fraction

The passage around this formula

the standard two-level exchange solution, exact for all t under this Hamiltonian. From it, $\langle$ $H_S$ $\rangle(t)$ = ($\hbar$$\omega$/2)[$\cos^2(gt)$ - $\sin^2(gt)$] = ($\hbar$$\omega$/2)$\cos(2gt)$ , while $\langle$ $H_f$ $\rangle$ is exactly time-independent — a general fact for any state evolving under its own generator, not special to this one — and equal here to $\langle$ e,g|$H_f$|e,g$\rangle$ = $\hbar$$\omega$/2 - $\hbar$$\omega$/2 + 0 = 0 , since $H_{SB}$ is purely off-diagonal in this basis. The ledger reading follows immediately: $\Delta E_\Lambda(t) = \langle H_f\rangle - \langle H_S\rangle(t) = -\frac{\hbar\omega}{2}\cos(2gt)$. At t=0 this equals -$\hbar$$\omega$/2 , exactly the bath’s own ground-state energy, matching the uncorrelated-bath limit derived above. At the swap time $t^\ast$ = $\pi$/(2g) ,…

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