Fig. 2. Er³⁺ electron spin coherence time T₂.

(A) Measured normalized Hahn echo integral A_e (green full circles) as a function of the delay 2τ between the first pulse and the echo, at 10 mK cryostat temperature, φ = 47^∘ and high pulse power (β = 700 ns^−1/2). Each data point is magnitude averaged over 60 measurements with a repetition time of 4 s. The solid black line is a fit to $A_e^2=exp\{-2{(2\mathrm{\tau }/T_2)}^x\}+C$ (with the offset C subtracted from the data and the fit), yielding T₂ = 23.2 ± 0.5 ms and x = 2.4 ± 0.1. Open red circles are the result of a cluster-correlation expansion (CCE) simulation of the nuclear spin bath for the same field orientation. The dashed black line is a fit to the simulation, yielding T_{2, sim} = 27.2 ms and x_sim = 2.74. (B) Measured coherence time T₂ and exponent x (green diamonds) as a function of the cryostat temperature. The red dashed line is the result of the CCE simulation. The blue squares result from a second fit of the data as $A_e^2=exp\{-2[{(2\mathrm{\tau }/T_{2,\text{sim}})}^{x_{\text{sim}}}+{(2\mathrm{\tau }/T_2)}^x]\}+C$ to extract the net decoherence effect of spectral diffusion due to paramagnetic impurities.