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The Journal of Chemical Physics

Analytical unitary bounds on quantum dynamics: Design of optimum NMR experiments in two-spin-12 systems

Thomas S. Untidt, Niels Chr. Nielsen

Published Nov 15, 2000• 15 citations
Publisher

Abstract

We report analytical solutions to the unitary bound problem for coherence/polarization transfer in IS two-spin-12 systems by means of unitary operations. Theoretical upper bounds for the transfer efficiency along with the associated optimum transformation operators are obtained analytically by decomposing the unitary operator as a product of exponentials in the special unitary Lie group SU(4). Addressing NMR spectroscopy as a specific example, the method is demonstrated for the non-Hermitian transfers I−→S− and 2I−Sz→S− being relevant for heteronuclear single-quantum coherence (HSQC) experiments as well as the double- to single-quantum transfer I−S−→I−Sβ+IβS− being representative for coherence-order and spin-state-selective transfer in INADEQUATE CR experiments. Furthermore, using a Lagrangian function approach it is demonstrated how the method enables analytical description of two-dimensional bounds for Iz→Sz cross polarization.

DOI
10.1063/1.1318756

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We report analytical solutions to the unitary bound problem for coherence/polarization transfer in IS two-spin-12 systems by means of unitary operations. Theoretical upper bounds for the transfer efficiency along with the associated optimum transformation operators are obtained analytically by decomposing the unitary operator as a product of exponentials in the special unitary Lie group SU(4). Addressing NMR spectroscopy as a specific example, the method is demonstrated for the non-Hermitian transfers I−→S− and 2I−Sz→S− being relevant for heteronuclear single-quantum coherence (HSQC) experiments as well as the double- to single-quantum transfer I−S−→I−Sβ+IβS− being representative for coherence-order and spin-state-selective transfer in INADEQUATE CR experiments. Furthermore, using a Lagrangian function approach it is demonstrated how the method enables analytical description of two-dimensional bounds for Iz→Sz cross polarization.
We report a general bound for the efficiency of polarization or coherence transfer between quantized states under unitary transformations. This bound is more general than the so-called universal bound on spin dynamics [O. W. S\o{}rensen, J. Magn. Reson. 86, 435 (1990)] that applies only to transformations between Hermitian states. The new bound on transfer efficiency under unitary transformations between general non-Hermitian operators depends exclusively on the singular values of the matrix representations of these operators.
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.
The effect of spin permutation symmetry on the maximum achievable efficiency of coherence transfer between pairs of Hermitian or non-Hermitian spin operators is investigated. It is shown that spin permutation symmetry may tighten significantly the quantum mechanical bounds hitherto known to apply for coherence transfer processes. This feature is demonstrated for the heteronuclear coherence transfer processes S x → F x , S - → F - and 2F z S x → F x , 2F z S - → F - (F = Σ n i=1 I i ) in I2S and I3S (n = 2,3) spin-1/2 systems. The bounds arising from permutation symmetry apply for non-selective multidimensional NMR experiments.
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.
The effect of restricted control of unitary quantum evolution is investigated with specific attention to NMR spectroscopy. It is demonstrated that in cases where the Hamiltonian through commutation fails to span the entire Lie algebra su(n) for an n-level quantum system, the maximum transfer efficiency may be reduced significantly relative to previously known unitary bounds on spin dynamics. The paper describes methods to determine the degree of controllability and the conditional unitary bounds induced by restricted control. These features are exemplified in relation to heteronuclear coherence transfer by planar and isotropic mixing in liquid state NMR.
Modern applications of quantum control in quantum information science and technology require the precise characterization of quantum states and quantum channels. In particular, high-performance quantum state engineering often demands that quantum states are transferred with optimal efficiency via realizable controlled evolution, the latter often modeled by quantum channels. When an appropriate quantum control model for an interested system is constructed, the exploration of optimal bounds on state transfer for the underlying quantum channel is then an important task. In this work, we analyze the state transfer efficiency problem for different classes of quantum channels, including unitary, mixed unitary, and Markovian. We then apply the theory to nuclear magnetic resonance (NMR) experiments. We show that two most commonly used control techniques in NMR, namely gradient field control and phase cycling, can be described by mixed unitary channels. Then we show that employing mixed unitary channels does not extend the unitarily accessible region of states. Also, we present a strategy of optimal experiment design, which incorporates coherent radio-frequency field control, gradient field control, and phase cycling, aiming at maximizing state transfer efficiency and meanwhile minimizing the number of experiments required. Finally, we perform pseudopure state preparation experiments on two- and three-spin systems, in order to test the bound theory and to demonstrate the usefulness of nonunitary control means.