  Superconducting states that break space-group symmetries of the underlying
crystal can exhibit nontrivial spatial modulation of the order parameter.
Previously, such remarkable states were intimately associated with the breaking
of translational symmetry, giving rise to the density-wave orders, with
wavelengths spanning several unit cells. However, a related basic concept has
been long overlooked: when only intra-unit-cell symmetries of the space group
are broken, the superconducting states can display a distinct type of
nontrivial modulation preserving long-range lattice translation. Here, we refer
to this new concept as the pair density modulation (PDM), and report the first
observation of a PDM state in exfoliated thin flakes of iron-based
superconductor FeTe$_{\text{0.55}}$Se$_{\text{0.45}}$. Using scanning tunneling
microscopy, we discover robust superconducting gap modulation with the
wavelength corresponding to the lattice periodicity and the amplitude exceeding
30% of the gap average. Importantly, we find that the observed modulation
originates from the large difference in superconducting gaps on the two
nominally equivalent iron sublattices. The experimental findings, backed up by
model calculations, suggest that in contrast to the density-wave orders, the
PDM state is driven by the interplay of sublattice symmetry breaking and a
peculiar nematic distortion specific to the thin flakes. Our results establish
new frontiers for exploring the intertwined orders in strong-correlated
electronic systems and open a new chapter for iron-based superconductors.
