Application of optimal control to CPMG refocusing pulse design

A continuous-wave and pulsed X-band electron spin resonance spectrometer operating in ultra-high vacuum for the study of low dimensional spin ensembles

We report the development of a continuous-wave and pulsed X-band electron spin resonance (ESR) spectrometer for the study of spins on ordered surfaces down to cryogenic temperatures. The spectrometer operates in ultra-high vacuum and utilizes a half-wavelength microstrip line resonator realized using epitaxially grown copper films on single crystal Al2O3 substrates. The one-dimensional microstrip line resonator exhibits a quality factor of more than 200 at room temperature, close to the upper limit determined by radiation losses. The surface characterizations of the copper strip of the resonator by atomic force microscopy, low-energy electron diffraction, and scanning tunneling microscopy show that the surface is atomically clean, flat, and single crystalline. Measuring the ESR spectrum at 15 K from a few nm thick molecular film of YPc2, we find a continuous-wave ESR sensitivity of 2.6 × 1011 spins/G · Hz1/2, indicating that a signal-to-noise ratio of 3.9 G · Hz1/2 is expected from a monolayer of YPc2 molecules. Advanced pulsed ESR experimental capabilities, including dynamical decoupling and electron-nuclear double resonance, are demonstrated using free radicals diluted in a glassy matrix.

Controlling NMR spin systems for quantum computation

Nuclear magnetic resonance is arguably both the best available quantum technology for implementing simple quantum computing experiments and the worst technology for building large scale quantum computers that has ever been seriously put forward. After a few years of rapid growth, leading to an implementation of Shor's quantum factoring algorithm in a seven-spin system, the field started to reach its natural limits and further progress became challenging. Rather than pursuing more complex algorithms on larger systems, interest has now largely moved into developing techniques for the precise and efficient manipulation of spin states with the aim of developing methods that can be applied in other more scalable technologies and within conventional NMR. However, the user friendliness of NMR implementations means that they remain popular for proof-of-principle demonstrations of simple quantum information protocols.

Optimization of TROSY- and anti-TROSY-based 15N CPMG relaxation dispersion experiments through phase cycling

CPMG relaxation dispersion studies of biomolecular dynamics on the μs-ms timescale can provide detailed kinetic, thermodynamic, and structural insights into function. Frequently, the 15N spin serves as the probe of choice, as uniform incorporation of the 15N isotope is facile and cost-effective, and the interpretation of the resulting data is often relatively straightforward. In conventional CPMG relaxation dispersion experiments the application of CPMG pulses with constant radiofrequency (RF) phase can lead to artifactual dispersion profiles that result from off-resonance effects, RF field inhomogeneity, and pulse miscalibration. The development of CPMG experiments with the [0013]-phase cycle has significantly reduced the impact of pulse imperfections over a greater bandwidth of frequency offsets in comparison to constant phase experiments. Application of 15N-TROSY-based CPMG schemes to studies of the dynamics of large molecules is necessary for high sensitivity, yet the correct incorporation of the [0013]-phase cycle is non-trivial. Here we present TROSY- and anti-TROSY-based 15N CPMG experiments with the [0013]-phase cycling scheme and demonstrate, through comprehensive numerical simulations and experimental validation, enhanced resistance to pulse imperfections relative to traditional schemes utilizing constant phase CPMG pulses. Notably, exchange parameters derived from the new experiments are in good agreement with those obtained using other, more established, 15N-based CPMG approaches.

Dynamical Uncertainty Propagation with Noisy Quantum Parameters

Many quantum technologies rely on high-precision dynamics, which raises the question of how these are influenced by the experimental uncertainties that are always present in real-life settings. A standard approach in the literature to assess this is Monte Carlo sampling, which suffers from two major drawbacks. First, it is computationally expensive. Second, it does not reveal the effect that each individual uncertainty parameter has on the state of the system. In this Letter, we evade both these drawbacks by incorporating propagation of uncertainty directly into simulations of quantum dynamics, thereby obtaining a method that is orders of magnitude faster than Monte Carlo simulations and directly provides information on how each uncertainty parameter influences the system dynamics. Additionally, we compare our method to experimental results obtained using the IBM quantum computers.

