Resonance Offsets Research Articles

Pulsed dynamic nuclear polarization: a comprehensive Floquet description.

Dynamic nuclear polarization (DNP) experiments using microwave (mw) pulse sequences are one approach to transfer the larger polarization on the electron spin to nuclear spins of interest. How the result of such experiments depends on the external magnetic field and the excitation power is part of an ongoing debate and of paramount importance for applications that require high chemical-shift resolution. To date numerical simulations using operator-based Floquet theory have been used to predict and explain experimental data. However, such numerical simulations provide only limited insight into parameters relevant for efficient polarization transfer, such as transition amplitudes or resonance offsets. Here we present an alternative method to describe pulsed DNP experiments by using matrix-based Floquet theory. This approach leads to analytical expressions for the transition amplitudes and resonance offsets. We validate the method by comparing computations by these analytical expressions to their numerical counterparts and to experimental results for the XiX, TOP and TPPM DNP sequences. Our results explain the experimental data and are in very good agreement with the numerical simulations. The analytical expressions allow for the discussion of the scaling behaviour of pulsed DNP experiments with respect to the external magnetic field. We find that the transition amplitudes scale inversely with the external magnetic field.

Tidal interactions shape period ratios in planetary systems with three-body resonant chains

Aims. These last years several STIPs (Systems with Tightly packed Inner Planets) in the super-Earth mass regime have been discovered harboring chains of resonances. It is generally believed that planet pairs get trapped in MMR (mean-motion resonance) during the migration phase in the protoplanetary disk, while the tides raised by the host star provide a source of dissipation on very long timescales. In this work, we aim to study the departure from exact commensurabilities observed among the STIPs which harbor 3-planet resonances and analyze how tides play an important role in shaping the resonance offsets for the STIPs. Methods. We analyzed the resonance offsets between adjacent pairs for five multi-planetary systems, namely Kepler-80, Kepler-223, K2-138, TOI-178, and TRAPPIST-1, highlighting the existence of different trends in the offsets. On the one hand, we derived analytical estimates for the offsets, which confirm that the departure of the planetary pairs from the nominal MMRs are due to the 3-planet resonant dynamics. On the other hand, we performed N-body simulations including both orbital migration and tidal dissipation from the host star with simple prescriptions in order to test the effectiveness of this mechanism at shaping the observed trend in the offsets, focusing our study on the preservation of the resonant patterns in the different systems with the same general setup. Results. We found that the trends in the offsets of the five detected systems can be produced by tidal damping effects, regardless of the considered value for the tidal factor. It is a robust mechanism that relaxes the system towards equilibrium while efficiently moving it along 3-planet resonances, which induces the observed resonance offset for each planet pair. In addition, we showed that for Kepler-80, K2-138, and TOI-178, the amplitudes of the resonant offsets can also be reproduced with an appropriate tidal factor, for the estimated age of the systems.

A robust Freeman-Hill-inspired pulse protocol for ringdown-free T1 relaxation measurements

A new difference-spectroscopy method is introduced for measuring T1 relaxation times. It is inspired by the earlier work of Freeman and Hill and eliminates the need for recording signal intensities at thermodynamic equilibrium. The new method is termed SIP-R (Split-Inversion Pulse and Recovery) and reduces the number of refinable parameters in the curve fitting process of relaxation-delay-dependent signal intensities by using two instead of the three parameters typically used in the standard inversion-recovery sequence. The SIP-R method preserves the dynamic range of measurement of the standard inversion-recovery method but converts the rise-to-maximum mathematical functionality of the recorded data into a decay-to-zero functionality. The decay-to-zero functionality renders the SIP-R sequence advantageous for inverse Laplace transformation numerical optimizations. The new technique proves to be extremely robust with respect to pulse imperfections, pulse-power changes during the pulse sequence, pulse-width miscalibrations, resonance offsets, and radiofrequency field variations. It also compensates for acoustic ring-down effects and proves reliable for measurements with inhomogeneously broadened signals up to several kilohertz linewidth. 1H NMR experiments with methane gas at pressures up to 50 atm in toroid-cavity pressure vessel probes and in the presence of the methane-to-methanol conversion catalyst Cu-ZnO/Al2O3 are used to show the usefulness of the new method for relaxation time investigations under pressure, at strong radiofrequency field gradients, and in the presence of paramagnetic materials.

