  The nature of dielectric echoes in amorphous solids at low temperatures is
investigated. It is shown that at long delay times the echo amplitude is
determined by a small subset of two level systems (TLS) having negligible
relaxation and decoherence because of their weak coupling to phonons. The echo
decay can then be described approximately by power law time dependencies with
different powers at times longer and shorter than the typical TLS relaxation
time. The theory is applied to recent measurements of two and three pulse
dipolar echo in borosilicate glass BK7 and provides a perfect data fit in the
broad time and temperature ranges under the assumption that there exist two TLS
relaxation mechanisms due to TLS-phonons and TLS-TLS interaction. This
interpretation is consistent with the previous experimental and theoretical
investigations. Further experiments verifying the theory predictions are
suggested.
