Neutral atom entangling gate in the ultrastrong coupling regime

We propose a method to deterministically entangle qubits or ensembles of qubits interacting with a shared bosonic mode in the ultrastrong coupling regime. We show that the resulting gate is a product of two unitaries: one unitary acts only on the quantum state of the qubits and entangles them, while the other acts only on the quantum state of the boson, producing a phase shift. We find that the gate time is inversely proportional to the qubit-boson interaction strength, and by tuning the qubit-boson interaction strength, one can prepare a maximally entangled state or a squeezed state. Applying the quantum gate to multiple qubit ensembles, we show that the quantum gate prepares a Schr\"odinger cat state. We also examine imperfections such as including free evolution of the qubits, and show that this produces an effective mixing. Our proposal is feasible for ultrastrong coupling experiments.