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∣g,e⟩|g,e\rangle

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Because HSBH_{SB} conserves total excitation number, the dynamics starting from a single excitation stays confined to the two-dimensional subspace spanned by |e,g⟩\rangle (system excited, bath in its ground state) and |g,e⟩\rangle (the reverse). On resonance, both basis states carry the same bare energy, +ℏ\hbarω\omega/2 - ℏ\hbarω\omega/2 = 0 , so HfH_f restricted to this subspace is proportional to the swap operator between the two, and the Schrödinger equation solves in closed form. Starting from |ψ(0)\psi(0)⟩\rangle = |e,g⟩\rangle ,

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ee

Symbol e

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∣g,e⟩|g,e\rangle

Equation 82 · Evolutionary Physics

The Entry a Relabeling Cannot Write

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

Because HSBH_{SB} conserves total excitation number, the dynamics starting from a single excitation stays confined to the two-dimensional subspace spanned by |e,g⟩\rangle (system excited, bath in its ground state) and |g,e⟩\rangle (the reverse). On resonance, both basis states carry the same bare energy, +ℏ\hbarω\omega/2 - ℏ\hbarω\omega/2 = 0 , so HfH_f restricted to this subspace is proportional to the swap operator between the two, and the Schrödinger equation solves in closed form. Starting from |ψ(0)\psi(0)⟩\rangle = |e,g⟩\rangle ,

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∣g,e⟩|g,e\rangle

Equation 96 · Evolutionary Physics

The Entry a Relabeling Cannot Write

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

At t=0 this equals -ℏ\hbarω\omega/2 , exactly the bath’s own ground-state energy, matching the uncorrelated-bath limit derived above. At the swap time t∗t^\ast = π\pi/(2g) , the state is |g,e⟩\rangle up to an overall phase — the excitation has moved entirely into the bath — and Δ\Delta EΛ(t∗)E_\Lambda(t^\ast) = +ℏ\hbarω\omega/2 , exactly the bath’s excited-state energy with ⟨\langle HSBH_{SB}⟩\rangle = 0 at that instant, matching the closed form by direct substitution rather than by coincidence. A system-only observer watching only ⟨\langle HSH_S⟩(t)\rangle(t) across this half-period would see the tracked energy swing by a full ℏ\hbarω\omega with no term in HSH_S alone to explain it; the ledger accounts for every joule of…

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