\section{Introduction}

Quantum measurement takes place at the interface between the quantum world
and macroscopic reality. According to quantum mechanics, as soon as we
observe a system we actually perturb it. Quantum non-demolition (QND)
measurement aims to evade the back action noise arising from the measurement
\cite{1,2}, which has given us a window on a variety of quantum systems,
including mechanical oscillators \cite{3,4}, trapped ions \cite{5,6},
solid-state spin qubits \cite{7,8}, circuit quantum electrodynamics \cite%
{9,10}, and photons \cite{11,12}.

In the quantum optics community, Imoto \textit{et al.} \cite{13} proposed an
optical interferometer for QND measurement scheme of the photon number using
the optical Kerr effect, in which the cross-Kerr interaction encodes the
photon number of the signal state onto a phase shift of the probe state.
Subsequently, some works about the photon number QND measurement are studied
\cite{14,15,16,17,44,18}. However, the main obstacles to optical QND
measurement using cross-phase modulation based on third-order nonlinear
susceptibilities $\chi^{(3)}$ are the small value of nonlinearity in the
available media and the absorption of photons. Munro \textit{et al.} \cite%
{19} presented an implementation of the QND measurement scheme with the
required nonlinearity provided by the giant Kerr effect achievable with
AC-Stark electromagnetically transparency. In addition, the schemes using
cavity or circuit quantum electrodynamical systems have enabled photon
number QND measurements that do not rely on the material nonlinearity
through strong light-matter interactions \cite{20,21,22,23,24}.

\begin{figure*}[tb]
\centering{\includegraphics[scale=0.32,angle=0]{fig1.eps}}
\caption{ (a) Schematic of QND measurement of photon number. The probe system
consists of an atom-light hybrid interferometer with active correlation
output readout. In the SU(2) interferometer in the middle box, the LRP is utilized to realize the splitting and combination of
the atomic spin wave and the optical wave. $\hat{a}_{W}^{(0)}$ is in
coherent state, $\hat{S}_{a}^{(0)}$ is an atomic collective excitation which
is prepared by the NRP1. The atomic spin wave
$\hat{S}_{a}^{(1)}$ experiences a phase modulation $\protect\phi_{AC}$ via
the AC-Stark effect by signal light $\hat{b}^{(in)}$. The generated atomic
spin wave $\hat{S}_{a}^{(3)}$ of the SU(2) interferometer and the optical
wave $\hat{a}_{S}^{(1)}$ which is correlated with $\hat{S}_{a}^{(0)}$ are
combined to realize active correlation output readout via the NRP2. $%
\mathrm{LRP}$, linear Raman process; $\mathrm{NRP}$, nonlinear Raman
process. (b) Energy levels of the atom. The lower two energy states $|g\rangle$ and
$|m\rangle$ are the hyperfine split ground states. The higher-energy state $|e\rangle$
are the excited state. The strong pump beams $A_{p_{1}}$ ($A_{p_{2}}$) and $E_{p_{1}}$ ($E_{p_{2}}$)
couple the transitions $|g\rangle$ $\rightarrow$ $|e\rangle$ and $|m\rangle$ $\rightarrow$ $|e\rangle$,
respectively. $\hat{b}^{(in)}$ is far off resonance with the transition $|m\rangle$ $\rightarrow$ $|e\rangle$
by a large detuning. }
\label{fig1}
\end{figure*}

Recently, the atom-light hybrid interferometer has drawn considerable
interest because it is sensitive to both optical and atomic phase shift \cite%
{25,26,27,28,29,30}. There are two types of atom-light hybrid
interferometers, which have been also demonstrated experimentally by our
group. The first type is the SU(2) linear interferometer with linear Raman
process (LRP) replacing regular beam splitter for linear superposition of
atomic wave and optical wave \cite{27}. The second type is the SU(1,1)
nonlinear interferometer utilizing nonlinear Raman process (NRP) for wave splitting and recombination \cite%
{28}. This new type interferometer can be applied to QND measurement of
photon number based on the AC-Stark effect \cite{31,32}.  However, due to the requirements of detection of laser source stability in LIGO, the intensity stabilization justments are very strict across the gravitational wave band \cite{36} and higher accuracy is required for QND measurement of high photon number. 

