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.     

<Superscript>29</Superscript>Si nuclear spin relaxation in microcrystals of plastically deformed Si: B samples

Single crystals and microcrystals Si: B enriched with ²⁹Si isotopes have been studied using nuclear magnetic resonance and electron paramagnetic resonance (EPR) in the temperature range from 300 to 800 K. It has been found that an increase in the temperature from 300 to 500 K leads to a change in the kinetics of the relaxation of the saturated nuclear spin system. At 300 K, the relaxation kinetics corresponds to direct electron–nuclear interaction with inhomogeneously distributed paramagnetic centers introduced by the plastic deformation of the crystals. At 500 K, the spin relaxation occurs through the nuclear spin diffusion and electron–nuclear interaction with an acceptor impurity. It has been revealed that the plastic deformation affects the EPR spectra at 9 K.     

Relaxation of protons by radicals in rotationally immobilized proteins

Proton spin-lattice relaxation by paramagnetic centers may be dramatically enhanced if the paramagnetic center is rotationally immobilized in the magnetic field. The details of the relaxation mechanism are different from those appropriate to solutions of paramagnetic relaxation agents. We report here large enhancements in the proton spin-lattice relaxation rate constants associated with organic radicals when the radical system is rigidly connected with a rotationally immobilized macromolecular matrix such as a dry protein or a cross-linked protein gel. The paramagnetic contribution to the protein-proton population is direct and distributed internally among the protein protons by efficient spin diffusion. In the case of a cross-linked-protein gel, the paramagnetic effects are carried to the water spins indirectly by chemical exchange mechanisms involving water molecule exchange with rare long-lived water molecule binding sites on the immobilized protein and proton exchange. The dramatic increase in the efficiency of spin relaxation by organic radicals compared with metal systems at low magnetic field strengths results because the electron relaxation time of the radical is orders of magnitude larger than that for metal systems. This gain in relaxation efficiency provides completely new opportunities for the design of spin-lattice relaxation based contrast agents in magnetic imaging and also provides new ways to examine intramolecular protein dynamics.     

NMR Spin Trapping: Insight into the Hidden Life of Free Radical Adducts

In our original work (V. Khramtsov, L.J. Berliner and T.L. Clanton, Magn. Reson. Med. 42:228??34, 1999), a ³¹P nuclear magnetic resonance spin trapping (ST NMR) approach aimed to overcome comparatively short lifetimes of the paramagnetic adducts by detecting stable diamagnetic adducts of the degradation of phosphorous-containing nitrone was proposed. High stability of the NMR adducts of the nitrone spin traps with C-centered radicals and the ability to track their origin from paramagnetic adducts make ST NMR a valuable tool for the studies of these radicals in chemical and biological systems. Complementary ST NMR and electron paramagnetic resonance studies of the nitrone adducts derived from the addition of O- and S-centered radicals and nucleophiles revealed new insights into the pathways of the paramagnetic and diamagnetic adduct formation and degradation. In particular, use of ST NMR allowed for observation of reversible nucleophilic addition to the nitrones. On one hand, nucleophilic addition may result in a possible spin trapping artifact via the Forrester–Hepburn mechanism; whereas, on the other hand, the reverse reaction provides the key step in the “recycling??mechanism that is important for the antioxidant actions of the nitrones.     

Modulation of the distance dependence of paramagnetic relaxation enhancements by CSA x DSA cross-correlation

Paramagnetic metal ions with fast-relaxing electronic spin and anisotropic susceptibility tensor provide a rich source of structural information that can be derived from pseudo-contact shifts, residual dipolar couplings, dipole-dipole Curie spin cross-correlation, and paramagnetic relaxation enhancements. The present study draws attention to a cross-correlation effect between nuclear relaxation due to anisotropic chemical shielding (CSA) and due to the anisotropic dipolar shielding (DSA) caused by the electronic Curie spin. This CSA x DSA cross-correlation contribution seems to have been overlooked in previous interpretations of paramagnetic relaxation enhancements. It is shown to be sufficiently large to compromise the 1/r6 distance dependence usually assumed. The effect cannot experimentally be separated from auto-correlated DSA relaxation. It can increase or decrease the observed paramagnetic relaxation enhancement. Under certain conditions, the effect can dominate the entire paramagnetic relaxation, resulting in nuclear resonances narrower than in the absence of the paramagnetic center. CSAxDSA cross-correlation becomes important when paramagnetic relaxation is predominantly due to the Curie rather than the Solomon mechanism. Therefore the effect is most pronounced for relaxation by metal ions with large magnetic susceptibility and fast-relaxing electron spin. It most strongly affects paramagnetic enhancements of transverse relaxation in macromolecules and of longitudinal relaxation in small molecules.     

