Mølmer–Sørensen Gate: Phonon-Bus Entanglement

Trapped-ion quantum processors entangle qubits without ever letting them touch: two ions in a linear trap share one collective vibration, and a pair of off-resonant laser tones pushes that shared "phonon bus" around a loop in phase space whose direction depends on the ions' spins. This simulator renders the trap, the two ions and their laser drive in 3D alongside a live phase-space trace of α(t), so you can watch the loop open, sweep out a geometric phase, and close again — the exact mechanism behind the Mølmer–Sørensen gate used in real trapped-ion hardware (IonQ, Quantinuum). Tune the sideband detuning and coupling strength, flip the spin pair to mirror the loop, and track the accumulated entangling phase against the π/4 target for a maximally entangling two-qubit gate.