In this paper, we present an actively correlated atom-light hybrid
interferometer, which is combination of SU(2) and SU(1,1) type atom-light
hybrid interferometers. We theoretically study the SNR
and estimate the quality of this scheme as a QND measurement using the
criteria introduced by Holland \textit{et al.} \cite{33}, and the condition for
a perfect correlation is given. In the presence of losses, the measurement
quality is reduced. However, we can adjust the gain parameter of the NRP in readout stage to reduce the impact due to losses. Compared to the previous works \cite{31,32}, the actively correlated atom-light hybrid interferometer can be realized QND measurement of photon number with higher precision.

\section{Atom-light hybrid interferometer with active correlation output
readout}

The schematic of QND measurement of photon number is shown in Fig.~\ref{fig1} (a)%
. The probe system consists of an actively correlated atom-light hybrid interferometer,
and when the atoms are illuminated by the off-resonant signal light $\hat{b}^{(in)}$, the
interaction between atom and light will lead to an AC-Stark shift in atomic level without atomic
absorption. The energy levels of atom are given in Fig. ~\ref{fig1} (b).
In the NRP1 (NRP2) and LRP1 (LRP2), the strong pump beams $A_{p_{1}}$ ($A_{p_{2}}$) and $E_{p_{1}}$ ($E_{p_{2}}$)
couple the transitions $|g\rangle$ $\rightarrow$ $|e\rangle$ and $|m\rangle$ $\rightarrow$ $|e\rangle$,
respectively. $|g\rangle$ and $|m\rangle$ are the hyperfine split ground states, $|e\rangle$ denotes the excited state.
We can express the output operators of the probe system as a function of
the input operators in three steps: the state preparation, the SU(2)
interferometer, the readout. In the first step, we prepare the initial
atomic spin wave $\hat{S}_{a}^{(0)}$ and a correlated optical wave $\hat{a}%
_{S}^{(1)}$ via the NRP1. It can be described as%
\begin{equation}
\hat{S}_{a}^{(0)}=G_{1}\hat{S}_{a}^{(in)}+g_{1}e^{i\theta_{1}}\hat{a}%
_{S}^{\dagger(in)},\text{ }\hat{a}_{S}^{(1)}=G_{1}\hat{a}_{S}^{(in)}+g_{1}e^{i%
\theta_{1}}\hat{S}_{a}^{\dagger(in)},
\end{equation}
where $e^{i\theta_{1}}=\eta A_{p_{1}}/|\eta A_{p_{1}}|$, $A_{p_{1}}$ is the
amplitude of the pump beam and $\eta$ is the coupling constant. $%
G_{1}=\cosh(|\eta A_{p_{1}}|\tau)$ and $g_{1}=\sinh(|\eta A_{p_{1}}|\tau)$ are
the gains of the NRP1 with $G_{1}^{2}-g_{1}^{2}=1$,
$\tau$ is the pulse duration of pump beam.

Next, the beam splitter in a SU(2) interferometer is provided by the
LRP, which can split and mix the atomic spin wave and the
optical wave coherently for interference. When the atoms are illuminated by
an off-resonant signal light $\hat{b}^{(in)}$, the atomic spin wave $\hat{S}%
_{a}^{(1)}$ experiences an AC-Stark shift $\phi _{AC}$ \cite{34,40}. $\hat{a}_{W}^{(0)}$ is in coherent state, $\hat{S}%
_{a}^{(0)}$ is an atomic collective excitation which is prepared by the
NRP1. The relationship between input and output in
the SU(2) interferometer is given by%
\begin{equation}
\hat{S}_{a}^{(3)}=t\hat{S}_{a}^{(0)}+r\hat{a}_{W}^{(0)},\text{ }\hat{a}_{W}^{(2)}=t%
\hat{a}_{W}^{(0)}+r\hat{S}_{a}^{(0)},
\end{equation}%
where $t=e^{i\phi /2}\cos (\phi /2)$, $r=ie^{i\phi /2}\sin (\phi /2)$. $\phi
$ includes the phase difference $\phi _{0}$ of the interferometer and the
phase difference $\phi _{AC}$ caused by the AC-Stark effect with
$\phi_{AC}=\kappa\hat{n}_{b}$ \cite{34,40}, $\kappa$ and $\hat{n}_{b}$ are the AC-Stark coefficient and
photon number operator of the signal light $\hat{b}^{(in)}$, respectively.

