Bogoliubov vacuum ($b_k|\Omega ⟩=0$)

A Bogoliubov vacuum is annihilated by the quasiparticle linear operators that diagonalize a stable quadratic bosonic Hamiltonian. If $a_k=u_k{b}_k-v_k{b}_{-k}^{†}$, its original-mode occupation number is $|v_k{|}^2$. It is generally a multimode squeezed vacuum rather than the original Fock vacuum. In infinitely many modes, existence in the original Fock space requires a separate convergence analysis.