Improved design of frequency-swept pulse sequences

Frequency-swept pulses are extensively used in magnetic resonance spectroscopic techniques for the robust manipulation of spins across wide ranges of offset frequencies in the presence of B₁ field variations. Nevertheless, designing pulse sequences consisting of multiple frequency-swept pulses can be challenging, as they often require specific timings and parameter tweaking. In the present work we discuss a simple and general approach for constructing such sequences. We present new and improved pulse sequences for applications including broadband B₁-tolerant CPMG (CHORUS-CPMG), broadband chirped excitation with suppression of homonuclear J-modulation (PROCHORUS), and the further compression of frequency-swept pulse sequences by superposition of pulses which reduces pulse sequence durations by 25-40%. All sequence design strategies are accompanied by mathematical presentations, experimental results, and supporting simulations.

On the use of frequency-swept pulses and pulses designed with optimal control theory for the acquisition of ultra-wideline NMR spectra

Frequency-swept (FS) pulses, such as wideband uniform-rate smooth-truncation (WURST) pulses, have found much success for the acquisition of ultra-wideline (UW) solid-state NMR spectra. In this preliminary study, new pulses and pulse sequences are explored in simulation and experimentally for several nuclei exhibiting UWNMR powder patterns under static conditions, including ¹¹⁹Sn (I = 1/2), ¹⁹⁵Pt (I = 1/2), ²H (I = 1), and ⁷¹Ga (I = 3/2). First, hyperbolic secant (HS) and tanh/tan (THT) pulses are tested and implemented as excitation and refocusing pulses in spin-echo and Carr-Purcell/Meiboom Gill (CPMG)-type sequences, and shown to have comparable performances to analogous WURST pulses. Second, optimal control theory (OCT) is utilized for the design of new Optimal Control Theory Optimized Broadband Excitation and Refocusing (OCTOBER) pulses, using carefully parameterized WURST, THT, and HS pulses as starting points. Some of the new OCTOBER pulses used in spin-echo sequences are capable of efficient broadband excitation and refocusing, in some cases resulting in spectra with increased signal enhancements over those obtained in experiments using conventional FS pulses. Finally, careful consideration of the spin dynamics of several systems, by monitoring of the time evolution of the density matrix via the Liouville-von Neumann equation and analysis of the time-resolved Fourier transforms of the pulses, lends insight into the underlying mechanisms of the FS and OCTOBER pulses. This is crucial for understanding their performance in terms of generating uniformly excited patterns of high signal intensity, and for identifying trends that may offer pathways to generalized parameterization and/or new pulse shapes.

Quantitative nanoscale MRI with a wide field of view

Novel magnetic sensing modalities using quantum sensors or nanoscale probes have drastically improved the sensitivity and hence spatial resolution of nuclear magnetic resonance imaging (MRI) down to the nanoscale. Recent demonstrations of nuclear magnetic resonance (NMR) with paramagnetic colour centres include single molecule sensitivity, and sub-part-per-million spectral resolution. Mostly, these results have been obtained using well-characterised single sensors, which only permit extended imaging by scanning-probe microscopy. Here, we enhance multiplexed MRI with a thin layer of ensemble spin sensors in an inhomogeneous control field by optimal control spin manipulation to improve ensemble sensitivity and field of view (FOV). We demonstrate MRI of fluorine in patterned thin films only 1.2 nm in thickness, corresponding to a net moment of 120 nuclear spins per sensor spin. With the aid of the NMR signal, we reconstruct the nanoscale depth distribution of the sensor spins within the substrate. In addition, we exploit inhomogeneous ensemble control to squeeze the point spread function of the imager to about 100 nm and show that localisation of a point-like NMR signal within 40 nm is feasible. These results pave the way to quantitive NMR ensemble sensing and magnetic resonance microscopy with a resolution of few ten nanometers.

