Linking the rotation of a rigid body to the Schrödinger equation: The quantum tennis racket effect and beyond.

The design of efficient and robust pulse sequences is a fundamental requirement in quantum control. Numerical methods can be used for this purpose, but with relatively little insight into the control mechanism. Here, we show that the free rotation of a classical rigid body plays a fundamental role in the control of two-level quantum systems by means of external electromagnetic pulses. For a state to state transfer, we derive a family of control fields depending upon two free parameters, which allow us to adjust the efficiency, the time and the robustness of the control process. As an illustrative example, we consider the quantum analog of the tennis racket effect, which is a geometric property of any classical rigid body. This effect is demonstrated experimentally for the control of a spin 1/2 particle by using techniques of Nuclear Magnetic Resonance. We also show that the dynamics of a rigid body can be used to implement one-qubit quantum gates. In particular, non-adiabatic geometric quantum phase gates can be realized based on the Montgomery phase of a rigid body. The robustness issue of the gates is discussed.

Time-optimal excitation of maximum quantum coherence: Physical limits and pulse sequences.

Here we study the optimum efficiency of the excitation of maximum quantum (MaxQ) coherence using analytical and numerical methods based on optimal control theory. The theoretical limit of the achievable MaxQ amplitude and the minimum time to achieve this limit are explored for a set of model systems consisting of up to five coupled spins. In addition to arbitrary pulse shapes, two simple pulse sequence families of practical interest are considered in the optimizations. Compared to conventional approaches, substantial gains were found both in terms of the achieved MaxQ amplitude and in pulse sequence durations. For a model system, theoretically predicted gains of a factor of three compared to the conventional pulse sequence were experimentally demonstrated. Motivated by the numerical results, also two novel analytical transfer schemes were found: Compared to conventional approaches based on non-selective pulses and delays, double-quantum coherence in two-spin systems can be created twice as fast using isotropic mixing and hard spin-selective pulses. Also it is proved that in a chain of three weakly coupled spins with the same coupling constants, triple-quantum coherence can be created in a time-optimal fashion using so-called geodesic pulses.

Optimal control of the signal-to-noise ratio per unit time of a spin 1/2 particle: the crusher gradient and the radiation damping cases.

We show to which extent the signal to noise ratio per unit time of a spin 1/2 particle can be maximized. We consider a cyclic repetition of experiments made of a measurement followed by a radio-frequency magnetic field excitation of the system, in the case of unbounded amplitude. In the periodic regime, the objective of the control problem is to design the initial state of the system and the pulse sequence which leads to the best signal to noise performance. We focus on two specific issues relevant in nuclear magnetic resonance, the crusher gradient and the radiation damping cases. Optimal control techniques are used to solve this non-standard control problem. We discuss the optimality of the Ernst angle solution, which is commonly applied in spectroscopic and medical imaging applications. In the radiation damping situation, we show that in some cases, the optimal solution differs from the Ernst one.

Gradient ascent pulse engineering for rapid exchange saturation transfer.

Efforts in the clinical translation of the paraCEST contrast agent Yb-HPDO3A have prompted an investigation into saturation pulse optimality under energy constraints. The GRAPE algorithm has been adapted and implemented for saturation pulse optimization with chemical exchange. The flexibility of the methodology, both in extracting the microscopical parameter ensemble for the algorithm as well as in determining the characteristics of this new class of rising amplitude waveforms allows rapid testing and implementation. Optimal pulses achieve higher saturation efficiencies than the continuous wave gold standard for rapid and especially for variable exchange rates, as brought about by pH-catalysis. Gains of at least 5-15% without any tradeoff have been confirmed both on a spectrometer and on a clinical imager. Pool specific solutions, with pulses optimized for a specific exchange rate value, additionally increase the flexibility of the CEST ratiometric analysis. A simple experimental approach to determine close to optimal triangular pulses is presented.

Visualization and analysis of modulated pulses in magnetic resonance by joint time-frequency representations.

We study the utility of joint time-frequency representations for the analysis of shaped or composite pulses for magnetic resonance. Such spectrograms are commonly used for the visualization of shaped laser pulses in optical spectroscopy. This intuitive representation provides additional insight compared to conventional approaches, which exclusively show either temporal or spectral information. We focus on the short-time Fourier transform, which provides not only amplitude but also phase information. The approach is illustrated for broadband inversion pulses, multiple quantum excitation and broadband heteronuclear decoupling. The physical interpretation and validity of the approach is discussed.

Apparent rate constant mapping using hyperpolarized [1-(13)C]pyruvate.

