Spin quantum computation in silicon nanostructures

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 this review a particularly important spin decoherence channel, and bandstructure effects on the exchange gate control. Specifically, we review our work on donor electron spin spectral diffusion due to background nuclear spin flip-flops, and how isotopic purification of silicon can significantly enhance the electron spin dephasing time. We then review our calculation of donor electron exchange coupling in the presence of degenerate silicon conduction band valleys. We show that valley interference leads to orders of magnitude variations in electron exchange coupling when donor configurations are changed on an atomic scale. These studies illustrate the substantial potential that donor electron/nuclear spins in silicon have as candidates for qubits and simultaneously the considerable challenges they pose. In particular, our work on spin decoherence through spectral diffusion points to the possible importance of isotopic purification in the fabrication of scalable solid state quantum computer architectures. We also provide a critical comparison between the two main proposed spin-based solid state quantum computer architectures, namely, shallow donor bound states in Si and localized quantum dot states in GaAs.     

High-Field Phenomena of Qubits

Electron and nuclear spins are very promising candidates to serve as quantum bits (qubits) for proposed quantum computers, as the spin degrees of freedom are relatively isolated from their surroundings and can be coherently manipulated, e.g., through pulsed electron paramagnetic resonance (EPR) and nuclear magnetic resonance (NMR). For solid-state spin systems, impurities in crystals based on carbon and silicon in various forms have been suggested as qubits, and very long relaxation rates have been observed in such systems. We have investigated a variety of these systems at high magnetic fields in our multifrequency pulsed EPR/ENDOR (electron nuclear double resonance) spectrometer. A high magnetic field leads to large electron spin polarizations at helium temperatures, giving rise to various phenomena that are of interest with respect to quantum computing. For example, it allows the initialization of both the electron spin as well as hyperfine-coupled nuclear spins in a well-defined state by combining millimeter and radio-frequency radiation. It can increase the T ₂ relaxation times by eliminating decoherence due to dipolar interaction and lead to new mechanisms for the coherent electrical readout of electron spins. We will show some examples of these and other effects in Si:P, SiC:N and nitrogen-related centers in diamond.     

An all-silicon linear chain NMR quantum computer

An updated version of our all-silicon quantum computing scheme [T.D. Ladd, J.R. Goldman, F. Yamaguchi, Y. Yamamoto, E. Abe, K.M. Itoh, Phys. Rev. Lett. 89 (2002) 017901. [3]] and the experimental progress towards its realization are discussed. We emphasize the importance of revisiting a wide range of isotope effects which have been explored over the past several decades for the construction of solid-state silicon quantum computers. Using RF decoupling techniques [T.D. Ladd, D. Maryenko, Y. Yamamoto, E. Abe, K.M. Itoh, Phys. Rev. B. 71 (2005) 014401] phase decoherence times T₂=25 s of ²⁹Si nuclear spins in single-crystal Si have been obtained at room temperature. We show that a linear chain of ²⁹Si stable isotopes with nuclear spin I=1/2 embedded in a spin free ²⁸Si stable isotope matrix can form an ideal building block for solid-state quantum information processors, especially, in the form of a quantum memory which requires a large number of operations within T₂ for the continuous error correction.     

Solid-state quantum computation—a new direction for nanotechnology

We review current proposals for six types of solid-state quantum computers. We discuss the general requirements for solid-state quantum computers and describe proposals which employ superconducting junctions, electron orbitals in quantum dots, electron spin resonance, nuclear spins of impurity atoms, and nuclear spins in a crystal lattice. We also describe our proposed nuclear spin quantum computer based on magnetic resonance force microscopy. Finally, we describe our numerical method for modeling quantum transformations with a large number (up to 1000) of qubits.     

Ab initio determination of crystal lattice constants and thermal expansion for germanium isotopes

The temperature dependence of the equilibrium lattice constants for five isotopically pure Ge crystals (⁷⁰Ge, ⁷²Ge, ⁷³Ge, ⁷⁴Ge and ⁷⁶Ge) and naturally occurring Ge were calculated from ab initio electronic theory within the quasiharmonic approximation. It is shown that at very low temperature, calculations including zero point motions contribution to the free energy correctly reproduced the experimental unit cell parameters for the five isotopes. For ⁷⁰Ge, ⁷⁴Ge, ⁷⁶Ge and naturally occurring Ge, the predicted thermal expansion coefficients agree very well with experiments. The calculated thermal expansitivity for ⁷³Ge does not agree well with experiment. It is speculated that this anomaly may due to the non-zero nuclear spin of the ⁷³Ge isotope.     

