text:
\section{Introduction}
Initially presented as a frequency converter that produces tunable coherent radiation \cite{giordmaine1965tunable},
the simplest configuration of an Optical Parametric Oscillator (OPO) was showed to be a source of multiple non-classical states of light. The long list includes squeezed states \cite{Wu1986,Fabre1997}, quantum correlated beams \cite{Heidmann1987, Cassemiro:07}, entangled thermal states \cite{Peng1992}, and multicolor entanglement for two  \cite{Polzik2002, Villar2005}, three  \cite{Coelho823}, and up to six modes of the field \cite{barbosa2018hexapartite}.
This versatile source has found applications in quantum metrology \cite{vahlbruch2007}
and quantum communication protocols, such as quantum key distribution \cite{pereira2000}
and quantum teleportation \cite{furusawa1998}. Moreover, extensions over the basic configuration allowed the generation of cluster states with arbitrarily large number of modes \cite{furusawa2013,pfister2014} as a possible resource for quantum computation \cite{vanLoock2009}.

The usual configuration consists of a non-linear medium inside an optical cavity. The nonlinear medium will couple the pump field to a pair of modes named signal and idler, often through a second ($\chi^{(2)}$) or third ($\chi^{(3)}$) order non-linearity. Energy exchange among these three fields will follow both energy and momentum conservation \cite{giordmaine1962}.
The nonlinear medium acts as a parametric amplifier and, if amplification matches the cavity losses, the oscillation  threshold is reached and we have the
generation of intense output fields.
Even for this simple case we may have distinct cavity configurations \cite{Fabre_Cohadon}. The cavity  may be either resonant for all three modes (triply resonant OPO - TROPO), for both converted fields (doubly resonant OPO - DROPO), or just for one of them (singly resonant OPO - SROPO).

Quantum treatment of a TROPO using a $\chi^{(2)}$ medium was extensively done in both operation regimes (below \cite{graham1968quantum,
Drummond1990} and above threshold \cite{
Reid1989,Fabre_1990}). This treatment applies for the  DROPO as well, as far as the pump depletion is negligible. That is the case far below the oscillation threshold,  where pump is treated just as a classical field. To the best of our knowledge, a full quantum  treatment for the DROPO and the effects of pump depletion on the quantum noise is missing. Here we use the multimode analysis of the cavity evolution \cite{MUNOZ2018} to compare the noise and the quantum correlations of DROPO and TROPO in above threshold operation, demonstrating that multipartite entanglement and the squeezing of the pump noise are present even in a single pass regime.

The model has the advantage of being applicable to open cavities, beyond the closed cavity regime adopted in typical input/output formalism \cite{yurke1984,yurke1985}. That is particularly interesting for the study of  $\chi^{(3)}$ OPOs, and we compare their behavior to the usual $\chi^{(2)}$ oscillator above threshold.
The small gain, typically found in $\chi^{(2)}$ amplifiers based on   parametric down conversion (PDC), leads to high reflectance mirrors (typically greater than 95\%) for continuous operation even for the best available crystals. On the other hand, although many oscillators based on four wave mixing (4WM)  will present a closed cavity \cite{vahala2004},
this process can be much stronger in atomic vapors \cite{PhysRevA.78.043816}, where the  $\chi^{(3)}$ process is enhanced close to the atomic transitions.
This enable the development of OPOs with higher transmittance mirrors (open cavity) \cite{yu2010,alvaro2020}.

We begin by a description of the classical behavior of an OPO considering the DROPO and the TROPO case without any approximation for the reflections coefficients of the cavity (section~\ref{ClassAppr}), accounting for the evolution of the mean fields along the gain medium (both $\chi^{(2)}$ and $
\chi^{(3)}$). We go beyond the studies of the classical behavior of $\chi^{(2)}$ \cite{Debuisschert:93} that accounts only for the first order of the expansion to compute the mean fields, which is not appropriate to simulate a $\chi^{(3)}$ OPO with a higher gain \cite{alvaro2020}.
Afterwards, we provide a quantum description of the systems considering the quantum treatment of the fluctuations in terms of the symmetric and antisymmetric basis of the electromagnetic field yielding a full description of the state of the fields in terms of a covariance matrix (section~\ref{Qcorr_sec}). Next, we evaluate the quantum features generated by different types of OPO (section~\ref{secent}), squeezing, bipartite and multipartite entanglement, showing the main differences that appear on each configuration.

\section{Classical Approach}\label{ClassAppr}
We evaluate the evolution of the  mean field during the parametric amplification in both cases, PDC and 4WM, before considering the steady state conditions inside a cavity. The cavity feedback will lead to a dramatic effect of gain saturation, completely modifying the response of the free propagating process.

Using a medium with second (third) order non-linearity, one (two) photon(s) of the pump beam (of frequency $\omega_1$) might be converted into two photons, signal and idler (with respective frequencies $\omega_1$ and $\omega_2$). The interaction Hamiltonians $\hat{H}^{{(2)}}$, which represents the PDC, and $\hat{H}^{{(3)}}$, representing the 4WM, are given by:
\begin{eqnarray}
\hat{H}^{{(2)}}&=&i\hbar \chi^{(2)}\hat{a}_{0}(t)\hat{a}_{1}^{\dagger}(t)\hat{a}_{2}^{\dagger}(t)-h.c.,
\label{hintchi2}
\\
\hat{H}^{{(3)}}&=&i\hbar\chi^{(3)} \hat{a}_0^2(t) \hat{a}_1^{\dagger}  (t)\hat{a}_2^{\dagger}(t) + h.c.,
\label{HamiltonianoInteracaoTemporal}
\end{eqnarray}
in the interaction picture. Here $\hat{a}_n$, with $n=\{0,1,2\}$, represents the annihilation operators of the pump, signal and idler modes, respectively, and the parameter $\chi^{(m)}$ with $m=\{2,3\}$ is associated with the non-linear susceptibility coefficient of each gain medium.

In order to determine the mean amplitudes of the output fields as a function of the system parameters, the operators evolution $\hat{a}_n(t)$ through the gain medium are evaluated through the Heisenberg equations for the field operators, $\frac{d}{d t}\hat{a}_{n}(t)=(i/\hbar)[ \hat{H}^{(m)},\hat{a}_{n}(t)]$.
Linearizing the field operators as $\hat{a}_{n}(t) = \alpha_{n} + \delta\hat{a}_{n}(t)$, where $\alpha_{n}$ is the mean field amplitude and $\delta\hat{a}_{n}(t)$ is the field fluctuations, the set of expressions that describe the mean value evolution of the fields $\alpha_{n}$ through the gain medium can be written as:
\begin{subequations}
\label{eq_heisemberg_amplitudes1}
\begin{align}
  \frac{d \alpha_{0}}{d t} &= -(m-1)\chi^{(m)}\alpha_{0}^{*(m-2)}\alpha_{1}\alpha_{2}, \label{eq_HA_1} \\
\frac{d \alpha_{1}}{d t} &= \chi^{(m)}\alpha_{0}^{(m-1)}\alpha_{2}^{*}, \label{eq_HA_2} \\
\frac{d \alpha_{2}}{d t} &= \chi^{(m)}\alpha_{0}^{(m-1)}\alpha_{1}^{*}, \label{eq_HA_3}
\end{align}
\end{subequations}
where $\alpha^{*}_{n}$ is the complex conjugate of $\alpha_{n}$. The mean value of the field is a complex number that can be explicitly written in terms of real amplitude and phase: $\alpha_{n}=\sqrt{P_n}e^{i \theta_n}$.
The parameter $P_n=\alpha_{n}\alpha_{n}^*$ is proportional to the photon number in the field $n$ and hence to the field power.
Differentiating $P_n$ in time we have
\begin{align}
\frac{d}{dt}P_n&= \alpha_n \frac{d \alpha_n^*}{dt}+\alpha_n^*\frac{d \alpha_n}{dt},
\label{power}
\end{align}
and with the help of Eq.(\ref{eq_heisemberg_amplitudes1}),
we have a set of differential equations that describe the evolution of $P_n
\begin{align}
\label{difpower}
\frac{d P_0}{dt} &= -2(m-1) \chi^{(m)}
\sqrt{P_{0}^{(m-1)}P_1 P_2} \,\cos{\theta}, \nonumber  \\
\frac{d P_1}{dt} &= 2 \chi^{(m)} \sqrt{P_{0}^{(m-1)}P_1 P_2} \,\cos{\theta}, \nonumber \\
\frac{d P_2}{dt} &= 2 \chi^{(m)} \sqrt{P_{0}^{(m-1)}P_1 P_2} \,\cos{\theta},
\end{align}
depending on a global phase $\theta= \theta_1 + \theta_2-(m-1)\theta_0$.  In practice, the exact value will depend on the phase matching condition, involving the value of $\chi^{(m)}$ and on the reflection coefficient of the mirrors \cite{Debuisschert:93}, but the general effect will be the modulation of the coupling. Therefore, we will chose $\theta$ that maximizes the coupling, thus $\cos(\theta)= 1$.
A detailed evaluation of the field evolution for the case of a $\chi^{(2)}$ medium is given in \cite{rosencher2002}.

As the fields propagate along the gain medium, we may evaluate the power transfer from the pump to the converted modes.
The number of photons of the pump will be reduced, leading to $P_0(t)=P_0(0)-(m-1)p(t)$. This photon depletion leads to a change in the photon number of the converted fields as $P_1(t)=P_1(0)+p(t)$ and $P_2(t)=P_2(0)+p(t)(t)$.
Considering a balanced power for converted modes, $P_1(t)=P_2(t)$, therefore the set of Eqs. (\ref{difpower}) can be used to obtain the evolution of $p(t)$, related with the power transfer along the path. For convenience,
\begin{equation}  \label{eq_dx/dt}
\frac{dp}{dt}=2\chi^{(m)}(P_0(0)-(m-1)p)^{\frac{1}{2}(m-1)}(P_1(0)+p).
\end{equation}\par

The total power variation of signal field, $\Delta P_{1}^{(m)}$, for each gain medium $m$, will be computed integrating  Eq. (\ref{eq_dx/dt}) in the limits of the entrance $(t_{1},p(t_{1}))=(0,0)$, and the end $(t_2,p(t_2)) = (L/(\nn c), \Delta P_{1}^{(m)})$ of the gain medium of length $L$, where $c$ is the velocity of light in the vacuum and $\nn$ is the refractive  index.
Considering the $\chi^{(2)}$ gain medium we have
\begin{eqnarray}\label{gainchi2}
\Delta P_{1}^{(2)}&=& P_0(0)-(P_0(0)+P_1(0)) \nonumber \\
&\times& tanh^2 \left[\kappa^{(2)} \sqrt{P_0(0)+P_1(0)} \right.  \nonumber \\ &-& arctanh \left.\left[ \sqrt{\frac{P_0(0)}{P_0(0)+P_1(0)}}\right]\right],
\end{eqnarray}
with $\kappa^{(m)} = \chi^{(m)} L/(\nn c)$. Although this equation is not simple, in the limit of a weak coupling we have $\Delta P_1^{(2)} =  \kappa^{(2)} /P_1(0)\sqrt{P_{0}}$, recovering the situation observed in \cite{Debuisschert:93}.
For a $\chi^{(3)}$ gain medium we obtain
\begin{equation}\label{resultado_ganho_1}
\Delta P_{1}^{(3)} = \frac{P_0(0) P_1(0) \left(e^{2 \kappa^{(3)} (P_0(0)+2P_1(0))}-1\right)}{2 P_1(0) e^{2 \kappa^{(3)} (P_0(0)+2 P_1(0))}+P_0(0)}.
\end{equation}

In the $\chi^{(3)}$ scenario, it is interesting  to evaluate the signal (and idler) relative gain $G^{(3)}$. From  Eq.  (\ref{resultado_ganho_1}) we have
\begin{equation}\label{resultado_ganho_2}
G^{(3)}=\frac{\Delta P_{1}^{(3)}}{P_1(0)} =\frac{ g^{(3)}-1}{1+2 g^{(3)} P_1(0)/P_0(0)},
\end{equation}
where  $g^{(3)}=\exp{[2 \kappa^{(3)} (P_0(0)+2P_1(0))]}$ corresponds to the unsaturated amplification. For a completely open cavity and a weak seed ($P_1(0)\ll P_0(0)$),  Eq. (\ref{resultado_ganho_2})  simplifies as
\begin{equation}\label{resultado_ganho_3}
G^{(3)}+1=e^{2 \kappa^{(3)} P_0(0)}=g^{(3)}.
\end{equation}
The amplification then increases exponentially with the pump power and the medium length, which is proportional to $\kappa^{(3)}$. This is the typical situation of unsaturated parametric amplifier as used in \cite{PhysRevA.78.043816}.
It also becomes evident the role of an increasing seed: it will lead to a saturation of the power transfer process, and therefore a reduction of the gain in Eq. (\ref{resultado_ganho_2}).

