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... (A cure is offered by the "shortcuts-to-adiabaticity" approach [5], but it comes with the necessity of accurate pulse shaping or additional fields.) Composite pulses -trains of pulses with well-defined relative phases used as control parameters [6,7] -sit somewhere in the "sweet spot" as they feature extreme accuracy and robustness, while being significantly faster than adiabatic methods (but slower than resonant excitation by a factor of 2-3 or more). ...
... The dominant majority of composite pulses in the literature are designed to produce specific rotations on the Bloch sphere, typically at angles π (generating complete population transfer), π/2 (half population transfer), π/4 and 3π/4, as reviewed in Refs. [6,7]. There exist just a few composite sequences which produce general rotations at arbitrary angles [41][42][43][44][45][46]. ...
... Composite rotations are broadly divided into two large groups called variable and constant rotations. The variable rotations [6,44,45] feature well-defined transition probability but not well-defined phases of the propagator. Constant (or phase-distortionless rotations) feature both well-defined populations and well-defined phases of the propagator [41][42][43]. ...
In some applications of quantum control, it is necessary to produce very weak excitation of a quantum system. Such an example is presented by the concept of single-photon generation in cold atomic ensembles or doped solids, e.g. by the DLCZ protocol, for which a single excitation is shared among thousands and millions atoms or ions. Another example is the possibility to create huge Dicke state of $N$ qubits sharing a single or a few excitations. Other examples are using tiny rotations to tune high-fidelity quantum gates or using these tiny rotations for testing high-fidelity quantum process tomography protocols. Ultrasmall excitation of a quantum transition can be generated by either a very weak or far-detuned driving field. However, these two approaches are sensitive to variations in the experimental parameters, e.g. the transition probability varies with the square of the pulse area. Here we propose a different method for generating a well-defined pre-selected very small transition probability -- of the order of $10^{-2}$ to $10^{-8}$ -- by using composite pulse sequences. The method features high fidelity and robustness to variations in the pulse area and the pulse duration.
... In principle, pulses should be as short as possible in order to minimize relaxation effects, but in practice, limited available power limits the maximum achievable pulse amplitude which in turn limits the frequency sweep rate and therefore the pulse length. In response, optimal control methods such as composite pulses [14,15], adiabatic pulses [16,17], optimal control theory (OCT) pulses using different numerical algorithms [18][19][20][21][22], including gradient ascent pulse engineering (GRAPE) [23][24][25][26][27][28], were developed to accomplish broadband excitation and uniform inversion across a given bandwidth. In practice, optimal shaped pulses are distorted by nonlinearities in the power amplifier and by the resonator transfer function, moving them away from the extremum in the optimization landscape. ...
In this paper, we numerically optimize broadband pulse shapes that maximize Hahn echo amplitudes. Pulses are parameterized as neural networks (NN), nonlinear amplitude limited Fourier series (FS), and discrete time series (DT). These are compared to an optimized choice of the conventional hyperbolic secant (HS) pulse shape. A power constraint is included, as are realistic shape distortions due to power amplifier nonlinearity and the transfer function of the microwave resonator. We find that the NN, FS, and DT parameterizations perform equivalently, offer improvements over the best HS pulses, and contain a large number of equivalent optimal solutions, implying the flexibility to include further constraints or optimization goals in future designs.
... This conceptual framework involves combining conventional wave plates with the optical axis of individual retarders rotated at specific angles. Leveraging the mathematical analogy between the description of polarization rotation and the dynamics of a light-driven two-level quantum system (akin to the concept of composite pulses [20]), we identified optimal rotation sequences for the wave plates. In contrast to single wave plates, these composite sequences can mitigate the chromatic dependence of retardation, enabling operation at broad bandwidths. ...
We theoretically propose a type of tunable polarization retarder, which is composed of sequences of half-wave and quarter-wave polarization retarders, allowing operation at broad spectral bandwidth. The constituent retarders are composed of stacked standard half-wave retarders and quarter-wave retarders rotated at designated angles relative to their fast polarization axes. The proposed composite retarder (CR) can be tuned to an arbitrary value of the retardance by varying the middle retarder alone while maintaining its broadband spectral bandwidth intact.
... Composite pulse technology was originally utilized in NMR [22], and quantum optimal control technic has been applied on composite pulses in NMR system [23,24]. Recently, the composite pulse is also used in quantum coherent manipulations to enhance such as gate fidelity and robust [25][26][27][28]. ...
Measuring the phonon number of the laser-cooled ions is an indispensable step in evaluating whether an ion is in ground state. At present, commonly used methods in the experiments are red-to-blue sideband ratios and adiabatic evolution red-sideband methods. We theoretically propose a method using composite pulses which does not need a fit of state evolution and can directly measure the population of the selected Fock state. It can measure higher Fock state population more directly comparing with the adiabatic evolution red-sideband method. We use quantum optimal control method to improve the fidelity of unitary operation of the composite pulses. With quantum optimal control technology, we can discuss the situation where the laser strength is strong, and many approximations will not be necessary, where the gate fidelity can be further improved. Then we give a method to modify the measurement result for a higher accuracy which has a good performance, and we give an example to illustrate its application on high Fock state measurement.
... As radio frequency pulses on hydrogen can achieve Rabi frequencies above 100 kHz≫ ∆ω particle , composite pulse techniques (see e.g. Ref. [89,90]) offer a potential path to compensate for detunings and Rabi frequency errors that are up to approximately half of the Rabi frequency. ...
We introduce pentacene-doped naphthalene as a material for diamagnetic levitation, offering compelling applications in matter-wave interferometry and nuclear magnetic resonance. Pentacene-doped naphthalene offers remarkable polarizability of its nuclear spin ensemble, achieving polarization rates exceeding 80 % at cryogenic temperatures with polarization lifetimes extending weeks. We design a multi-spin Stern-Gerlach-type interferometry protocol which, thanks to the homogeneous spin distribution and the absence of a preferential nuclear-spin quantization axis, avoids many of the limitations associated with materials hosting electronic spin defects, such as nanodiamonds containing NV centers. We assess the potential of our interferometer to enhance existing bounds on the free parameters of objective collapse models. Beyond matter-wave interferometry, we analyze the prospects for implementing magic angle spinning at frequencies surpassing the current standard in NMR, capitalizing on the exceptional rotational capabilities offered by levitation. Additionally, we outline a novel protocol for measuring spin ensemble polarization via the position of the nanoparticle and conduct an analysis of dominant noise sources, benchmarking the required isolation levels for various applications.
