Estimation of covariance matrix for Stokes-like measurement under phase-space deformation

Abstract

We investigate the signatures of phase-space noncommutativity in an anisotropic oscillator and propose an optomechanical protocol to experimentally characterize it through Stokes-like measurements of an optical field. Starting from a deformed phase-space algebra characterized by the position-position and momentum-momentum noncommutativity parameters $θ$ and $η$, we employ the Darboux transformation to construct an equivalent Hamiltonian in canonical variables. The resulting effective dynamics exhibits correlations between the oscillator degrees of freedom. We derive the corresponding covariance matrix and examine how the noncommutative parameters enter the mechanical dynamics and the optical response. By coupling the oscillator to a coherent optical probe, we obtain analytical expressions for the time-dependent optical coherence attenuation and the nonlinear Kerr phase. We then formulate an interferometric detection scheme through Stokes-like measurements, in which the outgoing optical field is combined with a phase-controlled reference beam at a balanced beam splitter, and the resulting photocurrents are measured using photodetectors. The optical response provides access to the modified normal-mode frequencies, which enables us to estimate experimental values of $θ$ and $η$. Our results establish a theoretical connection between NC-space dynamics, Gaussian-state correlations, and experimentally accessible optical observables, suggesting an optomechanical route toward testing signatures of phase-space noncommutativity.