  Two-level systems (TLS) in amorphous materials limit coherence times of a
number of solid-state quantum devices. Interactions between TLS become
prominent below 100 mK, but the coupling mechanism and statistical properties
are still unclear. Here we determine the homogeneous linewidth of ytterbium
ions (Yb$ ^{3+} $) in silica glass at 10-80 mK by using photon echo techniques
as a probe of TLS. First, the homogeneous linewidth can be reduced by applying
a magnetic field of 0.3 T. This effect is due to reduced magnetic interactions
between adjacent Yb$ ^{3+} $. Secondly, we observe saturation of the linewidth
below 50 mK to a level of approximately 30 kHz, which is much larger than the
lifetime-limited value of 0.2 kHz. This saturation behavior is in conflict with
the coupling to independent TLS. We show that this effect can be explained by
coherently coupled TLS.