... Software suppression of their effects will be necessary because it is difficult to mitigate them only by hardware calibration. Error-robust control sequences compensating for the effects of systematic errors, which are known as "composite pulses", have been developed in the field of nuclear magnetic resonance (NMR) [25][26][27]. These errorrobust sequences are composed of several operations such that errors in the operations are canceled each other. ...
Precise control of quantum systems is one of the most important milestones for achieving practical quantum technologies, such as computation, sensing, and communication. Several factors deteriorate the control precision and thus their suppression is strongly demanded. One of the dominant factors is systematic errors, which are caused by discord between an expected parameter in control and its actual value. Error-robust control sequences, known as composite pulses, have been invented in the field of nuclear magnetic resonance (NMR). These sequences mainly focus on the suppression of errors in one-qubit control. The one-qubit control, which is the most fundamental in a wide range of quantum technologies, often suffers from detuning error. As there are many possible control sequences robust against the detuning error, it will practically be important to find ``optimal" robust controls with respect to several cost functions such as time required for operation, and pulse-area during the operation, which corresponds to the energy necessary for control. In this paper, we utilize the Pontryagin's maximum principle (PMP), a tool for solving optimization problems under inequality constraints, to solve the time and pulse-area optimization problems. We analytically obtain pulse-area optimal controls robust against the detuning error. Moreover, we found that short-CORPSE, which is the shortest known composite pulse so far, is a probable candidate of the time optimal solution according to the PMP. We evaluate the performance of the pulse-area optimal robust control and the short-CORPSE, comparing with that of the direct operation.
... As a promising tool, the composite pulse (CP) technique [54][55][56][57][58][59][60][61][62][63][64][65] has come into being and provides sufficient support for the realization of reliable quantum computations in three-level systems. The CP sequence, initially introduced in nuclear magnetic resonance [66,67], refers to a train of pulses with precisely designated phases. In general, these sequences are constructed to compensate for deviations in physical parameters through some controllable variables, such as phase, detuning, and duration. ...
We propose a robust and high-fidelity scheme for realizing universal quantum gates by optimizing short pulse sequences in a three-level system. To alleviate the sensitivity to the errors, we recombine all elements of error matrices to construct a cost function with three types of weight factors. The modulation parameters are obtained by searching for the minimum value of this cost function. The purposes of introducing the weight factors are to reduce the detrimental impact of high-order error matrices, suppress population leakage to the third state, correct the operational error in the qubit space, and optimize the total pulse area of short pulse sequences. The results demonstrate that the optimized sequences exhibit strong robustness against errors and effectively reduce the total pulse area. Therefore, this work presents a valuable method for achieving exceptional robustness and high speed in quantum computations.
... Thus, 'software' reduction of errors by control methods will be important as well as 'hardware' reduction by calibrations. A control method that compensates for effects of systematic errors is the composite quantum gate (CQG), which was invented and has been developed in the field of nuclear magnetic resonance (NMR) [21][22][23]. A CQG that is robust against a systematic error replaces a single operation with a sequence of operations such that the errors in the operations are canceled each other. ...
Precise control of quantum systems is a cornerstone for realizing high-quality quantum technology such as quantum computing and quantum communication. The performance of control of systems often deteriorates due to systematic errors. In one-qubit control, the pulse length error is a typical systematic error, which is often caused by deviation of the strength of the control field. A composite quantum gate is a method for suppressing effects of such systematic errors at the cost of a long operation time. A longer operation time implies stronger decoherence, and thus a shorter composite quantum gates is preferable from the viewpoint of noise immunity. However, it has not been clear how short composite quantum gate can be implemented. This problem can be regarded as an optimisation problem under constraints: optimising the operation time while requiring the error robustness. In this paper, we find a lower bound on the operation time of all the composite quantum gates with the first-order robustness against the pulse length error, in which effects of the error are eliminated up to its first order. The derivation of this bound is based on a geometric property of robustness against the pulse length error. This can be used for search after high-performance composite quantum gates.
... However, the major disadvantage of the resonant pulse is its high susceptibility to systematic errors. To significantly enhance the robustness against systematic errors, one can turn to the composite pulses [67][68][69], a train of pulses with identical amplitudes and relative phases to be addressed. A general expression for the total evolution operator of composite pulses can be formulated as follows: ...
In this work, we develop a supervised learning model for implementing robust quantum control in composite-pulse systems, where the training parameters can be either phases, detunings, or Rabi frequencies. This model exhibits great resistance to all kinds of systematic errors, including single, multiple, and time-varying errors. We propose a modified gradient descent algorithm for adapting the training of phase parameters, and show that different sampling methods result in different robust performances. In particular, there is a tradeoff between high-fidelity and robustness for a given number of training parameters, and both of them can be simultaneously enhanced by increasing the number of training parameters (pulses). For its applications, we demonstrate that the current model can be used for achieving high-fidelity arbitrary superposition states and universal quantum gates in a robust manner. This work provides a highly efficient learning model for fault-tolerant quantum computation by training various physical parameters.
... Furthermore, the CEBS technique allows for shorter pulse trains than many other acceleration methods. The increase in the momentum transfer rate is related to the destructive interference of non-adiabatic losses in close connection with the concepts of shortcuts to adiabaticity [47] and composite pulses [48]. Therefore pulse sequence shaping based on optimal control protocols is a potential of improvement in efficiency and robustness. ...
We report here on the realization of light-pulse atom interferometers with Large-momentum-transfer atom optics based on a sequence of Bragg transitions. We demonstrate momentum splitting up to 200 photon recoils in an ultra-cold atom interferometer. We highlight a new mechanism of destructive interference of the losses leading to a sizeable efficiency enhancement of the beam splitters. We perform a comprehensive study of parasitic interferometers due to the inherent multi-port feature of the quasi-Bragg pulses. Finally, we experimentally verify the phase shift enhancement and characterize the interferometer visibility loss.
... π -pulse technique is sensitive to the resonance condition and also the exact area of the pulses and deviation from these conditions reduces the population transfer efficiency. In recent years, various methods have been proposed to optimize the π -pulse method, which the most important of them is the composite pulse method [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47]. In composite pulse 32 technique in two-state systems [42,43], an odd number of pulses with specific phases and the same pulse area is used for population transfer and by increasing the number of pulses, the sensitivity of the system to the conditions of π -pulse technique decreases. ...
