Dependence on crystal parameters of the correlation time between signal and idler beams in parametric down conversion calculated in the Wigner representation

From Stochastic Optics to the Wigner Formalism: The Role of the Vacuum Field in Optical Quantum Communication Experiments

“…Furthermore, the theory of PDC was developed starting from the quantised Maxwell equations and passing to the Wigner formalism, in order to calculate the correlation time between the signal and idler fields [22] and the spectrum of light [23] in terms of typical experimental parameters. A similar treatment was given to the study of parametric down and up conversion [24,25].…”

Wigner representation for experiments on quantum cryptography using two-photon polarization entanglement produced in parametric down-conversion

“…The process of PDC in the Wigner representation, along with the theory of detection, was developed within the Heisenberg picture in a series of papers [8][9][10][11][12][13]. The whole formalism was applied to the experimental arena of Bell's inequalities and non-classical aspects of light, showing that the Wigner formalism provides an interesting and efficient tool, not only for practical calculations, but also for conceptual questions.…”

Local Realistic Interpretation of Entangled Photon Pairs in the Weyl-Wigner Formalism

“…Within the WW formalism the interaction between the fields (either signals or vacuum fields) and macroscopic bodies may be treated as in classical electrodynamics. This is for instance the case for the action of a laser on a crystal with nonlinear susceptibility, studied elsewhere [12,26].…”

“…We see that going from Equation (26) to Equation (24) the signal terms (those of order |D| 2 ) are multiplied times 2, whilst those coming from the vacuum (of order unity) are eliminated. This may be seen as a subtraction of the vacuum (ZPF) and multiplication of the signal times 2, which leads to the following rule for the single detection rate in the WW formalism:…”

“…Another example is the down-conversion process in a nonlinear crystal. Avoiding a detailed study of the physics inside the crystal [12,26] we shall study a single mode problem with the model Hamiltonian [27]…”