Conditional quantum operation of two exchange-coupled single-donor spin qubits in a MOS-compatible silicon device
Mateusz T. Mądzik, Arne Laucht, Fay E. Hudson, Alexander M. Jakob, Brett C. Johnson, David N. Jamieson, Kohei M. Itoh, Andrew S. Dzurak, Andrea Morello
备注：
10 pages, 5 figures, plus Supplementary Information. v2 contains additional references, and a simpler explanation of two-qubit CROT gates for donors in silicon
Silicon nanoelectronic devices can host single-qubit quantum logic operations with fidelity better than 99.9%. For the spins of an electron bound to a single donor atom, introduced in the silicon by ion implantation, the quantum information can be stored for nearly 1 second. However, manufacturing a scalable quantum processor with this method is considered challenging, because of the exponential sensitivity of the exchange interaction that mediates the coupling between the qubits. Here we demonstrate the conditional, coherent control of an electron spin qubit in an exchange-coupled pair of $^{31}$P donors implanted in silicon. The coupling strength, $J = 32.06 \pm 0.06$ MHz, is measured spectroscopically with unprecedented precision. Since the coupling is weaker than the electron-nuclear hyperfine coupling $A \approx 90$ MHz which detunes the two electrons, a native two-qubit Controlled-Rotation gate can be obtained via a simple electron spin resonance pulse. This scheme is insensitive to the precise value of $J$, which makes it suitable for the scale-up of donor-based quantum computers in silicon that exploit the Metal-Oxide-Semiconductor fabrication protocols commonly used in the classical electronics industry.
Selenium impurities in silicon are deep double donors and their optical and electronic properties have been recently investigated due to their application for infrared detection. However, a singly-ionised selenium donor (Se$^{+}$) possesses an electron spin which makes it a potential candidate as a silicon-based spin qubit, with significant potential advantages compared to the more commonly studied group V donors. Here we study the electron spin relaxation ($T_1$) and coherence ($T_2$) times of Se$^{+}$ in isotopically purified 28-silicon, and find them to be up to two orders of magnitude long…
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 donor qubits with separations ~15nm is challenging. Here 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…
Nuclear spin registers in the vicinity of electron spins in solid state systems offer a powerful resource to address the challenge of scalability in quantum architectures. We investigate here the properties of $^{29}$Si nuclear spins surrounding donor atoms in silicon, and consider the use of such spins, combined with the donor nuclear spin, as a quantum register coupled to the donor electron spin. We find the coherence of the nearby $^{29}$Si nuclear spins is effectively protected by the presence of the donor electron spin, leading to coherence times in the second timescale - over two orders …
Dopant atoms are ubiquitous in semiconductor technologies, providing the tailored electronic properties that underpin the modern digital information era. Harnessing the quantum nature of these atomic-scale objects represents a new and exciting technological revolution. In this article we describe the use of ion-implanted donor spins in silicon for quantum technologies. We review how to fabricate and operate single-atom spin qubits in silicon, obtaining some of the most coherent solid-state qubits, and we discuss pathways to scale up these qubits to build large quantum processors. Heavier group…
Donor impurity spins in silicon-28 are highly competitive qubits for upcoming solid-state quantum technologies, yet a proven scalable strategy for multi-qubit devices remains conspicuously absent. These CMOS-compatible, atomically identical qubits offer significant advantages including 3-hour coherence ($T_2$) lifetimes, as well as simultaneous qubit initialization, manipulation and readout fidelities near $\sim\!99.9\%$. These properties meet the requirements for many modern quantum error correction protocols, which are essential for constructing large-scale universal quantum technologies. Ho…
Shifts from the expected nuclear magnetic resonance frequencies of antimony and bismuth donors in silicon of greater than a megahertz are observed in electrically detected magnetic resonance spectra. Defects created by ion implantation of the donors are discussed as the source of effective electric field gradients generating these shifts via quadrupole interaction with the nuclear spins. The experimental results are modeled quantitatively by molecular orbital theory for a coupled pair consisting of a donor and a spin-dependent recombination readout center.