Modulation of the distance dependence of paramagnetic relaxation enhancements by CSA × DSA cross-correlation

Paramagnetic metal ions with fast-relaxing electronic spin and anisotropic susceptibility tensor provide a rich source of structural information that can be derived from pseudo-contact shifts, residual dipolar couplings, dipole-dipole Curie spin cross-correlation, and paramagnetic relaxation enhancements. The present study draws attention to a cross-correlation effect between nuclear relaxation due to anisotropic chemical shielding (CSA) and due to the anisotropic dipolar shielding (DSA) caused by the electronic Curie spin. This CSA x DSA cross-correlation contribution seems to have been overlooked in previous interpretations of paramagnetic relaxation enhancements. It is shown to be sufficiently large to compromise the 1/r6 distance dependence usually assumed. The effect cannot experimentally be separated from auto-correlated DSA relaxation. It can increase or decrease the observed paramagnetic relaxation enhancement. Under certain conditions, the effect can dominate the entire paramagnetic relaxation, resulting in nuclear resonances narrower than in the absence of the paramagnetic center. CSAxDSA cross-correlation becomes important when paramagnetic relaxation is predominantly due to the Curie rather than the Solomon mechanism. Therefore the effect is most pronounced for relaxation by metal ions with large magnetic susceptibility and fast-relaxing electron spin. It most strongly affects paramagnetic enhancements of transverse relaxation in macromolecules and of longitudinal relaxation in small molecules.     

13C spin-lattice relaxation in natural diamond: Zeeman relaxation at 4.7 T and 300 K due to fixed paramagnetic nitrogen defects

13C spin-lattice relaxation times in the laboratory frame, ranging from 1.4 to 36 h, have been measured on a suite of five natural type Ia and Ib diamonds at 4.7 T and 300 K. Each of the diamonds contains two types of fixed paramagnetic centers with overlapping inhomogeneous electron paramagnetic resonance (EPR) lines. EPR techniques have been employed to identify these defects and to determine their concentrations and relaxation times at X-band. Spin-lattice relaxation behavior of 13C in diamonds containing paramagnetic P1, P2, N2. and N3 centers are discussed. Depending on the paramagnetic impurity types and concentrations present in each diamond, three different nuclear spin-lattice relaxation (SLR) paths exist, namely that due to electron SLR mechanisms and two types of three-spin processes (TSPs). The one three-spin process (TSP1) involves a simultaneous transition of two electron spins belonging to the same hyperfine EPR line and a flip of a 13C spin, while the other process (TSP2) involves two electron spins belonging to different hyperfine EPR lines and a 13C spin. It is shown that the thermal contact between the 13C nuclear Zeeman and electron dipole-dipole interaction reservoirs is field dependent, thus forming a bottleneck in the 13C relaxation path due to TSP1 at high magnetic fields.     

Multifrequency time-resolved EPR (9.5GHz and 95GHz) on covalently linked porphyrin-quinone model systems for photosynthetic electron transfer: effect of molecular dynamics on electron spin polarization

