Mølmer–Sørensen Gate — Phase-Space Diagram

Bichromatic Drive

Spin-dependent force Ω_eff/2π 40 kHz

Sideband detuning δ/2π 50 kHz

Gate Loops

Motional Mode

Phonon number ⟨n⟩

0.00

Loop phase δt

0.00 π

Two-Qubit State

Entangling phase Θ

0.00 π

Bell fidelity

0.0%

|ψ⟩ = 1.00|gg⟩ + 0.00i|ee⟩

How it works

This is a 2D phase-space rendering of the same Mølmer–Sørensen gate: instead of a 3D ion trap, the shared motional mode's coherent amplitude α(t) = (Ω_eff/δ)(1 − e^iδt) is plotted directly as a point moving in the (Re α, Im α) plane. It traces a circle of radius Ω_eff/δ through the origin — closing exactly when δt = 2πk.

α(t) = (Ω_eff/δ)(1 − e^(iδt))
⟨n(t)⟩ = |α(t)|²  (right ring gauge)
Θ(t) = (Ω_eff/δ)² (δt − sin δt)

The shaded region swept by the line from the origin to α(t) is tracked with a running shoelace-formula area integral; its magnitude doubled equals the analytic geometric phase Θ(t) exactly — a real numerical identity, not just an illustration. The population disk on the right plots the qubit amplitude vector (cos Θ, sin Θ): as Θ sweeps to π/4 the vector reaches 45°, the point where |gg⟩ and |ee⟩ become equally populated and the two ions are maximally entangled.

Ω_eff — 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 Ω_eff, δ.

Loop 1

Phase-space trajectory α(t) Swept geometric-phase area Population disk (|gg⟩ / |ee⟩)