23219044
Anomalous behavior of control pulses in presence of noise with singular autocorrelation.
We report on the anomalous behavior of control pulses for spins under spin-spin relaxation and subject to classical noise with a singular autocorrelation function. This behavior is not detected for noise with analytic autocorrelation functions. The effect is manifest in the different scaling behavior of the deviation of a real pulse to the ideal, instantaneous one. While a standard pulse displays scaling ∝τp(1), a first-order refocusing pulse normally shows scaling ∝τp(2). But in presence of cusps in the noise autocorrelation the scaling ∝τp(3/2) occurs. Cusps in the autocorrelation are characteristic for fast fluctuations in the noise with a spectral density of Lorentzian shape. We prove that the anomalous exponent cannot be avoided; it represents a fundamental limit. On the one hand, this redefines the strategies one has to adopt to design refocusing pulses. On the other hand, the anomalous exponent, if found in experiment, provides important information on the noise properties.
39106783
Effects of Diffusion on Free Precession in Nuclear Magnetic Resonance Experiments
Nuclear resonance techniques involving free precession are examined, and, in particular, a convenient variation of Hahn's spin-echo method is described. This variation employs a combination of pulses of different intensity or duration ("90-degree" and "180-degree" pulses). Measurements of the transverse relaxation time ${T}_{2}$ in fluids are often severely compromised by molecular diffusion. Hahn's analysis of the effect of diffusion is reformulated and extended, and a new scheme for measuring ${T}_{2}$ is described which, as predicted by the extended theory, largely circumvents the diffusion effect. On the other hand, the free precession technique, applied in a different way, permits a direct measurement of the molecular self-diffusion constant in suitable fluids. A measurement of the self-diffusion constant of water at 25\ifmmode^\circ\else\textdegree\fi{}C is described which yields $D=2.5(\ifmmode\pm\else\textpm\fi{}0.3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$ ${\mathrm{cm}}^{2}$/sec, in good agreement with previous determinations. An analysis of the effect of convection on free precession is also given. A null method for measuring the longitudinal relaxation time ${T}_{1}$, based on the unequal-pulse technique, is described.
19156575
Modified Spin‐Echo Method for Measuring Nuclear Relaxation Times
A spin echo method adapted to the measurement of long nuclear relaxation times (T2) in liquids is described. The pulse sequence is identical to the one proposed by Carr and Purcell, but the rf of the successive pulses is coherent, and a phase shift of 90° is introduced in the first pulse. Very long T2 values can be measured without appreciable effect of diffusion.
95360601
New, compensated Carr-Purcell sequences
Abstract New, compensated Carr-Purcell pulse sequences are reported, all based on x and y phase alternation of the π pulses. The sequences compensate cumulative pulse errors for all three components of magnetization. Applications include the measurement of homonuclear dipole coupling in the presence of chemical shifts and the measurement of heteronuclear dipole coupling in magic-angle spinning experiments (REDOR). The performance of the new pulse sequences is compared experimentally to previously reported schemes.
34939261
DYNAMICAL SUPPRESSION OF DECOHERENCE IN TWO-STATE QUANTUM SYSTEMS
The dynamics of a decohering two-level system driven by a suitable control Hamiltonian is studied. The control procedure is implemented as a sequence of radio-frequency pulses that repetitively flip the state of the system, a technique that can be termed quantum ``bang-bang'' control after its classical analog. Decoherence introduced by the system's interaction with a quantum environment is shown to be washed out completely in the limit of continuous flipping and greatly suppressed provided the interval between the pulses is made comparable to the correlation time of the environment. The model suggests a strategy to fight against decoherence that complements existing quantum error-correction techniques.
121233468
Photon-echo technique for reducing the decoherence of a quantum bit
Abstract A useful method for reducing the decoherence of a quantum bit (qubit) is proposed, which is based on the photon-echo phenomenon in coherent optical transient. The decoherence of the qubit is caused by the dephasing coupling with a bosonic thermal reservoir. In the method, successive coherent π pulses are applied with very short pulse widths to reduce the decoherence. It is found that the decoherence of the qubit disappears at a certain time which can be controlled by appropriately choosing the pulse separation time and the number π pulses.
2566091
DYNAMICAL DECOUPLING OF OPEN QUANTUM SYSTEMS
We propose a novel dynamical method for beating decoherence and dissipation in open quantum systems. We demonstrate the possibility of filtering out the effects of unwanted (not necessarily known) system-environment interactions and show that the noise-suppression procedure can be combined with the capability of retaining control over the effective dynamical evolution of the open quantum system. Implications for quantum information processing are discussed. {copyright} {ital 1999} {ital The American Physical Society}
119059810
Performance of Deterministic Dynamical Decoupling Schemes: Concatenated and Periodic Pulse Sequences
Dynamical decoupling can be used to preserve arbitrary quantum states despite undesired interactions with the environment, using control Hamiltonians affecting the system only. We present a system-independent analysis of dynamical decoupling based on leading order decoupling error estimates, valid for bounded-strength environments. Using as a key tool a renormalization transformation of the effective system-bath coupling Hamiltonian, we delineate the reliability domain of dynamical decoupling used for quantum state preservation, in a general setting for a single qubit. We specifically analyze and compare two deterministic dynamical decoupling schemes\char22{}periodic and concatenated\char22{}and distinguish between two limiting cases of fast versus slow environments. We prove that concatenated decoupling outperforms periodic decoupling over a wide range of parameters. These results are obtained for both ``ideal'' (zero-width) and realistic (finite-width) pulses This work extends and generalizes our earlier work [Phys. Rev. Lett. 95, 180501 (2005)].
