Spin–valley protected Kramers pair in bilayer graphene

Device fabrication

All the presented data were measured on a single device fabricated on a van der Waals heterostructure stacked using standard dry transfer techniques. The stack consists of a 35-nm-thick top hBN layer, the Bernal BLG sheet, a 28-nm-thick bottom hBN and a graphite back gate layer. The ohmic contacts to the BLG are one-dimensional edge contacts. To form QDs, we utilize the two overlapping layers of Cr/Au (3 nm/20 nm) metallic gates shown in Fig. 1a. The upper gate layer consists of finger gates, 20 nm wide and 60 nm apart. These gates are deposited on top of a 26-nm-thick insulating aluminium oxide layer. The widths of the channels defined by the split gates are 40 nm for the left channel and 75 nm for the right channel. We tune the dot into the few-hole regime as shown in Fig. 1b. We attribute the set of parallel discrete steps to the boundaries of the regions of the charge stability diagram with a constant N_h = 0, 1, 2, … as indicated in the figure. We identify the first hole transition around V_P1 ≈ 13 V (marked by the red circle), as no charge transition line appears at higher plunger gate voltages. Additional lines with markedly different couplings to MS and P1 gates correspond to unintended dots formed due to nearby charge inhomogeneities.

By applying a large negative voltage V_BG = −7.4 V to the global back gate we induce a large displacement field (D = −0.9 V nm⁻¹), which opens a band gap in BLG of the order of 100 meV. This field allows us to notably decouple the dot from the reservoirs, achieving tunnelling rates to the leads of just tens of hertz.

Measurement set-up

The sample is mounted on the mixing chamber of a Bluefors LD400 dilution refrigerator, which has a base temperature of 9 mK and an independently extracted electron temperature of around 30 mK. All the measurement and control electronics are located at room temperature and are connected to the device via 24 DC lines. Each line is low-pass filtered using RC filters mounted on the mixing chamber plate, with a time constant of approximately 10 μs. For DC biasing of gates and ohmic contacts, we use in-house-built low-noise voltage sources with a cutoff frequency of 7 Hz, except for the pulsing gate P1 line, which is left unfiltered and has a cutoff frequency of 1,200 Hz. The DC plunger gate voltage V_P1 is combined with the pulsing voltage V_pulse using a 2 MΩ/2.7 kΩ divider at room temperature. The sensor current is amplified using an in-house-built current-to-voltage converter with a 10 MΩ feedback resistor, followed by a ×100 analogue voltage amplifier and an analogue low-pass filter with a cutoff frequency of around 10 kHz. Sensor time traces are recorded using an NI-6251 data acquisition card with a sampling frequency of 20 kHz.

Elzerman sequence

We start the three-level single-shot readout sequence with the empty dot subjected to the Load phase (see Fig. 1c): the pulse level shifts all three states below the E_F of the leads. A hole from the lead can tunnel into any of the three states with almost equal probabilities¹⁹ of 1/3. During the Load phase of duration T_load, the sensor current drops abruptly, indicating that the dot has been filled, as shown in the typical time traces in Fig. 1d. In the following Read phase (see Fig. 1c), the second pulse level puts the ES of interest above E_F in the lead while the GS stays below. If a hole remains in the ES after loading and during the Read phase without relaxing, then, at some random time governed by the tunnelling-out rate, it will tunnel to the leads and thereafter the GS of the dot will be occupied again through a tunnelling-in process. This in-and-out tunnelling manifests itself as a step in the sensor current shown in Fig. 1d (green trace). In contrast, no step is observed when the hole is initially loaded into the GS, or when it relaxes before it can tunnel out during the Read phase (blue trace). Once a hole arrives in the GS, it blocks any further transitions since the number of resonant unoccupied lead states is exponentially suppressed by the low T_e ≈ 30 mK (ref. ¹⁹). The final pulse level empties the dot by moving all states above E_F of the leads (see Fig. 1c), and after that the sequence starts again.

Postselection of sensor traces

We digitize current sensor traces using the two-threshold procedure described in ref. ¹⁹. We divide the entire set of N digitized current traces into four groups, N = (N_e, N_g, N_nl, N_er). Here, N_nl is the number of traces where a hole was not loaded during the Load phase. N_g is the number of traces with zero steps in the Read phase, which we label as a hole being in the GS. N_e is the number of traces with a single step during the Read phase, which we define as a hole being in the ES. N_er is for the rest of the traces, which we label as errors, as they exhibit multiple steps in the Read phase. Extended Data Fig. 1 shows a typical sensor current trace for each group along with its digitized version. The typical proportions are as follows: 80.89% for the GS, 9.41% for the ES, 0.39% for combined errors and 0.58% for traces that are not loaded. The two most common types of error causing multiple-step traces are digitization errors and random thermal/charge noise steps. The first is due to the fact that the sensor trace does not exhibit enough points in both charge states to reliably determine the threshold for digitization. Thermal steps are exponentially suppressed by low electron temperature, while major charge jumps occur on a very long timescale of minutes and hours. We simply discard all error traces from the dataset, assuming that the errors are uncorrelated with whether the ES or GS is occupied in the beginning of the Read phase. Note that by discarding clearly ES counts, although marked as thermal step errors in Extended Data Fig. 1, we underestimate the ES probability.

