  We propose microwave-controlled rotations for qubits realized as Majorana
bound states. To this end we study an inhomogeneous Kitaev chain in a microwave
cavity. The chain consists of two topologically nontrivial regions separated by
a topologically trivial, gapped region. The Majorana bound states at the
interfaces between the left (right) regions and the central region are coupled,
and their energies are split by virtual cotunneling processes. The amplitude
for these cotunneling processes decreases exponentially in the number of sites
of the gapped region, and the decay length diverges as the gap of the
topologically trivial region closes. We demonstrate that microwave radiation
can exponentially enhance the coupling between the Majorana bound states, both
for classical and quantized electric fields. By solving the appropriate
Liouville equation numerically we show that microwaves can drive Rabi
oscillations in the Majorana sector. Our model emerges as an effective
description for a topological semiconductor nanowire in a microwave cavity.
Thus, our proposal provides an experimentally feasible way to obtain full
single-qubit control necessary for universal quantum computation with Majorana
qubits.
