Extended Data Fig. 10: Heralded entanglement between two ¹⁷¹Yb ions in the same nanophotonic cavity.

a, The entanglement protocol requires single-qubit Z rotations which cannot be achieved using global microwave driving. Instead, we utilize a differential a.c. Stark shift of the qubits generated by an optical pulse with Rabi frequency Ω, oppositely detuned from the ions’ optical transition frequencies. We embed these pulses in alternate periods of an XY-8 dynamical decoupling sequence, preserving qubit coherence whilst simultaneously accumulating a differential phase³⁹. The sequence consists of 32 π pulses separated by 2τ_s = 5.8 μs; we vary the differential phase by adjusting the a.c. Stark intensity which scales with Ω². b, We verify the a.c. Stark control in two ways. First, we independently initialize Ions 1 and 3 in superposition states, apply the a.c. Stark sequence, and measure the spin populations \(\langle \widehat{X}\rangle \) and \(\langle \widehat{Y}\rangle \). We calculate the difference between the qubits’ phase rotations, Φ_a.c., and plot this for different a.c. Stark intensities, Ω² (solid line). We also probe the differential phase using heralded Bell states, detailed in c (markers), these independently derived measurements exhibit close correspondence. c, After heralding an entangled state on Ions 1 and 3, we apply an a.c. Stark sequence prior to measurement. This adds a Bell state phase and corresponding shift to the parity oscillation of coherence with photon emission time, t₀. We measure this shift for different a.c. Stark intensity values, Ω²; the four parity oscillations correspond to the labelled markers in b. d, We dynamically adjust the a.c. Stark intensity in each experiment to counteract the stochastic heralding phase \(\Delta {\widetilde{\omega }}_{13}{t}_{0}\), rendering the entangled state coherence independent of t₀. We accept photons in a 500 ns window and perform maximum likelihood tomography on the quantum state. We plot the resulting density matrix which has a fidelity of \({\mathcal{F}}=0.763\,\pm \,0.005\), the heralding rate is \({\mathcal{R}}=4.5\) Hz.