Abstract
We calculate the biphoton state generated by spontaneous parametric down-conversion in a thin crystal and under collinear phase matching conditions using a pump consisting of any superposition of Laguerre–Gauss modes. The result has no restrictions on the angular or radial momenta or, in particular, on the width of the pump, signal and idler modes. We demonstrate the strong effect of the ratio of the pump width to the signal/idler widths on the composition of the down-converted entangled fields. The knowledge of this ratio is shown to be essential for calculating the maximally entangled states that can be produced using pumps with a complex spatial profile.
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GENERAL SCIENTIFIC SUMMARY Introduction and background. Entanglement is one of the most remarkable phenomena of quantum physics. The nonlinear process of spontaneous parametric down-conversion (SPDC), the generation of two lower-frequency photons when a pump field interacts with a nonlinear crystal, produces pairs of photons that are entangled, in particular in their orbital angular momentum. Understanding the structure of the entanglement between these photons is important for applications in quantum information, such as multidimensional quantum imaging, and for applications involving spatial light modulators.
Main results. We consider the bi-photon state generated by SPDC in a thin crystal and under collinear phase matching conditions and derive an expression for the simultaneous correlations in the orbital angular momentum and radial momentum of the down-converted Laguerre–Gauss modes for a pump consisting of any superposition of Laguerre–Gauss modes. We use our result to demonstrate the strong effect of the pump to signal/idler width ratio on the composition of the down-converted entangled fields. This impacts on the width of the modal expansion (also known as the spiral bandwidth), which relates to the amount of entanglement of the final state and is shown to be essential when calculating the maximally entangled states that can be produced using pumps with a complex spatial profile.
Wider implications. Our result provides a 'recipe' for calculating the down-converted state for any experimental configuration, with no restrictions on any of the defining parameters. It allows experimental results to be analysed and, indeed, experiments to be tailored to produce a given down-converted state.
Figure. Calculation of the down-converted state produced by a Laguerre–Gauss pump of orbital angular momentum ℓp allows us to see how the ratio of the pump:signal/idler size, γ, affects the modes present in the down-converted state. By choosing this ratio together with a particular pump configuration we can tailor the output state for different applications.