  Using the deterministic, on-demand generation of two entangled phonons, we
demonstrate a quantum eraser protocol in a phononic interferometer where the
which-path information can be heralded during the interference process.
Omitting the heralding step yields a clear interference pattern in the
interfering half-quanta pathways; including the heralding step suppresses this
pattern. If we erase the heralded information after the interference has been
measured, the interference pattern is recovered, thereby implementing a
delayed-choice quantum erasure. The test is implemented using a closed
surface-acoustic-wave communication channel into which one superconducting
qubit can emit itinerant phonons that the same or a second qubit can later
re-capture. If the first qubit releases only half of a phonon, the system
follows a superposition of paths during the phonon propagation: either an
itinerant phonon is in the channel, or the first qubit remains in its excited
state. These two paths are made to constructively or destructively interfere by
changing the relative phase of the two intermediate states, resulting in a
phase-dependent modulation of the first qubit's final state, following
interaction with the half-phonon. A heralding mechanism is added to this
construct, entangling a heralding phonon with the signalling phonon. The first
qubit emits a phonon herald conditioned on the qubit being in its excited
state, with no signaling phonon, and the second qubit catches this heralding
phonon, storing which-path information which can either be read out, destroying
the signaling phonon's self-interference, or erased.
