Mølmer–Sørensen Gate: Phase-Space Loop (2D)

Gate Drive

Sideband detuning δ/ν 0.55

Coupling strength g 0.45

Simulation speed 1.0×

Two-Qubit Spin Pair

Live Readouts

Loop displacement |α(t)|

0.000

Loops closed

Entangling phase Θ

0.000 rad

Target Θ = π/4

0.785 rad

Estimated Bell-state fidelity F

0.0%

How it works

Two trapped ions share a common motional (phonon) mode of the linear chain, oscillating at frequency ν. Two laser tones, detuned symmetrically by ±δ from ν (a bichromatic "red + blue sideband" drive), exert a state-dependent optical force on each ion proportional to its spin along x, σ_x. The force displaces the shared phonon mode along a closed loop in phase space:

α(t) = (g/δ) (1 − e^(iδt)),   s = s₁·s₂ = ±1
Displacement:  |α(t)| = (g/δ)·√(2 − 2cos δt)
Loop closes when δt = 2πn  →  |α| = 0

Because the force sign depends on the product of the two spins (s₁·s₂), each two-qubit basis state traces a loop of the same radius but opposite handedness. When the loop returns exactly to the origin, the qubits are fully disentangled from the shared motion — but they have picked up a spin-dependent geometric phase equal to the area swept out:

Θ(t) = s · (g/δ)² · (δt − sin δt)
Unitary:  U(t) ≈ exp[ i Θ(t) σx⊗σx ]

A maximally entangling gate — turning |00⟩ into a Bell state — is reached when Θ = π/4 at a moment where the loop has closed. Away from closure, residual spin–motion correlation reduces the achievable fidelity; the readout above combines both effects as a simplified pedagogical estimate, F ≈ e^−|α|²·cos²(2(Θ−π/4)), not a full open-system calculation.

δ/ν — sideband detuning; sets the loop's angular speed and the gate period T = 2π/δ.
g — coupling (laser strength × Lamb-Dicke factor); sets the loop radius g/δ.
Spin pair — flips the sign of s₁·s₂, mirroring the loop and the sign of Θ.

This bichromatic phonon-bus scheme is the entangling gate used in real trapped-ion quantum processors (IonQ, Quantinuum, university ion-trap groups) — the mechanism national and corporate quantum-computing programs are racing to scale to more qubits. This 2D view renders the same closed-form α(t) and Θ(t) as the 3D version, laid flat: a top-down ion-chain schematic above a full phase-space plot you can pan and zoom.

Ion (spin ↑) Ion (spin ↓) Phase-space trace α(t)

Gate period T = 2π/δ ≈ 11.4

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