  A transducer of single photons between microwave and optical frequencies can
be used to realize quantum communication over optical fiber links between
distant superconducting quantum computers. A promising scalable approach to
constructing such a transducer is to use ensembles of quantum emitters
interacting simultaneously with electromagnetic fields at optical and microwave
frequencies. However, inhomogeneous broadening in the transition frequencies of
the emitters can be detrimental to this collective action. In this article, we
utilise a gradient-based optimization strategy to design the temporal shape of
the laser field driving the transduction system to mitigate the effects of
inhomogeneous broadening. We study the improvement of transduction efficiencies
as a function of inhomogeneous broadening in different single-emitter
cooperativity regimes and correlate it with a restoration of superradiance
effects in the emitter ensembles. Furthermore, to assess the optimality of our
pulse designs, we provide certifiable bounds on the design problem and compare
them to the achieved performance.
