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The Wigner function of the thermal state is given as $W(\alpha)=\frac{1}{\pi(\langle n\rangle+ \frac{1}{2})} \exp \left( -\frac{|\alpha|^2}{\langle n\rangle+\frac{1}{2}} \right)$ The source is Quantum optics, the author is Girish S. Agarwal

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Energy loss by photons occurs in a dielectric medium when the photons interact with the atoms of which the dielectric is made. This interaction does not normally result in a red-shift of the photons, but rather an attenuation of a beam of photons due to photons being absorbed. Depending on the nature of the material and the energy of the photons, the ...

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I'm assuming your input photon has a known polarization (say horizontal). You won't see interference, because the polarizers act as a "which-path" measuring device. If you erase the polarization information, the interference pattern will appear.

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In general, degeneracy in quantum mechanics means that there are at least two states that have indistinguishable energy. In the case of a degenerate photon pair, the two photons have will the same frequency (as E = $\hbar$ $\omega$). This is, of course, oversimplifying things slightly. The above describes the case when the photons are each in a single mode....

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The photon pair (obtained in a spontaneous parametric down conversion WITHOUT post-selection) keeps the same filter angle, reducing crosstalk in quantum optic networks, therefore increasing channel manipulation.

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After searching I find these "closing operators" defined by the authors correspond to particular sets of operators mentioned in a part (on quantum regression theorem) of the book Statistical methods in quantum optics 1.

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This is done in several papers, e.g. here: http://cds.cern.ch/record/546624/files/0204052.pdf The relevant portion is Lemma 1. Given a state $\rho$ with covariance matrix in block form $$\gamma_{\rho}=\begin{pmatrix}{} A & B \\ B^T & C \end{pmatrix}$$ the covariance matrix after a measurement after a projection onto the pure Gaussian state with ...

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