \delta a_{out}(-\omega) \rangle
\right | \right ]\,,
\end{equation}

\noindent
corresponding to a phase
\begin{equation}
\label{eq:sqphase}
\theta_{m} = \frac{\nu(\omega) - \pi}{2} \,,
\end{equation}

\noindent
where $\nu(\omega)$ is the phase of
$\langle \delta a_{out}(\omega) \delta a_{out}(-\omega) \rangle$.
The spectrum of the conjugate quadrature (i.e.\, with a phase
$\nu(\omega)/2)$ corresponds to a plus sign in Eq.\ (\ref{eq:Sw3}) and
by virtue of the Heisenberg principle shows an excess noise above the
vacuum. Taking $C=1$
(corresponding to vacuum noise units) and splitting Eq.\ (\ref{eq:Sw3})
into a vacuum noise component plus a normally ordered part we finally
arrive to
\begin{equation}
\label{eq:Swf}
S_{-,+}(\omega) = 1 +
\langle : \delta a^{\dagger}_{out}(\omega) \delta a_{out}(-\omega) :
\rangle \mp \left |
\langle : \delta a_{out}(\omega) \delta a_{out}(-\omega) : \rangle
\right | \,,
\end{equation}

\noindent
for both the squeezing and the ``stretching'' spectra.
After tedious but simple algebra, the spectra of the fundamental
and second harmonic modes can be written as
\begin{mathletters}
\label{eq:spectrum}
\begin{eqnarray}
S^{a}_{-,+}(\omega) & = & 1 + 4 \gamma_{c} |B| \frac{N_{-,+}}{D}\,,
\label{eq:spectruma} \\
S^{b}_{-,+}(\omega) & = & 1 + 8 \mu n |B| \frac{N_{-,+}}{D}\,,
\label{eq:spectrumb}
\end{eqnarray}
\end{mathletters}

\noindent
where $B =2\sqrt{\mu} \,\beta_{in} -(\mu +i \Gamma) \alpha^{2}$ and
\begin{mathletters}
\label{eq:Sc}
\begin{eqnarray}
N_{-,+} & = & 2 |B| (\gamma + 2 \mu n) \mp \sqrt{\left [
(\gamma + 2 \mu n)^{2} - (\delta + 2 \Gamma n)^{2} + |B|^{2} +\omega^{2}
\right ]^{2} + 4 (\gamma + 2 \mu n)^{2}
(\delta + 2 \Gamma n)^{2}}\,,
\label{eq:ScN}\\
D & = & \left[(\gamma+2\mu n)^{2} + (\delta + 2 \Gamma n)^{2} -
|B|^{2} - \omega^{2} \right ]^{2} + 4 (\gamma+2\mu n)^{2} \omega^{2}\,.
\label{eq:ScD}
\end{eqnarray}
\end{mathletters}

\noindent
The correlations defining the squeezing phase $\nu(\omega)$ are given
by
\begin{mathletters}
\label{eq:corr}
\begin{eqnarray}
\label{eq:corra}
\langle \delta a_{out}(\omega)\delta a_{out}(-\omega) \rangle & = &
4\gamma_{c} B \left [\omega^{2} + |B|^{2} +
(\gamma + 2 \mu n)^{2} - (\delta + 2 \Gamma n)^{2} + i \,
2 (\gamma + 2 \mu n) (\delta + 2 \Gamma n)\right ]/D \,,\\
\label{eq:corrb}
\langle \delta b_{out}(\omega)\delta b_{out}(-\omega) \rangle & = &
8\mu\,\alpha^{2} B \left [\omega^{2} + |B|^{2} +
(\gamma + 2 \mu n)^{2} - (\delta + 2 \Gamma n)^{2} + i\,
2 (\gamma + 2 \mu n) (\delta + 2 \Gamma n)\right ]/ D \,.
\end{eqnarray}
\end{mathletters}

\noindent
The trigonometric equations for the corresponding phases are quite
complicated and rather useless. However, an interesting
consequence can directly be drawn from Eqs.\ (\ref{eq:corr}), namely,
for detuning-s such as $\delta+2 \Gamma n =0$ the phases
are independent of $\omega$ equaling those of $B$ and $\alpha^{2} B$
respectively.

\section{Squeezing performance}{\label{SP}}
In the previous sections we have developed the general raw formulae
regarding quantum noise in the system. This section is devoted to the
analysis of the quantum noise behavior implied by them an in
particular to proceed with our program of finding optimum quantum
noise reduction. However, before any specific assessment of the
quantum noise performance we will elaborate a little more on the
formulae (mainly by adequate normalizations) in order to gain
physical insight and ease our task. As an aftermath we shall obtain
general results going well beyond the specifics of the system addressed
here.

\subsection{General results concerning one-mode systems}
Let us begin defining a nonlinear
and a total decay rate as $\gamma_{nl} \equiv 2 \mu n$ and
$\gamma_{t} \equiv \gamma +\gamma_{nl}$ respectively.
We shall scale the evolution with this total decay rate defining an
dimensionless time $\tau \equiv \gamma_{t}\,t$.
In the spectra (\ref{eq:spectrum}) the only dependence on $\theta$ is
through $B$ disappearing for $\beta_{in}=0$.
It is also possible to restrict this dependence to such
a term directly in Eq.\ (\ref{eq:linquant}) and
Eq.\ (\ref{eq:linboundary}) by means of appropriate phase shifts of the
modes. All together account for
\begin{eqnarray}
\label{eq:normeq}
\frac{d\,\delta c}{d \tau} & = &
-\left [1 + i \Delta \right]\delta c +
\tilde{B}\,\delta c^{\dagger} \nonumber \\
& + & \sqrt{2 \tilde{\gamma}_{nl}}\;\delta d_{in} +
\sqrt{2 \tilde{\gamma}_{c}}\;\delta c_{in} +
\sqrt{2\tilde{\gamma}_{s}}\,s_{in}\,,
\end{eqnarray}

\noindent
where the tilde represents divided by $\gamma_{t}$,
$\Delta = \tilde{\delta} + 2 \tilde{\Gamma} n $, $\tilde{B} =
2\sqrt{\tilde{\mu}}\,\delta_{in} - (\tilde{\mu}+i \tilde{\Gamma})n$
and the modes are redefined as
\begin{mathletters}
\label{eq:nfields}
\begin{eqnarray}
c & \equiv &  a \, e ^{-i \theta}\,,
\label{eq:c}\\
c_{in,out} & \equiv &
\frac{a_{in,out}}{\sqrt{\gamma_{t}}} \, e ^{-i \theta}\,,
\label{eq:cinout}\\
 d_{in,out} & \equiv &
\frac{b_{in,out}}{\sqrt{\gamma_{t}}} \, e ^{-i 2 \theta}\,,
\label{eq:dinout}\\
s_{in} & \equiv & \frac{w_{in}}{\sqrt{\gamma_{t}}} \, e ^{-i \theta}\,.
\label{eq:sin}
\end{eqnarray}
\end{mathletters}

\noindent
In agreement with the previous notation $\delta_{in}$ denotes the mean
value of $d_{in}$. The boundary conditions of the new modes are
\begin{mathletters}
\label{eq:nlinboundary}
\begin{eqnarray}
\delta c_{out} & = & \sqrt{2 \tilde{\gamma}_{c}}\,\delta c - \delta c_{in}\,
\label{eq:nlinboundaryc} \\
\delta d_{out} & = & \sqrt{2 \tilde{\gamma}_{nl}}\, \delta c - \delta d_{in}\,.
\label{eq:nlinboundaryd}
\end{eqnarray}
\end{mathletters}

\noindent
For coherent states, the correlations of the new input modes remain
as white noise but in the scaled time $\tau$. We shall refer to the
previous formulae as the tilde normalization.

The evolution equation (\ref{eq:normeq}) encodes the dynamic response of
the intracavity system to a series of noisy input channels ($\delta
d_{in}$, $\delta c_{in}$ and $s_{in}$). Quantum Mechanical
consistency, i.e., conservation of equal-time conmutators, imposes a
fluctuation-dissipation relation which under this normalization reads
\begin{equation}
\label{eq:add1}
\tilde{\gamma}_{nl}+\tilde{\gamma}_{c}+\tilde{\gamma}_{s}=1\,.
\end{equation}

\noindent
The evolution equation (\ref{eq:normeq}) along with Eqs.\
(\ref{eq:nlinboundary}) are now written in such a way that the input-output
couplings are real-valued as in the standard input-output formalism
\cite{Coll84,Gar91}. This is a completely general result. Provided a well
defined linearized theory in the sense of preserving equal-time
conmutators we only need the adequate set of phase shifts of the input
channels (a trivial unitary transformation preserving conmutators) making the
couplings real-valued to obtain a theory formally equal to the standard
input-output formalism simply because this is the theory preserving
the equal-time conmutators when the couplings are real-valued. Thus,
for any system with only one effective mode there is a formulation in which the
intracavity field follows
\begin{equation}
\label{eq:geneqnorm}
\frac{d\,\delta c}{d \tau}  =
-\left [1 + i \Delta \right]\delta c +
\tilde{B}\,\delta c^{\dagger} + \sum_{n=1}^{N}
\sqrt{2 \tilde{\gamma}_{n}} \,\delta c_{in}^{n}\,,
\end{equation}
with
\begin{equation}
\label{eq:genadd1}
\sum_{n=1}^{N} \tilde{\gamma}_{n} = 1 \,.
\end{equation}

\noindent
The frequency scale $\gamma_{t}$ defining the dimensionless time $\tau$
is just the real part of the factor multiplying $\delta c$
after the phase shifts. N-1 of the input channels will have a time-reversed
counterpart corresponding to the outgoing channels fulfilling
\begin{equation}
\label{eq:gennlinboundary}
\delta c_{out}^{n}  =
\sqrt{2 \tilde{\gamma}_{n}}\,\delta c - \delta c_{in}^{n}\,.
\end{equation}

\noindent
The remaining input channel will account for the irreversible losses.
The corresponding spectra are related with the intracavity spectra by
\begin{equation}
\label{eq:genspectra}
S_{-,+}^{n}(\tilde{\omega}) = 1 + :S_{-,+}^{n}(\tilde{\omega}):
\; = \; 1 + 2 \tilde{\gamma}_{n} :S_{-,+}(\tilde{\omega}): \,,
\end{equation}
where $:S_{-,+}(\tilde{\omega}):$ denotes the intracavity spectra and
we have made use of the proportionality of normally ordered
intracavity and outgoing correlations \cite{Gar91}. The spectra
$S_{-,+}^{n}(\tilde{\omega})$ coincide with the spectra of the original
formulation as the new outgoing modes are just a phase sift of the
originals.