14729824
Keeping a quantum bit alive by optimized pi-pulse sequences.
A general strategy to maintain the coherence of a quantum bit is proposed. The analytical result is derived rigorously including all memory and backaction effects. It is based on an optimized pi-pulse sequence for dynamic decoupling extending the Carr-Purcell-Meiboom-Gill cycle. The optimized sequence is very efficient, in particular, for strong couplings to the environment.
14729824
Keeping a quantum bit alive by optimized pi-pulse sequences.
A general strategy to maintain the coherence of a quantum bit is proposed. The analytical result is derived rigorously including all memory and backaction effects. It is based on an optimized pi-pulse sequence for dynamic decoupling extending the Carr-Purcell-Meiboom-Gill cycle. The optimized sequence is very efficient, in particular, for strong couplings to the environment.
122851920
Exact results on dynamical decoupling by π pulses in quantum information processes
The aim of dynamical decoupling consists in the suppression of decoherence by appropriate coherent control of a quantum register. Effectively, the interaction with the environment is reduced. In particular, a sequence of ? pulses is considered. Here we present exact results on the suppression of the coupling of a quantum bit to its environment by optimized sequences of ? pulses. The effect of various cut-offs of the spectral density of the environment is investigated. As a result we show that the harder the cut-off is, the better an optimized pulse sequence can deal with it. For cut-offs which are neither completely hard nor very soft we advocate iterated optimized sequences.
12935324
Universality of Uhrig dynamical decoupling for suppressing qubit pure dephasing and relaxation.
The optimal N-pulse dynamical decoupling discovered by Uhrig for a spin-boson model [Phys. Rev. Lett. 98, 100504 (2007)10.1103/PhysRevLett.98.100504] is proved to be universal in suppressing to O(T;{N+1}) the pure dephasing or the longitudinal relaxation of a qubit (or spin 1/2) coupled to a generic bath in a short-time evolution of duration T. For suppressing the longitudinal relaxation, a Uhrig pi-pulse sequence can be generalized to be a superposition of the ideal Uhrig pi-pulse sequence as the core and an arbitrarily shaped pulse sequence satisfying certain symmetry requirements. The generalized Uhrig dynamical decoupling offers the possibility of manipulating the qubit while simultaneously combating the longitudinal relaxation.
118450707
How to Enhance Dephasing Time in Superconducting Qubits
We theoretically investigate the influence of designed pulse sequences in restoring quantum coherence lost due to background noise in superconducting qubits. We consider both $1/f$ noise and random telegraph noise and show that the qubit coherence time can be substantially enhanced by carefully engineered pulse sequences. Conversely, the time dependence of qubit coherence under external pulse sequences could be used as a spectroscopic tool for extracting the noise mechanisms in superconducting qubits, i.e., by using Uhrig's pulse sequence [Phys. Rev. Lett. 98, 100504 (2007)], one can obtain information about moments of the spectral density of noise. We also study the effect of pulse sequences on the evolution of the qubit affected by a strongly coupled fluctuator and show that the non-Gaussian features in decoherence are suppressed by the application of pulses.
32355508
Optimal pulse spacing for dynamical decoupling in the presence of a purely dephasing spin bath
Maintaining quantum coherence is a crucial requirement for quantum computation; hence protecting quantum systems against their irreversible corruption due to environmental noise is an important open problem. Dynamical decoupling (DD) is an effective method for reducing decoherence with a low control overhead. It also plays an important role in quantum metrology, where, for instance, it is employed in multiparameter estimation. While a sequence of equidistant control pulses [the Carr-Purcell-Meiboom-Gill (CPMG) sequence] has been ubiquitously used for decoupling, Uhrig recently proposed that a nonequidistant pulse sequence [the Uhrig dynamic decoupling (UDD) sequence] may enhance DD performance, especially for systems where the spectral density of the environment has a sharp frequency cutoff. On the other hand, equidistant sequences outperform UDD for soft cutoffs. The relative advantage provided by UDD for intermediate regimes is not clear. In this paper, we analyze the relative DD performance in this regime experimentally, using solid-state nuclear magnetic resonance. Our system qubits are {sup 13}C nuclear spins and the environment consists of a {sup 1}H nuclear spin bath whose spectral density is close to a normal (Gaussian) distribution. We find that in the presence of such a bath, the CPMG sequence outperforms the UDD sequence. An analogy betweenmore » dynamical decoupling and interference effects in optics provides an intuitive explanation as to why the CPMG sequence performs better than any nonequidistant DD sequence in the presence of this kind of environmental noise.« less
4353251
Optimized dynamical decoupling in a model quantum memory
Any quantum system, such as those used in quantum information or magnetic resonance, is subject to random phase errors that can dramatically affect the fidelity of a desired quantum operation or measurement. In the context of quantum information, quantum error correction techniques have been developed to correct these errors, but resource requirements are extraordinary. The realization of a physically tractable quantum information system will therefore be facilitated if qubit (quantum bit) error rates are far below the so-called fault-tolerance error threshold, predicted to be of the order of 10-3–10-6. The need to realize such low error rates motivates a search for alternative strategies to suppress dephasing in quantum systems. Here we experimentally demonstrate massive suppression of qubit error rates by the application of optimized dynamical decoupling pulse sequences, using a model quantum system capable of simulating a variety of qubit technologies. We demonstrate an analytically derived pulse sequence, UDD, and find novel sequences through active, real-time experimental feedback. The latter sequences are tailored to maximize error suppression without the need for a priori knowledge of the ambient noise environment, and are capable of suppressing errors by orders of magnitude compared to other existing sequences (including the benchmark multi-pulse spin echo). Our work includes the extension of a treatment to predict qubit decoherence under realistic conditions, yielding strong agreement between experimental data and theory for arbitrary pulse sequences incorporating nonidealized control pulses. These results demonstrate the robustness of qubit memory error suppression through dynamical decoupling techniques across a variety of qubit technologies.
