  We show how Floquet Majorana fermions may be experimentally realized by
periodic driving of a solid-state platform. The system comprises a planar
Josephson junction made of a proximitized heterostructure containing a 2D
electron gas (2DEG) with Rashba spin-orbit coupling and Zeeman field. We map
the sub-gap Andreev bound states of the junction to an effective
one-dimensional Kitaev model with long range hopping and pairing terms. Using
this effective model, we study the response of the system to periodic driving
of the chemical potential, as applied via microwaves through a top-gate. We use
the bulk Floquet topological invariants to characterize the system in terms of
the number of zero and $\pi$ Majorana modes for experimentally realistic
parameters. We present signatures of these modes on differential conductance
and local density of states. A notable feature is subharmonic response to the
drive when both Majorana zero and $\pi$ modes are present, thus opening up the
possibility of using this feature to clearly identify Majorana modes. We
highlight the robustness of these features to disorder, and show that despite
hybridization due to long localization lengths, the main features of the
subharmonic response are still identifiable in an experimental system that is
readily accessible.
