Controllability of quantum mechanical systems by root space decomposition of su(<i>N</i>)

Altafini, Claudio

doi:10.1063/1.1467611

Cited by 165 publications

(200 citation statements)

References 20 publications

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“…The proof of Proposition 1 was essentially relying on the Jurdjevic-Quinn sufficient condition for stabilizability [18], opportunely modified in order to deal with skew-symmetric infinitesimal generators. The key tool is the so-called "ad-condition", i.e., a particular type of Lie brackets often used for testing controllability, see [4]. The following proposition shows that this condition is never satisfied by the system (9).…”

Section: Propositionmentioning

confidence: 99%

Feedback Control of Spin Systems

Altafini

2006

Quantum Inf Process

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The feedback stabilization problem for ensembles of coupled spin 1/2 systems is discussed from a control theoretic perspective. The noninvasive nature of the bulk measurement allows for a fully unitary and deterministic closed loop. The Lyapunov-based feedback design presented does not require spins that are selectively addressable. With this method, it is possible to obtain control inputs also for difficult tasks, like suppressing undesired couplings in identical spin systems.

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Section: Propositionmentioning

confidence: 99%

Feedback Control of Spin Systems

Altafini

2006

Quantum Inf Process

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“…This condition is equivalent to the requirement that the Lie algebra generated from H 0 and µ forms a complete set of operators [32] and T is large enough to avoid hindering the dynamics. In general, we may assume controllability of an arbitrary quantum system, as uncontrollable quantum systems have been shown to constitute a null set in the space of Hamiltonians [62]. Upon satisfaction of the controllability requirement, analysis of the global control landscape topology of Eq.…”

Section: Formulation Of the Control Objectivementioning

confidence: 99%

Exploring quantum control landscapes: Topology, features, and optimization scaling

2011

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Quantum optimal control experiments and simulations have successfully manipulated the dynamics of systems ranging from atoms to biomolecules. Surprisingly, these collective works indicate that the effort (i.e., the number of algorithmic iterations) required to find an optimal control field appears to be essentially invariant to the complexity of the system. The present work explores this matter in a series of systematic optimizations of the state-to-state transition probability on model quantum systems with the number of states N ranging from 5 through 100. The optimizations occur over a landscape defined by the transition probability as a function of the control field. Previous theoretical studies on the topology of quantum control landscapes established that they should be free of sub-optimal traps under reasonable physical conditions. The simulations in this work include nearly 5000 individual optimization test cases, all of which confirm this prediction by fully achieving optimal population transfer of at least 99.9% upon careful attention to numerical procedures to ensure that the controls are free of constraints. Collectively, the simulation results additionally show invariance of required search effort to system dimension N . This behavior is rationalized in terms of the structural features of the underlying control landscape. The very attractive observed scaling with system complexity may be understood by considering the distance traveled on the control landscape during a search and the magnitude of the control landscape slope. Exceptions to this favorable scaling behavior can arise when the initial control field fluence is too large or when the target final state recedes from the initial state as N increases.

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“…(2) and each of the control Hamiltonians are a closed Lie group SU(2 N +1 ) [38] [11,12]. An equivalent diagrammatic representation relies on graph connectivity for assessing the controllability of quantum systems represented as Lie algebras [13,14].…”

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Universal control of nuclear spins via anisotropic hyperfine interactions

et al. 2008

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We show that nuclear spin subsystems can be completely controlled via microwave irradiation of resolved anisotropic hyperfine interactions with a nearby electron spin. Such indirect addressing of the nuclear spins via coupling to an electron allows us to create nuclear spin gates whose operational time is significantly faster than conventional direct addressing methods. We experimentally demonstrate the feasibility of this method on a solid-state ensemble system consisting of one electron and one nuclear spin.Coherent control of quantum systems promises optimal computation [1], secure communication [2], and new insight into the fundamental physics of many-body problems [3]. Solid-state proposals [4,5,6,7,8] for such quantum information processors employ isolated spin degrees of freedom which provide Hilbert spaces with long coherence times. Here we show how to exploit a local, isolated electron spin to coherently control nuclear spins. Moreover, we suggest that this approach provides a fast and reliable means of controlling nuclear spins and enables the electron spins of such solid-state systems to be used for state preparation and readout [9] of nuclear spin states, and additionally as a spin actuator for mediating nuclear-nuclear spin gates.Model System. The spin Hamiltonian of a single local electron spin with angular momentum, S = 1 2 and N nuclear spins, each with angular momentum I k = 1 2 , in the presence of a magnetic field B is [10]:Here β e is the Bohr magneton, γ k n is the gyromagnetic ratio;Ŝ andÎ k are the spin-1 2 operators. The secondrank tensors g, A k , δ, and D kl represent the electron gfactor, the hyperfine interaction, the chemical shift, and the nuclear dipole-dipole interaction respectively.In the regime where the static magnetic field B = B 0ẑ provides a good quantization axis for the electron spin, the Hamiltonian can be simplified by dropping the nonsecular terms which corresponds to keeping only electron interactions involving S z . The quantization axis of any nuclear spin depends on the magnitudes of the hyperfine interaction and the main magnetic field, as well as their relative orientations. When these two fields are comparable in magnitude [37] H 0 can be approximated by: (2) with N=1. The electron spin state is in an eigenstate of purely the Zeeman interaction, while the nuclear spin state is not an eigenfunction of the Zeeman interaction alone due to the anisotropic hyperfine interaction. Because α0|β1 = 0 and α0|β0 = 0 the electron spin operator (Ŝx) has finite probabilities between all levels (dashed arrows). This allows for universal control of the entire spin system. The filled and unfilled circles represent the relative spin state populations of the ensemble at thermal equilibrium. In our experimental setup the energy differences are ω12/2π = 7.8 MHz, ω34/2π = 40 MHz, ω14/2π = 12.005 GHz, ω23/2π = 11.954GHzThe nuclear dipole-dipole interaction is neglected as it is typically 10 2 times weaker than the hyperfine terms.As described in Figure 1 (N=1), the nuclear spin is qu...

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Controllability of quantum mechanical systems by root space decomposition of su(N)

Cited by 165 publications

References 20 publications

Feedback Control of Spin Systems

Feedback Control of Spin Systems

Exploring quantum control landscapes: Topology, features, and optimization scaling

Universal control of nuclear spins via anisotropic hyperfine interactions

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