4389322
Preserving electron spin coherence in solids by optimal dynamical decoupling
To exploit the quantum coherence of electron spins in solids in future technologies such as quantum computing, it is first vital to overcome the problem of spin decoherence due to their coupling to the noisy environment. Dynamical decoupling, which uses stroboscopic spin flips to give an average coupling to the environment that is effectively zero, is a particularly promising strategy for combating decoherence because it can be naturally integrated with other desired functionalities, such as quantum gates. Errors are inevitably introduced in each spin flip, so it is desirable to minimize the number of control pulses used to realize dynamical decoupling having a given level of precision. Such optimal dynamical decoupling sequences have recently been explored. The experimental realization of optimal dynamical decoupling in solid-state systems, however, remains elusive. Here we use pulsed electron paramagnetic resonance to demonstrate experimentally optimal dynamical decoupling for preserving electron spin coherence in irradiated malonic acid crystals at temperatures from 50 K to room temperature. Using a seven-pulse optimal dynamical decoupling sequence, we prolonged the spin coherence time to about 30 μs; it would otherwise be about 0.04 μs without control or 6.2 μs under one-pulse control. By comparing experiments with microscopic theories, we have identified the relevant electron spin decoherence mechanisms in the solid. Optimal dynamical decoupling may be applied to other solid-state systems, such as diamonds with nitrogen-vacancy centres, and so lay the foundation for quantum coherence control of spins in solids at room temperature.
44725432
Performance comparison of dynamical decoupling sequences for a qubit in a rapidly fluctuating spin-bath
Avoiding the loss of coherence of quantum mechanical states is an important prerequisite for quantum information processing. Dynamical decoupling (DD) is one of the most effective experimental methods for maintaining coherence, especially when one can access only the qubit system and not its environment (bath). It involves the application of pulses to the system whose net effect is a reversal of the system-environment interaction. In any real system, however, the environment is not static, and therefore the reversal of the system-environment interaction becomes imperfect if the spacing between refocusing pulses becomes comparable to or longer than the correlation time of the environment. The efficiency of the refocusing improves therefore if the spacing between the pulses is reduced. Here, we quantify the efficiency of different DD sequences in preserving different quantum states. We use C-13 nuclear spins as qubits and an environment of H-1 nuclear spins as the environment, which couples to the qubit via magnetic dipole-dipole couplings. Strong dipole-dipole couplings between the proton spins result in a rapidly fluctuating environment with a correlation time of the order of 100 mu s. Our experimental results show that short delays between the pulses yield better performance if they are compared with the bath correlation time. However, as the pulse spacing becomes shorter than the bath correlation time, an optimum is reached. For even shorter delays, the pulse imperfections dominate over the decoherence losses and cause the quantum state to decay.
16826595
Optimized, unequal pulse spacing in multiple echo sequences improves refocusing in magnetic resonance.
A recent quantum computing paper (G. S. Uhrig, Phys. Rev. Lett. 98, 100504 (2007)) analytically derived optimal pulse spacings for a multiple spin echo sequence designed to remove decoherence in a two-level system coupled to a bath. The spacings in what has been called a "Uhrig dynamic decoupling (UDD) sequence" differ dramatically from the conventional, equal pulse spacing of a Carr-Purcell-Meiboom-Gill (CPMG) multiple spin echo sequence. The UDD sequence was derived for a model that is unrelated to magnetic resonance, but was recently shown theoretically to be more general. Here we show that the UDD sequence has theoretical advantages for magnetic resonance imaging of structured materials such as tissue, where diffusion in compartmentalized and microstructured environments leads to fluctuating fields on a range of different time scales. We also show experimentally, both in excised tissue and in a live mouse tumor model, that optimal UDD sequences produce different T(2)-weighted contrast than do CPMG sequences with the same number of pulses and total delay, with substantial enhancements in most regions. This permits improved characterization of low-frequency spectral density functions in a wide range of applications.
27833612
Enhanced refocusing of fat signals using optimized multipulse echo sequences
Endogenous magnetic resonance contrast based on the localized composition of fat in vivo can provide functional information. We found that the unequal pulse timings of the Uhrig's dynamical decoupling multipulse echo sequences significantly alter the signal intensity compared to conventional, equal‐spaced Carr–Purcell–Meiboom–Gill sequences. The signal increases and decreases depending on the tissue and sequence parameters, as well as on the interpulse spacings; particularly strong differences were observed in fatty tissues, which have a highly structured morphology and a wide range of chemical shifts and J‐couplings. We found that the predominant mechanism for fat refocusing under multipulse echo sequences is the chemical structure, with stimulated echoes playing a pivotal role. As a result, specialized pulse sequences can be designed to optimize refocusing of the fat chemical shifts and J‐couplings, where the degree of refocusing can be tailored to specific types of fats. To determine the optimal time delays, we simulated various Uhrig dynamical decoupling and Carr–Purcell–Meiboom–Gill pulse sequence timings, and these results are compared to experimental results obtained on excised and in vivo fatty tissue. Applications to intermolecular multiple quantum coherence imaging, where the improved echo refocusing translates directly into signal enhancements, are presented as well. Magn Reson Med 69:1044–1055, 2013. © 2012 Wiley Periodicals, Inc.
121189272
Broadband Population Inversion
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.
