Two-dimensional electron systems in GaAs/AlGaAs heterostructures have provided a platform for investigating numerous phenomena in condensed matter physics. The quantum Hall effect is a particularly remarkable phenomenon due to its topological properties, including chiral edge states with quantized conductance. This report describes progress made in interference measurements of these edge states in electronic Fabry-Perot interferometers. Previous interference experiments in the quantum Hall regime have been stymied by Coulomb charging effects and poor quantum coherence. These Coulomb charging effects have been dramatically suppressed by the implementation of a novel GaAs/AlGaAs heterostructure which utilizes auxiliary screening wells in addition to the primary GaAs quantum well. Using this heterostructure, Aharonov-Bohm interference is measured in very small devices which have greatly improved coherence. Robust Aharonov-Bohm interference is reported at fractional quantum Hall states nu = 1/3 and nu = 2/3. Discrete jumps in phase at nu = 1/3 consistent with anyonic braiding statistics are observed. The report concludes with proposed future experiments, including extending these results to possible non-Abelian quantum Hall states.

<div>Strong electronic correlations in many-body systems are cradles of new physics. They give birth to novel collective states hosting emergent quasiparticles as well as intriguing geometrical charge patterns. Two-dimensional electron gas in GaAs/AlGaAs under perpendicular magnetic field is one of the most well-known hosts in condensed matter physics where a plethora of the collective states appear. In the strong magnetic field regime, strong Coulomb interactions among the electrons create emergent quasiparticles, i.e. composite fermions and Cooper-paired composite fermions. In the weak magnetic field regime, modified Coulomb interactions drive electron solid phases having geometrical charge patterns in the shape of stripes and bubbles and lower the spatial symmetry of the states.</div><div><br></div><div>The fascinating charge order in bubble geometry is the electron bubble phase predicted first by the Hartree-Fock theory. In a bubble phase, certain number of electrons cluster as an entity called bubble and the bubbles order into a crystal of triangular lattice. In addition to the Hartree-Fock theory, the density matrix renormalization group and the exact diagonalization methods further support the formation of electronic bubbles.</div><div><br></div><div>Reentrant integer quantum Hall states are commonly accepted as the manifestations of the bubble phases in transport experiment. Soon after the first prediction of the Hartree-Fock theory, the reentrant integer quantum Hall states were observed in the third and higher Landau levels. Since then, the association to the bubble phases has been tested with different experimental techniques for decades.</div><div><br></div><div>Although the experimental results from different methods support the bubble phase picture of the reentrant integer quantum Hall states, the electron confinement under the quantum well structure hindered direct scanning of bubble morphology. Thus none of the experiments could showcase the bubble morphology of the reentrant integer quantum Hall states. Meanwhile, a significant discrepancy still remained in between the bubble theories and the experiments. Even though the bubble theories predict the proliferation of bubble phases with increasing orbital index, none of the experiments could observe multiple reentrant integer quantum Hall states in a high Landau level, which signify the multiple bubble formation. Therefore, the proliferation of bubble phases with increasing Landau level index was pessimistic. </div><div><br></div><div>In this Dissertation, I present my research on solving this discrepancy. In chapter 4, we performed a magnetotransport measurement of reentrant integer quantum Hall states in the third and higher Landau levels at various different temperatures. Then, we scrutinized how each of the reentrant integer quantum Hall states develops with the gradual increase of the temperature. As a result, we observed multiple reentrant integer quantum Hall states in the fourth Landau level which are associated with the two- and three-electron bubble phases. This result strongly supports the bubble phase picture of the reentrant integer quantum Hall states by confirming the possibility of the proliferation of bubble phases in high Landau levels.</div><div><br></div><div>In chapter 5, I analyzed the energetics of newly resolved two- and three-electron bubble phases in the fourth Landau level as well as those of two-electron bubble phases in the third Landau level. Here, I first found, in the fourth Landau level, the three-electron bubbles are more stable than the two-electron bubbles indicating that the multi-electron bubbles with higher electron number are more stable within a Landau level. Secondly, I found distinct energetic features of two- and three-electron bubble phases which are independent of Landau level index throughout the third and the fourth Landau levels. These results highlight the effect of the number of electrons per bubble on the energetics of multi-electron bubble phases and are expected to contribute on improving the existing Hartree-Fock theories.</div>