Calculation of ES probability

We perform relaxation time measurements in a regime with low Γ_out ≈ Γ_in ≈ 15 Hz, which are comparable to the measured T₁ at certain magnetic fields. In this regime, the occupation of the ES after the short loading times T_load ≈ Γ_in⁻¹ is limited not by the relaxation but by the tunnelling-in rate. Nevertheless, even for one of the shortest extracted T₁ ≈ 40 ms, we demonstrate that the decay of the calculated renormalized ES probability P^e can be fitted by a simple exponent. We consider the state readout of two double-degenerate Kramers pairs at B_∥ = 900 mT, as shown in Extended Data Fig. 2a. As established in previous measurements^9,26, the total occupations of the ES, GS and non-loaded states after T_load are

where \({\varGamma }_{{\rm{in}}}^{{\rm{e}}}\) and \({\varGamma }_{{\rm{in}}}^{{\rm{g}}}\) are tunnelling-in rates to the ES and the GS respectively. We extract the sum of tunnelling-in rates \({\varGamma }_{{\rm{in}}}^{{\rm{e}}}+{\varGamma }_{{\rm{in}}}^{{\rm{g}}}=4{\varGamma }_{{\rm{in}}}=57.6\,{\rm{Hz}}\) and the tunnelling-out rate \({\varGamma }_{{\rm{out}}}^{{\rm{e}}}={\varGamma }_{{\rm{out}}}^{{\rm{g}}}={\varGamma }_{{\rm{out}}}=12\,{\rm{Hz}}\) by fitting the exponential decay and rise of the average dot occupation during the Load and Empty phases respectively, as shown in Extended Data Fig. 2a. During the Read phase, we measure the ESs with efficiency^9,26

where T_read is the time spent in the Read phase and \({\varGamma }_{{\rm{out}}}^{{\rm{e}}}\) the intrinsic tunnelling rate.

The resulting full and renormalized probabilities are simply products of the two efficiencies:

In Extended Data Fig. 2b, we plot the measured P^e = N_e/(N_e + N_g) along with theoretical curves from equation (3), using \({\varGamma }_{{\rm{in}}}^{{\rm{e}}}=22.2\,{\rm{Hz}}\) and T₁ = 41 ms. As expected, at short T_load, the full and renormalized occupations markedly differ. However, as long as \({T}_{{\rm{load}}} > 60\,{\rm{ms}}\approx 3/({\varGamma }_{{\rm{in}}}^{{\rm{e}}}+{\varGamma }_{{\rm{in}}}^{{\rm{g}}})\) the two curves match well, and their downward trend can be successfully approximated by the simple exponential function \({P}_{\exp }^{\,{\rm{e}}}({T}_{{\rm{load}}}) \approx {\mathrm{e}}^{-{T}_{{\rm{load}}}/{T}_{1}}\) as shown with the dashed line. With increasing relaxation time, the renormalized probability becomes closer to the exponent. The best least-square weighted exponential fit of the experimental points yields T₁ = 45 ± 3 ms, which is within 10% of the value given by the correct renormalized probability expression.

Dark counts

In Extended Data Fig. 3, we rule out potential thermally activated and flicker noise origin (dark counts) of steps by comparing the step distribution at notably different T_load = 0.2 s and 12.8 s. The average of all the single-shot I_sens traces identified as the GS (purple) shows a flat behaviour during the Read phase. In contrast, the average of ESs (green) exhibits a bump that rises on a timescale of 1/Γ_out and decays within 1/Γ_in (ref. ²⁵). The exponential decrease in step density to zero, governed by the tunnelling-out rate, allows us to eliminate the charge noise origin of the steps, as one would anticipate a constant distribution of random charge jumps over the Read phase. Another potential source of false-positive counts could be thermally activated steps. Since we only count single steps, the distribution might appear similar. However, with average tunnelling-in and out times of 1/Γ_in/out = 75 ms and a Read duration of T_read = 350 ms, we expect a Poissonian distribution P(k) = λ^k e^−λ/k! for the probability of a certain number of tunnelling events k, with an average of λ = 4.7. The probability of having a single step (corresponding to two tunnelling events) is P(k = 2) ≈ 10%, while the probability of having no step is P(k = 0) ≈ 1%, meaning that the remaining 88% of shots should be discarded as errors with around 1% of all shots being not-loaded. In fact, for T_load = 0.2 s, we identify 45.9% of shots with a single step, 51.9% of shots with no steps, 0.9% as not-loaded shots and only 1.3% as multiple-step errors, effectively ruling out thermal activation as the origin of the steps.