Population transfer of two-state nuclei interacting with a train of composite X-ray free electron laser (XFEL) pulses has been investigated theoretically. In this study, we calculate the effective intensity of the XFEL pulse for each nucleus so that the time temporal pulse area of Rabi frequency is equal to π. We show that with increasing the number of composite pulses, even with a significant deviation of the effective intensity of the laser beam from the calculated value, the population is completely transferred from the ground state to the excited state. For numerical study, nuclei with a high lifetime in the excited state, compared to the XFEL laser pulse duration, have been selected so that the effect of spontaneus emission can be neglected. Finally, it has been shown that despite the detuning effects, by increasing the number of XFEL composite pulses as well as the effective intensity of the laser pulse, the population is completely transferred to the excited state.
... Error correction and noise mitigation strategies are crucial for quantum computation. Two important and distinct strategies have been suppression of systematic errors using composite pulses [1][2][3] and correction of random errors using classical and quantum codes [4][5][6]. Both of them are fine-grained approaches that aim to make more perfect gates from imperfect ones. ...
The intrinsic probabilistic nature of quantum systems makes error correction or mitigation indispensable for quantum computation. While current error-correcting strategies focus on correcting errors in quantum states or quantum gates, these fine-grained error-correction methods can incur significant overhead for quantum algorithms of increasing complexity. We present a first step in achieving error correction at the level of quantum algorithms by combining a unified perspective on modern quantum algorithms via quantum signal processing (QSP). An error model of under- or over-rotation of the signal processing operator parameterized by $\epsilon < 1$ is introduced. It is shown that while Pauli $Z$-errors are not recoverable without additional resources, Pauli $X$ and $Y$ errors can be arbitrarily suppressed by coherently appending a noisy `recovery QSP.' Furthermore, it is found that a recovery QSP of length $O(2^k c^{k^2} d)$ is sufficient to correct any length-$d$ QSP with $c$ unique phases to $k^{th}$-order in error $\epsilon$. Allowing an additional assumption, a lower bound of $\Omega(cd)$ is shown, which is tight for $k = 1$, on the length of the recovery sequence. Our algorithmic-level error correction method is applied to Grover's fixed-point search algorithm as a demonstration.
... Finally, one can also combine these techniques with other advanced coherent control methods such as optimal control [26][27][28][29], composite pulses [13,[30][31][32][33][34][35] and shortcuts to adiabaticity [36][37][38] to improve robustness, increase speed and suppress unwanted transitions. Expansion of the methods to multilevel systems is also envisioned. ...
Efficient initialization and manipulation of quantum states is important for numerous applications and it usually requires the ability to perform high fidelity and robust swapping of the populations of quantum states. Stimulated Raman adiabatic passage (STIRAP) has been known to perform efficient and robust inversion of the ground states populations of a three-level system. However, its performance is sensitive to the initial state of the system. In this contribution we demonstrate that a slight modification of STIRAP, where we introduce a non-zero single-photon detuning, allows for efficient and robust population swapping for any initial state. The results of our work could be useful for efficient and robust state preparation, dynamical decoupling and design of quantum gates in ground state qubits via two-photon interactions.
We construct a fault-tolerant quantum error-correcting protocol based on a qubit encoded in a large spin qudit using a spin-cat code, analogous to the continuous-variable cat encoding. With this, we can correct the dominant error sources, namely processes that can be expressed as error operators that are linear or quadratic in the components of angular momentum. Such codes tailored to dominant error sources can exhibit superior thresholds and lower resource overheads when compared to those designed for unstructured noise models. A key component is the gate that preserves the rank of spherical tensor operators. Categorizing the dominant errors as phase and amplitude errors, we demonstrate how phase errors, analogous to phase-flip errors for qubits, can be effectively corrected. Furthermore, we propose a measurement-free error-correction scheme to address amplitude errors without relying on syndrome measurements. Through an in-depth analysis of logical gate errors, we establish that the fault-tolerant threshold for error correction in the spin-cat encoding surpasses that of standard qubit-based encodings. We consider a specific implementation based on neutral-atom quantum computing, with qudits encoded in the nuclear spin of 87 Sr , and show how to generate the universal gate set, including the rank-preserving gate, using quantum control and the Rydberg blockade. These findings pave the way for encoding a qubit in a large spin with the potential to achieve fault tolerance, high threshold, and reduced resource overhead in quantum information processing.
Published by the American Physical Society 2024
Decoherence and imperfect control are crucial challenges for quantum technologies. Common protection strategies rely on noise temporal autocorrelation, which is not optimal if other correlations are present. We develop and demonstrate experimentally a strategy that uses the cross-correlation of two noise sources. Utilizing destructive interference of cross-correlated noise extends the coherence time tenfold, improves control fidelity, and surpasses the state-of-the-art sensitivity for high frequency quantum sensing, significantly expanding the applicability of noise protection strategies.
Published by the American Physical Society 2024
A number of composite pulse (CP) sequences for four basic quantum phase gates—the Z, S, T, and general phase gates—are presented. The CP sequences contain up to 18 pulses and can compensate up to eight orders of experimental errors in the pulse amplitude and duration. The short CP sequences (up to eight pulses) are calculated analytically and the longer ones numerically. The results demonstrate the remarkable flexibility of CPs accompanied by extreme accuracy and robustness to errors. These CP sequences, in particular the Z, S, and T gates, can be very useful quantum control tools in quantum information applications, because they provide a variety of options to find the optimal balance among ultrahigh fidelity, error range, and speed, which may be different in different physical applications.
In this work, we develop a supervised learning model for implementing robust quantum control in composite-pulse systems, where the training parameters can be either phases, detunings, or Rabi frequencies. This model exhibits great resistance to all kinds of systematic errors, including single, multiple, and time-varying errors. We propose a modified gradient descent algorithm for adapting the training of phase parameters, and show that different sampling methods result in different robust performances. In particular, there is a trade-off between high fidelity and robustness for a given number of training parameters, and both can be simultaneously enhanced by increasing the number of training parameters (pulses). For its applications, we demonstrate that the current model can be used for achieving high-fidelity arbitrary superposition states and universal quantum gates in a robust manner. This work provides a highly efficient learning model for fault-tolerant quantum computation by training various physical parameters.
