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⟩. Pushing charge between the (1,1) and (0,2) configurations with the interdot detuning ε hybridizes S(1,1) with S(0,2) — but Pauli exclusion forbids the triplet from ever occupying (0,2), so T0 stays fixed while S is pulled down. Diagonalizing the 2×2 charge Hamiltonian gives the exchange splitting:
H_charge = [ 0 √2 t ] (basis: S(1,1), S(0,2))
[ √2 t ε ]
J(ε) = √((ε/2)² + 2t²) − ε/2
In the {|S⟩,|T0⟩} basis the qubit obeys an effective Bloch equation with an external magnetic-field gradient ΔBz (from a micromagnet or nuclear polarization) providing a second, non-commuting axis:
H_qubit = (J/2) σ_z + (gμ_BΔB_z/2) σ_x
dS/dt = Ω × S , Ω = (ΔB_z, 0, J) / ħ
- Detuning ε and tunnel coupling t — set J(ε) via the charge anticrossing formula above; large negative ε pushes the singlet toward (0,2) and opens a large exchange splitting.
- Field gradient ΔBz — tilts the precession axis off the Bloch-sphere pole, so exchange alone can no longer fully rotate the qubit — exactly the mechanism used to build a universal single-qubit gate set (Petta et al., Science 2005; Foletti et al., Nat. Phys. 2009).
- Prepare |S⟩ / |T₀⟩ — reinitializes the Bloch vector to a pole; a real device does this via rapid adiabatic passage deep into (0,2), which projects onto the singlet ground state.
The rotating cloud on the left shows the two-electron charge distribution reshaping as ε sweeps the ground state from evenly split (1,1) toward doubly-occupied (0,2) — the same charge motion that makes J tunable electrically, with no oscillating magnetic field required.