Matrix method for analysis of selective NMR pulses

Tesiram, Yasvir A.; Separovic, Frances

doi:10.1002/cmr.a.20028

Cited by 5 publications

(4 citation statements)

References 40 publications

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“…It is generally not possible to compute analytically ψ 1 and ψ 2 due to the complexity of the radius r(t) which can be expressed in terms of the inverse of elliptic integrals [Eq. (10)].…”

Section: Dynamics Of the Bloch Vectormentioning

confidence: 99%

“…This paragraph is aimed at showing how to derive analytically the regular arcs which are solutions of Eq. ( 7) and (10).…”

Section: A Derivation Of the Regular Arcsmentioning

confidence: 99%

“…Performing efficient and selective quantum state transfer by external electromagnetic fields remains a challenge of practical and fundamental interest with applications extending from atomic physics to magnetic resonance and quantum information science [1,2,3,4,5]. Different analytical and numerical methods have been proposed up to date to design control fields [6,7,8,9,10,11,12,13,14,15,16,17]. Among others, we can mention optimal control techniques [1] for which the selectivity problem has been addressed numerically with standard iterative algorithms [18,19,21,22,23,24,20,25,26].…”

Section: Introductionmentioning

confidence: 99%

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Time-optimal selective pulses of two uncoupled spin-1/2 particles

et al. 2018

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We investigate the time-optimal solution of the selective control of two uncoupled spin 1/2 particles. Using the Pontryagin Maximum Principle, we derive the global time-optimal pulses for two spins with different offsets. We show that the Pontryagin Hamiltonian can be written as a one-dimensional effective Hamiltonian. The optimal fields can be expressed analytically in terms of elliptic integrals. The time-optimal control problem is solved for the selective inversion and excitation processes. A bifurcation in the structure of the control fields occurs for a specific offset threshold. In particular, we show that for small offsets, the optimal solution is the concatenation of regular and singular extremals. 1The PMP and the geometric techniques have been recently applied with success to a series of fundamental problems in quantum control, such as, to cite a few, the state to state transfer [33,34,35,36,37,38], the implementation of unitary gates [40,39,41,42,43], the simultaneous control of different systems [44,45,46,47,48] and the control of two-level quantum systems or spin systems in presence of relaxation [49,50,51,52,53,54,55,58,57,56,59].In magnetic resonance, a benchmark example for the selective control of spins is given by an inhomogeneous ensemble of spin 1/2 particles with different offsets [6,8]. In this paper, we propose to investigate the simplest selectivity problem, that is the simultaneous time-optimal control of two uncoupled spins by means of magnetic fields. The two spins are assumed to be initially at the thermal equilibrium state, i.e. the north pole of the Bloch sphere. We consider in this work both the selective excitation and inversion processes for which the goal is to steer one of the two spins towards the equator or the south pole, while bringing back the other to the initial state. We derive the global time-optimal solution with a constraint on the maximum available field intensity. For a large offset difference, the optimal pulse is regular of maximum intensity. We show the existence of a bifurcation for a specific offset threshold. For smaller offset difference, the optimal solution is the concatenation of regular and singular arcs, the singular solution corresponding to zero field.The article is organized as follows. In Sec. 2, we define the model system and we show how to apply the PMP in this case. Section 3 is dedicated to the presentation of the results. We derive the time-optimal solutions for the selective excitation and inversion of spins. We discuss how this minimum time varies as a function of the offset difference. A comparison with a direct numerical optimization is made in Sec. 5. A summary of the different results obtained and prospective views are presented in Sec. 6. Technical computations are reported in the appendices A, B and C.

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Section: Dynamics Of the Bloch Vectormentioning

confidence: 99%

“…This paragraph is aimed at showing how to derive analytically the regular arcs which are solutions of Eq. ( 7) and (10).…”

Section: A Derivation Of the Regular Arcsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Time-optimal selective pulses of two uncoupled spin-1/2 particles

et al. 2018

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“…They can be viewed as a simultaneous state-to-state control protocol for different copies of the system characterized by different values of parameters [33,34,36,[46][47][48][49][50]. Selective control has been used extensively in Nuclear Magnetic Resonance [51][52][53][54][55]. In this framework, the goal is to find a control that allows us to reach (possibly as fast as possible) different target states for each copy of the system, the target states being chosen specifically to minimize measurement error.…”

Section: Introductionmentioning

confidence: 99%

Optimal control strategies for parameter estimation of quantum systems

Ansel,

Dionis,

Sugny

2024

SciPost Phys.

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Optimal control theory is an effective tool to improve parameter estimation of quantum systems. Different methods can be employed for the design of the control protocol. They can be based either on Quantum Fischer Information (QFI) maximization or selective control processes. We describe the similarities, differences, and advantages of these two approaches. A detailed comparative study is presented for estimating the parameters of a spin-1/2 system coupled to a bosonic bath. We show that the control mechanisms are generally equivalent, except when the decoherence is not negligible or when the experimental setup is not adapted to the QFI. In this latter case, the precision achieved with selective controls can be several orders of magnitude better than that given by the QFI.

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