Ensemble Control of Bloch Equations

Li, Jr-Shin; Khaneja, Navin

doi:10.1109/tac.2009.2012983

Cited by 213 publications

(181 citation statements)

References 26 publications

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“…From now on, the argument is similar to that in [27]: the formula lim n→∞ { e −χ/n e −ψ/n e χ/n e ψ/n } n 2 = e [χ,ψ] ,…”

Section: An Example Of Perturbed Maxwell-bloch Equationmentioning

confidence: 91%

“…The controllability of a system consisting of an infinite collection of finite-dimensional systems has been analyzed for the situation of the Bloch equation (G = S O(3)) in [25][26][27]32]. To the best of our knowledge no general results are available for generic systems and values of N; moreover the counter-example in Theorem 4 in [27] warns that general results may be impossible to obtain.…”

Section: Extensions To An Infinite Set Of Perturbationsmentioning

confidence: 99%

“…Problems of simultaneous control of a finite collection of systems have been addressed recently in applications related to quantum control [20][21][22][23][24][25][26][27][28][29][30][31]. In such circumstances, the system is a collection of molecules or atoms or spin systems and the control is a magnetic field (in NMR) or a laser.…”

Section: Introductionmentioning

confidence: 99%

“…The infinite dimensional version (an infinite number of systems A ϵ = ϵA with ϵ taking arbitrary values in an interval ]ϵ * , ϵ * [) was treated in [26,27,32] for the specific situation of the Bloch equations.…”

Section: Introductionmentioning

confidence: 99%

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Ensemble controllability and discrimination of perturbed bilinear control systems on connected, simple, compact Lie groups

Belhadj

Salomon

Turinici

2015

European Journal of Control

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The controllability of bilinear systems is well understood for finite dimensional isolated systems where the control can be implemented exactly. However when perturbations are present some interesting theoretical questions are raised. We consider in this paper a control system whose control cannot be implemented exactly but is shifted by a time independent constant in a discrete list of possibilities. We prove under general hypothesis that the collection of possible systems (one for each possible perturbation) is simultaneously controllable with a common control. The result is extended to the situations where the perturbations are constant over a common, long enough, time frame. We apply the result to the controllability of quantum systems. Furthermore, some examples and a convergence result are presented for the situation where an infinite number of perturbations occur. In addition, the techniques invoked in the proof allow to obtain generic necessary and sufficient conditions for ensemble controllability.

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“…From now on, the argument is similar to that in [27]: the formula lim n→∞ { e −χ/n e −ψ/n e χ/n e ψ/n } n 2 = e [χ,ψ] ,…”

Section: An Example Of Perturbed Maxwell-bloch Equationmentioning

confidence: 91%

Section: Extensions To An Infinite Set Of Perturbationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Ensemble controllability and discrimination of perturbed bilinear control systems on connected, simple, compact Lie groups

Belhadj

Salomon

Turinici

2015

European Journal of Control

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“…We first present a general mathematical model for pulse design as an optimal control problem of a continuum of bilinear systems. Employing Lie algebra tools from our prior theoretical work (20), we show the control problem of pulse design can be mapped to a problem of polynomial approximation. This notion guides us to develop a unified computational method for optimal pulse design based on multidimensional pseudospectral (PS) approximations, by which a continuous-time optimal control problem of pulse design can be discretized to a constrained optimization problem using interpolating polynomials.…”

mentioning

confidence: 99%

Optimal pulse design in quantum control: A unified computational method

Ruths

et al. 2011

Proc. Natl. Acad. Sci. U.S.A.

130

109

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Many key aspects of control of quantum systems involve manipulating a large quantum ensemble exhibiting variation in the value of parameters characterizing the system dynamics. Developing electromagnetic pulses to produce a desired evolution in the presence of such variation is a fundamental and challenging problem in this research area. We present such robust pulse designs as an optimal control problem of a continuum of bilinear systems with a common control function. We map this control problem of infinite dimension to a problem of polynomial approximation employing tools from geometric control theory. We then adopt this new notion and develop a unified computational method for optimal pulse design using ideas from pseudospectral approximations, by which a continuous-time optimal control problem of pulse design can be discretized to a constrained optimization problem with spectral accuracy. Furthermore, this is a highly flexible and efficient numerical method that requires low order of discretization and yields inherently smooth solutions. We demonstrate this method by designing effective broadband π∕2 and π pulses with reduced rf energy and pulse duration, which show significant sensitivity enhancement at the edge of the spectrum over conventional pulses in 1D and 2D NMR spectroscopy experiments.pseudospectral methods | ensemble control | Lie algebra | broadband excitation C ompelling applications for quantum control have received particular attention and have motivated seminal studies in wide-ranging areas from coherent spectroscopy and MRI to quantum optics. Designing and implementing time-varying excitations (rf pulses) to manipulate complex dynamics of a large quantum ensemble on the order of Avogadro's number is a long-standing problem and an indispensable step that enables every application of quantum control (1). For example, magnetic resonance applications often suffer from imperfections such as inhomogeneity in the static magnetic field (B 0 inhomogeneity) and in the applied rf field (rf inhomogeneity). In addition, there is dispersion in the Larmor frequency of spins due to chemical shifts. A good pulse design strategy must be robust to these effects, and such variations need to be considered in the modeling and pulse design stages in order to match theoretical predictions to experimental outcomes. As difficult experiments with more demanding performance specifications have emerged, the complexity of finding optimal pulse sequences has drastically increased. For example, as high-field NMR spectrometers are increasingly more accessible and required, broadband excitation pulses are needed to cover a wide 13 C chemical-shift range (e.g., up to 40 kHz). In addition, to design excitation and inversion pulses that are practical for a typical NMR spectrometer, methods must accommodate realistic maximum rf power and pulse duration while accomplishing the desired spin evolution. In the majority of cases, the length of an rf pulse is constrained by the fixed delays that dictate a certain coherence transfer. S...

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Open‐loop control of parameter‐dependent discrete‐time systems

Scherlein

Schönlein

Helmke

2014

Proc Appl Math and Mech

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We consider parameter-dependent linear time-invariant discrete-time single input systems, where the system matrix and the input vector are assumed to depend continuously on a parameter varying over a compact interval. We face the problem of steering the zero state simultaneously arbitrarily close towards a given continuous family of desired terminal states with a finite parameter independent open-loop input sequence. Starting from existing sufficient conditions, which include simplicity of the eigenvalues of the system matrices, we examine the case of multiple eigenvalues.

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Ensemble Control of Bloch Equations

Cited by 213 publications

References 26 publications

Ensemble controllability and discrimination of perturbed bilinear control systems on connected, simple, compact Lie groups

Ensemble controllability and discrimination of perturbed bilinear control systems on connected, simple, compact Lie groups

Optimal pulse design in quantum control: A unified computational method

Open‐loop control of parameter‐dependent discrete‐time systems

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