Proximity of exoplanets to first-order mean-motion resonances

ABSTRACT Planetary formation theories and, more specifically, migration models predict that planets can be captured in mean-motion resonances (MMRs) during the disc phase. The distribution of period ratios between adjacent planets shows an accumulation in the vicinity of the resonance, which is not centred on the nominal resonance but instead presents an offset slightly exterior to it. Here, we extend on previous works by thoroughly exploring the effect of different disc and planet parameters on the resonance offset during the disc migration phase. The dynamical study is carried out for several first-order MMRs and for both low-mass Earth-like planets undergoing type-I migration and giant planets evolving under type-II migration. We find that the offset varies with time during the migration of the two-planet system along the apsidal corotation resonance family. The departure from the nominal resonance increases for higher planetary masses and stronger eccentricity damping. In the Earth to super-Earth regime, we find offset values in agreement with the observations when using a sophisticated modelling for the planet–disc interactions, where the damping time-scale depends on the eccentricity. This dependence causes a feedback that induces an increase of the resonance offsets. Regarding giant planets, the offsets of detected planet pairs are well reproduced with a classical K-factor prescription for the planet–disc interactions when the eccentricity damping rate remains low to moderate. In both regimes, eccentricities are in agreement with the observations too. As a result, planet–disc interactions provide a generic channel to generate the offsets found in the observations.

Mechanism of chirp excitation

In design of a chirp inversion pulse, we keep the sweep rate a≪ω12, where ω1 is the amplitude of the pulse. This is the adiabaticity condition for the inversion to work. We can convert a chirp inversion pulse to an excitation pulse by keeping the chirp rate high, with a>ω12. To be precise a=2.8ω12. The analysis of such a pulse breaks the evolution into three phases. The first and third phase are adiabatic phases while the second phase is non-adiabatic. Starting from north pole such a pulse brings the magnetization to equator, however there is a nonuniform phase which depends on the resonance offsets. We show how by following this pulse with a chirp inversion pulse at twice the sweep rate of excitation pulse, we can refocus this uniform phase. We find there is still some phase dispersion. This can be further eliminated by bringing in a second inversion pulse. The combination of these three chirp pulses allows us to excite arbitrary large bandwidth without increasing the peak amplitude of the pulse. Refocusing properties of pair of chirp has been studied before but our description is very different.

Application of the small-tip-angle approximation in the toggling frame for the design of analytic robust pulses in quantum control

We apply the Small Tip-Angle Approximation in the Toggling Frame in order to analytically design robust pulses against resonance offsets for state to state transfer in two-level quantum systems. We show that a broadband or a local robustness up to an arbitrary order can be achieved. We provide different control parameterizations to satisfy experimental constraints and limitations on the amplitude or energy of the pulse. A comparison with numerical optimal solutions is made.

Numerical Engineering of Robust Adiabatic Operations

Adiabatic operations are powerful tools for robust quantum control in numerous fields of physics, chemistry and quantum information science. The inherent robustness due to adiabaticity can, however, be impaired in applications requiring short evolution times. We present a single versatile gradient-based optimization protocol that combines adiabatic control with effective Hamiltonian engineering in order to design adiabatic operations tailored to the specific imperfections and resources of an experimental setup. The practicality of the protocol is demonstrated by engineering a fast, 2.3 Rabi cycle-long adiabatic inversion pulse for magnetic resonance with built-in robustness to Rabi field inhomogeneities and resonance offsets. The performance and robustness of the pulse is validated in a nanoscale force-detected magnetic resonance experiment on a solid-state sample, indicating an ensemble-averaged inversion accuracy of $\sim 99.997\%$. We further showcase the utility of our protocol by providing examples of adiabatic pulses robust to spin-spin interactions, parameter-selective operations and operations connecting arbitrary states, each motivated by experiments.