In the final step, the generated atomic spin wave $\hat{S}_{a}^{(3)}$ of the
SU(2) interferometer and the optical wave $\hat{a}_{S}^{(1)}$ are combined
to realize active correlation output readout via the NRP2. It can be expressed as%
\begin{equation}
\hat{S}_{a}^{(out)}=G_{2}\hat{S}_{a}^{(3)}+g_{2}e^{i\theta_{2}}\hat{a}%
_{S}^{\dagger(1)},\text{ }\hat{a}_{S}^{(out)}=G_{2}\hat{a}_{S}^{(1)}+g_{2}e^{i%
\theta_{2}}\hat{S}_{a}^{\dagger(3)},
\end{equation}
where $e^{i\theta_{2}}=\eta A_{p_{2}}/|\eta A_{p_{2}}|$, $A_{p_{2}}$ is the
amplitude of the pump beam and $\eta$ is the coupling constant. $%
G_{2}=\cosh(|\eta A_{p_{2}}|\tau)$ and $g_{2}=\sinh(|\eta A_{p_{2}}|\tau)$ are
the gains of the NRP2 with $G_{2}^{2}-g_{2}^{2}=1$.

And thus, the full input-output relation of the actively correlated atom-light hybrid interferometer is%
\begin{align}
\hat{a}_{S}^{(out)} & =A\hat{a}_{S}^{(in)}+B\hat{S}_{a}^{\dagger(in)}+C\hat{a%
}_{W}^{\dagger(0)},  \notag \\
\hat{S}_{a}^{(out)} & =D\hat{a}_{S}^{\dagger(in)}+E\hat{S}_{a}^{(in)}+F\hat{a%
}_{W}^{(0)},
\end{align}
where%
\begin{align}
A & =G_{2}G_{1}+g_{2}g_{1}e^{i(\theta_{2}-\theta_{1})}t^{\ast},  \notag \\
B & =G_{2}g_{1}e^{i\theta_{1}}+G_{1}g_{2}e^{i\theta_{2}}t^{\ast},  \notag \\
D & =G_{1}g_{2}e^{i\theta_{2}}+G_{2}g_{1}e^{i\theta_{1}}t,  \notag \\
E & =g_{2}g_{1}e^{i(\theta_{2}-\theta_{1})}+G_{2}G_{1}t,  \notag \\
C & =g_{2}e^{i\theta_{2}}r^{\ast},F=G_{2}r.
\end{align}

\section{QND measurement of photon number}

\subsection{SNR analysis}

SNR analysis is helpful to examine the performance of
the QND measurement process. Given the homodyne detection, the SNR is defined
as%
\begin{equation}
R=\frac{\langle\hat{X}_{S}^{(out)}\rangle^{2}}{\Delta^{2}\hat{X}_{S}^{(out)}%
},\label{eq1}%
\end{equation}
where $\langle\hat{X}_{S}^{(out)}\rangle$ and $\Delta^{2}\hat{X}_{S}^{(out)}$
denote the quantum expectation and variance of the amplitude quadrature,
respectively. In our scheme, they are given by%
\begin{align}
\langle\hat{X}_{S}^{(out)}\rangle &  =\langle\hat{a}_{S}^{(out)}+\hat{a}%
_{S}^{\dagger(out)}\rangle\nonumber\\
&  =g_{2}N_{\alpha}^{1/2}[\cos(\theta_{2}-\theta_{\alpha}-\phi)-\cos(\theta_{2}%
-\theta_{\alpha})].\label{eq2}%
\end{align}%
\begin{align}
\Delta^{2}\hat{X}_{S}^{(out)} &  =G_{2}^{2}G_{1}^{2}+G_{2}^{2}g_{1}^{2}%
+g_{2}^{2}(1-\cos\phi)/2\nonumber\\
&  +g_{2}^{2}g_{1}^{2}(1+\cos\phi)/2+G_{1}^{2}g_{2}^{2}(1+\cos\phi
)/2\nonumber\\
&  +2G_{2}G_{1}g_{2}g_{1}\cos(\theta_{2}-\theta_{1}-\phi)\nonumber\\
&  +2G_{2}G_{1}g_{2}g_{1}\cos(\theta_{2}-\theta_{1}).\label{eq3}%
\end{align}
Here $\hat{a}_{S}^{(in)}$ and $\hat{S}_{a}^{(in)}$ are in vacuum states,
$\hat{a}_{W}^{(0)}$ is in a coherent state $|\alpha\rangle$ with
$\alpha=N_{\alpha}^{1/2}e^{i\theta_{\alpha}}$ where $N_{\alpha}$ and $\theta_{\alpha}$ are the photon number and initial
phase of the coherent state, respectively. Under the condition of $\theta_{1}=0$, $\theta_{2}=\pi$, $\theta
_{\alpha}=\pi/2$, and $\phi_{0}=0$ with a small $\phi_{AC}$ around
$\phi_{0}$, assuming the signal light is in a number state $|n\rangle$ with
$\hat{n}_{b}|n\rangle=n_{b}|n\rangle$ and substituting $\phi_{AC}$ with
$\kappa\hat{n}_{b}$, Eq.~(\ref{eq2}) and Eq.~(\ref{eq3}) are rewritten in
terms of $n_{b}$, such that%
\begin{equation}
\langle\hat{X}_{S}^{(out)}\rangle=g_{2}\kappa N_{\alpha}^{1/2}n_{b},
\end{equation}
and%
\begin{equation}
\Delta^{2}\hat{X}_{S}^{(out)}=(G_{2}G_{1}-g_{2}g_{1})^{2}+(G_{2}g_{1}%
-g_{2}G_{1})^{2}+G_{1}^{2}g_{2}^{2}\kappa^{2}n_{b}^{2}/2,
\end{equation}
respectively.\ \ \ \ \

