Ultrafast coherent control of a hole spin qubit in a germanium quantum dot

Operation speed and coherence time are two core measures for the viability of a qubit. Strong spin-orbit interaction (SOI) and relatively weak hyperfine interaction make holes in germanium (Ge) intriguing candidates for spin qubits with rapid, all-electrical coherent control. Here we report ultrafast single-spin manipulation in a hole-based double quantum dot in a germanium hut wire (GHW). Mediated by the strong SOI, a Rabi frequency exceeding 540 MHz is observed at a magnetic field of 100 mT, setting a record for ultrafast spin qubit control in semiconductor systems. We demonstrate that the strong SOI of heavy holes (HHs) in our GHW, characterized by a very short spin-orbit length of 1.5 nm, enables the rapid gate operations we accomplish. Our results demonstrate the potential of ultrafast coherent control of hole spin qubits to meet the requirement of DiVincenzo's criteria for a scalable quantum information processor.


Introduction
Perfecting the quality of qubits hinges on high fidelity and fast single-and two-qubit gates.
Electron spin qubits in Silicon (Si) quantum dots (QD) are considered promising building blocks for scalable quantum information processing 1-5 , with long coherence times and high gate fidelities already demonstrated [6][7][8][9][10][11][12][13][14] . The conventional approach of using magnetic fields to operate singlequbit gates results in relatively low Rabi frequencies 12 , spurring the development of electrically driven spin resonance based on the spin-orbit interaction 15 as an alternative. This all-electrical approach promises faster Rabi rotations and reduced power consumption, as well as paving the way towards scalability, since electric fields are much easier to apply and localize than magnetic fields. In a Si QD, with relatively weak intrinsic SOI, a synthetic SOI has been introduced to provide fast and high-fidelity gates: a Rabi frequency above 10 MHz and gate fidelity of 99.9% have been reported in an isotopically enriched dot 6 . However, the magnetic field gradient enabling the synthetic SOI also exposes the system to charge-noise-induced spin dephasing 6,16 , posing a formidable technical challenge. As such, the search for a high quality spin qubit with fast manipulation and slow decoherence remains open 17 .
Hole spins provide an intriguing alternative for encoding qubits as compared to conduction electrons 18-21 , in particular in group IV materials such as Si and Ge 22-34 . Thanks to their underlying atomic P orbitals, which carry a finite angular momentum and have odd parity, holes experience an inherently strong SOI and weak hyperfine interaction 35,36 . The strong spin-orbital hybridization of hole states opens the door to fast all-electrical spin control. To date, several works on hole spin qubits have been reported in a multitude of systems, such as Si metal-oxide-semiconductor (MOS) 37 , undoped strained Ge quantum well 38,39 and GHW 40  Here we advance the ultrafast control of hole spin qubits by performing faster spin rotations than any reported to date. By applying microwave bursts to one gate of a GHW hole double quantum dot (DQD) 31 and utilizing Pauli spin blockade (PSB) for spin-to-charge conversion and measurement, we observe multiple electric dipole spin resonance (EDSR) signals in the DQD. At one of these resonances we achieve a Rabi frequency exceeding 540 MHz, with a dephasing time of 84 ns and a quality factor of ~ 45. This ultrafast driving is enabled by a very strong SOI, with an equivalent hole spin-orbit length of 1.5 nm. The driving speed is a strong function of the EDSR peaks we study, hence even higher quality factors are likely as qubit encoding is optimized.