Let us now define a sort of ``reference'' system with only one
time-reversible input channel, i.e.,
\begin{equation}
\label{eq:refeq}
\frac{d\,\delta c}{d \tau} =
-\left [1 + i \Delta \right]\delta c + \tilde{B}\,\delta c^{\dagger}
+ \sqrt{2}\, \delta c_{in}^{ref}\,,
\end{equation}

\noindent
and
\begin{equation}
\label{eq:refbc}
\delta c_{out}^{ref} = \sqrt{2}\, \delta c - \delta c_{in}^{ref} \,.
\end{equation}

\noindent
Obviously $:S_{-,+}^{ref}(\tilde{\omega}): \; = 2
:S_{-,+}(\tilde{\omega}):$ so that we get finally
\begin{equation}
\label{eq:Scentral}
 S_{-,+}^{n}(\tilde{\omega}) = 1 +
 \tilde{\gamma}_{n} :S_{-,+}^{ref}(\tilde{\omega}): \,.
\end{equation}

\noindent
This is the central result of this section. Let us elaborate a little
about its interpretation. Squeezing in a given output channel
means that for a certain range of phase shifts the corresponding quadratures
show an intensity of their fluctuations below that of the associated
incoming channel (assumed in a coherent state). In view of
Eqs.\ (\ref{eq:gennlinboundary}), the amplitude of the outgoing fluctuations
is a coherent superposition of the intracavity and the incoming
fluctuations. Squeezing is possible if an adequate correlation between
$\delta c$ and the relevant input channel is established. But the intracavity
field is nothing else than the dynamic response of the intracavity system to
the incoming channels. The input channels are uncorrelated and so the
dynamic response of the intracavity system to them. A given input
channel can consequently correlate only with the dynamic response to itself.
The presence of any other input channel can only degrade the effect.
The great advantage of the tilde normalization is that makes this fact
explicit. Indeed, Eq.\ (\ref{eq:Scentral}) express the output spectra
as the dynamic response of the system to an isolated input channel, i.e.,
$:S_{-,+}^{ref}(\tilde{\omega}):$, scaled down by the ``static''
contribution to the noise owing to the presence of extra input
channels. The scale factor $\tilde{\gamma}_{n}$ is just the ratio
between the coupling constant of the chosen output channel and the
sum of all of them.

Eq.\ (\ref{eq:Scentral}) greatly simplifies our task of finding the
optimum path to maximum noise reduction as we can center our efforts onto the
simple reference system described by Eqs.\ (\ref{eq:refeq}) and
(\ref{eq:refbc}). Even more interesting the results concerning the
reference system will be of general applicability to any one-mode
system including as such any multiply resonant system under
adiabatic elimination of all the modes but one. The normally ordered spectra
of the reference system are easily calculated as
\begin{equation}
\label{eq:SN}
:S^{ref}_{-,+}(\tilde{\omega}):\; = 4 |\tilde{B}| \frac{
2|\tilde{B}| \mp \sqrt{(1+\tilde{\omega}^{2}+|\tilde{B}|^{2}-\Delta^{2})^{2}
+4 \Delta^{2}}}{(1-\tilde{\omega}^{2}-|\tilde{B}|^{2}+\Delta^{2})^{2}
+4 \tilde{\omega}^{2}} \,.
\end{equation}

\noindent
Our first step is to determine if the dynamic response is capable of
a total noise suppression. Perfect squeezing can only occur at a dynamic
instability. Equaling to zero the l.h.s.\ of Eq.\ (\ref{eq:eigen})
(the only possible unstable eigenvalue) and after proper normalization an
equation determining the instability can be written as

\begin{equation}
\label{eq:refins}
1 + \Delta^{2} = |\tilde{B}|^{2}  \,.
\end{equation}

\noindent
Written in this way an interesting parallelism with the standard OPO
below threshold shows up, i.e.\, an instability appears when the modulus of
the ``losses'' coefficient equals that of the ``parametric''
coefficient, a sort of natural extension of the condition for the
instability in the conventional OPO for which the coefficients are real.
Inserting the instability condition (\ref{eq:refins}) in
$:S^{ref}_{-}(\tilde{\omega}):$ results in
\begin{equation}
\label{eq:Si}
:S^{ref}_{I}(\tilde{\omega}):\; = 4 |\tilde{B}| \frac{
2|\tilde{B}|-\sqrt{4 |\tilde{B}|^{2} +
\tilde{\omega}^{2}(\tilde{\omega}^{2}+4)}}{
\tilde{\omega}^{2}(\tilde{\omega}^{2}+4)} \,.
\end{equation}

\noindent
Applying L'Hopital's rule with respect to $\tilde{\omega}^{2}$,
$:S^{ref}_{I}(\tilde{\omega}):$ equals -1 at
$\tilde{\omega}=0$, that is, perfect squeezing is obtained at the instability, again in
parallel with OPO. In other words, the dynamic response of the system
assuming that the condition Eq.\ (\ref{eq:refins}) is reachable,
is capable of a complete suppression of quantum noise.

Spectrum (\ref{eq:SN}) is simple enough to permit analytical
optimization. Taking
partial derivative in $:S^{ref}_{-}(\tilde{\omega}):$ with respect to
$\tilde{\omega}$ and equaling to zero, $\tilde{\omega}=0$ appears as
the optimum point whatever the values of $\Delta$ and $|\tilde{B}|$.
The same applies to $\Delta = 0$ when taking partial derivative with
respect to $\Delta$. Notice that this last condition implies also a
squeezing phase independent of the frequency. The optimized noise obtained
imposing these two conditions simplifies to
\begin{equation}
\label{eq:Sopt}
:S_{opt}:\; = -\frac{4 |\tilde{B}|}{(1+|\tilde{B}|)^{2}}\,,
\end{equation}

\noindent
with a minimum at the instability $|\tilde{B}| = 1$ approached
monotonically. These conditions ($\tilde{\omega}=0$, $\Delta=0$ and
$|\tilde{B}| = 1$) will help us in finding optimum paths. In
particular, moving $|\tilde{B}|$ from zero to one while maintaining
$\Delta=\tilde{\omega}=0$ defines an optimum path reaching the
instability for the ``reference'' model.

An optimum path is defined solely by the squeezing spectrum leaving
aside the ``stretching'' one. It is important to study also the accompanying
excess noise on the conjugate quadrature for it could invalidate in
practice the optimum path if this excess noise is unbearable high.
The minimal excess noise production imposed by the Heisenberg
principle corresponds to $S_{-}(\omega)S_{+}(\omega)=1$. It is a perfect
complementary relation between quadratures: the deamplification of
fluctuations in a given quadrature must equal the amplification of
fluctuations in the conjugate. In that case we are dealing with a Minimum
Uncertainty State (MUS) for those quadratures, the text-book
definition of a squeezed state. Adding 1 to Eq.\ (\ref{eq:SN}) and
after some minor algebra
\begin{equation}
\label{eq:SS}
S_{-,+}^{ref}(\tilde{\omega}) =
\frac{\left (
2 |\tilde{B}| \pm \sqrt{(\tilde{\omega}^{2}+|\tilde{B}|^{2}+1-\Delta^{2})^{2}
+4 \Delta^{2}}
\right )^{2}}
{(1-\tilde{\omega}^{2}-|\tilde{B}|^{2}+\Delta^{2})^{2}
+4 \tilde{\omega}^{2}} \,.
\end{equation}

\noindent
Straightforward algebra leads to $S_{-}^{ref}(\tilde{\omega})
S_{+}^{ref}(\tilde{\omega}) =1$. The excess noise is minimum again
in parallel to the standard OPO system with real coefficients.

