Exchange anisotropies in microwave-driven singlet-triplet qubits

Abstract

Hole spin qubits are emerging as the workhorse of semiconducting quantum processors because of their large spin-orbit interaction, enabling fast, low-power, all-electric operations. However, this interaction also causes non-uniformities, resulting in site-dependent qubit energies and anisotropies. Although these anisotropies enable single-spin control, if not properly harnessed, they can hinder scalability. Here, we report on microwave-driven singlet-triplet qubits in planar germanium and use them to investigate spin anisotropies. For in-plane magnetic fields, the spins are largely anisotropic and electrically tunable, allowing access to all transitions and coherence times exceeding 3 μs are extracted. For out-of-plane fields they have an isotropic response. Even in this field direction, where the qubit lifetime is strongly affected by nuclear spins, we find 400 ns coherence times. Our work adds a valuable tool to investigate and harness the spin anisotropies, applicable to two-dimensional devices, facilitating the path towards scalable quantum processors.

Fig. 1. Spin blockade and microwave spectroscopy in a two-spin system.

a Device schematics showing the gate layout. The two bottom rightmost gates are kept at 0 V. Fast pulses are applied to gates PL and PR to change the charge occupation of the DQD, while microwave bursts are applied to the right barrier BR to induce spin transitions. The in-plane magnetic field B_y is applied perpendicular to the axis of the DQD. b Stability diagram of the investigated transition. The dark blue triangle in the (2,2) regime corresponds to the Pauli Spin Blockade region. This colour code is maintained throughout the paper, where dark corresponds to blocked states and light to unblocked. The red dashed line indicates the device’s operational detuning line. c, d Charge sensor amplitude as a function of B_y, ε and f_MW. The increase in the signal corresponds to a higher triplet return probability, reflecting a change in the system’s level population. In c the dependence is shown at detuning ε = 4.6 meV and in d at B_y = 0 mT. The solid lines are the transition frequencies obtained when using the extracted parameters $ḡ$, δg_∥ and δg_⊥. The inset in c shows a low-field high-resolution zoom-in. In both cases, the duration of the microwave burst is 10 μs and the readout time 1 μs. From the transition frequency in d we can extract the exchange interaction versus detuning by converting the voltage amplitude of the pulse into energy with the lever arms (see Supplementary Fig. 9a). By fitting (orange dotted line) to the expression $J=\left(-\epsilon +\sqrt{{\epsilon }^2+8t^2}\right)/2$, we obtain a tunnel coupling of t_c/h = 6.6 ±  0.1 GHz. e Energy diagram of the four investigated states versus detuning. The anticrossing between the $∣S⟩$ and $∣T_{-}⟩$ states occurs at ε_AC, having two consequences: the ground state changes and defines the minimum energy difference between the two lowest energy states, ${\mathrm{\Delta }}_{S{T}_{-}}$. f Schematic explaining the quantization axes (grey and brown), the addition and difference of the Zeeman vectors (black) as well as the projections δb_∥ and δb_⊥ (orange), which enable spin driving via exchange modulation.

Fig. 2. Electrical tunability of ḡ, δg_⊥, δg_∥ and the quantization axes tilt.

a, b Magnetic field spectroscopy for two different voltage configurations of the electrostatic gates. The coloured solid lines are the transitions corresponding to the parameters written in the legend. Panels c, d show the voltage differences of each configuration with respect to the voltage configuration 1 (Fig. 1c). e Extracted parameters for each electrostatic configuration, including the misalignment δθ, demonstrating electrical tunability.

Fig. 3. Detuning spectroscopy and Rabi frequency dependences.

a Plot showing the transition frequency f_MW versus detuning at an in-plane magnetic field of 70 mT. The anticrossing (black arrow) occurs at ε_AC = 1.8 meV and has an amplitude of ${\mathrm{\Delta }}_{S{T}_{-}}=175$ MHz. We note that after the avoided crossing, the green transition corresponds to a double spin-flip from T₋ → T₊, which usually is not observed in spin qubit systems. At low ε, we observe subharmonic transitions from T₋−S and T₋−T₀ indicated by the red and purple arrows, respectively. This indicates that the non-linearities of the driving mechanism are non-negligible. b and c Rabi frequency versus magnetic field and detuning, respectively, following the same colour code as in Fig. 1e. The solid lines in b, c represent the exact solution for the Rabi frequencies with the parameters extracted from the Hamiltonian. The analytical expressions in the limits J ≪ ∣δb∣ and J  ≫ ∣δb∣ can be found in the “Methods” section. In b and for higher fields, we note a discrepancy between the experimental data and the numerical solution as the field increases, which can be at least partially attributed to the larger cable attenuation, i.e. smaller power arriving at the device, at higher frequencies.

Fig. 4. Out-of-plane magnetic field spectroscopy and Rabi frequency dependences.

a Out-of-plane magnetic field spectroscopy at ε = 2.3 meV. The solid lines show the spin transitions from the singlet state to the triplets. The dashed lines correspond to the subharmonic transitions with half and a third of the frequency of the transitions. b and c show the measured Rabi frequencies as a function of B_z and ε, respectively. The solid lines are the predicted f_Rabi considering only exchange modulation. The error bars representing the standard deviation are smaller than the markers.

Fig. 5. Inhomogeneous dephasing times.

a Inhomogeneous dephasing time for the three transitions at in-plane field as a function of detuning. The colour code of each transition remains the same as previously described. Solid lines represent the fitted $T_2^{*}$ values for each transition, while the dashed lines indicate the dephasing time expected if only the detuning noise contribution is considered. The purple and green dashed lines overlap and the peak predicted by the red fit (corresponding to the anticrossing) is not observed experimentally. Each measurement for extracting $T_2^{*}$ has been integrated for 20 min. The fitting parameters can be found in the Supplementary Note 9. b Detuning susceptibility extracted from the derivative of the Larmor frequency of each transition with respect to ε. The orange colour refers to the out-of-plane S−T₀ transition at 12.5 mT. The inset shows a zoom-in at large detunings. c Inhomogeneous dephasing time for the S−T₀ transition at out-of-plane field as a function of detuning, highlighting that the dephasing time is limited either by Zeeman or detuning noise. d Inhomogeneous dephasing time for the S−T₀ transition at ε = 7.4 meV as a function of B_z, demonstrating an insensitivity to the magnetic field value. The error bars correspond to the standard deviation of the fit.