Results
Measurement techniques and EDSR spectrum. A scanning electron microscope (SEM) image of the DQD device is shown in Fig. 1a (Supplementary Fig. 1a shows a schematic of the device).
The device consists of an insulating layer and five electrodes above a GHW grown using the Stranski-Krastanow (S-K) method 42,43 . The charge stability diagram of the DQD is mapped out and given in Supplementary Fig. 1b, with a zoom-in to one particular triple point given in Fig. 1b. Charge occupations of the DQD are about 5 holes in the left dot and 10 holes in the right dot. When measuring the current through the DQD at a d.c. bias of 3 mV ( Fig. 1b and Supplementary Fig. 1c) and −3 mV ( Supplementary Fig. 1d), a clear signature of PSB is observed: The zero-detuning current drops to 1 pA in the forward biased ( sd = 3 mV), blocked configuration (dash base line of the triangle), compared to 30 pA ( sd = −3 mV) in the reversed biased, non-spin-blocked regime. While PSB 44 is usually detected in the (0,2) or (2,0) to (1,1) charge configurations, it has been observed in other charge configurations as well 45 . With this in mind we conjecture that the transition we observe occurs near the (n+1, m+1) to (n, m+2) charge transition, which can be equivalently described in terms of two-hole states near the (1,1) to (0,2) transition.
In the PSB regime, with a magnetic field B perpendicular to the substrate, we generate EDSR by applying a microwave pulse to gate R. The a.c. electric field displaces the hole wave function around its equilibrium position periodically, leading to spin rotation mediated by the strong SOI ( Fig. 1d). When the microwave frequency matches the resonant frequency of the spin states and causes spin flips, PSB is lifted and an increase in the transport current is observed (a pure orbital transition without spin flip cannot lift the PSB and cannot affect the current). By mapping out the current as a function of B and microwave frequency , we find multiple spin resonances, as shown in Fig. 1c and Supplementary Fig. 2.
The major observed resonances in Fig.1c are well described by a two-hole model built upon a single singlet in the (0,2) charge configuration (S 02 ) and two-spin states |↓↑⟩, |↑↓⟩), |↑↑⟩ and |↓↓⟩ in the (1,1) charge configuration (Supplementary Note 2), as evidenced by Fig.1e and 1f. The large number of resonances and different slopes in Fig.1c are clear hints of different -factors for the two dots. Indeed, to generate the theoretical spectrum in Fig.1f, we use -factors of 7 and 3.95 for the two dots. With such different -factors 30,31,46 , the two-spin states in the (1,1) regime should be spin product states for any magnetic field above 0.1 T. In the following we focus on two of these resonances, denoted as mode A and mode B in Fig. 1c. Within our model, the corresponding transitions involve single-spin-flip in the left (A, between |↓↓⟩ and |↑↓⟩) and the right (B, between |↓↓⟩ and |↓↑⟩) dot.

Rabi oscillations.
To demonstrate coherent control of a hole spin qubit, we apply a three-step pulse sequence on gate R (Fig. 2a) to generate Rabi oscillations for mode A at = 100 mT. The probability of a parallel spin state (spin blocked) or anti-parallel state (unblocked) is measured by the averaged current through the DQD as a function of the microwave burst duration burst and microwave frequency (Fig. 2b). We can resolve up to seven oscillations within 180 ns, and the standard chevron pattern helps us to pinpoint the qubit Larmor frequency at 7.92 GHz. To investigate how fast the qubit can be driven coherently, we vary the microwave power of the driving field from 0 dBm to 9 dBm and measure the Rabi frequency of mode A (Fig. 2d). Rabi oscillations at = 7.92 GHz with a fit to • cos(2π R burst + ) • exp(−( burst 2 R ⁄ ) 2 ) + 0 (An offset of 0.5 pA is set between two oscillations for clarity) are shown in Fig. 2c at three different microwave power P = -5, 0, 6 dBm. At the strongest driving with = 9 dBm (Fig. 4b), we achieve a Rabi frequency of Rabi = 542 ± 2 MHz for mode A and 291 ± 1 MHz for mode B. Alternatively, two-axis control can be achieved by varying the relative phase ∆ of the microwave modulation between the two pulses 6,15,37 . The results of relative phase (∆ ) in cycles identify the control of the rotation axis in addition to ∆ (Fig. 3c). From the Ramsey experiment of mode A, dephasing times of 2 * = 84 ± 9 ns and 2 * = 42 ± 4 ns are extracted at = −10 dBm (Fig. 3d) and = 9 dBm (Fig. 3e), respectively. The former is a better representation of hole spin dephasing, while the latter reflects coherence degradation from the onset of microwave induced heating. The measured coherence times can be extended by performing a Hahn echo pulse sequence. For instance, the coherence time is extracted to be 2 * = 66 ± 6 ns in mode B at = 0 dBm (Fig. 3f), while an enhanced coherence time of 523 ns is obtained using Hahn echo (Fig.   3g). The leading SOI for a 2D HH gas is the Rashba term 47 where ± = ( ± )/2, ± = ± and the Rashba constant 2 arises from the spherical component of the Luttinger Hamiltonian 48 . We have determined that the cubic-symmetry term ∝ 6 / 19 electric field at the QD generated by the microwave pulses applied to gate R, and so ∝ 1/ 2 the spin-orbit length defined in the Supplementary Note 10.
We can obtain the spin-orbit coupling strength according to Eq. (2) from observations of Rabi oscillations of both mode B ( Fig. 4a and Supplementary Fig. 7.1) and mode A ( Fig. 4a and Supplementary Fig. 6) in the range of ≤ 9 dBm. In mode A, nine oscillation periods are observed within 16 ns at = 9 dBm (Fig. 4b), with a Rabi frequency of Rabi = 542 ± 2 MHz at = 7.92 GHz. When the microwave power is further increased ( Supplementary Fig. 6 49 . In comparison, our hole system has, inherently, a much stronger spin-orbit coupling (thus a much smaller so ), which is the determining factor for the ultrafast operation of our qubit.

