where 1M is the identity for all matter degrees of freedom, and {|n kx,kz ⟩} are the Fock states for a given k x and k z . To avoid gauge ambiguities, this mode truncation can be performed as discussed in Ref. 81, where the Pkx enters into the exponential of the PZW operator (See Eq. 73). Then, for each k x , this truncated Hamiltonian is diagonalized to find the dispersion plots and corresponding Hopfield 132 coefficients as a function of k x .