Readout performance

For quantum information applications, it is crucial to identify the factors limiting the readout performance. We analyse 2,000 single-shot traces to extract the histogram of the peak value of the sensor current during the Read phase, as illustrated in the inset of Fig. 3b. The well-separated Gaussian peaks representing the detection of the GS and the ES, with a signal-to-noise ratio of approximately 4.9, result in an electrical readout fidelity exceeding 99.9% (refs. ^25,27). The negligible electrical readout error suggests that the readout performance is limited by the spin/valley-to-charge conversion²⁷. We find that, despite the deliberate reduction of the tunnelling rates, the observed notably long spin–valley relaxation time still just meets all the minimum requirements²⁷ for achieving a fault-tolerant 99% readout visibility threshold⁴⁰, as follows. (1) A large energy splitting more than 13 times larger than electron temperature. In our experiment Δ₁ ≈ 55 μeV ≳ 13k_BT ≈ 52 μeV. (2) A tunnelling-out time 100 times faster than the relaxation time. In our experiment, \({T}_{1}^{({\rm{sv}})}=30\,{\rm{s}}\gtrsim 100/{\Gamma }_{{\rm{out}}}=8\,{\rm{s}}\). (3) A sampling rate for data acquisition 12 times larger than the reloading rate. In our experiment, Γ_s = 10 kHz > 12Γ_out = 150 Hz.

Ruling out pure spin relaxation

Here we explicitly rule out the argument that the observed long spin–valley relaxation times could be interpreted as fully dominated by pure spin relaxation. Indeed, spin T₁ times from a few seconds to almost a minute¹⁰ have also been observed in both silicon and GaAs QDs⁴¹. Moreover, previous studies do indeed report strong power-law dependences⁴¹, which could, at first sight, explain the marked drop in relaxation times as we transition from the pure spin to the spin–valley blockade regime while simultaneously shrinking the energy splitting.

To eliminate this argument, in Extended Data Fig. 4 we plot the data from Fig. 1e as a function of the energy splitting between the first ES and the GS. Additionally, we include the spin relaxation rate T₁⁻¹ data at higher magnetic fields (1.5–3 T) from ref. ⁴, measured in an analogous weakly coupled (Γ ≈ 350 Hz) single-QD BLG device using the same Elzerman technique. The energy splitting dependence of T₁ extracted from both experiments is best described by a power law, T₁⁻¹ ∝ ΔE^2.5 (highlighted by the green solid line). The observed power is in line with ∝ΔE³⁻⁷, as seen in GaAs and silicon spin qubits, and originates from a combination of electron–phonon relaxation and various spin-mixing mechanisms such as hyperfine interaction, spin–orbit coupling and spin–valley mixing⁴¹. In the case of BLG, spin-mixing mechanisms are not well known, nor has any theoretical prediction for spin T₁ dependence on the magnetic field been made while considering the two-dimensional nature of phonons. However, calculations for single-layer graphene show that the power law T₁⁻¹ ∝ ΔE^2–4 should not differ much from the mentioned three-dimensional platforms. Taking this into account, if we assume that long spin–valley relaxation is solely dominated by spin relaxation, the trend correlating spin and Kramers points reveals a remarkable power of ∝ΔE²⁰ as marked by the dashed blue line, although fitting a power law on such a restricted energy range should be approached with caution. Hence, a more plausible explanation for such a marked change in T₁ would be the dual protection afforded by simultaneous spin–valley blocking when operating within the Kramers doublet. The continuous connection between the spin–valley (blue dots) and spin (green dots) relaxation data points can be attributed to the finite valley mixing term. Indeed, even small values of \({\Delta }_{{{\rm{K}}}^{+}{{\rm{K}}}^{-}} < 2\,{\rm{neV}} \approx 0.5\,{\rm{MHz}}\) (ref. ³) can provide effective mixing, considering that our shortest loading times are in the tens of milliseconds. Additionally, the spin or valley relaxation channel data points align with the pure spin trend, indicating that valley relaxation is significantly longer than spin relaxation, in agreement with previous observations³.