From Eq. (\ref{gainchi2}) and (\ref{resultado_ganho_1}), we can relate the added power on the converted fields to the coupling coefficients $\kappa^{(m)}$, and the field power at the input of the amplifier, $P_0(0)$ and $P_1(0)$. This result is used now to evaluate the intracavity steady state.
First, we compare the solutions for both gain media in a doubly resonant cavity in Section \ref{DROPO_R}, where pump beam makes a single pass through the cavity while signal and idler beams are resonant.
Next, in Section \ref{TROPO_R}, we consider a triply resonant OPO, exploring different reflections of the input mirror for the pump field.

\subsection{Doubly Resonant Optical Parametrical Oscillator}\label{DROPO_R}
The first system being studied is a DROPO (Fig. \ref{DROPO}). For convenience, we will label the power at the input and the output of the medium by the positions $0$ and $L$.
In this case, the pump beam makes a single pass through the cavity, therefore the pump power injected in the cavity defines $P_{0in}=P_0(0)$. When it leaves the cavity, the output power
is given by
$P_{0out}=P_0(L)=P_{0in}-(m-1)\Delta P_{1}^{(m)}$.

In the steady state, we can relate the input power of the signal on the amplifier to the output of the amplifier using the reflectance $R_{1}$ on the output coupler $M_{PT}$, as $P_1(0)=R_{1}P_1(L)$. The output of the OPO will be related to the output of the amplifier as well $P_{1out}=(1-R_{1})P_1(L)$. From the definition of $\Delta P_1^{(m)}=P_1(L)-P_1(0)$
we can calculate the output power $P_{1out}$ as
\begin{equation}
P_{1out}=\frac{(1-R_{1})P_{1}(0)}{R_{1}}= \Delta P_{1}^{(m)}.
\label{eq_P_1out}
\end{equation}
This result is already expected from a cavity in equilibrium due to energy conservation: the energy added to a given mode will match the losses through the output coupler.

\begin{figure}[t]
	\begin{center}
		\includegraphics[scale=1.3]{Cavitywithgainmedium.pdf}
		\caption{OPO model using a ring cavity with two highly reflective mirrors ($M_{1}=M_{2}=1$), an output coupler $M_{PT}$ and a gain medium with length $L$. The $M_{PT}$ reflectante coefficient for the pump beam is $R_{0}$ and for both, signal and idler beams, $R_{1}$.}
 		\label{DROPO}
	\end{center}
\end{figure}

Numerical solution of Eq.~ (\ref{eq_P_1out}), combined with ~Eq. (\ref{gainchi2}) (for $\chi^{(2)}$) and~Eq. (\ref{resultado_ganho_1}) (for $\chi^{(3)}$)  gives the value of $P_1(0)$ as a function of the pump power $P_{0in}$. Evaluation of the output power  $P_{1out}$ is immediate. The result is presented in
Fig. ~\ref{chi2chi3pth}(a), for the  $\chi^{(2)}$ gain medium, giving an output power close to the parabolic curve deduced in \cite{rosencher2002}. On the other hand, in Fig.~\ref{chi2chi3pth}(b), for the  $\chi^{(3)}$ gain medium, we can observe that the curve approaches a proportional response for sufficiently high pump power. Even close to the threshold, as showed in the inset, the evolution could be closely approximated by a linear response when the cavity coupling is higher, as observed in \cite{alvaro2020}.
The asymptotic behavior is similar for the distinct couplings, and very different from the one observed with  $\chi^{(2)}$.
The insets put in evidence the reduction of the threshold power for a reduction of the cavity losses.
The chosen value of the nonlinearity  $\kappa^{(3)}=3 W^{-1}$ is based on the observed amplification in  \cite{PhysRevA.78.043816}. The value of $\kappa^{(2)}=0.5 W^{-1/2}$ is chosen to match the threshold power for both media with $R_1=0.85$. We will adopt this value for the simulations along the remaining of the article.

Although the power has some dramatic changes for distinct coupling, a better comparison can be done using the conversion efficiency
\begin{equation}
\eta=\frac{\hbar\omega_1 P_{1out}+\hbar\omega_2 P_{2out}}{\hbar\omega_0 P_{0in}}.
\end{equation}
The efficiency is showed in Fig.~\ref{chi2chi3ef}, with the pump power normalized to the threshold power. The conversion efficiency increases monotonically in both cases, but the most relevant difference between $\chi^{(2)}$ and $\chi^{(3)}$ amplifiers comes from the fact that a maximum (unitary) efficiency is observed at a pump power $\simeq 4$ times above threshold for $\chi^{(2)}$ oscillators, while $\chi^{(3)}$ oscillators evolve asymptotically to unitary gain. Notice as well that while the unitary efficiency for $\chi^{(2)}$ is reached at $\sigma_{max} \simeq 4$ for a closed cavity, the position of the maximum is reduced when cavity losses are increased, as observed in \cite{rosencher2002}.

\begin{figure}[ht]
	\begin{center}
		\includegraphics[scale=0.3]{DROPO_chi2andchi3_P1out.pdf}
		\caption{DROPO: $P_{1out}$ as a function of $P_{0in}$ for different reflectivity coefficients $R_{1} = \{75\%, 85\%, 95\%\}$. Simulations considering (a) a $\chi^{(2)}$ gain medium with $\kappa^{(2)}=0.5 W^{-1/2}$ and in (b) a $\chi^{(3)}$ gain medium with $\kappa^{(3)}=3 W^{-1}$.}
		\label{chi2chi3pth}
	\end{center}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[scale=0.3]{DROPO_chi2andchi3_eff.pdf}
\caption{\label{chi2chi3ef} DROPO: Conversion efficiency $\eta$ as a function of pump power (normalized by the threshold power) $\sigma$ for $R_{1} = \{75\%, 85\%, 95\%\}$. Simulations with a) a $\chi^{(2)}$ gain medium and b) a $\chi^{(3)}$ gain medium.}
 \end{figure}

\subsection{Triply Resonant Optical Parametrical Oscillator}\label{TROPO_R}
We have now a cavity for the pump that enhances its power, what makes this system being called pump enhanced DROPO as well.
Self-consistency equation for the converted field, Eq. (\ref{eq_P_1out}), is still valid.
However the self consistency equations will differ for the pump field that now includes a beam splitter transformation for the input coupler:
\begin{align}
\sqrt{P_{0}(0)}&= t_0\sqrt{P_{0in}} - r_0 \sqrt{P_{0}(L)} \label{eq:sqrtP00}\\
\sqrt{P_{0out}}&= t_0\sqrt{P_{0}(L)} + r_0 \sqrt{P_{0in}}\label{eq:sqrtP0out},
\end{align}
where $r_0=\sqrt{R_0}$ is the reflection coefficient and $t_0=\sqrt{1-R_0}=\sqrt{T_0}$ is the transmission coefficient.

The output pump power in terms of the input pump power $P_{0in}$ and the intracavity pump power $P_{0}(0)$ is obtained by combining Eqs. (\ref{eq:sqrtP00}) and (\ref{eq:sqrtP0out}):
\begin{equation}\label{eq:P0out}
P_{0out}=\frac{P_{0in}-2t_0 \sqrt{P_{0in}P_{0}(0)}+T_0 P_0(0)}{R_0}.
\end{equation}

We would like to evaluate the behavior of $P_{1out}$ in terms of $P_{0in}$. A numerical solution for this quantity can be obtained by evaluating the intracavity fields $P_{0}(0)$ and $P_{1}(0)$ as a function of the input power, the coupling constant $\kappa^{(m)}$ and the mirror reflectances $R_0$, $R_1=R_2$. As an example, a detailed evaluation of the output power for $\chi^{(3)}$ TROPO is performed in the Appendix \ref{App_TROPORdif}.

For the moment we will consider that the cavity mirrors have the same reflectance for pump, signal and idler beams, $R_0=R_1=R_2=R$. While this situation is quite unusual for $\chi^{(2)}$ OPOs, it is common for the $\chi^{(3)}$ condition, when all the resonant fields may be nearly degenerate in frequency, leading to balanced losses. In this situation a direct equation for the intracavity fields and the pump power can be obtained.

In a steady state the total intracavity power, $P_{T0}$, is constant at any point inside the cavity, resulting in the relation $P_{T0} =P_{0}+(m-1)P_{1}$. Due to energy conservation, the relation between the input pump field and the output fields of the cavity is $P_{0in}=P_{0out}+(m-1)P_{1out}$, where $P_{1out}$ is given by  Eq. (\ref{eq_P_1out}). Now it is possible to rewrite the intracavity fields, $P_{0}(0)$ and $P_{1}(0)$,  in terms of the input pump power $P_{0in}$, and the total intracavity pump power, $P_{T0}$. From the above description, combined
with  Eqs. (\ref{eq:sqrtP00})
and (\ref{eq:P0out}), we obtain
\begin{eqnarray}
	P_{0}(0)&=&\frac{1-R}{4}\frac{(P_{0in}+P_{T0})^2}{P_{0in}} \label{eq:P00},  \\
		P_{1}(0)&=& \frac{R-1}{4(m-1)}\frac{(P_{0in}-P_{T0})^2}{P_{0in}}+\frac{RP_{T0}}{m-1}\label{eq:P10},
\end{eqnarray}
where  Eq. (\ref{eq:P00}) and  Eq. (\ref{eq:P10}) describe the behavior of $P_{0}(0)$  and $P_{1}(0)$ as a function of $P_{T0}$ and $P_{0in}$.

 \begin{figure}[t]
	\centering
	\includegraphics[scale=0.3]{TROPO_chi2andchi3_P1out.pdf}
 \caption{\label{fig:TROPOMEDIO} TROPO: $P_{1out}$ in function of $P_{0in}$ for different reflectivity coefficients $R_{1} = \{75\%, 85\%, 95\%\}$. Simulations considering a) a $\chi^{(2)}$ gain medium and in b) a $\chi^{(3)}$ gain medium.}
\end{figure}

 \begin{figure}[t]
	\centering
	\includegraphics[scale=0.3]{TROPO_chi2andchi3_eff.pdf}
 \caption{\label{fig:TROPOQE}TROPO: $\eta$ in function of $\sigma$ for $R_{1} = \{75\%, 85\%, 95\%\}$. Simulations with a) a $\chi^{(2)}$ gain medium and b) a $\chi^{(3)}$ gain medium.}
\end{figure}

The problem now becomes writing $P_{T0}$ as a function of the input pump power, $P_{0in}$. In the $\chi^{(2)}$ TROPO  configuration, by equalling  Eq. (\ref{gainchi2}) and Eq. (\ref{eq_P_1out}) results in
\begin{align}\label{eq:PT0Pin2}
\frac{(1-R)(P_{0in}-P_{T0})^2}{4 R P_{0in} P_{T0}} =&
tanh^2 \left[\kappa^{(2)} \sqrt{P_{T0}} \right.\\
- arctanh &\left.\left[ \sqrt{(1-R)\frac{(P_{0in}+P_{T0})^2}{4P_{0in}P_{T0}}}\right]\right],  \nonumber
\end{align}
which is used to find a numerical solution to $P_{T0}$ as a function of $P_{0in}$.
Considering the $\chi^{(3)}$ TROPO, the treatment consists of equalling  Eq. (\ref{resultado_ganho_1})  to Eq. (\ref{eq_P_1out}). Rearranging the terms we obtain $\frac{P_1(0)}{P_0(0)} = \frac{R e^{2\kappa_{3} P_{T0}}-1}{2(1-R) e^{2\kappa_{3} P_{T0}}} $
and substituting   Eq. (\ref{eq:P00}) and  Eq. (\ref{eq:P10}) in the later expression results in:
\begin{align}\label{eq:PT0Pin3}
(1-R)(P_{0in}-P_{T0})^2 &- 4 R P_{0in}P_{T0}\\ \nonumber
 =&(e^{-2\kappa_{3} P_{T0}}-R)(P_{0in}+P_{T0})^2,
\end{align}
enabling a numerical solution for $P_{T0}$ as a function of $P_{0in}$.
Using equations  Eq. (\ref{eq:PT0Pin2}) and  Eq. (\ref{eq:PT0Pin3}) as an input in equation  Eq. (\ref{eq:P10}) we have $P_{1}(0)$ as a function of $P_{0in}$ for $\chi^{(2)}$ and $\chi^{(3)}$ TROPO, respectively. In order to evaluate the behavior of $P_{1out}$ as a function of $P_{0in}$ we just replace $P_{1}(0)$ in  Eq. (\ref{eq_P_1out}).