Broadband RF-amplitude-dependent flip angle pulses with linear phase slope

Pulse sequences in NMR spectroscopy sometimes require the application of pulses with effective flip angles different from 90° and 180°. Previously (Magn. Reson. Chem. 2015, 53, 886-893), offset-compensated broadband excitation pulses with RF-amplitude-dependent effective flip angles (RADFA) were introduced that are applicable in such cases. However, especially RF-amplitude-restricted RADFA pulses turned out to perform not as good as desired in terms of achievable bandwidths. Here, a class of RF-amplitude-restricted RADFA pulses with linear phase slope is introduced that allows excitation over much larger bandwidths with better performance. In this theoretical work, the basic principle of the pulse class is explained, their physical limits explored, and their properties, also compared with other pulse classes, discussed in detail. Copyright © 2017 John Wiley & Sons, Ltd.

An optimal control approach to design entire relaxation dispersion experiments

A general approach is introduced to optimize experiments for the analysis of spin systems in the presence of chemical exchange. Rather than optimizing individual pulse sequence elements, such as refocusing pulses, entire relaxation dispersion sequences are optimized in the form of a single shaped pulse. This is achieved by defining a performance index that is only based on the remaining signal after the relaxation dispersion sequence for a range of exchange, relaxation, offset, and rf inhomogeneity parameters. The approach is demonstrated by optimizing energy-limited broadband relaxation dispersion sequences that closely approach the overall effect of ideal CPMG sequences. As illustrated both theoretically and experimentally, significant improvements are found compared to standard amplitude or energy-limited CPMG sequences.

Environmental noise spectroscopy with qubits subjected to dynamical decoupling

A qubit subjected to pure dephasing due to classical Gaussian noise can be turned into a spectrometer of this noise by utilizing its readout under properly chosen dynamical decoupling (DD) sequences to reconstruct the power spectral density of the noise. We review the theory behind this DD-based noise spectroscopy technique, paying special attention to issues that arise when the environmental noise is non-Gaussian and/or it has truly quantum properties. While we focus on the theoretical basis of the method, we connect the discussed concepts with specific experiments, and provide an overview of environmental noise models relevant for solid-state based qubits, including quantum-dot based spin qubits, superconducting qubits, and NV centers in diamond.

Absolute phase effects on CPMG-type pulse sequences

We describe and analyze the effects of transients within radio-frequency (RF) pulses on multiple-pulse NMR measurements such as the well-known Carr-Purcell-Meiboom-Gill (CPMG) sequence. These transients are functions of the absolute RF phases at the beginning and end of the pulse, and are thus affected by the timing of the pulse sequence with respect to the period of the RF waveform. Changes in transients between refocusing pulses in CPMG-type sequences can result in signal decay, persistent oscillations, changes in echo shape, and other effects. We have explored such effects by performing experiments in two different low-frequency NMR systems. The first uses a conventional tuned-and-matched probe circuit, while the second uses an ultra-broadband un-tuned or non-resonant probe circuit. We show that there are distinct differences between the absolute phase effects in these two systems, and present simple models that explain these differences.

Broadband excitation pulses with variable RF amplitude-dependent flip angle (RADFA)

Pulse sequences in NMR spectroscopy sometimes require the adjustment of effective flip angles with respect to experiment-specific or sample-specific parameters. Here, we present a quality factor for efficient optimization of offset-compensated broadband excitation pulses with RF amplitude-dependent effective flip angles (RADFA). After proof of principle, physical limits of RF amplitude-restricted and RF power-restricted broadband RADFA pulses are explored and corresponding pulse shapes and performances characterized in detail.

Robust INEPT and refocused INEPT transfer with compensation of a wide range of couplings, offsets, and B1-field inhomogeneities (COB3)

Following the two-step optimization procedure previously introduced with the COB-INEPT (Ehni and Luy, 2012), a corresponding inphase-to-antiphase transfer element with close to optimal transfer efficiencies over a coupling range comprising approximately J-6J has been derived. The hard pulse sequence length is only 5.5 ms for coupling constants within 125-750 Hz. Robustness with respect to an offset range of 37.5 kHz on carbon (corresponding to 250 ppm on a 600 MHz spectrometer) and 10 kHz on protons (16.6 ppm at 600 MHz) is achieved with corresponding BUBI and BURBOP broadband pulses. As the sequence achieves a three times higher upper limit of J-compensation compared to the COB-INEPT, we name the transfer element COB3-INEPT. Next to the description of optimization and pulse sequence details, the performance of the resulting element is demonstrated on a test sample and partially aligned sample with actual total couplings in the range of 134 Hz⩽(1)TCH⩽391 Hz. The sequence can also be used for inphase-to-antiphase transfer starting from carbon, where the upper limit of J-compensation is 6J for CH-groups, 3J for CH2-groups, and slightly less than 2J for CH3. Theoretical transfers and experimental verification for the different multiplicities in an refocused INEPT are given.

