Theory of nonlinear optical spectroscopy of electron spin coherence in quantum dots

We study in theory the generation and detection of electron spin coherence in nonlinear optical spectroscopy of semiconductor quantum dots doped with single electrons. In third-order differential transmission spectra, the inverse width of the ultra-narrow peak at degenerate pump and probe frequencies gives the spin relaxation time ($T_1$), and that of the Stoke and anti-Stoke spin resonances gives the effective spin dephasing time due to the inhomogeneous broadening ($T_2^{\ast }$). The spin dephasing time excluding the inhomogeneous broadening effect ($T_2$) is measured by the inverse width of ultra-narrow hole-burning resonances in fifth-order differential transmission spectra.