Photon statistics, parity measurements, and Heisenberg-limited interferometry: example of the two-mode SU(1,1) SU(1,1) coherent states
Christopher C. Gerry, Richard J. Birrittella, Andrae Raymond, Raul Carranza · Journal of Modern Optics · 2011
Parity measurement-based quantum optical interferometry and metrology is known to be sensitive to the joint photon number distribution, , of the two-mode quantum states inside the interferometer. In cases where the statistics display a clustering of the populations around the borders (i.e. along the axes ) with a plateau over most of the interior, Heisenberg-limited (or super-sensitive) measurements of phase shifts are enabled. This follows from the heuristic number-phase uncertainty relation because of the uncertainty in the location of the photons being ΔN ∼ . As an example of the connection between the photon statistics and Heisenberg-limited phase uncertainty, we study parity measurement-based interferometry of the two-mode Barut–Girardello coherent states for SU(1,1)SU(1,1) obtained from the coupling together of two representations of SU(1,1) each associated with single-mode boson realizations with Bargmann indices as discussed by Gerry and Benmoussa [Phys. Rev. A 2000, 62, 033812]. These states generally have distributions with clustered populations along the borders. We demonstrate that these states do, indeed, lead to super-sensitive measurements and also lead to super-resolved interference fringes.