Optical detection and ionization of donors in specific electronic and nuclear spin States

We resolve the remarkably sharp bound exciton transitions of highly enriched 28Si using a single-frequency laser and photoluminescence excitation spectroscopy, as well as photocurrent spectroscopy. Well-resolved doublets in the spectrum of the 31P donor reflect the hyperfine coupling of the electronic and nuclear donor spins. The optical detection of the nuclear spin state, and selective pumping and ionization of donors in specific electronic and nuclear spin states, suggests a number of new possibilities which could be useful for the realization of silicon-based quantum computers.     

Preparation of quantum states of two spin-12 particles in the form of the Schmidt decomposition

We consider a system of two spins that are coupled via an isotropic Heisenberg Hamiltonian. For the first time, a two-step method for the preparation of an arbitrary quantum state of two qubits in the form of the Schmidt decomposition is proposed. The simplified version of this method is applied to the physical system of an atom with a nuclear spin 1/2 and one valence electron. As an example, the preparation of two-spin quantum states in the ³¹P system is considered.     

Crystal lattice quantum computer

31 P nucleus can be used to represent a quantum bit (‘qubit’) with a relatively long relaxation time. In a CeP crystal lattice, ³¹P nuclei are periodically situated in three dimensions at distances of about 6 Å. The application of a static magnetic field gradient in one direction causes differences in the Zeemanfrequencies of separate nuclei. This allows thousands of distinct qubits to be individually addressed. Initializations of the qubits can be done efficiently by the Pound–Overhauser double resonance effect on the nuclear spins and the antiferromagnetically ordered 4f electron spins of cerium ions. Logic operations can be performed by simple pulse sequences, and computational results after logic operations can be measured by the nuclear magnetic resonance of neighboring nuclei, or the electron resonance of neighboring 4f electrons of cerium ions. Received: 26 October 1998/Accepted: 29 October 1998     

Characterization of ZnO and Zn0.95Co0.05O prepared by sol-gel method using PAC spectroscopy

We report on the first NMR study of ⁷³Ge nuclear spin decoherence in germanium single crystals with different abundance of the ⁷³Ge isotope. Hahn echo decays are well fit by a superposition of two exponentials. The deviation from the single exponential is more pronounced in the more spin-diluted sample, causing long-lived echoes. We show that the decay of these echoes becomes slower with the reduction of ⁷³Ge abundance and is therefore caused by dipole–dipole interaction, reflecting the fundamental decoherence process in the spin system. The fast decay at the beginning of the relaxation process is shown to be mainly caused by the quadrupole interaction. Our experimental findings are supported by the calculations of Hahn echo decays in the germanium crystals under study. Quite good agreement between the theory and experiment is demonstrated.     

Electrical detection of coherent nuclear spin oscillations in phosphorus-doped silicon using pulsed ENDOR

We demonstrate the electrical detection of pulsed X-band electron nuclear double resonance (ENDOR) in phosphorus-doped silicon at 5 K. A pulse sequence analogous to Davies ENDOR in conventional electron spin resonance is used to measure the nuclear spin transition frequencies of the (31)P nuclear spins, where the (31)P electron spins are detected electrically via spin-dependent transitions through Si/SiO(2) interface states, thus not relying on a polarization of the electron spin system. In addition, the electrical detection of coherent nuclear spin oscillations is shown, demonstrating the feasibility to electrically read out the spin states of possible nuclear spin qubits.     

Magnetic effects in the oxidation of silicon

The mass spectrometry study has indicated that the magnetic field accelerates the oxidation of the surface of silicon crystals. The oxidation rate also depends on the nuclear spin of silicon: the oxidation rate of atoms with magnetic nuclei (²⁹Si) is almost twice as high as that of atoms with spinless, unmagnetized nuclei (²⁸Si and ³⁰Si). Both effects—magnetic field and magnetic isotope—reliably prove that the oxidation of silicon is a spin-selective reaction involving radicals and radical pairs as intermediate paramagnetic particles. A spin-selective magnetic sensitive oxidation mechanism is discussed.     