Graph states are a broad family of entangled quantum states, each defined by a graph composed of edges representing the correlations between subsystems. Such states constitute versatile resources for quantum computation and quantum-enhanced measurement. Their generation and engineering require a high level of control over entanglement. Here we report on the generation of continuous-variable graph states of atomic spin ensembles, which form the nodes of the graph. We program the entanglement structure encoded in the graph edges by combining global photon-mediated interactions in an optical cavity with local spin rotations. By tuning the entanglement between two subsystems, we either localize correlations within each subsystem or enable Einstein–Podolsky–Rosen steering—a strong form of entanglement that enables the extraction of precise information from one subsystem through measurements on the other. We further engineer a four-mode square graph state, highlighting the flexibility of our approach. Our method is scalable to larger and more complex graphs, laying groundwork for measurement-based quantum computation and advanced protocols in quantum metrology.
We propose a protocol for robust quantum state engineering using composite pulses (CPs) in four-level systems. The analytical expression of the propagator is derived for the implementation of universal single-qubit gates and the maximum superposition state. By carefully designing the relative phases between pulses, the CP sequences can compensate for the pulse area error to any desired order. We present two classes of CP sequences, one generating robust population inversion and the other generating robust superposition states. As applications, we employ the well-designed CP sequences to achieve the conversion of the W and Greenberger-Horne-Zeilinger states with high fidelity in a Rydberg atomic system. It is shown that the CP sequences yield excellent robustness with respect to the pulse area errors, and they possess a short evolution time.
Coherent control errors, for which ideal Hamiltonians are perturbed by unknown multiplicative noise terms, are a major obstacle for reliable quantum computing. In this paper we present a framework for analyzing the robustness of quantum algorithms against coherent control errors using Lipschitz bounds. We derive worst-case fidelity bounds which show that the resilience against coherent control errors is mainly influenced by the norms of the Hamiltonians generating the individual gates. These bounds are explicitly computable even for large circuits and they can be used to guarantee fault tolerance via threshold theorems. Moreover, we apply our theoretical framework to derive a guideline for robust quantum algorithm design and transpilation, which amounts to reducing the norms of the Hamiltonians. Using the three-qubit quantum Fourier transform as an example application, we demonstrate that this guideline targets robustness more effectively than existing ones based on circuit depth or gate count. Furthermore, we apply our framework to study the effect of parameter regularization in variational quantum algorithms. The practicality of the theoretical results is demonstrated via implementations in simulation and on a quantum computer.
Because processors based on superconducting qubits are inherently noisy, schemes for increased performance that yield higher fidelity, robustness, or improved error correction could be beneficial. Focusing on leakage, seepage, and robustness, we implemented single-qubit gates from composite and adiabatic pulses on a transmon qubit and assessed their performance relative to default pulses in terms of robustness and seepage and leakage rates using interleaved and leakage randomized benchmarking. Unsurprisingly, these pulses did not lead to marked reductions in leakage or seepage rates because they were not designed to do so. However, they were able to compensate for a broader range of systematic drive amplitude and off-resonance errors compared with standard gates. In some cases, using these pulses improved robustness by nearly an order of magnitude. These pulses could be useful for improving quantum error correction protocols or in contexts where cross-talk and calibration drift are problematic.
In this work, we propose a comprehensive design for narrowband and passband composite pulse sequences by involving the dynamics of all states in the three-state system. The design is quite universal as all pulse parameters can be freely employed to modify the coefficients of error terms. Two modulation techniques, the strength and phase modulations, are used to achieve arbitrary population transfer with a desired excitation profile, while the system keeps minimal leakage to the third state. Furthermore, the current sequences are capable of tolerating inaccurate waveforms, detuning errors, and work well when rotating wave approximation is not strictly justified. Therefore, this work provides versatile adaptability for shaping various excitation profiles in both narrowband and passband sequences.
Precise control of hyperfine matterwaves via Raman excitations is instrumental to a class of atom-based quantum technology. We investigate the Raman matterwave control technique for alkaline-like atoms in an intermediate regime characterized by the single-photon detuning where a choice can be made to balance the Raman excitation power efficiency with the control speed, excited-state adiabatic elimination, and spontaneous-emission suppression requirements. Within the regime, rotations of atomic spinors by the Raman coupling are biased by substantial light shifts. Taking advantage of the fixed bias angle, we show that composite biased rotations can be optimized to enable precise ensemble spinor matterwave control within nanoseconds, even for multiple Zeeman pseudospins defined on the hyperfine ground states and when the laser illumination is strongly inhomogeneous. Our scheme fills a technical gap in light pulse atom interferometry for achieving high-speed Raman spinor matterwave control with moderate laser power.
Using optimal control, we establish and link the ultimate bounds in time (referred to as the quantum speed limit) and energy of two- and three-level quantum nonlinear systems which feature 1:2 resonance. Despite the unreachable complete inversion, by using the Pontryagin maximum principle, we determine the optimal time, pulse area, or energy for a given arbitrary accuracy. We show that the third-order Kerr terms can be absorbed in the detuning in order to lock the dynamics to the resonance. In the two-level problem, we determine the nonlinear counterpart of the optimal π-pulse inversion for a given accuracy. In the three-level problem, we obtain an intuitive pulse sequence similar to the linear counterpart but with different shapes. We prove the (slow) logarithmic increasing of the optimal time as a function of the accuracy.
We report here on the realization of light-pulse atom interferometers with large-momentum-transfer atom optics based on a sequence of Bragg transitions. We demonstrate momentum splitting up to 200 photon recoils in an ultracold atom interferometer. We highlight a new mechanism of destructive interference of the losses leading to a sizable efficiency enhancement of the beam splitters. We perform a comprehensive study of parasitic interferometers due to the inherent multiport feature of the quasi-Bragg pulses. Finally, we experimentally verify the phase shift enhancement and characterize the interferometer visibility loss.
Over the past few decades, quantum information processing research has focused heavily on error-mitigation schemes and error-correcting codes. However, while many proposed schemes have been successful in mitigating errors, most of them are perturbative and assume deterministic systematic errors, leaving studies of the problem considering the full noise and errors distribution scarce. In this work we introduce an error-mitigation scheme for robust single-qubit unitary gates based on a composite segmented design that accounts for the full distribution of the physical noise and errors in the system. We provide two optimization approaches to construct these robust segmented gates, perturbative and nonperturbative, which address all orders of errors. We demonstrate the effectiveness of our scheme in the photonics realm for the realization of dual-rail directional couplers. Specifically, we show that the three-segment composite design for the fundamental single-qubit unitary operations reduces the error by an order of magnitude for a realistic distribution of errors. Moreover, we demonstrate that the two approaches are compatible for small errors and significantly reduce the overhead of modern error-correction codes. Our methods are rather general and can be applied to other realizations of quantum information processing units.
