Practical schemes for the measurement of angular-momentum covariance matrices in quantum optics

Angular-momentum variables represent basic observables both in classical and quantum optics, specially in three fundamental areas: polarization, interferometry and light-matter interaction [1–10]. For definiteness, throughout we focus on intrinsic (not orbital) angular momenta. This is for example the case of the Stokes parameters, which provide a complete account of secondorder (in complex amplitudes) statistical properties of two-mode polarization and interference. Moreover, spin operators are basic in atomic physics such as in the case of ensembles of two-level atoms described individually as spin 1/2 systems. Second-order statistics of angular-momentum variables are crucial in diverse areas. This is the case of quantum metrology, where angular-momentum statistics determine the ultimate limit to the resolution of interferometric and spectroscopic measurements [1–3]. Moreover, angular-momentum covariance matrices enter in the analysis of many-body entanglement [4], in continuousvariable polarization entanglement [5], and for lightmediated detection of atomic-spin correlations [6]. Recently we have proposed an SU(2)-invariant characterization of angular-momentum fluctuations via the diagonalization of the covariance matrix [11]. Invariance under SU(2) transformations is a desirable property since two states connected by a deterministic SU(2) transformation should be statistically equivalent. Similar invariance ideas are at the heart of current investigations about coherence between classical vectorial waves [12]. In this work we develop simple practical schemes to determine experimentally the angular-momentum covariance matrix of a given system in an unknown state. We particularize the method to diverse optical two-mode po-