Discrete quadrature spectrum of composite bosons: Weyl relation, intensity correlations, and binned homodyne detection

Abstract

A composite boson made of two fermions can look elementary under one measurement and composite under another. In an exactly solvable paired-mode model we compare three probes: classical drives, periodically binned quadrature statistics, and intensity correlations $g^{(2)}$. The Weyl defect has an exact product formula and a controlled dilute expansion in the Schmidt moments. The binned observables are instead piecewise constant in the Schmidt weights and sensitive to single eigenvalues crossing bin edges; on the flat-spectrum ladder they converge slowly and nonmonotonically over the computed range. A spectral filter and a polar correction recover an exact Pauli algebra; a heralded joint projection then extracts a logical singlet from the vacua of two independent copies of the composite, with probability set by the accepted weight and the mixedness of the filtered reduced qubit. Finite-mode examples with Schmidt number about three succeed at the percent level despite order-one Weyl defects; near-unit $g^{(2)}$ does not control the binned algebra. For flat spectra the construction converges strongly to its oscillator counterpart, whose vacuum overlap is strictly positive at every scale; for infinite spectra with summable Schmidt amplitudes it converges at all but countably many filter cutoffs, positivity at the cutoffs studied remaining open.