  We develop a general microscopic theory describing the phonon decoherence of
quantum dots and indistinguishability of the emitted photons in photonic
structures. The coherence is found to depend fundamentally on the
dimensionality of the structure resulting in vastly different performance for
quantum dots embedded in a nano-cavity (0D), waveguide (1D), slab (2D), or bulk
medium (3D). In bulk, we find a striking temperature dependence of the
dephasing rate scaling as $T^{11}$ implying that phonons are effectively
'frozen out' for $T \lesssim 4 \mathrm{K}$. The phonon density of states is
strongly modified in 1D and 2D structures leading to a linear temperature
scaling for the dephasing strength. The resulting impact on the photon
indistinguishability can be important even at sub-Kelvin temperatures. Our
findings provide a comprehensive understanding of the fundamental limits to
photon indistinguishability in photonic structures.
