  Detecting the transmission phase of a quantum dot via interferometry can
reveal the symmetry of the orbitals and details of electron transport.
Crucially, interferometry will enable the read-out of topological qubits based
on one-dimensional nanowires. However, measuring the transmission phase of a
quantum dot in a nanowire has not yet been established. Here, we exploit recent
breakthroughs in the growth of one-dimensional networks and demonstrate
interferometric read-out in a nanowire-based architecture. In our two-path
interferometer, we define a quantum dot in one branch and use the other path as
a reference arm. We observe Fano resonances stemming from the interference
between electrons that travel through the reference arm and undergo resonant
tunnelling in the quantum dot. Between consecutive Fano peaks, the transmission
phase exhibits phase lapses that are affected by the presence of multiple
trajectories in the interferometer. These results provide critical insights for
the design of future topological qubits.