\subsection{Standard normalization}
As discussed in the introduction we are here principally interested in
the squeezing behavior with respect to the photon number $n$.
Unfortunately the tilde normalization is inappropriate to such a task
as the frequency scale depends on $n$ itself.
It is far more convenient to use $\gamma^{-1}$ as the time scale instead of
$\gamma_{t}^{-1}$ and to normalize the photon number as $m = \nu
n/\gamma$. In complete parallelism to the tilde normalization we have
then,
\begin{eqnarray}
\label{eq:normeqhat}
\frac{d\,\delta c}{d \tau} & = &
-\left [1 + 2 m K_{r} + i (\hat{\delta} + 2 m K_{i}) \right]\delta c
+ \left [\sqrt{K_{r}}\,\eta_{in} - (K_{r}+i K_{i}) m \right ]\,
\delta c^{\dagger} \nonumber \\
& + & 2 \sqrt{m K_{r}}\;\delta d_{in} +
\sqrt{2 \hat{\gamma}_{c}}\;\delta c_{in} +
\sqrt{2\hat{\gamma}_{s}}\,s_{in}\,,
\end{eqnarray}
where the hat represents divided by $\gamma$ and
\begin{equation}
\label{eq:etain}
\eta_{in} \equiv \frac{2 \sqrt{\nu}}{\gamma}\,\beta_{in}e^{-i2\theta}\,,
\end{equation}

\noindent
which represents the harmonic mode input amplitude normalized to the
value at the standard OPO threshold. The spectra (\ref{eq:spectrum})
become now
\begin{mathletters}
\label{eq:hatspectrum}
\begin{eqnarray}
S^{a}_{-,+}(\hat{\omega}) & = & 1 + 4 \hat{\gamma}_{c}
 |\hat{B}| \frac{\hat{N}_{-,+}}{\hat{D}}\,,
\label{eq:hatspectruma} \\
S^{b}_{-,+}(\hat{\omega}) & = &
1 + 8 K_{r}m |\hat{B}| \frac{\hat{N}_{-,+}}{\hat{D}}\,,
\label{eq:hatspectrumb}
\end{eqnarray}
\end{mathletters}

\noindent
with
\begin{mathletters}
\label{eq:hatSc}
\begin{eqnarray}
\hat{N}_{-,+} & = & 2 |\hat{B}| (1 + 2 K_{r} m) \mp
\sqrt{\left [ (1 + 2 K_{r} m)^{2} - (\hat{\delta} +
2 K_{i} m)^{2}+ |\hat{B}|^{2} +\hat{\omega}^{2} \right ]^{2} +
4 (1 + 2 K_{r} m)^{2}
(\hat{\delta} +2 K_{i} m)^{2}}\,,
\label{eq:hatScN}\\
\hat{D} & = &  \left [(1+2 K_{r} m)^{2}+(\hat{\delta}+ 2 K_{i} m)^{2}
- |\hat{B}|^{2} - \hat{\omega}^{2} \right ]^{2}+
4 (1+2 K_{r} m)^{2} \hat{\omega}^{2} \,,
\label{eq:hatScD}
\end{eqnarray}
\end{mathletters}

\noindent
and $\hat{B} = \sqrt{K_{r}}\,\eta_{in} - (K_{r}+i K_{i}) m$.
We shall refer to the above formulae
as the hat normalization. To refer the frequency to the cavity decay
constant with the subsequent re-normalization of the system
parameters, i.e.\, the hat normalization, is quite a standard procedure
in the literature.

Some care must be taken when studying the quantum noise behavior as a
function of $n$ (or $m$). It is not a free parameter of the problem
as would be the input fields or the phase mismatch but it is in a nonlinear
relation with them. We need, therefore, to check that the proposed
values of $n$ are indeed a solution of Eq.\ (\ref{eq:neqn}).
Fortunately the spectra (\ref{eq:spectrum}) do not depend on the overall
sign of $\delta + 2 \Gamma n$ and therefore the conclusions reached in section
\ref{LEE} about the existence of $|\alpha_{in}|$ permit a safe
variation of $n$ in search of strong noise reduction provided the
stability of the corresponding fixed points.

\subsection{Squeezing at the fundamental mode}
Applying Eq.\ (\ref{eq:Scentral}) to the fundamental mode
\begin{equation}
\label{eq:Satilde}
S^{a}_{-,+}(\tilde{\omega}) = 1 + \tilde{\gamma}_{c}
:S^{ref}_{-,+}(\tilde{\omega}): \,.
\end{equation}

\noindent
It is clear that the best
performance corresponds to $\gamma_{nl} = 0$, that is, either $n=0$ or $\mu =
0$, as then $\tilde{\gamma}_{c}$ maximizes to
$\eta \equiv \gamma_{c}/(\gamma_{c}+\gamma_{s})$
(the escape efficiency of the cavity). The case $n=0$ corresponds to
the very well known case of squeezed vacuum generation. For $\mu =0$
and finite $n$ the system is formally equivalent to a resonant optical
Kerr effect system whose quantum noise behavior has been amply studied
previously \cite{Rey89}. The condition (\ref{eq:refins}) reduces for $\mu =0$
to $\delta = -2 n \Gamma \pm \sqrt{n^{2}\Gamma^{2}-\gamma^{2}}$, the
well known turning points of optical dispersive bistability \cite{Rey89}
but with the nonlinear dispersion induced by cascading. Indeed, such cascading
induced bistability has been experimentally demonstrated in \cite{Whi96a}.
Rewriting it within the hat normalization the condition reads
\begin{equation}
\label{eq:insKerr}
\hat{\delta}_{\pm} = -2 m K_{i} \pm \sqrt{m^{2} K_{i}^{2} - 1} \,,
\end{equation}

\noindent
where $K_{i}=-1/\pi$ as it is evaluated at $\Delta k L_{m} = 2 \pi$.
Once $\tilde{\gamma}_{c}$ is independent of $\Delta k$ and $n$,
the optimum path with respect to $m$ (the only remaining free parameter)
is determined solely by the reference system. It corresponds to increase
$m$ till $m=\pi$ (where the condition (\ref{eq:insKerr}) is reached)
while maintaining $\hat{\delta} = 2 m / \pi$ and $\omega = 0$.
Fig.\ \ref{fg:SaKerr} displays the evolution of
both the maximum squeezing and the maximum excess noise following
such a path for three values of the escape
efficiency, namely, 0.9, 0.99 and the ideal 1. The noise is expressed in
dB's with respect to the vacuum noise. A Heisenberg limited excess noise
appears in such a case as a specular image of the squeezing. The
instability is signaled by the divergence in the excess noise.
Above it, the curves shown are not physical as they correspond to
unstable fixed points.
The case $\eta = 0.99$ in Fig.\ \ref{fg:SaKerr} shows an excellent
behavior with an almost Heisenberg limited excess noise till
near the instability.
\begin{figure}
\centerline{\includegraphics*[scale=0.5]{SaKerr.eps}}
\caption{Noise spectra at zero frequency of the fundamental mode following and optimum
path for three escape efficiencies of the cavity including the ideal
case $\eta = 1$. The curves above the
divergences are not physical.}
\label{fg:SaKerr}
\end{figure}

Fig.\ \ref{fg:SaKcomp} illustrates the idea of optimum path by comparing
the $\eta=0.99$ plot of Fig.\ \ref{fg:SaKerr} against various cases
with fixed values of $\hat{\delta}$. Below $m=\pi$, for a given $m$
the maximum squeezing is obtained when $\Delta = 0$ as expected.
Above $m=\pi$ it is not possible to reach the minimum noise of
$1-\eta$ fulfilling $\Delta = 0$.
\begin{figure}
\centerline{\includegraphics*[scale=0.5]{SaKcomp.eps}}
\caption{Comparison among noise spectra at zero frequency (fundamental
mode) following various paths in the parameter space.
The optimum one corresponds to $\Delta = 0$. Above the divergences
the results are not physical.}
\label{fg:SaKcomp}
\end{figure}

Being $\mu =0$ the formulae simplify enough for allowing a simple
expression for the squeezing phase. More specifically, from
$B=-i\Gamma \alpha^{2}$, $\theta_{m}=\theta + \pi/2$. On the other
hand substituting Eq.\ (\ref{eq:state}) in Eq.\ (\ref{eq:boundarya}) results in
$\sqrt{2 \gamma_{c}}\alpha_{out} = \alpha (\gamma_{c}-\gamma_{s} + i \Gamma
n)$, giving a squeezing phase relative to that of the output field of
\[
\frac{\pi}{2} - \arctan \left (\frac{\Gamma n}{\gamma_{c}-\gamma_{s}} \right
)\,.
\]

\noindent
At the instability $\Gamma n = \gamma$ and for low
$\gamma_{s}$ it approaches $45^{o}$. There is a possibility of an extra
control of the squeezing phase not present in the conventional Kerr
effect system by making use of the harmonic mode. Taking a finite
$\mu$ but low enough so that $\mu n \ll \gamma$ and at the same
time a $\beta_{in}$ high enough to imply $2\sqrt{\mu} \beta_{in} \approx
\gamma$, still we will have $\gamma_{nl} \approx \gamma$ while the
squeezing phase relative to the output field will depend on both the
modulus and the relative phase between the input fields. In practice, however,
maintaining $\mu n$ very low could imply a exceedingly high $\beta_{in}$
in order to have a $2\sqrt{\mu}\beta_{in}$ intense
enough for a significant influence on the final phases. Of course, the
instability point would accordingly depend on $\beta_{in}$.

