  We predict that phonon subband quantization can be detected in the non-linear
electron current through double quantum dot qubits embedded into nano-size
semiconductor slabs, acting as phonon cavities. For particular values of the
dot level splitting $\Delta$, piezo-electric or deformation potential
scattering is either drastically reduced as compared to the bulk case, or
strongly enhanced due to phonon van Hove singularities. By tuning $\Delta$ via
gate voltages, one can either control dephasing, or strongly increase emission
into phonon modes with characteristic angular distributions.
