This repo is used to find the optimal parameters of the pump laser to get high purity quantum light by SPDC process. This project is based on Jax. To use Jax, Jaxlib as well as Jax need to be downloaded. A way to download Jax for windows is to use the unstable build. For Linux and MacOS users, please refer to installation section here.

Theory of high-gain spontaneous parametric down-conversion

The numerical optimization model is from the paper by Quesada et al Phys. Rev. A, (2020). The dynamics describing SPDC can be seen as

$$\frac{\partial }{\partial z}a_S(z, \omega) = \text{i}\omega \frac{1}{\Delta v_S} a_S(z, \omega) + \frac{\text{i}\gamma (z)}{\sqrt{2\pi}}\int \beta (\omega + \omega ')a_I^\dagger (z, \omega ')d\omega ',$$ $$\frac{\partial }{\partial z}a_I^\dagger(z, \omega) = -\text{i}\omega \frac{1}{\Delta v_I} a_I ^\dagger(z, \omega) - \frac{\text{i}\gamma ^* (z)}{\sqrt{2\pi}}\int \beta ^* (\omega + \omega ')a_S(z, \omega),$$

where $\frac{1}{\Delta v_j} = \frac{1}{v_j} - \frac{1}{v_p}$, where $v_j$ is the group velocity of the mode $j \in {S, I}$, $\omega$ the frequency, $\beta(\omega)$ the slowly varying envelope of the pump in frequency domain, $\gamma (z)$ a tophat function associated to second order nonlinearity and $a_j (z, \omega)$ the annihilation operator for the mode $j$. The equations can be described by discretization into normal modes as

$$\frac{\partial}{\partial z}\begin{bmatrix} \bar{a}_S(z, \omega)\\\ \bar{a}_I^\dagger(z, \omega) \end{bmatrix}= \text{i}\begin{bmatrix} \frac{1}{\Delta v_S} \omega & \frac{\text{i}\gamma}{\sqrt{2\pi}}\beta \\\ -\frac{\text{i}\gamma ^*}{\sqrt{2\pi}}\beta ^\dagger & -\frac{1}{\Delta v_I} \omega \end{bmatrix} \begin{bmatrix} \bar{a}_S(z, \omega')\\\ \bar{a}_I^\dagger(z, \omega') \end{bmatrix} = \text{i}A\begin{bmatrix} \bar{a}_S(z, \omega')\\\ \bar{a}_I^\dagger(z, \omega') \end{bmatrix} ,$$

where $\frac{1}{\Delta v_j} \omega$ is a diagonal matrix for containing information about phase matching for state $j$ and $\beta$ a Hankel matrix representing $\beta (\omega + \omega')$. The matrix $A$ is an element of the Lie algebra and is used to generate the propagator associate to the dynamics which will be an element of the group $SU(1,1)$.\

\noindent The spectral purity of SPDC dynamics in frequency and space domain for a waveguide can be represented by the Schmidt number $K$, which can be expressed by

$$\begin{align} K = \frac{\text{Tr}\left[U_{SI}^*U_{SI}^{T}\right]^2}{\text{Tr}\left[U_{SI}^*U_{SI}^{T}U_{SI}^*U_{SI}^{T}\right]}, \end{align}$$

where $U_{SI}$ is a block of the matrix $U$, which is an element of the group $SU(1,1)$. The constraint that the average photon number pairs is equal to $n$, which in this notebook will be 3, can be written as

$$\begin{align} \text{Tr}\left[U_{SI}^* U_{SI}^T\right] - n = 0. \end{align}$$

The matrix $U$ can be obtained by exponentiating $A$ such as

$$U = \begin{bmatrix} U_{SS} & U_{SI}\\\ U_{IS}^* & U_{II}^* \end{bmatrix} = \text{exp}\left(\text{i}\begin{bmatrix} \frac{1}{\Delta v_S} \omega & \frac{\gamma}{\sqrt{2\pi}} \beta \\\ -\frac{\gamma ^*}{\sqrt{2\pi}}\beta ^\dagger & -\frac{1}{\Delta v_I} \omega \end{bmatrix}\right).$$

