Violation of a Bell inequality in two-dimensional orbital angular momentum state-spaces

J. Leach; B. Jack; J. Romero; M. Ritsch-Marte; R.W. Boyd; A.K. Jha; S.M. Barnett; S. Franke-Arnold; M. J. Padgett

doi:10.1364/OE.17.008287

Optics Express
Vol. 17,
Issue 10,
pp. 8287-8293
(2009)
•https://doi.org/10.1364/OE.17.008287

Violation of a Bell inequality in two-dimensional orbital angular momentum state-spaces

J. Leach, B. Jack, J. Romero, M. Ritsch-Marte, R.W. Boyd, A.K. Jha, S.M. Barnett, S. Franke-Arnold, and M. J. Padgett

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J. Leach, B. Jack, J. Romero, M. Ritsch-Marte, R.W. Boyd, A.K. Jha, S.M. Barnett, S. Franke-Arnold, and M. J. Padgett, "Violation of a Bell inequality in two-dimensional orbital angular momentum state-spaces," Opt. Express 17, 8287-8293 (2009)

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- Quantum Optics
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About this Article
History
- Original Manuscript: February 23, 2009
- Revised Manuscript: March 20, 2009
- Manuscript Accepted: April 20, 2009
- Published: April 30, 2009

Abstract

We observe entanglement between photons in controlled superposition states of orbital angular momentum (OAM). By drawing a direct analogy between OAM and polarization states of light, we demonstrate the entangled nature of high order OAM states generated by spontaneous downconversion through violation of a suitable Clauser Horne Shimony Holt (CHSH)-Bell inequality. We demonstrate this violation in a number of two-dimensional subspaces of the higher dimensional OAM Hilbert space.

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Fig. 1. Bloch spheres for (a) polarization and (b) OAM (∣±2〉) states. The sphere for OAM states shows the phase structure (grayscale) of the relevant modes. The states on the sphere are expressed solely in terms of their azimuthal phase (i.e. ℓ-value) without any reference to their radial dependence.

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Fig. 2. Schematic of experimental apparatus for using spatial light modulators to measure the correlations in orbital states of down-converted photons

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Fig. 3. The relative coincidence count as a function of relative orientation of the sector apertures. The measured count rates were normalized by D, the denominator in equ 6. It is the coincidence counts at orientations θ_A = 0, θ_B =

\frac{π}{8 ℓ}

, θ'_A =

\frac{π}{4 ℓ}

and θ'_B =

\frac{3 π}{8 ℓ}

which show the largest violation of the Bell inequality.

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Tables (1)

Table 1. Different entangled states |ψ〉_l and the associated measured value of S. A value of |S| > 2 violates the CHSH inequality.

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Equations (6)

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∣ θ 〉 = \frac{1}{\sqrt{2}} (e^{i ℓ θ} ∣ + ℓ + e^{- i ℓ θ} ∣ - ℓ 〉)

∣ ψ 〉 = \sum_{n = - \infty}^{n = + \infty} c_{n} {| n 〉}_{s} {∣ - n 〉}_{i} .

C (θ_{A}, θ_{B}) = {∣ 〈 θ_{A} ∣ 〈 θ_{B} ∣ ∣ ψ 〉 ∣}^{2} \propto \cos^{2} [ℓ (θ_{A} - θ_{B})] .

{∣ ψ 〉}_{ℓ} = \frac{1}{\sqrt{2}} [∣ ℓ 〉_{s} ∣ - ℓ 〉_{i} + ∣ - ℓ 〉_{s} ∣ ℓ 〉_{i}]

S = E (θ_{A}, θ_{B}) - E (θ_{A}, {θ'}_{B}) + E ({θ'}_{A}, θ_{B}) + E ({θ'}_{A}, {θ'}_{B})

E (θ_{A}, θ_{B}) = \frac{C (θ_{A}, θ_{B}) + C (θ_{A} + \frac{π}{2 ℓ}, θ_{B} + \frac{π}{2 ℓ}) - C (θ_{A} + \frac{π}{2 ℓ}, θ_{B}) - C (θ_{A}, θ_{B} + \frac{π}{2 ℓ})}{C (θ_{A}, θ_{B}) + C (θ_{A} + \frac{π}{2 ℓ}, θ_{B} + \frac{π}{2 ℓ}) + C (θ_{A} + \frac{π}{2 ℓ}, θ_{B}) + C (θ_{A}, θ_{B} + \frac{π}{2 ℓ})} .

Entangled state	S	Violation by σ
\|ψ〉₂	2.69 ± 0.02	35
\|ψ〉₃	2.55 ± 0.04	14
\|ψ〉₄	2.33 ± 0.07	5

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Tables (1)

Equations (6)

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