In some applications of quantum control, it is necessary to produce very weak excitation of a quantum system. Such an example is presented by the concept of single-photon generation in cold atomic ensembles or doped solids, e.g., by the Duan-Lukin-Cirac-Zoller (DLCZ) protocol, for which a single excitation is shared among thousands and millions of atoms or ions. Another example is the possibility to create a huge Dicke state of N qubits sharing a single or a few excitations. Other examples are using tiny rotations to tune high-fidelity quantum gates, or using these tiny rotations to test high-fidelity quantum process tomography protocols. Ultrasmall excitation of a quantum transition can be generated by either a very weak or far-detuned driving field. However, these two approaches are sensitive to variations in the experimental parameters. Here we propose a different method for generating a well-defined arbitrary preselected very small transition probability—in the range from 10−2 to 10−8—by using composite pulse sequences. Contrary to other methods, it features both high fidelity and robustness to variations in the pulse area and the pulse duration.
We theoretically study a technique for a broadband nonreciprocal wave retarder whose quarter-wave plate phase retardation is the same in forward and backward directions. The system is built using a number of sequential nonreciprocal wave plates. The proposed device can also be utilized to create a broadband optical diode, which consists of two achromatic quarter-wave plates, one reciprocal and the other non-reciprocal, that are sandwiched between two polarizers aligned in parallel.
Atoms are ideal quantum sensors and quantum light emitters. Interfacing atoms with nanophotonic devices promises nanoscale sensing and quantum optical functionalities. But precise optical control of atomic states in these devices is challenged by the spatially varying light-atom coupling strength, which is generic to nanophotonic devices. We demonstrate numerically that despite the inhomogeneity, composite picosecond optical pulses with optimally tailored phases are able to evanescently control the atomic electric dipole transitions nearly perfectly, with fidelity f>99% across volumes large enough for, e.g., controlling cold atoms confined in near-field optical lattices. Our proposal is followed by a proof-of-principle demonstration with a 85Rb-vapor–optical-nanofiber interface, where the excitation by an N=3 sequence of guided picosecond D1 control pulses reduces the absorption of a coguided nanosecond D2 probe pulse by up to approximately 70%. The close-to-ideal performance is corroborated by comparing the absorption data across the parameter space with first-principles modeling of the mesoscopic atomic vapor response. Extension of the composite technique to N≥5 appears highly feasible to support arbitrary local control of atomic dipoles with exquisite precision. This unprecedented ability would allow error-resilient atomic spectroscopy and open up a variety of research opportunities in nanophotonic quantum optics.
Nuclear magnetic resonance (NMR) spectroscopy is a powerful technique for analyzing chemical and biological systems. However, in complex solutions with similar molecular components, NMR signals can overlap, making it challenging to distinguish and quantify individual species. In this paper, we introduce new spectral editing sequences that exploit the differences in nuclear spin interactions (J-couplings) between weakly- and strongly-coupled two-spin systems. These sequences selectively attenuate or nullify undesired spin magnetization while they preserve the desired signals, resulting in simplified NMR spectra and potentially facilitating single-species imaging applications. We demonstrate the effectiveness of our approach using a 31P spectral filtration method on a model system of nicotinamide dinucleotide (NAD), which exists in oxidized (NAD+) and reduced (NADH) forms. The presented sequences are robust to field inhomogeneity, do not require additional sub-spectra, and retain a significant portion of the original signal.
The generation and manipulation of ultracold atomic ensembles in the quantum regime require the application of dynamically controllable microwave fields with ultra-low noise performance. Here, we present a low-phase-noise microwave source with two independently controllable output paths. Both paths generate frequencies in the range of 6.835 GHz ± 25 MHz for hyperfine transitions in 87Rb. The presented microwave source combines two commercially available frequency synthesizers: an ultra-low-noise oscillator at 7 GHz and a direct digital synthesizer for radio frequencies. We demonstrate a low integrated phase noise of 480 µrad in the range of 10 Hz to 100 kHz and fast updates of frequency, amplitude, and phase in sub-µs time scales. The highly dynamic control enables the generation of shaped pulse forms and the deployment of composite pulses to suppress the influence of various noise sources.
Strontium clock atom interferometry is a promising new technology, with multiple experiments under development around the world to explore its potential for dark matter and gravitational wave detection. In these detectors, large momentum transfer using sequences of many laser pulses is necessary, and thus high-fidelity pulses are important since small errors become magnified. Quantum optimal control (QOC) is a framework for developing control pulse waveforms that achieve high fidelity and are robust against experimental imperfections. Resonant single-photon transitions using the narrow clock transition of strontium involve significantly different quantum dynamics than more established atom interferometry methods based on far-detuned two-photon Raman or Bragg transitions, which leads to new opportunities and challenges when applying QOC. Here, we study in simulation QOC pulses for strontium clock interferometry and demonstrate their advantage over basic square pulses (primitive pulses) and composite pulses in terms of robustness against multiple noise channels. This could improve the scale of large momentum transfer in Sr clock interferometers, paving the way to achieving these scientific goals.
Purpose
The aim of this study is to develop and optimize an adiabatic T1ρ$$ {\mathrm{T}}_{1\uprho} $$ (T1ρ,adiab$$ {\mathrm{T}}_{1\uprho, \mathrm{adiab}} $$) mapping method for robust quantification of spin‐lock (SL) relaxation in the myocardium at 3T.
Methods
Adiabatic SL (aSL) preparations were optimized for resilience against B0$$ {\mathrm{B}}_0 $$ and B1+$$ {\mathrm{B}}_1^{+} $$ inhomogeneities using Bloch simulations. Optimized B0$$ {\mathrm{B}}_0 $$‐aSL, Bal‐aSL and B1$$ {\mathrm{B}}_1 $$‐aSL modules, each compensating for different inhomogeneities, were first validated in phantom and human calf. Myocardial T1ρ$$ {\mathrm{T}}_{1\uprho} $$ mapping was performed using a single breath‐hold cardiac‐triggered bSSFP‐based sequence. Then, optimized T1ρ,adiab$$ {\mathrm{T}}_{1\uprho, \mathrm{adiab}} $$ preparations were compared to each other and to conventional SL‐prepared T1ρ$$ {\mathrm{T}}_{1\uprho} $$ maps (RefSL) in phantoms to assess repeatability, and in 13 healthy subjects to investigate image quality, precision, reproducibility and intersubject variability. Finally, aSL and RefSL sequences were tested on six patients with known or suspected cardiovascular disease and compared with LGE, T1$$ {\mathrm{T}}_1 $$, and ECV mapping.
