We study multiple-spin coherence transfers in linear Ising spin chains with nearest-neighbor couplings. These constitute a model for efficient information transfers in future quantum computing devices and for many multidimensional experiments for the assignment of complex spectra in nuclear magnetic resonance spectroscopy. We complement prior analytic techniques for multiple-spin coherence transfers with a systematic numerical study where we obtain strong evidence that a certain analytically motivated family of restricted controls is sufficient for time optimality. In the case of a linear three-spin system, additional evidence suggests that prior analytic pulse sequences using this family of restricted controls are time optimal even for arbitrary local controls. In addition, we compare the pulse sequences for linear Ising spin chains to pulse sequences for more realistic spin systems with additional long-range couplings between nonadjacent spins. We experimentally implement the derived pulse sequences in three- and four-spin systems and demonstrate that they are applicable in realistic settings under relaxation and experimental imperfections---in particular---by deriving broadband pulse sequences which are robust with respect to frequency offsets.
We study multiple-spin coherence transfers in linear Ising spin chains with nearest-neighbor couplings. These constitute a model for efficient information transfers in future quantum computing devices and for many multidimensional experiments for the assignment of complex spectra in nuclear magnetic resonance spectroscopy. We complement prior analytic techniques for multiple-spin coherence transfers with a systematic numerical study where we obtain strong evidence that a certain analytically motivated family of restricted controls is sufficient for time optimality. In the case of a linear three-spin system, additional evidence suggests that prior analytic pulse sequences using this family of restricted controls are time optimal even for arbitrary local controls. In addition, we compare the pulse sequences for linear Ising spin chains to pulse sequences for more realistic spin systems with additional long-range couplings between nonadjacent spins. We experimentally implement the derived pulse sequences in three- and four-spin systems and demonstrate that they are applicable in realistic settings under relaxation and experimental imperfections---in particular---by deriving broadband pulse sequences which are robust with respect to frequency offsets.
Experiments in coherent spectroscopy correspond to control of quantum-mechanical ensembles guiding them from initial to final target states. The control inputs (pulse sequences) that accomplish these transfers should be designed to minimize the effects of relaxation and to optimize the sensitivity of the experiments. For example in Nuclear Magnetic Resonance (NMR) spectroscopy, a question of fundamental importance is what is the maximum efficiency of polarization or coherence transfer between coupled spins in the presence of relaxation. Furthermore, what is the optimal pulse sequence which achieves this efficiency? This letter is a continuation of our previous work on the above problems. Here, we attempt to answer these questions for an Ising spin chain, under the presence of transverse relaxation. The methods presented in this paper are expected to find applications not only in coherent spectroscopy but also in quantum information processing.
In this thesis, we study methods for optimal manipulation of coupled spin dynamics in the presence of relaxation. We use these methods to compute analytical upper bounds for the efficiency of coherence and polarization transfer between coupled nuclear spins in multidimensional nuclear magnetic resonance (NMR) experiments, under the presence of relaxation. We derive relaxation optimized pulse sequences which achieve or approach these bounds and maximize the sensitivity of the experiments. The two systems examined in detail in this thesis, arising in the context of protein NMR spectroscopy, are a coupled heteronuclear spin pair and an Ising spin chain. New schemes for coherence and polarization transfer are found, which yield substantial gains in transfer efficiency, compared to the traditionally used pulse sequences. From the standpoint of mathematical control theory, the problems arising in connection with optimal manipulation of dissipative quantum dynamics, motivate the study of a class of dissipative bilinear control systems and the computation of their reachable sets. For these systems we show that the optimal solution, corresponding to the maximization of transfer efficiency, and the reachable set can be found by solving a semidefinite program.
In this paper, we study the design of pulse sequences for nuclear magnetic resonance spectroscopy as a problem of time optimal control of the unitary propagator. Radio-frequency pulses are used in coherent spectroscopy to implement a unitary transfer between states. Pulse sequences that accomplish a desired transfer should be as short as possible in order to minimize the effects of relaxation and to optimize the sensitivity of the experiments. Here, we give an analytical characterization of such time optimal pulse sequences applicable to coherence transfer experiments in multiple-spin systems. We have adopted a general mathematical formulation, and present many of our results in this setting, mindful of the fact that new structures in optimal pulse design are constantly arising. From a general control theory perspective, the problems we want to study have the following character. Suppose we are given a controllable right invariant system on a compact Lie group. What is the minimum time required to steer the system from some initial point to a specified final point? In nuclear magnetic resonance (NMR) spectroscopy and quantum computing, this translates to, what is the minimum time required to produce a unitary propagator? We also give an analytical characterization of maximum achievable transfer in a given time for the two-spin system.
Experiments in coherent spectroscopy correspond to control of quantum mechanical ensembles guiding them from initial to final target states by unitary transformations. The control inputs (pulse sequences) that accomplish these unitary transformations should take as little time as possible so as to minimize the effects of relaxation and to optimize the sensitivity of the experiments. Here, we present a novel approach for efficient control of dynamics in spin chains of arbitrary length. The approach relies on creating certain three spin encoded states, which can be efficiently transferred through a spin chain. The methods presented are expected to find applications in control of spin dynamics in coherent spectroscopy and quantum information processing.
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.
Radio-frequency pulses are used in nuclear-magnetic-resonance spectroscopy to produce unitary transfer of states. Pulse sequences that accomplish a desired transfer should be as short as possible in order to minimize the effects of relaxation, and to optimize the sensitivity of the experiments. Many coherence-transfer experiments in NMR, involving a network of coupled spins, use temporary spin decoupling to produce desired effective Hamiltonians. In this paper, we demonstrate that significant time can be saved in producing an effective Hamiltonian if spin decoupling is avoided. We provide time-optimal pulse sequences for producing an important class of effective Hamiltonians in three-spin networks. These effective Hamiltonians are useful for coherence-transfer experiments in three-spin systems and implementation of indirect swap and ${\ensuremath{\Lambda}}_{2}(U)$ gates in the context of NMR quantum computing. It is shown that computing these time-optimal pulses can be reduced to geometric problems that involve computing sub-Riemannian geodesics. Using these geometric ideas, explicit expressions for the minimum time required for producing these effective Hamiltonians, transfer of coherence, and implementation of indirect swap gates, in a three-spin network are derived (Theorems 1 and 2). It is demonstrated that geometric control techniques provide a systematic way of finding time-optimal pulse sequences for transferring coherence and synthesizing unitary transformations in quantum networks, with considerable time savings (e.g., 42.3% for constructing indirect swap gates).