  We predict a spin pure dephasing channel in electron relaxation between
states with unequal Zeeman splittings, exemplified by a spin-preserving
electron tunneling between quantum dots in a magnetic field. The dephasing is
caused by a mismatch in electron $g$-factors in the dots leading to
distinguishability of phonons emitted during tunneling with opposite spins.
Combining multiband $\boldsymbol{k}{\cdot}\boldsymbol{p}$ modeling and
dynamical simulations via a Master equation we show that this fundamental
effect of spin measurement effected by the phonon bath may be widely controlled
by the size and composition of the dots or on demand, via tuning of external
fields. By comparing the numerically simulated degree of dephasing with the
predictions of general theory based on distinguishability of environment
states, we show that the proposed mechanism is the dominant phonon-related spin
dephasing channel and may limit spin coherence time in tunnel-coupled
structures at cryogenic temperatures.