Considering a balanced case $G_{2}=G_{1}=G$ and $g_{2}=g_{1}=g$, the SNR is
approximately $g^{2}\kappa^{2}N_{\alpha}n_{b}^{2}$. To have single-photon
resolution, we need AC-Stark coefficient $\kappa\sim1/g N_{\alpha}^{1/2}$. Compared to
the traditional SU(2) interferometer \cite{31}, the required coefficient is smaller by a
factor of $1/g$ . This is due to the method of active correlation
output readout for the SU(2) interferometer \cite{35}, in which the SNR is 
greater than the SU(2) interferometer by a factor of $g$ of NRP.

\subsection{Quality estimation}

In this section, we estimate the quality of the actively correlated
atom-light hybrid interferometer as a QND measurement using the criteria
introduced by Holland \textit{et al.} \cite{33}, which are%
\begin{align}
C_{S^{in}S^{out}}^{2}& =\frac{|\langle S^{in}S^{out}\rangle -\langle
S^{in}\rangle \langle S^{out}\rangle |^{2}}{\Delta ^{2}S^{in}\Delta
^{2}S^{out}},  \label{eq4} \\
C_{S^{in}P^{out}}^{2}& =\frac{|\langle S^{in}P^{out}\rangle -\langle
S^{in}\rangle \langle P^{out}\rangle |^{2}}{\Delta ^{2}S^{in}\Delta
^{2}P^{out}},  \label{eq5} \\
C_{S^{out}P^{out}}^{2}& =\frac{|\langle S^{out}P^{out}\rangle -\langle
S^{out}\rangle \langle P^{out}\rangle |^{2}}{\Delta ^{2}S^{out}\Delta
^{2}P^{out}},  \label{eq6}
\end{align}%
with%
\begin{align}
\Delta ^{2}S^{in}& =\langle (S^{in})^{2}\rangle -\langle S^{in}\rangle ^{2},
\notag \\
\Delta ^{2}S^{out}& =\langle (S^{out})^{2}\rangle -\langle S^{out}\rangle
^{2},  \notag \\
\Delta ^{2}P^{in}& =\langle (P^{in})^{2}\rangle -\langle P^{in}\rangle ^{2},
\notag \\
\Delta ^{2}P^{out}& =\langle (P^{out})^{2}\rangle -\langle P^{out}\rangle
^{2},
\end{align}%
where $S^{in}$ is the input signal incident on the scheme and $P^{out}$ is
the output probe measured by a detector. Here $S^{in}$ is the photon number
of the input signal $\hat{N}^{(in)}=\hat{b}^{\dagger (in)}\hat{b}^{(in)}$,
and the probe is the amplitude quadrature $\hat{X}_{S}^{(out)}$. Eq.~(\ref{eq4}) is about how much the probe system degrades the signal of the measured
system. Eq.~(\ref{eq5}) is about how good the probe system is a measurement
device. Eq.~(\ref{eq6}) is that how good the probe system is a state preparation
device. For a ideal QND measurement device the correlation coefficients $%
