Wigner functional theory for quantum optics

Based on the recent derivation of quadrature bases that incorporate the spatiotemporal degrees of freedom [Phys. Rev. A 98, 043841 (2018)], we develop a Wigner functional theory for quantum optics. Since these spatiotemporal quadrature bases span a large Hilbert space that does not require factorization as a tensor product of discrete Hilbert spaces, the Wigner functions associated with such a space become functionals and operations are expressed by functional integrals. The resulting formalism enables tractable calculations for scenarios where both spatiotemporal degrees of freedom and particle-number degrees of freedom are relevant. We compute examples of Wigner functionals for a few states and operators.