PhD Work

To investigate the orbital angular momentum (OAM) spectrum of entangled photons in spontaneous parametric down conversion

Currently, as part of my PhD research work, I am working in the field of Quantum Optics. Specifically, I am studying the properties of the orbital angular momentum of light.

1. Introduction:

Spontaneous parametric down-conversion (SPDC) is a second-order nonlinear process with respect to the response of polarisation to the electric field. The SPDC process involves the incidence of high-intensity ‘pump’ photons on a nonlinear, non-centrosymmetric crystal (for example, beta barium borate). The pump photon is destroyed which results in the creation of two new photons with lower energy (down conversion). These newly created (‘signal’ and ‘idler’) photons are entangled in various degrees of freedom like position-momentum, angular position-angular momentum bases.

2. Diagrammatic Representation of SPDC:

Schematic diagram of SPDC (non-collinear)

3. Two- Photon Wavefunction:

Upon down-conversion, the two-photon state of the down-converted photons ( $\psi_{t p}$ ) can be written in the transverse momentum basis ( $\boldsymbol{q}_{s}, \boldsymbol{q}_{i}$ which are the transverse components of the wavevectors $\boldsymbol{k}_{s}, \boldsymbol{k}_{i}$ ) as the product of pump field amplitude (V_p) and the phase-matching function ( $\phi$ ).

$\begin{aligned} \psi_{t p}\left(\boldsymbol{q}_{s}, \boldsymbol{q}_{i} \right) &=V_{p}\left(\boldsymbol{q}_{p}\right) \Phi\left(\boldsymbol{q}_{s}, \boldsymbol{q}_{i}\right) \\&=V_{p}\left(\boldsymbol{q}_{s}+\boldsymbol{q}_{i}\right) \textrm{sinc}\left(\frac{L}{2} \Delta k_{z}\right) \end{aligned}$

The same can also be written in the Laguerre-Gaussian (LG) basis as:

$\left|\psi_{t p}\right\rangle=\sum_{l_{s}} \sum_{l_{i}} \sum_{p_{s}} \sum_{p_{i}} C_{l_{i}, p_{i}}^{l_{s}, p_{s}}\left|l_{s}, p_{s}\right\rangle\left|l_{i}, p_{i}\right\rangle$

where $l$ and $p$ represent the orbital angular momentum (OAM) mode index and radial index respectively. OAM mode index $l$ signifies that the photon carries orbital angular momentum $l\hbar$ . In SPDC process, OAM is conserved, i.e., $l_{p} = l_{s}+l_{i}$ . Hence, if the incident LG pump beam (V_p) has OAM mode index $l_{p}=0$ , then for the down converted photons, we have $l_{s} = -l_{i} = l$ . In such a case, the two-photon wavefunction reduces to:

$\left|\psi_{t p}\right\rangle=\sum_{l=-\infty}^{\infty} \sqrt{S_{l}}|l\rangle_{s}|-l\rangle_{i}$

where $S_{l}$ produces the Schmidt spectrum. It is the probability of the signal and idler photons being detected with OAM $l\hbar$ and $-l\hbar$ respectively.

4. Expression for Schmidt spectrum $\boldsymbol{S_{l}}$

$S_{l}=\frac{1}{4 \pi^{2}} \iint_{0}^{\infty}\left|\int_{-\pi}^{\pi} V_{p}\left(\rho_{s}, \rho_{i}, \phi_{s}, \phi_{i}\right) \Phi\left(\rho_{s}, \rho_{i}, \phi_{s}, \phi_{i}\right) e^{i l\left(\phi_{s}-\phi_{i}\right)} d \phi_{s} d \phi_{i}\right|^{2} \rho_{s} \rho_{i} d \rho_{s} d \rho_{i}$

where $\boldsymbol {q} \equiv\left(q_{x}, q_{y}\right)=(\rho \cos \phi, \rho \sin \phi)$

5. Project Objectives

The project objective is to combine different LG modes (in the pump beam) to obtain a flat rectangular spectrum, denoting a maximally entangled state. Later, a generalised method to reproduce any desired spectrum using a combination of LG modes would be devised. Experimental verification of the same would provide confirmation regarding the applicability of the proposed scheme.

$V_{p}\left(\rho_{p}, \phi_{p}\right)=\sum \alpha_{p} L G(l=0, p) \stackrel{\text { down-conversion }}{\longrightarrow} S_{l}= \begin{cases}1 / 2 L ; & l \epsilon[-L, L] \\ 0 ; & \text { otherwise }\end{cases}$