  Two-level system (TLS) defects constitute a major decoherence source of
quantum information science, but they are generally less understood at material
interfaces than in deposited films. Here we study surface TLSs at the metal-air
interface, by probing them using a quasi-uniform field within vacuum-gap (VG)
capacitors of resonators. The VG capacitor has a nano-gap which creates an
order-of-magnitude larger contribution from the metal-air interface than
typical resonators used in circuit QED. We measure three phenomena and find
qualitative agreement with an interacting TLS model, where near-resonant TLSs
experience substantial frequency jitter from the state switching of far-detuned
low-frequency TLSs. First, we find that the loss in all of our VG resonators is
weakly or logarithmically power dependent, in contrast to data from deposited
dielectric films. Second, we add a saturation tone with power $P_{in}$ to a
transmission measurement and obtain the TLS Rabi frequency $\Omega_{0}$. These
data show a substantially weaker $P_{in}$ dependence of $\Omega_{0}$ than the
prediction from the standard non-interacting TLS model. Lastly, we increase the
temperature and find an increased TLS jitter rate and dephasing rate from
power-dependent loss and phase noise measurements, respectively. We also anneal
samples, which lowers the low-frequency TLS density and jitter rate, but the
single-photon loss is found to be unchanged. The results are qualitatively
consistent with a fast-switching interacting-TLS model and they contrast the
standard model of TLSs which describes TLSs independently.