Direct optimization of signal-to-noise ratio of CPMG-like sequences in inhomogeneous fields

The performance of the standard CPMG sequence in inhomogeneous fields can be improved with the use of broadband excitation and refocusing pulses. In previous work we have developed short composite broadband refocusing pulses together with practical excitation pulses to realize such performance gains, and quantified them using the ratio of signal to noise power (SNR). In this work we systematically explore the performance of refocusing pulses as a function of the overall pulse length up to ten times the length of the regular 180° pulse. This is in the regime of non-adiabatic pulses. We introduce a new performance functional for numerical pulse optimization that directly maximizes SNR and study the effect of symmetry constraints on the pulses. We find that for the optimal pulses, the SNR per asymptotic echo increases with pulse length but, for typical echo spacings, the SNR per unit time is maximized for refocusing pulses that are between two and four times longer than the duration of the standard rectangular 180° pulse. The performance is limited by the control bandwidth and the inability of finding the global maximum. The best performance was obtained with symmetric phase-alternating (SPA) refocusing pulses optimized with our novel performance functional. To test them in the CPMG sequence, we also developed axis-matching excitation (AMEX) pulses for use with these SPA refocusing pulses and tested the new AMEX-SPA sequences experimentally in a grossly inhomogeneous field, observing excellent agreement with the theoretical expectations. One of these sequences produced over 4.5 times higher SNR per asymptotic echo and 3.9 times higher SNR per unit time than the standard CPMG sequence with the same instantaneous RF power level. We also found that the new sequences are at least as robust to changes in the RF field strength as the standard CPMG sequence.

Adaptive hybrid optimal quantum control for imprecisely characterized systems

Optimal quantum control theory carries a huge promise for quantum technology. Its experimental application, however, is often hindered by imprecise knowledge of the input variables, the quantum system's parameters. We show how to overcome this by adaptive hybrid optimal control, using a protocol named Ad-HOC. This protocol combines open- and closed-loop optimal control by first performing a gradient search towards a near-optimal control pulse and then an experimental fidelity estimation with a gradient-free method. For typical settings in solid-state quantum information processing, adaptive hybrid optimal control enhances gate fidelities by an order of magnitude, making optimal control theory applicable and useful.

Chirped CPMG for well-logging NMR applications

In NMR well-logging, the measurement apparatus typically consists of a permanent magnet which is inserted into a bore, and the sample is the rock surrounding the borehole. When compared to the conditions of standard NMR experiments, this application is thus challenged by relatively weak and invariably inhomogeneous B0 and B1 fields. Chemical shift information is not generally obtained in these measurements. Instead, diffusivity, porosity and permeability information is collected from multi-echo decay measurements - most often using a Carr-Purcell Meiboom-Gill (CPMG) pulse sequence to enhance the experiment's limited sensitivity. In this work, we explore the consequences of replacing the hard square pulses used in a typical CPMG sequence with chirped pulses sweeping a range of frequencies. The greater bandwidths that for a maximum B1 level can be excited by chirped pulses translates into marked expansion of the detection volume, and thus significant signal-to-noise improvements when compared to standard CPMG acquisitions using hard pulses. This improvement, usually amounting to signal enhancements ⩾3, can be used to reduce the experimental time of NMR well-logging measurements, for measuring T2 even when B0 and B1 inhomogenieties complicate the measurements, and opening new opportunities in the determination of diffusional properties.

Cavity cooling of an ensemble spin system

We describe how sideband cooling techniques may be applied to large spin ensembles in magnetic resonance. Using the Tavis-Cummings model in the presence of a Rabi drive, we solve a Markovian master equation describing the joint spin-cavity dynamics to derive cooling rates as a function of ensemble size. Our calculations indicate that the coupled angular momentum subspaces of a spin ensemble containing roughly 10(11) electron spins may be polarized in a time many orders of magnitude shorter than the typical thermal relaxation time. The described techniques should permit efficient removal of entropy for spin-based quantum information processors and fast polarization of spin samples. The proposed application of a standard technique in quantum optics to magnetic resonance also serves to reinforce the connection between the two fields, which has recently begun to be explored in further detail due to the development of hybrid designs for manufacturing noise-resilient quantum devices.

