Two electrons confined to a gate-defined double quantum dot form a two-level "singlet-triplet" spin qubit spanned by the spin singlet |S⟩ and the ms=0 triplet |T0⟩. Diagonalizing the 2×2 charge Hamiltonian in the basis {S(1,1), S(0,2)} gives the exchange splitting:
J(ε) = √((ε/2)² + 2t²) − ε/2
This 2D build shows that Hamiltonian two ways at once. The left panel is the anticrossing energy diagram every quantum-dot paper actually plots: the flat |T0⟩ line (untouched by charge, since Pauli exclusion forbids the triplet from ever occupying (0,2)) against the lower hybridized-singlet branch ES−(ε) = ε/2 − √((ε/2)²+2t²), which works out to exactly −J(ε) — so the vertical gap between the two curves at the current ε is exactly J(ε), the same quantity driving the precession on the right. (A faint upper branch ES+ is drawn too, purely to show the avoided-crossing shape; it plays no role in the qubit's own dynamics.) This is the same anticrossing that the 3D version's rotating "electron cloud" was a stand-in for.
The right panel is a meridian projection of the qubit's full 3D Bloch vector onto the (bx, bz) plane. The qubit obeys dS/dt = Ω × S with Ω = (ΔBz, 0, J)/ħ, computed with the exact same Rodrigues rotation used by the 3D version — this build tracks all three Bloch components internally, it just draws two of them. When ΔBz = 0 the rotation axis is pure z, so a state prepared at a pole is an eigenvector of the rotation and stays exactly fixed — a genuine special case, reproduced exactly. Once ΔBz ≠ 0 the tilted axis carries the state out of the x-z plane into a precession cone (the two-axis control mechanism itself), which is why the arrow tip visibly shrinks inside the unit circle — that shrinking is physically meaningful, not projection noise: it shows the state has acquired a by component. P(singlet) stays exact regardless, since it depends only on bz = 2·P(|S⟩) − 1.
- Detuning ε and tunnel coupling t — set J(ε) via the anticrossing formula; large negative ε pushes the singlet toward (0,2) and opens a large exchange splitting (wider gap on the left panel).
- Field gradient ΔBz — tilts the precession axis (dashed line) off the vertical S/T₀ pole axis on the right panel, so exchange alone can no longer fully rotate the qubit — the two-axis control used for universal single-qubit gates (Petta et al., Science 2005; Foletti et al., Nat. Phys. 2009). Watch the arrow tip pull inward off the unit circle as you raise ΔBz — that's the state tipping into the third (by) dimension.
- Prepare |S⟩ / |T₀⟩ — resets the Bloch vector to a pole; with ΔBz=0 it stays there, otherwise it immediately starts precessing off-axis.