Electric fields can be used to tune donor spins in silicon using the Stark shift, whereby the donor electron wave function is displaced by an electric field, modifying the hyperfine coupling between the electron spin and the donor nuclear spin. We present a technique based on dynamic decoupling of the electron spin to accurately determine the Stark shift, and illustrate this using antimony donors in isotopically purified silicon-28. We then demonstrate two different methods to use a DC electric field combined with an applied resonant radio-frequency (RF) field to conditionally control donor nu…
The influence of the electric field created by a gate potential of the silicon quantum computer on the hyperfine interaction constant (HIC) is obtained. The errors due to technological inaccuracy of location of donor atoms under a gate are evaluated. The energy spectra of electron-nuclear spin system of two interacting donor atoms with various values of HIC are calculated. The presence of two pairs of anticrossing levels in the ground electronic state is shown. Parameters of the structure at which errors rate can be greatly minimized are found.
We experimentally study the coupling of Group V donor spins in silicon to mechanical strain, and measure strain-induced frequency shifts which are linear in strain, in contrast to the quadratic dependence predicted by the valley repopulation model (VRM), and therefore orders of magnitude greater than that predicted by the VRM for small strains $|\varepsilon| < 10^{-5}$. Through both tight-binding and first principles calculations we find that these shifts arise from a linear tuning of the donor hyperfine interaction term by the hydrostatic component of strain and achieve semi-quantitative agre…
Donor spin in silicon have achieved record values of coherence times and single-qubit gate fidelities. The next stage of development involves demonstrating high-fidelity two-qubit logic gates, where the most natural coupling is the exchange interaction. To aid the efficient design of scalable donor-based quantum processors, we model the two-electron wave function using a full configuration interaction method within a multi-valley effective mass theory. We exploit the high computational efficiency of our code to investigate the exchange interaction, valley population, and electron densities for…
For donor nuclear spins in silicon, we show how to deterministically prepare various symmetric and asymmetric Dicke states which span a complete basis of the many-body Hilbert space. The state preparation is realized by cooperative pumping of nuclear spins by coupled donor electrons, and the required controls are in situ to the prototype Kane proposal for quantum computation. This scheme only requires a sub-gigahertz donor exchange coupling which can be readily achieved without atomically precise donor placement, hence it offers a practical way to prepare multipartite entanglement of spins in …
Gate control of donor electrons near interfaces is a generic ingredient of donor-based quantum computing. Here, we address the question: how is the phonon-assisted qubit relaxation time $T_1$ affected as the electron is shuttled between the donor and the interface? We focus on the example of the `flip-flop qubit' [Tosi et al., arXiv:1509.08538v1], defined as a combination of the nuclear and electronic states of a phosphorous donor in silicon, promising fast electrical control and long dephasing times when the electron is halfway between the donor and the interface. We theoretically describe or…
We propose a novel optical and electrical hybrid scheme for the measurement of nuclear spin qubits in silicon. By combining the environmental insensitivity of the integer quantum Hall effect with the optically distinguishable hyperfine states of phosphorus impurities in silicon, our system can simultaneously offer nuclear spin measurement and robustness against environmental defects. 31P donor spins in isotopically purified 28Si are often discussed as very promising quantum memory qubits due to their extremely long decoherence times, and our proposed device offers an effective implementation f…
Donors in silicon can now be positioned with an accuracy of about one lattice constant, making it possible in principle to form donor arrays for quantum computation or quantum simulation applications. However the multi-valley character of the silicon conduction band combines with central cell corrections to the donor state Hamiltonian to translate atomic scale imperfections in donor placement into strongly disordered inter-donor hybridization. We present a simple model that is able to account accurately for central-cell corrections, and use it to assess the impact of donor-placement disorder o…
Donors in silicon are now demonstrated as one of the leading candidates for implementing qubits and quantum information processing. Single qubit operations, measurements and long coherence times are firmly established, but progress on controlling two qubit interactions has been slower. One reason for this is that the inter donor exchange coupling has been predicted to oscillate with separation, making it hard to estimate in device designs. We present a multivalley effective mass theory of a donor pair in silicon, including both a central cell potential and the effective mass anisotropy intrins…
Opto-Electrical Detection of Donor Bound Excitons in Silicon-on-Insulator Substrate
