  Extreme confinement of electromagnetic energy by phonon polaritons holds the
promise of strong and new forms of control over the dynamics of matter. To
bring such control to the atomic-scale limit, it is important to consider
phonon polaritons in two-dimensional (2D) systems. Recent studies have pointed
out that in 2D, splitting between longitudinal and transverse optical (LO and
TO) phonons is absent at the $\Gamma$ point, even for polar materials. Does
this lack of LO--TO splitting imply the absence of a phonon polariton in polar
monolayers? Here, we derive a first-principles expression for the conductivity
of a polar monolayer specified by the wavevector-dependent LO and TO phonon
dispersions. In the long-wavelength (local) limit, we find a universal form for
the conductivity in terms of the LO phonon frequency at the $\Gamma$ point, its
lifetime, and the group velocity of the LO phonon. Our analysis reveals that
the phonon polariton of 2D is simply the LO phonon of the 2D system. For the
specific example of hexagonal boron nitride (hBN), we estimate the confinement
and propagation losses of the LO phonons, finding that high confinement and
reasonable propagation quality factors coincide in regions which may be
difficult to detect with current near-field optical microscopy techniques.
Finally, we study the interaction of external emitters with two-dimensional hBN
nanostructures, finding extreme enhancement of spontaneous emission due to
coupling with localized 2D phonon polaritons, and the possibility of multi-mode
strong and ultra-strong coupling between an external emitter and hBN phonons.
This may lead to the design of new hybrid states of electrons and phonons based
on strong coupling.
