Dicke-state preparation through global transverse control of Ising-coupled qubits

We consider the problem of engineering the two-excitation Dicke state $\ket{D^{3}_{2}}$ in a three-qubit system with long-ranged (all-to-all) Ising-type qubit-qubit interaction, which is also subject to global transverse (Zeeman-type) control fields. The theoretical underpinning for our envisioned state-preparation scheme, in which $|000\rangle$ is adopted as the initial state of the system, is provided by a Lie-algebraic result that guarantees state-to-state controllability of this system for an arbitrary choice of initial- and final states that are invariant with respect to permutations of qubits. This state-preparation scheme is envisaged in the form of a pulse sequence that involves three instantaneous ($\delta$-shaped) control pulses, which are equivalent to global qubit rotations, and two Ising-interaction pulses of finite durations between consecutive control pulses. The design of this pulse sequence -- whose total duration is $T\approx 0.95\:\hbar/J$, where $J$ is the Ising-coupling strength -- leans heavily on the concept of the symmetric sector -- a four-dimensional, permutationally-invariant subspace of the three-qubit Hilbert space. We demonstrate the feasibility of the proposed state-preparation scheme by carrying out a detailed numerical analysis of its robustness to systematic errors, i.e. deviations from the optimal values of the eight parameters that characterize the underlying pulse sequence.