Derivation — C011 (identification of $H_A$ as a bichromatic MS drive)

Standard bichromatic MS Hamiltonian

A single laser tone resonant with a carrier $|\!\downarrow\rangle\!\leftrightarrow\!|e\rangle$ transition, in the Lamb-Dicke regime, couples to a motional mode of frequency $\omega_m$ with strength scaled by the Lamb-Dicke parameter $\eta$. The Molmer-Sorensen drive uses two tones symmetrically detuned by $\pm\delta$ from the red and blue motional sidebands of that transition. In the interaction picture with respect to the motional mode and the carrier, and to first order in $\eta$ (Lamb-Dicke expansion), the two sideband tones produce $$H_A=\tfrac12\hbar\eta\Omega\,\sigma_x^{(de)}\big(a\,e^{i\delta t}+a^\dagger e^{-i\delta t}\big)$$ per ion, where $\sigma_x^{(de)}=|\!\downarrow\rangle\langle e|+|e\rangle\langle\downarrow|$ is the Pauli-$x$ on the $\{|\!\downarrow\rangle,|e\rangle\}$ two-level subspace, $\Omega$ the carrier Rabi frequency, and $\delta$ the symmetric detuning. (This is the textbook MS form; see Sorensen-Molmer 1999 and Kim 2010.)

Collective operator $S_{x,e}$

For two ions sharing the same motional mode and equal Rabi frequencies, the single-ion $\sigma_x^{(de)}$ add coherently into the collective operator $$S_{x,e}=\sigma_x^{(de)}\otimes\mathbb 1+\mathbb 1\otimes\sigma_x^{(de)},$$ which is exactly protocol.S_xe in the self-contained code. Hence $$H_A=\tfrac12\hbar\eta\Omega\,S_{x,e}\big(a\,e^{i\delta t}+a^\dagger e^{-i\delta t}\big),$$ the claimed form.

"Force whose phase depends on the eigenstate of $S_{x,e}$"

$H_A=\tfrac12\hbar\eta\Omega\,S_{x,e}\,(a e^{i\delta t}+a^\dagger e^{-i\delta t})$. $S_{x,e}$ commutes with the bosonic factor, so in any eigenbasis of $S_{x,e}$ with eigenvalue $s$ the motional part is $$H_A^{(s)}=\tfrac12\hbar\eta\Omega\,s\,(a e^{i\delta t}+a^\dagger e^{-i\delta t}),$$ a time-dependent linear drive (displacement force) on the oscillator whose sign/amplitude is set by the eigenvalue $s$ of $S_{x,e}$. Eigenvalues of $S_{x,e}$ (a sum of two $\sigma_x$, each with spectrum $\{+1,-1,0$ on the third level$\}$) are $\{+2,0,-2\}$ on the $\{\downarrow,e\}^{\otimes2}$ sector; each drives a coherent displacement of opposite phase. This is precisely the state-dependent force that the paper describes, and it is the mechanism that in C012/C013 generates the $\exp(i\Phi S_{x,e}^2)$ collective coupling.

Conclusion

$H_A$ as written is the standard bichromatic Molmer-Sorensen interaction in the Lamb-Dicke regime, restricted to the $\{|\!\downarrow\rangle,|e\rangle\}$ transition and promoted to the two-ion collective operator $S_{x,e}$. The structure matches the literature MS form term-by-term ($\tfrac12\hbar\eta\Omega$ prefactor, $S_{x,e}$ spin operator, $a e^{i\delta t}+a^\dagger e^{-i\delta t}$ bichromatic motional factor) and the state-dependent-force interpretation is correct. This is a definition/identification, confirmed structurally; the downstream propagator (C012) provides the quantitative corroboration.