The behavior of $P_{1out}$ as a function of $P_{0in}$ is showed for a $\chi^{(2)}$ gain medium in Fig.~\ref{fig:TROPOMEDIO}~(a) and  for a $\chi^{(3)}$ gain medium in Fig.~\ref{fig:TROPOMEDIO}~(b). The behavior of both systems are analysed for three different reflection coefficients and the increase in the threshold power can be observed when $R$ changes from $R=95\%$ to $R=75\%$. Furthermore, the converted fields power increases with the input pump power, but differently from the DROPO, both curves present a parabolic-like shape.

A better comparison of the curves can be obtained from the conversion efficiency, as showed in Fig.  \ref{fig:TROPOQE}. The maximum conversion efficiency occurs approximately in $\sigma=4$ decreasing from this point on to approximately 50\% for $\sigma \gg 4$. The behavior for the different reflectivity coefficient is very similar in all $\sigma$ analysed for both  $\chi^{(2)}$ and $\chi^{(3)}$ TROPO in the presented range.

These results are similar to those obtained for $\chi^{(2)}$ OPOs, as presented in \cite{Debuisschert:93,rosencher2002}. The main point of the present treatment is to obtain a detailed evolution of the mean fields inside the gain medium. That is a fundamental part in the evaluation of the noise for a cavity in the open limit, beyond the first order approximation for the mean field, as we demonstrate now.

\section{Theoretical description of the field quantum fluctuations}\label{Qcorr_sec}

In order to evaluate the quantum fluctuations as a function of the oscillator parameters in the spectral domain,
we follow the formalism described in~\cite{MUNOZ2018}. We begin by writing the time-dependent annihilation operator in terms of the annihilation operators acting on the modes of the sideband frequencies of the central carrier frequency $\omega_n$
\begin{eqnarray}
\hat{a}_{n}(t)&=&\int_{-\omega_{n}}^{\infty} e^{-i\Omega t} \hat{a}_{\Omega}d\Omega,
\label{eq_TFBL}
\end{eqnarray}
where $\hat{a}_{\Omega}$ is the photon annihilation operator in the mode of frequency $\Omega=(\omega-\omega_{n})$, $\Omega$ represents the sideband frequency and $\omega_{n}$ is the carrier frequency of mode $n$. Considering a narrow optical spectra for the carrier, $\omega_{n}\gg |\Omega|$, the integral limit can be approximated as $\omega_{n} \rightarrow \infty$.
In the linearized form, the annihilation operator is
can be rewritten as
\begin{eqnarray}
\hat{a}_{n}(t)=\langle\hat{a}_{n}\rangle + \delta\hat{a}_{n}(t)= \alpha_{ n}+ \int^{'} e^{-i\Omega t}\hat{a}_{\Omega ,n}d\Omega,
\label{mn}
\end{eqnarray}
where $\alpha_{n}$ represents the mean value of the carrier at frequency $\omega_n$ and the symbol $^{'}$ in the integral represents the integration limits between $-\infty$ to $\infty$ relative to sidebands frequency disregarding the carrier term. This integral will give rise to the fluctuation term $\delta \hat a_n(t)$.

The interaction Hamiltonian of each interaction process,  Eq. (\ref{hintchi2}) and~(\ref{HamiltonianoInteracaoTemporal}), can now be rewritten with the help of  Eq.~(\ref{mn}). Higher order terms in fluctuation and rapidly oscillating terms, that don't satisfy energy conservation, are neglected. The constant part having only the mean fields is removed as well. We are left only with the contributions of the sidebands, given by
\begin{eqnarray}
\hat H^{(m)}=\int_{\epsilon}^{\infty} d \Omega \hat H^{(m)}(\Omega)
\label{hbl1}
\end{eqnarray}
where the sum is taken form a lower frequency component $\epsilon$ defined by the bandwidth of the pump field. The contributions of the sidebands add linearly, and are described by
\begin{eqnarray}
\hat H^{(m)}(\Omega)&=&i\hbar\chi^{(m)} \Omega[\alpha_{{0}}^{*(m-1)}(\hat{a}_{\Omega,1} \hat{a}_{-\Omega,2}+\hat{a}_{-\Omega,1} \hat{a}_{\Omega,2}) \nonumber \\
&+&(m-1)\alpha_{0}^{*(m-2)}\alpha_{1}(\hat{a}^{\dagger}_{\Omega,0}\hat{a}_{\Omega,2}+\hat{a}^{\dagger}_{-\Omega,0}
\hat{a}_{-\Omega,2})\nonumber \\
&+&(m-1)\alpha_{0}^{*(m-2)}\alpha_{2}(\hat{a}^{\dagger}_{\Omega,0}\hat{a}_{\Omega,1}+\hat{a}^{\dagger}_{-\Omega,0}
\hat{a}_{-\Omega,1}) \nonumber \\
&+&(m-2)\alpha_{1}\alpha_{2}(\hat{a}^{\dagger}_{\Omega,0}\hat{a}^{\dagger}_{-\Omega,0}+\hat{a}^{\dagger}_{-\Omega,0}\hat{a}^{\dagger}_{\Omega,0})-h.c.]. \nonumber \\
\label{hbl2}
\end{eqnarray}

A convenient form of writing this Hamiltonian is using a
symmetric and antisymmetric combination of modes, defined as:
$\hat{a}_{ns/a}=[\hat{a}_{\Omega,{n}} \pm \hat{a}_{-\Omega,{n}}]/\sqrt{2}$, where $+~(-)$ signal refers to symmetric (antisymmetric) sideband operators thus simplifying the Hamiltonian into two terms, $\hat H^{(m)}(\Omega)=\hat H^{(m)}_{ \mathit{s}}(\Omega)+\hat H^{(m)}_{ \mathit{a}}(\Omega)$, given by:
\begin{eqnarray}
\hat H^{(m)}_{ \mathit{s/a}}(\Omega)&=&\pm \alpha_{0}^{*(m-1)}\hat{a}_{1s/a}\hat{a}_{2s/a}\nonumber\\&+&(m-1)\alpha^{*(m-2)}_{0}\alpha_{1}\hat{a}^{\dagger}_{0s/a}\hat{a}_{2s/a}\nonumber \\
&+&(m-1)\alpha_{0}^{*(m-2)}\alpha_{2}\hat{a}_{0s/a}^{\dagger}\hat{a}_{1s/a}\nonumber \\
&\pm &(m-2)\alpha_{1}\alpha_{2}\hat{a}^{\dagger 2}_{0s/a}-h.c.\qquad.
\label{hamsimantissim}
\end{eqnarray}

As showed in details in the reference~\cite{MUNOZ2018}, this Hamiltonian represents a two-mode squeezing process on the  twin beams in the presence of an intense pump field, and a pair of beam splitter process between the pump and one of the generated fields in the presence of an intense mean field related with the conjugated mode.
In the case of Hamiltonian, related to the $\chi^{(3)}$ medium, one additional term is present, which is related with a squeezing process in the pump field in the presence of a pair of intense converted fields.

We can use the  Heisenberg equation $d\hat{a}_{n(s/a)}(t)/dt =-i/\hbar[H_{\chi^{(m)}(s/a)},\hat{a}_{n(s/a)}^{\dagger}(t)]$ to evaluate the transformation of the field operators from the input to the output of the amplifier.
Calculation can be performed using an  auxiliary variable $\xi_m=\chi^{(m)} t$ giving a compact form for the Heisenberg equation
\begin{equation} \label{eq_heisemberg2}
	\frac{d \vec{\textbf{A}}_{(s/a)}}{d\xi_m} = \textbf{M}_{(s/a)} (\xi_m)\vec{\textbf{A}}_{(s/a)}, \mbox{where}
\end{equation}
$	\vec{\textbf{A}}_{s/a} = (\hat{a}_{0(s/a)}~~ \hat{a}_{0(s/a)}^{\dagger} ~~\hat{a}_{1(s/a)} ~~\hat{a}_{1(s/a)}^{\dagger}~~ \hat{a}_{2(s/a)}~~ \hat{a}_{2(s/a)}^{\dagger} )^T$. The evolution matrix has an explicit dependence on the amplitude of the fields inside the crystal, that evolve along the propagation, as evaluated in the previous section,

\begin{widetext}
 \begin{footnotesize}
 \setlength{\arraycolsep}{0.1pt}
\renewcommand{\arraystretch}{0.1}
 \begin{eqnarray}
 \textbf{M}_{(s/a)}&=&\\&\xi_{m}&\left(
\begin{array}{cccccc}
 0 & \mp 2(m-2)\alpha_{\omega 1}\alpha_{\omega 2} &  -(m-1)\alpha^{*(m-2)}_{\omega 0}\alpha_{\omega 2}  & 0 & - (m-1)\alpha^{*(m-2)}_{\omega 0}\alpha_{\omega 1}  & 0  \\
 \mp 2(m-2)\alpha_{\omega 0}\alpha^{*}_{\omega 2} & 0 & 0  & -(m-1)\alpha^{(m-2)}_{\omega 0}\alpha^{*}_{\omega 2}  & 0  & - (m-1)\alpha^{*(m-2)}_{\omega 0}\alpha^{*}_{\omega 1}  \\
(m-1)\alpha^{(m-2)}_{\omega 0}\alpha^{*}_{\omega 2}  & 0 & 0  & 0  & 0  & \pm \alpha^{m-1}_{\omega 0}\\
0  & (m-1)\alpha^{*(m-2)}_{\omega 0}\alpha_{\omega 2}  & 0 & 0  & \pm \alpha^{m-1}_{\omega 0}  & 0  \\
 (m-1)\alpha^{(m-2)}_{\omega 0}\alpha^{*}_{\omega 1}  &0 &0 & \pm \alpha^{*(m-1)}_{\omega 0}  & 0  & 0 \\
 0  & (m-1)\alpha^{*(m-2)}_{\omega 0}\alpha_{\omega 1} & \pm \alpha^{*(m-1)}_{\omega 0}  & 0  & 0 & 0  \nonumber \\
\end{array}
 \right).
\end{eqnarray}
\end{footnotesize}
\end{widetext}
Solving the differential  Eq. (\ref{eq_heisemberg2}), the result can be written as:
\begin{equation}\label{gain}
\vec{\textbf{A}}_{(s/a)}\mid_{\xi_m=\kappa^{(m)}} = \textbf{G}_{m(s/a)} \vec{\textbf{A}}_{(s/a)}\mid_{\xi_m=0},
\end{equation}
with
\begin{equation}
	\textbf{G}_{m(s/a)}= exp\left(\int_0^{\kappa^{(m)}} d\xi_m\textbf{M}_{m(s/a)}(\xi_m)\right).
 \label{eq_ganho}
\end{equation}

Most of the works until now considered that the evolution of the field inside the crystal is negligible due to the low total power variation of signal of this systems~\cite{Debuisschert:93}, a valid situation for a small gain, typical situation found in closed cavities. The integration of~ Eq. (\ref{eq_ganho}) allows us to study the behavior of the fields inside two different gain media without this consideration enabling the accurate study in the open cavity regime \cite{alvaro2020}.

\begin{figure}[b]
	\centering
	\includegraphics[scale=1.3]{CavityRT.pdf}

\caption{\label{cavfp} Representation of the fields inside the cavity with a gain medium $\chi^{(m)}$. $M_1$ and  $M_2$ are the input and output mirrors, with reflectivities (transmissivities) coefficients $\mathbf{R}$ ($\mathbf{T}$) and $\mathbf{R}^{'}$($\mathbf{T}^{'}$).}
\end{figure}

In order to evaluate the behavior of the field fluctuations in a round trip inside the cavity (Fig. \ref{cavfp}) the procedure described in~\cite{MUNOZ2018} was adapted to a format that can be used considering both $\chi^{(2)}$ and $\chi^{(3)}$ as the gain medium inside the cavity (see Appendix \ref{transform} for further details).
The output field $\bf{\hat{A}_{R}}$ from the cavity is directly related to the incident field $\bf{\hat{A}_{in}}$ and the additional vacuum field $\bf{\hat{A}_v}$, associated to spurious losses of the cavity, by the relation
\begin{equation}\label{cavityBS}
   \vec{\textbf{A}}_R = \textbf{R}_{\kappa}\vec{\textbf{A}}_{in} + \textbf{T}_{\kappa}\vec{\textbf{A}}_{\nu}.
\end{equation}
where $\textbf{R}_{\kappa}$ and $\textbf{T}_{\kappa}$ are the effective reflection and transmission matrices of the OPO cavity, accounting for all the mirror coupling and the gain transformation described by Eq. (\ref{eq_ganho}).