Assessment of new symmetry-based dipolar recoupling schemes for homonuclear magnetization exchange between quadrupolar nuclei in two-dimensional correlation MAS NMR

We provide an extensive experimental and numerical evaluation of MQ-phase (S)M supercycles with M={3,4} of three groups of symmetry-based homonuclear dipolar recoupling rf-pulse sequences, S={SR22p1,SR441,SRNNN/2}, for establishing proximities among half-integer spin quadrupolar nuclei under moderately fast magic-angle-spinning (MAS) conditions in single-quantum–single-quantum (1Q–1Q) correlation NMR experiments. The relative merits of the (S)M schemes for variations in resonance offsets and rf-amplitude errors were assessed by numerically simulated magnetization transfers in spin-3/2 pairs with variable isotropic chemical shifts and quadrupolar coupling constants. Experimental demonstrations of 23Na (spin-3/2) NMR on Na2MoO4·2H2O and 27Al (spin-5/2) NMR on AlPO-CJ19 [(NH4)2Al4(PO4)4HPO4·H2O] are presented at 14.1 T and 24 kHz MAS. We recommend using the (SR221)3 or (SR221)4 supercycles for samples that exhibit small chemical-shift dispersions (<3 kHz), and any (SRNNN/2)3 scheme with N⩾10 for larger spreads of isotropic chemical shifts. However, because the (SRNNN/2)3 sequences recouple heteronuclear dipolar interactions, their application to proton-bearing samples requires high-power proton decoupling during the mixing period. Alternatively, the (SR241)3 and (SR241)4 schemes may be employed in the absence of proton decoupling, but with poorer compensation to resonance-offsets and rf-amplitude errors.

Improved Magnetization Transfers among Quadrupolar Nuclei in Two-Dimensional Homonuclear Correlation NMR Experiments Applied to Inorganic Network Structures.

We demonstrate that supercycles of previously introduced two-fold symmetry dipolar recoupling schemes may be utilized successfully in homonuclear correlation nuclear magnetic resonance (NMR) spectroscopy for probing proximities among half-integer spin quadrupolar nuclei in network materials undergoing magic-angle-spinning (MAS). These (SR2) (SR2) and (SR2)M recoupling sequences with and offer comparably efficient magnetization transfers in single-quantum–single-quantum (1Q–1Q) correlation NMR experiments under moderately fast MAS conditions, as demonstrated at 14.1 T and 24 kHz MAS in the contexts of B NMR on a NaO–CaO–BO–SiO glass and Al NMR on the open framework aluminophosphate AlPO-CJ19 [(NH)Al(PO)HPOHO]. Numerically simulated magnetization transfers in spin–3/2 pairs revealed a progressively enhanced tolerance to resonance offsets and rf-amplitude errors of the recoupling pulses along the series (SR2) (SR2) (SR2)M for increasing differences in chemical shifts between the two nuclei. Nonetheless, for scenarios of a relatively minor chemical-shift dispersions ( kHz), the (SR2)M supercycles perform best both experimentally and in simulations.

Detuning-modulated composite pulses for high-fidelity robust quantum control

We introduce a novel control method for robust quantum information processing suited for quantum integrated photonics. We utilize off-resonant detunings as control parameters to derive a new family of composite pulses for high-fidelity population transfer within the quantum error threshold. By design, our detuning-modulated N-piece composite sequences correct for any control inaccuracies including pulse strength, duration, resonance offsets errors, Stark shifts, unwanted frequency chirp, etc. We reveal the symmetries of the pulses that allow for straightforward scaling with minimal pulse overhead for an arbitrary N. Furthermore, we implement the composite solutions in coupled waveguides allowing a complete light transfer that is robust to fabrication errors.

Low-distortion diffusion tensor MRI with improved phaseless encoding.