119093212
Use of composite rotations to correct systematic errors in NMR quantum computation
We implement an ensemble quantum counting algorithm on three NMR spectrometers with 1 H resonance frequencies of 500, 600 and 750 MHz. At higher frequencies, the results deviate markedly from naive theoretical predictions. These systematic errors can be attributed almost entirely to off-resonance effects, which can be substantially corrected for using fully compensating composite rotation pulse sequences originally developed by Tycko. We also derive an analytic expression for generating such sequences with arbitrary rotation angles.
119420671
Tackling systematic errors in quantum logic gates with composite rotations
We describe the use of composite rotations to combat systematic errors in single-qubit quantum logic gates and discuss three families of composite rotations which can be used to correct off-resonance and pulse length errors. Although developed and described within the context of nuclear magnetic resonance quantum computing, these sequences should be applicable to any implementation of quantum computation.
15557332
Arbitrary precision composite pulses for NMR quantum computing.
We discuss the implementation of arbitrary precision composite pulses developed using the methods of Brown et al. [K.R. Brown, A.W. Harrow, I.L. Chuang, Arbitrarily accurate composite pulse sequences, Phys. Rev. A 70 (2004) 052318]. We give explicit results for pulse sequences designed to tackle both the simple case of pulse length errors and the more complex case of off-resonance errors. The results are developed in the context of NMR quantum computation, but could be applied more widely.
118586549
Optimization of short coherent control pulses
Experimentally, however, this idealized situation is not realistic. The real pulses are always of finite amplitude and of finite length. So the question must be addressed which effect a finite pulse duration p has. We want to elucidate this issue in the present work theoretically in a fully quantum mechanical framework. The issue of pulses of finite duration and finite amplitude has been considered in other papers. Viola and Knill generalized their previous approach of averaging over a symmetry group 16 by instantaneous changes of the effective Hamiltonian to continuous changes requiring only finite, bounded control amplitudes 17. But the presence of the coupling to the bath during the pulses has not been considered. Khodjasteh and Lidar considered the effect of finite pulse duration in a comprehensive comparison of various schemes for pulse sequences 18. They did not consider to make the detrimental effects of the finite pulse duration vanish on the level of the individual pulse, but they discussed possible cancellations on the higher hierarchical level of the pulse sequence. In the present work, we will choose another route and aim at reducing the decoherence due to finite pulse duration already on the level of the single pulse. As a side remark, we note that the statement in Ref. 18 on optimized pulse sequences 8 that they hold only for bosonic bath operators is very likely not to be correct. It has been conjectured very recently by Lee et al. 19 that the optimized sequences apply to the most general phase decoherence model.
116945712
FAST TRACK COMMUNICATION: Generalization of short coherent control pulses: extension to arbitrary rotations
We generalize the problem of the coherent control of small quantum systems to the case where the quantum bit (qubit) is subject to a fully general rotation. Following the ideas developed in Pasini et al (2008 Phys. Rev. A 77 032315), the systematic expansion in the shortness of the pulse is extended to the case where the pulse acts on the qubit as a general rotation around an axis of rotation varying in time. The leading and the next-leading corrections are computed. For certain pulses we prove that the general rotation does not improve on the simpler rotation with fixed axis.
118553455
Optimized pulses for the perturbative decoupling of a spin and a decoherence bath
In the framework of nuclear magnetic resonance, we consider the general problem of the coherent control of a spin coupled to a bath by means of composite or continuous pulses of duration ${\ensuremath{\tau}}_{\text{p}}$. We show explicitly that it is possible to design the pulse in order to achieve a decoupling of the spin from the bath up to the third order in ${\ensuremath{\tau}}_{\text{p}}$. The evolution of the system is separated in the evolution of the spin under the action of the pulse and of the bath times correction terms. We derive the correction terms for a general time dependent axis of rotation and for a general coupling between the spin and the environment. The resulting corrections can be made to vanish by an appropriate design of the pulse. For $\ensuremath{\pi}$ and $\ensuremath{\pi}/2$ pulses, we demonstrate explicitly that pulses exist which annihilate the first- and the second-order corrections even if the bath is fully quantum mechanical, i.e., it displays internal dynamics. Such pulses will also be useful for quantum information processing.
53349343
Frequency modulated pulses for quantum bits coupled to time-dependent baths
p. By shaping the pulse, we perturbatively decouple the dynamics of the bath from the dynamics of the quantum bit during the pulse. Such shaped pulses provide single quantum bit gates robust against decoherence which are useful for quantum-information processing. We extend previous results in two ways: (i) we treat frequency modulated pulses and (ii) we pass from time-independent baths to analytically time-dependent baths. First- and second-order solutions for and = 2 pulses are presented. They are useful in experiments where amplitude modulation is dicult to realize.
41456215
Dynamically error-corrected gates for universal quantum computation.
Scalable quantum computation in realistic devices requires that precise control can be implemented efficiently in the presence of decoherence and operational errors. We propose a general constructive procedure for designing robust unitary gates on an open quantum system without encoding or measurement overhead. Our results allow for a low-level error correction strategy solely based on Hamiltonian engineering using realistic bounded-strength controls and may substantially reduce implementation requirements for fault-tolerant quantum computing architectures.
8601928
Arbitrarily accurate dynamical control in open quantum systems.
We show that open-loop dynamical control techniques may be used to synthesize unitary transformations in open quantum systems in such a way that decoherence is perturbatively compensated for to a desired (in principle arbitrarily high) level of accuracy, which depends only on the strength of the relevant errors and the achievable rate of control modulation. Our constructive and fully analytical solution employs concatenated dynamically corrected gates, and is applicable independently of detailed knowledge of the system-environment interactions and environment dynamics. Explicit implications for boosting quantum gate fidelities in realistic scenarios are addressed.