Knowing the OPO output fields, the correlation between the output quadratures can be analyzed. Following the analysis in the reference~\cite{MUNOZ2018} we performed a complete description of the covariance matrix of the Hermitian operators $\hat{p}_{\omega_{n}}$ and $\hat{q}_{\omega_{n}}$ of pump, signal and idler modes, that satisfies the commutation relation $[\hat{p}_{\omega}, \hat{q}_{\omega^{'}}^{'}]= 2 i \delta (\omega-\omega^{'})$  and are related with the operators $\hat{a}_{\omega}$ and $\hat{a}_{\omega}^\dagger$ by $\hat{p}_{\omega}= (\hat{a}_{\omega}+\hat{a}_{\omega}^{\dagger})/2$ and $\hat{q}_{\omega}= i(\hat{a}_{\omega}-\hat{a}_{\omega}^{\dagger})/2$. The covariance matrix of the reflected field is given by:
\begin{equation}\label{eq:matCovar_main}
\textbf{V}_R =\tilde{ \textbf{R}}_\kappa \textbf{V}_{in}\tilde{\textbf{R}}_\kappa^{-1}+\tilde{\textbf{T}}_\kappa \textbf{V}_{\nu}\tilde{\textbf{T}}_\kappa^{-1},
\end{equation}
where $\textbf{V}_{in}$ is the input covariance matrix of the pump, signal and idler fields, and $\textbf{V}_{\nu}$ is the covariance matrix related with the input vacuum modes. We considered that the input covariance matrix represents a coherent state, $\textbf{V}_{\nu}=\textbf{V}_{in}=\textbf{I}$. Details of the calculation procedure can be found in  \cite{MUNOZ2018}.

\section{Analysis of the covariance matrix\label{secent}}

The covariance matrix $\textbf{V}_R$ gives a unique description of the system, and for a Gaussian state is equivalent to the determination of the density operator \cite{Simon}. It gives all the possible information about the system, including squeezing and entanglement of the fields. In what follow, we will make a detailed analysis of the quantum features that can be found in the different configurations, DROPO and TROPO, $\chi^{(2)}$ and $\chi^{(3)}$.

In order to give a general view, we have chosen a reflectance of $R=85\%$ for the output coupler, and no additional loss in the cavity. Analysis frequency is chosen to be half of the cavity bandwidth $BW$: $\Omega=0.5 (1-R)/\tau$, where $\tau$ is the round trip time of the wave inside the cavity. TROPO is chosen to be with all the reflectances identical.

\subsection{Source of squeezed states}

If we want to observe the noise compression, as evaluated for instance in \cite{Fabre1997}, we may restrict the study to the covariance of the symmetric variables \cite{PhysRevA.88.052113}. From the symmetry between signal and idler modes, only the presentation of the variances of one of these fields is necessary.
The quadratures associates to the amplitude and phase are represented by $\Delta p_{si}$ and $\Delta q_{si}$, with $i=\{0,1,2\}$ for pump, signal and idler fields.

\begin{figure}[b]
	\centering
	\includegraphics[scale=0.30]{DROPO_chi2andchi3_Var.pdf}
\caption{\label{VarSimchi23} DROPO: Diagonal terms of $V_{s}$ in terms of $\sigma$ for $R_{1}=85\%$ and $\Omega= 0.5 BW$. Considering in a) a $\chi^{(2)}$ gain medium and in b) a $\chi^{(3)}$ gain medium.}
\end{figure}

 \begin{figure}[hb]
	\centering
	\includegraphics[scale=0.30]{TROPO_chi2andchi3_Var.pdf}
 \caption{\label{fig:TROPOVarSim} TROPO: Diagonal terms of $V_{s}$ in terms of $\sigma$ for $R_{1}=85\%$ and $\Omega= 0.5 BW$. Considering in a) a $\chi^{(2)}$ gain medium and in b) a $\chi^{(3)}$ gain medium.}
 \end{figure}

Individual variances for the DROPO are presented in~Fig. \ref{VarSimchi23}, for  $\chi^{(2)}$ and $\chi^{(3)}$ gain medium.
Equivalent results for the TROPO are presented in Fig.~\ref{fig:TROPOVarSim}.
There are many common features on these curves. The compression of the phase noise of the pump, $\Delta^2 q_{s0}$, as observed in \cite{Fabre1997}, is verified not only in Fig.~\ref{fig:TROPOVarSim}a, but in all the distinct configurations. It is interesting to notice that while the compression is limited to 0.5 in the $\chi^{(2)}$ TROPO \cite{Fabre_1990}, the $\chi^{(2)}$ DROPO can beat this value. On the other hand, while compression of this quadrature in presented in $\chi^{(3)}$ OPO, they are not so effective as squeezers for the pump. In fact, for the $\chi^{(3)}$ DROPO, noise compression is even limited to the range of $\sigma < 2.2$. But then we have a curious feature: for $\sigma < 2.3$, the pump amplitude $\Delta^2p_{s0}$ becomes squeezed. This difference in behavior between $\chi^{(2)}$ and $\chi^{(3)}$ OPOs can be explained by the additional term in Eq.~(\ref{hamsimantissim}), giving the compression operator acting on the pump, associated to the mean converted fields inside the cavity. This effect should compete with the usual dynamics of the phase noise compression provided by the back conversion of the signal and idler fields into the pump mode described in \cite{Fabre1997}. For a strong field, it should beat the phase compression.

As for the converted fields, they present a nearly perfect thermal state right above the threshold, but for an increasing pump power, while phase noise $\Delta^2q_{s1}$ grows smoothly, the amplitude noise $\Delta^2p_{s1}$ presents a strong peak, that is much more pronounced for the TROPO, and almost coinciding with the peak noise for the pump amplitude $\Delta^2p_{s0}$. On the other hand, above a certain value, the noise drops and we eventually have noise compression for this field. While this effect was already predicted in the literature for the $\chi^{(2)}$ TROPO, to the best of our knowledge is was not observed yet. A good reason could be the fact that is should appear above $\sigma=3$, situation where the thermal effects will become dramatic in optical crystals, and the intense fields that are produced will elude the usual homodyne techniques for noise measurement. It would be necessary to use self-homodyning, as done in \cite{Villar2005}, for its observation. Nevertheless, the use of $\chi^{(3)}$ gain medium on a DROPO reduces the value of the necessary pump power for reaching the squeezed output. In fact, for a $\chi^{(2)}$ DROPO we reach a significant noise reduction to the shot noise level, that was even observed in preliminary studies using an OPO with atomic vapor as the gain medium \cite{alvaro2020}, although squeezing was not yet verified. The strong compression for the  amplitudes of pump, signal and idler field is a dramatic demonstration of the role of pump depletion even in a single pass of the beam through the crystal.

\begin{figure}[b]
\centering
\includegraphics[scale=0.30]{DROPO_chi2andchi3_Duan.pdf}
\caption{\label{Duanchi23} DROPO: EPR inequality (continuum, orange), $\Delta^{2} p_{-}$ (dashed,purple) and $\Delta^{2} q_{+}$ (dashed, green) in terms of $\sigma$ for $R_{1}=85\%$ and $\Omega=0.5 BW$. Considering in a) a $\chi^{(2)}$ gain medium and in b) a $\chi^{(3)}$ gain medium. The grey solid line represents the limit value of the inequality given by~ Eq. (\ref{eq_duan}).}
\end{figure}

\begin{figure}[hb]
\begin{center}
\includegraphics[scale=0.30]{TROPO_chi2andchi3_Duan.pdf}
\caption{TROPO: EPR inequality (continuum, orange), $\Delta^{2} p_{-}$ (dashed,purple) and $\Delta^{2} q_{+}$ (dasehd, green) in terms of $\sigma$ for $R_{1}=85\%$ and $\Omega=0.5 BW$. Considering in a) a $\chi^{(2)}$ gain medium and in b) a $\chi^{(3)}$ gain medium. The grey solid line represents the limit value of inequality~ Eq. (\ref{eq_duan}).}
\label{fig:TROPODuan}
\end{center}
\end{figure}

\subsection{Bipartite Entanglement}

Two mode entanglement, the basic resource for quantum information processing, can be directly observed from the second order momenta \cite{Reid1989,Duan2000}.
In fact, noise compression in the Einstein-Podoslky-Rosen type operators criteria~\cite{Duan2000} is a sufficient condition for a successful teleportation of a quantum state between two sites \cite{furusawa1998}. This DGCZ criterion can be expressed as an inequality of the form
\begin{equation}
\Delta^{2}p_{-}+\Delta^{2} q_{+}>2,
\label{eq_duan}
\end{equation}
where the EPR-type variables are $\Delta^{2} p_{-}=(p_{s1}- p_{s2})/\sqrt{2}$ and $\Delta^{2} q_{+}=(q_{s1}+ q_{s2})/\sqrt{2}$. If the variances of these lines combinations of quadratures violates the inequality, the bipartition $\{1,2\}$ is necessarily entangled.

Twin beams produced by OPOs are a regular source of entangled states \cite{Peng1992,Villar2005}. In what follows, we will evaluate the noise compression of the correlated intensities and the anti-correlated phases of the fields generated in distinct cavity configurations.
Fig.~\ref{Duanchi23} presents the behavior of~ Eq. (\ref{eq_duan}) considering the DROPO, while Fig.~\ref{fig:TROPODuan} present the results for the TROPO.
An outstanding result is the robustness of the twin beam correlation \cite{Heidmann1987}. The subtraction of the amplitudes for all the four configurations is the same, and directly related only to the cavity bandwidth and detection efficiency - the squeezing level depends on the fraction of the twin photons, that are generated by the parametric conversion, that is detected. Therefore, this variance is independent of the pump power, even though the variance of each field may change dramatically, from excess noise to squeezing, as seen in the previous section.

On the other hand, phase anti-correlation, associated to the noise compression in $\Delta^2 q_+$, is more fragile, and depends strongly on the pump power \cite{Villar2005}. Starting from the same level as the $\Delta^2 p_-$ close to the threshold, it had a monotonic increase. Here the effect of the gain medium is very relevant: while $\chi^{(2)}$ OPOs have a limit where this variance asymptotically reaches the vacuum level for increasing pump power ($\Delta^2 q_+<1$), $\chi^{(3)}$ OPOs will cross this limit at very low pump power, $\sigma\simeq1.5$. Moreover, the loss of noise compression for growing pump power is more pronounced in the DROPOs, when compared to TROPOs, for both gain medium, and should be considered on the development of entangled bipartite sources.

