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arXiv:quant-ph/0102017 · Journal of Physics A Mathematical and General

Complete controllability of finite-level quantum systems

Hongyan Fu, S. G. Schirmer, A. I. Solomon

Published Feb 19, 2001• 88 citations
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Abstract

Complete controllability is a fundamental issue in the field of control of quantum systems, not least because of its implications for dynamical realizability of the kinematical bounds on the optimization of observables. In this paper we investigate the question of complete controllability for finite-level quantum systems subject to a single control field, for which the interaction is of dipole form. Sufficient criteria for complete controllability of a wide range of finite-level quantum systems are established and the question of limits of complete controllability is addressed. Finally, the results are applied to give a classification of complete controllability for four-level systems.

DOI
10.1088/0305-4470/34/8/313

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Complete controllability is a fundamental issue in the field of control of quantum systems, not least because of its implications for dynamical realizability of the kinematical bounds on the optimization of observables. In this paper we investigate the question of complete controllability for finite-level quantum systems subject to a single control field, for which the interaction is of dipole form. Sufficient criteria for complete controllability of a wide range of finite-level quantum systems are established and the question of limits of complete controllability is addressed. Finally, the results are applied to give a classification of complete controllability for four-level systems.
Sufficient conditions for complete controllability of N-level quantum systems subject to a single control pulse that addresses multiple allowed transitions concurrently are established. The results are applied in particular to Morse and harmonic-oscillator systems, as well as some systems with degenerate energy levels. Controllability of these model systems is of special interest since they have many applications in physics, e.g., Morse and harmonic oscillators serve as models for molecular bonds, and the standard control approach of using a sequence of frequency-selective pulses to address a single transition at a time is either not applicable or only of limited utility for such systems.
In this work, we study operator controllability of finite-dimensional closed quantum systems driven by a single coherent control. Two general Lie-algebraic results on controllability of such systems are proved. The first gives sufficient conditions under which controllability of the system is preserved when controlled energy-level shifts are added. The second gives a sufficient basis-element condition under which the dynamical Lie algebra contains su(N)\mathfrak{su}(N)su(N) and the system is therefore operator controllable. These results, together with known controllability conditions, are applied to obtain a complete classification of operator controllability of all three-level quantum systems in terms of their free and interaction Hamiltonian matrix elements. This classification covers all combinations of energy-level and level-spacing degeneracies, two or three allowed transitions, as well as the presence or absence of controlled energy-level shifts, and gives necessary and sufficient conditions for operator controllability in each case.
Complete controllability of finite-dimensional quantum systems with energy level degeneracy is investigated using two different approaches. One approach is to apply a weak constant field to eliminate the degeneracy and then control it using techniques developed for non-degenerate quantum systems. Conditions for the elimination of degeneracy are found and the issues of influence of relaxation time of a constant external field on the target state are addressed through the fidelity. Another approach is to control the degenerate system by a single control field directly. It is found that the system with twofold degenerate excited states and non-degenerate ground state is completely controllable except for the two-level system. Conditions of complete controllability are found for both systems with different energy gaps and with equal energy gaps.
In this paper we investigate the limits of control for mixed-state quantum systems. The constraint of unitary evolution for nondissipative quantum systems imposes kinematical bounds on the optimization of arbitrary observables. We summarize our previous results on kinematical bounds and show that these bounds are dynamically realizable for completely controllable systems. Moreover, we establish improved bounds for certain partially controllable systems. Finally, the question of dynamical realizability of the bounds for arbitrary partially controllable systems is shown to depend on the accessible sets of the associated control system on the unitary group U(N)U(N)U(N) and the results of a few control computations are discussed briefly.
We investigate the problem of control of quantum systems. Various notions of controllability of quantum systems are discussed and Lie algebraic techniques are employed to derive necessary and sufficient conditions for the resulting degrees of controllability. The results are employed to study the degree of controllability for various model systems. The problem of optimal control is formulated as a problem of maximisation of the expectation value of an observable of the system subject to dynamical constraints and costs. A class of iterative algorithms is discussed, and shown to converge to a solution of the Euler-Lagrange equations, which is a necessary condition for optimality of a solution. A numerical realisation of the algorithm is presented and potential implementation of the resulting control fields in the laboratory is discussed. The algorithm is applied to find optimal control fields for a variety of applications including selective and simultaneous excitation of individual quantum dots in a globally addressed ensemble, creation of superposition states, and selective excitation of certain vibrational modes of a simple molecule such as Hydrogen Fluoride. In each case we systematically explore the parameter space of the algorithm to select solutions with desirable physical features from the multitude of possible choices. As there are many possible optimal control fields which give similar yields the question of whether some are more robust than others is investigated. A number of optimal fields are analysed quantitatively in terms of robustness with regard to temporal and spectral noise affecting the fields, uncertainty in the model parameters and the effect of experimental limitations such as restrictions on the bandwidth of the fields. It is shown that whilst the solutions tend to be sensitive to uncertainty in the energy levels chosen for the model, they appear to be surprisingly tolerant to even severe bandwidth limitations.
The problem of identifying the dynamical Lie algebras of finite-level quantum systems subject to external control is considered, with special emphasis on systems that are not completely controllable. In particular, it is shown that the dynamical Lie algebra for an N -level system with symmetrically coupled transitions, such as a system with equally spaced energy levels and uniform transition dipole moments, is a subalgebra of so ( N ) if N = 2ℓ + 1, and a subalgebra of sp(ℓ) if N = 2ℓ. General criteria for obtaining either so (2ℓ + 1) or sp (ℓ) are established.