# generation / destruction of fermions by phonons

my Hamiltonian consists of 1D free fermions coupled to a bosonic bath. The interaction is dictated both by scattering terms

$H^{scatt}=\sum_{kq}\alpha^S_{kq}c^\dagger_kc_{k+q}X_q+h.c.$

as well as generation/destruction terms such as

$H^{gd}=\sum_{kq}\alpha^{gd}_{kq}c^\dagger_kc^\dagger_{-k-q}X_q+h.c.$

Conventional second quantization convention is implied ($c$ are fermionic operators, $\alpha$ is just a momentum coupling matrix). I am setting up a BBGKY hyerarchy equation for the correlation functions to study the kinetics of this problem. Has anyone any reference about this? I see a lot for the case of scattering with phonons but the $H^{gd}$ is never there, but I suppose these kind of models could have been studied in chemical kinetics or superconductivity. I would like to see some example of master equation, or equation of motions with these processes taken into account...

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Well, in a semiconductor you'd expect electron-hole pair creation, but that's already covered by the first term. I have a hard time visualizing processes described by the second term. Could you expand a bit on that? –  Lagerbaer Sep 18 '13 at 4:33
of course, the semiconductor is just an analogy. the model comes up from a more complicated 1D spin quantum statistical model after a jordan wigner transform... I am trying to find physical analogies to my transformed hamiltonian (superconductivity?) –  ventu Sep 19 '13 at 0:49