Resonant microwave-mediated interactions between distant electron spins.

Nonlocal qubit interactions are a hallmark of advanced quantum information technologies1-5. The ability to transfer quantum states and generate entanglement over distances much larger than qubit length scales greatly increases connectivity and is an important step towards maximal parallelism and the implementation of two-qubit gates on arbitrary pairs of qubits6. Qubit-coupling schemes based on cavity quantum electrodynamics2,7,8 also offer the possibility of using high-quality-factor resonators as quantum memories3,9. Extending qubit interactions beyond the nearest neighbour is particularly beneficial for spin-based quantum computing architectures, which are limited by short-range exchange interactions10. Despite the rapidly maturing device technology for silicon spin qubits11-16, experimental progress towards achieving long-range spin-spin coupling has so far been restricted to interactions between individual spins and microwave photons17-20. Here we demonstrate resonant microwave-mediated coupling between two electron spins that are physically separated by more than four millimetres. An enhanced vacuum Rabi splitting is observed when both spins are tuned into resonance with the cavity, indicating a coherent interaction between the two spins and a cavity photon. Our results imply that microwave-frequency photons may be used to generate long-range two-qubit gates between spatially separated spins.

Shuttling a single charge across a one-dimensional array of silicon quantum dots.

Significant advances have been made towards fault-tolerant operation of silicon spin qubits, with single qubit fidelities exceeding 99.9%, several demonstrations of two-qubit gates based on exchange coupling, and the achievement of coherent single spin-photon coupling. Coupling arbitrary pairs of spatially separated qubits in a quantum register poses a significant challenge as most qubit systems are constrained to two dimensions with nearest neighbor connectivity. For spins in silicon, new methods for quantum state transfer should be developed to achieve connectivity beyond nearest-neighbor exchange. Here we demonstrate shuttling of a single electron across a linear array of nine series-coupled silicon quantum dots in ~50 ns via a series of pairwise interdot charge transfers. By constructing more complex pulse sequences we perform parallel shuttling of two and three electrons at a time through the array. These experiments demonstrate a scalable approach to physically transporting single electrons across large silicon quantum dot arrays.

Erratum: Microwave Detection of Electron-Phonon Interactions in a Cavity-Coupled Double Quantum Dot [Phys. Rev. Lett. 120, 097701 (2018)].

This corrects the article DOI: 10.1103/PhysRevLett.120.097701.

Quadrupolar Exchange-Only Spin Qubit.

We propose a quadrupolar exchange-only spin qubit that is highly robust against charge noise and nuclear spin dephasing, the dominant decoherence mechanisms in quantum dots. The qubit consists of four electrons trapped in three quantum dots, and operates in a decoherence-free subspace to mitigate dephasing due to nuclear spins. To reduce sensitivity to charge noise, the qubit can be completely operated at an extended charge noise sweet spot that is first-order insensitive to electrical fluctuations. Because of on-site exchange mediated by the Coulomb interaction, the qubit energy splitting is electrically controllable and can amount to several GHz even in the "off" configuration, making it compatible with conventional microwave cavities.

Realizing a Circuit Analog of an Optomechanical System with Longitudinally Coupled Superconducting Resonators.

We realize a superconducting circuit analog of the generic cavity-optomechanical Hamiltonian by longitudinally coupling two superconducting resonators, which are an order of magnitude different in frequency. We achieve longitudinal coupling by embedding a superconducting quantum interference device into a high frequency resonator, making its resonance frequency depend on the zero point current fluctuations of a nearby low frequency LC resonator. By applying sideband drive fields we enhance the intrinsic coupling strength of about 15 kHz up to 280 kHz by controlling the amplitude of the drive field. Our results pave the way towards the exploration of optomechanical effects in a fully superconducting platform and could enable quantum optics experiments with photons in the yet unexplored radio frequency band.

Microwave Detection of Electron-Phonon Interactions in a Cavity-Coupled Double Quantum Dot.

Quantum confinement leads to the formation of discrete electronic states in quantum dots. Here we probe electron-phonon interactions in a suspended InAs nanowire double quantum dot (DQD) that is electric-dipole coupled to a microwave cavity. We apply a finite bias across the wire to drive a steady state population in the DQD excited state, enabling a direct measurement of the electron-phonon coupling strength at the DQD transition energy. The amplitude and phase response of the cavity field exhibit oscillations that are periodic in the DQD energy level detuning due to the phonon modes of the nanowire. The observed cavity phase shift is consistent with theory that predicts a renormalization of the cavity center frequency by coupling to phonons.

