Spin-lattice relaxation times of single donors and donor clusters in silicon

Engineering Spin-Orbit Interactions in Silicon Qubits at the Atomic-Scale

Spin-orbit interactions arise whenever the bulk inversion symmetry and/or structural inversion symmetry of a crystal is broken providing a bridge between a qubit's spin and orbital degree of freedom. While strong interactions can facilitate fast qubit operations by all-electrical control, they also provide a mechanism to couple charge noise thereby limiting qubit lifetimes. Previously believed to be negligible in bulk silicon, recent silicon nano-electronic devices have shown larger than bulk spin-orbit coupling strengths from Dresselhaus and Rashba couplings. Here, it is shown that with precision placement of phosphorus atoms in silicon along the [110] direction (without inversion symmetry) or [111] direction (with inversion symmetry), a wide range of Dresselhaus and Rashba coupling strength can be achieved from zero to 1113 × 10-13eV-cm. It is shown that with precision placement of phosphorus atoms, the local symmetry (C2v, D2d, and D3d) can be changed to engineer spin-orbit interactions. Since spin-orbit interactions affect both qubit operation and lifetimes, understanding their impact is essential for quantum processor design.

Atomic Engineering of Molecular Qubits for High-Speed, High-Fidelity Single Qubit Gates

Universal quantum computing requires fast single- and two-qubit gates with individual qubit addressability to minimize decoherence errors during processor operation. Electron spin qubits using individual phosphorus donor atoms in silicon have demonstrated long coherence times with high fidelities, providing an attractive platform for scalable quantum computing. While individual qubit addressability has been demonstrated by controlling the hyperfine interaction between the electron and nuclear wave function in a global magnetic field, the small hyperfine Stark coefficient of 0.34 MHz/MV m-1 achieved to date has limited the speed of single quantum gates to ∼42 μs to avoid rotating neighboring qubits due to power broadening from the antenna. The use of molecular 2P qubits with more than one donor atom has not only demonstrated fast (0.8 ns) two-qubit SWAP gates and long spin relaxation times of ∼30 s but provides an alternate way to achieve high selectivity of the qubit resonance frequency. Here, we show in two different devices that by placing the donors with comparable interatomic spacings (∼0.8 nm) but along different crystallographic axes, either the [110] or [310] orientations using STM lithography, we can engineer the hyperfine Stark shift from 1 MHz/MV m-1 to 11.2 MHz/MV m-1, respectively, a factor of 10 difference. NEMO atomistic calculations show that larger hyperfine Stark coefficients of up to ∼70 MHz/MV m-1 can be achieved within 2P molecules by placing the donors ≥5 nm apart. When combined with Gaussian pulse shaping, we show that fast single qubit gates with 2π rotation times of 10 ns and ∼99% fidelity single qubit operations are feasible without affecting neighboring qubits. By increasing the single qubit gate time to ∼550 ns, two orders of magnitude faster than previously measured, our simulations confirm that >99.99% single qubit control fidelities are achievable.

The Use of Exchange Coupled Atom Qubits as Atomic-Scale Magnetic Field Sensors

Phosphorus atoms in silicon offer a rich quantum computing platform where both nuclear and electron spins can be used to store and process quantum information. While individual control of electron and nuclear spins has been demonstrated, the interplay between them during qubit operations has been largely unexplored. This study investigates the use of exchange-based operation between donor bound electron spins to probe the local magnetic fields experienced by the qubits with exquisite precision at the atomic scale. To achieve this, coherent exchange oscillations are performed between two electron spin qubits, where the left and right qubits are hosted by three and two phosphorus donors, respectively. The frequency spectrum of exchange oscillations shows quantized changes in the local magnetic fields at the qubit sites, corresponding to the different hyperfine coupling between the electron and each of the qubit-hosting nuclear spins. This ability to sense the hyperfine fields of individual nuclear spins using the exchange interaction constitutes a unique metrology technique, which reveals the exact crystallographic arrangements of the phosphorus atoms in the silicon crystal for each qubit. The detailed knowledge obtained of the local magnetic environment can then be used to engineer hyperfine fields in multi-donor qubits for high-fidelity two-qubit gates.

Elastodynamically Induced Spin and Charge Pumping in Bulk Heavy Metals

Analogous to the spin-Hall effect (SHE), ab initio electronic structure calculations reveal that acoustic phonons can induce charge (spin) current flowing along (normal to) its propagation direction. Using the Floquet approach we have calculated the elastodynamically induced charge and spin pumping in bulk Pt and demonstrate that (i) the longitudinal charge pumping originates from the Berry curvature, while the transverse pumped spin current is an odd function of the electronic relaxation time and diverges in the clean limit. (ii) The longitudinal charge current is of nonrelativstic origin, while the transverse spin current is a relativistic effect that to lowest order scales linearly with the spin-orbit coupling strength. (iii) Both charge and spin pumped currents have parabolic dependence on the amplitude of the elastic wave.

