Carr-Purcell Sequence Research Articles

Deep learning enhanced noise spectroscopy of a spin qubit environment

The undesired interaction of a quantum system with its environment generally leads to a coherence decay of superposition states in time. A precise knowledge of the spectral content of the noise induced by the environment is crucial to protect qubit coherence and optimize its employment in quantum device applications. We experimentally show that the use of neural networks (NNs) can highly increase the accuracy of noise spectroscopy, by reconstructing the power spectral density that characterizes an ensemble of carbon impurities around a nitrogen-vacancy (NV) center in diamond. NNs are trained over spin coherence functions of the NV center subjected to different Carr–Purcell sequences, typically used for dynamical decoupling (DD). As a result, we determine that deep learning models can be more accurate than standard DD noise-spectroscopy techniques, by requiring at the same time a much smaller number of DD sequences.

Floquet Engineering Using Pulse Driving in a Diamond Two-Level System Under Large-Amplitude Modulation

The nitrogen-vacancy center (NVC) in a diamond is a promising platform for Floquet engineering. Using the synchronized readout, we investigate the NVC's Floquet state driven by the Carr-Purcell sequence in a large-amplitude ac magnetic field. We observe the dynamics represented as Bessel functions up to 211th order in a systematic and quantitative agreement with the theoretical model. Furthermore, numerical calculations show that the effect of finite pulse duration and error limits the modulation amplitude available for Floquet engineering. This work provides an approach to precisely investigate Floquet engineering, showing the extendable range of modulation amplitude for two-level systems.

Complex B1+ mapping with Carr-Purcell spin echoes and its application to electrical properties tomography.

PurposeTo present a new complex‐valued B1 + mapping method for electrical properties tomography using Carr‐Purcell spin echoes.MethodsA Carr‐Purcell (CP) echo train generates pronounced flip‐angle dependent oscillations that can be used to estimate the magnitude of B1 +. To this end, a dictionary is used that takes into account the slice profile as well as T2 relaxation along the echo train. For validation, the retrieved B1 + map is compared with the actual flip angle imaging (AFI) method in a phantom (79 ε0, 0.34 S/m). Moreover, the phase of the first echo reflects the transceive phase. Overall, the CP echo train yields an estimate of the complex‐valued B1 +, allowing electrical properties tomography with both permittivity and conductivity. The presented method is evaluated in phantom scans as well as for in vivo brain at 3 T.ResultsIn the phantom, the obtained magnitude B1 + maps retrieved from the CP echo train and the AFI method show excellent agreement, and both the reconstructed estimated permittivity (79 ± 3) ε0 and conductivity (0.35 ± 0.04) S/m values are in accordance with expectations. In the brain, the obtained electrical properties are also close to expectations. In addition to the retrieved complex B1 + information, the decay of the CP echo trains also yields an estimate for T2.ConclusionThe CP sequence can be used to simultaneously provide both B1 + magnitude and phase estimations, and therefore allows for full reconstruction of the electrical properties.

Preserving entanglement and nonlocality in solid-state qubits by dynamical decoupling

In this paper, we study how to preserve entanglement and nonlocality under dephasing produced by classical noise with large low-frequency components, such as $1/f$ noise, using dynamical decoupling techniques. We first show that quantifiers of entanglement and nonlocality satisfy a closed relation valid for two independent qubits locally coupled to a generic environment under pure dephasing and starting from a general class of initial states. This result allows us to assess the efficiency of pulse-based dynamical decoupling for protecting nonlocal quantum correlations between two qubits subject to pure-dephasing local random telegraph and $1/f$ noise. We investigate the efficiency of an ``entanglement memory'' element under two-pulse echo and under sequences of periodic, Carr-Purcell, and Uhrig dynamical decoupling. The Carr-Purcell sequence is shown to outperform the other sequences in preserving entanglement against both random telegraph and $1/f$ noise. For typical $1/f$ flux-noise figures in superconducting nanocircuits, we show that entanglement and its nonlocal features can be efficiently stored up to times one order of magnitude longer than natural entanglement disappearance times employing pulse timings of current experimental reach.