Due to the motion-related instability of the signal phase, diffusion MRI is usually performed with single-shot techniques such as the echo-planar imaging (EPI), which are resolution-limited and suffer from distortions caused by resonance offsets. Multi-shot methods may improve the images but require time-consuming navigators or a trade-off of the sensitivity encoding to measure shot-dependent phase errors. We have recently introduced an alternative approach to multi-shot MRI called phaseless encoding, which, by analogy to optical super-resolution methods, relies on the magnitude value of images taken in different shots thus discarding the phase error without navigators, and demonstrated its capability to perform diffusion MRI at sub-millimeter scale on a standard 3T scanner. In this work, we apply phaseless encoding in a routine diffusion tensor imaging (DTI) protocol with a moderately high resolution that is still within reach of single-shot EPI with the same hardware, and compare both techniques with respect to image distortions. A qualitative comparison of the phaseless encoding with the established navigator-based readout-segmented EPI is also presented. Several technical improvements are proposed to make phaseless encoding compatible with the routine scanning mode. The tagging radiofrequency pulses used in the encoding sequence are made slice-selective to avoid artefacts caused by saturation effects in multi-slice scans and their flip angle is optimized to reduce the intrinsic SNR loss. The super-resolution reconstruction algorithm is also improved to better suppress Gibbs ringing and to correct for possible signal amplitude fluctuations. Our study shows that the phaseless encoding is a promising approach to diffusion weighted imaging. It can easily be implemented in multi-slice sequences and produces less distorted images than the single-shot EPI at the same resolution and hardware parameters. It provides similar results to readout-segmented EPI but without the need of navigators.

Analysis of Artifacts Caused by Pulse Imperfections in CPMG Pulse Trains in NMR Relaxation Dispersion Experiments

Nuclear magnetic resonance relaxation dispersion (rd) experiments provide kinetics and thermodynamics information of molecules undergoing conformational exchange. Rd experiments often use a Carr-Purcell-Meiboom-Gill (CPMG) pulse train equally separated by a spin-state selective inversion element (U-element). Even with measurement parameters carefully set, however, parts of 1H–15N correlations sometimes exhibit large artifacts that may hamper the subsequent analyses. We analyzed such artifacts with a combination of NMR measurements and simulation. We found that particularly the lowest CPMG frequency (νcpmg) can also introduce large artifacts into amide 1H–15N and aromatic 1H–13C correlations whose 15N/13C resonances are very close to the carrier frequencies. The simulation showed that the off-resonance effects and miscalibration of the CPMG π pulses generate artifact maxima at resonance offsets of even and odd multiples of νcpmg, respectively. We demonstrate that a method once introduced into the rd experiments for molecules having residual dipolar coupling significantly reduces artifacts. In the method the 15N/13C π pulse phase in the U-element is chosen between x and y. We show that the correctly adjusted sequence is tolerant to miscalibration of the CPMG π pulse power as large as ±10% for most amide 15N and aromatic 13C resonances of proteins.

Resonance capture and dynamics of three-planet systems

We present a series of dynamical maps for fictitious three-planet systems in initially circular coplanar orbits. These maps have unveiled a rich resonant structure involving two or three planets, as well as indicating possible migration routes from secular to double resonances or pure three-planet commensurabilities. These structures are then compared to the present-day orbital architecture of observed resonant chains. In a second part of the paper, we describe N-body simulations of type-I migration. Depending on the orbital decay time-scale, we show that three-planet systems may be trapped in different combinations of independent commensurabilities: (i) double resonances, (ii) intersection between a two-planet and a first-order three-planet resonances, and (iii) simultaneous libration in two first-order three-planet resonances. These latter outcomes are found for slow migrations, while double resonances are almost always the final outcome in high-density discs. Finally, we discuss an application to the TRAPPIST-1 system. We find that, for low migration rates and planetary masses of the order of the estimated values, most three-planet sub-systems are able to reach the observed double resonances after following evolutionary routes defined by pure three-planet resonances. The final orbital configuration shows resonance offsets comparable with present-day values without the need of tidal dissipation. For the 8/5 resonance proposed to dominate the dynamics of the two inner planets, we find little evidence of its dynamical significance; instead, we propose that this relation between mean motions could be a consequence of the interaction between a pure three-planet resonance and a two-planet commensurability between planets c and d.

Orbital migration and Resonance Offset of the Kepler-25 and K2-24 systems

Based on the model described in [1], we present an analytical+numerical study of the resonance capture under Type-I migration for the Kepler-25 [2] and K2-24 [3] Kepler systems, both close to a 2/1 mean-motion resonance. We find that, depending on the flare index and the proximity to the central star, the average value of the period-ratio between two consecutive planets show a significant deviation with respect to the resonant nominal value, up to values well in agreement with the observations.