C_{S^{in}S^{out}}^{2}$, $C_{S^{in}P^{out}}^{2}$ and $C_{S^{out}P^{out}}^{2}$
are unity. In our paper, the signal light leading to the AC-Stark shift is
far off resonance with a large detuning, i.e., the photon number of the
measured signal light is not changed before and after measurement. So the
first criterion is satisfied, the second and the third criteria become same:
$C_{\hat{N}^{(in)}\hat{X}_{S}^{(out)}}^{2}=C_{\hat{N}^{(out)}\hat{X}%
_{S}^{(out)}}^{2}$, that is,%
\begin{equation}
C^{2}=\frac{|\langle \hat{N}^{(in)}\hat{X}_{S}^{(out)}\rangle -\langle \hat{N%
}^{(in)}\rangle \langle \hat{X}_{S}^{(out)}\rangle |^{2}}{\Delta ^{2}\hat{N}%
^{(in)}\Delta ^{2}\hat{X}_{S}^{(out)}}.  \label{eq7}
\end{equation}%
For brevity, we omit the subscript of $C$. We obtain%
\begin{align}
\langle \hat{N}^{(in)}\rangle & =N_{\beta },\text{ }\Delta ^{2}\hat{N}%
^{(in)}=N_{\beta },  \notag \\
\langle \hat{X}_{S}^{(out)}\rangle & =g_{2}\kappa N_{\alpha }^{1/2}N_{\beta
},\text{ }\langle \hat{N}^{(in)}\hat{X}_{S}^{(out)}\rangle =g_{2}\kappa N_{\alpha
}^{1/2}N_{\beta }(N_{\beta }+1),  \notag \\
\Delta ^{2}\hat{X}_{S}^{(out)}&
=(G_{2}G_{1}-g_{2}g_{1})^{2}+(G_{2}g_{1}-G_{1}g_{2})^{2}  \notag \\
& +g_{2}^{2}\kappa ^{2}N_{\alpha }N_{\beta }+G_{1}^{2}g_{2}^{2}\kappa
^{2}N_{\beta }(N_{\beta }+1)/2.  \label{eq8}
\end{align}%
Here the signal light is in a coherent state $|\beta \rangle $ with photon
number $N_{\beta }$. Substituting Eq.~(\ref{eq8}) into Eq.~(\ref{eq7}), the
criteria can be written as%
\begin{equation}
C^{2}=\frac{1}{1+\frac{%
(G_{2}G_{1}-g_{2}g_{1})^{2}+(G_{2}g_{1}-G_{1}g_{2})^{2}}{g_{2}^{2}\kappa
^{2}N_{\alpha }N_{\beta }}+\frac{G_{1}^{2}(N_{\beta }+1)}{2N_{\alpha }}}.
\label{eq9}
\end{equation}%
As seen in Eq.~(\ref{eq9}), a perfect correlation $C^{2}\approx 1$ can be
satisfied under the condition of $g_{2}^{2}\kappa ^{2}N_{\alpha }N_{\beta
}\gg (G_{2}G_{1}-g_{2}g_{1})^{2}+(G_{2}g_{1}-G_{1}g_{2})^{2}$ and $%
2N_{\alpha }\gg G_{1}^{2}(N_{\beta }+1)$. In a balanced case, the condition
for a perfect correlation is $g^{2}\kappa ^{2}N_{\alpha }N_{\beta }\gg 1$
and $2N_{\alpha }\gg G^{2}(N_{\beta }+1)$.

\subsection{Optimized C in the presence of losses}

Next, we investigate the effects of losses on the correlation coefficient $C$
consisting of photon losses and atomic decoherence losses \cite{30}.

