Theoretical analysis of an atomic spin self-oscillator

We present the analytic solutions for atomic spin self-oscillator with and without rotating wave approximation. The spin ensemble is driven by a linear magnetic field which is produced by its output amplitude multiplied by k and phase shifted by $\varphi $. In appropriate condition, the spin will precess self-sustainably. We obtained analytic solutions for amplitude and frequency of the spin self-oscillator with slow-varying amplitude and phase approximation. Some interesting results are found. First, the setup time of the spin self-oscillator has a characteristic time of 2T$_{\mathrm{1}}$, and T$_{\mathrm{1\thinspace }}$is the longitudinal relaxation time. Second, the oscillating frequency is a complicated function of parameters, including $\varphi $, transverse relaxation time T$_{\mathrm{2}}$, k, oscillating frequency $\omega $ and longitudinal component of magnetic moment Mz. When $\varphi $ is optimized, the oscillating frequency has nothing to do with T$_{\mathrm{2}}$, k, Mz at both transient and equilibrium state. On the other hand, the frequency shift is reverse proportional to T$_{\mathrm{2}}$ if $\varphi $ is not optimized.