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

TL;DR

This paper provides new insights into the Bogoliubov theory of interacting bosons by exactly diagonalizing a simplified Hamiltonian, revealing novel collective excitations called vacuons and clarifying their relation to quasiphonons and known Bogoliubov excitations.

Contribution

It introduces the concept of vacuons as new collective excitations and connects them to the existing Bogoliubov theory, offering a deeper understanding of bosonic interactions.

Findings

Exact diagonalization of the simplified Hamiltonian

Identification of vacuons as bosonic Cooper pairs

Clarification of the relation between quasiphonons and Bogoliubov excitations

Abstract

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.