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Zeeman Degenerate Sideband Cooling in $^{176}$Lu$^+$

We explore degenerate Raman sideband cooling in which neighboring Zeeman states of a fixed hyperfine level are coupled via a two-photon Raman transition. The degenerate coupling between $|F,m_F\rangle\rightarrow |F,m_F-1\rangle$ facilitates the removal of multiple motional quanta in a single cycle. This method greatly reduces the number of cooling cycles required to reach the ground state compared to traditional sideband cooling. We show that near ground state cooling can be achieved with a pulse number as low as $\bar{n}$ where $\bar{n}$ is the average phonon number in the initial thermal state. We demonstrate proof-of-concept in $^{176}\mathrm{Lu}^+$ by coupling neighboring Zeeman levels on the motional sideband for the $F=7$ hyperfine level in $^3D_1$. Starting from a thermal distribution with an average phonon number of 6, we demonstrate near ground-state cooling with $\sim10$ pulses. A theoretical description is given that applies to any $F$ level and demonstrates how effective this approach can be.