  We present a theoretical model to show that the transition from an
antiferromagnetic (AFM) phase to the $d_{x^2-y^2}$-wave superconductivity
occurs through a robust companion of a finite momentum pairing between $({\bf
k},\uparrow)$ and $(-{\bf k}-{\bf Q},\downarrow)$ electrons, namely
pair-density wave (PDW) state, where ${\bf Q}=(\pi,\pi)$ is the AFM wavevector.
Interestingly, the spatial structure of the PDW is constrained to be a
$p+ip$-wave symmetry, which follows $p_{\bf k}=p_{-{\bf k}-{\bf Q}}$ under
fermions exchange in cuprates, dictating a spin-singlet pairing. Furthermore,
the PDW state produces a fully gapped quasiparticle spectrum which explains
recent observations of the fully gapped quasiparticle structure of lightly
doped cuprates in the hole doping side. We study the stability of all three
phases within the self-consistent mean-field theory. Finally, we calculate the
superfluid density to propose its exponential temperature dependence as a test
to the SC origin of this fully gapped state, while the phase modulation of the
PDW state can be visualized by scanning probes.
