Electron-hole coherent states for the Bogoliubov-de Gennes equation

We construct a new set of generalized coherent states, the electron-hole coherent states, for a (quasi-)spin particle on the infinite line. The definition is inspired by applications to the Bogoliubov-de Gennes equations where the quasi-spin refers to electron- and hole-like components of electronic excitations in a superconductor. Electron-hole coherent states generally entangle the space and the quasi-spin degrees of freedom. We show that the electron-hole coherent states allow obtaining a resolution of unity and form minimum uncertainty states for position and velocity where the velocity operator is defined using the Bogoliubov-de Gennes Hamiltonian. The usefulness and the limitations of electron-hole coherent states and the phase space representations built from them are discussed in terms of basic applications to the Bogoliubov-de Gennes equation such as Andreev reflection.