University of Jyväskylä（于韦斯屈莱大学）；QuTech（奎泰克）；Netherlands Organisation for Applied Scientific Research (TNO)（荷兰应用科学研究组织）；Kavli Institute of Nanoscience, Delft University of Technology（代尔夫特理工大学卡弗里纳米科学研究所）
Substitutional donor atoms in silicon are promising qubits for quantum computation with extremely long relaxation and dephasing times demonstrated. One of the critical challenges of scaling these systems is determining inter-donor distances to achieve controllable wavefunction overlap while at the same time performing high fidelity spin readout on each qubit. Here we achieve such a device by means of scanning tunnelling microscopy lithography. We measure anti-correlated spin states between two donor-based spin qubits in silicon separated by 16${\pm}1$ nm. By utilizing an asymmetric system with…
We present two strategies for performing two-qubit operations on the electron spins of an exchange-coupled pair of phosphorus donors in silicon, using the ability to set the donor nuclear spins in arbitrary states. The effective magnetic detuning of the two electron qubits is provided by the hyperfine interaction when the $^{31}$P nuclei are prepared in opposite spin states. This can be exploited to switch on and off SWAP operations with modest tuning of the electron exchange interaction. Furthermore, the hyperfine detuning enables high-fidelity conditional rotation gates based on selective re…
Last year, Salfi et al. made the first direct measurements of a donor wave function and found extremely good theoretical agreement with atomistic tight-binding [Salfi et al., Nat. Mater. 13, 605 (2014)]. Here, we show that multi-valley effective mass theory, applied properly, does achieve close agreement with tight-binding and hence gives reliable predictions. To demonstrate this, we variationally solve the coupled six-valley Shindo-Nara equations, including silicon's full Bloch functions. Surprisingly, we find that including the full Bloch functions necessitates a tetrahedral, rather than sph…
Individual donors in silicon chips are used as quantum bits with extremely low error rates. However, physical realizations have been limited to one donor because their atomic size causes fabrication challenges. Quantum dot qubits, in contrast, are highly adjustable using electrical gate voltages. This adjustability could be leveraged to deterministically couple donors to quantum dots in arrays of qubits. In this work, we demonstrate the coherent interaction of a $^{31}$P donor electron with the electron of a metal-oxide-semiconductor quantum dot. We form a logical qubit encoded in the spin sin…
The nuclear spins of ionized donors in silicon have become an interesting quantum resource due to their very long coherence times. Their perfect isolation, however, comes at a price, since the absence of the donor electron makes the nuclear spin difficult to control. We demonstrate that the quadrupolar interaction allows us to effectively tune the nuclear magnetic resonance of ionized arsenic donors in silicon via strain and determine the two nonzero elements of the S tensor linking strain and electric field gradients in this material to $S_{11}=1.5\times10^{22}$ V/m$^2$ and $S_{44}=6\times 10…
A high degree of controllability and long coherence time make the nuclear spin of a phosphorus donor in isotopically purified silicon a promising candidate for a quantum bit. However, long-distance two-qubit coupling and fast, robust gates remain outstanding challenges for these systems. Here, following recent proposals for long-distance coupling via dipole-dipole interactions, we present a simple method to implement fast, high-fidelity arbitrary single- and two-qubit gates in the absence of charge noise. Moreover, we provide a method to make the single-qubit gates robust to moderate levels of…
Ensembles of bismuth donor spins in silicon are promising storage elements for microwave quantum memories due to their long coherence times which exceed seconds. Operating an efficient quantum memory requires achieving critical coupling between the spin ensemble and a suitable high-quality factor resonator -- this in turn requires a thorough understanding of the lineshapes for the relevant spin resonance transitions, particularly considering the influence of the resonator itself on line broadening. Here, we present pulsed electron spin resonance measurements of ensembles of bismuth donors in n…
We show how to achieve optical, spin-selective transitions from the ground state to excited orbital states of group-V donors (P, As, Sb, Bi) in silicon. We consider two approaches based on either resonant, far-infrared (IR) transitions of the neutral donor or resonant, near-IR excitonic transitions. For far-IR light, we calculate the dipole matrix elements between the valley-orbit and spin-orbit split states for all the goup-V donors using effective mass theory. We then calculate the maximum rate and amount of electron-nuclear spin-polarization achievable through optical pumping with circularl…
In this work we perform direct single-shot readout of the singlet-triplet states in exchange coupled electrons confined to precision placed donor atoms in silicon. Our method takes advantage of the large energy splitting given by the Pauli-spin blockaded (2,0) triplet states, from which we can achieve a single-shot readout fidelity of 98.4$\pm$0.2%. We measure the triplet-minus relaxation time to be of the order 3s at 2.5T and observe its predicted decrease as a function of magnetic field, reaching 0.5s at 1T.