Covalently linked porphyrin–quinone model systems for photosynthetic electron transfer were examined by using time-resolved electron paramagnetic resonance (TREPR) at intermediate magnetic field and microwave frequency (0.34T/9.5GHz, X-band) and high field and frequency (3.4T/95GHz, W-band). The paramagnetic transients studied were the light-induced spin-correlated radical pair states of the donor–acceptor complex in polar solvents below the melting point and in the soft glass phase of a liquid crystal. It is shown that the systems form strongly exchange-coupled radical pairs, whose TREPR lineshapes are determined mainly by fast electron recombination together with both spin–lattice relaxation and modulation of the exchange interaction. Below the melting point the spin–lattice relaxation rate naturally slows down, but that of the spin on the quinone site is still of the order of 10⁶ s^-1. Most probably this is due to contributions from spin–rotation interaction, and dependent on the molecular orientation with respect to the magnetic field. This relaxation anisotropy is related to anisotropic motion of the quinone site in the solvent cage. The results allow conclusions to be drawn concerning the molecular dynamics and flexibility of the systems. To yield long-lived radical pair states that would mimic photosynthetic electron transfer, the two mechanisms described, modulation of exchange and spin–rotation interactions, have to be suppressed by reducing the molecular flexibility of the complex.     

Towards Determination of Distances Between Nanoparticles and Grafted Paramagnetic Ions by NMR Relaxation

We developed an approach for determining distances between carbon nanoparticles and grafted paramagnetic ions and molecules by means of nuclear spin–lattice relaxation data. The approach was applied to copper-, cobalt- and gadolinium-grafted nanodiamonds, iron-grafted graphenes, manganese-grafted graphene oxide and activated carbon fibers that adsorb paramagnetic oxygen molecules. Our findings show that the aforementioned distances vary in the range of 2.7–5.4 Å and that the fixation of paramagnetic ions to nanoparticles is most likely implemented by means of the surface functional groups. The nuclear magnetic resonance data data are compared with the results of electron paramagnetic resonance measurements and density functional theory calculations.     

End-to-end correlation for a C-12 hydrocarbon chain

The 19F nuclear spin-lattice relaxation rate constants were measured as a function of magnetic field strength for 1,12-diaminododecane labeled at one end with a nitroxide radical and at the other with a trifluoromethyl group. The magnetic relaxation dispersion profile (MRD) reports the spectral density function appropriate to the end-to-end correlation function for the doubly labeled molecule. After extrapolation to zero concentration to eliminate the intermolecular relaxation contribution to relaxation, the resulting intramolecular MRD profile was compared with several model approaches. The rotational model for the spectral density functions as included in the Solomon-Bloembergen-Morgan equations does not describe the data well. The earlier model of Freed for nuclear spin relaxation induced by a freely diffusing paramagnetic co-solute is not rigorous for this case because the paramagnet is tethered to the observed nuclear spin and only a restricted space in the immediate vicinity of the nuclear spin is accessible for pseudo-translational diffusion of one end of the molecule with respect to the other. A generalization of the Torrey model for magnetic relaxation by translational diffusion developed by Nevzorov and Freed, which includes the effect of restrictions imposed by the finite length of the chain, describes the experiment within experimental errors. A simple modification of the Hwang-Freed model that does not specifically include the dynamical effects of the finite tether also provides a good approximation to the data when the tether chain is sufficiently long.     

Longitudinal muon spin relaxation in muonium-like systems

Nuclear spin-lattice relaxation in paramagnetic systems is treated using the classic expression for transition probability between the coupled electron and nuclear spin states. The rate equations governing the incoherent occupancies of these states are solved analytically (where possible) and numerically (where not) to construct the relaxation function for the nuclear spin. The method is illustrated for muonium, and the muonium-substituted molecular radicals, for the case of perturbation due to fluctuation of the local field,i.e. modulation of the interaction with a third spin. A slight departure from single exponential behaviour is demonstrated for slow fluctuations.     

Slow-motion theory of nuclear spin relaxation in paramagnetic low-symmetry complexes: A generalization to high electron spin

The slow-motion theory of nuclear spin relaxation in paramagnetic low-symmetry complexes is generalized to comprise arbitrary values of S. We describe the effects of rhombic symmetry in the static zero-field splitting (ZFS) and allow the principal axis system of the static ZFS tensor to deviate from the molecule-fixed frame of the nuclear-electron dipole–dipole tensor. We show nuclear magnetic relaxation dispersion (NMRD) profiles for different illustrative cases, ranging from within the Redfield limit into the slow-motion regime with respect to the electron spin dynamics. We focus on S = 3/2 and compare the effects of symmetry-breaking properties on the paramagnetic relaxation enhancement (PRE) in this case with that of S = 1, which we have treated in a previous paper. We also discuss cases of S = 2, 5/2, 3, and 7/2. One of the main objectives of this investigation, together with the previous papers, is to provide a set of standard calculations using the general slow-motion theory, against which simplified models may be tested.     

