Fig. 5.

Measured grAl self-Kerr nonlinearity. The measured self-Kerr coefficients of 14 grAl samples are plotted versus $f_1^2\mathrm{∕}{j}_{\mathrm{c}}{V}_{\mathrm{grAl}}$, where f₁ = ω₁/2π is the frequency of the first-mode, j_c is the critical current density, and V_grAl is the sample volume, with values listed for each sample in the legend. Hilbert-shaped resonator samples are represented in blue, stripline samples in green, and Al-shunted stripline resonators in red. The error bars for the blue points show the standard deviation of the measured K₁₁ for nominally identical resonators. In the legend, the samples are listed in decreasing K₁₁ order, within each group. The blue up-oriented triangle corresponds to sample grAl#2 (highlighted in blue), and the green circle corresponds to grAl#1 (highlighted in green). The black line shows the calculated self-Kerr from Eq. (4), for a grain size a = 4 nm, which includes the 1 nm thickness of the aluminum oxide barrier. We estimate the main error source to be the photon number calibration, which can only be estimated within a factor of 10