A complete classification of operator controllability for closed three-level quantum systems driven by a single control

Abstract

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 $\mathfrak{su}(N)$ 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.

The paper asks when a closed three-level system driven by one real coherent control, $H+f(t)V$, can implement every unitary up to global phase. It gives necessary and sufficient conditions in terms of Hamiltonian matrix elements, covering energy and transition-frequency degeneracies, two or three allowed transitions, and arbitrary control-induced diagonal shifts. For a connected transition graph, systems with distinct nonzero transition frequencies are controllable. With three distinct equally spaced energies, three allowed transitions always give controllability. With only the two adjacent transitions, controllability holds exactly when their coupling magnitudes differ or the diagonal shifts are not equally spaced. For two degenerate energies, a two-transition system is controllable unless both transitions connect the degenerate pair to the third level and the pair has equal diagonal shifts. With three transitions and $E_i=E_j$, noncontrollability occurs exactly when $r=v_{ij}/(v_{ik}v_{kj})$ is real and $r(|v_{kj}|^2-|v_{ik}|^2)=v_j-v_i$. Disconnected graphs and scalar drift Hamiltonians are uncontrollable. The proofs use dynamical Lie-algebra generation, explicit commutators, and invertible coefficient matrices. Two general sufficient results establish preservation of controllability under specified additions of diagonal shifts, and generation of $\mathfrak{su}(N)$ from a connected graph of available basis elements with both quadratures on at least one edge. Counterexamples show why the shift-preservation assumptions cannot simply be dropped. The complete classification is restricted to closed three-level systems with a single control; admissible controls are assumed to include piecewise-constant functions. It concerns reachability, not control synthesis or implementation resources.