  Dissipationless nonlinearities for three-wave mixing are a key component of
many superconducting quantum devices, such as amplifiers and bosonic qubits. So
far, such third-order nonlinearities have been primarily achieved with circuits
of concatenated Josephson tunnel junctions. In this work, we theoretically
develop an alternative approach to realize third-order nonlinearities from
gate-tunable and intrinsically symmetry-broken quantum material Josephson
junctions. We illustrate this approach on two examples, an Andreev
interferometer and a magnetic Josephson junction. Our results show that both
setups enable Kerr-free three-wave mixing for a broad range of frequencies, an
attribute that is highly desirable for amplifier applications. Moreover, we
also find that the magnetic junction constitutes a paradigmatic example for
three-wave mixing in a minimal single-junction device without the need for any
external biases. We hope that our work will guide the search of dissipationless
nonlinearities in quantum material superconducting devices and inspire new ways
of characterizing symmetry-breaking in quantum materials with microwave
techniques.
