Analytic solutions of Bloch equations with phase term for two-pulse photon echo by the phase modulation pulses

In this paper, a method is proposed to obtain the analytical solutions of Bloch equations with phase term by using the elementary matrix transform. The effect of the initial phase on the components of Bloch vector is discussed. The result shows that by considering the initial phase of the input pulse, the amplitudes of the in-phase component and the in-quadrature component can be dynamically controlled, meanwhile the population inversion is almost immune to the initial phase. Additionally, the signal electric field expressions of the phase effect of two-pulse photon echo (2PE) is derived, which is also modulated by the initial phase.