Doppler Amplification of Motion of a Trapped Three-Level Ion

The system of a trapped ion translationally excited by a blue-detuned near-resonant laser, sometimes described as an instance of a phonon laser, has recently received attention as interesting in its own right and for its application to non-destructive readout of internal states of non-fluorescing ions. Previous theoretical work has been limited to cases of two-level ions. Here, we perform simulations to study the dynamics of a phonon laser involving the $\mathrm{\Lambda }$-type ${}^{138}B{a}^{+}$ ion, in which coherent population trapping effects lead to different behavior than in the previously studied cases. We also explore optimization of the laser parameters to maximize amplification gain and signal-to-noise ratio for internal state readout.