Use of Carr–Purcell pulse sequence with low refocusing flip angle to measure T1 and T2 in a single experiment

The Carr–Purcell pulse sequence, with low refocusing flip angle, produces echoes midway between refocusing pulses that decay to a minimum value dependent on T 2 ∗ . When the refocusing flip angle was π/2 (CP 90) and τ > T 2 ∗ , the signal after the minimum value, increased to reach a steady-state free precession regime (SSFP), composed of a free induction decay signal after each pulse and an echo, before the next pulse. When τ < T 2 ∗ , the signal increased from the minimum value to the steady-state regime with a time constant ( T ∗) = 2 T 1 T 2/( T 1 + T 2) , identical to the time constant observed in the SSFP sequence, known as the continuous wave free precession (CWFP). The steady-state amplitude obtained with M CP90 = M 0 T 2/( T 1 + T 2) was identical to CWFP. Therefore, this sequence was named CP-CWFP because it is a Carr–Purcell sequence that produces results similar to the CWFP. However, CP-CWFP is a better sequence for measuring the longitudinal and transverse relaxation times in single scan, when the sample exhibits T 1 ∼ T 2. Therefore, this sequence can be a useful method in time domain NMR and can be widely used in the agriculture, food and petrochemical industries because those samples tend to have similar relaxation times in low magnetic fields.

Herman Yaggi Carr, PhD (1924–2008): A tribute

Herman Yaggi Carr, PhD (1924–2008): A tribute

Which pulse sequence is optimal for myo‐inositol detection at 3T?

Optimized myo-inositol (mI) detection is important for diagnosing and monitoring a multitude of pathological conditions of the brain. Simulations are presented in this work, performed to decide which pulse sequence has the most significant advantage in terms of improving repeatability and accuracy of mI measurements at 3T over the pulse sequence used typically in the clinic, a TE = 35 ms PRESS sequence. Five classes of pulse sequences, four previously suggested for optimized mI detection (a short TE PRESS, a Carr-Purcell PRESS sequence, an optimized STEAM sequence, an optimized zero quantum filter), and one optimized for mI detection in this work (a single quantum filter) were compared to a standard, TE = 35 ms pulse sequence. While limiting the SNR of an acquisition to the equivalent SNR of a spectrum acquired in 5 min from an 8 cc voxel, it was found through simulations that the most repeatable mI measurements would be obtained with a Carr-Purcell sequence. This sequence was implemented in a clinical scanner, and improved mI measurements were demonstrated in vivo.

Multiecho sequence for velocity imaging in inhomogeneous rf fields

The unambiguous determination of velocities with spatial resolution in a multiecho PFG NMR sequence strongly depends on the homogeneity of the B 1 field. This affects, in particular, the use of surface coils that bear considerable potential for on-line flow monitoring where a fast-imaging sequence can become vital. However, even with most rf coils dedicated for imaging applications, B 1 inhomogeneities are sufficiently large to generate severe problems in performing velocity-imaging experiments. In this paper, the use of a combination of different phase cycles in Carr–Purcell sequences is discussed. The suggested phase cycling scheme tolerates large flip angle imperfections arising in inhomogeneous B 1 fields, and thus allows acquisition of a maximum number of echoes within a pulse train. The performance of the velocity-imaging sequence is proven by using phantom samples developing known laminar flow patterns.