53691360
Arbitrary quantum control of qubits in the presence of universal noise
We address the problem of deriving analytic expressions for calculating universal decoherence-induced errors in qubits undergoing arbitrary, unitary, time-dependent quantum control protocols. We show that the fidelity of a control operation may be expressed in terms of experimentally relevant spectral characteristics of the noise and of the control, over all Cartesian directions. We formulate control matrices in the time domain to capture the effects of piecewise-constant control, and convert them to generalized Fourier-domain filter functions. These generalized filter functions may be derived for complex temporally modulated control protocols, accounting for susceptibility to rotations of the qubit state vector in three dimensions. Taken together, we show that this framework provides a computationally efficient means to calculate the effects of universal noise on arbitrary quantum control protocols, producing results comparable with those obtained via time-consuming simulations of Bloch vector evolution. As a concrete example, we apply our method to treating the problem of dynamical decoupling incorporating realistic control pulses of arbitrary duration or form, including the replacement of simple π-pulses with complex dynamically corrected gates.
122150162
Effects of arbitrary laser or NMR pulse shapes on population inversion and coherence
We present a new perturbation expansion for calculating the effects of arbitrary pulse shapes in two‐level systems, even when the effects are grossly nonlinear. The first two terms have simple physical interpretations. This expansion converges rapidly for all values of resonance offset with simple shapes, and for any pulse shape far from resonance. We generate very simple, symmetric, single phase pulse shapes which produce uniform inversion or polarization and which can be combined into multiple pulse sequences. We also show that pulse shape modification is superior to construction of composite pulse sequences, since such sequences must become erratic far from resonance.
93820434
Theoretical Analysis of Relaxation during Shaped Pulses. II. The Effects of Cross Relaxation
Most treatments of solution-state pulse FT NMR ignore relaxation effects during pulses. Since the length of nonselective 90° pulses is on the order of 10 μs, while proton T1 values (in solution) are generally >0.5 s, this simplification is valid. The advent of highly selective shaped pulses, however, has resulted in practical pulse lengths of over 100 ms. When using these pulses on samples where T1 or T2 is of the same order of magnitude as the pulse length, one cannot ignore relaxation during the pulse. Here, a description of the buildup of the NOE during the selective pulse and a quantitative analysis of ID selective NOE difference experiments where shaped pulses have been used is presented. Although this experiment itself is not new, a quantitative analysis of the data has not been presented previously. It is shown that longitudinal, transverse, and cross relaxation during the selective pulse cannot be ignored for samples where any of these relaxation times is on the order of the pulse length. This is especially significant in applications to large molecules, such as proteins, where spectral crowding requires high selectivity, i.e., long pulses, and the T1 values are frequently less than 0.5 s.
33231158
Application of optimal control theory to the design of broadband excitation pulses for high-resolution NMR.
Optimal control theory is considered as a methodology for pulse sequence design in NMR. It provides the flexibility for systematically imposing desirable constraints on spin system evolution and therefore has a wealth of applications. We have chosen an elementary example to illustrate the capabilities of the optimal control formalism: broadband, constant phase excitation which tolerates miscalibration of RF power and variations in RF homogeneity relevant for standard high-resolution probes. The chosen design criteria were transformation of I(z)-->I(x) over resonance offsets of +/- 20 kHz and RF variability of +/-5%, with a pulse length of 2 ms. Simulations of the resulting pulse transform I(z)-->0.995I(x) over the target ranges in resonance offset and RF variability. Acceptably uniform excitation is obtained over a much larger range of RF variability (approximately 45%) than the strict design limits. The pulse performs well in simulations that include homonuclear and heteronuclear J-couplings. Experimental spectra obtained from 100% 13C-labeled lysine show only minimal coupling effects, in excellent agreement with the simulations. By increasing pulse power and reducing pulse length, we demonstrate experimental excitation of 1H over +/-32 kHz, with phase variations in the spectra <8 degrees and peak amplitudes >93% of maximum. Further improvements in broadband excitation by optimized pulses (BEBOP) may be possible by applying more sophisticated implementations of the optimal control formalism.
2986447
Exploring the limits of broadband excitation and inversion pulses.
The design of broadband excitation and inversion pulses with compensation of B(1)-field inhomogeneity is a long standing goal in high resolution NMR spectroscopy. Most optimization procedures used so far have been restricted to particular pulse families to keep the scale of the problem within manageable limits. This restriction is unnecessary using efficient numerical algorithms based on optimal control theory. A systematic study of rf-limited broadband excitation by optimized pulses and broadband inversion by optimized pulses with respect to bandwidth and B(1)-field is presented. Upper limits on minimum pulse lengths are set for different degrees of pulse performance.
17334983
Scalable design of tailored soft pulses for coherent control.
We present a scalable scheme to design optimized soft pulses and pulse sequences for coherent control of interacting quantum many-body systems. The scheme is based on the cluster expansion and the time-dependent perturbation theory implemented numerically. This approach offers a dramatic advantage in numerical efficiency, and it is also more convenient than the commonly used Magnus expansion, especially when dealing with higher-order terms. We illustrate the scheme by designing 2nd-order self-refocusing pi pulses and a 6th-order 8-pulse refocusing sequence for a chain of qubits with nearest-neighbor couplings. We also discuss the performance of soft-pulse refocusing sequences in suppressing decoherence due to low-frequency environment.