Design strategies for improved velocity-selective pulse sequences

Over the years, a variety of MRI methods have been developed for visualizing or measuring blood flow without the use of contrast agents. One particular class of methods uses flow-encoding gradients associated with an RF pulse sequence to distinguish spins in flowing blood from stationary spins. While a strength of these particular methods is that, in general, they can be tailored to capture a desired range of blood flow, such sequences either do not provide a sharp transition from stationary spins to flowing spins, or else are long, generating relaxation losses and undesirable SAR, and have limited immunity to resonance offsets and to RF inhomogeneity. This article provides design methods for improving these longer RF pulse sequences, especially to provide improved immunity to RF inhomogeneity, and also to improve immunity to resonance offsets, as well as to minimize RF sequence length. These design methods retain the flexibility to capture a desired range of blood flow, with sharp transitions between stationary spins and flowing blood. These improvement strategies are demonstrated through Bloch equations simulations of examples of these new sequences in the presence of blood flow. Examples of improved sequences that should prove suitable for use at 3.0Tesla are presented.

PT-symmetric metasurfaces: wave manipulation and sensing using singular points

We investigate here active metasurfaces obeying parity–time (PT) symmetry and their sensing applications, taking advantage of singularities unique to non-Hermitian systems, such as the spontaneous PT-symmetry-breaking point (exceptional point or EP) and the coherent perfect absorber–laser (CPAL) point. We show theoretically that a PT-symmetric metasurface sensor may provide enhanced sensitivities compared to traditional passive sensors based on metamaterial/metasurface resonators, because the singular point of one-way zero reflection arising from the EP or the CPAL-related sharp resonance may result in dramatically modulated scattering responses or resonance offsets. We demonstrate the proposed concept with realistic metasurface sensors based on photopumped graphene metasurfaces that simultaneously offer terahertz optical gain and (bio)chemical sensing functions. The proposed PT-symmetric metasurfaces may impact not only loss compensation and extraordinary manipulation of electromagnetic waves, but also practical sensing and detection applications.

Wideband frequency-swept excitation in pulsed EPR spectroscopy

Excitation of electron spins with monochromatic rectangular pulses is limited to bandwidths that are smaller than the spectral widths of most organic radicals and much smaller than the spectral widths of transition and rare earth metal ions. With frequency-swept pulses, bandwidths of up to 800MHz have previously been attained for excitation and detection of spin packets at frequencies of about 9.6GHz and bandwidths of up to 2.5GHz in a polarization transfer experiment at frequencies of about 34GHz. The remaining limitations, mainly due to resonator bandwidth and due to pulse length restrictions are discussed. Flip angles for state-space rotations on passage of a transition can generally be computed from the critical adiabaticity by the Landau-Zener-Stückelberg-Majorana expression. For hyperbolic secant pulses, the Demkov-Kunike model describes excitation for spin packets within and outside the sweep range. Well within the sweep range, the Bloch-Siegert phase shift is proportional to critical adiabaticity to a very good approximation. Because of the dependence of both flip angle and coherence phase on critical adiabaticity, it is advantageous to use pairs of amplitude and frequency modulation functions that provide such offset-independent adiabaticity. Compensation for the resonator response function should restore offset-independent adiabaticity. Whereas resonance offsets and Bloch-Siegert phase can be refocused at certain pulse length ratios, phase dispersion in coupled spin systems cannot generally be refocused. Based on the bandwidth limitations that arise from spin dynamics, requirements are derived for a spectrometer that achieves precise spin control over wide bands. The design of such a spectrometer and hardware characterization by EPR experiments are discussed.