Single spin qubits based on phosphorus donors in silicon are a promising candidate for a large-scale quantum computer. Despite long coherence times, achieving uniform magnetic control remains a hurdle for scale-up due to challenges in high-frequency magnetic field control at the nanometre-scale. Here, we present a proposal for a flopping-mode electric dipole spin resonance qubit based on the combined electron and nuclear spin states of a double phosphorus donor quantum dot. The key advantage of utilising a donor-based system is that we can engineer the number of donor nuclei in each quantum do…
Spins of donor electrons and nuclei in silicon are promising quantum bit (qubit) candidates which combine long coherence times with the fabrication finesse of the silicon nanotechnology industry. We outline a potentially scalable spin qubit architecture where donor nuclear and electron spins are coupled to spins of electrons in quantum dots and discuss requirements for donor placement aligned to quantum dots by single ion implantation.
An atomistic method of calculating the spin-lattice relaxation times ($T_1$) is presented for donors in silicon nanostructures comprising of millions of atoms. The method takes into account the full band structure of silicon including the spin-orbit interaction. The electron-phonon Hamiltonian, and hence the deformation potential, is directly evaluated from the strain-dependent tight-binding Hamiltonian. The technique is applied to single donors and donor clusters in silicon, and explains the variation of $T_1$ with the number of donors and electrons, as well as donor locations. Without any ad…
Over the past decade donor spin qubits in isotopically enriched $^{28}$Si have been intensely studied due to their exceptionally long coherence times. More recently bismuth donor electron spins have become popular because Bi has a large nuclear spin which gives rise to clock transitions (first-order insensitive to magnetic field noise). At every clock transition there are two nearly degenerate transitions between four distinct states which can be used as a pair of qubits. Here it is experimentally demonstrated that these transitions are excited by microwaves of opposite helicity such that they…
Electric control of individual atoms or molecules in a solid-state system offers a promising way to bring quantum mechanical functionalities into electronics. This idea has recently come into the reach of the established domain of silicon technology, leading to the realization of single-atom transistors and to the first measurements of electron spin dynamics in single donors. Here we show that we can electrically couple two donors embedded in a multi-gate silicon transistor, and induce coherent oscillations in their charge states by means of microwave signals. We measure single-electron tunnel…
We present a complete theoretical treatment of Stark effects in doped silicon, whose predictions are supported by experimental measurements. A multi-valley effective mass theory, dealing non-perturbatively with valley-orbit interactions induced by a donor-dependent central cell potential, allows us to obtain a very reliable picture of the donor wave function within a relatively simple framework. Variational optimization of the 1s donor binding energies calculated with a new trial wave function, in a pseudopotential with two fitting parameters, allows an accurate match of the experimentally det…
We study the charge transfer dynamics between a silicon quantum dot and an individual phosphorous donor using the conduction through the quantum dot as a probe for the donor ionization state. We use a silicon n-MOSFET (metal oxide field effect transistor) biased near threshold in the SET regime with two side gates to control both the device conductance and the donor charge. Temperature and magnetic field independent tunneling time is measured. We measure the statistics of the transfer of electrons observed when the ground state D0 of the donor is aligned with the SET states.
