Continuous-variable spatial entanglement for bright optical beams

Abstract

A light beam is said to be position squeezed if its position can be determined to an accuracy beyond the standard quantum limit. We identify the position and momentum observables for bright optical beams and show that position and momentum entanglement can be generated by interfering two position, or momentum, squeezed beams on a beam splitter. The position and momentum measurements of these beams can be performed using a homodyne detector with local oscillator of an appropriate transverse beam profile. We compare this form of spatial entanglement with split detection-based spatial entanglement.

Transverse angular momentum of photons

“…Finally, when α 00 ∈ R, Eqs. (94-95) reproduce the well-known result that either α 10 or α 01 must have an imaginary part to guarantee a nonzero tilting angle θ µ [24]. In a similar manner we can evaluate the expectation value of the angular momentum operator obtaining…”

“…(98) and (99) imply that in order to have a nonzero transverse orbital angular momentum either α 10 or α 01 must have a real part. However, it is well known that a superposition with real coefficients of the fundamental mode ψ 00 with either ψ 10 or ψ 01 describes approximatively a displaced Gaussian beam along the x-or the y-axis, respectively [24]. In other words, a lateral displacement of a Gaussian beam changes its transverse angular momentum or, vice versa, the occurrence of non zero transverse components of the angular momentum cause a transverse displacement of the beam [12].…”

Spatial-state Stokes-operator squeezing and entanglement for optical beams

“…The quadrature amplitudes are associated with the transverse linear momentum of the field, whilst the polarization states and spatial states considered to date are respectively associated to the spin angular momentum and the transverse angular momentum. We immediately see why interaction of a polarization squeezed field with an atomic ensemble can yield atomic spin squeezing [7], and why spatially entangled light can be used to test the EPR paradox with position and momentum variables [5] as was originally proposed by Einstein, Podolsky and Rosen [8].…”

“…[5] proposed EPR entanglement of the positionmomentum observables of optical fields, and this has recently been experimentally observed [6]. In these references, entanglement was present only for one transverse axis of the beam, and analysis only required consideration of the TEM 00 and TEM 01 modes.…”

“…To date, the vast majority of research has been focused on fields which exhibit non-classical features on their amplitude and phase quadratures. Recently, however, a number of papers have been published on squeezing and entanglement of other field variables, and in particular the polarization [2] and spatial structure [3,4,5,6]. These variables can be directly related to momentum components of the field.…”