  Quantum state transfer by propagating wave packets of electromagnetic
radiation requires tunable couplings between the sending and receiving quantum
systems and the propagation channel or waveguide. The highest fidelity of state
transfer in experimental demonstrations so far has been in superconducting
circuits. Here, the tunability always comes together with nonlinear
interactions, arising from the same Josephson junctions that enable the
tunability. The resulting non-linear dynamics correlates the photon number and
spatio-temporal degrees of freedom and leads to a multi-mode output state, for
any multi-photon state. In this work, we study as a generic example the release
of complex quantum states from a superconducting resonator, employing a flux
tunable coupler to engineer and control the release process. We quantify the
multi-mode character of the output state and discuss how to optimize the
fidelity of a quantum state transfer process with this in mind.
