Preparation of Dicke States in an Ion Chain

We have investigated theoretically and experimentally a method for preparing Dicke states in trapped atomic ions. We consider a linear chain of $N$ ion qubits that is prepared in a particular Fock state of motion, $|m>$. The $m$ phonons are removed by applying a laser pulse globally to the $N$ qubits, and converting the motional excitation to $m$ flipped spins. The global nature of this pulse ensures that the $m$ flipped spins are shared by all the target ions in a state that is a close approximation to the Dicke state $\D{N}{m}$. We calculate numerically the fidelity limits of the protocol and find small deviations from the ideal state for $m = 1$ and $m = 2$. We have demonstrated the basic features of this protocol by preparing the state $\D{2}{1}$ in two $^{25}$Mg$^+$ target ions trapped simultaneously with an $^{27}$Al$^+$ ancillary ion.