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Physical Review A

Fourier relationship between the angle and angular momentum of entangled photons

Anand K. Jha, Barry Jack, E. Yao, Jonathan Leach, Robert W. Boyd, Gerald S. Buller, Stephen Mark Barnett, Sonja Franke‐Arnold, Miles J. Padgett

Published Oct 8, 2008• 70 citations
Publisher

Abstract

We study the Fourier relationship between angle and orbital angular momentum of entangled photons. Spatial light modulators allow us to define and control the spatial mode measurement state. We observe strong quantum correlations, establishing that angular position and momentum distributions between the photons are related as conjugate Fourier pairs.

DOI
10.1103/physreva.78.043810

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We study the Fourier relationship between angle and orbital angular momentum of entangled photons. Spatial light modulators allow us to define and control the spatial mode measurement state. We observe strong quantum correlations, establishing that angular position and momentum distributions between the photons are related as conjugate Fourier pairs.
The angular profile and the orbital angular momentum of a light mode are related by Fourier transform. Any modification of the angular distribution, e. g. via diffraction off a suitably programmed spatial light modulator, influences the orbital angular momentum spectrum of the light. This holds true even at the single photon level. We observe the influence of various angular masks on the orbital angular momentum spectrum, both in the near and the far field, and describe the resulting patterns in terms of angular diffraction. If photons are entangled in their orbital angular momentum, diffraction of one photon affects the orbital angular momentum spectrum of its partner photon, and angular ghost diffraction can be measured in the coincidence counts. We highlight the role of the angular Fourier relationship for these effects.
The entanglement of optical angular position and orbital angular momentum has become one of the important degrees of freedom for quantum information science. Quantum control of the two conjugate variables may provide a gate to control the production of the entanglement. Here, we theoretically and experimentally show that by controlling the transverse spatial coherence of the pump beam, one can modulate the strength of the quantum correlations between the angular position and orbital angular momentum in down-converted photon pairs. By determining the relation of the Einstein, Podolsky, and Rosen correlations with the transverse spatial coherence of the pump, the boundary of the entanglement can be determined. Our work shows that the transverse spatial coherence of the pump can act as a switch to control the production of quantum entanglement both for discrete and continuous conjugate variables and may provide another platform for the quantum control research field.
The generation of paired photons entangled in the spatial degree of freedom, i.e., in orbital angular momentum, offers a convenient physical resource to investigate the nature of entanglement in a multidimensional Hilbert space with controllable dimensionality. The two main physical processes that generate pairs of photons which show correlations in orbital angular momentum are (a) spontaneous parametric down-conversion (SPDC), and (b) Raman transitions induced in atomic ensembles. One question naturally arises: what kinds of correlations exist between the orbital angular momentum of the generated photons? The answer might be different if we consider the whole quantum state of the generated photons, i.e., all possible directions where the pairs of photons can be emitted, or if we consider only a small section of the full set of directions.