Manifestations of the absence of spin diffusion in multipulse NMR experiments on diluted dipolar solids

Puzzling anomalies previously observed in multipulse NMR experiments in natural abundance 29Si [A.E. Dementyev, D. Li, K. MacLean, and S.E. Barrett, Phys. Rev. B 68, 153302 (2003)] such as long-lived spin echoes and even-odd asymmetries, are also found in polycrystalline C60. Further experiments controlling the phases and tilting angles of the pulse trains, as well as analytical and numerical calculations allowed us to explain the origin of these anomalies. We prove that the observation of long magnetization tails requires two conditions: i) an rf field inhomogeneity able to produce different tilting angles in different sites of the sample and ii) the absence of spin diffusion (non-effective flip-flop interactions). The last requirement is easily satisfied in diluted dipolar solids, where the frequency differences between sites, caused by disorder or other sources, are usually at least one order of magnitude larger than the dipolar couplings. Both conditions lead to the generation of stimulated echoes in Carr-Purcell (CP) and Carr-Purcell-Meiboom-Gill (CPMG) pulse trains. We show, both experimentally and theoretically, that the stimulated echoes interfere constructively or destructively with the normal (Hahn) echoes depending on the alternation or not of the pi pulse phases in the CP and the CPMG sequences. Constructive interferences occur for the CP and CPMG sequences with and without phase alternation respectively, which are the cases where long magnetization tails are observed. Sequences with two, three and four pi pulses after the pi/2 pulse allow us to disentangle the contributions of the different echoes and show how the stimulated echoes originate the even-odd asymmetry observed in both 29Si and C60 polycrystalline samples.

Transverse relaxation in the rotating frame induced by chemical exchange

In the presence of radiofrequency irradiation, relaxation of magnetization aligned with the effective magnetic field is characterized by the time constant T 1 ρ . On the other hand, the time constant T 2 ρ characterizes the relaxation of magnetization that is perpendicular to the effective field. Here, it is shown that T 2 ρ can be measured directly with Carr–Purcell sequences composed of a train of adiabatic full-passage (AFP) pulses. During adiabatic rotation, T 2 ρ characterizes the relaxation of the magnetization, which under adiabatic conditions remains approximately perpendicular to the time-dependent effective field. Theory is derived to describe the influence of chemical exchange on T 2 ρ relaxation in the fast-exchange regime, with time constant defined as T 2 ρ,ex . The derived theory predicts the rate constant R 2ρ, ex (=1/T 2ρ, ex ) to be dependent on the choice of amplitude- and frequency-modulation functions used in the AFP pulses. Measurements of R 2 ρ,ex of the water/ethanol exchanging system confirm the predicted dependence on modulation functions. The described theoretical framework and adiabatic methods represent new tools to probe exchanging systems.

Transverse Nuclear Spin Relaxation Studies of Viscoelastic Properties of Membrane Vesicles. I. Theory

Transverse nuclear spin relaxation measurements employing Carr-Purcell (CP) pulse sequences can provide detailed information on the slow-motional dynamics in biomembranes. In this paper, a comprehensive relaxation model is developed for the analysis of such experiments performed on unilamellar quasi-spherical vesicles. The basis of the model is the stochastic Liouville equation in which two different relaxation processes are considered (i.e., vesicle shape fluctuations and molecular translational diffusion). It is shown that for vesicle radii R 0 ≥ 200 nm, translational diffusion of the lipid molecules along the vesicle shell is too slow to contribute significantly to transverse spin relaxation in the kHz range, whereas vesicle shape fluctuations constitute the dominant transverse relaxation process. The theory is employed in model calculations for pulse frequency-dependent transverse 3 1 P nuclear spin relaxation rates. R C P 2 , ∞ (ω), from CP sequences. The analysis reveals that R C P 2 , ∞ (ω), induced by vesicle fluctuations, depends linearly on ω - 1 over a wide frequency range in the kHz regime. Notably, within this linear dispersion regime, the bending elastic modulus κ is the only relevant parameter because the magnitude of R C P 2 , ∞ (ω) does not depend on the size of the vesicle R 0 , the effective lateral tension σ, or the viscosity of the surrounding fluid η. On the other hand, R 0 , σ, ij, and K determine the frequency at which R C P 2 , ∞ (ω) levels off to a constant plateau value independent of ω. Thus, analysis of the R C P 2 , ∞ (ω) dispersion profiles is a direct way to determine the bending elastic modulus and other viscoelastic parameters of membrane vesicles.

