Closed-form photon-number amplitudes for cubic squeezed vacua

Derive explicit closed-form expressions for the photon-number amplitudes \(c_n(\gamma,r)=\langle n|\Gamma_\gamma S_{r,\pi}|0\rangle\) of cubic squeezed vacuum states, and thereby obtain closed-form expressions for their photon-number probabilities.

Background

The paper analyzes photon statistics of cubic Gaussian states through moments of creation and annihilation operators because direct photon-number distributions are difficult to evaluate. The authors state that no closed-form analytical expressions are available even in the pure cubic squeezed-vacuum case.

Such formulas would make it possible to study the full photon-number distribution and its asymptotic behavior directly, rather than relying on moment calculations or numerically truncated Fock expansions.

References

We note, however, that we do not have closed-form analytical expressions for $$ p_n=\langle n|\hat\rho_{\mathrm{G},\gamma}|n\rangle. $$ This is true even for cubic squeezed vacua: indeed, we do not know of explicit expressions for \begin{equation} c_n(\gamma, r)=\langle n|\Gamma_\gamma S_{r,\pi}|0\rangle, \end{equation} because of the presence of the cubic factor.

— Tradeoff between Wigner negativity and decoherence time for cubic Gaussian states  (2610.10073 - Arnhem et al., 7 Oct 2026) in Section 7, “Moments of \(\hat a\) and \(\hat a^\dagger\) in cubic squeezed vacua”