Spin squeezed states have applications in metrology and quantum information processing. While there has been significant progress in producing spin squeezed states and understanding their properties, most spin squeezing research to date has focused on ensembles of qubit spins. We explore squeezed state production in an ensemble of spin f>1/2 alkali atoms (qudits). Collective interactions are achieved through coherent quantum feedback of a laser probe, interacting with the ensemble through the Faraday interaction. This process can be enhanced with further control of the atomic qudits, both before and after the collective interaction. Initial preparation increases the collective squeezing parameter through enhancement of resolvable quantum fluctuations, but comes at a price of increased decoherence. After the collective interaction, qudit control maps the generated entanglement to different pseudo-spin subspaces where it ismetrologically useful, e.g., the clock transition or the stretched state formagnetometry. In the latter case, additional internal control can be used to squeeze the individual atoms, further enhancing the total squeezing in a multiplicative manner. These considerations highlight the unique capabilities of our platform: we are able to transfer coherences and correlations between subspaces and integrate control tools to explore a wider variety of nonclassical states, with ultimate application in sensors or other quantum information processors.
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