Pulse Imperfections Research Articles

Nutation-based longitudinal-sensing protocols for high-field NMR with nitrogen-vacancy centers in diamond

Nitrogen-vacancy (N-V) centers in diamond enable nuclear magnetic resonance (NMR) spectroscopy of samples at the nano- and microscale. However, at typical tesla-scale NMR magnetic field strengths, N-V–NMR protocols become difficult to implement due to the challenge of driving fast N-V pulse sequences sensitive to nuclear Larmor frequencies above a few megahertz. We perform simulations and theoretical analysis of the experimental viability of N-V NMR at tesla-scale magnetic fields using a measurement protocol called DRACAERIS (Double Rewound ACquisition Amplitude Encoded Radio Induced Signal). DRACAERIS detects the longitudinal magnetization of the NMR sample at a much lower driven Rabi frequency, more suitable technically for N-V detection. We discuss how pulse errors, finite pulse lengths, and nuclear spin-spin couplings affect the resulting NMR spectra. We find that DRACAERIS is less susceptible to pulse imperfections and off-resonance effects than previous protocols for longitudinal magnetization detection. We also identify reasonable parameters for experimental implementation. Published by the American Physical Society 2024

Comparison of driven equilibrium and standard spin-echo sequence in MR microscopy: Analysis of signal dependence on RF pulse imperfection and diffusion

Rapid MR imaging of slowly relaxing samples is often challenging. The most commonly used solutions are found in multi spin-echo (RARE) sequences or gradient-echo (GE) sequences, which allow faster imaging of such samples with multiple acquisitions of k-space lines per excitation or imaging with very short repetition times (TRs). Another solution is the use of a spin-echo (SE) sequence superimposed with a driven equilibrium Fourier transform (DEFT) method. Such a (DE-SE) imaging sequence has two refocusing RF pulses that produce two spin-echoes. In the first echo, the signal is acquired from the k-space line, and in the second echo, a 90° RF pulse is applied, typically 180° out of phase with respect to the excitation RF pulse. This last RF pulse allows almost complete magnetization reversal back to the longitudinal orientation with minimal magnetization loss. The DE-SE sequence and its RARE variant are widely used in clinical imaging, but its use in MR microscopy has some peculiarities related to the usually less perfect RF pulse flip angles and diffusion. In this study, their effects are first theoretically analyzed and later verified by experiments on test samples performed on a 9.4 T system for MR microscopy. Experiments on a water-filled tube for TE = 3.4 ms and TR = 25–200 ms showed that the DE-SE sequence produces about 10 times more signal than the SE sequence in this TR range. Finally, the performance of the DE-SE sequence compared to the SE sequence was demonstrated on a biological sample. The presented DE-SE sequence has been shown to be effective for rapid imaging of samples with long T1 relaxation times in MR microscopy and can also be considered as a suitable method for rapid proton density weighed imaging of materials.

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.

Echo modulations under homonuclear decoupling

A simple Hahn echo in a single-spin system with a static Hamiltonian can lead to echo modulations if the static Hamiltonian contains a component along the direction of the echo pulse. These modulations manifest as side bands in the Fourier transform of the echo decay. Experimentally, echo modulations that can be explained by such a model have been observed under homonuclear decoupling in solids where pulse imperfections can lead to residual effective fields in the interaction frame that have arbitrary orientations in space. We show analytically that such echo modulations are significantly reduced in intensity using double-echo sequences in agreement with experimental observations. Using pulse shapes for homonuclear decoupling that mimic and amplify the pulse distortions expected from pulse transients, we show that these effective fields can be one explanation for the observed reduction in echo modulations going from a single to a double Hahn-echo sequence.

Single-shot T2 mapping via multi-echo-train multiple overlapping-echo detachment planar imaging and multitask deep learning.