Hyperpolarization of [1-13C]pyruvate in solution allows real-time measurement of uptake and metabolism using MR spectroscopic methods. After injection and perfusion, pyruvate is taken up by the cells and enzymatically metabolized into downstream metabolites such as lactate, alanine, and bicarbonate. In this work, we present comprehensive methods for the quantification and interpretation of hyperpolarized 13C metabolite signals. First, a time-domain spectral fitting method is described for the decomposition of FID signals into their metabolic constituents. For this purpose, the required chemical shift frequencies are automatically estimated using a matching pursuit algorithm. Second, a time-discretized formulation of the two-site exchange kinetic model is used to quantify metabolite signal dynamics by two characteristic rate constants in the form of (i) an apparent build-up rate (quantifying the build-up of downstream metabolites from the pyruvate substrate) and (ii) an effective decay rate (summarizing signal depletion due to repetitive excitation, T1-relaxation and backward conversion). The presented spectral and kinetic quantification were experimentally verified in vitro and in vivo using hyperpolarized [1-13C]pyruvate. Using temporally resolved IDEAL spiral CSI, spatially resolved apparent rate constant maps are also extracted. In comparison to single metabolite images, apparent build-up rate constant maps provide improved contrast by emphasizing metabolically active tissues (e.g. tumors) and suppression of high perfusion regions with low conversion (e.g. blood vessels). Apparent build-up rate constant mapping provides a novel quantitative image contrast for the characterization of metabolic activity. Its possible implementation as a quantitative standard will be subject to further studies.

Effects of pyruvate dose on in vivo metabolism and quantification of hyperpolarized ¹³C spectra.

Real-time in vivo measurements of metabolites are performed by signal enhancement of [1-(13)C]pyruvate using dynamic nuclear polarization, rapid dissolution and intravenous injection, acquisition of free induction decay signals and subsequent quantification of spectra. The commonly injected dose of hyperpolarized pyruvate is larger than typical tracer doses, with measurement before complete dilution of the injected bolus. Pyruvate is in exchange with its downstream metabolites lactate, alanine and bicarbonate. A transient exposure to high pyruvate blood concentrations may cause the saturation of cellular uptake and metabolic conversion. The goal of this study was to examine the effects of a [1-(13)C]pyruvate bolus on metabolic conversion in vivo. Spectra were quantified by three different methods: frequency-domain fitting with LCModel, time-domain fitting with AMARES and simple linear least-squares fitting in the time domain. Since the simple linear least-squares approach showed bleeding artifacts and LCModel produced noisier time signals. AMARES performed best in the quantification of in vivo hyperpolarized pyruvate spectra. We examined pyruvate doses of 0.1-0.4 mmol/kg (body mass) in male Wistar rats by acquiring slice-selective free induction decay signals in slices dominated by heart, liver and kidney tissue. Dose effects were noted in all cases, except for alanine in the cardiac slice below the dose of 0.2 mmol/kg. Our results indicate unlimited cellular uptake of pyruvate up to this dose and limited enzymatic activity of lactate dehydrogenase. In the cardiac slice above 0.2 mmol/kg and in liver and kidney slices, reflect limited cellular uptake or enzymatic activity, or a combination of both effects. The results indicate that the dose of pyruvate must be recognized as an important determinant for metabolic tissue kinetics, and saturation effects must be taken into account for the quantitative interpretation of the observed results.

Second order gradient ascent pulse engineering.

We report some improvements to the gradient ascent pulse engineering (GRAPE) algorithm for optimal control of spin ensembles and other quantum systems. These include more accurate gradients, convergence acceleration using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton algorithm as well as faster control derivative calculation algorithms. In all test systems, the wall clock time and the convergence rates show a considerable improvement over the approximate gradient ascent.

Time-optimal control of spin 1/2 particles in the presence of radiation damping and relaxation.

We consider the time-optimal control of an ensemble of uncoupled spin 1/2 particles in the presence of relaxation and radiation damping effects, whose dynamics is governed by nonlinear equations generalizing the standard linear Bloch equations. For a single spin, the optimal control strategy can be fully characterized analytically. However, in order to take into account the inhomogeneity of the static magnetic field, an ensemble of isochromats at different frequencies must be considered. For this case, numerically optimized pulse sequences are computed and the dynamics under the corresponding optimal field is experimentally demonstrated using nuclear magnetic resonance techniques.

Singular extremals for the time-optimal control of dissipative spin 1/2 particles.

We consider the time-optimal control by magnetic fields of a spin 1/2 particle in a dissipative environment. This system is used as an illustrative example to show the role of singular extremals in the control of quantum systems. We analyze a simple case where the control law is explicitly determined. We experimentally implement the optimal control using techniques of nuclear magnetic resonance. To our knowledge, this is the first experimental demonstration of singular extremals in quantum systems with bounded control amplitudes.

Measurement of 2J(H,C)- and 3J(H,C)-coupling constants by alpha/beta selective HC(C)H-TOCSY.

A new heteronuclear NMR pulse sequence for the measurement of nJ(C,H) coupling constants, the alpha/beta selective HC(C)H-TOCSY, is described. It is shown that the S3E element (Meissner et al., 1997a,b) can be used to obtain spin state selective coherence transfer in molecules, in which adjacent CH moieties are labeled with 13C. Application of the alpha/beta selective HC(C)H-TOCSY to a 10 nt RNA tetraloop 5'-CGCUUUUGCG-3', in which the four uridine residues are 13C labeled in the sugar moiety, allowed measurement of two bond and three bond J(C,H) coupling constants, which provide additional restraints to characterize the sugar ring conformation of RNA in cases of conformational averaging.

