  The symmetry-breaking first-order phase transition between superfluid phases
$^3$He-A and $^3$He-B can be triggered extrinsically by ionising radiation or
heterogeneous nucleation arising from the details of the sample cell
construction. However, the role of potential homogeneous intrinsic nucleation
mechanisms remains elusive. Discovering and resolving the intrinsic processes
may have cosmological consequences, since an analogous first-order phase
transition, and the production of gravitational waves, has been predicted for
the very early stages of the expanding Universe in many extensions of the
Standard Model of particle physics. Here we introduce a new approach for
probing the phase transition in superfluid $^3$He. The setup consists of a
novel stepped-height nanofluidic sample container with close to atomically
smooth walls. The $^3$He is confined in five tiny nanofabricated volumes and
assayed non-invasively by NMR. Tuning of the state of $^3$He by confinement is
used to isolate each of these five volumes so that the phase transitions in
them can occur independently and free from any obvious sources of heterogeneous
nucleation. The small volumes also ensure that the transitions triggered by
ionising radiation are strongly suppressed. Here we present the preliminary
measurements using this setup, showing both strong supercooling of $^3$He-A and
superheating of $^3$He-B, with stochastic processes dominating the phase
transitions between the two. The objective is to study the nucleation as a
function of temperature and pressure over the full phase diagram, to both
better test the proposed extrinsic mechanisms and seek potential parallel
intrinsic mechanisms.
