Deterministic carving of quantum states with Grover's algorithm

We show that iteration of a few ( $\sim N^{1/4}$) unitary steps of Grover's algorithm suffices to perfectly prepare a Dicke state of $N$ atoms in a cavity. We also show that a few subsequent Grover steps can be employed to generate GHZ and Cat states. The Grover iteration is physically realized by global qubit rotations and by the phase shift of single photons reflected on the cavity. Our protocols are deterministic and require no individual addressing of the atoms. A detailed error analysis accounting for spatial mode matching of the photon to the cavity, spontaneous emission, mirror scattering, and the finite bandwidth of the photon mode is used to predict the fidelity of the prepared states as a function of system parameters and atom-cavity cooperativity. The fidelity can be increased by heralding on detection of the reflected photon.