The loss of the two arms inside the SU(2) interferometer is called the internal loss, as shown in Fig.~\ref{fig3}(a). The loss at the output of SU(2) and the associated optical field is called external loss, as shown in Fig.~\ref{fig3}(b). Two fictitious beam splitters are introduced to mimic the loss of photons into the
environment, then the optical waves $\hat{a}%
_{W}^{(1)}$ and $\hat{a}_{S}^{(1)}$ experience losses as%
\begin{align}
\hat{a}_{W,l}^{(1)} &  =\sqrt{\eta_{1}}\hat{a}_{W}^{(1)}+\sqrt{1-\eta_{1}%
}\hat{V}_{1},\\
\hat{a}_{S,l}^{(1)} &  =\sqrt{\eta_{2}}\hat{a}_{S}^{(1)}+\sqrt{1-\eta_{2}%
}\hat{V}_{2},
\end{align}
where subscript $l$ indicates the loss, $\eta_{1}$ ($\eta_{2}$) and $\hat{V}_{1}$ ($\hat{V}_{2}$) represent the
transmission rate and vacuum, respectively. The spin waves $\hat{S}_{a}^{(2)}$
and $\hat{S}_{a}^{(3)}$ also undergo the collisional dephasing $e^{-\Gamma
_{1}\tau_{1}}$ ($e^{-\Gamma_{2}\tau_{2}}$), then the spin waves are described
by%
\begin{align}
\hat{S}_{a,l}^{(2)} &  =\hat{S}_{a}^{(2)}e^{-\Gamma_{1}\tau_{1}}+\hat{F}%
_{1},\\
\hat{S}_{a,l}^{(3)} &  =\hat{S}_{a}^{(3)}e^{-\Gamma_{2}\tau_{2}}+\hat{F}%
_{2},
\end{align}
where $\langle\hat{F}_{1}\hat{F}_{1}^{\dagger}\rangle=1-e^{-2\Gamma_{1}%
\tau_{1}}$ and $\langle\hat{F}_{2}\hat{F}_{2}^{\dagger}\rangle=1-e^{-2\Gamma
_{2}\tau_{2}}$ guarantees the consistency of the operator properties of
$\hat{S}_{a,l}^{(2)}$ and $\hat{S}_{a,l}^{(3)}$, respectively. The
input-output relation for loss case of $\hat{a}_{S}^{(out)}$ becomes%
\begin{align}
\hat{a}_{S,l}^{(out)}  & =\hat{a}_{S}^{(in)}\mathcal{A}+\hat{S}_{a}%
^{\dagger(in)}\mathcal{B}+\hat{a}_{W}^{\dagger(0)}\mathcal{C}+\hat{V}%
_{1}^{\dagger}\mathcal{D}+\hat{V}_{2}\mathcal{E}\nonumber\\
& +\hat{F}_{1}^{\dagger}\mathcal{F}+\hat{F}_{2}^{\dagger}\mathcal{G},
\end{align}
where%
\begin{align}
\mathcal{A} &  =[\sqrt{\eta_{2}}G_{2}G_{1}+g_{2}g_{1}e^{i(\theta_{2}%
-\theta_{1})}(e^{-\Gamma_{1}\tau_{1}}e^{-i\phi}+\sqrt{\eta_{1}})e^{-\Gamma
_{2}\tau_{2}}/2],\nonumber\\
\mathcal{B} &  =[\sqrt{\eta_{2}}G_{2}g_{1}e^{i\theta_{1}}+G_{1}g_{2}%
e^{i\theta_{2}}(e^{-\Gamma_{1}\tau_{1}}e^{-i\phi}+\sqrt{\eta_{1}}%
)e^{-\Gamma_{2}\tau_{2}}/2],\nonumber\\
\mathcal{C} &  =g_{2}e^{i\theta_{2}}(e^{-\Gamma_{1}\tau_{1}}e^{-i\phi}%
-\sqrt{\eta_{1}})e^{-\Gamma_{2}\tau_{2}}/2,\nonumber\\
\mathcal{D} &  =-g_{2}e^{i\theta_{2}}\sqrt{1-\eta_{1}}e^{-\Gamma_{2}\tau_{2}%
}/\sqrt{2},\nonumber\\
\mathcal{E} &  =G_{2}\sqrt{1-\eta_{2}},\mathcal{F}=g_{2}e^{i\theta_{2}%
}e^{-\Gamma_{2}\tau_{2}}/\sqrt{2},\mathcal{G}=g_{2}e^{i\theta_{2}}.
\end{align}

\begin{figure}[t]
\centering{\includegraphics[scale=0.5,angle=0]{fig2.eps}}\caption{ A lossy interferometer model with (a) internal loss and (b) external loss.}%
\label{fig3}%
\end{figure}

\begin{figure}[t]
\centering{\includegraphics[scale=0.6,angle=0]{fig3.eps}}\caption{ Correlation coefficient $C$ as a function of $\eta_1$ and $\eta_2$, where $e^{-\Gamma_{1}\tau_{1}}=e^{-\Gamma_{2}\tau_{2}}=0.9$, $\kappa=10^{-10}%
$, $g_{1}=g_{2}=3$, $N_{\alpha}=10^{12}$, and $N_{\beta}=10^{8}$. The correlation coefficient in the area of upper right corner and within the $C_{0}$ lines can be kept above 0.6.}%
\label{fig4}%
\end{figure}