Quantitative magnetic resonance imaging provides robust biomarkers in clinics. Nevertheless, the lengthy scan time reduces imaging throughput and increases the susceptibility of imaging results to motion. In this context, a single-shot T2 mapping method based on multiple overlapping-echo detachment (MOLED) planar imaging was presented, but the relatively small echo time range limits its accuracy, especially in tissues with large T2 . In this work we proposed a novel single-shot method, Multi-Echo-Train Multiple OverLapping-Echo Detachment (METMOLED) planar imaging, to accommodate a large range of T2 quantification without additional measurements to rectify signal degeneration arisen from refocusing pulse imperfection. Multiple echo-train techniques were integrated into the MOLED sequence to capture larger TE information. Maps of T2 , B1 , and spin density were reconstructed synchronously from acquired METMOLED data via multitask deep learning. A typical U-Net was trained with 3000/600 synthetic data with geometric/brain patterns to learn the mapping relationship between METMOLED signals and quantitative maps. The refocusing pulse imperfection was settled through the inherent information of METMOLED data and auxiliary tasks. Experimental results on the digital brain (structural similarity (SSIM) index=0.975/0.991/0.988 for MOLED/METMOLED-2/METMOLED-3, hyphenated number denotes the number of echo-trains), physical phantom (the slope of linear fitting with reference T2 map=1.047/1.017/1.006 for MOLED/METMOLED-2/METMOLED-3), and human brain (Pearson's correlation coefficient (PCC)=0.9581/0.9760/0.9900 for MOLED/METMOLED-2/METMOLED-3) demonstrated that the METMOLED improved the quantitative accuracy and the tissue details in contrast to the MOLED. These improvements were more pronounced in tissues with large T2 and in application scenarios with high temporal resolution (PCC=0.8692/0.9465/0.9743 for MOLED/METMOLED-2/METMOLED-3). Moreover, the METMOLED could rectify the signal deviations induced by the non-ideal slice profiles of refocusing pulses without additional measurements. A preliminary measurement also demonstrated that the METMOLED is highly repeatable (mean coefficient of variation (CV)=1.65%). METMOLED breaks the restriction of echo-train length to TE and implements unbiased T2 estimates in an extensive range. Furthermore, it corrects the effect of refocusing pulse inaccuracy without additional measurements or signal post-processing, thus retaining its single-shot characteristic. This technique would be beneficial for accurate T2 quantification.

Spatially encoded polarization transfer for improving the quantitative aspect of 1H–13C HSQC

Peak overlap hampers quantification in one-dimensional (1D) 1H NMR. 2D 1H -13C HSQC spectrum provides resolution superior to 1D 1H NMR. However, quantifying the components in a complex mixture with HSQC is not straightforward as in 1D 1H NMR. Quantification using HSQC could open up new avenues for studying metabolism. The variations in 1H–13C scalar couplings, T1, T2, and pulse imperfections contribute to this problem. Although T1 and T2 can be suitably chosen to minimize their deleterious effects, the differential polarization transfer for different resonances owing to large variations in 1H -13C couplings does not allow the cross-peak intensities to be directly correlated to the quantity of metabolites. Existing approaches are time-consuming. We show that spatial encoding of the polarization transfer delays in HSQC using sweep frequency pulses in the presence of a magnetic field gradient allows one to have a transfer of polarization from 1H to 13C insensitive to variations in 1H -13C couplings improving the quantitative aspect of HSQC. Comparisons to other QHSQC and perfected HSQC variants are also provided.

Process tomography of robust dynamical decoupling with superconducting qubits

Dynamical decoupling is a technique that protects qubits against noise. The ability to preserve quantum coherence in the presence of noise is essential for the development of quantum devices. Here the Rigetti quantum computing platform was used to test different dynamical decoupling sequences. The performance of the sequences was characterized by Quantum Process Tomography and analyzed using the quantum channels formalism. It is shown that dynamical decoupling can reduce qubit dephasing but cannot protect against spontaneous emission. Furthermore, from process tomography results, it was also possible to conclude that the action of dynamical decoupling cannot be understood as a simple modification of the qubit coherence time. It is also shown here that the performance of dynamical decoupling on the Rigetti's qubits is limited by pulse imperfections. However, the performance can be improved using robust dynamical decoupling, i.e. sequences that are robust against experimental imperfections. The sequences tested here outperformed previous dynamical decoupling sequences tested in the same platform.

