Abstract
We report an experimental test of Leggett's non-local hidden variable theory in an orbital angular momentum (OAM) state space of light. We show that the correlations we observe are in conflict with Leggett's model, thus excluding a particular class of non-local hidden variable theories for the first time in a non-polarization state space. It is known that the violation of the Leggett inequality becomes stronger as more detection settings are used. The required measurements become feasible in an OAM subspace, and we demonstrate this by testing the inequality using three and four settings. We observe excellent agreement with quantum predictions and a violation of five and six standard deviations, respectively, compared to Leggett's non-local hidden variable theory.
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GENERAL SCIENTIFIC SUMMARY Introduction and background. Quantum mechanics is generally accepted to be correct, but it is not the only theory that can explain correlations between measurements on two separated systems, such as signal and idler photons from a nonlinear crystal. Correlations may also arise from 'prior agreement' between the two photons that are embedded in local hidden variables. Violations of the Bell inequality, which puts a limit to the correlations achievable with local hidden variable theories, have ruled out this class of theories. More recently, Leggett introduced a theory that involved a class of nonlocal hidden variables and an inequality with which to test it. The Leggett inequality has been generalised, and violated in the polarisation state space of photons.
Main results. We extend the violation of Leggett inequalities to the orbital angular momentum (OAM) state space of photons, which is associated with their helical wavefronts. We define our measurements in a Bloch sphere for OAM and measure the Leggett parameter LN (where N is the number of settings for the signal photon) as we change the angle χ (see figure). We observe excellent agreement with quantum mechanical predictions (red line), and show a violation of five and six standard deviations for N = 3 and N = 4, respectively.
Wider implications. This experiment supports quantum mechanics against Leggett's class of nonlocal hidden variable theories. The OAM of photons offers a highly accessible and rich state space with which to test other theories.
Figure. Measurements are defined on an OAM Bloch sphere, where the states correspond to coherent superpositions of OAM states (insets). We test the Leggett inequality for the cases where the signal photon has three (a) and four (b) settings, labeled ai. Measurements bi and b'i are made on the idler photon, and the resulting coincidences are used to calculate LN. The correlations we observe (black dots) are greater than is allowed by Leggett's theory (green line) and are in close agreement with quantum predictions (red line).