Quantum logic via the exchange blockade in ultracold collisions

A nuclear spin can act as a quantum switch that turns on or off ultracold collisions between atoms even when there is neither interaction between nuclear spins nor between the nuclear and electron spins. This "exchange blockade" is a new mechanism for implementing quantum logic gates that arises from the symmetry of composite identical particles, rather than direct coupling between qubits. We study the implementation of the entangling sqrt SWAP gate based on this mechanism for a model system of two atoms, each with ground electronic configuration 1S0, spin 1/2 nuclei, and trapped in optical tweezers. We evaluate a proof-of-principle protocol based on adiabatic evolution of a one-dimensional double Gaussian well, calculating fidelities of operation as a function of interaction strength, gate time, and temperature.     

Sideband cooling while preserving coherences in the nuclear spin state in group-II-like atoms

We propose a method for laser cooling group-II-like atoms without changing the quantum state of their nuclear spins, thus preserving coherences that are usually destroyed by optical pumping in the cooling process. As group-II-like atoms have a (1)S(0) closed-shell ground state, nuclear spin and electronic angular momentum are decoupled, allowing for their independent manipulation. The hyperfine interaction that couples these degrees of freedom in excited states can be suppressed through the application of external magnetic fields. Our protocol employs resolved-sideband cooling on the forbidden clock transition, (1)S(0) --> (3)P(0), with quenching via coupling to the rapidly decaying (1)P(1) state, deep in the Paschen-Back regime. This makes it possible to laser cool neutral atomic qubits without destroying the quantum information stored in their nuclear spins, as shown in two examples, (171)Yb and (87)Sr.     

73Ge NMR spectra in germanium single crystals with different isotopic composition

We have studied the influence of isotopic disorder on the local deformations in Ge single crystals from both experimental and calculation points of view. The nuclear magnetic resonance (NMR) spectra of⁷³Ge nuclei (the nuclear spin equals 9/2) in perfect single crystals of germanium with different isotopic content were measured at temperatures 80, 300 and 450 K. Abnormal broadening of the spectrum was found to occur when the magnetic field was aligned along the [111] axis of a crystal. The observed specific angular dependence of the quadrupole broadening was attributed to isotopic disorder among atoms of germanium sited around the⁷³Ge NMR probe. Local lattice deformations in germanium crystal lattice due to isotopic impurity atoms were calculated in the framework of the adiabatic bond charge model. The results obtained were applied to study random noncubic crystal field interactions with the nuclear quadrupole moments and corresponding effects in NMR spectra. Simulated second and fourth moments of resonance frequency distributions caused by the magnetic dipole-dipole and electric quadrupole interactions are used to analyze the lineshapes, theoretical predictions agree qualitatively with the experimental data.     

Quantum computation using the 13C nuclear spins near the single NV defect center in diamond

We discuss the possibility of realizing quantum computation on the basis of a cluster of single interacting nuclear spins in solids. This idea seems to be feasible because of the combination of two techniques—Single Molecule Spectroscopy and Optically Detected Electron Nuclear Double Resonance. Compared to the well-known bulk Nuclear Magnetic Resonance (NMR), the proposed method of quantum computation has the advantage that quantum computation is performed with pure spin states and the quantum processor is more easily scalable. At the same time, the advantages of NMR quantum computation are kept: long coherence time and easy construction of quantum gates. As a specific system to implement the above idea, we discuss the ¹³C-nuclear spins in the nearest vicinity of a single nitrogen-vacancy (NV) defect center in diamond, which can be optically detected using the technique of scanning confocal microscopy. Owing to the hyperfine coupling of the ground state electron paramagnetic spin S=1 of the center to ¹³C nuclear spins in a diamond lattice, the states of nuclear spins in the vicinity of the defect-center can be addressed individually. Preliminary consideration shows that it should be possible to address up to 12 individual ¹³C nuclear spins. The dephasing time of the nuclear spin states at low temperatures allows realization up to 10⁵ gates.     