Results
The highest T1ρ,adiab$$ {\mathrm{T}}_{1\uprho, \mathrm{adiab}} $$ preparation efficiency was obtained in simulations for modules comprising 2 HS pulses of 30 ms each. In vivo T1ρ,adiab$$ {\mathrm{T}}_{1\uprho, \mathrm{adiab}} $$ maps yielded significantly higher quality than RefSL maps. Average myocardial T1ρ,adiab$$ {\mathrm{T}}_{1\uprho, \mathrm{adiab}} $$ values were 183.28 ±$$ \pm $$ 25.53 ms, compared with 38.21 ±$$ \pm $$ 14.37 ms RefSL‐prepared T1ρ$$ {\mathrm{T}}_{1\uprho} $$. T1ρ,adiab$$ {\mathrm{T}}_{1\uprho, \mathrm{adiab}} $$ maps showed a significant improvement in precision (avg. 14.47 ±$$ \pm $$ 3.71% aSL, 37.61 ±$$ \pm $$ 19.42% RefSL, p < 0.01) and reproducibility (avg. 4.64 ±$$ \pm $$ 2.18% aSL, 47.39 ±$$ \pm $$ 12.06% RefSL, p < 0.0001), with decreased inter‐subject variability (avg. 8.76 ±$$ \pm $$ 3.65% aSL, 51.90 ±$$ \pm $$ 15.27% RefSL, p < 0.0001). Among aSL preparations, B0$$ {\mathrm{B}}_0 $$‐aSL achieved the better inter‐subject variability. In patients, B1$$ {\mathrm{B}}_1 $$‐aSL preparations showed the best artifact resilience among the adiabatic preparations. T1ρ,adiab$$ {\mathrm{T}}_{1\uprho, \mathrm{adiab}} $$ times show focal alteration colocalized with areas of hyper‐enhancement in the LGE images.
Conclusion
Adiabatic preparations enable robust in vivo quantification of myocardial SL relaxation times at 3T.
Objective:
Temperature controlled T1 and T2 relaxation times are measured on NiCl2 and MnCl2 solutions from the ISMRM/NIST system phantom at low magnetic field strengths of 6.5 mT, 64 mT and 550 mT.
Materials and methods:
The T1 and T2 were measured of five samples with increasing concentrations of NiCl2 and five samples with increasing concentrations of MnCl2. All samples were scanned at 6.5 mT, 64 mT and 550 mT, at sample temperatures ranging from 10 °C to 37 °C.
Results:
The NiCl2 solutions showed little change in T1 and T2 with magnetic field strength, and both relaxation times decreased with increasing temperature. The MnCl2 solutions showed an increase in T1 and a decrease in T2 with increasing magnetic field strength, and both T1 and T2 increased with increasing temperature.
Discussion:
The low field relaxation rates of the NiCl2 and MnCl2 arrays in the ISMRM/NIST system phantom are investigated and compared to results from clinical field strengths of 1.5 T and 3.0 T. The measurements can be used as a benchmark for MRI system functionality and stability, especially when MRI systems are taken out of the radiology suite or laboratory and into less traditional environments.
Recent progress in quantum signal processing (QSP) and its generalization, quantum singular value transformation, has led to a grand unification of quantum algorithms. However, inherent experimental noise in quantum devices severely limits the length of realizable QSP sequences. We consider a model of QSP with generic perturbative noise in the signal processing basis and present a diagrammatic notation useful for analyzing such errors. To demonstrate our technique, we study a specific coherent error, that of under- or overrotation of the signal processing operator parametrized by ε≪1. For this coherent error model, it is shown that while Pauli Z errors are not recoverable without additional resources, Pauli X and Y errors can be arbitrarily suppressed by coherently appending a noisy recovery QSP without the use of additional resources or ancillas. Furthermore, through a careful accounting of errors using our diagrammatic tools, we provide an upper and lower bound on the length of this recovery QSP operator. We anticipate that the perturbative technique and the diagrammatic notation proposed here will facilitate future study of generic noise in QSP and quantum algorithms.
This chapter discusses control and classification of inhomogeneous quantum ensembles. A sampling-based learning control method is systematically developed for ensemble control and classification. Inhomogeneous quantum ensembles are introduced in Sect. 3.1, and a sampling-based learning control method is presented in Sect. 3.2. The results on control of inhomogeneous quantum ensembles are presented in Sect. 3.3\(^\text {a}\). Section 3.4 through Sect. 3.7 present the results on quantum discrimination and ensemble classification using the sampling-based learning control method\(^\text {b}\). Section 3.8 includes the conclusion and further reading.
Emulsion stability is important in many environmental and industrial applications. Unstable emulsions tend to have a layered structure which will evolve in space and time. Magnetic Resonance (MR) is a nondestructive method that can be used to study emulsion instability based on relaxation times (T1, T2), and/or diffusion (D). The bulk transverse relaxation lifetime (T2) is the most useful parameter for analysis. Since breaking of emulsions creates different layers and phases, spatially resolved T2 data is more valuable than bulk whole sample data. Previous literature studies were principally based on bulk T1, T2 or self-diffusion measurements without spatial resolution. In this study, for the first time, spatially resolved T2 distributions were used to examine oil and water behavior during emulsion breaking.
Diluted bitumen was used as the oil phase in a synthetic emulsion. Measurements undertaken included spatially resolved T2 distribution measurements and T1-T2 relaxation correlation measurements. The results reveal changes in the relaxation times of oil and water caused by changes in the dynamics of the water and oil during emulsion breaking. In each layer of the sample, comparing peak T2 values with bulk T2 values of different components gives an insight into phase environments. Water and oil content in each layer was resolved by using peak areas of the T2 distribution. The results of T2 distribution imaging were consistent with optical microscopy images, which showed W/O emulsions in the oil dominant region and complex W/O/W emulsions in the water dominant region of the sample. With the above information one can obtain full spatial and temporal information on evolving phases as well as kinetics of phase dynamics in emulsion breaking.