Proposed silicon-based quantum-computer architectures have attracted attention because of their promise for scalability and their potential for synergetically utilizing the available resources associated with the existing Si technology infrastructure. Electronic and nuclear spins of shallow donors (e.g. phosphorus) in Si are ideal candidates for qubits in such proposals because of their long spin coherence times due to their limited interactions with their environments. For these spin qubits, shallow donor exchange gates are frequently invoked to perform two-qubit operations. We discuss in thi…
The silicon-based quantum computer proposal has been one of the intensely pursued ideas during the past three years. Here we calculate the donor electron exchange in silicon and germanium, and demonstrate an atomic-scale challenge for quantum computing in Si (and Ge), as the six (four) conduction band minima in Si (Ge) lead to inter-valley electronic interferences, generating strong oscillations in the exchange splitting of two-donor two-electron states. Donor positioning with atomic scale precision within the unit cell thus becomes a decisive factor in determining the strength of the exchange…
We consider a quantum network of mid-infrared, graphene plasmons coupled to the hydrogen-like excited states of group-V donors in silicon. First, we show how to use plasmon-enhanced light-matter interactions to achieve single-shot spin readout of the donor qubits via optical excitation and electrical detection of the emitted plasmons. We then show how plasmons in high mobility graphene nanoribbons can be used to achieve high-fidelity, two-qubit gates and entanglement of distant Si donor qubits. The proposed device is readily compatible with existing technology and fabrication methods.
Kinetics of electron-phonon scattering in silicon resolved by Rydberg transitions of donors
Radboud University（拉德堡德大学）；Paul Scherrer Institute（保罗谢勒研究所）；ETH Zurich（苏黎世联邦理工学院）；Institute of Space Research, German Aerospace Center (DLR)（德国航空航天中心空间研究所）；Leibniz-Institut für Kristallzüchtung (IKZ)（莱布尼茨晶体生长研究所）；University of Iowa（爱荷华大学）；Eindhoven University of Technology（埃因霍温理工大学）；University of Surrey（萨里大学）
Donor electron spins in semiconductors make exceptional quantum bits because of their long coherence times and compatibility with industrial fabrication techniques. Despite many advances in donor-based qubit technology, it remains difficult to selectively manipulate single donor electron spins. Here, we show that by replacing the prevailing semiconductor host material (silicon) with germanium, donor electron spin qubits can be electrically tuned by more than an ensemble linewidth, making them compatible with gate addressable quantum computing architectures. Using X-band pulsed electron spin re…
We propose a method to electrically control electron spins in donor-based qubits in silicon. By taking advantage of the hyperfine coupling difference between a single-donor and a two-donor quantum dot, spin rotation can be driven by inducing an electric dipole between them and applying an alternating electric field generated by in-plane gates. These qubits can be coupled with exchange interaction controlled by top detuning gates. The qubit device can be fabricated deep in the silicon lattice with atomic precision by scanning tunneling probe technique. We have combined a large-scale full band a…
Single donors in semiconductor nanostructures represent a key element to develop spin related quantum functionalities in atomic scale devices. Quantum transport through a single Arsenic donor in the channel of a Silicon nano-field effect transistor under microwave irradiation is investigated. The device is characterized at mK temperatures in the regime of Coulomb-blockade. Photon assisted tunneling and microwave induced electron pumping regimes are revealed respectively at low and high microwave power. At sufficiently high power, the microwave irradiation induces tunneling through the first ex…
We measure a large valley-orbit splitting for shallow isolated phosphorus donors in a silicon gated nanowire. This splitting is close to the bulk value and well above previous reports in silicon nanostructures. It was determined using a double dopant transport spectroscopy which eliminates artifacts induced by the environment. Quantitative simulations taking into account the position of the donors with respect to the Si/SiO$_2$ interface and electric field in the wire show that the values found are consistent with the device geometry.
