Higher (2nd)-order Wigner distribution function in quantum phase space for entangled bi-modal coherent states, a representative of higher (2nd)-order optical-polarization, is introduced by generalizing kernel (transiting) operator in Cahill-Glauber C(s)-correspondence rule. The nature is analyzed which reveals the occurrence of oscillating three peaks: 'two' for individual bi-modes and third for interference between modes. Also, the graphics of 2nd-order polarization-Wigner distribution function, incisively, demonstrates that it is of non-Gaussian nature attaining non-negative values in quantum phase space.
We reveal a sharp form of complementarity in the \emph{Janus state}—a coherent superposition of two squeezed vacua. Using a compact generating-function formalism, we derive closed-form expressions for
g(k) at all orders and introduce \emph{squeezing polynomials}
Pk(z) that factorize the moments and expose phase-dependent scaling. The Janus state operates as a phase-tunable quantum switch for multi-photon correlations: when which-path/which-state information is available (trivial interference), the higher-order coherences
g(k) reproduce the single-squeezed limit and diverge; when path information is erased and genuine interference from coherent superposition is enabled, multi-photon events are extinguished, driving
g(k)→0 for all
k≥3. We obtain a hierarchy of suppression with exponents set by the interference phase and the parity of
k, and we visualize the mechanism via pronounced Wigner-function negativity. The framework is general and experimentally accessible, enabling engineered nonclassical light with tailored higher-order statistics.