Robust and high-fidelity control of electron spins in solids is the cornerstone for facilitating applications of solid-state spins in quantum information processing and quantum sensing. However, precise control of spin systems is always challenging due to the presence of various noises originating from the thermal environment and control fields. Here, noise-resilient quantum gates, designed with robust optimal control (ROC) algorithms, are demonstrated experimentally with nitrogen-vacancy centers in diamond to realize tailored robustness against detunings and Rabi errors simultaneously. In the presence of both 10% off-resonance detuning and 10% deviation of a Rabi frequency, we achieve an average single-qubit gate fidelity of up to 99.89%. Our experiments also show that, ROC-based multipulse quantum sensing sequences can suppress spurious responses resulting from finite widths and imperfections of microwave pulses, which provides an efficient strategy for enhancing the performance of existing multipulse quantum sensing sequences.
A single bichromatic field near resonant to a qubit transition is typically used for σ̂x or σ̂y Mølmer-Sørensen-type interactions in trapped-ion systems. Using this field configuration, it is also possible to synthesize a σ̂z spin-dependent force by merely adjusting the beat-note frequency. Here, we expand on previous work and present a comprehensive theoretical and experimental investigation of this scheme with a laser near resonant to a quadrupole transition in Sr+88. Further, we characterize its robustness to optical phase and qubit frequency offsets, and demonstrate its versatility by entangling optical, metastable, and ground-state qubits.
In this work, we put forward an approach of constructing the optimal composite pulse sequence for robust population transfer in two-level systems. This approach is quite universal and applicable to a variety of systems, because the modulation parameters of composite pulses are obtained by minimizing the homemade cost function rather than nullifying the error coefficients of the transition probability. Specifically, we design different forms of the cost function for implementing optimal robustness with respect to the single or multiple errors. When slightly adjusting the constraints of the cost function, this approach can be easily extended to achieve arbitrary population transfer. The numerical results demonstrate that the optimized sequences are immune from various systematic errors, allowing us to produce a broadening excitation range of the transition probability. Therefore, this work offers an optimal robust design for controlling quantum states in error-prone environments.
Une nouvelle methode de decouplage en large bande basee sur l'inversion repetee, des spins protoniques par une sequence d'impulsions rf composees concue pour etre relativement insensibles aux effets de compensation de resonance et ete decrite anterieurement. Les auteurs developpent ici la theorie de cette methode et illustrent comment les cycles d'impulsions optimaux sont construit et comparent la theorie et l'experience
An approach to high-resolution NMR of deuterium in solids is described. The m=1→-1 transition is excited by a double-quantum process and the decay of coherence Q(τ) is monitored. Fourier transformation yields a deuterium spectrum devoid of quadrupole broadening, and if the deuterium nuclei are dilute, the double-quantum spectrum is a high-resolution one. The technique is applied to a single crystal of oxalic acid dihydrate enriched to ∼ 10% in deuterium, and the carboxyl and the water deuterium shifts are indeed resolved.
The possibilities for the extension of spectroscopy to two dimensions are discussed. Applications to nuclear magnetic resonance are described. The basic theory of two‐dimensional spectroscopy is developed. Numerous possible applications are mentioned and some of them treated in detail, including the elucidation of energy level diagrams, the observation of multiple quantum transitions, and the recording of high‐resolution spectra in inhomogenous magnetic fields. Experimental results are presented for some simple spin systems.
It is demonstrated theoretically and experimentally that double-quantum coherence can be excited with nearly maximal efficiency over a large range of spin coupling strengths in NMR with the use of composite excitation sequences. Such sequences are shown to be directly constructible from existing composite pulses. In addition, a general procedure is given for converting a composite excitation sequence to an equivalent, reduced form that contains fewer rf pulses and simpler rf phase shifts. Experiments are performed on the proton pair of CH2Cl2 oriented in a liquid crystal solvent. Applications to quadrupolar spin-1 nuclei and other dynamically simple multilevel systems are discussed.
On montre quelques proprietes des impulsions composites d'inversion de spin deduites de la sequence de depart Ca=(4―a)(4―a) dans laquelle le parametre a peu varier de facon continue sur un certain domaine de valeurs tel que les elements de la sequence n'aient pas besoin d'etre des multiples d'impulsions de 90°
A simple procedure is described for constructing composite pulses of arbitrary flip angle which are compensated to an arbitrarily high degree for pulse imperfections. The performance of some population inversion pulse sequences is tested experimentally.
It is shown experimentally and by calculations, using average Hamiltonian theory, that the decoupling efficiency of the composite decoupling schemes MLEV-4 and MLEV-16, introduced by Levitt, Freeman, and Frenkiel, can be improved by inserting delays and changing one pulse length of the composite pulse.
We derive iterative schemes for generating rf pulse sequences that invert nuclear spin populations over either broad or narrow ranges of resonance offsets and rf amplitudes. The schemes employ only phase shifting and concatenation operations. Iterative application of a scheme produces a series of pulse sequences with successive improvements in population inversion performance.
A theory is presented for construction of sequences of phase-shifted radiation pulses for coherently inverting populations over a broad band of transition frequencies. Such sequences have applications in nuclear magnetic resonance and coherent optics. Examples of sequences are derived, together with computer simulations of their inversion properties.
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The role of symmetry in two-dimensional NMR is examined. It is shown that symmetry considerations may lead to selection rules restricting the number of allowed coherence transfer pathways. We introduce a new experiment, bilinear correlation spectroscopy (bilinear COSY) In the limit of short mixing time, this method provides fewer peaks in comparison with an ordinary correlation experiment, reducing spectral crowding. The patterns formed by the summetry-allowed peaks give information as to relative signs of coupling constants.
An alternative definition of single transition operators is given for the description of selective excitation and detection experiments in multilevel spin systems. This definition has the virtues of a simple physical interpretation and easy application to arbitrarily complicated systems. Some applications to the excitation and detection of multiple quantum transitions in spin 1 and spin 3/2 systems as well as in coupled spin systems are described.