Fast entangling gates in long ion chains

We present a model for implementing fast entangling gates (${\sim}1~\mu$s) with ultra-fast pulses in arbitrarily long ion chains, that requires low numbers of pulses and can be implemented with laser repetition rates well within experimental capability. We demonstrate that we are able to optimise pulse sequences that have theoretical fidelities above $99.99\%$ in arbitrarily long ion-chains, for laser repetition rates on the order of $100-300$~MHz. Notably, we find higher repetition rates are not required for gates in longer ion chains, which is in contrast to scaling analyses with other gate schemes. When pulse imperfections are considered in our calculations, we find that achievable gate fidelity is independent of the number of ions in the chain. We also show that pulse control requirements do not scale up with the number of ions. We find that population transfer efficiencies of above $99.9\%$ from individual ultra-fast pulses is the threshold for realising high-fidelity gates, which may be achievable in near-future experiments.

Coherent transport of spin by adiabatic passage in quantum dot arrays

We introduce an adiabatic transfer protocol for spin states in large quantum dot arrays that is based on time-dependent modulation of the Heisenberg exchange interaction in the presence of a magnetic field gradient. We refer to this protocol as spin-CTAP (coherent transport by adiabatic passage) in analogy to a related protocol developed for charge state transfer in quantum dot arrays. The insensitivity of this adiabatic protocol to pulse imperfections has potential advantages for reading out extended spin qubit arrays. When the static exchange interaction varies across the array, a quantum-controlled version of spin-CTAP is possible, where the transfer process is conditional on the spin states in the middle of the array. This conditional operation can be used to generate N-qubit entangled GHZ states. Using a realistic noise model, we analyze the robustness of the spin-CTAP operations and find that high-fidelity (>95%) spin eigenstate transfer and GHZ state preparation is feasible in current devices.

Robust Dynamic Hamiltonian Engineering of Many-Body Spin Systems

We introduce a new approach for the robust control of quantum dynamics of strongly interacting many-body systems. Our approach involves the design of periodic global control pulse sequences to engineer desired target Hamiltonians that are robust against disorder, unwanted interactions and pulse imperfections. It utilizes a matrix representation of the Hamiltonian engineering protocol based on time-domain transformations of the Pauli spin operator along the quantization axis. This representation allows us to derive a concise set of algebraic conditions on the sequence matrix to engineer robust target Hamiltonians, enabling the simple yet systematic design of pulse sequences. We show that this approach provides an efficient framework to (i) treat any secular many-body Hamiltonian and engineer it into a desired form, (ii) target dominant disorder and interaction characteristics of a given system, (iii) achieve robustness against imperfections, (iv) provide optimal sequence length within given constraints, and (v) substantially accelerate numerical searches of pulse sequences. Using this systematic approach, we develop novel sets of pulse sequences for the protection of quantum coherence, optimal quantum sensing and quantum simulation. Finally, we experimentally demonstrate the robust operation of these sequences in a system with the most general interaction form.

Optimized fast gates for quantum computing with trapped ions

We present an efficient approach to optimising pulse sequences for implementing fast entangling two-qubit gates on trapped ion quantum information processors. We employ a two-phase procedure for optimising gate fidelity, which we demonstrate for multi-ion systems in linear Paul trap and microtrap architectures. The first phase involves a global optimisation over a computationally inexpensive cost function constructed under strong approximations of the gate dynamics. The second phase involves local optimisations that utilise a more precise ODE description of the gate dynamics, which captures the non-linearity of the Coulomb interaction and the effects of finite laser repetition rate. We propose two novel gate schemes that are compatible with this approach, and we demonstrate that they outperform existing schemes in terms of achievable gate speed and fidelity for feasible laser repetition rates. In optimising sub-microsecond gates in microtrap architectures, the proposed schemes achieve orders of magnitude higher fidelities than previous proposals. Finally, we investigate the impact of pulse imperfections on gate fidelity and evaluate error bounds for a range of gate speeds.