A coherent spin-photon interface in silicon.

Electron spins in silicon quantum dots are attractive systems for quantum computing owing to their long coherence times and the promise of rapid scaling of the number of dots in a system using semiconductor fabrication techniques. Although nearest-neighbour exchange coupling of two spins has been demonstrated, the interaction of spins via microwave-frequency photons could enable long-distance spin-spin coupling and connections between arbitrary pairs of qubits ('all-to-all' connectivity) in a spin-based quantum processor. Realizing coherent spin-photon coupling is challenging because of the small magnetic-dipole moment of a single spin, which limits magnetic-dipole coupling rates to less than 1 kilohertz. Here we demonstrate strong coupling between a single spin in silicon and a single microwave-frequency photon, with spin-photon coupling rates of more than 10 megahertz. The mechanism that enables the coherent spin-photon interactions is based on spin-charge hybridization in the presence of a magnetic-field gradient. In addition to spin-photon coupling, we demonstrate coherent control and dispersive readout of a single spin. These results open up a direct path to entangling single spins using microwave-frequency photons.

Resonantly driven CNOT gate for electron spins.

Single-qubit rotations and two-qubit CNOT operations are crucial ingredients for universal quantum computing. Although high-fidelity single-qubit operations have been achieved using the electron spin degree of freedom, realizing a robust CNOT gate has been challenging because of rapid nuclear spin dephasing and charge noise. We demonstrate an efficient resonantly driven CNOT gate for electron spins in silicon. Our platform achieves single-qubit rotations with fidelities greater than 99%, as verified by randomized benchmarking. Gate control of the exchange coupling allows a quantum CNOT gate to be implemented with resonant driving in ~200 nanoseconds. We used the CNOT gate to generate a Bell state with 78% fidelity (corrected for errors in state preparation and measurement). Our quantum dot device architecture enables multi-qubit algorithms in silicon.

High-Resolution Valley Spectroscopy of Si Quantum Dots.

We study an accumulation mode Si/SiGe double quantum dot (DQD) containing a single electron that is dipole coupled to microwave photons in a superconducting cavity. Measurements of the cavity transmission reveal dispersive features due to the DQD valley states in Si. The occupation of the valley states can be increased by raising the temperature or applying a finite source-drain bias across the DQD, resulting in an increased signal. Using the cavity input-output theory and a four-level model of the DQD, it is possible to efficiently extract valley splittings and the inter- and intravalley tunnel couplings.

Threshold Dynamics of a Semiconductor Single Atom Maser.

We demonstrate a single atom maser consisting of a semiconductor double quantum dot (DQD) that is embedded in a high-quality-factor microwave cavity. A finite bias drives the DQD out of equilibrium, resulting in sequential single electron tunneling and masing. We develop a dynamic tuning protocol that allows us to controllably increase the time-averaged repumping rate of the DQD at a fixed level detuning, and quantitatively study the transition through the masing threshold. We further examine the crossover from incoherent to coherent emission by measuring the photon statistics across the masing transition. The observed threshold behavior is in agreement with an existing single atom maser theory when small corrections from lead emission are taken into account.

Electron Spin Resonance at the Level of 10

We report on electron spin resonance measurements of phosphorus donors localized in a 200 µm^{2} area below the inductive wire of a lumped element superconducting resonator. By combining quantum limited parametric amplification with a low impedance microwave resonator design, we are able to detect around 2×10^{4} spins with a signal-to-noise ratio of 1 in a single shot. The 150 Hz coupling strength between the resonator field and individual spins is significantly larger than the 1-10 Hz coupling rates obtained with typical coplanar waveguide resonator designs. Because of the larger coupling rate, we find that spin relaxation is dominated by radiative decay into the resonator and dependent upon the spin-resonator detuning, as predicted by Purcell.

Strong coupling of a single electron in silicon to a microwave photon.

Silicon is vital to the computing industry because of the high quality of its native oxide and well-established doping technologies. Isotopic purification has enabled quantum coherence times on the order of seconds, thereby placing silicon at the forefront of efforts to create a solid-state quantum processor. We demonstrate strong coupling of a single electron in a silicon double quantum dot to the photonic field of a microwave cavity, as shown by the observation of vacuum Rabi splitting. Strong coupling of a quantum dot electron to a cavity photon would allow for long-range qubit coupling and the long-range entanglement of electrons in semiconductor quantum dots.

