$$H_{SB} = \hbar g\big(\sigma_+^S \sigma_-^B + \sigma_-^S \sigma_+^B\big)$$

Equation 79 · 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.

A single exactly solvable case makes the ledger’s behavior concrete, and it is worth being explicit that what follows is an analytic evaluation of a stated model, not a simulation and not a measurement. Let S and B each be a two-level system — a spin, in the sense this literature uses the word for any two-state quantum degree of freedom — with bare Hamiltonians $H_S$ = ($\hbar$$\omega$/2)$\sigma_z^S$ and $H_B$ = ($\hbar$$\omega$/2)$\sigma_z^B$ tuned to the same frequency $\omega$ , coupled by an excitation-conserving exchange term, $H_{SB} = \hbar g\big(\sigma_+^S \sigma_-^B + \sigma_-^S \sigma_+^B\big)$. Because $H_{SB}$ conserves total excitation number, the dynamics starting from a single excitation stays confined to the two-dimensional subspace spanned by…

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