Planetary migration and the origin of the 2:1 and 3:2 (near)-resonant population of close-in exoplanets

We present an analytical and numerical study of the orbital migration and resonance capture of fictitious two-planet systems with masses in the super-Earth range undergoing Type-I migration. We find that, depending on the flare index and proximity to the central star, the average value of the period ratio, $P_2/P_1$, between both planets may show a significant deviation with respect to the nominal value. For planets trapped in the 2:1 commensurability, offsets may reach values on the order of $0.1$ for orbital periods on the order of $1$ day, while systems in the 3:2 mean-motion resonance (MMR) show much smaller offsets for all values of the semimajor axis. These properties are in good agreement with the observed distribution of near-resonant exoplanets, independent of their detection method. We show that 2:1-resonant systems far from the star, such as HD82943 and HR8799, are characterized by very small resonant offsets, while higher values are typical of systems discovered by Kepler with orbital periods approximately a few days. Conversely, planetary systems in the vicinity of the 3:2 MMR show little offset with no significant dependence on the orbital distance. In conclusion, our results indicate that the distribution of Kepler planetary systems around the 2:1 and 3:2 MMR are consistent with resonant configurations obtained as a consequence of a smooth migration in a laminar flared disk, and no external forces are required to induce the observed offset or its dependence with the commensurability or orbital distance from the star.

ENDOR with band-selective shaped inversion pulses

Electron Nuclear DOuble Resonance (ENDOR) is based on the measurement of nuclear transition frequencies through detection of changes in the polarization of electron transitions. In Davies ENDOR, the initial polarization is generated by a selective microwave inversion pulse. The rectangular inversion pulses typically used are characterized by a relatively low selectivity, with full inversion achieved only for a limited number of spin packets with small resonance offsets. With the introduction of pulse shaping to EPR, the rectangular inversion pulses can be replaced with shaped pulses with increased selectivity. Band-selective inversion pulses are characterized by almost rectangular inversion profiles, leading to full inversion for spin packets with resonance offsets within the pulse excitation bandwidth and leaving spin packets outside the excitation bandwidth largely unaffected. Here, we explore the consequences of using different band-selective amplitude-modulated pulses designed for NMR as the inversion pulse in ENDOR. We find an increased sensitivity for small hyperfine couplings compared to rectangular pulses of the same bandwidth. In echo-detected Davies-type ENDOR, finite Fourier series inversion pulses combine the advantages of increased absolute ENDOR sensitivity of short rectangular inversion pulses and increased sensitivity for small hyperfine couplings of long rectangular inversion pulses. The use of pulses with an almost rectangular frequency-domain profile also allows for increased control of the hyperfine contrast selectivity. At X-band, acquisition of echo transients as a function of radiofrequency and appropriate selection of integration windows during data processing allows efficient separation of contributions from weakly and strongly coupled nuclei in overlapping ENDOR spectra within a single experiment.

Spin-locking of half-integer quadrupolar nuclei in NMR of solids: The far off-resonance case

Spin-locking of spin I=3/2 and I=5/2 nuclei in the presence of large resonance offsets has been studied using both approximate and exact theoretical approaches and, in the case of I=3/2, experimentally. We show the variety of coherences and population states produced in a far off-resonance spin-locking NMR experiment (one consisting solely of a spin-locking pulse) and how these vary with the radiofrequency field strength and offset frequency. Under magic angle spinning (MAS) conditions and in the “adiabatic limit”, these spin-locked states acquire a time dependence. We discuss the rotor-driven interconversion of the spin-locked states, using an exact density matrix approach to confirm the results of the approximate model. Using conventional and multiple-quantum filtered spin-locking 23Na (I=3/2) NMR experiments under both static and MAS conditions, we confirm the results of the theoretical calculations, demonstrating the applicability of the approximate theoretical model to the far off-resonance case. This simplified model includes only the effects of the initial rapid dephasing of coherences that occurs at the start of the spin-locking period and its success in reproducing both experimental and exact simulation data indicates that it is this dephasing that is the dominant phenomenon in NMR spin-locking of quadrupolar nuclei, as we have previously found for the on-resonance and near-resonance cases. Potentially, far off-resonance spin-locking of quadrupolar nuclei could be of interest in experiments such as cross polarisation as a consequence of the spin-locking pulse being applied to a better defined initial state (the thermal equilibrium bulk magnetisation aligned along the z-axis) than can be created in a powdered solid with a selective radiofrequency pulse, where the effect of the pulse depends on the orientation of the individual crystallites.