Discussion
Ultrafast control of hole spins has also been achieved in a Ge/Si core/shell nanowire 41 , though that hole system is quite different from ours. The key difference stems from the respective geometries.
Our GHW has a confinement potential that is much stronger in one direction, akin to a quantum well, for which theory predicts a spin-3/2 (i.e. 'heavy hole') ground state. Ge/Si core/shell nanowires have cylindrical symmetry, where theory predicts a spin-1/2 ('light-hole') ground state 5 .
Due to possible mass reversal, the magnitude of the effective mass is not a reliable indicator of spin character. On the other hand, the two systems do share the feature of a strong spin-orbit coupling, which enables the fast control.

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The large Rabi frequencies in our system are achieved with a small driving electrical field ~10 4 V/m, compared to ~10 6 V/m static electrical field used for QD confinement (Supplementary Fig.   8 Fig. 4). We are thus optimistic that even faster Rabi operation is achievable after optimization.
A high quality qubit requires both fast manipulation and slow decoherence. We have thus investigated dephasing for both modes A and B (Supplementary Fig. 6d and Fig. 3f). The dephasing rate appears approximately uniform across the two modes, so that the qubit quality factor = 2 Rabi 2 Rabi is roughly determined by the Rabi frequency of the different modes. We thus obtain a lower bound estimate of the quality factor ~ 45 using Rabi = 542 MHz and 2 * = 42 ns at = 9 dBm . This value predicts a fidelity of the gate to be −1/ = 97.8% . A benchmark for Rabi frequency of QD spin qubits is set in Supplementary Fig. 10. Our hole spin qubit has one of the fastest Rabi frequency, with a quality factor around 45 that we believe can be further improved.
In conclusion, we have achieved ultrafast spin manipulation in a Ge HW. We report a Rabi frequency of up to 540 MHz at a small magnetic field of 100 mT, and obtain a dephasing time of 84 ns from a Ramsey fringe experiment. A hole spin qubit with a quality-factor of 45 is thus realized in our experiment. As dephasing appears to change little across different modes, higherquality qubits could be achievable for state combinations with stronger spin-orbit coupling. We report a small spin-orbit length in a smaller Ge double quantum dot compared to existing work on GHW in the literature 40 with narrower electrodes. We attribute the ultrafast control of a hole spin qubit that we have observed to an overall strong spin-orbit coupling, possibly assisted by a nearby excited state, even though the relative smaller longitude dot size (along y) may have reduced the Rabi frequency. In other words, our results demonstrate that hole spins in GHW QDs are intriguing candidates for semiconductor quantum computing, providing the ability of all-electrical ultrafast control without the need for a micromagnet 13 or a co-planar stripline 14 .

Methods
Device fabrication. Our hut wire was grown on Si (001) by means of a catalyst-free method based on molecular beam epitaxy. A Ge layer (1.5 nm) was deposited by S-K growth mode on a Si buffer layer (100 nm). A 3.5-nm-thick Si cap was then grown on top of the Ge layer to protect the nanowire with a width of 20 nm (Details in Supplementary Fig. 1a).  Fig. 2a, we apply two-stage pulses to gate R for spin initialization, control and readout. The length of one cycle is fixed at 640 ns and 320 ns of it is for spin control. The average transport current is measured by a digital multimeter after a low-noise current preamplifier SR570.

Data availability
All the data that support the findings of this study are available from the corresponding author upon reasonable request.