Understanding Overhauser Dynamic Nuclear Polarisation through NMR relaxometry

Overhauser dynamic nuclear polarisation (DNP) represents a potentially outstanding tool to increase the sensitivity of solution and solid state NMR experiments, as well as of magnetic resonance imaging. DNP signal enhancements are strongly linked to the spin relaxation properties of the system under investigation, which must contain a paramagnetic molecule used as DNP polariser. In turn, nuclear spin relaxation can be monitored through NMR relaxometry, which reports on the field dependence of the nuclear relaxation rates, opening a route to understand the physical processes at the origin of the Overhauser DNP in solution. The contributions of dipole–dipole and Fermi-contact interactions to paramagnetic relaxation are here described and shown to be responsible to both the relaxometry profiles and the DNP enhancements, so that the experimental access to the former can allow for predictions of the latter.     

Outer-sphere nuclear spin relaxation in paramagnetic systems: a low-field theory

A low-field theory for paramagnetic relaxation enhancement (PRE), appropriate for the outer-sphere relaxation, is presented for the electron spin quantum number S = 1, 3/2, 2, 5/2, 3 and 7/2. The theory is used to calculate the PRE at low magnetic field, as a function of the translational diffusion coefficient, for various values of the electron spin quantum number, for small and fairly large values of the static zero-field splitting (ZFS), and for a given set of parameters determining the electron spin relaxation. We have found earlier that the static ZFS has a profound influence on the electron spin relaxation; such effects are also evident in the present study. Comparisons are made with other existing models for the outer-sphere PRE, and significant differences are found for slowly diffusing systems with large ZFS. The theory is also used to obtain a novel interpretation of experimental data for an acetone solution of a Mn(III) complex.     

Cross correlation between the dipole-dipole interaction and the Curie spin relaxation: the effect of anisotropic magnetic susceptibility

Cross-correlated relaxation caused by the interference of nuclear dipole-dipole interaction and the Curie spin relaxation (DD-CSR cross relaxation) is generalized to treat the case of anisotropic magnetic susceptibility, including the important case where the latter originates from zero-field splitting. It is shown that the phenomenon of DD-CSR cross relaxation is absolutely general and to be expected under any electronic configuration. The results of the generalization are presented for a model system, and the consequences for paramagnetic metalloproteins are illustrated with an example of cerium(III)-substituted calbindin. The effects of the magnetic anisotropy are found to be substantial.     

Development of the Earth’s Field NMR Spectrometer for Liquid Screening

Earth??s field nuclear magnetic resonance (NMR) can be used for non-destructive in-bottle liquid screening by evaluating the spin?Clattice relaxation times of protons in various liquids. An Earth??s field NMR apparatus has been developed and optimized to measure the spin?Clattice relaxation times of various liquids contained in 500?ml PET bottles. Two methods to generate 90-degree readout pulses using transient oscillating signals generated in a resonator are reported. The lower detection limit of pure water was 1?ml. The minimum measurable spin?Clattice relaxation time was 50?ms.     

Electron paramagnetic resonance in YbNiAl<Subscript>2</Subscript> single crystals with strong magnetic anisotropy

Anisotropy in the magnetic properties of YbNiAl₂ intermetallide has been studied. Electron paramagnetic resonance (EPR) signals assigned to the localized magnetic moments of trivalent ytterbium have been detected at temperatures below 20 K. Spin–lattice relaxation processes like the Orbach–Aminov process with participation of the first excited Stark sublevel of the Yb³⁺ ion with an energy of 96 K govern electron–spin dynamics and the disappearance of spectrum lines with a further increase in temperature. Strong magnetic anisotropy effects are discussed as a main reason for the appearance of electron paramagnetic resonance.