A numerical method is introduced to determine the nuclear magnetic resonance frequency of a donor $(^{31}\mathrm{P})$ doped inside a silicon substrate under the influence of an applied electric field. This phosphorus donor has been suggested for operation as a qubit for the realization of a solid-state scalable quantum computer. The operation of the qubit is achieved by a combination of the rotation of the phosphorus nuclear spin through a globally applied magnetic field and the selection of the phosphorus nucleus through a locally applied electric field. To realize the selection function, it is required to know the relationship between the applied electric field and the change of the nuclear magnetic resonance frequency of phosphorus. In this study, based on the wave functions obtained by the effective-mass theory, we introduce an empirical correction factor to the wave functions at the donor nucleus. Using the corrected wave functions, we formulate a first-order perturbation theory for the perturbed system under the influence of an electric field. In order to calculate the potential distributions inside the silicon and the silicon dioxide layers due to the applied electric field, we use the multilayered Green's functions and solve an integral equation by the moment method. This enables us to consider more realistic, arbitrary shape, and three-dimensional qubit structures. With the calculation of the potential distributions, we have investigated the effects of the thicknesses of silicon and silicon dioxide layers, the relative position of the donor, and the applied electric field on the nuclear magnetic resonance frequency of the donor.
A numerical method is introduced to determine the nuclear magnetic resonance frequency of a donor (31P) doped inside a silicon substrate under the influence of an applied electric field. This phosphorus donor has been suggested for operation as a qubit for the realization of a solid-state scalable quantum computer. The operation of the qubit is achieved by a combination of the rotation of the phosphorus nuclear spin through a globally applied magnetic field and the selection of the phosphorus nucleus through a locally applied electric field. To realize the selection function, it is required to know the relationship between the applied electric field and the change of the nuclear magnetic resonance frequency of phosphorus. In this study, based on the wave functions obtained by the effective-mass theory, we introduce an empirical correction factor to the wave functions at the donor nucleus. Using the corrected wave functions, we formulate a first-order perturbation theory for the perturbed system under the influence of an electric field. In order to calculate the potential distributions inside the silicon and the silicon dioxide layers due to the applied electric field, we use the multilayered Green's functions and solve an integral equation by the moment method. This enables us to consider more realistic, arbitrary shape, and three-dimensional qubit structures. With the calculation of the potential distributions, we have investigated the effects of the thicknesses of silicon and silicon dioxide layers, the relative position of the donor, and the applied electric field on the nuclear magnetic resonance frequency of the donor.
The nuclear magnetic resonance (NMR) frequency of a single qubit structure of Kane’s solid-state quantum computer is investigated by using the perturbation theory. With higher-order excited states (up to 3d modes) included in our calculation, the perturbation frequencies and energies are obtained numerically. To compute for arbitrary A gate geometries, the perturbation potential inside the qubit structure is determined through an electromagnetic simulation method. Calculations show that the potential distributions for realistic A gate geometries are far from linear ones. Our results show that the A gate voltage has a much more effective control over the NMR frequency of the phosphorus nucleus than that previously shown. Using our method, arbitrary A gate structures of any shapes or geometries can be engineered for the realization of a solid-state scalable quantum computer. We also investigate an alternative A gate structure using SiGe as the insulation barrier. Our study shows that this A gate structure offers a much more efficient utilization of the control voltage than the original A gate structure using SiO2 as the insulation barrier.