Analytical planar mixing transfer functions for two coupled spin-1 nuclei.

Analytical coherence transfer functions are presented for spin systems consisting of two spins 1 under planar mixing conditions. Compared to isotropic mixing experiments, larger transfer amplitudes are found and differences in the multiplet patterns are discussed.

Superposition of scalar and residual dipolar couplings: analytical transfer functions for three spins 1/2 under cylindrical mixing conditions.

The superposition of scalar and residual dipolar couplings gives rise to so-called cylindrical mixing Hamiltonians in dipolar coupling spectroscopy. General analytical polarization and coherence transfer functions are presented for three cylindrically coupled spins 12 under energy-matched conditions. In addition, the transfer efficiency is analyzed as a function of the relative coupling constants for characteristic special cases.

Structure of the Bacillus subtilis peptide antibiotic subtilosin A determined by 1H-NMR and matrix assisted laser desorption/ionization time-of-flight mass spectrometry.

Subtilosin A produced by Bacillus subtilis is a macrocyclic peptide antibiotic which comprises 35 amino acids. Its molecular mass (3399.7 Da), determined by matrix-assisted laser desorption/ionization time-of-flight mass spectrometry, and chemical properties gave experimental support for unusual intramolecular linkages. The three-dimensional fold of native subtilosin in dimethylsulfoxide was determined from two-dimensional 1H-NMR spectra recorded at 600 MHz. Based on the backbone conformation, a structure for subtilosin A is presented which is characterized by three inter-residue bridges where two cysteines are linked with two phenylalanine residues, respectively, and a third cysteine is bound to a threonine residue.

Analytical polarization and coherence transfer functions for three dipolar coupled spins 12.

Analytical polarization and coherence transfer functions are presented for a spin system consisting of three dipolar coupled homonuclear spins 12 under energy matched conditions. Based on these transfer functions, optimal durations of Hartmann-Hahn mixing periods can be determined for arbitrary dipolar coupling constants D(12), D(13), and D(23). In addition, the dependence of the transfer efficiency on the relative size of the dipolar coupling constants is illustrated.

Broadband heteronuclear Hartmann-Hahn sequences with short cycle times.

For some applications, broadband heteronuclear Hartmann-Hahn sequences with very short cycle times are required. A quality factor is presented that makes it possible to assess the relative sizes of cycle time, bandwidth, and maximum RF amplitude for any given multiple-pulse sequence. This quality factor is determined for multiple-pulse sequences that are commonly used in HEHAHA experiments and for some favorable sequences that were so far only discussed in the context of heteronuclear decoupling.

Single-shot experiments for the acquisition of coherence-transfer functions in real time.

Simple pulse sequences are introduced that make it possible to acquire experimental Hartmann-Hahn transfer functions for arbitrary multiple-pulse sequences in a single shot. With this approach it is possible to study the detailed dependence of coherence-transfer functions on experimental parameters in real time.

Structural and dynamical properties of a denatured protein. Heteronuclear 3D NMR experiments and theoretical simulations of lysozyme in 8 M urea.

Oxidized and reduced hen lysozyme denatured in 8 M urea at low pH have been studied in detail by NMR methods. 15N correlated NOESY and TOCSY experiments have provided near complete sequential assignment for both 1H and 15N resonances. Over 900 NOEs, including 130 (i, i + 2) and 23 (i, i + 3) NOEs, could be identified by analysis of the NOESY spectra of the denatured states, and 3J(HN, Halpha) coupling constants and 15N relaxation rates have been measured. The coupling constant and NOE data were analyzed by comparisons with theoretical predictions from a random coil polypeptide model based on amino acid specific phi,psi distributions extracted from the protein data bank. There is significant agreement between predicted and experimental NMR parameters suggesting that local conformations of the denatured states are largely determined by short-range interactions within the polypeptide chain. This result is supported by the observation that the chemical shift, coupling constant, and NOE data are little affected by whether or not the four disulfide bridge cross-links are formed in the denatured protein. The relaxation data, however, show significant differences between the oxidized and reduced protein. Analysis of the relaxation data in terms of simple dynamics models provides evidence for weak clustering of hydrophobic groups near tryptophan residues and increased barriers to motion in the more compact conformers formed when the polypeptide chain is cross-linked by the disulfide bridges. Using this information, a structural description of these denatured states is given in terms of an ensemble of conformers, which have a complex relationship between their local and global characteristics.

Directed TOCSY, a method for selection of directed correlations by optimal combinations of isotropic and longitudinal mixing.

The directed TOCSY pulse sequence element transfers coherence predominantly into "forward-directed" antiphase coherences while simultaneously suppressing in-phase and "backward-directed" antiphase coherences. This novel selection principle, based on the "direction" of the target coherences, provides a new approach for the simplification of crowded spectra. In this article, the theory of directed TOCSY is presented for linear spin systems that are frequently found in carbon-labeled biomolecules.