C056 - With qubit-frequency error epsilon_q and motional-frequency error epsilon_m, drive (A) ha...
Verdict: verified
Location: Supp. Mat. S3, Eq. (S6)
Type / expected artifact: math / math
Claim: With qubit-frequency error epsilon_q and motional-frequency error epsilon_m, drive (A) has H_A = (eta hbar Omega/2) S_{e,phi} (a e^{i(delta+epsilon_m+epsilon_q)t} e^{i phi_m} + a^dag e^{-i(delta+epsilon_m-epsilon_q)t} e^{-i phi_m}), reducing to the main-text H_A for phi_s=phi_m=0, epsilon_m=epsilon_q=0.
Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain verified -> verified.
Source location(s): source/supp_content.tex:90-99 (Eq. (S6)).
Conclusion
Symbolic check: at phi_s=0 S_{e,phi}=sigma_x; at eps_m=eps_q=phi_m=0 the sideband exponents reduce to +/- i delta t, matching main-text H_A; prefactor etahbarOmega/2 identical to 1/2 hbar eta Omega. eps_m shifts both sidebands equally (d/d eps_m=+1,+1; common motional offset), eps_q oppositely (+1,-1; qubit offset splits sidebands), exactly as written. All 6 symbolic checks pass.
Verification details
Derivation excerpt: 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.
Executable rerun: sympy_check.py exited 0 in 0.598s; log verification/C056/attempts/R002/sympy_check.log.
Output excerpt:
S_{e,phi}(phi_s=0) == sigma_x : True
blue exponent limit: I*delta*t expected: I*delta*t -> True
red exponent limit: -I*delta*t expected: -I*delta*t -> True
d(blue)/d eps_m = 1 d(red)/d eps_m = 1 -> same sign (common motional offset): True
d(blue)/d eps_q = 1 d(red)/d eps_q = -1 -> opposite sign (qubit offset splits sidebands): True
prefactor match: True
ALL CHECKS PASS: True