35902664
High-fidelity one-qubit operations under random telegraph noise (9 pages)
We address the problem of implementing high-fidelity one-qubit operations subject to time-dependent noise in the qubit energy splitting. We show with explicit numerical results that high-fidelity bit flip operations may be generated by imposing bounded control fields. For noise correlation times shorter than the time for a {pi} pulse, the time-optimal {pi} pulse itself yields the highest fidelity. For very long correlation times, fidelity loss is approximately due to systematic error, which is efficiently tackled by compensation for off resonance with a pulse sequence (CORPSE). For intermediate ranges of the noise correlation time, we find that short CORPSE, which is less accurate than CORPSE in correcting systematic errors, yields higher fidelities. Numerical optimization of the pulse sequences using gradient ascent pulse engineering results in noticeable improvement of the fidelity for a bit flip operation on the computational basis states and a small but still positive fidelity enhancement for the NOT gate.
119298791
Soft-pulse dynamical decoupling in a cavity
Dynamical decoupling is a coherent control technique where the intrinsic and extrinsic couplings of a quantum system are effectively averaged out by application of specially designed driving fields (refocusing pulse sequences). This entails pumping energy into the system, which can be especially dangerous when it has sharp spectral features like a cavity mode close to resonance. In this work we show that such an effect can be avoided with properly constructed refocusing sequences. To this end we construct the average Hamiltonian expansion for the system evolution operator associated with a single ``soft'' pi-pulse. To second order in the pulse duration, we characterize a symmetric pulse shape by three parameters, two of which can be turned to zero by shaping. We express the effective Hamiltonians for several pulse sequences in terms of these parameters, and use the results to analyze the structure of error operators for controlled Jaynes-Cummings Hamiltonian. When errors are cancelled to second order, numerical simulations show excellent qubit fidelity with strongly-suppressed oscillator heating.
18540313
Second-order shaped pulses for solid-state quantum computation
We present the construction and detailed analysis of highly optimized self-refocusing pulse shapes for several rotation angles. We characterize the constructed pulses by the coefficients appearing in the Magnus expansion up to second order. This allows a semianalytical analysis of the performance of the constructed shapes in sequences and composite pulses by computing the corresponding leading-order error operators. Higher orders can be analyzed with the numerical technique suggested by us previously. We illustrate the technique by analyzing several composite pulses designed to protect against pulse amplitude errors, and on decoupling sequences for potentially long chains of qubits with on-site and nearest-neighbor couplings.
1576440
Robust dynamical decoupling
Quantum computers, which process information encoded in quantum mechanical systems, hold the potential to solve some of the hardest computational problems. A substantial obstacle for the further development of quantum computers is the fact that the lifetime of quantum information is usually too short to allow practical computation. A promising method for increasing the lifetime, known as dynamical decoupling (DD), consists of applying a periodic series of inversion pulses to the quantum bits. In the present review, we give an overview of this technique and compare different pulse sequences proposed earlier. We show that pulse imperfections, which are always present in experimental implementations, limit the performance of DD. The loss of coherence due to the accumulation of pulse errors can even exceed the perturbation from the environment. This effect can be largely eliminated by a judicious design of pulses and sequences. The corresponding sequences are largely immune to pulse imperfections and provide an increase of the coherence time of the system by several orders of magnitude.
118451489
Phenomenological Study of Decoherence in Solid-State Spin Qubits due to Nuclear Spin Diffusion
We present a study of the prospects for coherence preservation in solid-state spin qubits using dynamical decoupling protocols. Recent experiments have provided the first demonstrations of multipulse dynamical decoupling sequences in this qubit system, but quantitative analyses of potential coherence improvements have been hampered by a lack of concrete knowledge of the relevant noise processes. We present simulations of qubit coherence under the application of arbitrary dynamical decoupling pulse sequences based on an experimentally validated semiclassical model. This phenomenological approach bundles the details of underlying noise processes into a single experimentally relevant noise power spectral density. Our results show that the dominant features of experimental measurements in a two-electron singlet-triplet spin qubit can be replicated using a $1/\omega^{2}$ noise power spectrum associated with nuclear-spin-flips in the host material. Beginning with this validation we address the effects of nuclear programming, high-frequency nuclear-spin dynamics, and other high-frequency classical noise sources, with conjectures supported by physical arguments and microscopic calculations where relevant. Our results provide expected performance bounds and identify diagnostic metrics that can be measured experimentally in order to better elucidate the underlying nuclear spin dynamics.
118522585
Dynamics and decoherence in the central spin model in the low-field limit
We present a combination of analytic calculations and a powerful numerical method for large spin baths in the low-field limit. The hyperfine interaction between the central spin and the bath is fully captured by the density matrix renormalization group. The adoption of the density matrix renormalization group for the central spin model is presented and a proper method for calculating the real-time evolution at infinite temperature is identified. In addition, we study to which extent a semiclassical model, where a quantum spin-1/2 interacts with a bath of classical Gaussian fluctuations, can capture the physics of the central spin model. The model is treated by average Hamiltonian theory and by numerical simulation.
117001367
Spin noise in a quantum dot ensemble: from a quantum mechanical to a semi-classical description
Spin noise spectroscopy is a promising technique for revealing the microscopic nature of spin dephasing processes in quantum dots (QDs). We compare the spin-noise in an ensemble of singly charged QDs calculated by two complementary approaches. The Chebyshev polynomial expansion technique (CET) accounts for the full quantum mechanical fluctuation of the nuclear spin bath and a semi-classical approach (SCA) is based on the averaging the electron spin dynamics over all different static Overhauser field configurations. We observe a remarkable agreement between both methods in the high-frequency part of the spectra determined by static nuclear fields. The low-frequency part is determined by the long time fluctuations of the Overhauser field. We find small differences in the spectra depending on the distribution of hyperfine couplings. The spin-noise spectra in strong enough magnetic fields where the nuclear dynamics is quenched calculated by two complimentary approaches are in perfect agreement.
