Single-Shot Readout of Hole Spins in Ge

The strong atomistic spin–orbit coupling of holes makes single-shot spin readout measurements difficult because it reduces the spin lifetimes. By integrating the charge sensor into a high bandwidth radio frequency reflectometry setup, we were able to demonstrate single-shot readout of a germanium quantum dot hole spin and measure the spin lifetime. Hole spin relaxation times of about 90 μs at 500 mT are reported, with a total readout visibility of about 70%. By analyzing separately the spin-to-charge conversion and charge readout fidelities, we have obtained insight into the processes limiting the visibilities of hole spins. The analyses suggest that high hole visibilities are feasible at realistic experimental conditions, underlying the potential of hole spins for the realization of viable qubit devices.

hydrofluoric acid was performed in order to remove the native oxide. For the gate electrodes Ti/Pd (5/20 nm) were deposited on top of ≈8 nm hafnium oxide created by atomic layer deposition.
All measurements were performed in a dilution refrigerator with a base temperature of 15 mK. A Tektronix AWG5014C arbitrary wave generator was used to apply voltage pulses to the gates and the Zurich Instruments UHFLI lock-in amplifier was used for the readout.
The sample was mounted onto a printed circuit board incorporating RC filters (R = 10 kΩ, C = 10 nF) for the DC lines, the bias tees for the reflectometry (R = 10 kΩ, C = 10 nF) and the fast gate lines (R = 1.8 MΩ, C = 10 nF). The matching circuit consisted of an 820 nH inductor (1206CS-821XJLB) and a varactor (MA46H070-1056) which was biased with 3 V.
The frequency used for the reflectometry readout was 169.5 MHz. The fast gate and the input reflectometry lines were attenuated by 27 dB and 42 dB, respectively. Attenuators were distributed at the different stages of the dilution refrigerator and at room temperature.
The reflected signal was amplified at two stages, once at 4K and once at room temperature.
We used a CITLF2 cryogenic amplifier, a ZX60-33LN-S+ room temperature amplifier and a ZX30-9-4-S+ directional coupler. The power of the RF signal on the lock-in output was -35 dBm.

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Relaxation rate vs magnetic field -logarithmic plots log (B (mT)) Figure S1: Spin relaxation rate vs magnetic field -logarithmic plots. a)-b) Log-log plots of the spin relaxation rate vs magnetic field for the first (a)) and the second break (b)), together with the linear fit. The extracted slope of 1.5 ± 0.1 (1.4 ± 0.3) for the first (second) breaks gives the spin relaxation rate dependence on the magnetic field.

Averaged spin relaxation measurements
The spin relaxation time can be also extracted from the averaged single-shot measurements.
For each loading time we integrate the averaged charge sensor RA (light blue area in the Fig. S2a)). The integrated values follow as well an exponential decay as the loading time is increasing and the obtained relaxation times are in good agreement with those obtained with the single-shot analysis (Fig. S2b)).

Deterministic loading of the spin-down state
If already during the load stage the electrochemical potentials are in the configuration that µ ↓ < µ SHT < µ ↑ , only the spin-down hole can be loaded. With a slightly modified pulsing sequence (Fig. S3a)) we show that we are able to deterministically load the spin-down state. 1 In this experiment the pulsing sequence has four different stages: load, plunge, read and empty and it is similar to the three-stage pulsing sequence used for the read level calibration described in the main text. The difference is that now the amplitude of the load stage is S-3 varied and an additional stage is added after the load stage in order to be sure that the dot is loaded. Read and empty stages stay the same. The load stage is varied from the configuration where the electrochemical potentials of both spin-up (µ ↑ ) and spin-down (µ ↓ ) holes are above the electrochemical potential of the SHT (µ SHT ) to the configuration where both µ ↑ and µ ↓ are below the µ SHT . In the configuration where µ ↑ > µ SHT > µ ↓ , only a spin-down hole can be loaded. Accordingly, no spin signature is measured in the read phase as can be seen in Fig. S3b). The voltage range for which only the spin-down state is loaded corresponds to the Zeeman splitting.

Lever arm and effective hole temperature
In similar systems 1,2 the lever arm was extracted by mapping the occupation probability in the reservoir. In this method the SHT current is measured along a Coulomb peak break (dashed line in Figure S4b function where V eff = ∆V 2 G(QD) + ∆V 2 G(SHT) and a,b,c and d are fitting parameters. The fitting parameter c has units of energy and when divided with the Boltzmann constant k B gives a parameter in the units of temperature, containing the lever arm α. Repeating this fit for different temperatures leads to the plot shown in Fig. S4c) from which the effective hole temperature and the lever arm can be extracted.
Despite the rather long averaging (integrating the current value at each point for 200 ms), the measurements were very noisy for higher temperatures, making the fit rather difficult.
To overcome this, each line trace was smoothed with the Savitzky-Golay filter (first order polynomial, for a window size =3). The filtering caused additional broadening of lines, leading to large errors as can be seen in Fig. S4c). Nevertheless an effective hole temperature of 300-400mK and and a lever arm between 0.02 and 0.03 can be estimated.  readout fidelity as will be seen below.

Charge readout fidelity
The model consisted of the following steps: (i) Random assignment of the spin state For a spin in the excited state (spin-up), a pulse with a height and width which corresponds to the experimental RA was generated. Its width was determined by the experimentally extracted tunnelling rates: the tunnelling-out rate of a spin-up hole (Γ ↑,out ) and the tunnelling-in rate of a spin-down hole (Γ ↓,in ). The former was extracted from the fit to the histogram (Fig. S5b)) representing the detection times of the spin-up hole (t ↑ (det) in Figure S5a), where t ↑ (decay) is equal to 1/(Γ ↑ + T −1 1 ). The exponential fit gives a decay rate t ↑ (decay) −1 equal to (0.125 ± 0.007) µs −1 , which gave us finally Γ ↑,out = (0.113 ± 0.007) µs −1 . Γ ↓,in was extracted from the fit to the histogram (Fig. S5c)) representing the peak widths (t ↓ (det) in Figure S5a)): Γ ↓,in = (0.125 ± 0.005) µs −1 . All values were extracted for the normalized threshold = 0.70.
After the pulse was generated, Gaussian noise was added to this trace with the noise S-6 amplitude being a free parameter.
For a spin in the ground state (spin-down), no pulse was generated; only Gaussian noise is added to the trace.
(ii) Application of a numerical filter of 4th order to the above trace.
(iii) Extraction of the peak value of the reflection amplitude. This value was then added to an array which was corresponding either to a spin-down or a spin-up state.
(v) From each array a normalized histogram was created.
(vi) An optimal Gaussian function was found for each of the two calculated histograms, giving the spin-up (N ↑ ) and the spin-down probabilities (N ↓ ) (pink and purple traces in Fig. S5d)).
In order to be able to compare the two simulated Gaussian functions with the experimental data, a histogram of the experimental reflection amplitude peak values was plotted. Also here the data were normalized. To the bimodal experimental distribution two optimized Gaussian functions were fitted. Those were then compared to the two Gaussian functions obtained from the model (Fig. S5d)). The charge readout fidelities as a function of the threshold value (see the main text, Figure 4c inset) can be extracted from where F ↓ is the charge readout fidelity for a spin-down state and F ↑ is a charge readout S-9