Heisenberg Exchange in Quantum Dots

Once charge fluctuations are integrated out, neighboring quantum-dot spin qubits reduce to the exchange Hamiltonian

$H_{\mathrm{ex}}(t)=J(t){\vec{S}}_1\cdot {\vec{S}}_2.$

This note is intentionally the Hamiltonian-first companion to exchange-interaction-in-quantum-dots. Use it when the question is what rotation an exchange pulse implements, how encoded spin manifolds inherit logical axes, or why $\sqrt{\text{SWAP}}$ keeps reappearing. For barrier-vs-detuning control, Hubbard-model intuition, and charge-noise tradeoffs, go back to the companion note.

Two-spin algebra

Exchange conserves total spin and splits the singlet from the triplet manifold by an amount set by $J$. That symmetry makes the interaction simultaneously useful and constrained: a pulse changes the relative phase between singlet and triplet sectors, so exchange alone does not give arbitrary two-qubit control, but it naturally generates the SWAP / sqrt-swap-as-universal-gate family once combined with local phases.

The entangling angle is set by the pulse area $\int J(t)dt$. Short pulses give partial swaps; the canonical half-swap pulse is the native entangler in the Loss-DiVincenzo picture.

Encoded-spin projections

• loss-divincenzo-qubit / spin-qubit: exchange is the direct nearest-neighbor entangling resource.
• singlet-triplet-qubit: after projection into the $\{|S⟩,|T_0⟩\}$ manifold, exchange acts primarily as the logical energy splitting between the encoded basis states.
• exchange-only-qubit / rx-qubit / aeon-qubit: pairwise terms like $J_{12}{\mathbf{S}}_1\cdot {\mathbf{S}}_2+J_{23}{\mathbf{S}}_2\cdot {\mathbf{S}}_3$ become non-collinear logical axes, always-on splittings, or resonant drive matrix elements inside a three-spin encoded subspace.

Scope boundary

• Keep microscopic origin, barrier-vs-detuning tuning, and materials-specific charge sensitivity in exchange-interaction-in-quantum-dots.
• Keep operating-point protection in charge-noise-sweet-spot.
• Use this note as the compact algebraic bridge between those device stories and gate-level control.

Key relationships

• sqrt-swap-as-universal-gate — gate-level consequence of exchange pulses
• singlet-triplet-qubit — logical $Z$ splitting from exchange in the $m_S=0$ subspace
• exchange-only-qubit — universal control from non-collinear exchange axes in a three-spin encoding
• rx-qubit — always-on exchange operated in a resonantly driven regime
• aeon-qubit — always-on exchange at a double sweet spot
• exchange-interaction-in-quantum-dots — microscopic origin and tuning knobs for $J$

References

• loss-divincenzo-1998-quantum-dots — direct two-spin exchange as a native entangler
• petta-2005-singlet-triplet — exchange control in the two-spin encoded manifold
• divincenzo-2000-exchange-only — three-spin encoded control built from pairwise exchange
• shim-2016-aeon — always-on exchange at a double sweet spot