Coherence Scaling in High-Dimensional Qudits - Experimental Results with Barium-137

The angular momentum eigenstates of the unpaired electron in trapped ¹³⁷Ba⁺ ions offer a promising pathway toward high-fidelity qudit (d > 2) encoding. Due to the non-zero nuclear spin (I = 3/2), the 6S_1/2 and 5D_5/2 manifolds contain, respectively, 8 and 24 non-degenerate levels at intermediate magnetic fields (~ few Gauss).

Leveraging recent developments in state preparation techniques for trapped ion quantum computing, we demonstrate high-fidelity (>99.5%) state preparation and measurement results over 25 basis states, the maximal measurable qudit encoding across the 6S_1/2 and 5D_5/2 manifolds in ¹³⁷Ba⁺. We demonstrate coherent control over this enlarged Hilbert space by performing Ramsey-type coherence probing measurements (generating many-state superpositions and probing phase-sensitive population recovery) and benchmark the scaling of decohering effects with increasing qudit dimension. We discuss the largest contributors to error in our system and the steps needed to maintain high-dimensional coherence (as measured via the contrast of Ramsey-type measurements) in this qudit implementation.

This work establishes the feasibility of using trapped ions for large-qudit (d>10) quantum computation, which is a promising alternative approach to expanding the Hilbert space in trapped ion quantum computing.