Low-field permanent magnets for industrial process and quality control

In this review we focus on the technology associated with low-field NMR. We present the current state-of-the-art in low-field NMR hardware and experiments, considering general magnet designs, rf performance, data processing and interpretation. We provide guidance on obtaining the optimum results from these instruments, along with an introduction for those new to low-field NMR. The applications of lowfield NMR are now many and diverse. Furthermore, niche applications have spawned unique magnet designs to accommodate the extremes of operating environment or sample geometry. Trying to capture all the applications, methods, and hardware encompassed by low-field NMR would be a daunting task and likely of little interest to researchers or industrialists working in specific subject areas. Instead we discuss only a few applications to highlight uses of the hardware and experiments in an industrial environment. For details on more particular methods and applications, we provide citations to specialized review articles.

Axis-matching excitation pulses for CPMG-like sequences in inhomogeneous fields

The performance of the standard CPMG sequence in inhomogeneous fields can be improved with the use of broadband excitation and refocusing pulses. Here we introduce a new class of excitation pulses, so-called axis-matching excitation pulses, that optimize the response for a given refocusing pulse. These new excitation pulses are tailored to the refocusing pulses and take their imperfections into account. Rather than generating purely transverse magnetization, these pulses are designed to generate magnetization pointing along the axis of the effective rotation of the refocusing cycle. This approach maximizes the CPMG component and minimizes the CP component of the signal. Replacing a standard 90° pulse with a new excitation pulse matched to the 180° refocusing pulse increases the signal bandwidth and improves the echo amplitudes by 30% in inhomogeneous fields in comparison to the standard CPMG sequence. Larger gains are obtained with more advanced refocusing pulses. Recent work demonstrated that it is possible to increase the signal to noise ratio (SNR) of individual echoes by more than a factor of 1.5 (in power units) without increasing the duration or amplitude of the refocusing pulses. This was achieved by replacing the standard 180° refocusing pulse by a short phase alternating pulse and the standard 90° excitation pulse by a broadband excitation pulse. We show here that with suitable axis-matching excitation pulses, the SNR further increases by over a factor of 2. We discuss the underlying theory and present several practical implementations of purely phase modulated axis-matching excitation pulses for a number of different refocusing pulses that were derived using methods of optimal control. To gain the full benefit of these new excitation pulses, it is essential to replace the standard phase cycling scheme based on 180° phase shifts by a new scheme involving phase inversion. We tested the new pulses experimentally and observe excellent agreement with the theoretical expectations. We also demonstrate that an additional benefit of axis-matching excitation pulses is the decrease of the transient that appears in the amplitudes of the first few echoes, thus enabling better measurements of short relaxation times.

BEBE(tr) and BUBI: J-compensated concurrent shaped pulses for 1H-13C experiments

Shaped pulses designed for broadband excitation, inversion and refocusing are important tools in modern NMR spectroscopy to achieve robust pulse sequences especially in heteronuclear correlation experiments. A large variety of mostly computer-optimized pulse shapes exist for different desired bandwidths, available rf-field strengths, and tolerance to B1-inhomogeneity. They are usually derived for a single spin 1/2, neglecting evolution due to J-couplings. While pulses with constant resulting phase are selfcompensated for heteronuclear coupling evolution as long as they are applied exclusively on a single nucleus, the situation changes for concurrently applied pulse shapes. Using the example of a (1)H,(13)C two spin system, two J-compensated pulse pairs for the application in INEPT-type transfer elements were optimized: a point-to-point pulse sandwich called BEBE(tr), consisting of a broadband excitation and time-reversed excitation pulse, and a combined universal rotation and point-to-point pulse pair called BUBI, which acts as a refocusing pulse on (1)H and a corresponding inversion pulse on (13)C. After a derivation of quality factors and optimization protocols, a theoretical and experimental comparison with conventionally derived BEBOP, BIBOP, and BURBOP-180° pulses is given. While the overall transfer efficiency of a single pulse pair is only reduced by approximately 0.1%, resulting transfer to undesired coherences is reduced by several percent. In experiments this can lead to undesired phase distortions for pairs of uncompensated pulse shapes and even differences in signal intensities of 5-10% in HSQC and up to 68% in more complex COB-HSQC experiments.