Enhancing the Robustness of Dynamical Decoupling Sequences with Correlated Random Phases

We show that the addition of correlated phases to the recently developed method of randomized dynamical decoupling pulse sequences can improve its performance in quantum sensing. In particular, by correlating the relative phases of basic pulse units in dynamical decoupling sequences, we are able to improve the suppression of the signal distortion due to π pulse imperfections and spurious responses due to finite-width π pulses. This enhances the selectivity of quantum sensors such as those based on NV centers in diamond.

Achieving pure spin effects by artifact suppression in methyl adiabatic relaxation experiments.

Recent methyl adiabatic relaxation dispersion experiments provide examination of conformational dynamics across a very wide timescale (102-105s-1) and, particularly, provide insight into the hydrophobic core of proteins and allosteric effects associated with modulators. The experiments require efficient decoupling of 1H and 13C spin interactions, and some artifacts have been discovered, which are associated with the design of the proton decoupling scheme. The experimental data suggest that the original design is valid; however, pulse sequences with either no proton decoupling or proton decoupling with imperfect pulses can potentially exhibit complications in the experiments. Here, we demonstrate that pulse imperfections in the proton decoupling scheme can be dramatically alleviated by using a single composite π pulse and provide pure single-exponential relaxation data. It allows the opportunity to access high-quality methyl adiabatic relaxation dispersion data by removing the cross-correlation between dipole-dipole interaction and chemical shift anisotropy. The resulting high-quality data is illustrated with the binding of an allosteric modulator (G2BR) to the ubiquitin conjugating enzyme Ube2g2.

Characterizing spin-bath parameters using conventional and time-asymmetric Hahn-echo sequences

Spin-bath noise characterization, which is typically performed by multi-pulse control sequences, is essential for understanding most spin dynamics in the solid-state. Here, we theoretically propose a method for extracting the characteristic parameters of a noise source with a known spectrum, using a single modified Hahn-echo sequence. By varying the application time of the pulse, measuring the coherence curves of an addressable spin, and fitting the decay coefficients to a theoretical function derived by us, we extract parameters characterizing the physical nature of the noise. Assuming a Lorentzian noise spectrum, we illustrate this method for extracting the correlation time of a bath of nitrogen paramagnetic impurities in diamond, and its coupling strength to the addressable spin of a nitrogen-vacancy center. Considering a realistic experimental scenario with $5\%$ measurement error, the parameters can be extracted with an accuracy of $\sim 10 \%$. The scheme is effective for samples having a natural homogeneous coherence time ($T_2$) up to two orders of magnitude greater than the inhomogeneous coherence time ($T_2^*$), and mitigates technical noise when further averaging is irrelevant. Beyond its potential for reducing experiment times by an order-of-magnitude, such single-pulse noise characterization could minimize the effects of long time-scale drifts and accumulating pulse imperfections and numerical errors.

An improved differential evolution algorithm for learning high-fidelity quantum controls

Precisely and efficiently designing control pulses for the preparation of quantum states and quantum gates are the fundamental tasks for quantum computation. Gradient-based optimal control methods are the routine to design such pulses. However, the gradient information is often difficult to calculate or measure, especially when the system is not well calibrated or in the presence of various uncertainties. Gradient-free evolutionary algorithm is an alternative choice to accomplish this task but usually with low-efficiency. Here, we design an efficient mutation rule by using the information of the current and the former individuals together. This leads to our improved differential evolution algorithm, called daDE. To demonstrate its performance, we numerically benchmark the pulse optimization for quantum states and quantum gates preparations on small-scale NMR system. Further numerical comparisons with conventional differential evolution algorithms show that daDE has great advantages on the convergence speed and robustness to several uncertainties including pulse imperfections and measurement errors.

Randomization of Pulse Phases for Unambiguous and Robust Quantum Sensing.

We develop theoretically and demonstrate experimentally a universal dynamical decoupling method for robust quantum sensing with unambiguous signal identification. Our method uses randomization of control pulses to simultaneously suppress two types of errors in the measured spectra that would otherwise lead to false signal identification. These are spurious responses due to finite-width π pulses, as well as signal distortion caused by π pulse imperfections. For the cases of nanoscale nuclear-spin sensing and ac magnetometry, we benchmark the performance of the protocol with a single nitrogen vacancy center in diamond against widely used nonrandomized pulse sequences. Our method is general and can be combined with existing multipulse quantum sensing sequences to enhance their performance.