Sisyphus Thermalization of Photons in a Cavity-Coupled Double Quantum Dot.

We investigate the nonclassical states of light that emerge in a microwave resonator coupled to a periodically driven electron in a nanowire double quantum dot (DQD). Under certain drive configurations, we find that the resonator approaches a thermal state at the temperature of the surrounding substrate with a chemical potential given by a harmonic of the drive frequency. Away from these thermal regions we find regions of gain and loss, where the system can lase, or regions where the DQD acts as a single-photon source. These effects are observable in current devices and have broad utility for quantum optics with microwave photons.

Phonon-assisted gain in a semiconductor double quantum dot maser.

We develop a microscopic model for the recently demonstrated double-quantum-dot maser. In characterizing the gain of this device we find that, in addition to the direct stimulated emission of photons, there is a large contribution from the simultaneous emission of a photon and a phonon, i.e., the phonon sideband. We show that this phonon-assisted gain typically dominates the overall gain, which leads to masing. Recent experimental data are well fit with our model.

Semiconductor double quantum dot micromaser.

The coherent generation of light, from masers to lasers, relies upon the specific structure of the individual emitters that lead to gain. Devices operating as lasers in the few-emitter limit provide opportunities for understanding quantum coherent phenomena, from terahertz sources to quantum communication. Here we demonstrate a maser that is driven by single-electron tunneling events. Semiconductor double quantum dots (DQDs) serve as a gain medium and are placed inside a high-quality factor microwave cavity. We verify maser action by comparing the statistics of the emitted microwave field above and below the maser threshold.

Injection Locking of a Semiconductor Double Quantum Dot Micromaser.

Emission linewidth is an important figure of merit for masers and lasers. We recently demonstrated a semiconductor double quantum dot (DQD) micromaser where photons are generated through single electron tunneling events. Charge noise directly couples to the DQD energy levels, resulting in a maser linewidth that is more than 100 times larger than the Schawlow-Townes prediction. Here we demonstrate a linewidth narrowing of more than a factor 10 by locking the DQD emission to a coherent tone that is injected to the input port of the cavity. We measure the injection locking range as a function of cavity input power and show that it is in agreement with the Adler equation. The position and amplitude of distortion sidebands that appear outside of the injection locking range are quantitatively examined. Our results show that this unconventional maser, which is impacted by strong charge noise and electron-phonon coupling, is well described by standard laser models.

Photon emission from a cavity-coupled double quantum dot.

We study a voltage biased InAs double quantum dot (DQD) that is coupled to a superconducting transmission line resonator. Inelastic tunneling in the DQD is mediated by electron phonon coupling and coupling to the cavity mode. We show that electronic transport through the DQD leads to photon emission from the cavity at a rate of 10 MHz. With a small cavity drive field, we observe a gain of up to 15 in the cavity transmission. Our results are analyzed in the context of existing theoretical models and suggest that it may be necessary to account for inelastic tunneling processes that proceed via simultaneous emission of a phonon and a photon.

Fast room-temperature phase gate on a single nuclear spin in diamond.

Nuclear spins support long lived quantum coherence due to weak coupling to the environment, but are difficult to rapidly control using nuclear magnetic resonance as a result of the small nuclear magnetic moment. We demonstrate a fast ∼500 ns nuclear spin phase gate on a (14)N nuclear spin qubit intrinsic to a nitrogen-vacancy center in diamond. The phase gate is enabled by the hyperfine interaction and off-resonance driving of electron spin transitions. Repeated applications of the phase gate bang-bang decouple the nuclear spin from the environment, locking the spin state for up to ∼140 µs.

Extreme harmonic generation in electrically driven spin resonance.

We report the observation of multiple harmonic generation in electric dipole spin resonance in an InAs nanowire double quantum dot. The harmonics display a remarkable detuning dependence: near the interdot charge transition as many as eight harmonics are observed, while at large detunings we only observe the fundamental spin resonance condition. The detuning dependence indicates that the observed harmonics may be due to Landau-Zener transition dynamics at anticrossings in the energy level spectrum.

Charge relaxation in a single-electron Si/SiGe double quantum dot.

We measure the interdot charge relaxation time T1 of a single electron trapped in an accumulation mode Si/SiGe double quantum dot. The energy level structure of the charge qubit is determined using photon assisted tunneling, which reveals the presence of a low-lying excited state. We systematically measure T1 as a function of detuning and interdot tunnel coupling and show that it is tunable over four orders of magnitude, with a maximum of 45 µs for our device configuration.