An endohedral fullerene-based nuclear spin quantum computer

We propose a new scalable quantum computer architecture based on endohedral fullerene molecules. Qubits are encoded in the nuclear spins of the endohedral atoms, which posses even longer coherence times than the electron spins which are used as the qubits in previous proposals. To address the individual qubits, we use the hyperfine interaction, which distinguishes two modes (active and passive) of the nuclear spin. Two-qubit quantum gates are effectively implemented by employing the electronic dipolar interaction between adjacent molecules. The electron spins also assist in the qubit initialization and readout. Our architecture should be significantly easier to implement than earlier proposals for spin-based quantum computers, such as the concept of Kane [B.E. Kane, Nature 393 (1998) 133].     

Quantum computing by optical control of electron spins

We review the progress and main challenges in implementing large-scale quantum computing by optical control of electron spins in quantum dots (QDs). Relevant systems include self-assembled QDs of III–V or II–VI compound semiconductors (such as InGaAs and CdSe), monolayer fluctuation QDs in compound semiconductor quantum wells, and impurity centres in solids, such as P-donors in silicon and nitrogen-vacancy centres in diamond. The decoherence of the electron spin qubits is discussed and various schemes for countering the decoherence problem are reviewed. We put forward designs of local nodes consisting of a few qubits which can be individually addressed and controlled. Remotely separated local nodes are connected by photonic structures (microcavities and waveguides) to form a large-scale distributed quantum system or a quantum network. The operation of the quantum network consists of optical control of a single electron spin, coupling of two spins in a local nodes, optically controlled quantum interfacing between stationary spin qubits in QDs and flying photon qubits in waveguides, rapid initialization of spin qubits and qubit-specific single-shot non-demolition quantum measurement. The rapid qubit initialization may be realized by selectively enhancing certain entropy dumping channels via phonon or photon baths. The single-shot quantum measurement may be in situ implemented through the integrated photonic network. The relevance of quantum non-demolition measurement to large-scale quantum computation is discussed. To illustrate the feasibility and demand, the resources are estimated for the benchmark problem of factorizing 15 with Shor's algorithm.     

Hall effect measurements on transmutation doped semiconductors

Transmutation doping of semiconductors by neutron irradiation is a well known technique, mainly applied to achieve an extremely uniform low level n-doping of large Si crystals via the ³⁰Si(n,γ)³¹Si→³¹P nuclear reaction. Similar experiments in other semiconductors never gained a comparable importance. In the last years, however, it has been shown that the doping of semiconductors by implanting radioactive isotopes can yield valuable information about the processes occurring during the incorporation of dopant atoms into the lattice as well as the defect–dopant interactions occurring after the decay of the unstable isotope to a daughter isotope with usually different elemental properties. In this contribution, Hall effect measurements carried out so far on implanted radioactive dopants will be reviewed. The specific problems and the potential of the method will be discussed. This revised version was published online in July 2006 with corrections to the Cover Date.     

Storage of multiple coherent microwave excitations in an electron spin ensemble

Strong coupling between a microwave photon and electron spins, which could enable a long-lived quantum memory element for superconducting qubits, is possible using a large ensemble of spins. This represents an inefficient use of resources unless multiple photons, or qubits, can be orthogonally stored and retrieved. Here we employ holographic techniques to realize a coherent memory using a pulsed magnetic field gradient and demonstrate the storage and retrieval of up to 100 weak 10?GHz coherent excitations in collective states of an electron spin ensemble. We further show that such collective excitations in the electron spin can then be stored in nuclear spin states, which offer coherence times in excess of seconds.     

Measurement of the electric quadrupole moments of 26–29Na

The nuclear electric quadrupole moments of the isotopes ²⁶Na, ²⁷Na, ²⁸Na and ²⁹Na were measured by -NMR spectroscopy in single crystals of LiNbO₃ and NaNO₃. High degrees of nuclear polarization were produced by optical pumping of the sodium atoms in a fast beam with a collinear laser beam. The polarized nuclei were implanted into the crystals and NMR signals were observed in the -decay asymmetries. Preparatory measurements also yielded improved values for the magnetic moments of ^27-31Na and confirmed the spin I=3/2 for ³¹Na. The results are discussed in comparison with large-basis shell model calculations. Received: 1 February 2000 / Accepted: 3 April 2000