  The theory of angular momentum connects physical rotations and quantum spins
together at a fundamental level. Physical rotation of a quantum system will
therefore affect fundamental quantum operations, such as spin rotations in
projective Hilbert space, but these effects are subtle and experimentally
challenging to observe due to the fragility of quantum coherence. Here we
report a measurement of a single-electron-spin phase shift arising directly
from physical rotation, without transduction through magnetic fields or
ancillary spins. This phase shift is observed by measuring the phase difference
between a microwave driving field and a rotating two-level electron spin
system, and can accumulate nonlinearly in time. We detect the nonlinear phase
using spin-echo interferometry of a single nitrogen-vacancy qubit in a diamond
rotating at 200,000rpm. Our measurements demonstrate the fundamental
connections between spin, physical rotation and quantum phase, and will be
applicable in schemes where the rotational degree of freedom of a quantum
system is not fixed, such as spin-based rotation sensors and trapped
nanoparticles containing spins.