We study the effect of losses under the condition of $\theta_{1}=0$,
$\theta_{2}=\pi$, $\theta_{\alpha}=\pi/2$, and $\phi_{0}=0$ with a small
$\phi_{AC}$ around $\phi_{0}$. Considering losses the terms of $\langle\hat
{X}_{S}^{(out)}\rangle_{l}$, $\langle\hat{N}^{(in)}\hat{X}_{S}^{(out)}%
\rangle_{l}$, and $(\Delta^{2}\hat{X}_{S}^{(out)})_{l}$ in Eq.~(\ref{eq8}) are
given by
\begin{equation}
\langle\hat{X}_{S}^{(out)}\rangle_{l}=g_{2}e^{-\Gamma_{1}\tau_{1}}%
e^{-\Gamma_{2}\tau_{2}}\kappa N_{\alpha}^{1/2}N_{\beta},
\end{equation}%
\begin{equation}
\langle\hat{N}^{(in)}\hat{X}_{S}^{(out)}\rangle_{l}=g_{2}e^{-\Gamma_{1}%
\tau_{1}}e^{-\Gamma_{2}\tau_{2}}\kappa N_{\alpha}^{1/2}N_{\beta}(N_{\beta}+1),
\end{equation}
and%
\begin{align}
&  (\Delta^{2}\hat{X}_{S}^{(out)})_{l}\nonumber\\
&  =(\sqrt{\eta_{2}}G_{2}G_{1}-g_{2}g_{1}e^{-\Gamma_{1}\tau_{1}}e^{-\Gamma
_{2}\tau_{2}}/2-g_{2}g_{1}e^{-\Gamma_{2}\tau_{2}}\sqrt{\eta_{1}}%
/2)^{2}\nonumber\\
&  +(\sqrt{\eta_{2}}G_{2}g_{1}-G_{1}g_{2}e^{-\Gamma_{1}\tau_{1}}e^{-\Gamma
_{2}\tau_{2}}/2-G_{1}g_{2}e^{-\Gamma_{2}\tau_{2}}\sqrt{\eta_{1}}%
/2)^{2}\nonumber\\
&  +g_{2}^{2}[(1-e^{-2\Gamma_{1}\tau_{1}})e^{-2\Gamma_{2}\tau_{2}%
}/2+(1-e^{-2\Gamma_{2}\tau_{2}})]\nonumber\\
&  +g_{2}^{2}(2g_{1}^{2}+1)\kappa^{2}N_{\beta}(N_{\beta}+1)e^{-2\Gamma_{1}%
\tau_{1}}e^{-2\Gamma_{2}\tau_{2}}/4\nonumber\\
&  +g_{2}^{2}\kappa^{2}N_{\beta}[N_{\alpha}+(N_{\beta}+1)/4]e^{-2\Gamma
_{1}\tau_{1}}e^{-2\Gamma_{2}\tau_{2}}\nonumber\\
&  +(g_{2}e^{-\Gamma_{2}\tau_{2}}\sqrt{\eta_{1}}/2-g_{2}e^{-\Gamma_{1}\tau
_{1}}e^{-\Gamma_{2}\tau_{2}}/2)^{2}\nonumber\\
&  +g_{2}^{2}(1-\eta_{1})e^{-2\Gamma_{2}\tau_{2}}/2+G_{2}^{2}(1-\eta_{2}),
\end{align}
where subscript $l$ denotes the loss. Then the QND measurement criterion for loss case can be obtained according to
Eq.~(\ref{eq7}).

In our scheme, the atomic spin wave stays in the atomic ensemble while the
optical field travels out of the atomic ensemble. Here, in the short coherence
time the atomic collisional dephasing loss $\Gamma_{1}\tau_{1}$ ($\Gamma
_{2}\tau_{2}$) is very small, then we set $e^{-\Gamma_{1}\tau_{1}}%
=e^{-\Gamma_{2}\tau_{2}}=0.9$. The correlation coefficient $C$ as a function
of $\eta_{1}$ and $\eta_{2}$ in balanced case is shown in Fig.~\ref{fig4}. It
is shown that the reduction in the correlation coefficient $C$ increases as
the loss increases. The correlation coefficient in the area of upper right
corner and within the $C_{0}$ lines can be kept above $0.6$. It is shown that
our scheme is more tolerant with the internal photon loss compared to the
photon loss outside the SU(2) interferometer. The reason behind the
phenomenon is that large external photon loss affects quantum correlation
between the light wave and atomic spin wave, which destroys the active
correlation output readout.

