Deterministic carving of quantum states with Grover's algorithm*

We show that iteration of a few ( 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.