  Quantum two-level systems (TLSs) are present in the materials of qubits and
are considered defects because they limit qubit coherence. For superconducting
qubits, the quintessential Josephson junction barrier is made of amorphous
alumina, which hosts TLSs. However, TLSs are not understood generally -- either
structurally or in atomic composition. In this study, we greatly extend the
quantitative data available on TLSs by reporting on the physical dipole moment
in two alumina types: polycrystalline
$\mathrm{\mathrm{\gamma-Al}_{2}\mathrm{O}_{3}}$ and amorphous
$\mathrm{a-Al}\mathrm{O_{x}}$. To obtain the dipole moments $p_z$, rather from
the less-structural coupling parameter g, we tune individual TLSs with an
external electric field to extract the $p_z$ of the TLSs in a cavity QED
system. We find a clear difference in the dipole moment distribution from the
film types, indicating a difference in TLS structures. A large sample of
approximately 400 individual TLSs are analyzed from the polycrystalline film
type. Their dipoles along the growth direction $p_z$ have a mean value of
2.6$\pm$0.3 Debye (D) and standard deviation $\sigma$ = 1.6$\pm$0.2 D . The
material distribution fits well to a single Gaussian function. Approximately
200 individual TLSs are analyzed from amorphous films. Both the mean $p_z$
=4.6$\pm$0.5 D and $\sigma$ =2.5$\pm$0.3 D are larger. Amorphous alumina also
has some very large $p_z$, > 8.6 D, in contrast to polycrystalline which has
none of this moment. These large moments agree only with oxygen-based TLS
models. Based on data and the candidate models (delocalized O and
hydrogen-based TLSs), we find polycrystalline alumina has smaller ratio of
O-based to H-based TLS than amorphous alumina.
