  Electron spins in quantum dots under coherent control exhibit a number of
novel feedback processes. Here, we present experimental and theoretical
evidence of a feedback process between nuclear spins and a single electron spin
in a single charged InAs quantum dot, controlled by the coherently modified
probability of exciting a trion state. We present a mathematical model
describing competition between optical nuclear pumping and nuclear
spin-diffusion inside the quantum dot. The model correctly postdicts the
observation of a hysteretic sawtooth pattern in the free-induction-decay of the
single electron spin, hysteresis while scanning a narrowband laser through the
quantum dot's optical resonance frequency, and non-sinusoidal fringes in the
spin echo. Both the coherent electron-spin rotations, implemented with
off-resonant ultrafast laser pulses, and the resonant narrow-band optical
pumping for spin initialization interspersed between ultrafast pulses, play a
role in the observed behavior. This effect allows dynamic tuning of the
electron Larmor frequency to a value determined by the pulse timing,
potentially allowing more complex coherent control operations.
