Fast Hole Tunneling Times in Germanium Hut Wires Probed by Single-Shot Reflectometry

Heavy holes confined in quantum dots are predicted to be promising candidates for the realization of spin qubits with long coherence times. Here we focus on such heavy-hole states confined in germanium hut wires. By tuning the growth density of the latter we can realize a T-like structure between two neighboring wires. Such a structure allows the realization of a charge sensor, which is electrostatically and tunnel coupled to a quantum dot, with charge-transfer signals as high as 0.3 e. By integrating the T-like structure into a radiofrequency reflectometry setup, single-shot measurements allowing the extraction of hole tunneling times are performed. The extracted tunneling times of less than 10 μs are attributed to the small effective mass of Ge heavy-hole states and pave the way toward projective spin readout measurements.

: Schematic showing the setup of the simulated experiment was generated from the UHFLI out 1 port and fed to the LO port of the mixer in order to imitate the reflectometry signal. The result of the multiplication of these two signals, was fed to the UHFLI input 2 from the mixer RF port and measured with its scope (see Figure S1). The scope sampling rate was set to 900 MHz. To observe the influence of the UHFLI demodulator, the signal was fed through it with a low pass filter of 20 kHz (4th order) (Figure S2 c) and d)). The threshold for the 'tunnel time measurement' for this simulation was set to 100 mV. We can see that an additional delay of 15 µs , originating from the slow rise time, is added for both the left (5 µs) and the right (7 µs) tunnel event equally, proving that the filter behavior is constant and simply adds a constant offset in the measurement.
However, due to the fact that the pulse duration, marked as L, in the experiment described in Figure 4 of our manuscript is more than 30 times the rise time, all tunneling events are observed. Note: The amplitudes of all measurement simulations shown here are 5-6 orders of magnitude higher than in the real experiment. The reason for this is that for the proper operation of the mixer, the signal amplitudes on it's inputs need to have certain value.
We have repeated the same simulation for 30 kHz, 50 kHz and 100 kHz low pass filter bandwidths and the final difference between the two signals simulating the tunneling events was always 2 µs (see Figure S3), as it would be for an infinite bandwidth setup.

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Voltage ( c) d) Figure S2: Measurements of the simulated experiment proving the validity of our experimental approach. a) In blue the square waveform corresponding to the pulse for loading a hole is shown. The second square waveform, multiplied with a 100 MHz sinusoidal signal, simulating the tunneling event is shown in green. The high level of this second waveform is delayed for 5 µs simulating the time it takes for a hole to tunnel into the QD after the blue square waveform has been applied. b) Similar to a) but here, the second waveform is delayed for 7 µs. In the inset a zoom-in of the 'reflectometry carrier signal' of 100 MHz modulated with the square waveform, is shown. c) -d) Demodulated signal measured after the low pass filter of 20 kHz (4th order), simulating tunnel events which take place after delay times of 5 µs and 7 µs, respectively. The simulation reveals that the slow rise time adds and additional delay time of 15 µs for both tunnel events.

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Summarizing, the low-pass filter bandwidth does not influence our measurement, since we measure only the relative delay. The consequence of measuring with a smaller bandwidth is just a shift of the whole histogram on the x-axis. The shift of the fitted histograms is inversely proportional to the the low-pass filter bandwidth: the higher the bandwidth, the closer to zero time is the histogram. The histograms shift to smaller delay times for higher bandwidth but the extracted tunneling times are the same. This can be seen in Figure 4 of the manuscript.