Spin is a quantum mechanical property that describes the magnetic orientation of elementary and composite particles such as electrons and nuclei. For spin-half particles placed in a magnetic field, two states with differing energies are defined. Proposals exist for developing a revolutionary kind of computer that encodes bits of information on the states of a simple quantum two-level system - like those of a spin in a magnetic field. These quantum computers exploit the fundamental principles of quantum mechanics in order to perform certain calculations exponentially faster than their classical counterparts. Key to the development of such a device is the identification of potential systems that are suitable for employment as a quantum bit (qubit). This thesis explores quantum bits constructed from the spin states of single atoms hosted in silicon semiconductor devices. Silicon is an exceptional platform for spin-based quantum computing - in addition to being the dominant material in the microelectronic industry, it is also predominantly composed of nuclear spin-zero isotopes, providing very little background magnetic disturbance. The donor-atom phosphorus (31P) is investigated here. It possesses a nucleus that binds an additional electron to its positive charge, providing two distinct spin qubits. Here, several world-first experimental milestones in spin-based quantum information processing are reported. First, the readout of the electron spin is demonstrated in single-shot, using a silicon device which combines implanted phosphorus donors with a sensitive nanoscale electrometer. Then, the electron spin is coherently operated as a qubit by delivering on-chip microwave excitations. Combining the electron spin detection and control capabilities enables a quantum non-demolition measurement of the single 31P nuclear spin. Experiments yield exceptional nuclear readout fidelities - the highest for any solid-state qubit. Through an application of nuclear magnetic resonance, high-fidelity quantum control of the nuclear spin qubit is demonstrated. Finally, a single spin-bearing silicon nucleus (29Si) coupled to the electron spin is detected in single-shot and operated as a qubit. These results, along with the clear pathways identified for improving device performance, illustrate that single-atom electron and nuclear spin qubits in silicon constitute an excellent basis on which to construct a quantum computer.
The nuclear spin state of a phosphorus donor (31P) in isotopically enriched silicon-28 is an excellent host to store quantum information in the solid state. The spin's insensitivity to electric fields yields a solid-state qubit with record coherence times but also renders coupling to other quantum systems very challenging. Here, we describe how to generate a strong electric dipole (>100 D) at microwave frequencies for the nuclear spin. This is achieved by applying a magnetic drive to the spin of the donor-bound electron while simultaneously controlling its charge state with electric fields. Under certain conditions, the microwave magnetic drive also renders the nuclear spin resonance frequency and electric dipole strongly insensitive to electrical noise, yielding long (>1 ms) dephasing times and robust gate operations. The nuclear spin could then be strongly coupled to microwave resonators, with a vacuum Rabi splitting of the order of 1 MHz, or to other nuclear spins, nearly half a micrometer apart, via strong electric dipole-dipole interaction. This work brings the 31P nuclear qubit into the realm of hybrid quantum systems and opens up new avenues in quantum information processing.
In the Kane silicon-based electron-mediated nuclear spin quantum computer architecture, phosphorous is doped at precise positions in a silicon lattice, and the P donor nuclear spins act as qubits. Logical operations on the nuclear spins are performed using externally applied magnetic and electric fields. There are two important interactions: the hyperfine and exchange interactions, crucial for logical qubit operations. Single qubit operations are performed by applying radio frequency magnetic fields resonant with targeted nuclear spin transition frequencies, tuned by the gate-controlled hyperfine interaction. Two qubit operations are mediated through the exchange interaction between adjacent donor electrons. It is important to examine how these two interactions vary as functions of experimental parameters. Here we provide such an investigation. First, we examine the effects of varying several experimental parameters: gate voltage, inter donor separation, donor depth below the silicon oxide interface and back gate depth, to explore how these variables affect the donor electro density. Second, we calculate the hyperfine interaction and the exchange coupling as a function of these parameters. These calculations were performed using an anisotropic effective mass Hamiltonian. The electric field potential was obtained using Technology Computer Aided Design software, and the interfaces were modelled as a barrier using a step function. We aim to provide relevant information for the experimental design of these devices and highlight the significance of environmental factors other than gate potential that affect the donor electron.