In the unbalanced case ($g_{1}\neq g_{2}$), we can adjust the gain ratio
$g_{2}/g_{1}$ of the beam recombination process to reduce the reduction in
correlation coefficient. After optimizing the $g_{2}$, the correlation coefficient as a
function of $\eta_{1}$ and $\eta_{2}$ can also be obtained, where the line
with $C$ equal to $0.6$ is denoted as $C_{1}$. The contour line of optimized
$g_{2}/g_{1}$ as a function of $\eta_{1}$ and $\eta_{2}$ are shown in Fig.
\ref{fig5}, where the position of $C_{0}$ (before optimization) and $C_{1}$ (after optimization) in the contour figure of the correlation coefficient versus
transmission rates has changed. It is demonstrated that for a given $g_{1}$
by optimizing $g_{2}/g_{1}$, the small area between the $C_{0}$ and $C_{1}$
can still kept above $0.6$. That is within a certain losses range, $C$ can
continue to beat the criteria (such as $C$ equal to $0.6)$ after optimizing
$g_{2}/g_{1}$. The newly added area after optimization is divided into two parts, one of which is the ratio $g_{2}/g_{1}$ within the very small area above is less than 1, and the other part the ratio $g_{2}/g_{1}$ needs to be greater than 1.

\begin{figure}[t]
\centering{\includegraphics[scale=0.6,angle=0]{fig4.eps}}
\caption{ Contour line of optimized
$g_{2}/g_{1}$ as a function of $\eta_{1}$ and $\eta_{2}$ with $e^{-\Gamma
_{1}\tau_{1}}=e^{-\Gamma_{2}\tau_{2}}=0.9$, where $\kappa=10^{-10}$, $g_{1}%
=3$, $N_{\alpha}=10^{12}$, and $N_{\beta}=10^{8}$. The correlation coefficient
in the area of upper right corner and within the line $C_{0}$ (before optimization) or $C_{1}$ (after optimization) can
be kept above $0.6$.}%
\label{fig5}%
\end{figure}

\section{Discussion and Conclusion}

Our scheme can also be thought of as inserting an SU(2) interferometer into one of the arms of the SU(1,1) interferometer.
Compared to a conventional SU(1,1) interferometer, the number of phase-sensing particles is further increased due to  input field $\hat{a}_{W}^{(0)}$.
In the previous scheme \cite{31}, a strong stimulated Raman process can indeed be used to excite most of the atoms to the $|m\rangle$ state to improve the number of phase-sensing particles. However, this will produce additional nonlinear effects and noise light fields \cite{41,42,43}.  Therefore, in the current scheme, firstly a spontaneous Raman process is used, and then a LRP is used to generate the Rabi-like superposition oscillation between light and the atom. By controlling the interaction time, more atoms can be generated to the  $|m\rangle$ state to improve the number of phase-sensing particles to enhance the QND measurement.

Based on the previous experimental works \cite{27, 28}, the actively correlated atom-light hybrid interferometer could be implemented in the rubidium atomic system.  Moreover, the scheme is a multi-arm interferometer and it provides an option for the simultaneous estimation of more than two parameters with a wide range of applications,  such as phase imaging \cite{37}, quantum sensing networks \cite{38},
the detection of vector fields \cite{39} and so on.

In conclusion, we have proposed an actively correlated atom-light hybrid
interferometer and used it for QND measurement of photon number via the
AC-Stark effect. In the scheme, the atomic spin wave of the SU(2)
interferometer is prepared via a NRP and the output is
detected with the method of active correlation output readout via another
NRP. Benefiting from that, the SNR in
the balanced case is improved by a factor of $g$ compared to the traditional
SU(2) interferometer. The condition for a perfect correlation is given.
The measurement quality is reduced in the presence of losses. We
can adjust the gain parameter of the NRP in readout stage to reduce
the impact of losses.

\section*{Funding}

This work is supported by Development Program of China Grant No.
2016YFA0302001; National Natural Science Foundation of China Grants No.
11974111, No. 11874152, No. 91536114, No. 11574086, No. 11974116, No.
11654005; Shanghai Rising-Star Program Grant No. 16QA1401600; the Shanghai
talent program, and the Fundamental Research Funds for the Central
Universities.

\section*{Disclosures}

The authors declare no conflicts of interest.