Artefact-free broadband 2D NMR for separation of quadrupolar and paramagnetic shift interactions

Two new two-dimensional, broadband, solid-state NMR experiments for separating and correlating the quadrupolar and shift interactions of spin I=1 nuclei in paramagnetic systems are proposed. The new pulse sequences incorporate the short, high-power adiabatic pulses (SHAPs) into the shifting d-echo experiment of Walder et al. [J. Chem. Phys., 142, 014201 (2015)], in two different ways, giving double and quadruple adiabatic shifting d-echo sequences. These new experiments have the advantage over previous methods of both suppressing spectral artefacts due to pulse imperfections, and exhibiting a broader excitation bandwidth. Both experiments are analysed with theoretical calculations and simulations, and are applied experimentally to the 2H NMR of deuterated CuCl2 ⋅2H2O, and two deuterated samples of the ion conductor oxyhydride BaTiO3−xHy prepared using two different methods. For the CuCl2 ⋅2H2O sample, both new methods obtain very high-quality spectra from which the parameters describing the shift and quadrupolar interaction tensors, and their relative orientation, were extracted. The two BaTiO3−xHy samples exhibited different local hydride environments with different tensor parameters. The 2H spectra of these oxyhydrides exhibit inhomogeneous broadening of the 2H shifts, and so whilst the quadrupolar interaction parameters were easily extracted, the measurement of the shift parameters was more complex. However, effective shift parameters were extracted, which combine the effects of both the paramagnetic shift tensor and the inhomogeneous broadening.

Improved sensitivity and quantification for 29Si NMR experiments on solids using UDEFT (Uniform Driven Equilibrium Fourier Transform)

We demonstrate the possibility to use UDEFT (Uniform Driven Equilibrium Fourier Transform) technique in order to improve the sensitivity and the quantification of one-dimensional 29Si NMR experiments under magic-angle spinning (MAS). We derive an analytical expression of the signal-to-noise ratios of UDEFT and single-pulse (SP) experiments subsuming the contributions of transient and steady-state regimes. Using numerical spin dynamics simulations and experiments on 29Si-enriched amorphous silica and borosilicate glass, we show that 59180298059180 refocusing composite π-pulse and the adiabatic inversion using tanh/tan modulation improve the robustness of UDEFT technique to rf-inhomogeneity, offset, and chemical shift anisotropy. These pulses combined with a two-step phase cycle limit the pulse imperfections and the artifacts produced by stimulated echoes. The sensitivity of SP, UDEFT and CPMG (Carr-Purcell-Meiboom-Gill) techniques are experimentally compared on functionalized and non-functionalized mesoporous silica. Furthermore, experiments on a flame retardant material prove that UDEFT technique provides a better quantification of 29Si sites with higher sensitivity than SP method.

Stabilization and manipulation of multispin states in quantum-dot time crystals with Heisenberg interactions

A discrete time crystal is a recently discovered non-equilibrium phase of matter that has been shown to exist in disordered, periodically driven Ising spin chains. In this phase, if the system is initially prepared in one of a certain class of pure multispin product states, it periodically returns to this state over very long time scales despite the presence of interactions, disorder, and pulse imperfections. Here, we show that this phase occurs in GaAs quantum dot spin arrays containing as few as three quantum dots, each confining one electron spin, for naturally occurring levels of nuclear spin noise and charge noise. Although the physical interaction in these arrays is a nearest-neighbor Heisenberg exchange interaction, we show that this can be effectively converted into an Ising interaction by applying additional pulses during each drive period. We show that by changing the rotation axis of these pulses, we can select the direction of the Ising interaction and, consequently, the quantization axis of the stabilized multispin states. Moreover, we demonstrate that it is possible to perform coherent rotations of the stabilized states while remaining in the time crystal phase. These findings open up the intriguing possibility of using time crystal phases to extend the lifetime of quantum states for information applications.

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.