Iterative schemes for NMR have been developed by several groups. A theoretical framework based on mathematical dynamics is described for such iterative schemes in nonlinear NMR excitation. This is applicable to any system subjected to coherent radiation or other experimentally controllable external forces. The effect of the excitation, usually a pulse sequence, can be summarized by a propagator or superpropagator (U). The iterative scheme (F) is regarded as a map of propagator space into itself, Un+1=FUn. One designs maps for which a particular propagator U¯ or set of propagators {U¯} is a fixed point or invariant set. The stability of the fixed points along various directions is characterized by linearizing F around the fixed point, in analogy to the evaluation of an average Hamiltonian. Stable directions of fixed points typically give rise to broadband behavior (in parameters such as frequency, rf amplitude, or coupling constants) and unstable directions to narrowband behavior. The dynamics of the maps are illustrated by ‘‘basin images’’ which depict the convergence of points in propagator space to the stable fixed points. The basin images facilitate the optimal selection of initial pulse sequences to ensure convergence to a desired excitation. Extensions to iterative schemes with several fixed points are discussed. Maps are shown for the propagator space SO(3) appropriate to iterative schemes for isolated spins or two-level systems. Some maps exhibit smooth, continuous dynamics whereas others have basin images with complex and fractal structures. The theory is applied to iterative schemes for broadband and narrowband π (population inversion) and π/2 rotations, MLEV and Waugh spin decoupling sequences, selective n-quantum pumping, and bistable excitation.
On propose une famille de schemas de preparation qui peut preceder n'importe quel type de sequence d'impulsions selectives dans l'espace de facon a supprimer ou fortement attenuer des reponses harmoniques non desirees
Etude de la relation entre les elements quaternions, et les angles d'Eulerian, les cosinus de direction des axes de rotation et les angles de rotation respectifs. Comme exemple illustratif une impulsion z de groupe est evaluee en detail
Many phase-alternated decoupling sequences have been tested for broadband decoupling in liquid. A number of sequences having effective decoupling bandwidths comparable or larger than that of WALTZ-16 have been found. The acronym PAR (Phase Alternated Rotation of magnetization) is used for these sequences.
The refocusing pulse RY(π) of a Carr-Purcell spin-echo experiment may be replaced with a composite sequence of three pulses, RX(π/2)RY(π)RX(π/2), which compensates the effects of pulse length errors due to spatial inhomogeneity of the radiofrequency field. An analysis in terms of rotation operators indicates that the overall effect of the sequence RX(π/2 + Δ)RY(π + 2Δ)RX(π/2 + Δ) is equivalent to an exact π pulse applied about an axis in the XY plane which is phase shifted by an angle Δ with respect to the Y axis. Good spin state inversion is therefore achieved, but there is a phase shift of 2Δ for odd-numbered echoes, which is canceled on even-numbered echoes. For spin systems with no homonuclear coupling, the degree of compensation exceeds that of the well-known Meiboom-Gill modificaation, while for coupled spin systems the composite pulse scheme provides effective compensation whereas the Meiboom-Gill method breaks down. The technique has been tested on the proton spectrum of 1,1,2-trichloroethane with deliberately misset refocusing pulses; gross fluctuations in intensity are observed unless the composite pulses are used. A slightly different composite pulse sequence is proposed for situations where radiofrequency offset effects are the dominant source of error.
A discussion is given of the effects of a pair of pulses of rf magnetic field (at the NMR frequency) on the Zeeman and dipolar energies of the spin system in a solid. Zeeman order is partly transformed into dipolar order when the two pulses have different rf phases. A theoretical calculation provides a very simple relation between the efficiency of this transfer of energy (as a function of the characteristics of the pulses and their time separation) and the magnitude and shape of the dipolar component of the free-precession signal. This prediction has been verified quantitatively in the case of the F19 spins in a CaF2 crystal with the large magnetic field in a [100] direction. A maximum efficiency of transfer of Zeeman energy into dipolar energy of 56% has been obtained with a pi2 phase shift between a first pi2 pulse and a second pi4 pulse, separated by a time of the order of T2. (An ideal adiabatic demagnetization in the rotating frame would have an efficiency of 100%.) The experimental results also indicate that the F19 spin system reaches internal quasiequilibrium in a time of the order of a few T2 after the pulse pair. It is also shown that the use of a pair of rf pulses (both of the same rf phase this time) leads to a new method for measuring the complete shape of free-precession signals, which avoids the usual difficulties due to the finite recovery time of the observation circuit.
Above a critical power threshold for a given pulse width, a short pulse of coherent traveling-wave optical radiation is observed to propagate with anomalously low energy loss while at resonance with a two-quantum-level system of absorbers. The line shape of the resonant system is determined by inhomogeneous broadening, and the pulse width is short compared to dissipative relaxation times. A new mechanism of self-induced transparency, which accounts for the low energy loss, is analyzed in the ideal limit of a plane wave which excites a resonant medium with no damping present. The stable condition of transparency results after the traversal of the pulse through a few classical absorption lengths into the medium. This condition exists when the initial pulse has evolved into a symmetric hyperbolic-secant pulse function of time and distance, and has the area characteristic of a "2pi pulse." Ideal transparency then persists when coherent induced absorption of pulse energy during the first half of the pulse is followed by coherent induced emission of the same amount of energy back into the beam direction during the second half of the pulse. The effects of dissipative relaxation times upon pulse energy, pulse area, and pulse delay time are analyzed to first order in the ratio of short pulse width to long damping time. The analysis shows that the 2pi pulse condition can be maintained if losses caused by damping are compensated by beam focusing. In an amplifying inhomogeneously broadened medium an analytic "pi pulse area" solution is presented in the limit of a sharp leading edge of the pulse. The dynamics of self-induced transparency are studied for the particular effects of Doppler velocities upon a resonant gas. The analysis of transparency for random orientations of dipole moments associated with degenerate rotational states yields modified forms of self-induced transparency behavior, which indicates a finite pulse energy loss as a function of distance in some cases. The effect of self-induced transparency on the photon echo is considered. Experimental observations of self-induced transparency have been made in a ruby sample at resonance with a pulsed ruby-laser beam. Single and multiple 2pi pulse outputs have been observed, and pulse areas measured in the range of 2pi. The experimental results are compared with the predictions of the ideal-plane-wave theory. Deviations from the ideal-plane-wave theory are discussed. An analysis is made of the effect of a transverse mode of the propagating beam upon the transparency properties of the pulse.
A new method for the separation of coherent and incoherent magnetization transfer in two-dimensional (2D) NOE and chemical exchange spectroscopy is described. The new experiment differs from previous 2D exchange experiments, in that the mixing time τm is incremented systematically together with the evolution time t1. This permits the distinction of the different orders of multiple-quantum coherence during the mixing time and allows the separation of coherent and incoherent transfer processes. The resulting clarification of chemical exchange and 2D NOE spectra is illustrated with experiments on small molecules and on a globular protein, the basic pancreatic trypsin inhibitor.