There is no hope of any behavior similar to the reported in \cite{Cab97}
as a competition between second and third order nonlinearities needs
$\mu \neq 0$, opening the fundamental mode to the fluctuations of the
input harmonic mode with strong deleterious effects. At most, some
remnants of the
enhanced efficiency coming from the competition between nonlinearities can be
observed for low $\gamma_{nl}$. Then, as shown in
Fig.\ \ref{fg:Sa}, the best working point is not necessarily located at
$\mu = 0$, i.e.\, maximum squeezing is obtained with a finite mismatch.
\begin{figure}
\centerline{\includegraphics*[scale=0.5]{Sa.eps}}
\caption{Squeezing in the fundamental mode at zero frequency as a
function of the phase mismatch for a low intracavity photon number.}
\label{fg:Sa}
\end{figure}

\subsection{Squeezing at the harmonic mode}
For the harmonic mode Eq.\ (\ref{eq:Scentral}) yields
\begin{equation}
\label{eq:Sbtilde}
S^{b}_{-,+}(\tilde{\omega}) = 1 + \tilde{\gamma}_{nl}
:S^{ref}_{-,+}(\tilde{\omega}): \,.
\end{equation}

\noindent
Now, the situation is the complete opposite: the performance is favored
by a finite $\mu$ in order to have a non-zero $\gamma_{nl}$ and a large
$n$ to approach the ratio $\tilde{\gamma}_{nl}$ to one. In fact, under
ideal conditions of perfect dynamic noise suppression and no absorption
and scattering losses ($\gamma_{s}=0$), the squeezing in both modes
are complementary in the sense of
\begin{equation}
\label{eq:abcomple}
S^{a}_{-}+S^{b}_{-} = 2
-\frac{\gamma_{c}}{\gamma_{t}}-\frac{\gamma_{nl}}{\gamma_{t}} \;=\; 1\,,
\end{equation}

\noindent
a direct consequence of the fluctuation-dissipation relation
(\ref{eq:genadd1}). This complementarity has been previously reported for
the doubly resonant degenerate parametric oscillator \cite{Fab90}.
The maximum squeezing available for the harmonic mode whatever the
dynamic response of the system is easily obtained by setting
$:S^{ref}_{-,+}(\tilde{\omega}):$ to -1 in Eq.\ (\ref{eq:Sbtilde}), that is,
\begin{equation}
\label{eq:ilimit}
S_{M} = 1 - \frac{2 m K_{r}}{1+2 m K_{r}} \; = \;
\frac{1}{1+2 m K_{r}} \,.
\end{equation}

\noindent
This static contribution to the noise is now nonlinear in the sense
that it depends on the phase mismatch and $m$. An immediate consequence
of Eq.\ (\ref{eq:ilimit}) is the possibility of an arbitrarily large
quantum noise reduction for any finite value of $K_{r}$. The 1/9 limit
of the conventional phase-matched SHG is therefore due to a failure
of the setup to maximize the dynamic response of the system. Let us
center then, firstly in the SHG-like case with $\beta_{in}=0$ as it
includes the above mentioned conventional setup (the experiments in
\cite{Pas94} and \cite{Tsu95}). The instability
points are now given by (directly in the hat normalization)
\begin{equation}
\label{eq:hatinsb}
\hat{\delta}_{\pm} = - 2 m K_{i} \pm \sqrt{m^{2}
(K_{i}^{2}-3 K_{r}^{2}) - 4 K_{r} m - 1} \,.
\end{equation}

\noindent
Both kinds of nonlinearities (dispersive and absorptive) are in this case
necessary as the factor $K_{i}^{2}-3 K_{r}^{2}$ needs to be positive to
allow $\hat{\delta}_{\pm}$ to be real. The phase-matched case is
therefore excluded.
\begin{figure}
\centerline{\includegraphics*[scale=0.5]{Ki2-3Kr2.eps}}
\caption{The value of $K_{i}-3 K_{r}$ as a function of the phase
mismatch.}
\label{fg:g-3f}
\end{figure}

\noindent
Fig.\ \ref{fg:g-3f} shows $K_{i}^{2}-3 K_{r}^{2}$ as a function of the
phase mismatch and indeed near above $\pi$ it is positive. Optimum
approaches to (\ref{eq:hatinsb}) are now more difficult to
evaluate than in the fundamental mode as both the dynamic processing of
the noise and the static contribution from the noise inputs (encoded
in $\tilde{\gamma}_{nl}$) depend on $m$ and $\Delta k$. With respect to
to $m$ is clear that the static part is optimized at $m \rightarrow
\infty$. This limit can be approached letting $\tilde{\omega} = 0$
and $\hat{\delta} = - 2 K_{i} m$ (i.e.\, $\Delta = 0$). $|\tilde{B}|$
reduces in this case of $\beta_{in}=0$ to $m
\sqrt{K_{r}^{2}+K_{i}^{2}}/(1+2 m K_{r})$ showing a monotonic
increasing behavior with respect to $m$ from 0 to the maximum
(at $m \rightarrow \infty$)
\begin{equation}
\label{eq:Bmax}
|\tilde{B}| = \frac{1}{4}\left [1+\left (\frac{K_{i}}{K_{r}}\right )^{2}
\right ] \,.
\end{equation}
Notice that for $K_{i}^{2}=3 K_{r}^{2}$ it consistently equals 1.
We have then, both $\tilde{\gamma}_{nl}=1$  and the fastest approach to 1 of
$|\tilde{B}|$ when $m \rightarrow \infty$. Therefore, the squeezing
along an optimum path with respect to $\Delta k$ is given by substituting
Eq.\ (\ref{eq:Bmax}) in the spectrum (\ref{eq:Sopt}) and then the
obtained $:S_{opt}:$ in Eq.\ (\ref{eq:Sbtilde}). Obviously, in a
real experiment $m$ can be large but always finite. Let us take as a
``large'' $m$ one giving a $S_{M}$ around 20 dB as in the
case $\eta = 0.99$ of Fig.\ \ref{fg:SaKerr}. This corresponds to
$m=50$. Figure \ref{fg:Sbkm50} displays $S^{b}_{-,+}$ as a function
of the phase mismatch in such a case.
To illustrate the modulation exerted by $S_{M}$ we have take this time
$\hat{\delta}$ equal to the real part of Eq.\ (\ref{eq:hatinsb}) plus a
very small number. In this way the plot remains valid for the whole range
of the $\Delta k L_{m}$. While Eq.\ (\ref{eq:hatinsb}) is complex the condition
$\Delta =0$ is almost fulfilled and above the instability the noise
suppression reduction follows $S_{M}$. Again the pernicious effect of
the instability regarding the excess noise has a very short range.
For comparison $S_{M}$ is also depicted.
\begin{figure}
\centerline{\includegraphics*[scale=0.5]{Sbkm50.eps}}
\caption{Noise spectra at zero frequency of the harmonic mode
following a nearly optimum path with respect to the phase mismatch for the
SHG like case.}
\label{fg:Sbkm50}
\end{figure}

The optimum path with respect to $m$ is much more complicated to find
because the intricate dependence of $K_{i}$ and $K_{r}$ with respect
to the phase mismatch. Figure \ref{fg:Sboptbin0} has been generated
finding numerically the minima of $S_{-}^{b}(0)$ while scanning
the range of $m$. For comparison the phase-matched case is also depicted
showing an asymptotic behavior towards $-10 \log 9$. For low values
of $m$ the effect of $\tilde{\gamma}_{nl}$ overwhelms the dynamic
response so that the best value corresponds to maximize $K_{r}$.
As soon as the two curves depart from each other the dynamic response
dominates the behavior and the minimum noise is at the instability as in
Fig.\ \ref{fg:Sbkm50}. At this stage the optimum path begins to follow
the instability all the time. It should be taken then, as a
mathematical limit. However, in view of Fig. \ref{fg:Sbkm50}, before
reaching it, bearable values of the excess noise are accessible with
a slight diminution of the squeezing.
\begin{figure}
\centerline{\includegraphics*[scale=0.5]{Sboptbin0.eps}}
\caption{Squeezing in the harmonic mode along an optimum path with
respect to the normalized intracavity photon number ($m$) for the SHG
like case compared with the phase-matched SHG case.}
\label{fg:Sboptbin0}
\end{figure}

It is worth to mention that a squeezing as large as
48\% induced by cascading has been very recently reported \cite{Kas97}.
The cascading was due, however, to a detuning of the pump mode in
a triply resonant non-degenerate OPO with a much lower finesse for the
pump mode rather than by phase mismatch.
Under such conditions a cascaded $\chi^{(3)}$ is also
induced leading ideally to perfect squeezing in the pump mode.

Although a finite mismatch allows to reach $S_{M}$,
the overall optimum working point corresponding to
$\Delta k = 0$ is out of reach. The question arises then, of if it is
possible to fulfill Eq.\ (\ref{eq:refins}) at $\Delta k = 0$. A glance
at the definition of $B$ suggests it should be possible adding a
driving to the harmonic mode. In such a case $\tilde{\gamma}_{nl}$ simplifies
to $2m/(1+2m)$ obviously independent of $\eta_{in}$. Constructing an optimum path
with respect to the harmonic input reduces then, to set $\Delta=\tilde{\omega}=0$.
Phase matching along with $\Delta = 0$ implies $K_{r} = 1$, $K_{i} = 0$ and
$\hat{\delta} = 0$ so that the instability condition (\ref{eq:refins})
simplifies to
\begin{equation}
\label{eq:etains}
1 + 2 m = |\eta_{in} - m| \,,
\end{equation}

\noindent
a perfectly achievable condition.
We can further optimize by choosing the phase of $\eta_{in}$
adequately to approach $\tilde{B}$ to one as much as possible.
The extreme cases correspond to $\eta_{in}$ real, i.e.,
$\eta_{in}= - (1 + m)$ and $\eta_{in}= 1 + 3 m$.
From Eqs.\ (\ref{eq:phases}) and (\ref{eq:ain}) it is easy to check
that they correspond respectively to
$\phi-\varphi/2 = \pi$ and $\phi-\varphi/2=0$. The negative case
maximizes $\tilde{B}$. It has been previously reported in \cite{Sch95}.
Taking squared modulus of Eq.\ (\ref{eq:boundaryb})
the negative case appears as promoting harmonic output power
while the converse is valid for the positive.
The squeezing phase is also easy to calculate in this case.
In particular, given the correlation (\ref{eq:corrb}), $\nu(\omega)$
is determined by the phase of $\alpha^{2} B$ (independent of $\omega$
as $\Delta =0$), something proportional to
\[
(\eta_{in}-m) e^{i 4 \theta} \,.
\]

