  We investigate theoretically the spin echo signal of an electron localized in
a quantum dot and interacting with a bath of nuclear spins. We consider the
regime of very low magnetic fields (corresponding to fields as low as a
militesla in realistic GaAs and InGaAs dots). We use both the exact numerical
simulations and the analytical theory employing the effective pure dephasing
Hamiltonian. The comparison shows that the latter approach describes very well
the spin echo decay at magnetic fields larger than the typical Overhauser
field, and that the timescale at which this theory works is larger than
previously expected. The numerical simulations are also done for very low
values of electron spin splitting at which the effective Hamiltonian based
theory fails quantitatively. Interestingly, the qualitative difference in the
spin echo decay between the cases of a homonuclear and a heteronuclear bath
(i.e. bath containing nuclear isotopes having different Zeeman energies),
predicted previously using the effective Hamiltonian approach, is still visible
at very low fields outside the regime of applicability of the analytical
theory. We have found that the spin echo signal for a homonuclear bath
oscillates with a frequency corresponding to the Zeeman splitting of the single
nuclear isotope present in the bath. The physics behind this feature is similar
to that of the electron spin echo envelope modulation (ESEEM). While purely
isotropic hyperfine interactions are present in our system, the tilting of the
electron precession axis at low fields may explain this result.
