Generalized measurement of the non-normal two-boson operator $Z_{\gamma }=a_1+\gamma a_2^{†}$

Abstract

We address the generalized measurement of the two-boson operator $Z_{\gamma }=a_1+\gamma a_2^{†}$ which, for $|\gamma {|}^2\ne 1$, is not normal and cannot be detected by a joint measurement of quadratures on the two bosons. We explicitly construct the minimal Naimark extension, which involves a single additional bosonic system, and present its decomposition in terms of two-boson linear SU(2) interactions. The statistics of the measurement and the added noise are analyzed in details. Results are exploited to revisit the Caves-Shapiro concept of generalized phase observable based on heterodyne detection.

8 pages, 2 figures, slightly revised version