\noindent
The corresponding squeezing phases are $\theta_{m} = 2\theta + \pi$
for the negative case while for the positive case it changes from
$\theta_{m} = 2\theta + \pi$ to $\theta_{m} = 2\theta + \pi/2$ at
$\eta_{in}=m$. On the other hand, the output harmonic amplitude is
proportional to (see Eq.\ (\ref{eq:boundaryb}))
\begin{equation}
\label{eq:bout}
b_{out} \propto (\eta_{in} - 2 m) e^{i (2 \theta + \pi)} \,.
\end{equation}

\noindent
Consequently, the relative squeezing phase for the negative
$\eta_{in}$ is $\pi$, i.e., amplitude squeezing. The positive case is
more complicated. It remains equal $\pi$ (amplitude squeezing) till
$\eta=m$. Above this value it changes to $\pm \pi/2$ depending on the
sign of $\eta_{in}/2 - m$ yielding in any case phase squeezing. At a
first glance, it appears there is a sudden change from amplitude to
phase squeezing when the input phases are fix to $\phi-\varphi/2=0$
and $|\eta_{in}|$ passes through $m$. It is not so however, as at
this point $B=0$ and the state collapses to a coherent state with no
squeezing. The situation is clearly depicted in figure \ref{fg:Sbin50}
where $S_{-,+}^{b}(0)$ are displayed as a function of $\eta_{in}$
assumed real. The r.h.s.\ of the plot corresponds to $\phi-\varphi/2 = 0$
while the l.h.s.\ to $\phi-\varphi/2 = \pi$ and for negative ordinates
it should be considered as an optimum path with respect to $\eta_{in}$.
The behavior is completely symmetric with respect to $\eta_{in}=m$
where both the squeezing and the excess noise equal that of the vacuum.
\begin{figure}
\centerline{\includegraphics*[scale=0.5]{Sbin50.eps}}
\caption{Noise spectra at zero frequency (harmonic mode) following an
optimum path with respect to the normalized input harmonic amplitude
($\eta_{in}$). The curves are not physical
above the divergences.}
\label{fg:Sbin50}
\end{figure}

The optimum path with respect to $m$ is now given by
$S_{M}$ at $K_{r}=1$. Figure \ref{fg:SM} is the equivalent to
Fig.\ \ref{fg:Sboptbin0} for the new situation. It represents the
maximum efficiency as far as quantum noise reduction is concerned the
system can yield in any way with respect to $m$. The
improvement with respect to the standard phase-matched SHG as well as
to the optimized SHG is certainly high.
\begin{figure}
\centerline{\includegraphics*[scale=0.5]{SM.eps}}
\caption{Maximum squeezing (harmonic mode) as a function of the
normalized intracavity photon number ($m$) in nonlinear second order singly
resonant device. For comparison the phase-matched and optimized SHG
cases are also shown.}
\label{fg:SM}
\end{figure}

\section{Discussion and conclusions}{\label{DC}}
Two are the main purposes of the present work. On one side, to gain
physical insight about the origins of quantum noise in singly
resonant systems. On the other, to explore their potential as
squeezed light sources. In such a task we have used a model including
all the relevant physics we wanted to address but simple enough to
be tractable. The results shown in the previous sections
certainly reveal a high potential of the studied configurations.
An evaluation of the limits of the model in reproducing the real
physical situation as well as a discussion of possible implementations,
seems, therefore, in order.

One obvious idealization of the model is to assume perfectly coherent
inputs neglecting the excess noise of real lasers something
expected to have deleterious effects at low frequencies.  White and
coworkers \cite{Whi96b} have developed an analytical approach to
this problem resulting in an impressive agreement with the experiments.
As expected, the excess noise completely destroys the
squeezing at low frequencies. In their experiments, however,
the deleterious effect was restricted to only 7 MHz by adding
a mode cleaner to the system, the spectrum coinciding with the ideal
one out of this range. Even better, in \cite{Bre97} the laser noise
was shot-noise-limited  down to 1 MHz, again using an external mode cleaner.

Considering as sensible the assumption of coherent states for the input
modes as well as a value of $m$ around 3 (we will see below it looks
like the case) our main concern about the fundamental mode
results summarized in Fig.\ \ref{fg:SaKerr} refers to the feasibility
of the chosen escape efficiencies.  The ratio
$\gamma_{c}/(\gamma_{c}+\gamma_{s})$ is difficult to maximize in a resonant
mode because, by its own resonant nature, $\gamma_{c}$ must be rather low.
Thus, in \cite{Whi96a} it was
only of 0.52, while in \cite{Pas94} it was 0.36. Even in \cite{Kur93}, a
doubly resonant system specifically designed to squeeze the fundamental
mode, the escape efficiency was around 0.9, limiting the maximum
squeezing achievable to 90\% (in practice, a 52\% of noise reduction was
reached). It appears, then, that nowadays the $\eta=0.99$ should be taken
rather as an ideal illustrative case.

In contrast, the ultimate limit for the noise
suppression in the harmonic mode (Eq.\ (\ref{eq:ilimit})) is pushed up
by the fundamental mode photon number, opening a way to bypass the
usual untouchable limit imposed by the escape efficiency of the cavity
(as in the fundamental mode). Therefore, the squeezing in the harmonic mode
can be arbitrarily large under the ideal assumption that the energy
load inside the cavity can be also arbitrarily large. However, this is
not totally true as the model does not take into
account the losses in the harmonic mode which necessarily limit the
degree of noise suppression. We can estimate this
limitation assuming the absorption in one single pass through the
nonlinear material equivalent to the effect of a beam splitter with
the adequate reflectivity. Taking an absorption of 0.6\%/cm as in
\cite{Kur93} and a length of 1 cm, the equivalent reflectivity would be
of $6 \;10^{-3}$. The spectrum after the beam splitter is given by
$S_{out}=1 + T :S_{in}:$. Setting $:S_{in}: = -1$ and $T = 1-R$,
the ultimate squeezing achievable is precisely $R = 6\;10^{-3}$, i.e.\,
- 22 dB. In other words, the chosen value of $m=50$ in
Figs.\ \ref{fg:Sbkm50} and \ref{fg:Sbin50} represents
more or less the maximum the model can stand without the inclusion
of the harmonic mode losses.

Of course, we still cannot assume $m=50$ as a realistic limit for
the state of the art devices as $m$ depends not only on the
intra-cavity photon number but on the ratio $\nu/\gamma$
between the nonlinearity and losses. This ratio must be high enough in
order to prevent a degradation of the nonlinear optical response in
the system as commented in the introduction.
Besides, this ratio scales down the power
available in the external sources. In view of these complications, probably
the most reliable way of setting the physical scale of $m$ is to
compare the results with the reported experiments. In \cite{Tsu95}
the quoted noise reduction was of -5.2 dB. Setting to zero
$\Delta k$ and $\beta_{in}$ in Eq.\ (\ref{eq:hatspectrumb}) corresponding
to phase-matched SHG, a -5.2 dB squeezing results at $m=2.5$, far from
the $m=50$ limit. Fortunately, the limit
(\ref{eq:ilimit}) grows up quite quickly for low $m's$
(see Fig.\ \ref{fg:SM}). Thus, 10 dB of noise suppression are
reached at $m = 5$, no such unthinkable value.
However, -15 dB of noise reduction requires $m=15$, while a
-20 dB figure is at the $m=50$ limit, an order of magnitude higher.
New nonlinear materials seem the only possibility for such high
squeezing degrees. A promising via consists in the use
of resonant nonlinearities in asymmetric quantum wells (AQW).
Huge nonlinearities have been demonstrated in frequency
doubling experiments and even a tuning of the nonlinearity with a d.c.
field \cite{Sir92}. Obviously, also the absorption is enhanced by the
resonance. This can be a problem as the ratio $\nu/\gamma$ could at
the end of the day not be increased. To asses this possibility requires
quite a detailed analysis out of the scope of the present work.
We can foreseen, however, a promising advantage in the fact that the
losses in the harmonic mode have little influence on the
performance. By maintaining a strong two photon resonance but
relaxing the one photon counterpart (tuning
with a d.c. field or by an adequate energy level engineering), the
nonlinearity would be certainly enhanced while the losses at the
fundamental mode would not increase so strongly, thus enhancing
$\nu/\gamma$. With only one passage through the cavity of the harmonic
mode and taking into account that a very thin layer of material is capable of SHG
\cite{Sir92}, the corresponding deleterious effect cannot be very large.
Even a more exciting possibility comes from the recent experimental
demonstrations of absorption inhibition in AQW induced by quantum
interference \cite{Fai96,Schm97,Fai97}. The absorption
transparency and the resonant enhancement can be combined using an
adequate quantum well engineering leading to very efficient frequency
doublers (see \cite{Schm96}, where precisely a scheme only resonant at
the harmonic mode is proposed).