122578897
Spectral Diffusion Decay in Spin Resonance Experiments
In spin resonance experiments random flipping by ${T}_{1}$ or ${T}_{2}$ processes of nearby, nonresonant spins introduces fluctuations into the precessional frequency of the observed spins. These fluctuations may be described by means of a stochastic model, and for wide classes of both Markoffian and non-Markoffian distributions we make predictions for the line shape, for the free induction decay, and for various spin-echo signals. If the homogeneous broadening of the line is due to a dipolar interaction term, then we find that the conditional distribution for the precessional frequency has the shape of a Lorentzian with a cutoff on the wings, rather than a Gaussian shape as commonly assumed. The causes and consequences of Lorentzian diffusion are analyzed in detail for samples in which ${T}_{1}$ processes control the source of local frequency fluctuations and for samples in which ${T}_{2}$ processes dominate. Recent two- and three-pulse spin-echo experiments of Mims et al. dramatically confirm the predictions of Lorentzian diffusion for electron paramagnetic resonances in samples with temperature-dependent diffusion, as well as with temperature-independent diffusion. "Instantaneous" diffusion caused by the action of the applied pulses is predicted by our model and explains features of Mims' data. The generality of our principal results still permits the outcome of various resonance experiments to be predicted, even when a simple dipolar interaction is no longer an adequate model.
26812150
Universal Dynamical Decoupling of a Single Solid-State Spin from a Spin Bath
Avoiding Loss in a Quantum System Single electron spins in solid-state environments have been explored as candidates for quantum information storage and computation; however, they often interact strongly with their surroundings and lose the stored information on the time scale of pico- to milliseconds. Dynamical decoupling schemes have been introduced to “undo” the effects of this interaction by applying a sequence of control pulses that reverse the undesirable evolution of the system. De Lange et al. (p. 60, published online 9 September) tested several decoupling schemes on a nitrogen vacancy center in diamond and found that a scheme with evenly spaced pulses with double-axis decoupling could prolong the coherence time of an arbitrary spin state up to 25-fold. The coherence time of single spins is extended by a sequence of microwave pulses. Controlling the interaction of a single quantum system with its environment is a fundamental challenge in quantum science and technology. We strongly suppressed the coupling of a single spin in diamond with the surrounding spin bath by using double-axis dynamical decoupling. The coherence was preserved for arbitrary quantum states, as verified by quantum process tomography. The resulting coherence time enhancement followed a general scaling with the number of decoupling pulses. No limit was observed for the decoupling action up to 136 pulses, for which the coherence time was enhanced more than 25 times compared to that obtained with spin echo. These results uncover a new regime for experimental quantum science and allow us to overcome a major hurdle for implementing quantum information protocols.
119229694
Efficient coherent control by sequences of pulses of finite duration
Reliable long-time storage of arbitrary quantum states is a key element of quantum information processing. In order to dynamically decouple a spin or quantum bit from a dephasing environment by non-instantaneous pulses, we introduce an optimized sequence of N control π pulses that are realistic in the sense that they have a finite duration and a finite amplitude. We show that optimized dynamical decoupling is still applicable and that higher-order decoupling can be reached if shaped pulses are implemented. The sequence suppresses decoherence up to the order , with T being the total duration of the sequence and τmx the maximum length of the pulses. The exponent depends on the shape of the pulse. Based on existing experiments, a concrete setup for the verification of the properties of the advocated sequence is proposed.
119290335
High Order Coherent Control Sequences of Finite-Width Pulses
The performance of sequences of designed pulses of finite length $\tau$ is analyzed for a bath of spins and it is compared with that of sequences of ideal, instantaneous pulses. The degree of the design of the pulse strongly affects the performance of the sequences. Non-equidistant, adapted sequences of pulses, which equal instantaneous ones up to $\mathcal{O}(\tau^3)$, outperform equidistant or concatenated sequences. Moreover, they do so at low energy cost which grows only logarithmically with the number of pulses, in contrast to standard pulses with linear growth.
115177329
The Magnus expansion and some of its applications
Abstract Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem, shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an elegant setting to build up approximate exponential representations of the solution of the system. It provides a power series expansion for the corresponding exponent and is sometimes referred to as Time-Dependent Exponential Perturbation Theory. Every Magnus approximant corresponds in Perturbation Theory to a partial re-summation of infinite terms with the important additional property of preserving, at any order, certain symmetries of the exact solution. The goal of this review is threefold. First, to collect a number of developments scattered through half a century of scientific literature on Magnus expansion. They concern the methods for the generation of terms in the expansion, estimates of the radius of convergence of the series, generalizations and related non-perturbative expansions. Second, to provide a bridge with its implementation as generator of especial purpose numerical integration methods, a field of intense activity during the last decade. Third, to illustrate with examples the kind of results one can expect from Magnus expansion, in comparison with those from both perturbative schemes and standard numerical integrators. We buttress this issue with a revision of the wide range of physical applications found by Magnus expansion in the literature.
35139068
High-order commutator-free exponential time-propagation of driven quantum systems
We discuss the numerical solution of the Schrodinger equation with a time-dependent Hamilton operator using commutator-free time-propagators. These propagators are constructed as products of exponentials of simple weighted sums of the Hamilton operator. Owing to their exponential form they strictly preserve the unitarity of time-propagation. The absence of commutators or other computationally involved operations allows for straightforward implementation and application also to large scale and sparse matrix problems. We explain the derivation of commutator-free exponential time-propagators in the context of the Magnus expansion, and provide optimized propagators up to order eight. An extensive theoretical error analysis is presented together with practical efficiency tests for different problems. Issues of practical implementation, in particular the use of the Krylov technique for the calculation of exponentials, are discussed. We demonstrate for two advanced examples, the hydrogen atom in an electric field and pumped systems of multiple interacting two-level systems or spins that this approach enables fast and accurate computations.
