Bogoliubov theory of interacting bosons: new insights from an old problem

In a gas of $N$ interacting bosons, the Hamiltonian $H_c$, obtained by dropping all the interaction terms between free bosons with moment $\hbar\mathbf{k}\ne\mathbf{0}$, is diagonalized exactly. The resulting eigenstates $|\:S,\:\mathbf{k},\:\eta\:\rangle$ depend on two discrete indices $S,\:\eta=0,\:1,\:\dots$, where $\eta$ numerates the \emph{quasiphonons} carrying a moment $\hbar\mathbf{k}$, responsible for transport or dissipation processes. $S$, in turn, numerates a ladder of \textquoteleft vacua\textquoteright$\:|\:S,\:\mathbf{k},\:0\:\rangle$, with increasing equispaced energies, formed by boson pairs with opposite moment. Passing from one vacuum to another ($S\rightarrow S\pm1$), results from creation/annihilation of new momentless collective excitations, that we call \emph{vacuons}. Exact quasiphonons originate from one of the vacua by \textquoteleft creating\textquoteright$\:$an asymmetry in the number of opposite moment bosons. The well known Bogoliubov collective excitations (CEs) are shown to coincide with the exact eigenstates $|\:0,\:\mathbf{k},\:\eta\:\rangle$, i.e. with the quasiphonons created from the lowest-level vacuum ($S=0$). All this is discussed, in view of existing or future experimental observations of the vacuons (PBs), a sort of bosonic Cooper pairs, which are the main factor of novelty beyond Bogoliubov theory.