C056 derivation

Claim

The generalized drive-(A) Hamiltonian, Eq. (S6), $$ H_A = \frac{\eta \hbar \Omega}{2} S_{e,\phi}\left(\hat a\, e^{i(\delta+\epsilon_m+\epsilon_q)t}e^{i\phi_m} + \hat a^\dagger\, e^{-i(\delta+\epsilon_m-\epsilon_q)t}e^{-i\phi_m}\right), $$ with $S_{e,\phi}=|e\rangle\langle\downarrow|\,e^{i\phi_s}+|\downarrow\rangle\langle e|\,e^{-i\phi_s}$ (collective: sum over two ions), reduces to the main-text $$ H_A^{\rm main} = \tfrac12 \hbar\eta\Omega\, S_{x,e}\left(\hat a\, e^{i\delta t} + \hat a^\dagger\, e^{-i\delta t}\right) $$ for $\phi_s=\phi_m=0,\ \epsilon_m=\epsilon_q=0$, and $\epsilon_q,\epsilon_m$ enter the sideband phases as written.

Spin part

At $\phi_s=0$, $S_{e,\phi}=|e\rangle\langle\downarrow|+|\downarrow\rangle\langle e|=\sigma_{x,e}$ on the $\{|\downarrow\rangle,|e\rangle\}$ subspace; summed over ions this is the collective $S_{x,e}$. Verified symbolically (matrix on $\{\downarrow,e\}$ equals $\sigma_x$).

Sideband phases

The blue-sideband term (multiplying $\hat a$) has exponent $i(\delta+\epsilon_m+\epsilon_q)t+i\phi_m$; the red ($\hat a^\dagger$) term has $-i(\delta+\epsilon_m-\epsilon_q)t-i\phi_m$. Setting $\epsilon_m=\epsilon_q=\phi_m=0$: $$ +i\delta t,\qquad -i\delta t, $$ exactly the main-text bracket $\hat a e^{i\delta t}+\hat a^\dagger e^{-i\delta t}$. The prefactor $\eta\hbar\Omega/2$ equals $\tfrac12\hbar\eta\Omega$ identically.

Structure of the error terms

Writing each effective sideband frequency as $\delta+(\partial/\partial\epsilon)$: - $\partial(\text{blue})/\partial\epsilon_m = \partial(\text{red})/\partial\epsilon_m = +1$: a motional-frequency error $\epsilon_m$ shifts BOTH sidebands by the same amount (a common offset of the bichromatic field relative to the mode) — correct interpretation of a motional-frequency error. - $\partial(\text{blue})/\partial\epsilon_q = +1$ but $\partial(\text{red})/\partial\epsilon_q = -1$: a qubit-frequency error $\epsilon_q$ shifts the blue and red sidebands oppositely (the carrier moves, splitting the symmetric detuning) — correct interpretation of a qubit-frequency error.

All six symbolic checks pass (see sympy_check.log). The claim is confirmed exactly.

Verdict

verified — the generalized Eq. (S6) reduces to the main-text $H_A$ in the stated limit and the error terms $\epsilon_q,\epsilon_m$ enter the sideband phases exactly as written.