Unpolarized, incoherent repumping light for prevention of dark states in a trapped and laser-cooled single ion

Many ion species commonly used for laser-cooled ion-trapping studies have a low-lying metastable ²D_3/2 state that can become populated due to spontaneous  emission from the ²P_1/2 excited state. This requires a repumper laser to maintain the ion in the Doppler cooling cycle. Typically, the ²D_3/2 state, or some of its hyperfine components if the ion has nuclear spin, has a higher multiplicity than the upper state of the repumping transition. This can lead to dark states, which have to be destabilized by an external magnetic field or by modulating the polarization of the repumper laser. We propose using unpolarized, incoherent amplified spontaneous emission (ASE) to drive the repumping transition. An ASE source offers several advantages compared to a laser. It prevents the buildup of dark states without external polarization modulation even in zero magnetic field, it can drive multiple hyperfine transitions simultaneously, and it requires no frequency stabilization. These features make it very compact and robust, which is essential for the development of practical, transportable optical ion clocks. We construct a theoretical model for the ASE radiation, including the possibility of the source being partially polarized. Using ⁸⁸Sr⁺ as an example, the performance of the ASE source compared to a single-mode laser is analyzed by numerically solving the eight-level density-matrix equations for the involved energy levels. Finally, a reduced three-level system is derived, yielding a simple formula for the excited-state population and scattering rate, which can be used to optimize the experimental parameters. The required ASE power spectral density can be obtained with current technology.