Precision tomography of a three-qubit donor quantum processor in silicon

Nuclear spins were among the first physical platforms to be considered for quantum information processing^1,2, because of their exceptional quantum coherence³ and atomic-scale footprint. However, their full potential for quantum computing has not yet been realized, owing to the lack of methods with which to link nuclear qubits within a scalable device combined with multi-qubit operations with sufficient fidelity to sustain fault-tolerant quantum computation. Here we demonstrate universal quantum logic operations using a pair of ion-implanted ³¹P donor nuclei in a silicon nanoelectronic device. A nuclear two-qubit controlled-Z gate is obtained by imparting a geometric phase to a shared electron spin⁴, and used to prepare entangled Bell states with fidelities up to 94.2(2.7)%. The quantum operations are precisely characterized using gate set tomography (GST)⁵, yielding one-qubit average gate fidelities up to 99.95(2)%, two-qubit average gate fidelity of 99.37(11)% and two-qubit preparation/measurement fidelities of 98.95(4)%. These three metrics indicate that nuclear spins in silicon are approaching the performance demanded in fault-tolerant quantum processors⁶. We then demonstrate entanglement between the two nuclei and the shared electron by producing a Greenberger-Horne-Zeilinger three-qubit state with 92.5(1.0)% fidelity. Because electron spin qubits in semiconductors can be further coupled to other electrons^7-9 or physically shuttled across different locations^10,11, these results establish a viable route for scalable quantum information processing using donor nuclear and electron spins.

Spin read-out in atomic qubits in an all-epitaxial three-dimensional transistor

The realization of the surface code for topological error correction is an essential step towards a universal quantum computer^1-3. For single-atom qubits in silicon^4-7, the need to control and read out qubits synchronously and in parallel requires the formation of a two-dimensional array of qubits with control electrodes patterned above and below this qubit layer. This vertical three-dimensional device architecture⁸ requires the ability to pattern dopants in multiple, vertically separated planes of the silicon crystal with nanometre precision interlayer alignment. Additionally, the dopants must not diffuse or segregate during the silicon encapsulation. Critical components of this architecture-such as nanowires⁹, single-atom transistors⁴ and single-electron transistors¹⁰-have been realized on one atomic plane by patterning phosphorus dopants in silicon using scanning tunnelling microscope hydrogen resist lithography^11,12. Here, we extend this to three dimensions and demonstrate single-shot spin read-out with 97.9% measurement fidelity of a phosphorus dopant qubit within a vertically gated single-electron transistor with <5 nm interlayer alignment accuracy. Our strategy ensures the formation of a fully crystalline transistor using just two atomic species: phosphorus and silicon.

Addressable electron spin resonance using donors and donor molecules in silicon

Phosphorus donor impurities in silicon are a promising candidate for solid-state quantum computing due to their exceptionally long coherence times and high fidelities. However, individual addressability of exchange coupled donors with separations ~15 nm is challenging. We show that by using atomic precision lithography, we can place a single P donor next to a 2P molecule 16 ± 1 nm apart and use their distinctive hyperfine coupling strengths to address qubits at vastly different resonance frequencies. In particular, the single donor yields two hyperfine peaks separated by 97 ± 2.5 MHz, in contrast to the donor molecule that exhibits three peaks separated by 262 ± 10 MHz. Atomistic tight-binding simulations confirm the large hyperfine interaction strength in the 2P molecule with an interdonor separation of ~0.7 nm, consistent with lithographic scanning tunneling microscopy images of the 2P site during device fabrication. We discuss the viability of using donor molecules for built-in addressability of electron spin qubits in silicon.

Atomically engineered electron spin lifetimes of 30 s in silicon

Scaling up to large arrays of donor-based spin qubits for quantum computation will require the ability to perform high-fidelity readout of multiple individual spin qubits. Recent experiments have shown that the limiting factor for high-fidelity readout of many qubits is the lifetime of the electron spin. We demonstrate the longest reported lifetimes (up to 30 s) of any electron spin qubit in a nanoelectronic device. By atomic-level engineering of the electron wave function within phosphorus atom quantum dots, we can minimize spin relaxation in agreement with recent theoretical predictions. These lifetimes allow us to demonstrate the sequential readout of two electron spin qubits with fidelities as high as 99.8%, which is above the surface code fault-tolerant threshold. This work paves the way for future experiments on multiqubit systems using donors in silicon.

Characterizing Si:P quantum dot qubits with spin resonance techniques

Quantum dots patterned by atomically precise placement of phosphorus donors in single crystal silicon have long spin lifetimes, advantages in addressability, large exchange tunability, and are readily available few-electron systems. To be utilized as quantum bits, it is important to non-invasively characterise these donor quantum dots post fabrication and extract the number of bound electron and nuclear spins as well as their locations. Here, we propose a metrology technique based on electron spin resonance (ESR) measurements with the on-chip circuitry already needed for qubit manipulation to obtain atomic scale information about donor quantum dots and their spin configurations. Using atomistic tight-binding technique and Hartree self-consistent field approximation, we show that the ESR transition frequencies are directly related to the number of donors, electrons, and their locations through the electron-nuclear hyperfine interaction.