Carr-Purcell sequences with composite pulses.

We present novel Carr-Purcell-like sequences using composite pulses that exhibit improved performance in strongly inhomogeneous fields. The sequences are designed to retain the intrinsic error correction of the standard Carr-Purcell-Meiboom-Gill (CPMG) sequence. This is achieved by matching the excitation pulse with the refocusing cycle such that the initial transverse magnetization lies along the axis n(Beta) characterizing the overall rotation of the refocusing cycle. Such sequences are suitable for relaxation measurements. It is shown that in sufficiently inhomogeneous fields, the echo amplitudes have an initial transient modulation that is limited to the first few echoes and then decay with the intrinsic relaxation time of the sample. We show different examples of such sequences that are constructed from simple composite pulses. Sequences of the form 90 degrees (0)-(90 degrees (90-theta/2)-theta(180-theta/2)-90 degrees (90-theta/2))(n) with theta approximately 90 degrees and 270 degrees generate signal over a bandwidth larger than that of the conventional CPMG sequence, resulting in an improved signal-to-noise ratio in inhomogeneous fields. The new sequence 127 degrees (x,y)-(127 degrees (x)-127 degrees (-x))(n) only excites signal off-resonance with a spectrum that is bimodal, peaking at Delta omega(0)=+/-omega(1). Depending on the phase and exact timing of the first pulse, symmetric or antisymmetric excitation is obtained. We also demonstrate several new sequences with improved dependence on the RF field strength. The sequence (22.5 degrees (67.5)-90 degrees (-22.5))-(90 degrees (67.5)-45 degrees (157.5)-90 degrees (67.5))(n) has the property that the phase of the signal depends on B(1), allowing coarse B(1) imaging in a one-dimensional experiment.

Differentiation of hepatic malignancies from hemangiomas and cysts by T2 relaxation times: early experience with multiply refocused four-echo imaging at 1.5 T.

The purpose of this study was to examine hepatic lesions with a sequence designed to yield improved T2 measurements and evaluate the clinical utility of these measurements in distinguishing malignant from benign disease. Using a modified Carr-Purcell sequence incorporating features designed to compensate for imperfections in the imaging system, including a train of refocusing pulses emitted in an MLEV pattern oriented in composite fashion along all three coordinate axes, and a single spatially selective pulse placed immediately before a spiral readout, 14 benign lesions and 13 malignant lesions were evaluated prospectively with a conventional 1.5 T imager. The maximum, minimum, and mean T2 values of malignant lesions, hemangiomas, and cysts exceeded corresponding published values from spin-echo and echoplanar studies. The mean T2 value of the malignant lesions differed significantly (P < 0.0001) from those of hemangiomas and cysts. All malignant lesions and all benign lesions were distinguishable by their T2 values, which had ranges of no greater than 118.6 msec and no less than 134.3 msec, respectively. This early experience suggests that improved T2 measurements can facilitate the differentiation of hepatic malignancies from hemangiomas and cysts.

Analysis of Carr–Purcell Sequences with Nonideal Pulses

Analysis of Carr–Purcell Sequences with Nonideal Pulses

13C Dipolar NMR Study of Benzene Chemisorbed on High-Surface Alumina-Supported Platinum

NMR spectroscopy of dipole-coupled pairs of 13 C nuclei is employed to probe the molecular structure and dynamics of benzene molecules chemisorbed on a Pt/η-Al 2 O 3 catalyst. A modified Carr-Purcell sequence, which suppresses inhomogeneous broadening, permits good spectral resolution over a broad temperature range. A comparison between spectral data and computer simulations indicates more than one type of site for the chemisorbed molecules. The molecular structure appears to be indistinguishable from that in polycrystalline benzene but with a vastly different motional behavior