Broadband CPMG sequence with short composite refocusing pulses

We demonstrate that CPMG sequences with phase-modulated refocusing pulses of the same duration as the standard 180° pulses can generate echo trains with significantly increased amplitudes compared to the standard CPMG sequence in the case when there is a large range of Larmor frequencies across the sample. The best performance is achieved with symmetric phase-alternating (SPA) composite refocusing pulses of the form α-yβ+yα-y. In comparison to standard 180° pulses, we show that with SPA refocusing pulses with α≈27° and β≈126°, it is possible to double the signal-to-noise ratio without increasing the total pulse duration or power consumption of the refocusing pulses. The increased bandwidth of these pulses more than compensates for the decrease in performance in the vicinity of resonance. To achieve the full benefit of the broadband nature of the SPA pulses in a CPMG sequence, it is necessary to combine these refocusing pulses with a broadband excitation pulse. When it is not possible to use a short, high amplitude excitation pulse, we show that phase-alternating (PA) excitation pulses are suitable for this purpose. We present a detailed analysis of the underlying spin dynamics of these new pulse sequences and confirm the simulations with experiments. We show that for samples with T1/T2>1, the new sequences in grossly inhomogeneous fields do not only generate echoes with an increased amplitude, but also with an increased decay time. Finally, we analyze the diffusion properties and show quantitatively that the broadband sequences have a substantially higher diffusion sensitivity compared with the standard CPMG sequence.

Design of non-selective refocusing pulses with phase-free rotation axis by gradient ascent pulse engineering algorithm in parallel transmission at 7T

At ultra-high magnetic field (≥ 7T), B1 and ΔB0 non-uniformities cause undesired inhomogeneities in image signal and contrast. Tailored radiofrequency pulses exploiting parallel transmission have been shown to mitigate these phenomena. However, the design of large flip angle excitations, a prerequisite for many clinical applications, remains challenging due the non-linearity of the Bloch equation. In this work, we explore the potential of gradient ascent pulse engineering to design non-selective spin-echo refocusing pulses that simultaneously mitigate severe B1 and ΔB0 non-uniformities. The originality of the method lays in the optimization of the rotation matrices themselves as opposed to magnetization states. Consequently, the commonly used linear class of large tip angle approximation can be eliminated from the optimization procedure. This approach, combined with optimal control, provides additional degrees of freedom by relaxing the phase constraint on the rotation axis, and allows the derivative of the performance criterion to be found analytically. The method was experimentally validated on an 8-channel transmit array at 7T, using a water phantom with B1 and ΔB0 inhomogeneities similar to those encountered in the human brain. For the first time in MRI, the rotation matrix itself on every voxel was measured by using Quantum Process Tomography. The results are complemented with a series of spin-echo measurements comparing the proposed method against commonly used alternatives. Both experiments confirm very good performance, while simultaneously maintaining a low energy deposition and pulse duration compared to well-known adiabatic solutions.

A systematic approach for optimizing the robustness of pulse sequence elements with respect to couplings, offsets, and B1-field inhomogeneities (COB)

Robust experiments that cover a wide range of chemical shift offsets and J-couplings are highly desirable for a multitude of applications in small molecule NMR spectroscopy. Many attempts to improve individual aspects of the robustness of pulse sequence elements based on rational and numerical design have been reported, but a general optimization strategy to cover all necessary aspects for a fully robust sequence is still lacking. In this article, a viable optimization strategy is introduced that covers a defined range of couplings, offsets, and B(1)-field inhomogeneities (COB) in a time-optimal way. Individual components of the optimization strategy can be optimized in any adequate way. As an example for the COB approach, we present the (1)H -(13)C-COB-INEPT with transfer of approximately 99% over the full carbon and proton bandwidth and (1)J(CH) -couplings in the range of 120-250 Hz, which have been optimized using efficient algorithms derived from optimal control theory. The theoretical performance is demonstrated in a number of corresponding COB-HSQC experiments.

