I would like to input an state: $$|\psi\rangle_{in}=\frac{1}{\sqrt{n_H!n_V!m_H!m_V!}}(a_{H}^{\dagger})^{n_H}(a_{V}^{\dagger})^{n_V}(b_{H}^{\dagger})^{m_H}(b_{V}^{\dagger})^{m_V}|0000\rangle_{a_Ha_Vb_Hb_V}$$ to a beamsplitter, as shown in the image below (let's assume $\eta=\frac{1}{2}$ for simplicity), such that the output state calculated retains the polarization information (let's hold off on actual measurement for now).

As we have a 4-mode state at the input, its unclear to me what the accompanying beamsplitter matrix would look like. There are other helpful Stack Exchange answers (see: Output of a beamsplitter with photon number (Fock) state inputs and Matrix representation of beamsplitter, for numerical computation of output based on given photon number (Fock state) input), but to my knowledge they don't reveal how to apply the beamsplitter unitary for the case laid out here.

Hong-Ou-Mandel Interference

  • $\begingroup$ one can find the many-boson scattering matrix corresponding to a given linear operator via the permanents. Are you asking how this works? $\endgroup$ – glS Mar 11 at 11:39

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