# Bi-Photon Entangled Airy Beams through Unstable Ocean Turbulence
## 2 Methods  
The sketch of a bi-photon entangled UOC link is illustrated in Figure 1. For non-trivial subspace (\(m_{0}\
eq 0\)), where \(m_{0}\) is a general symbol for the OAM number, we assume that the initial state of the bi-photon Airy beams is an entangled Bell-like state [19; 34]:  
\[|\Psi\rangle=\mu|m_{1},m_{2}\rangle+\
u|-m_{1},-m_{2}\rangle, \tag{1}\]  
where \(m_{1}=m_{0}\), \(m_{2}=-m_{1}\) for the conversation of momentum, \(|m_{i}\rangle\) represents a single photon OAM eigenmode of an Airy mode, \(m_{i}\) is the OAM quantum number corresponding to the OAM \(m_{i}\hbar\), the subscript \(i(i=1,2)\) is used to label two different channels of the two photons and \(\mu\) and \(\
u\) denote probability amplitudes that are subject to \(|\mu|^{2}+|\
u|^{2}=1\). As the constituent photons launched with equal probability (\(\mu=\
u\)), the maximal entanglement occurs for the initial bi-photon. Figure 1: Sketch of a thought experiment involving bi-photon propagation through independent turbulence paths 1 and 2. Using cylindrical representation and inserting the completeness relation \(\int|\vec{r}_{i}\rangle\langle\vec{r}_{i}|\mathrm{d}\vec{r}_{i}=I\), one can obtain the probability of the joint OAM measurement at propagation distance \(\mathsf{z}\) as [35]  
\[ \begin{split}& p(m_{1}m_{2}|\Psi)=\langle m_{1}m_{2}|\Psi\rangle \langle\Psi|m_{1}m_{2}\rangle\\ &=\int\int\langle m_{1}m_{2}|\vec{r}_{1}\vec{r}_{2}\rangle \langle\vec{r}_{1}\vec{r}_{2}|\Psi\rangle\mathrm{d}\vec{r}_{1}\mathrm{d}\vec{r }_{2}\int\int\langle\Psi|\vec{r}_{1}^{\prime}\vec{r}_{2}^{\prime}\rangle \langle\vec{r}_{1}^{\prime}\vec{r}_{2}^{\prime}|m_{1}m_{2}\rangle\mathrm{d} \vec{r}_{1}^{\prime}\mathrm{d}\vec{r}_{2}^{\prime}\end{split}  \tag{2}\]  
with \(\vec{r}_{i}=(r_{i},\varphi_{i},z)\), \(r_{i}=|\vec{r}_{i}|\) is the radial distance from the axis and \(\vec{r}_{i}=(x_{i},y_{i})\) is the two-dimensional vector. The joint OAM detection probability is then given by the ensemble average [36] of Equation (2) as  
\[p_{m_{1}m_{2}}(z)=\int\cdots\int r_{1}r_{2}\mathrm{d}r_{1}\mathrm{d}r_{2} \mathrm{d}\varphi_{1}\mathrm{d}\varphi_{1}^{\prime}\mathrm{d}\varphi_{2} \mathrm{d}\varphi_{2}^{\prime}C_{\
abla}\prod\limits_{i=1}^{2}\frac{\exp[- \mathrm{i}m_{i}(\varphi_{i}-\varphi_{i}^{\prime})]}{2\pi} \tag{3}\]  
where \(C_{\
abla}=\langle\Psi(r_{1},r_{2},\varphi_{1},\varphi_{2},z)\Psi^{*}(r_{1}, r_{2},\varphi_{1}^{\prime},\varphi_{2}^{\prime},z)\rangle\) is defined as the rotational field correlation of the bi-photon. For convenience, we set \(\Delta\varphi_{i}=\varphi_{i}-\varphi_{i}^{\prime}\) and \(\bar{\varphi}_{i}=(\varphi_{i}+\varphi_{i}^{\prime})/2\) to rewrite Equation (3) as  
\[p_{m_{1}m_{2}}(z)=\int\cdots\int\biggl{(}\prod\limits_{i}r_{i}\mathrm{d}r_{i} \Delta\varphi_{i}\mathrm{d}\bar{\varphi}_{i}\biggr{)}C_{\
abla}(r_{1},r_{2}, \Delta\varphi_{1},\Delta\varphi_{2},\bar{\varphi}_{1},\bar{\varphi}_{2},z) \frac{\exp(-\mathrm{i}\sum_{i}m_{i}\Delta\varphi_{i})}{4\pi^{2}}. \tag{4}\]  
Generalizing the semi-classical approach [36] to bi-photon entangled case, the bi-photon entangled Airy beams \(\Psi(r_{1},r_{2},\varphi_{1},\varphi_{2},z)\) is given by  
\[ \begin{split}&\Psi(r_{1},r_{2},\varphi_{1},\varphi_{2},z)=R^{|m_{1 }|}(r_{1},z)R^{|m_{2}|}(r_{2},z)\frac{\exp[\mathrm{i}\phi(r_{1},r_{2},\varphi_ {1},\varphi_{2},z)]}{2\pi}\\ &\quad\times\{\mu\mathrm{exp}[\mathrm{i}m_{1}(\varphi_{1}-\varphi _{2})]+v\mathrm{exp}[\mathrm{i}m_{2}(\varphi_{1}-\varphi_{2})]\},\end{split}  \tag{5}\]  
where \(\phi(r_{1},r_{2},\varphi_{1},\varphi_{2},z)\) is the phase perturbation induced by turbulence, which gives rise to the crosstalk of OAM, and \(R^{|m_{i}|}(r_{i},z)\) is the radial component of Airy beam. Substituting Equation (5) into Equation (4), the final expression of the joint detection probability is derived as  
\[