Two trapped ions share a collective (center-of-mass) motional mode of frequency ν. A bichromatic laser field, detuned symmetrically by ±(ν+δ) from the two qubit-state carrier, applies a spin-dependent optical dipole force. In the interaction picture this gives the Mølmer–Sørensen Hamiltonian:
H = ħΩ_eff Ŝ_x (â e^(-iδt) + ↠e^(iδt))
Ŝ_x = σ_x^(1) + σ_x^(2) (collective spin operator)
This displaces the shared phonon mode along a spin-dependent circular trajectory in phase space:
α(t) = (Ω_eff/δ) (1 − e^(iδt))
⟨n(t)⟩ = |α(t)|² = 2(Ω_eff/δ)² (1 − cos δt)
Whenever δt = 2πk the loop closes — the ion returns to its initial motional state (⟨n⟩ → 0) with zero residual spin-motion entanglement, which is what makes the gate robust to thermal motional noise. While the loop is open, the state accumulates a purely geometric two-qubit phase equal to the area enclosed:
Θ(t) = (Ω_eff/δ)² (δt − sin δt)
|ψ(t)⟩ = cos Θ |gg⟩ − i sin Θ |ee⟩
Starting from both ions in |g⟩, a closed loop with Θ = π/4 leaves the qubits in the maximally entangled Bell state (|gg⟩ − i|ee⟩)/√2 — the entangling two-qubit gate used in real trapped-ion quantum computers (IonQ, Quantinuum, university groups worldwide).
- Ωeff — drive strength; sets the phase-space loop radius and how fast entanglement accumulates.
- δ — detuning from the motional sideband; sets how fast the loop is traced and where it closes.
- k — number of loops before the gate is stopped; more loops accumulate more phase at fixed Ωeff, δ.