Abstract
Abstract
We present another step in our efforts to expand far-field imaging beyond the diffraction limit and beyond the limitations imposed by fluorescent labels. The model system we examine is based on the exploitation of entangled photon pairs at 810 nm generated by spontaneous parametric down conversion (SPDC) of a 405 nm source to resolve the separations of the line pairs of a series of two-dimensional objects. A ghost-imaging optical system directs signal and idler photons along two paths, one of which contains the objects. Images are constructed using a covariance analysis of the entangled pairs using typically 106 frames collected with an EMCCD detector that was calibrated to account for the number of entangled pairs of photons, which is necessary for an accurate determination of the resolution and its comparison with the classical method. Conservation of momentum requires that there is inversion symmetry between the transverse k vectors of the photons passing through the target and their entangled counterparts directed along the targetless path. Our covariance analysis not only addresses this symmetry but also that the width of the covariance peak is broader than a single pixel owing to the nature of SPDC and the details of the optical arrangement. Consideration of this broadening provides significant improvement of the image quality. The Heisenberg limit (HL) is achieved within experimental error: using the same optical alignment, the resolution obtained with entangled pairs of 810 nm photons is, within experimental error, twice that obtained with classical 810 nm photons. More importantly, the error bars of measurements with 810 nm entangled pairs are quantified by the Cramér-Rao lower bound (CRLB) and compared with those of the classical measurements. While other workers have done quantum imaging on nonpoint sources, to our knowledge we are the first group that not only has done the quantum imaging but also has compared the CRLB and the measured error of the quantum and classical measurements for a series of different-sized objects. (Similar comparisons have, however, been made for systems that can be approximated as point sources, e.g., determining the separation of two laser beams).