These are certainly promising perspectives but we should not dismiss the
improvements arising at the range of the present nonlinear crystals
performances. Let us center then, around $m=2.5$.
As shown in the previous section the best strategy corresponds to drive
both modes with relative phases $\varphi-\phi/2 = \pi/2$ (negative
$\eta_{in}$) and $\Delta=0$.
\begin{figure}
\centerline{\includegraphics*[scale=0.5]{Sbm5.eps}}
\caption{Noise spectra at zero frequency (harmonic mode) for $\Delta
k = \hat{\delta} = 0$ as function of the normalized intracavity photon
number at various ``distances'' from the dynamic instability.}
\label{fg:sqm5}
\end{figure}

\noindent
In Fig.\ \ref{fg:sqm5} the noise behavior till $m=5$ is displayed for
various ``distances'' to the instability (\ref{eq:etains}).
Even at half the instability $\eta_{in}$ value,
the squeezing at $m=2.5$ grows from -5.1 dB (69\%)
to -7.2 dB (80\%). The excess noise, on the other hand, rapidly
increases at low $m's$ but it also saturates quickly to bearable values.
The improvement, although nothing spectacular is quite substantial.
In \cite{Sch95} it was not reckoned so because the noise suppression was
studied as function of the input power. Given its nonlinear relation
with $m$ the improvement is much slower with respect to this variable.
Besides the squeezing, the output power is also enhanced. Taking a
negative $\eta_{in}$ in Eq.\ (\ref{eq:bout}), the output power results in
\begin{equation}
\label{eq:Pw}
P_{out} \propto (2 m + |\eta_{in}|)^{2} \,,
\end{equation}

\noindent
and thus, the harmonic mode input contributes constructively to it.
As shown in Fig.\ \ref{fg:Pw} at half a way of the instability the
power is nearly doubled. Although from the theoretical point of view
the injection of a coherent signal in the harmonic mode looks quite
harmless, the experimental implementation is not trivial. However, the
remarkable achievements in \cite{Sch96a,Bre97} with the OPA strongly
support the feasibility of the idea.
\begin{figure}
\centerline{\includegraphics*[scale=0.5]{Pw.eps}}
\caption{The harmonic output power corresponding to the cases of
Fig.\ \ref{fg:sqm5}.}
\label{fg:Pw}
\end{figure}

Finally, a word of caution about the design of the device.
It is important to avoid the setting of oscillations out of
the fundamental mode (the so called subharmonic pumped
OPO \cite{Sch96b,Sch97}), something capable of
destroying the noise reduction \cite{Whi97}. At a first glance,
finite values of $\eta_{in}$ would favor the effect by promoting
the down conversion. But it is not necessarily so as the down
conversion is encouraged only for a given range of the relative phase
between the two driving fields. Thus, for the negative $\eta_{in}$
case studied above, being the harmonic output power maximized, the
down conversion is minimized.

To conclude, let us summarize the most relevant results.
Firstly, for any system with only
one effective mode we have given a systematic approach capable of
isolating the processing of quantum noise by the dynamic response of
the system. This dynamic processing is maximized
at zero frequency, zero generalized nonlinear detuning ($\Delta$ as
defined in section \ref{SP}) and at a dynamic instability.
The static contributions to the noise coming from the
different noisy inputs can in some cases, move the overall optimum
working point away from that corresponding to maximum dynamic noise
suppression. In spite of such, to have a rule to maximize the dynamic
quantum noise suppression resulted very useful to characterize the
squeezing behavior when applied to a specific optical system.
In particular, for the case of a singly resonant second
order nonlinear device, the squeezing
at the fundamental mode is limited by the escape efficiency of the
cavity, the best working point being within figures of merit
of conventional nonlinear crystals. In the harmonic
mode high squeezing requires new materials but it is only limited
by the losses in the non-resonant harmonic mode opening the possibility
of using multiple quantum wells with resonantly enhanced nonlinearities.
However, with standard nonlinear crystals still is possible a
substantial improvement with respect to the reported experiments
by injecting a coherent driving in the harmonic mode. Besides,
the output power is highly enhanced.

\acknowledgments
C.\ C.\ thanks S.\ Schiller and specially A.\ G.\ White for useful comments
and suggestions. Work supported in part by grants No.\ TIC95-0563-C05-03,
No.\ PB96-00819, CICYT (Spain) and Comunidad de Madrid 06T/039/96
(Spain).

redpajama_set_name: RedPajamaArXiv


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text:
The Honoured Dead: A Story of Friendship, Murder and the Search for Truth in the Arab World by Joseph Braude; published by Spiegel & Grau
Say 'Casablanca' and what reaction do you get? The Bergman and Bogart movie always rates first strike, coupled with an almost automatic ejection of 'Play it again Sam' and the humming of 'As Time Goes By'. Dig a little further, and a well-informed reader might remember the Casablanca Landings and maybe the subsequent Casablanca Conference between Churchill and Roosevelt. However it has always been one of the ironies of Moroccan history that none of these three enduring mental tags that have become attached to the word Casablanca were remotely connected to the Moroccan people. Not a single scene of the film Casablanca was shot in North Africa nor were any Moroccans used in the Hollywood studio, whilst the Casablanca landings were an Anglo-American military assault on a colony of Vichy France and the Casablanca Conference was totally dominated by the war strategy and characters of the two Western statesmen.
Only recently has a fourth 'Casablanca' been added to the memory bank of the West. The Casablanca bombings of 2003 are listed alongside the train bombings of Madrid of 2004, and the four London bomb blasts of July 7th 2007, as one of the after-shocks of militant Islamic terrorism in the wake of 9/11.
The Casablanca bombings were a true slice of Moroccan history, even if many of the targets were perceived to be connected with the West, - hotels, clubs, cultural centres, cemeteries or restaurants. A link was also quickly established between the notorious shanty-towns of Casablanca and the fourteen young Moroccan terrorists. It is also an affair that has smouldered on, albeit with less publicity, despite further bomb blasts and a dramatic prison break-out.
Which leads us to why an American journalist, Joseph Braude, is embedded in the Casablanca police force for a couple of months in the year 2008 and left free to follow his wit and his whims. The Honored Dead is his account of this experience. At first, I felt nothing but pained sympathy for his hosts in the police force, who had to endure the presence and questions of this loose cannon, with his neurotic fears and fantastic speculations, such as the danger of a worldwide conspiracy that would link Islamist terrorists with drug smuggling cartels in South America and northern Morocco. It also seemed evident that Braude, whilst trying to penetrate the mysterious silences that lurk behind a casual murder in a Casablanca warehouse, was equally as busy with creating a sense of mystery about himself, with references to youthful freelance work for the FBI, his Iraqi accent, his Sephardic background, his Middle-Eastern research or his lost Muslim friend, Ali. When he confesses to the reader, "maybe I need to relax. I've been in Morocco fifteen days" I began to fear that I was not exploring the dense social fabric of Morocco's metropolitan city in the company of an experienced expert, but was being fed the notebook jottings of a naive Walter Mitty-like Bogart-wannabe. The impression wasn't helped by some blood-chilling quotations from dubious Hadith, a crude understanding of Morocco's ancient Arab-Berber duality and some outright historical muddles about why the Portugese built fortresses in Morocco and the mystical role of the Sidar tree in Islamic belief.
But fortunately I read on and found, to my delight, that my first impressions gradually dissolved, and were replaced by fascination with the tale and then, finally, by grateful respect. For Joseph Braude has crafted an ingenious, moving, clever, respectful and ultimately honest book about Morocco and the Moroccan people. He has an acute ear for the elegant phrases that raise the dignity of Arabic life above the everyday scramble for funds, and a fine eye which succeeds in capturing scene after scene in memorably telling detail, whether it is the proud sitting room imbedded within the unpromising exterior of a slum hut, or a good cop at home, watching his team being beaten at football beside his son, or the curious balance of mood in a police station, caught halfway between violent assertion and a protective, engaged paternalism.
In retrospect, I now see that the initial flurry of conspiratorial speculation and confessional monologue (dressed in the borrowed language of TV cop shows or film noir scripts) were tools to capture the reader's interest. This is always a difficult task when it comes to North Africa. As the adage has it, the French and the Spanish are too involved to be dispassionate, while the Germans and the English are not involved enough to be interested. So, in this case, thank God for the unquiet American.
I also appreciated that Braude's idea of following one bizarre murder trail (rather than painting a broad anthropological picture of a year in the life of Casablanca) not only saves the general reader from drowning in a welter of confusing Muslim names but also allows him to gradually build up an emotional sympathy with one specific Moroccan family and its interconnections. As one is served up fascinating helpings of mystery, sexual deviancy, magic and political conspiracy, all wrapped up in a whodunnit, Braude also succeeds in instructing us about Moroccan diplomacy, its curious interface with the Israeli-American policy axis, the tactics available to the Moroccan government and its various police forces, not to mention the role of women, faith, corruption, employment, migration, patronage and education in modern Morocco.
When you say 'Casablanca' at the end of Braude's The Honored Dead, it is the living, pulsating, confusing, exasperating and fascinatingly complex city that now comes immediately to mind.