119183114
Numerical time propagation of quantum systems in radiation fields
Atoms, molecules or excitonic quasiparticles, for which excitations are induced by external radiation fields and energy is dissipated through radiative decay, are examples of driven open quantum systems. We explain the use of commutator-free exponential time propagators for the numerical solution of the associated Schr?dinger or master equations with a time-dependent Hamilton operator. These time propagators are based on the Magnus series but avoid the computation of commutators, which makes them suitable for the efficient propagation of systems with a large number of degrees of freedom. We present an optimized fourth-order propagator and demonstrate its efficiency in comparison to the direct Runge?Kutta computation. As an illustrative example we consider the parametrically driven dissipative Dicke model, for which we calculate the periodic steady state and the optical emission spectrum.
118868806
No-go theorems and optimization of dynamical decoupling against noise with soft cutoff
We study the performance of dynamical decoupling in suppressing decoherence caused by soft-cutoff Gaussian noise, using short-time expansion of the noise correlations and numerical optimization. For the noise with soft cutoff at high frequencies, there exists no dynamical decoupling scheme to eliminate the decoherence to arbitrary orders of the short time, regardless of the timing or pulse shaping of the control under the population conserving condition. We formulate the equations for optimizing pulse sequences that minimizes decoherence up to the highest possible order of the short time for the noise correlations with odd power terms in the short-time expansion. In particular, we show that the Carr-Purcell-Meiboom-Gill sequence is optimal in short-time limit for the noise correlations with a linear order term in the time expansion.
20846996
Measuring the spectrum of colored noise by dynamical decoupling.
Decoherence is one of the most important obstacles that must be overcome in quantum information processing. It depends on the qubit-environment coupling strength, but also on the spectral composition of the noise generated by the environment. If the spectral density is known, fighting the effect of decoherence can be made more effective. Applying sequences of inversion pulses to the qubit system, we developed a method for dynamical decoupling noise spectroscopy. We generate effective filter functions that probe the environmental spectral density without requiring assumptions about its shape. Comparing different pulse sequences, we recover the complete spectral density function and distinguish different contributions to the overall decoherence.
120926297
The Dipolar Broadening of Magnetic Resonance Lines in Crystals
In regular crystals, the width of the absorption lines arising from the magnetic moment of the electron or nucleus is caused primarily by the interaction between the magnetic dipoles. It is prohibitively difficult to determine the precise shape of the absorption line theoretically, but the invariance of the diagonal sum in quantum mechanics permits the calculation of the second moment of the frequency deviation, and hence the r.m.s. line breadth. The latter agrees excellently with the observations of Pake and Purcell on the magnetic absorption of the F nucleus in Ca${\mathrm{F}}_{2}$, both in absolute magnitude, and in the dependence on the direction between the magnetic field and the principal cubic axes. The fourth moment was also computed to examine how good an approximation is the conventional assumption of a Gaussian shape. As long as no exchange is present (the nuclear case) the Gaussian model is moderately good. For the 100 direction in a cubic crystal, the theoretical ratio of root mean fourth to root mean square breadth is 1.25. Pake and Purcell's measurements yield 1.24. A Gaussian model would require 1.32. The theory is extended to include crystals with two kinds of spin moments (two types of nuclei, or simultaneous nuclear and electronic spin). Coupling between unlike moments is less effective (by a factor ⅔ in the r.m.s. width) than that between like in broadening the lines.In the paramagnetic absorption caused by electronic spin, it is imperative to include the effect of exchange coupling. This interaction does not contribute to the second moment, but greatly increases the fourth. As a result, the lines are peaked much more sharply than one would compute from the second moment with the Gaussian model for line shape. This "exchange narrowing" explains why microwave paramagnetic absorption lines are much narrower than one first conjectures from the amount of dipolar coupling.The theoretical calculations are given in Sections II-IV. The final sections V-VI give the comparison with the experiments of Pake and Purcell, and with the model of Bloembergen, Purcell, and Pound, for r-f absorption in liquids.
120654734
Spectral densities and nuclear spin relaxation in solids
Abstract We investigate the properties of ten spectral densities relevant for nuclear spin relaxation studies in solids. This is preceded by a brief review of nuclear spin relaxation in solids which includes a discussion of the appropriate spin-dependent interactions and the various relaxation rates which can be measured. Also, the link between nuclear spin relaxation and dielectric relaxation is discussed. Where possible and/or appropriate each of the spectral densities is expressed as a continuous distribution of Bloembergen-Purcell-Pound (or Debye) spectral densities 2ξ /(1 + ξ 2 ω 2 ) for nuclear Larmor angular frequency ω and correlation time ξ. The spectral densities are named after their originators or the shape of the distributions of correlation times or both and are (1) Bloembergen-Purcell-Pound or δ-function, (2) Havriliak-Negami, (3) Cole-Cole, (4) Davidson-Cole, (5) Fang, (6) Fuoss-Kirkwood, (7) Bryn Mawr, (8) Wagner or log-Gaussian, (9) log-Lorentzian, and (10) Frohlich or energy box. The Havriliak-Negami spectral density is related to the Dissado-Hill theory for dielectric relaxation. The spectral densities are expressed in a way which makes them easy to compare with each other and with experimental data. Many plots of the distributions of correlation times and of the spectral densities vs. various correlation times characterizing the distributions are given.