Statement
The two-level result. Consider the ensemble of qubits
$$i\,\frac{d}{dt}M(\alpha,\delta,t) = \big((E+\alpha)\,\sigma_z + \delta\,u(t)\,\sigma_x\big)\,M(\alpha,\delta,t), \qquad M(\alpha,\delta,0) = I_2,$$
with \(E > 0\), an unknown detuning \(\alpha \in [v_0, v_1]\) (\(v_0 < 0 < v_1\)), an unknown coupling strength \(\delta \in [\delta_m, \delta_M]\) (\(0 < \delta_m \le \delta_M\)), and one real control \(u \in L^\infty([0,T],[-K,K])\), the same for every member of the ensemble. Robin, Augier, Boscain and Sigalotti proved that if \(3(E + v_0) > E + v_1\), then for every continuous target \(M_F\) from the parameter set to \(SU(2)\), every \(\varepsilon > 0\) and every \(K > 0\) there are \(T\) and \(u\) with \(\max_{\alpha,\delta}\lVert M(\alpha,\delta,T) - M_F(\alpha,\delta)\rVert < \varepsilon\) — approximate ensemble controllability in \(C^0\) [1].
Question. Consider a family of \(N\)-level systems, \(N \ge 3\),
$$i\,\frac{d}{dt}M(\theta,t) = \big(H_0(\theta) + u(t)\,H_1(\theta)\big)\,M(\theta,t), \qquad M(\theta,0) = I_N,$$
with parameters \(\theta\) in a compact set \(K\) containing a continuum of values (for example, a dispersion of the transition frequencies) and a single real control \(u\). Under which conditions on \(H_0(\theta)\) and \(H_1(\theta)\) is this ensemble approximately controllable in \(C^0(K, SU(N))\)? In particular, do multilevel analogues of the qubit result hold?
Why it matters
Ensemble controllability is the mathematical basis of robust control: one pulse that works for every value of an unknown or inhomogeneous parameter, without feedback. Most experiments drive a system with a single real field, but the classical ensemble results use two independent controls (the two quadratures of the field), and the step from two controls to one is not innocent: the rotating-wave approximation that turns one real field into two effective controls interacts with the adiabatic approximation that gives robustness [1, 2]. Multilevel systems — atoms with several relevant levels, qudits, molecules — are where robust pulses are increasingly needed.
Known results
Two controls. For the Bloch equations with dispersion in the Larmor frequency and in the field amplitude, ensembles are approximately controllable with two controls [3, 4]. Controllability questions for Bloch ensembles and other systems with continuous spectrum were also analysed by Beauchard, Coron and Rouchon [5].
Finitely many parameter values. If the parameter takes only finitely many values, simultaneous controllability of systems on \(SU(N)\) has been characterized by Lie-algebraic conditions (see the review in [2]). The difficulty of the problem above is the continuum of values.
General Lie-algebraic results. Lie-algebraic methods give ensemble controllability for classes of control-affine systems [6], and every semisimple Lie algebra admits parameter-dependent control Hamiltonians for which approximate ensemble controllability holds — but in general with many controls [7, 2].
Adiabatic methods. With two controls, robustness can be traced to conical intersections of the eigenvalues, which an adiabatic path can follow for every value of the parameter [8].
One control, one qubit. The theorem above is obtained by justifying rigorously the cascade of the rotating-wave and adiabatic approximations with high-order averaging; the condition \(3(E + v_0) > E + v_1\) keeps characteristic frequencies from overlapping [1].
One control, \(n\) levels: population transfer. A different cascade of rotating-wave and adiabatic approximations gives ensemble control of \(n\)-level systems with a scalar input that is robust to dispersion in the coupling strengths, but not in the transition frequencies [9]. With dispersion in the frequencies, Liang, Boscain and Sigalotti prove that one real control can invert the population between any two eigenstates \(e_p\) and \(e_q\) for the whole continuum, provided the gap \(\lambda_q - \lambda_p\) stays in a window \((v_0, v_1)\) that no other gap of the system enters, and the coupling between the two levels does not change sign. They give the control explicitly — a slowly chirped, slowly modulated sinusoid — and test numerically how sharp the conditions are [10].
Status
Partial progress. For \(N \ge 3\), a single control is now known to achieve robust population transfer between eigenstates, under conditions that keep the transition frequencies apart [10]. Approximate ensemble controllability in \(C^0(K, SU(N))\) — reaching every family of target unitaries — is not established for \(N \ge 3\), and the authors list weaker conditions and further controllability results (such as population splitting) as directions for future work.