Magnetic Resonance Imaging of Diffusion in the Presence of Background Gradients and Imaging of Background Gradients

Abstract The measurement of the self-diffusion coefficient D by an NMR technique that uses an applied gradient G A can be corrupted by systems that have a background magnetic field gradient G 0 and also by imaging gradients G 1 , when used in an imaging mode. In a nonimaging mode, the corrupting cross term G A · G 0 can be eliminated in the diffusion measurement by use of an alternating-pulse-field-gradient (APFG) sequence that allows an accurate and uncorrupted measurement of D . A Carr-Purcell echo train enables the measurement of the expectation value, 〈 D G 2 0 〉; assuming D and G 0 to be uncorrelated will allow 〈 G 2 0 〉 to be determined. An image of D or of 〈 D G 2 0 〉 may be obtained without the corrupting G A · G 0 and G A · G I terms by appending a standard imaging sequence to an APFG sequence or a Carr-Purcell sequence, respectively; assuming D and G 0 to be uncorrelated will allow (〈 G 2 0 〉 to be determined within each pixel. Measurements of D and 〈 G 2 0 〉 and their images are made in apple flesh in which minute air bubbles are shown to produce the large 〈 G 2 0 〉. Their values in an 81 g Golden Delicious apple at a measuring frequency of 100 MHz were D = 1.42 × 10 −5 cm 2 /s and [formula] = 8.9 G/cm.

Correlation of Chemical Bond Directions and Functional Group Orientations in Solids by Two-Dimensional NMR

We describe a new two-dimensional NMR technique for structural studies of polycrystalline and noncrystalline solids. The technique is a variant of 2D exchange spectroscopy applicable to organic molecules, macromolecules, or molecular complexes that are doubly 13C-labeled at a specific carboncarbon bond and singly 13C labeled at a specific functional group. A Carr-Purcell sequence is used to obtain dipolar spectra in the t 1 dimension. Spectra in the t 2 dimension are determined primarily by the chemical-shift anisotropy. With spin diffusion among the labeled sites between the t 1 and t 2 periods, the resulting 2D spectra reveal correlations between the direction of the labeled bond and the orientation of the labeled functional group. Experimental spectra of two polycrystalline model compounds, dimethyl succinate and diammonium succinate, are presented and compared with simulations to illustrate the structural information contained in the 2D spectra.

The Effect of Amplitude Imbalance on Compensated Carr-Purcell Sequences

The Effect of Amplitude Imbalance on Compensated Carr-Purcell Sequences

Practical T2 quantitation for clinical applications.

Many studies have investigated the use of magnetic resonance relaxation times for tissue characterization. A number have been performed in vivo with clinical whole-body imagers. Unfortunately, the results have yet to establish the role of quantitative tissue relaxation time measurements in clinical diagnosis. One of the major problems is that the techniques used in many of these studies are error prone, making the results inconclusive. In the present study, the problems associated with clinical T2 measurements were systematically evaluated in an attempt to obtain reliable in vivo quantitation. The authors demonstrate that spoiler gradients are the most effective technique for artifact suppression but that they render ineffective radio-frequency phase schemes such as the Meiboom-Gill modification to the Carr-Purcell sequence to compensate experimental imperfections. The present study results in a more reliable multi-echo sequence for T2 measurement. Preliminary clinical results in brain and cervix demonstrate the performance of the new technique.

Modification of the Carr-Purcell sequence for single-shot echo-planar imaging.

An artifact caused by the RF field inhomogeneity in echo-planar imaging (EPI) with the CPMG sequence is analyzed. A modified sequence is presented, using 90 degrees phase alternation of the pi pulses, which suppresses the spurious components of the signal. Thus the phase cycling procedure can be avoided and the pi-pulsed EPI experiment can be reduced to a single shot.