  We report the numerical discovery of a pair-density-wave (PDW) superconductor
from doping either (i) a spin-one Haldane chain or (ii) a two-leg ladder in the
rung singlet phase in which the doped charges occupy a single leg. We model
these systems using a generalized Kondo model. The itinerant electrons are
correlated and described by the $t-J$ model, and are further coupled to a spin
$1/2$ Heisenberg model through the Kondo coupling $J_K$. When the density of
electrons $x$ is one, the Mott insulator is in a Haldane phase or in a rung
singlet phase depending on whether $J_K$ is negative or positive. Upon doping,
a pair-density-wave with $\mathbf Q=\pi$ can emerge for both signs of $J_K$. In
the $J_K \rightarrow -\infty$ limit, the model reduces to the recently proposed
type II t-J model and we observes a continuous transition between the PDW
superconductor and an unconventional Luttinger liquid phase with doping. We
also identify a composite order parameter for the superconductor, which can be
understood as a Cooper pair formed by two nearby fermionic spin-polarons. Our
model and the predicted PDW phase can be experimentally realized by doping S=1
chains formed by Ni$^{2+}$ in a solid state system or a two-leg ladder of
fermionic cold atoms with a potential bias between legs, which preferentially
dopes carriers into a single leg.

Strongly Correlated Electrons