Bandwidth-limited control and ringdown suppression in high-Q resonators

We describe how the transient behavior of a tuned and matched resonator circuit and a ringdown suppression pulse may be integrated into an optimal control theory (OCT) pulse-design algorithm to derive control sequences with limited ringdown that perform a desired quantum operation in the presence of resonator distortions of the ideal waveform. Inclusion of ringdown suppression in numerical pulse optimizations significantly reduces spectrometer deadtime when using high quality factor (high-Q) resonators, leading to increased signal-to-noise ratio (SNR) and sensitivity of inductive measurements. To demonstrate the method, we experimentally measure the free-induction decay of an inhomogeneously broadened solid-state free radical spin system at high Q. The measurement is enabled by using a numerically optimized bandwidth-limited OCT pulse, including ringdown suppression, robust to variations in static and microwave field strengths. We also discuss the applications of pulse design in high-Q resonators to universal control of anisotropic-hyperfine coupled electron-nuclear spin systems via electron-only modulation even when the bandwidth of the resonator is significantly smaller than the hyperfine coupling strength. These results demonstrate how limitations imposed by linear response theory may be vastly exceeded when using a sufficiently accurate system model to optimize pulses of high complexity.

Exploring the limits of broadband 90° and 180° universal rotation pulses

90° and 180° universal rotation (UR) pulses are two of the most important classes of pulses in modern NMR spectroscopy. This article presents a systematic study characterizing the achievable performance of these pulses as functions of bandwidth, pulse length, and tolerance to B(1)-field inhomogeneity/miscalibration. After an evaluation of different quality factors employed in pulse design algorithms based on optimal control theory, resulting pulses are discussed in detail with a special focus on pulse symmetry. The vast majority of resulting BURBOP (broadband universal rotations by optimal control) pulses are either fully symmetric or have one symmetric and one antisymmetric Cartesian rf component, where the importance of the first symmetry has not been demonstrated yet and the latter one matches the symmetry that results from a previously derived construction principle of universal rotation pulses out of point-to-point pulses [3]. Optimized BURBOP pulses are shown to perform better than previously reported UR pulses, resulting in shorter pulse durations for the same quality of broadband rotations. From a comparison of qualities of effective universal rotations, we find that the application of a single optimal refocusing pulse matches or improves the performance of two consecutive inversion pulses in INEPT-like pulse sequence elements of the same total duration.

Recent advances in nuclear magnetic resonance quantum information processing

Quantum information processors have the potential to drastically change the way we communicate and process information. Nuclear magnetic resonance (NMR) has been one of the first experimental implementations of quantum information processing (QIP) and continues to be an excellent testbed to develop new QIP techniques. We review the recent progress made in NMR QIP, focusing on decoupling, pulse engineering and indirect nuclear control. These advances have enhanced the capabilities of NMR QIP, and have useful applications in both traditional NMR and other QIP architectures.

Designing optimal universal pulses using second-order, large-scale, non-linear optimization

Recently, RF pulse design using first-order and quasi-second-order pulses has been actively investigated. We present a full second-order design method capable of incorporating relaxation, inhomogeneity in B(0) and B(1). Our model is formulated as a generic optimization problem making it easy to incorporate diverse pulse sequence features. To tame the computational cost, we present a method of calculating second derivatives in at most a constant multiple of the first derivative calculation time, this is further accelerated by using symbolic solutions of the Bloch equations. We illustrate the relative merits and performance of quasi-Newton and full second-order optimization with a series of examples, showing that even a pulse already optimized using other methods can be visibly improved. To be useful in CPMG experiments, a universal refocusing pulse should be independent of the delay time and insensitive of the relaxation time and RF inhomogeneity. We design such a pulse and show that, using it, we can obtain reliable R(2) measurements for offsets within ±γB(1). Finally, we compare our optimal refocusing pulse with other published refocusing pulses by doing CPMG experiments.

