Extended Data Fig. 5: Hyperfine interaction between the qubits and the 73Ge nuclear spins.
From: Sweet-spot operation of a germanium hole spin qubit with highly anisotropic noise sensitivity
a,b, Normalized charge sensor signal for a Hahn echo experiment as a function of the total free evolution time 2τ (colour map in left subpanel) and the qubit frequency fQi (plot in right subpanel) as a function of the elevation angle of the magnetic field for Q1 (a) and Q2 (b). B = 89 mT is kept constant throughout the measurement. For any θB away from the \({x}^{{\prime} }{y}^{{\prime} }\) plane of \({\overleftrightarrow{g}}\), we observe the collapse-and-revival characteristic for a well-defined spectral component acting on the qubit. Due to the small tilt between the two qubit g-tensors, a common hyperfine sweet spot is defined by the intersection of two differently tiled ellipsoids in the lab frame, at ϕB = 97.5°, θB = 89.7°. Selecting this magnetic field orientation allows to operate both qubits in their respective hyperfine sweet planes simultaneously. Here, we find \({T}_{2,{{{\rm{Q}}}}1}^{\,{H}}=8.1(2)\,\mu {{{\rm{s}}}}\) and \({T}_{2,{{{\rm{Q}}}}2}^{\,{H}}=11.5(6)\,\mu {{{\rm{s}}}}\) for B as specified. c,d, Normalized charge sensor signal for a CPMG sequence with respectively 2 (c), and 8 (d) decoupling pulses, as a function of the spacing between two subsequent decoupling pulses τ and θB. Nτ is the total evolution time. The magnetic field strength is B = 133 mT and the azimuth angle is ϕB = 97.5°. The inset displays the fit to the data. These data complement the dataset displayed in Fig. 5c,d of the main text. e, Width of the hyperfine line σGe-73 as extracted from the CPMG coherence as a function of B (Extended Data Fig. 6, ϕB = 0°, blue markers), and as extracted from the CPMG coherence as a function of θB (Fig. 5c,d of the main text, ϕB = 97.5°, black markers). Data are presented as the fitted values of σGe-73 with error bars indicating the 1σ uncertainty of the fit. We find σGe-73 to be independent of θB in the experimental range, but observe an offset for the two different azimuth directions. Dashed lines correspond to the average \({\overline{\sigma }}_{{{{\rm{Ge}}}}{{\mbox{-}}}73}=9.4\) kHz for ϕB = 97.5° and \({\overline{\sigma }}_{{{{\rm{Ge}}}}{{\mbox{-}}}73}=17\) kHz for ϕB = 0°. f, The intensity of the 1/f component of the spectrum at 1 Hz extracted from the data presented in Extended Data Fig. 6. Data are presented as the fitted values of S0,E with error bars indicating the 1 s.d. uncertainty of the fit. We find \({S}_{0,{{{\rm{E}}}}}\propto {B}^{2}\propto {f}_{{{{\rm{Q}}}}2}^{\,{2}}\), as expected for charge noise with a 1/f spectrum.
