  We report on the direct measurement of the electron spin splitting and the
accompanying nuclear Overhauser (OH) field, and thus the underlying nuclear
spin polarization (NSP) and fluctuation bandwidth, in a single InAs quantum dot
under resonant excitation conditions with unprecedented spectral resolution.
The electron spin splitting is measured directly via resonant spin-flip single
photon Raman scattering detected by superconducting nanowires to generate
excitation-emission energy maps. The observed two-dimensional maps reveal an OH
field that has a non-linear dependence on excitation frequency. This study
provides new insight into earlier reports of so-called avoidance and tracking,
showing two distinct NSP responses directly by the addition of a emission
energy axis. The data show that the polarization processes depend on which
electron spin state is optically driven, with surprising differences in the
polarization fluctuations for each case: in one case, a stabilized field
characterized by a single-peaked distribution shifts monotonically with the
laser excitation frequency resulting in a nearly constant optical interaction
strength across a wide detuning range, while in the other case the previously
reported avoidance behavior is actually the result of a nonlinear dependence on
the laser excitation frequency near zero detuning leading to switching between
two distinct mesoscopic nuclear spin states. The magnitude of the field, which
is as large as 400 mT, is measured with sub-100 nuclear spin sensitivity.
Stable/unstable points of the OH field distribution are observed, resulting
from the non-linear feedback loop in the electron-trion-nuclear system. Nuclear
spin polarization state switching occurs between fields differing by 160 mT at
least as fast as 25 ms. Control experiments indicate that the strain-induced
quadrupolar interaction may explain the measured OH fields.
