  The pair density wave (PDW) superconducting state has been proposed to
explain the layer- decoupling effect observed in the compound
La$_{2-x}$Ba$_x$CuO$_4$ at $x=1/8$ (Phys. Rev. Lett. 99, 127003). In this state
the superconducting order parameter is spatially modulated, in contrast with
the usual superconducting (SC) state where the order parameter is uniform. In
this work, we study the properties of the amplitude (Higgs) modes in a
unidirectional PDW state. To this end we consider a phenomenological model of
PDW type states coupled to a Fermi surface of fermionic quasiparticles. In
contrast to conventional superconductors that have a single Higgs mode,
unidirectional PDW superconductors have two Higgs modes. While in the PDW state
the Fermi surface largely remains gapless, we find that the damping of the PDW
Higgs modes into fermionic quasiparticles requires exceeding an energy
threshold. We show that this suppression of damping in the PDW state is due to
kinematics. As a result, only one of the two Higgs modes is significantly
damped. In addition, motivated by the experimental phase diagram, we discuss
the mixing of Higgs modes in the coexistence regime of the PDW and uniform SC
states. These results should be observable directly in a Raman spectroscopy, in
momentum resolved electron energy loss spectroscopy, and in resonant inelastic
X-ray scattering, thus providing evidence of the PDW states.
