Why is only an x–z circle shown instead of a full 3D sphere?

The circle is a guide in the u_x–u_z plane. For **Δ = 0**, ω is along y and the trajectory lies entirely in that plane, so the dot follows a true great circle. For **Δ ≠ 0**, u_y is generally nonzero and the plotted point is a projection; the text readout still shows the full vector.

Bloch Sphere & Rabi Drive

A pure two-level state maps to a unit vector u on the Bloch sphere. The simulator integrates du/dt = u × ω with ω = (0, Ω, Δ), matching the usual rotating-frame picture of a σ_y-like drive (nutation Ω) plus σ_z detuning (Δ). The canvas plots (u_x, u_z) with a reference circle; when Δ = 0, u stays in the x–z plane and you see clean Rabi flopping between |0⟩ (north pole) and |1⟩. The side panel lists all three components.

Who it's for: Students learning two-level systems, Rabi oscillations, and the Bloch picture.

Key terms

Bloch Sphere
Rabi Frequency
Detuning
Two-Level System
Pauli Matrices

How it works

Two-level Bloch-vector dynamics with du/dt = u × ω and ω = (0, Ω, Δ): σ_y-like drive and σ_z detuning in a rotating-frame cartoon. The u_x–u_z plot shows Rabi flopping when Δ = 0.

Frequently asked questions

Why is only an x–z circle shown instead of a full 3D sphere?: The circle is a guide in the u_x–u_z plane. For Δ = 0, ω is along y and the trajectory lies entirely in that plane, so the dot follows a true great circle. For Δ ≠ 0, u_y is generally nonzero and the plotted point is a projection; the text readout still shows the full vector.

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