  We demonstrate how the dissipative interaction between a superconducting
qubit and a microwave photonic crystal can be used for quantum bath
engineering. The photonic crystal is created with a step-impedance transmission
line which suppresses and enhances the quantum spectral density of states,
influencing decay transitions of a transmon circuit. The qubit interacts with
the transmission line indirectly via dispersive coupling to a cavity. We
characterize the photonic crystal density of states from both the unitary and
dissipative dynamics of the qubit. When the qubit is driven, it dissipates into
the frequency dependent density of states of the photonic crystal. Our result
is the deterministic preparation of qubit superposition states as the
steady-state of coherent driving and dissipation near by the photonic crystal
band edge, which we characterize with quantum state tomography. Our results
highlight how the multimode environment from the photonic crystal forms a
resource for quantum control.
