Mesoscopic quantum superposition of the generalized cat state: A diffraction limit
Suranjana Ghosh, R.P. Sharma, Utpal Roy, Prasanta Kumar Panigrahi · Physical Review A · 2015
The orthogonality of cat and displaced cat states, underlying Heisenberg limited measurement in quantum metrology, is studied in the limit of a large number of states. The mesoscopic superposition of the generalized cat state is correlated with the corresponding state overlap function, controlled by the sub-Planck structures arising from phase-space interference. The asymptotic expression of this overlap function is evaluated, and the validity of large phase-space support and distinguishability of the constituent states, in which context the asymptotic limit is achieved, are discussed in detail. For a large number of coherent states, uniformly located on a circle, the overlap function significantly matches the diffraction pattern for a circular ring source with uniform angular strength. This is in accordance with the van Cittert--Zernike theorem, where the overlap function, similar to the mutual coherence function, matches a diffraction pattern. The physical situation under consideration is delineated in phase space by utilizing the Husimi $Q$ function.