As a result, DGCZ inequality is violate for all the value range for the $\chi^{(2)}$ OPOs, but it is satisfied only up to a certain level of pump power for $\chi^{(3)}$ OPOs. This is a main limitation of this system for bipartite entanglement, but since we are dealing here with pure states, the loss of entanglement in this two mode partition can be understood as their coupling to other modes of the system, as we should see in the next subsection.

Another important analysis is related with the behavior of the noise spectrum in terms of the analysis frequency normalized by the cavity bandwidth~$\Omega_{BW}$. As an example, we focus on the substraction of the signal and idler fields. It is very well known that in $\chi^{(2)}$ OPOs the behavior of the phase ($\Delta^{2}q_{s-}$) and amplitude ($\Delta^{2}p_{s-}$) are insensitive to the pump power, and depends only on the analysis frequency. Noise compression in  $\Delta^{2}p_{s-}$ will follow a Lorenztian, of width given by the cavity bandwidth, and for a lossless system, we have $\Delta^{2}p_{s-}\Delta^{2}q_{s-}=1$.
The same situation is verified for the DROPO, as can be observed in~Fig. \ref{fig:DROPOminusomega}. This response was recently been observed for a $\chi^{(3)}$ DROPO with atomic vapor \cite{alvaro2020}, where the twin beam generation has showed to be independent of the pump power, and follow a Lorentzian shape.

\begin{figure}[ht]
	\centering
	\includegraphics[scale=0.30]{DROPO_chi2andchi3_minus_omega.pdf}
 \caption{\label{fig:DROPOminusomega} DROPO: $\Delta^{2}p_{s-}$ and $\Delta^{2}p_{q-}$ in terms of $\Omega$ (normalized by the cavity bandwidth $BW$) for $R_{1}=85\%$ and $\Omega_{BW}=0.5$. Considering in a) a $\chi^{(2)}$ gain medium and in b) a $\chi^{(3)}$ gain medium.}
 \end{figure}

\subsection{Tripartite Entanglement}

While DGCZ criterion is a useful test for bipartite entanglement, it is not both necessary and sufficient on its usual form (Eq. \ref{eq_duan}). On the other hand, positivity under partial transposition (PPT) was showed to be a necessary and sufficient criterion not only for bipartite Gaussian states~\cite{Simon}, but for $1\times N$ bipartitions as well \cite{Werner2001}.

Partial transposition operation in a CV system is like a mirror reflection in the phase space, acting only in a partition. When a transposition operator is applied on the density operator, the corresponding Wigner function transforms as $W( {\bf{x}})\rightarrow  W(\mathbf{\Gamma} {\bf{x}})$, with $\mathbf{{x}}=({p}_{1},{q}_{1},{p}_{2},{q}_{2})$, we have the substitution of $\mathbf{{x}}\rightarrow\mathbf{\Gamma} {\bf{x}}$, with $\mathbf{\Gamma}=diag(1,1,1,-1)$.
If the new Wigner function does not correspond to physical density operator, the system is entangled. On the other hand, if it does and the Wigner function is Gaussian, then we know that the bipartition is separable, PPT can be immediately verified by the covariance matrix.
Writing the set of commutation rules as  $[\hat{x}_{i},\hat{x}_{j}]=i\mathbf{\Omega}_{i,j}$ where $\mathbf{\Omega}=\bigoplus_{k=1}^{N} J$ and $J= \begin{pmatrix} 0 & 1 \\-1 & 0 \end{pmatrix}$, the uncertainty relation can be expressed as  $\mathbf{V}+i\mathbf{\Omega} \geq 0$. Partial transposition implies in the transformation $\mathbf{V}  \rightarrow^{PT} \mathbf{\tilde{V}}=\mathbf{\Gamma} \mathbf{V} \mathbf{\Gamma}$.
Physicality of the partially transposed covariance,
$\mathbf{\tilde{V}}+i\mathbf{\Omega} \geq 0$.
can be verified by the evaluation of the symplectic eigenvalues $\nu_{k}$ of $\mathbf{\tilde{V}}$ \cite{ECVS(ADESSO2007), THESISADESSO, Williamson}.

\begin{equation}
\nu_{k}=\sqrt{\left[Eigenvalues(\mathcal{V})\right]_k}\,\, ,
\label{eq_PPT}
\end{equation}
where $\mathcal{V}=-(\mathbf{\tilde{V}\Omega})^{2}$. When $\nu_{k}\geq1~\forall~k$ the transformed matrix $\mathbf{\tilde{V}}$ is physical.
It follows from this conditions that if the minimum symplectic eigenvalue $\nu<1$, the covariance matrix $\mathbf{{V}}$ corresponds to an entangled state, and the minimum symplectic eigenvalue is an entanglement witness for the given bipartition.
The treatment can be extended to multipartite states, and for a $1\times N$ bipartition, this is a necessary and sufficient condition to demonstrate entanglement for Gaussian states \cite{Werner2001}.

\begin{figure}[t]
	\centering
	\includegraphics[scale=0.30]{DROPO_chi2andchi3_PPT_sym.pdf}
\caption{\label{pptchi23} DROPO: Minimum symplectic eigenvalues $\nu_{k}$ obtained from $\textbf{V}_{s}$ in terms of $\sigma$ for $R_{1}=85\%$ and $\Omega= 0.5 BW$. Considering in a) a $\chi^{(2)}$ gain medium and in b) a $\chi^{(3)}$ gain medium. The grey solid line represents the limit value of Eq.~ (\ref{eq_PPT}).}
\end{figure}

\begin{figure}[ht]
	\begin{center}
		\includegraphics[scale=0.30]{TROPO_chi2andchi3_PPT_sym.pdf}
		\caption{TROPO: Minimum symplectic eigenvalues $\nu_{k}$ obtained from $\textbf{V}_{s}$ in terms of $\sigma$ for $R_{1}=85\%$ and $\Omega= 0.5 {BW}$. Considering in a) a $\chi^{(2)}$ gain medium and in b) a $\chi^{(3)}$ gain medium. The grey solid line represents the limit value of Eq.~(\ref{eq_PPT}).}
		\label{fig:TROPOPPT}
	\end{center}
\end{figure}

For this subsection, we will keep our analysis on the subspace of the symmetric covariance matrix, as it will reflect the kind of entanglement usually observed when each beam issued from the OPO is treated as a single mode \cite{Coelho823}. For three modes, we have three possible bipartitions $1\times2$ and the tripartite entanglement is verified if  $\nu<1$ for all of them. Since the Simon-PPT criteria is necessary and sufficient, we will apply it as well to the subsystems formed by pairs of the beams, comparing the conclusions to those inferred from the DGCZ criterion.

In~Figs.~\ref{pptchi23}  and~ \ref{fig:TROPOPPT} the behavior of the symplectic eigenvalues for the DROPO  and TROPO are presented. For three modes, we have the eigenvalue for the transposition of the pump, $\nu_0$, and for signal or idler, $\nu_1=\nu_2$. We plot the transposition for the possible two mode subsystems, for pump and signal (or idler) $\nu_{01}=\nu_{02}$ and the pair formed by  the converted fields $\nu_{12}$.

It is evident that entanglement of each converted mode to the rest of the system is always verified ($\nu_1$), but the violation is reduced for increasing pump power. It is consistent with the fact that the converted fields are strongly entangled, as can be seen by $\nu_{12}$. If you compare this situation with the observed in Figs.~\ref{Duanchi23} and \ref{fig:TROPODuan}, it is clear that Eq.~(\ref{eq_duan}) fails in identifying some entangled states: as stated in \cite{Duan2000}, the condition is both necessary and sufficient only if the covariance matrix is in one of the standard form that they propose in the article. But a feature is common in both witnesses: the violation for the $\chi^{(3)}$ DROPO is rapidly reduced for a growing pump power.

Situation is more peculiar when we look at the pump mode. Entanglement of the pump with the pair of the converted fields $\nu_0$, or with just one of the fields $\nu_{01}$ is weaker than the observed for each of the down converted beams. Moreover, it can vanish in the region where we observe the peak in the amplitude fluctuations in Figs. \ref{VarSimchi23} and \ref{fig:TROPOVarSim}. This apparent loss of entanglement is not observed in other configurations of the TROPO \cite{villar2006, Coelho823}, where it remains entangled over the entire span of the pump power. The main difference in this case is the fact that all the modes have the same loss for the cavity. This loss of entanglement would be expected if we have loss of purity in the tripartite state \cite{purity}, but as we have found, the situation is more subtle, and looking in the details of the sideband modes, we have in fact hexapartite entanglement on the system \cite{barbosa2018hexapartite}.
As for the  $\chi^{(3)}$ DROPO, we can see that for sufficiently high pump power the pump is apparently disentangled from the twin pair. That is not necessarily true, if we consider now the correlation between the symmetric and the antisymmetric part of the covariance matrix.

\subsection{Multipartite entanglement}

So far, we have limited our analysis only to the symmetric combination of the sidebands, that was showed to be equivalent to the antisymmetric part \cite{PhysRevA.88.052113} in the OPO. As we have showed in \cite{barbosa2018hexapartite}, in the TROPO all the possible 31 bipartitions are entangled. Therefore, we will restrict the current analysis to the relevant features involving bipartitions in the symmetric/antisymmetric basis

\begin{figure}[b]
	\begin{center}
		\includegraphics[scale=0.30]{DROPO_chi2andchi3_PPT_sym_ant.pdf}
		\caption{DROPO: Minimum symplectic eigenvalues $\nu_{sa,k}$ considering $\textbf{V}_{s/a}$ in terms of $\sigma$ for $R_{1}=85\%$ and $\Omega_{BW}=0.5$. $k = \{0, 1, 01, 12\}$ represent the subsystems of pump, signal, pump and signal, signal and idler fields respectively, and $\nu_{sa}$ is obtained by the transposition of antisymmetric pump, signal and idler fields. Considering in a) a $\chi^{(2)}$ gain medium and in b) a $\chi^{(3)}$ gain medium.}
		\label{fig:DROPOPPThexa}
	\end{center}
\end{figure}

\begin{figure}[hb]
	\begin{center}
		\includegraphics[scale=0.30]{TROPO_chi2andchi3_PPT_sym_ant.pdf}
		\caption{TROPO: Minimum symplectic eigenvalues $\nu_{sa,k}$ considering $\textbf{V}_{s/a}$ in terms of $\sigma$ for $R_{1}=85\%$ and $\Omega_{BW}=0.5$. In which $k = \{0, 1, 01, 12\}$ represent the subsystems of pump, signal, pump and signal, signal and idler fields respectively, and $\nu_{sa}$ is obtained by the transposition of antisymmetric pump, signal and idler fields. Considering in a) a $\chi^{(2)}$ gain medium and in b) a $\chi^{(3)}$ gain medium.}
		\label{fig:TROPOPPThexa}
	\end{center}
\end{figure}
For the six modes involved, we will explore the bipartition involving all the modes of the pump ($\nu_{s0a0}$) and all the modes of the signal or idler ($\nu_{s1a1}= \nu_{s2a2}$). It will also be relevant to explore the $\times 3$ partition, involving all the symmetric $\times$ all the antisymmetric modes ($\nu_{sa}$).

As we can see in Figs. \ref{fig:DROPOPPThexa} (for the DROPO) and \ref{fig:TROPOPPThexa} (for the TROPO), entanglement for the pump is completely recovered once we account for the complete description of the state. The same is true for signal or idler modes. The reason for this effect is clear when we consider the shared information between symmetric and antisymmetric case: entanglement is maximized, with a  particularly strong violation for $\nu_{sa}$ in the TROPO (Fig. \ref{fig:TROPOPPThexa}). We have a minimum of $\nu_{sa}$ in the region where we have a peak in the noise of the amplitudes (Figs. \ref{VarSimchi23} and \ref{fig:TROPOVarSim}). This peak is associated with a strong correlation between the symmetric and antisymmetric modes, as is showed in the Appendix \ref{App_CV_sym_antsym}. This effect is dramatically enhanced in the TROPO when the three modes have balanced losses, and is less dramatic (yet still recognizable \cite{PhysRevA.88.052113}) for the usual $\chi^{(2)}$ TROPO, where the coupling of the cavity for the pump is $\approx$ 5 times that of the downconverted modes. On the other hand, for the $\chi^{3}$ OPO, we have a unitary value for $\nu_{sa}$, at $\sigma=2$ for the DROPO and $\sigma=4$ for the TROPO. That is associated to a vanishing correlation between the partitions, as can be seen in Appendix \ref{App_CV_sym_antsym}.

