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Strongly interacting two dimensional electron systems (2DESs) host a complexlandscape of broken symmetry states. The possible ground states are furtherexpanded by internal degrees of freedom such as spin or valley-isospin. Whiledirect probes of spin in 2DESs were demonstrated two decades ago, the valleyquantum number has only been probed indirectly in semiconductor quantum wells,graphene mono- and bilayers, and transition-metal dichalcogenides. Here, wepresent the first direct experimental measurement of valley polarization in atwo dimensional electron system, effected via the direct mapping of the valleyquantum number onto the layer polarization in bilayer graphene at high magneticfields. We find that the layer polarization evolves in discrete steps across 32electric field-tuned phase transitions between states of different valley,spin, and orbital polarization. Our data can be fit by a model that capturesboth single particle and interaction induced orbital, valley, and spinanisotropies, providing the most complete model of this complex system to date.Among the newly discovered phases are theoretically unanticipated orbitallypolarized states stabilized by skew interlayer hopping. The resulting roadmapto symmetry breaking in bilayer graphene paves the way for deterministicengineering of fractional quantum Hall states, while our layer-resolvedtechnique is readily extendable to other two dimensional materials where layerpolarization maps to the valley or spin quantum numbers, providing an essentialdirect probe that is a prerequisite for manipulating these new quantum degreesof freedom.

Direct measurement of discrete valley and orbital quantum numbers in a multicomponent quantum Hall system