Spin-orbit coupling in graphene can be increased far beyond its intrinsic value by proximity coupling to a transition metal dichalcogenide. In bilayer graphene, this effect was predicted to depend on the occupancy of both graphene layers, rendering it gate-tunable by an out-of-plane electric field. We experimentally confirm this prediction by studying magnetotransport in a dual-gated WSe$_2$/bilayer graphene heterostructure. Weak antilocalization, which is characteristic for phase-coherent transport in diffusive samples with spin-orbit interaction, can be strongly enhanced or suppressed at constant carrier density, depending on the polarity of the electric field. From the spin-orbit scattering times extracted from the fits, we calculate the corresponding Rashba and intrinsic spin-orbit parameters. They show a strong dependence on the transverse electric field, which is well described by a gate-dependent layer polarization of bilayer graphene.
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