The discovery of efficient quantum algorithms a decade ago has shown that a quantum computer encoding and processing information quantum mechanically, can solve important problems intractable with conventional computers. The invention of quantum error correction principle, makes quantum computation possibly fault-tolerant against the decoherence of information carriers. Thereafter, intensive research activities have been made toward the implementation of quantum computation with various realistic quantum systems. Among them, the most attractive implementation proposals are using silicon-based materials, which have the advantage of borrowing the existing ingenuity and resources accumulated during the development of modern microelectronics. In this dissertation we investigate several theoretical aspects of a silicon- based quantum computer in which qubits are represented by the spins of electrons bound to phosphorous donors in silicon. Encoding each qubit in terms of three neighboring donor electron spins, we can realize universal quantum gate operations with only the Heisenberg exchange coupling J S1 •S2 between neighboring donors. Therefore, studying the exchange coupling for a phosphorous donor pair in silicon is of central importance for providing the experimentalists with qualitative insights and quantitative guidance for building such a silicon quantum computer. After giving some general considerations on the quantum computer architecture, we develop the necessary theoretical tools. A multi-valley effective mass equation is derived and discussed, to handle impurities in a multi-valley semiconductor. Then we apply it to solve a single Si:P donor embedded in our quantum computer architecture. We show that the width of the silicon quantum well can significantly influence the energy splitting and charge distributions of the ground state. Oscillation of level splitting is observed as the quantum well width or donor position is varied at atomic scale. Elementary gate operations for 3-donor-spin qubits involve a neighboring pair of donors at each step. The exchange coupling in Heisenberg model hamiltonian, defined as the energy splitting of the lowest singlet and triplet states, has to be calculated from the realistic two-donor hamiltonian. We discuss several popular methods to solve the two-donor problem and develop an appropriate extended Hartree-Fock method that can give reliable results for a well-separated donor pair subject to tunable coupling. This method is first applied to a Si:P donor pair in the framework of hydrogenic effective mass theory, to study how to tune the exchange coupling with simple gate potentials. A followed study shows that under a parallel electric field the singlet and triplet states exhibit very different polarization behaviors. This difference can be exploited to measure the state of an electron spin. We also show that, a perpendicular electric field cannot tune the exchange coupling efficiently. Then we apply the realistic multi-valley effective mass equation, coupled with a realistic modeling of the potential generated by gate electrodes, to a pair of phosphorous donors in silicon quantum well. By varying the gate electrode voltages, the exchange coupling can be tuned with exponential efficiency in a wide range. We find that, for best performance, we need to set the quantum well width to be around 10 nm, and the donor separation to be around 10 a∗ b---- 24nm in the doping plane. We analyze in detail an adiabatic half-swap operation between neighboring donor spins. The gate operation time is estimated to be 0.2 ns, which satisfies the constraints put by the donor spin decoherence time and by the validity of adiabatic approximation. The interference between different valley components could lead to oscillations of exchange coupling as donor positions are shifted by occasion. We find the exchange oscillation persists even with only two relevant valleys for donors in a strongly-strained silicon quantum well. It is also shown that the oscillation induced by changing donor separation at atomic scale is strongly suppressed as donors approach each other. Finally, we study the entanglement issue in quantum computing context. We analyze the low-energy Hilbert space for a pair of qubits encoded by localized electron spins and suggest a suitable measure to describe the entanglement between qubits involving indistinguishable electrons and in the presence of leakage errors. The dynamics of inter-qubit entanglement during a gate operation is also studied.
The spins of atoms and atom-like systems are among the most coherent objects in which to store quantum information. However, the need to address them using oscillating magnetic fields hinders their integration with quantum electronic devices. Here, we circumvent this hurdle by operating a single-atom "flip-flop" qubit in silicon, where quantum information is encoded in the electron-nuclear states of a phosphorus donor. The qubit is controlled using local electric fields at microwave frequencies, produced within a metal-oxide-semiconductor device. The electrical drive is mediated by the modulation of the electron-nuclear hyperfine coupling, a method that can be extended to many other atomic and molecular systems and to the hyperpolarization of nuclear spin ensembles. These results pave the way to the construction of solid-state quantum processors where dense arrays of atoms can be controlled using only local electric fields.