Because the theoretical minimum of the Schmidt number is 1, the objective function becomes $K-1$ and the problem becomes

$$\begin{align} \text{min} \ & \frac{\text{Tr}\left[U_{SI}^*U_{SI}^{T}\right]^2}{\text{Tr}\left[U_{SI}^*U_{SI}^{T}U_{SI}^*U_{SI}^{T}\right]} - 1\\\ \text{s. t.} \ & \text{Tr}\left[U_{SI}^* U_{SI}^T\right] - n = 0,\\\ & \begin{bmatrix} U_{SS} & U_{SI}\\\ U_{IS}^* & U_{II}^* \end{bmatrix} = \text{exp}\left(\text{i}\begin{bmatrix} \frac{1}{\Delta v_S} \omega & \frac{\gamma}{\sqrt{2\pi}} \beta \\\ -\frac{\gamma ^*}{\sqrt{2\pi}}\beta ^\dagger & -\frac{1}{\Delta v_I} \omega \end{bmatrix}\right). \end{align}$$

If we are within symmetric group velocity matching regime and the pump has constant phase, then the differential equations can be written as

$$\frac{\partial }{\partial z}\hat{X}_+(z, \omega) = \text{i}\omega \frac{1}{\Delta v}\hat{X}_+(z, \omega) + \frac{|\gamma|}{\sqrt{2\pi}}\int d\omega ' \beta (\omega + \omega ') \hat{X}_+(z, \omega '),$$ $$\frac{\partial }{\partial z}\hat{X}_-(z, \omega) = \text{i}\omega \frac{1}{\Delta v}\hat{X}_-(z, \omega) - \frac{|\gamma|}{\sqrt{2\pi}}\int d\omega ' \beta (\omega + \omega ') \hat{X}_-(z, \omega '),$$

where

$$\hat{X}_+(z, \omega) = \frac{1}{\sqrt{2}}\left(\hat{a}_S(z, \omega) + \hat{a}^\dagger (z, \omega)\right),$$ $$\hat{X}_-(z, \omega) = \frac{1}{\sqrt{2}}\left(\hat{a}_S(z, \omega) - \hat{a}^\dagger (z, \omega)\right)$$

and $\frac{1}{\Delta v} = \frac{1}{v_S} - \frac{1}{v_P} = \frac{1}{v_P} - \frac{1}{v_I}$. Then, the propagators become

$$\hat{X}_+(z) = \text{exp}\left[\left(\text{i}\frac{1}{\Delta v}\omega + \frac{|\gamma|}{\sqrt{2\pi}}\beta \right)z\right]\hat{X}_+(0) = U_+(z)\hat{X}_+(0),$$ $$\hat{X}_-(z) = \text{exp}\left[\left(\text{i}\frac{1}{\Delta v}\omega - \frac{|\gamma|}{\sqrt{2\pi}}\beta \right)z\right]\hat{X}_-(0) = U_-(z)\hat{X}_-(0).$$

The Schmidt number and mean number of photons can then be expressed as

$$\begin{align} \text{min} \ & \frac{\text{Tr}\left[\left(U_+ - U_-\right)\left(U_+ - U_-\right)^\dagger \right]^2}{\text{Tr}\left[\left(\left(U_+ - U_-\right)\left(U_+ - U_-\right)^\dagger\right)\right]} - 1\\\ \text{s. t.} \ & \frac{1}{4}\text{Tr}\left[\left(U_+ - U_-\right)\left(U_+ - U_-\right)^\dagger \right] - n = 0,\\\ & U_+ = \text{exp}\left[\left(\text{i}\frac{1}{\Delta v}\omega + \frac{|\gamma|}{\sqrt{2\pi}}\beta \right)z\right],\\\ & U_- = \text{exp}\left[\left(\text{i}\frac{1}{\Delta v}\omega - \frac{|\gamma|}{\sqrt{2\pi}}\beta \right)z\right]. \end{align}$$