redpajama_set_name: RedPajamaCommonCrawl


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Over 3 years after RPWD Act, massive gaps in implementation persist, reveals RTI
News, News Articles, Uncategorized
Everyone agrees that the Rights of Persons with Disabilities Act 2016 is a game changing legislation that has the potential to transform the lives of India's disabled community. Why then over three years after it was passed by both Houses of Parliament, is it yet to show substantial impact? An RTI reveals the major gaps that persist. That's the focus on #StoryOfTheWeek.
Consider this :
Section 66, State Advisory Board on Disability – 26 States/Union Territories
Section 72, District Level Committees on Disability – 9.
Section 88, State Fund for Persons with Disabilities – 6.
Section 79 (7), State Advisory Committee to Assist Disability Commissioner – 9
A checklist that gives an insight into how critical provisions of the Rights of Persons with Disabilities (RPWD) Act 2016 remain poorly implemented over three years after this path breaking legislation was passed. These are the latest findings of an RTI filed by the Disability Rights Action Group (DRAG) into the status of the RPWD Act's implementation.
Compare this with a 2018 report released by the Disability Rights India Foundation (DRIF) in collaboration with the National Centre for Promotion of Employment for Disabled People (NCPEDP) and National Committee on the Rights of Persons with Disabilities (NCRPD) and you see how little has changed. The increase is marginal, by barely two to three. The only jump, if one can call it that, is in the rise in the number of state advisory boards – from 12 to 26.
Letter seeks Centre's intervention
The DRAG has now written to the Office of the Chief Commissioner for Persons with Disabilities to issue "strict directions to state government and union territories for speedy Implementation of the RPWD Act 2016".
The findings come at a time when India's disabled community is reeling under the effects of the COVID-19 pandemic and lockdown. The impact has been especially felt by people with disabilities and their families who are struggling to access basic services.
I filed the RTI on March 2020 to find out the status of the Act since the 2018 report. Only five to six states have set up advisory boards since 2018. I have now prepared a representation for the Ministry of Social Justice and Empowerment so they can issue directions to the states for speedy implementation. – Akeel Ahmed Ushmani, Disability Rights Advocacy Group
Strong reactions from PwD
The findings have caused outrage among many in the disabled community. There have been strong reactions on social media. "We are 2% of the equal population of India. The government has given us the RPWD ACT 2016. Then why we are not getting our rights?", asks Pranali Dhon.
The RTI findings become especially important when one looks at the struggles experienced during the lockdown. The lack of disability boards and committees in many states has meant that the needs of the disabled community were either not addressed at all initially or looked at in a piecemeal manner.
"The lack of disability-specific boards and committees has been felt deeply during the lockdown", says Ushmani. "If all the states had set up a specific fund for disability it could have been used for the disabled population. COVID-19 is a lesson for us. What is the point of having a law for the disabled population when it cannot come to our aid at a critical moment?", he asks.
Source: Newzhook
संगति फाउंडेशन : एक संस्था जो दिव्यांगो को स्वावलंबी बनाती है
A HIRING PLATFORM FOR THE SPECIAL
Sangati Foundation and Emosync | Webinars | Workshop | Individual Counseling sessions
Sangati Durlabh Toilets | First Accessible Public Toilet | International Day of Persons with Disabilities
The I'm Special calendar: A new-year revolution
Disabilities Act: Roll Back of Amendment is a Big Victory for Community
Bath Seats for the Elderly 1. Azadx Bath Chair
DEPwD organises visit of 14 youths with disabilities to experience accessible tourism at Statue of Unity in Gujarat
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I wasn't sure what I was going to write about today. I'd had some thoughts of writing about how awesome all the female Team GB Olympians have been, or maybe doing a round-up of some nice clothes I'd seen yesterday window shopping in town during my lunch break. And then I read this.
Regular readers will know that I am a fairly regular contributor to Any Other Woman and if you follow me on Twitter, you might know that I regularly chat with some of the other readers and writers. We congratulate each other on successes, we laugh at shared jokes, send virtual pieces of cake when someone needs a pick-me-up and show support constantly. I love it and genuinely feel a kinship with this group of funny, warm and interesting women.
Fee is one of those women. She's been commenting recently on AOW and Florence Finds about her pregnancy: offering advice to other readers suffering with morning sickness and telling us what it's like on the other side of the bridge (stealing Sarah's words there). I'm sure I'm not the only one that was looking forward to seeing the baby pictures and loved hearing how excited she was. And then, at 21 weeks, Fee's little son couldn't hold on and didn't make it. I once read that parents who have children who don't make it don't like the phrase 'loosing a baby' because they didn't 'loose' anything, but somehow I can't bring myself to use the D word.
I don't actually know Fee. I've never met her, I don't even know her husband's name. But as soon as I started reading her incredible post I had tears in my eyes. As she described her family coming home from holiday, the small acts of kindness strangers had shown her and them and the things her friends have done, those tears had begun to fall. I can still feel myself welling-up now. I'm so, so sad for her and her husband. There aren't any words that can make it better, and I'm not going to try here, but it's just so awful.
Now this terrible news is not about me (although perhaps one of the reasons this has affected me so much is because what has happened to Fee is my greatest fear) and I'm not going to make it so. However, when I recovered from my initial reaction from reading the post, my next thought was that I am genuinely part of a community at AOW. Sure enough, there are so many comments filled with kind words, real feelings of sadness and helplessness at the fact that one of our own is hurting. That has got to be a slight glimmer of hope at this time, hasn't it? We're very lucky that most of us not only have partners, family and good friends to help us along this sometimes awkward journey called life, but we also have this virtual community cheering us on. And I, for one, am very, very grateful for it.
Fee's post today made me cry also, like I was reading the news of someone I know well in real life. I found it odd to remember that I've never met her, and the comments show that I'm not the only one. If nothing else, the response shows us what an incredible community has formed from the growth of AOW. I'm honoured and lucky to be part of it.
We 'lost' our son Sam at 29 weeks. Esme was really too young to remember but I know that she and her siblings are aware of his place in our family because they see me fall apart every June 8th.
It is so unbelievebly hard for the parents of the child but we musn't underestimate how hard it hits the rest of the family, that they have lost someone too and that everyone needs to grieve in their own way and at their own pace. There is no right or wrong way.
time with him and give thanks to whichever force also gave 4 other beautiful, amazing and wonderful children.
As a foot note we did have a wonderful and suprising addition to our family 5 years later in the shape of Daisy Quinta who was also born at 29 weeks and is now a healthy strapping lass of 17 who wants (and WILL manage) to be a force in the West End in whatever Musical Theatre production will have her. Her sheer tenacity will see her through. BTW her middle name of Qunita was given to her to mark the fact that she is our fifth child and she very proud of it, so much in fact that she is thinking of using it as her stage name.
Basically, what I'm trying to say is that the loss of a child is the hardest most tragic thing that any family will have to endure but there really is life after this and we do and will survive. Take joy in what you have that makes you happy and know that you are not alone. We aren't the same people as before but maybe, just maybe we are stronger and better.
What a lovely post from your mum, Esme.
You're right, it is amazing how wonderful the community we have over on AnyOtherWoman is. Loving your new home here, though! Look forward to following your adventures.


It has been another exciting week here at QM. Our new T5 lighting system was powered up and our inverts have never been happier and arguably neither have our sales staff. Our team really did their homework on this one by finding the perfect combination of lamps and now the colors really pop. For more information about our lighting upgrade, check out the full press release here.
If you havent visited us recently, now is the time to stop in and check us out. Its always a pleasure having our valued clients stop by so you can see firsthand whats transpiring here at QM, theres almost always something new going on. Our Customer Service Representatives really enjoy showing our visitors around QMs state of the art facility, so come on by!
We received a fresh batch of Barnacle Blennies (Acanthemblemaria macrospilus) this past week and they look great. These are comical little blennies that love to perch themselves within the rock and watch everything going on within their little world, tirelessly waiting for a morsel of food to come by. Be sure to order in some of these interesting specimens as they will not be in stock long.
An excellent damsel from our East America supply line is the Purple & Yellow Chromis (Chromis insolatus). As with other Chromis species, these are a communal type with a vibrant contrasting color scheme. They make for a great schooling fish that isnt as commonly seen in the trade.
We have a very good shipment of Dragonface Pipefish (Corythoichthys intestinalis) from the South Pacific in stock. These make an interesting and useful addition to most SPS displays as they are a hunter of the dreaded Red Bugs that can plague Acropora.
Another cool little oddball from our East America supply line is the Sand Puffer (Torquigener hypselogeneion). These tend to be a more seasonal item and are now available in limited numbers. If youre into puffers these guys are pretty neat.
For those corallimorph collectors we have some awesome Super Orange Bullseye Mushrooms (Rhodactis sp). There is a very limited number available so be sure to have your Customer Service Representative select a cherry piece for you.


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Leigh Park reptile hypermarket celebrates first anniversary as excited kids come face to face with snakes, lizards and a turtle
The Aquatic Reptile Superstore in Greywell Road, Leigh Park, celebrated its first anniversary with live animal petting and in-store fun.
Stephen Day, the store manager, estimates that hundreds of people came to the store for the party on Saturday, September 18.
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The Aquatic Reptile Superstore in Greywell Road, Leigh Park, celebrates its first anniversary. Photo: Sam Stephenson
The Southsea resident said: "We had animals that people could hold.
"It was good to be able to do something in return for the community.
"We had very positive feedback, everyone was happy and really appreciated. Really well received. '
Balloons adorned the interior of the store as visitors were greeted to encounter the array of reptiles and aquatic animals.
Amy, 10, and Libby, 13, hang out with Winston. Photo: Sam Stephenson
The mother of seven, Wendy New, attended the event with her children.
She said: "Libby and Amy enjoyed seeing all the reptiles that day.
"Amy's favorite part was being able to touch the big turtle and Libby's was when we bought four more stick insects from the aquarium, as we already bought one a few months ago, they named it Betty. , who has now laid eggs.
"Plus my two other daughters, Rio and Elle, enjoyed the day holding the snake they had there."
Staff supervisor Carmel Landey showing off pets. Photo: Sam Stephenson
Wendy, who lives near the aquarium, runs the New's Creations store nearby.
She added: "My kids always jump in the aquarium and feed the fish for 20p, they love it."
The store offers a wide range of reptiles such as bearded dragons, leopard geckos, snakes, water dragons, chameleons, crested geckos, turtles and turtles, as well as tropical, marine and sea fish. 'Cold water.
Carmel Landey becomes comfortable with a pet. Photo: Sam Stephenson
Director Stephen joined the team in March 2020, ahead of the official opening of the Aquatic Reptile Superstore on September 16.
He added: "We are classified as essential, so we were able to open all the way.
"It hasn't been a bad year, it's been a good year, although it hasn't been amazing."
Plans for a new store in Gosport, slated to open on December 1, were also announced at the event as part of a competition.
A message from the editor, Mark Waldron
We have reduced the cost of digital subscriptions to our website by 50% for a limited time.
The aquatic reptile hypermarket. Photo: Sam Stephenson
You can now subscribe here for unlimited access to our online coverage, including Pompey, for less than 13p per day.


