Anomalous behavior of the $N =$ 2 bubble states in a tilted field

We have carried out the activation energy gap measurements of the $N =$ 2 bubble states using tilted-field technique in two GaAs/AlGaAs samples with Corbino geometry. The bubble states are expected to have decreasing gaps with the increasing tilted angle. In the conventional bubble states, surprisingly, we found that the energy gaps were nonlinear with the total magnetic field, which suggests that Zeeman effect isn't the only cause of the gap weakening. We also found a bubble state's energy gap with opposite in-plane field dependence, which occurred at an unusual filling factor for the bubble states.