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∣ψ(t)⟩=cos⁡(gt) ∣e,g⟩−isin⁡(gt) ∣g,e⟩|\psi(t)\rangle = \cos(gt)\,|e,g\rangle - i\sin(gt)\,|g,e\rangle

Why this formula appears here

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 , ∣ψ(t)⟩=cos⁡(gt) ∣e,g⟩−isin⁡(gt) ∣g,e⟩|\psi(t)\rangle = \cos(gt)\,|e,g\rangle - i\sin(gt)\,|g,e\rangle. the standard two-level exchange solution, exact for all t under this Hamiltonian. From it, ⟨\langle HSH_S ⟩(t)\rangle(t) = (ℏ\hbarω\omega/2)[cos⁡2(gt)\cos^2(gt) -…

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∣ψ(t)⟩=cos⁡(gt) ∣e,g⟩−isin⁡(gt) ∣g,e⟩,|\psi(t)\rangle = \cos(gt)\,|e,g\rangle - i\sin(gt)\,|g,e\rangle,

Equation 86 · Evolutionary Physics

The Entry a Relabeling Cannot Write

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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 , ∣ψ(t)⟩=cos⁡(gt) ∣e,g⟩−isin⁡(gt) ∣g,e⟩|\psi(t)\rangle = \cos(gt)\,|e,g\rangle - i\sin(gt)\,|g,e\rangle. the standard two-level exchange solution, exact for all t under this Hamiltonian. From it, ⟨\langle HSH_S ⟩(t)\rangle(t) = (ℏ\hbarω\omega/2)[cos⁡2(gt)\cos^2(gt) -…

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