Finally, instead of considering the bipartite case of pump and idler, or signal and idler, we may now consider the whole symmetric and antisymmetric combination for the pair of beams. We may compare now $\nu_{sa12}$ in Figs. \ref{fig:DROPOPPThexa} and \ref{fig:TROPOPPThexa} with $\nu_{s12}$ in
Figs. \ref{VarSimchi23} and \ref{fig:TROPOVarSim}. We cannot observe any difference, what is consistent with the absence of correlation between symmetric and antisymmetric part for signal and idler subsystems. That is not true for the pump mode, that presents relevant correlations involving the symmetric and antisymmetric part of the signal (or idler) modes as can be observed from $\nu_{sa01}$ in Figs. \ref{fig:DROPOPPThexa} and \ref{fig:TROPOPPThexa} compared to $\nu_{s01}$ in Figs. \ref{VarSimchi23} and \ref{fig:TROPOVarSim}. Entanglement between pump and signal in this case is recovered once that the correlation between symmetric and antisymmetric part are taken into account.

\section{Conclusion}

We can see clearly that the versatility of the
OPO as a source of non-classical states in the continuous variable regime is not an exclusivity of the PDC process, but is also present in the case of $\chi^{(3)}$ media. Moreover, even for an extremely open cavity, reaching the limit of a single-pass of the pump through the amplifier, as is the case of the DROPO, noise compression and pump entanglement are also present.
The method presented in~\cite{MUNOZ2018} we have successfully employed reproduces these features observed in the TROPO, and put in evidence the similarities of the DROPO in comparison with the TROPO.

Our analysis here is quite distinct from the one performed for the TROPO in \cite{MUNOZ2018,barbosa2018hexapartite}, that focus on the role of individual sidebands of each one of the beams. In our current approach, we kept the analysis for six modes, but heading back to the symmetric/antisymmetric basis of these sidebands. The reason is twofold: that is the usual measurement basis, leading to the image of entanglement involving individual beams (considered as carrier plus sidebands), and gives a greater evidence of the role of the correlations between the symmetric and antisymmetric spaces. Although identical in individual information, they share a strong correlation leading to relevant entanglement. A good amount of information is lost if this correlation is ignored.

This fact is particularly evident for the dynamics of a cavity with equal losses for the pump, signal and idler, and we can in this case observe the dramatic effect of the entanglement between the symmetric and the antisymmetric modes. It makes a clear difference between the tripartite case and the hexapartite analysis. We may conclude that a detailed analysis of the sidebands is much more than just a reproduction of two equivalent tripartite systems, but rather a rich system of six strongly entangled modes.

While the parametric amplification in the single pass regime \cite{PhysRevA.78.043816} can provide strong correlations in the 4WM process, the use of a cavity can provide a great enhancement of these effects: the twin beam correlation can be as perfect as the ratio of the coupling to the overall losses of the cavity approaches unit. In this case, open cavities, as showed in \cite{alvaro2020}, are a promising source of entangled states.

The linear treatment presented here can give further guidance for the transition between the two operational regimes. It is expected that the linearization should fail close to the oscillation threshold \cite{dechoum}. If that is the case, we may go beyond the treatment described in Eq.~(\ref{hbl1}), that gives only bi-linear operators, that keep the Gaussianity of the input states. But with open cavities, thanks to the high gain, this sudden transition may be smoother and a detailed investigation of the evolution of the sate, with the measurement of higher order momenta, could be performed.

\section{Acknowledgments}
BAFR and RBA contributed equally for the present work.
We thank C. Gonzalez-Arciniegas and P. Nussenzveig for fruitful discussions. This work was supported by FAPESP proc. 2015/18834-0, CAPES and CNPq. B. M. is suported by FAPESP proc. 2014/27223-2.

\bibliographystyle{unsrt}

redpajama_set_name: RedPajamaArXiv


We are coming on a boat from Playa del Carmen, how do we find you?
We set off at 9.30 am seven days a week and the tour last approximately four and a half hours.
However, if your party has a minimum of ten people, we can make a special trip, by prior arrangement, if you prefer at 1.30 pm any day of the week.
So – if you see a slot you want – book it right away. Because, with 80% of our bookings coming from the 20,000 daily cruise visitors, available slots tend to disappear as fast as a puddle in the desert.
As you are coming by cruise ship, please remember there will be plenty of other folks all trying to disembark at the same time as you. So pick a departure time at least 30 MINUTES AFTER you are supposed to dock to allow for the delays in embarkation.


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text:
ACTEGA ECOLEAF Reaches Milestones, Names New Leadership
The business predicts continued success in 2023 with Harald Jasper, managing director of ACTEGA Metal Print, set to take on this leadership position in January.
Harald Jasper, new managing director of ACTEGA Metal Print. (Source: ACTEGA)
ACTEGA has revealed the year's achievements and successes of its ACTEGA Metal Print business, responsible for bringing the ECOLEAF sustainable metallization technology to market.
The business predicts continued success in 2023 with Harald Jasper, managing director of ACTEGA Metal Print, set to take on this leadership position in January. With extensive industry experience, he will implement his own expert vision and strategies as well as maintain some of the innovative approaches to advancing ECOLEAF's success established this year by Stephan Plenz.
Thorsten Kröller, ACTEGA chairman, announced the appointment of Harald Jasper as the new general manager of ACTEGA Metal Print. A founding member and a key player in developing this innovative technology at ACTEGA Metal Print, Harald's expertise has been integral in establishing ECOLEAF's current structure. With an understanding of the market - alongside customers' needs - Jasper's deep knowledge promises strong success for the company going forward, the company says.
"Following years spent perfecting it prior to launch into an ever-evolving market, under Harald's leadership, we believe that the market acceleration for ECOLEAF is imminent and will change the industry forever," say Kröller, who also expressed his gratitude to Stephan Plenz, who has provided an invaluable outside-in look on ECOLEAF technology; with product and cost improvements made under his vision, ensuring that they are future-proofed against ever-changing customer demands.
According to the company, the success and appeal of ECOLEAF can be largely attributed to the sustainability benefits of its foil-less decoration process for metallic embellishments. Initial calculations from ACTEGA's Life Cycle Assessment find that ECOLEAF can achieve a carbon footprint reduction of more than 50 percent compared to cold and hot foil processes.
The metallization process not only delivers a more sustainable production model for labels and packaging printers and the brands they serve, but also positively impacts costs and production times without compromising quality. "It is all of these benefits combined that make ECOLEAF such an attractive and potentially transformative technology for the industry. This potential has been realized by many brands and printers across Europe in 2022, with many already incorporating ECOLEAF into their production facilities," said Kröller.
Paolo Grasso, sales director for ECOLEAF at ACTEGA, noted, "2022 has not been without its challenges but, despite this, it has been a year with a lot to be proud of. Some of the most significant developments on the road to ECOLEAF's market adoption was the announcement of our first digital partner, Springfield Solutions, and our first OEM partner, AB Graphic International (ABG). In addition to these significant and valuable collaborations helping us drive the conversion to sustainable metallization forward, numerous other forward-thinking companies committed to orders and installed ECOLEAF units in 2022."

redpajama_set_name: RedPajamaCommonCrawl


text:
Postmodern London landmark No 1 Poultry gets listed status
Architecture, News I by Betty Wood
No 1 Poultry in the City of London. Photography: Rob Low for The Spaces
James Stirling's Postmodern building No 1 Poultry was awarded Grade II*-listed status today, making it England's youngest listed building.
Completed in 1998, the candy-striped No 1 Poultry was architect Stirling's last project, finished five years after his death by Michael Wilford.
One of the City of London's most prominent examples of postmodern architecture – still deemed the 'marmite' of the design world – the building sits on a wedge-shaped site and features a pink and yellow limestone façade.
The building is famous for its 'candy-striped' limestone façade. Photography: Rob Low for The Spaces
Historic England's announcement follows a successful appeal by The 20th Century Society, which initially tried to list the building in December 2015. The Society objected to Buckley Gray Yeoman's planning application to fill in the building's colonnade and alter its windows.
Roger Bowdler, director of listing at Historic England, said: 'We are delighted that No 1 Poultry has now been listed. It is one of London's best-known modern buildings, a key work of late 20th century architecture, by one of England's leading post-war architects, James Stirling.'
The decision also follows hot on the heels of another Postmodern architecture landmark listed earlier this month: a pair of buildings within Sir Terry Farrell's 1980s Comyn Ching Triangle complex were given Grade II-listed status on 10 November.
Read next: Photographer Rob Low captures London's eccentric Postmodern landmarks
Tags: PoMoPostmodern architecture
A huge haul of Memphis design is up for auction at Wright
News, Travel
Saved By The Bell comes to life at this 90s-themed Airbnb in Dallas
Historic England lists a cache of radical Postmodern buildings
Janet Street-Porter's 'extroverted' Postmodern home is listed


HANOI (Viet Nam News/ANN) - Asean countries need to strengthen co-operation and share their experiences to ensure they can lead the way on next-generation 5G wireless technology.
So said Deputy Prime Minister Vu Duc Dam at the conference on 5G & Development of Digital ASEAN in Hanoi on Friday.
According to the Deputy PM, facing the changes and advancement of technology, the ones who are proactive and pioneering will be the more successful afterwards.
Speaking at the conference, Minister of Information and Communications Nguyen MAnh Hung highlighted the significant role of 5G network development.
According to the minister, Vietnam will be one of the first countries in the world to roll out 5G and is willing to co-operate with other ASEAN countries to establish joint ventures and factories to make the necessary IT equipment.
He said that from the beginning of 2019, the Ministry of Information and Communications had already licensed local carriers to test the technology in Hanoi and HCM City, noting that the country is also focusing on developing other 5G-based technologies and equipment.


Are you looking for a role where you're part of a collaborative team in an environment that truly makes a difference?
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text:
Admission open for MDS course 2022-23 session. Contact 8298899558
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The curriculum for the BDS has been designed to achieve the following objectives
To produce top-notch clinical practitioners in the field of dentistry.
To shape the careers of students by developing leadership qualities. The college provides approved degree of Bachelor of Dental Surgery (B. D. S.) as per the norms set up by Dental Council of India and Vinoba Bhave University. However, HCDSH constantly innovates and improves the education by periodically reviewing its curriculum, to ensure that it remains consistent in achieving its own vision and mission.
At HCDSH, the learning process is made interactive, making it more attention-grabbing and interesting. The training that the students go through at HCDS H provides them with a wide array of practical exposures which is not only required as a part of the curriculum but is also advantageous in providing them with the experience to deal with patients.
Description Eligibility Selection
Schedule of B.D.S. Course
The degree of The BDS is awarded by Vinoba Bhave University after successful completion of five years of the BDS course. The course consists of 4+1 (Internship) years. There is a University examination every year as per schedule.
Today's field of dentistry is much demanding. New areas of practice like Implantology, Cosmetic dentistry are the need of the day. HCDSH plans to include technology-rich aspects of clinical dentistry for its final year students.
Admission of BDS – General Information:
Hazaribag College of Dental Sciences & Hospital admits 100 students in the B.D.S. course every year. Admission for 100 seats in BDS (Bachelor of Dental Surgery)
The B.D.S. Program at HCDSH is recognized by Dental Council of India
The candidate must have passed 10+2 or equivalent examination from a recognized Board, Council or a University and must have obtained at least 50% marks in Physics, Chemistry and Biology taken together and must have passed in English. For candidates belonging to OBC minimum qualifying marks is 45% and SC/ST is minimum 40%. The candidate must attain the minimum age of 17 years at the time of admission.
Candidate should qualify in NEET (National Eliglbility Cum Entrance Test)of current year. Click here to check your result
(NEET) National Eligibility Cum Entrance Test Qualified Candidate only through counselling.