Parallel information transfer in a multinode quantum information processor

We describe a method for coupling disjoint quantum bits (qubits) in different local processing nodes of a distributed node quantum information processor. An effective channel for information transfer between nodes is obtained by moving the system into an interaction frame where all pairs of cross-node qubits are effectively coupled via an exchange interaction between actuator elements of each node. All control is achieved via actuator-only modulation, leading to fast implementations of a universal set of internode quantum gates. The method is expected to be nearly independent of actuator decoherence and may be made insensitive to experimental variations of system parameters by appropriate design of control sequences. We show, in particular, how the induced cross-node coupling channel may be used to swap the complete quantum states of the local processors in parallel.

New strategies for designing robust universal rotation pulses: application to broadband refocusing at low power

Optimizing pulse performance often requires a compromise between maximizing signal amplitude and minimizing spectral phase errors. We consider methods for the de novo design of universal rotation pulses, applied specifically but not limited to refocusing pulses. Broadband inversion pulses that rotate all magnetization components 180° about a given fixed axis are necessary for refocusing and mixing in high-resolution NMR spectroscopy. The relative merits of various methodologies for generating pulses suitable for broadband refocusing are considered. The de novo design of 180° universal rotation pulses (180(UR)(°)) using optimal control can provide improved performance compared to schemes which construct refocusing pulses as composites of existing pulses. The advantages of broadband universal rotation by optimized pulses (BURBOP) are most evident for pulse design that includes tolerance to RF inhomogeneity or miscalibration. Nearly ideal refocusing is possible over a resonance offset range of ± 170% relative to the nominal pulse B(1) field, concurrent with tolerance to B(1) inhomogeneity/miscalibration of ± 33%. We present new modifications of the optimal control algorithm that incorporate symmetry principles (S-BURBOP) and relax conservative limits on peak RF pulse amplitude for short time periods that pose no threat to the probe. We apply them to generate a set of low-power 180(BURBOP)(°) pulses suitable for widespread use in (13)C spectroscopy on the majority of available probes. A quantitative measure for the reduced spectral phase error provided by these symmetry principles is also derived. For pulses designed according to this symmetry, refocusing phase errors are virtually eliminated upon application of EXORCYCLE or an equivalent G-180(S-BURBOP)(°)-G gradient sandwich, independent of resonance offset and RF inhomogeneity. The magnitude of the refocused component is not significantly compromised in achieving such ideal phase performance.

Optimization of echo amplitudes resulting from a series of 90° pulses in an inhomogeneous static field

In an inhomogeneous static field, the pulse sequence 90°(x)-(τ-90°(y)-τ)(n) results in a train of echoes, the amplitudes of which settle to be proportional to M(0)/2 after a transient period. Analysis of the spin dynamics of the general τ-β°-τ refocusing cycle reveals that for ideal RF pulses, adding a preparation pulse followed by a delay to the above sequence can either eliminate the transient behavior, or increase the asymptotic echo intensity. This is achieved by controlling the alignment between the magnetization m and the rotation axis of the refocusing cycle. The effect of preparations pulses is demonstrated experimentally in the fringe field of a single sided magnet array. It is shown that for this instrument, transient effects in the echo train can be reduced, and asymptotic signal increased. Spin dynamics calculations indicate that the sequences are robust to finite RF pulse widths, but some discrepancy between theory and experiment is observed due to B(1) inhomogeneity. Refocusing sequences of the type studied here are useful in cases where experimental considerations, such as RF power limits, preclude the use of a 180° pulse in the refocusing cycle.

Robust slice-selective broadband refocusing pulses

Slice-selective broadband refocusing pulses are of great interest in localized MR spectroscopy for improving spatial selectivity, reducing chemical-shift displacement errors, and reducing anomalous J modulation. In practice the bandwidth of RF pulses is limited by the maximum available B1 amplitude. The goal of the present work is to design slice-selective and broadband refocusing pulses which are tolerant against B1 deviations. Pulse design is performed by numerical optimization based on optimal control theory. A comprehensive study of different cost functions and their effect on the optimization is given. The optimized slice-selective broadband refocusing pulses are compared to conventional Shinnar-Le Roux (SLR), broadband SLR, and hyperbolic secant pulses. In simulations and experiments optimized pulses were shown to fulfill broadband slice specifications over a range of ±20% B1 scalings. Experimental validation showed a reduction of chemical-shift displacement error by a factor of 3 compared to conventional SLR pulses.