Quantum electrodynamics and quantum optics

What's the expression of Wigner function they used

What measurement data is taken and how is it used to reconstruct the Wigner function

How does measurement efficiency impact the result

How does signal-LO mode-matching influence efficiency in homodyne detection

Why a single laser is used for local-oscillator and signal

Why do they use a doubler and down-converter for single photon sources

How does the spatial-temporal pulse shape of single photon match LO

How is the density matrix reconstructed? What are the values of the off-diagonal terms.

The paper mentions that within 14 hour of experiment 200 000 measurements corresponded to a vacuum state in the signal mode (no click of the PD), and only 14 000 corresponded to a single photon state in the signal mode (PD clicked). Can the fraction of useful measurement outcomes be increased by increasing the intensity of the pump laser impinging on the up conversion crystal thus increasing the amplitude of the coherent state in the single mode?