text:
Memphis: A Musical for Its Time
The hit Broadway musical Memphis, which recently aired on PBS with its original cast intact, has an irresistible swing. Joe DiPietro's book presents a simplified version of how rhythm and blues crossed over to attract white audiences in Memphis in the 1950s, with characters standing in for demographic groups and points of view. But you feel as you do at certain movies that dramatize key moments in the chronicle of popular music like American Hot Wax or Cadillac Records: the show gets it right emotionally even if you don't buy it as history, and you get caught up in the visceral excitement of the subject matter. It does a hundred things wrong and still delivers an awfully good time.
Chad Kimball plays Huey Calhoun, a white ne'er-do-well who ventures into a black club in his home town in 1951 to hear up close the glorious music he's been sneaking illicit listens to since he was a boy. There he discovers Felicia (Montego Glover), who looks like a black baby doll with bright button eyes and marcelled jet hair but sings gospel and R&B in an exuberant bourbon-and-molasses voice. Huey can't hold down a job as a department-store stock boy (he gets distracted and breaks the merchandise) but when he convinces his boss to let him try out in the music department he causes a small-time sensation playing "race records." The ones he sells to turned-on white customers are his own, of course; the store doesn't carry music that isn't white – and his employer is too scandalized to keep him on. But he talks his way into a DJ spot on a local radio station with a combination of sheer bravado and unarguable results: listeners who haven't heard anything like the kind of music he spins keep calling in to request more. Eventually he gets the station manager (Michael McGrath), a pragmatist with a terrible sidewall haircut, to let Felicia perform one of her songs live on the air, and her brother Delray (J. Bernard Calloway), who owns the club where Huey first saw her, starts producing her records. But Delray has trouble warming up to Huey. When he and Felicia start a romance, Delray foresees trouble, and he's right. Though they have to sneak around, mostly after dark, they aren't careful enough, and the first act ends with redneck thugs beating both of them up in the street. (They aim the worst blows, administered with a baseball bat, at Felicia.) The second act finds Huey as the host of an American Bandstand-style local TV show that gets away with showcasing African American singers and dancers, and Felicia setting her sights on a crossover career in New York City, where she and Huey can live safer lives.
J. Bernard Calloway and Montego Glover
The score – music by Bon Jovi's David Bryan, lyrics by Bryan and DiPietro – is undeniably crafted for Broadway but it includes some rousing big numbers. Huey gets two self-defining songs, "The Music of My Soul" (which explains what drew him to Delray's club) and "Radio" (which arrives close to intermission). "The Music of My Soul" enshrines an apple-of-temptation moment, like the song from the musical Jelly's Last Jam in which the child who will grow into Jelly Roll Morton violates the rules of his aristocratic Creole upbringing and mixes it up with lowdown New Orleans pop culture. The platter that lands Huey the radio-station gig is called "Everybody Wants to Be Black on a Saturday Night," which cleverly makes the show's point that black music is destined to cross the color bar. (Inevitably, there's a civics-class side to this idea: once Huey has introduced R&B to white kids, we get an interlude where a white girl persuades some black girls to let her skip rope with them. Fortunately Memphis doesn't include very much of this sort of thing.) Glover has most of the best songs: "Someday" (Felicia's first single), "Stand Up" (which ends on a spectacular gospel-style high note that she seems to hold onto for a sweet eternity) and especially "Colored Woman," which justly brings down the house. It's Felicia's self-defining ballad, summing up the limitations of black woman's existence in a white world as her mother laid them out for her but suggesting the eagerness of a woman from her generation to challenge those conventions. In addition to the marvelous Montego Glover, the cast features three African American performers with enormous voices: Calloway, James Monroe Iglehart as Bobby (janitor at the radio station and a Delray's regular) and especially Derrick Baskin as Gator, Delray's bartender. The role of Gator is one of those terrible symbolic concepts. He hasn't spoken a word since, as a little boy, he saw his daddy lynched, but at the end of act one, when Huey carries Felicia into the club, badly beaten, and Delray, enraged at him for putting his sister in danger, lunges at him, it's Gator who stops the fight with a song of hope and freedom, "Say a Prayer." God knows this idea shouldn't work; just describing the set-up is embarrassing. But it's a good number, and Baskin's singing catches you by the throat. Memphis certainly isn't subtle but at moments like these there's a kind of pop genius at work in it.
One song that gets audiences cheering is truly cringe-worthy, however: "Change Don't Come Easy," performed in the second act by Huey's mama (Cass Morgan) with Bobby, Gator and Delray as back-up. Mama Calhoun is a good ol' gal, a widow who lives low-rent with her son and holds down a job as a waitress. They have a warm relationship but when she finds Felicia in her home she throws her out, reminding Huey that she runs a good Christian house. But she grows more tolerant as the years go by (the musical roughly spans the fifties) and she begins to understand his love of soul music when she follows his advice and attends a black Baptist service, where the song we hear is called "Make Me Stronger." This whole subplot is phony as hell – a stage conceit built around the kind of cutesy cartoon character Broadway audiences always fall for. (It may not be Cass Morgan's fault entirely that she's so insufferable, but I wanted to throw something squishy at her.)
Chad Kimball and Montego Glover
Huey is a fascinating hero for a musical. He behaves instinctively with the kind of courage that comes from deep-seated democratic convictions so at odds with those of the culture he grew up in that he's freakish. Yet for the same reason he's dangerously shortsighted. He falls so utterly in love with R&B that he doesn't see any obstacle to turning on every other white boy and girl to the music of his soul, and his impulses are right. But he also falls passionately in love with Felicia and he figures that eventually the racist world around him will see things his way – and of course his impulses there are dead wrong. Even after she's beaten (the blows damage her badly enough to cut off any chance that she can ever get pregnant) he persists in thinking that his love can keep them both safe. When his TV show takes off he tells her that they can go public with their relationship because they're now too famous -- and he's too beloved in Memphis -- to draw any further violence from rednecks. The combination of his obstinate optimism and his refusal to imagine another life for himself outside Memphis dooms their romance.
Kimball is a charismatic performer but his acting is so stylized, both physically and vocally, that he seems almost Martian. He bends his knees and struts, throwing back his shoulders and throwing out his ass; he always seems to be at some peculiar angle with the stage floor. When he makes his first appearance on the stairs leading down to Delray's club, you figure Huey is hammered, but eventually you realize that he's playing every scene that way. And he has a way of emphasizing the last syllable of a word and then holding onto it for dear life, so that the last consonant stretches out for an extra couple of beats ("I'm gonna put you on the radi-oh-oh-oh"). He's a hell of a lot better when he sings, though, and he delivers the eleven o'clock number, "Memphis Lives in Me" (which is also hands down the best song in the score), with so much brio that he redeems all the unreasonable affectation in his performance. When I saw him sing it on Broadway last season he was a knockout. In the TV transcription he embroidered it a little too much, and his voice seemed a trifle ragged; he didn't execute the high trill, switching octaves on the last few bars, to bring it home. It still provided a lot of pleasure.
Memphis, which is directed energetically if not especially inventively by Christopher Ashley and choreographed by Sergio Trujillo, is upbeat in a way that doesn't make you feel foolish for going along with it. Its hopeful presentation of American cultural history feels distinctly of its time; it may be the first Obama-era Broadway musical.
– Steve Vineberg is Distinguished Professor of the Arts and Humanities at College of the Holy Cross in Worcester, Massachusetts, where he teaches theatre and film. He also writes for The Threepenny Review, The Boston Phoenix and The Christian Century and is the author of three books: Method Actors: Three Generations of an American Acting Style; No Surprises, Please: Movies in the Reagan Decade; and High Comedy in American Movies.
Labels: Steve Vineberg, Television, Theatre
Steve Fields February 21, 2013 at 3:14 PM
This looks like a very exciting production to watch. Are there any plans to bring it overseas?
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text:
Toronto Region Mayors and Chairs Appoint Toronto Global Board of Directors
The Toronto Global Mayors' and Chairs' Strategy Council introduced its new Board of Directors today, a diverse and enthusiastic group of community and business leaders tasked with selling the Toronto Region as an ideal location for foreign direct investment (FDI).
"We've recruited an all-star team of influencers and ambassadors to introduce the world to the investment opportunities in the Toronto Region, and to the benefits of working with our incredible range of talent and businesses," said Mark Cohon, Chair of the Board of Toronto Global. "I know I'm not alone when I say we can't wait to hit the ground running, and look forward to working with my fellow Board Members to show the world what we all know to be true – this is where they want to be."
With 14 community and business figures at the table, the Toronto Global Board of Directors draws from a deep pool of diverse professional and cultural backgrounds from across a variety of economic sectors. The Board will provide both professional and experienced strategic oversight to the management of Toronto Global, steering the agency's efforts to enhance the reputation of the Toronto Region and driving its success in promoting a premiere global business destination.
Launched in February 2017, Toronto Global is the product of all three levels of government coming together with a shared vision: to create an engine of growth for the Toronto Region by leveraging the multi-sector strengths of Toronto, Mississauga, Brampton and the municipalities of Halton, York and Durham. It marks the first time municipalities in the Toronto Region have come together to promote themselves to international investors as one region, under on Board, with one brand, and its work will help the region remain competitive on a global scale. Nations are competing fiercely to attract FDI and grow their economies, and now is the time for the Toronto Region to showcase its compelling business value proposition to the world.
"We have a very strong leadership foundation in place and we couldn't be more pleased with these appointments," said John Tory, Mayor of Toronto. "Each of these individuals' professional experience and passionate commitment to business in the Toronto Region will bring insightful perspectives to our Board."
Passionate and prominent business leaders from across the Toronto Region have stepped up to volunteer their time, knowledge, experience and contacts to drive the success of Toronto Global. They include Chair Mark Cohon, who transformed the CFL during his eight year stewardship as League Commissioner; Vice-Chair Janet Ecker, named a Member to the Order of Canada in 2016 for her leadership in the financial industry; Richard Florida, one of the world's leading urbanists and a renowned researcher, professor and author; Bilal Khan, founder of scale-up innovation hub OneEleven; Nitwin Kawale, recognized as Corporate Executive of the year by the Indo-Canada Chamber of Commerce; and Lisa De Wilde, CEO of TVO, and a recipient of the Queen's Diamond Jubilee Medal for her support of scientific literacy.
"These individuals are excellent choices for Toronto Global's inaugural Board of Directors," said Linda Jeffrey, Mayor of Brampton. "They bring demonstrated and relevant industry experience and strategic acumen to the table. We believe their combined experience will further support our goal to promote investment and sustained growth in the Toronto Region."
"The Directors' diverse backgrounds and wide-ranging experience across important industries will add valuable perspective to our Board," said Wayne Emmerson, Chairman and CEO of The Regional Municipality of York. "We appreciate their willingness to serve as ambassadors of Toronto Global and the Toronto Region, and we're looking forward to benefiting from their counsel and extensive expertise."
The new Board of Directors is comprised of the following appointees:
Mark Cohon, Chair
Executive Chairman, Georgian Bay Spirit Co and Chair of the Juno Awards
Janet Ecker, Vice-Chair
President and Chief Executive Officer of the Toronto Financial Services Alliance
Geneviève Bonin, Director
Partner, Management Consulting, PricewaterhouseCoopers Canada
Lisa de Wilde, Director
Chief Executive Officer, Ontario Educational Communications Authority (TVO)
Ted Egawa, Director
President and Chief Executive Officer, Canon Canada
Richard Florida, Director
Director, The Martin Prosperity Institute, Rotman School of Management, University of Toronto
Steve Howse, Director
President, Millington & Associates
Sriram Iyer, Director
President and Chief Executive Officer, ICICI Bank Canada Ltd.
Nitin Kawale, Director
Former President, Enterprise Business Unit, Rogers Communications
Bilal Khan, Director
Managing Director, OneEleven
Tina Lee, Director
Chief Executive Officer, T&T Supermarkets Inc.
Linda Maxwell, Director
Founder and Managing Director, Biomedical Zone, Ryerson University
Satish Rai, Director
Chief Investment Officer, OMERS
Catherine Roche, Director
Head, Office of Strategic Management, Bank of Montreal
The Toronto Global Board of Directors and their profiles are also available online at https://torontoglobal.ca/boardofdirectors
A message from Chair Mark Cohon can be found at the following link: https://youtu.be/enPpCjTATBE
About Toronto Global
Toronto Global is an arms-length, not-for-profit corporation representing municipalities in the Toronto Region. Its mandate is to market the entire region to international investors by promoting the Toronto Region's diversity of services, sectors and industries, diverse and innovative workforce and strong and stable economy to attract FDI from targeted industries around the world. Toronto Global's work will build upon the federal and provincial government's commitment to growing Canada's innovation and knowledge-based economies by driving targeted, region-wide economic growth to generate quality skills, jobs and opportunities for Canadians.
Toronto Global's team of experts will provide a comprehensive, customized service to ensure a smooth transition for international companies wishing to do business in the Toronto Region. Toronto Global will make investing in the region simpler by actively researching and targeting growth sectors such as advanced manufacturing, life sciences, financial services, technology and food and beverage, with a goal of bringing companies to the region to generate economic growth and create high value jobs.
About the Toronto Region
Comprised of the Cities of Toronto, Mississauga and Brampton and the Regions of Halton, York and Durham, the Toronto Region is the economic engine of Canada, generating 20 percent of the country's GDP and housing 40 percent of Canada's business headquarters. The region is a major player on the international stage, attracting business investment from all over the world. There has never been a better time to come together to build on this momentum and to bolster the region's stellar global reputation as a safe, stable and profitable place to invest.
The region is the most multicultural in the world and home to the most skilled and highly educated workforce among countries of the Organization for Economic Co-operation and Development (OECD). With our strong and diverse economy, competitive business costs, exceptional talent pool, strategic location and unmatched quality of life, the municipalities that make up the Toronto Region present a global business offering that is second to none. Each municipality within the Toronto Region is world renowned for its distinctive strengths in sectors that matter to global business leaders.


