Control of inhomogeneous quantum ensembles

Abstract

Finding control fields ͑pulse sequences͒ that can compensate for the dispersion in the parameters governing the evolution of a quantum system is an important problem in coherent spectroscopy and quantum information processing. The use of composite pulses for compensating dispersion in system dynamics is widely known and applied. In this paper, we make explicit the key aspects of the dynamics that makes such a compensation possible. We highlight the role of Lie algebras and noncommutativity in the design of a compensating pulse sequence. Finally, we investigate three common dispersions in NMR spectroscopy, namely the Larmor dispersion, rf inhomogeneity, and strength of couplings between the spins.Many applications in control of quantum systems involve controlling a large ensemble by using the same control field. In practice, the elements of the ensemble could show variation in the parameters that govern the dynamics of the system. For example, in magnetic resonance experiments, the spins of an ensemble may have large dispersion in their natural frequencies ͑Larmor dispersion͒, strength of applied rf field ͑rf inhomogeneity͒, and the relaxation rates of the spins ͓1͔. In solid-state NMR spectroscopy of powders, the random distribution of orientations of internuclear vectors of coupled spins within an ensemble leads to a distribution of coupling strengths ͓12͔. A canonical problem in control of quantum ensembles is to develop external excitations that can simultaneously steer the ensemble of systems with variation in their internal parameters from an initial state to a desired final state. These are called compensating pulse sequences as they can compensate for the dispersion in the system dynamics. From the standpoint of mathematical control theory, the challenge is to simultaneously steer a continuum of systems between points of interest with the same control signal. Typical applications are the design of excitation and inversion pulses in NMR spectroscopy in the presence of Larmor dispersion and rf inhomogeneity ͓1-10͔ or the transfer of coherence or polarization in a coupled spin ensemble with variations in the coupling strengths ͓13͔. In many cases of practical interest, one wants to find a control field that prepares the final state as some desired function of the parameter, for example slice selective excitation and inversion pulses in magnetic resonance imaging ͓14-17͔. The problem of designing excitations that can compensate for dispersion in the dynamics is a well studied subject in NMR spectroscopy, and extensive literature exists on the subject of composite pulses that correct for dispersion in system dynamics ͓1-7͔. Composite pulses have recently been used in quantum information processing to correct for systematic errors in single-and two-qubit operations ͓18-23͔. The focus of this paper is not to construct a new compensating pulse sequence but rather to highlight the aspects of system dynamics that make such a compensation possible and give proofs of the existence of a compensating pulse sequence. Our...

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

“…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.…”

“…Not all situations have favorable outcomes. For instance, using same arguments as in Remark page 030302-2 of [26], it is possible to show that for the controlled Hamiltonian σ z + ασ y + u(t)σ x the unknown perturbation α ∈]α * , α * [ cannot always be compensated. Indeed, the attainable propagators are of the form…”

“…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.…”

“…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.…”

High-fidelity quantum gates in the presence of dispersion

“…These numerical approaches are very powerful because the optimization algorithms they use are mostly platform independent and easily extendable to account for constraints such as those of ensemble control. Optimal control theory has been applied to the problem of robust or ensemble control in many different contexts like NMR [24][25][26][27], many-body entanglement [28], spin-chains [29], and spin systems [30]. The remainder of this paper is a follows.…”