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Q: Docker add php project on subdomain I have two projects. An API in Node.js and a web application in PHP.
I'm using jwilder/nginx-proxy and I want to add the API on api.example.com and the application on vault.example.com. On example.com I will have some static HTML pages.
The subdomain for the API works as expected.
Now my question is about the vault subdomain and about the static HTML pages.
To create an image for these projects I have to use Apache or Nginx, which means that the proxy will make a call to another web server to open the applications. Is this a correct way or the best option is to create volumes for these projects and add them inside the proxy with custom conf file?

A: The best practice for docker containers is to have a single process running in each container:

It is generally recommended that you separate areas of concern by using one service per container.

Source: https://docs.docker.com/config/containers/multi-service_container/
In your scenario, then, it is perfectly fine of you have 3 containers, one for the proxy, one for the node.js code and one for php code.

redpajama_set_name: RedPajamaStackExchange


Looks like all my friends in San Antonio are throwing down new Web edits. This video shows some really technical skills preformed by Brandon.
Looks like you're feelin' good! Check out our gear for people like you: fgfanatics.com. Follow us on twitter and like us on Facebook, too!


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Earlier this month, on the 9th February, the Mirboo North & District Community Foundation were delighted to celebrate the transition of a group of young people from our district into the next stage of their education. This included the award of $19,000 in total to nine recipients.
What is the Community Foundation's Educational Support Fund?
The Education Support Fund is a sub-fund of the Community Foundation which is dedicated to assisting the students in our area in accessing quality education at third level.
Rural students face many hurdles in transitioning to third level, not least of which is financial hardship.
As Kate Buxton, Director explained, "the reality is that before they even take their seat on the first day at Uni or TAFE, it's likely that they would have spent at least $5,000 on books, stationery and other materials, a laptop and accommodation bond."
To find out if you or a family member or friend would be eligible, read on…
L-R : Billie Buxton, Madison Brooks, Hayden Jones, Zac Porter, Alessandro Manuele, Briannon Dudek & Noah Mc Coll Absent: Michael Butler, Jake Porter
The Awards evening
The theme of the night was transitions and the two main speakers gave quite different but yet similarly themed speeches. Laurie Werrit, a local educator with broad experience, spoke about the transition both from the perspective of the students and their parents. He emphasised that first year of Uni can be very challenging and it is important that students build a support network of friends new and old as well as accessing any supports which the University provides.
To the parents he commented that this transition is only another step on a varied educational pathway and to keep supporting their young people in making the choices which are right for them, though you might be scratching your head in wonder.
Beau Linforth, a recipient of an award from the previous year, spoke about the Do's and Don'ts of a First Year student. With a wry smile he concluded:
"Learn to budget and differentiate between what you need and you want. You may want to go to the pub instead of doing your assignment but you don't need to and you may not be able to afford to!" There speaks experience.
Ruth Rogan, Executive Officer for the Community Foundation noted that, "these awards are about more than money, they are about the support of the community in the education of its young people, sending them out into the wider world to reach their full potential."
"It may be some years until they return to Mirboo North to share the benefits of their education, but in the meantime, they will be great ambassadors for Mirboo North, the strong, active and beautiful community from which they have come."
Diana Mueller, Chair of our Grants and Allocations committee then announced the following 9 recipients of the 2018 Education Support Fund awards.
Alessandro Manuele
Alessandro will be attending the Churchill campus of Federation University to study Business HR Management. He is a recipient of the Pierre de Coubertin Olympic Award, represented MNSC in athletics at the state level and was named Sporting All Round Age Champion at Graduation in December 2017.
Billie Buxton
Billie is heading to Swinburne University to study a Bachelor of Arts. Billie was a Monash Scholar from 2015 until 2017 and was also the recipient of the Southern Business Womens' Network Leadership initiative award 2016. She was Dux in German and Business Management in Year 12.
Briannon Dudek
Briannon will be attending her second year at La Trobe University to study Bachelor of Arts focusing on Archaeology, having taken a year off to save up some funds. Briannon has a keen interest in her course of study and was chosen to go to the Triabunna Barracks Archaelogical Field School in Tasmania.
Hayden Jones
Hayden will be attending Monash University to study a Bachelor of Nursing. Hayden attended Mary Mackillop CRC, where he was involved in the school sports teams and was a House Captain. Hayden has also been awarded the Rhonda Malone Memorial Bursary to go towards his studies.
Jake Porter
Jake will be attending Deakin University to study a Bachelor of Criminology/Cyber Security. Jake has been living in Melbourne and working to support going on to University. He is interested in becoming a Police officer and this course will help provide specialist skills. Jake previously studied at Federation University and was invited to join the Golden Key Honours Society.
Madison Brooks
Madison will be attending Deakin University to study a Bachelor of Art & Master of Teaching Secondary. While at MNSC Madison was a member of the school band and involved in Model United Nations Assembly. Madison has also been awarded the CWA Tertiary Award.
Michael will be attending Miller TAFE in Sydney to study Architectural Stonemasonry. Michael has a strong interest in Stonemasonary particularly architectural and restoration work. At MNSC he was School House Captain and received academic awards.
Noah McColl
Noah will be attending Monash University to study Bachelor of Science Monash University. Noah has a keen interested in the field of Zoology. He was a Monash Scholar from 2015 until 2017, a peer support leader, received academic excellence awards and was the Vice School Captain at MNSC. Noah has also been awarded a Monash Support Scholarship.
Zac Porter
Zac will be attending Victoria University to study a Bachelor of Paramedicine. Zac enjoys helping people and will be working and studying along with relocating to Melbourne. During his time at MNSC he was involved in the Peers Support Program.


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Q: MySQL EXISTS return 1 or 0 How can I return the 1 or 0 as true and false for my select query?  I've tried every combination of fetch_assoc, arrays, etc.  It usually returns as null, which doesn't make sense.
$querydetails = "select exists (select * from customer_det where id = $uid)";
$resultdetails = mysql_query($querydetails) or die(mysql_error());
while ($row = mysql_fetch_assoc($resultdetails)) {
    $resultdetails1 = $row[0];
}

That's the latest version of my query.  I guess it's not an array, but I'm not sure how to pull the value if there's no row to be named?
This is a post question to my previous thread: MySQL Update WHERE

A: SELECT IF(EXISTS(...),1,0) AS result

A: simple use COUNT and IF
SELECT IF(COUNT(*) = 0, 0, 1)
FROM table1
WHERE s = 2

*

*SQLFiddle Demo

A: SELECT count(*) FROM (
 SELECT id FROM table WHERE condition LIMIT 1
) AS result;

A: i am using mysql version 8.0.23 on AMS
SELECT
column1,
column2,
IF (exists(select * from MeasurementItemNos measureItem where measureItem.itemNo = ib.itemNo and measureItem.IsActive = 1),1,0) as isUploadedSizeSpec
from table

A: I would do:
select * from customer_det where id = $uid LIMIT 1

You will get either 1 row or zero rows.


text:
Q: Z3: Difference logic performance I have a problem to solve that can be translated into difference logic, and rather than implementing a decision procedure, I would like to use z3 for this purpose.
Nevertheless, I run a few examples and I had exponential-like runtimes (even though there is a polytime decision procedure for it). I am new to z3 and I dont know if I am doing something wrong. Here is the code that I am using (c++ api), varing this "max" variable.
int main(int argc, char **argv) {
context c;

solver s(c, "QF_IDL");

int max = 10000;
int prev = 0;
for(int i = 1; i < max; ++i){
    expr x = s.ctx().int_const(std::to_string(i).c_str());
    expr y = s.ctx().int_const(std::to_string(++i).c_str());
    expr pr = s.ctx().int_const(std::to_string(prev).c_str());
    s.add(pr < x);
    s.add(x < y);
    prev = i;
}

s.add(s.ctx().int_const(std::to_string(max-1).c_str()) < s.ctx().int_const(std::to_string(0).c_str()));

clock_t begin = clock();
switch (s.check()) {
    case unsat:   std::cout << "UNSAT"; break;
    case sat:     std::cout << "SAT"; break;
    case unknown: std::cout << "unknown\n"; break;
}

clock_t end = clock();
double elapsed_secs = double(end - begin) / CLOCKS_PER_SEC;

std::cout << "elapsed time: " << elapsed_secs;
}

Many thanks in advance,
Pedro

A: By default, Z3 uses the simplex engine and sometimes a Floyd Marshall engine to solve your constraints even when the logic is QF_IDL. In this case it uses the simplex engine, and the size of rows grows quadratically for this example.
You can force the sparce difference logic solver by inserting the following into your program:
 params p(c);
 p.set("auto_config", false);
 p.set("smt.arith.solver", (unsigned)1);
 solver s(c, "QF_IDL");
 s.set(p);

This sets the arithmetic solver to the sparse difference logic solver.
It does not suffer from space overhead. It still takes quadratic time,
or to be more precise time proportional to |V||E| where |V| is the number
of variables and |E| are the number of inequalities.
The main time bottleneck in this case is on big-num arithmetic, which is not
necessary in your case. I added an update to the unstable branch of Z3 so that it recognizes scenarios that only use small integers so that it can use a
more efficient representation. This takes time on the larger examples down by a factor of about 5. Nevertheless, the overhead is still |V||E|.