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I have always been generally fit through football however I never felt I was pushing myself to my limits. I love taking on a challenge and had done a couple of half marathons, but I wanted to push myself further. After scoping the internet I had come to the conclusion I wanted to be an Iron Man and take on the gruelling 2.4 mile swim, 112 mile bike followed by 26.2 mile marathon.
I was excited but apprehensive as after reading a few blogs on the internet, the training looked intense. I could only do 4 lengths front crawl without stopping so this was a real wakeup call as to what I was taking on. I got in touch with Health by Science as a friend recommended them and we sat down and worked backwards and pin pointed a realistic action plan of what was achievable.
I was impressed by their professionalism and their meticulous approach to training. I worked with Health by Science for 8 months and was over the moon to complete the Wales Ironman in 12 hours 48 minutes which far exceeded what I thought I was capable of.


This Oxford Shakespeare edition of Hamlet represents a radically new text of the best known and most widely discussed of all Shakespearean tragedies. Arguing that the text currently accepted is not, in fact, the most authoritative version of the play, this new edition turns to the First Folio of 1623--Shakespeare's "fair copy"--that has been preserved for us in the Second Quarto. Introducing fresh theatrical momentum, this revision provides, as Shakespeare intended, a better, more practical acting script.
Professor Hibbard's illuminating and original introduction explains the process by which variant texts were fused in the eighteenth century to create the most commonly used text of today. Drawing on both critical and theatrical history, he shows how this gusion makes Hamlet seem a much more 'problematic' play than it was when it originally appeared in the First Folio of 1623.


This is final blog in the 3-part series on IBM Db2 EventStore. Part 1 introduces IBM Db2 EventStore, and part 2 describes details on ingesting event data into IBM Db2 EventStore using the scenario of Clickstream analysis for a retail business. In this blog, we will look at analyzing the Clickstream data with IBM Db2 EventStore to derive timely insights on interests of retail customers.
Typically, ingesting streaming event data, persisting with low latency and analyzing it along with historical event data requires integrating multiple analytic systems. IBM Db2 EventStore is purpose built to simplify the complexity of harnessing event data with a single system. Its unique architecture enables analytics on both — the just arrived event data and historical event data.
As the web events are ingested and persisted they can be analyzed using Db2 EventStore OLAP API in Scala. IBM Db2 EventStore is integrated with IBM Data Science Experience in a single distribution. IBM Data Science Experience offers the Notebook environment which data scientists and data analysts can use to interactively explore, analyze and visualize the event data in IBM Db2 EventStore.
The Scala notebook built to query and visualize web metrics based on a sample Clickstream dataset is available in Github repository.
Here are the steps and insights derived from analyzing Clickstream data for CYBERSHOP.
The Customer browsing behavior is best understood with two metrics 'page views' and 'time on page'. The 'page views' indicate how many times customers have visited the page and 'time on page' indicates how much time was spent exploring the content.
The OLAP queries aggregate these metrics from Clickstream data of all users. Here are the visualized metrics from query results.
The visualized metrics show 'Smart Phones' are the leading category of interest across all product lines.
The OLAP query aggregates metrics for all products in 'Smart Phone' category. Here are the visualized metrics from query results.
The results indicate 'A-phone' is the leading product of interest in 'Smart Phones'.
Though 'X-Phone' trails in 'page views' metric compared to other phones, from the relatively high 'time on page' we can infer that there is high interest in X-phones among a section of customers.
The OLAP query aggregates metrics for all features in 'A-Phone'. Here are the visualized metrics from query results.
From the visualized metrics, we can infer 'Color' and 'Camera' are leading features of interest in 'A-Phones' among all customers.
Moving beyond aggregated metrics across all users, the queries can target data of individual users for different timelines. Drilling down to user level data helps understand the interests of the individual customer.
Here are visualized metrics of user 'David' for the past week.
The interactive visual allows to select timelines of each day to understand what product pages 'David' has visited and how much time he spent exploring these products. From the visualized results, we can infer user 'David' is a repeat visitor and has spent significant time exploring 'Smart Phones' in the last few days.
The analysis of recent and historical web events using IBM Db2 EventStore gives CYBERSHOP insights into the trending interests across the general customer base and individual users as well. These insights will help the business to target customers with personalized promotions.
IBM Db2 EventStore offers a well-integrated system to ingest, persist and analyze event data at scale. The system supports high speed ingests while enabling high performance analytics on most recent and historical event data. Delivered on Docker containers and Kubernetes engine, the system offers ease of deployment and elastic scale that is synonymous with the cloud experience. Integrated with IBM Data Science Experience, Db2 EventStore enables data scientists to use the familiar notebook interface to explore, analyze, visualize and build ML models with event data.
In summary, with IBM Db2 EventStore enterprises can harness their event data with ease in on-premises environments without the necessity to expose data to public Cloud services.
The sample dataset and the notebooks shared in the blog are available here. A short video demo of the use case is available on YouTube. For a free technology preview of IBM Db2 EventStore, download it from https://www.ibm.com/us-en/marketplace/project-eventstore.
Watch our on-demand webcast as Mike Gualtieri, Vice President and Principal Analyst at Forrester Research, Inc. talks with IBM about analyzing streaming data during an enlightening new webcast. Learn what business and technology drivers are central to the next wave of event-driven business applications and why fast data management is essential for the future of your business.
Thanks to my colleagues Loic Julien, Adam Storm, David Thomason and Avijit Chatterjee for their key contributions in implementing the use case.


FREE Chicago haunted Archer Avenue paranormal ghost tours, at your own pace along with when you desire to take it or even repeat it, all free.
APP Download for FREE - ' Drive Haunted Archer Ave Tour ', with GPS to take you from stop to stop (nearly 20), and includes audio historic & paranormal history guide of each stop with photos. It is like downloading a free audio photo book GPS tour for your iPhone 3G, iPhone 3GS, iPhone 4, and iPad Wi-Fi + 3G.
If your within 100 miles of Chicago - you can go to the App Store on your iPhone or iPad and search for Geotrio Tours and download the App and the ' Drive Haunted Archer Ave Tour ' will be downloaded (for free, of course).
You will see in on the upper left side of your screen - View in iTunes, click on that and then once on the iTune page- click on Free App and it will be downloaded.
This will allow you to explore and experience all the haunted and paranormal locations that are tied to the much written about Chicago's haunted road 'Archer Avenue' and surrounding areas starting just S.W. of Chicago at the gates of Resurrection Cemetery. Then it takes you to locations like the cemetery that has sculptures of the heads of some of those buried at the gravesite. The Spirit of Diane Masters, Resurrection Mary haunts, The Native American Healing Waters, Monks Castle, a location that is like stepping back in to time and is reported to be very haunted. This free paranormal ghost tour download includes six cemeteries in all, do to their paranormal stories or strange uniqueness.
In the early part of 2010, the company Geotrio came to Edward Shanahan and asked him to join him in this creation, a first to be done for the Chicago land area and Shanahan put his time and effort in to creating this App, mapping it out for the GPS. The photos are his personal ones and then the time adding the finishing touches to taping the audio to be added to the tour, as he is your tour guide at each of the stops. The decision to offer the App for free was his, as he knows people want to explore and experience the locations at their own pace and this allows it.
Edward Shanahan is a Chicago S.W. Suburban resident who has lived all of his 50 plus years within 10 minutes of these haunted locations. He is a Medium, Spirit Feeler, and Channeler, entertaining Psychic Reader, and the Paranormal Host for his Chicago Paranormal Nights with each night at a haunted location ending with a séance and is for both private groups and the public.
Shanahan is written about in four different paranormal books that includes information on Chicago's haunted S.W. Chicago suburban haunted locations, the newest being Ursula Beilski's 'Chicago Haunts 3' and he has assisted her in the past when her tours come to the Chicago S.W. Side.
The next Chicago area tour will be Bachelor's Grove Cemetery and surrounding locations.


A fantastic display and creative photos. Samsung Galaxy J4 Plus gives you both, on a budget.
Your Samsung Galaxy J4 Plus is easy on the eyes in terms of design, but even easier is the large, crystal-clear 6-inch HD+ display made for enjoying videos, games and apps on-the-go.
With a wide lens, the 13-megapixel camera on your Samsung Galaxy J4 Plus is able to take fantastic-looking photos even in low light. The floating shutter button allows you to hit capture from anywhere on-screen.
You can edit both your photos and videos to create a unique story, as well as organising your content by themes and categories, to make it easier to find what you're looking for.
Want to give your photos a more personal touch? You can use stamps, filters, AR stickers and selfie focus to create more unique and creative-looking results, perfect for your Instagram.


Over 55 million Americans lack access to a workplace retirement plan. It's therefore unsurprising that in recent years, public policy discussions around the retirement system in the US have focused on possible measures to expand coverage. Meanwhile, Australia and the UK have created near-universal coverage, but through different policy interventions.
In this webinar, Ida Rademacher will discuss parallels and potential lessons across the three systems with Steve Utkus from Vanguard's Center for Investor Research and Will Sandbrook from the UK's NEST program. Vanguard's 'How America Saves' publication has provided a critical benchmark for understanding retirement saving in the US. In recent years, they have published 'How Australia Saves' and this year, with NEST Insight as a partner, 'How the UK Saves'.
In conversation, Ida, Will and Steve will identify some of the points of similarity and difference that emerge from these publications, and use these as a jumping off point for a broader discussion encompassing other Vanguard, Aspen FSP and NEST research on retirement savings.
The Aspen Institute Financial Security Program and NEST Insight have come together in a transatlantic collaboration to help solve the next generation of retirement challenges. The partnership seeks to enhance understanding of, and provide new insight into, how the retirement system can be improved for low to moderate earners. Through research and discussion, it will help policy-makers and the industry to identify and test potential solutions that can form the next phase of innovation in this sector.
The Collaboration is seeking partners who are interested in providing feedback on these ideas to help us to develop our work program, and who can act as funding- and thought-partners as we embark on actualizing them. For more information read our Framing Paper: The Aspen-NEST Insight Transatlantic Collaboration.