Single electron transistor (SET) is an advanced tool to exploit in quantum devices. Working of such devices at room-temperature is essential for practical utilization. Dopant based single-electron devices are well studied at low-temperature although a few devices are developed for high-temperature operation with certain limitations. Here, we propose and theoretically exhibit that nitrogen (N) donor in silicon is an important candidate for effective designing of such devices. Theoretical calculation of density-of-states using semi-empirical DFT method indicates that N-donor in silicon has deep …
Deterministic control over the location and number of donors is crucial to donor spin quantum bits (qubits) in semiconductor based quantum computing. In this work, a focused ion beam is used to implant antimony donors close to quantum dots. Ion detectors are integrated next to the quantum dots to sense the implants. The numbers of donors implanted can be counted to a precision of a single ion. In low-temperature transport measurements, regular coulomb blockade is observed from the quantum dots. Charge offsets indicative of donor ionization are also observed in devices with counted donor implan…
We demonstrate how gradient ascent pulse engineering optimal control methods can be implemented on donor electron spin qubits in Si semiconductors with an architecture complementary to the original Kane's proposal. We focus on the high-fidelity controlled-NOT (CNOT) gate and explicitly find its digitized control sequences by optimizing its fidelity over the external controls of the hyperfine A and exchange J interactions. This high-fidelity CNOT gate has an error of about $10^{-6}$, below the error threshold required for fault-tolerant quantum computation, and its operation time of 100ns is ab…
Electronic and nuclear spins of shallow donors in Silicon are attractive candidates for qubits in quantum computer proposals. Shallow donor exchange gates are frequently invoked to preform two-qubit operations in such proposals. We study shallow donor electron properties in Si within the Kohn-Luttinger envelope function approach, incorporating the full Bloch states of the six band edges of Si conduction band, obtained from {\it ab initio} calculations within the density-functional and pseudopotential frameworks. Inter-valley interference between the conduction-band-edge states of Si leads to o…
We unravel theoretically a key intrinsic relaxation mechanism among the low-lying singlet and triplet donor-pair states in silicon, an important element in the fast-developing field of spintronics and quantum computation. Despite the perceived weak spin-orbit coupling (SOC) in Si, we find that our discovered relaxation mechanism, combined with the electron-phonon and inter-donor interactions, dominantly drives the transitions in the two-electron states over a large range of donor coupling regime. The scaling of the relaxation rate with inter-donor exchange interaction $J$ goes from $J^5$ to $J…
We report low frequency charge noise measurement on silicon substrates with different phosphorus doping densities. The measurements are performed with aluminum single electron transistors (SETs) at millikelvin temperatures where the substrates are in the insulating regime. By measuring the SET Coulomb oscillations, we find a gate voltage dependent charge noise on the more heavily doped substrate. This charge noise, which is seen to have a 1/f spectrum, is attributed to the electric field induced tunneling of electrons from their phosphorus donor potentials.
Developing devices that can reliably and accurately demonstrate the principles of superposition and entanglement is an on-going challenge for the quantum computing community. Modeling and simulation offer attractive means of testing early device designs and establishing expectations for operational performance. However, the complex integrated material systems required by quantum device designs are not captured by any single existing computational modeling method. We examine the development and analysis of a multi-staged computational workflow that can be used to design and characterize silicon…
A scaled quantum computer with donor spins in silicon would benefit from a viable semiconductor framework and a strong inherent decoupling of the qubits from the noisy environment. Coupling neighbouring spins via the natural exchange interaction according to current designs requires gate control structures with extremely small length scales. We present a silicon architecture where bismuth donors with long coherence times are coupled to electrons that can shuttle between adjacent quantum dots, thus relaxing the pitch requirements and allowing space between donors for classical control devices. …
Recent experimental efforts have led to considerable interest in donor-based localized electron spins in Si as viable qubits for a scalable silicon quantum computer. With the use of isotopically purified $^{28}$Si and the realization of extremely long spin coherence time in single-donor electrons, the recent experimental focus is on two-coupled donors with the eventual goal of a scaled-up quantum circuit. Motivated by this development, we simulate the statistical distribution of the exchange coupling $J$ between a pair of donors under realistic donor placement straggles, and quantify the error…
By varying the gate and substrate voltage in a short silicon-on-insulator trigate field effect transistor we control the ionization state of three arsenic donors. We obtain a good quantitative agreement between 3D electrostatic simulation and experiment for the control voltage at which the ionization takes place. It allows us observing the three doubly occupied states As- at strong electric field in the presence of nearby source-drain electrodes.
Proposed Silicon-based quantum computer architectures have attracted attention because of their promise for scalability and their potential for synergetically utilizing the available resources associated with the existing infrastructure of the powerful Si technology. Quantitative understanding of and precise physical control over donor (e.g. Phosphorus) exchange are crucial elements in the physics underlying the proposed Si-based quantum computer hardware. An important potential problem in this context is that inter-valley interference originating from the degeneracy in the Si conduction band …
