Violation of Bell’s inequality with continuous spatial variables

Spatial correlations in parametric down-conversion

“…These operators were originally formulated for single-modes of the electromagnetic field, and their realization presents considerable experimental challenges [234]. An insightful contribution was made by Abouraddy et al, who recognized that the operators (165) could be constructed for the spatial degrees of freedom of photons [218]. This is possible due to the isomorphism between the photon number Fock states |n , and the spatial degrees of freedom of a field described by a one-dimensional Hermite Gaussian mode with index n [218].…”

“…Yarnall et al [218,219] have used this type of parity entanglement to violate Bell's inequalities with parity measurements, which we describe in detail below in section 8.7. Using the expansion (158), the concurrence of the two-photon state can be calculated as a function of the pump beam profile and the length of the non-linear crystal.…”

“…An insightful contribution was made by Abouraddy et al, who recognized that the operators (165) could be constructed for the spatial degrees of freedom of photons [218]. This is possible due to the isomorphism between the photon number Fock states |n , and the spatial degrees of freedom of a field described by a one-dimensional Hermite Gaussian mode with index n [218]. To employ the full power of the analogy between the spatial version of the pseudo-operators (165) and the Pauli operators, one must be able to implement arbitrary rotations, composed of linear combinations of X, Y and Z.…”

Modal, Spectral, and Polarization Entanglement in Guided- Wave Parametric Down-Conversion

“…Another approach to binarization is based on constraining the parametric process in such a way that only two values of the continuous variable are permitted; such * Electronic address: mohsaleh@bu.edu configurations include narrowband spectral entanglement [12] and Type-II noncollinear down-conversion in which only two directions (as determined by intersecting rings) are permitted [13]. A third approach is to map the larger Hilbert space onto a binary space, as in spatial-parity entanglement [6,14].…”

“…Entanglement arises from the multiple possibilities for satisfying energy and momentum conservation, as required by the parametric interaction process. Spectral entanglement is exhibited equivalently in time or energy [2]; spatial entanglement may be manifested as entanglement in wavevector direction [3], transverse direction [4], orbital-angularmomentum (OAM) [5], or spatial parity [6,7]. A proper description of the quantum state of the photon pair (biphoton) requires a Hilbert space that includes all of the pertinent degrees of freedom.…”