  Tunneling two level systems affect damping, noise and decoherence in a wide
range of devices, including nanoelectromechanical resonators, optomechanical
systems, and qubits. Theoretically this interaction is usually described within
the tunneling state model. The dimensions of such devices are often small
compared to the relevant phonon wavelengths at low temperatures, and extensions
of the theoretical description to reduced dimensions have been proposed, but
lack conclusive experimental verification. We have measured the intrinsic
damping and the frequency shift in magnetomotively driven aluminum
nanoelectromechanical resonators of various sizes at millikelvin temperatures.
We find good agreement of the experimental results with a model where the
tunneling two level systems couple to flexural phonons that are restricted to
one or two dimensions by geometry of the device. This model can thus be used as
an aid when optimizing the geometrical parameters of devices affected by
tunneling two level systems.
