C078 - In the weak-field limit delta>>eta Omega, the continuous Hamiltonian is H_s = hbar J S_{x...
Verdict: verified
Location: Supp. Mat. S6, Eq. (S9)
Type / expected artifact: math / math
Claim: In the weak-field limit delta>>eta Omega, the continuous Hamiltonian is H_s = hbar J S_{x,e}^2 + hbar (Omega_C/2)(sigma_x tensor 1 + 1 tensor sigma_x) + hbar beta(|up>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:166-173 (Eq. (S9)).
Conclusion
Weak-field reduction of drive (A): the paper's exact closed-form Phi(t)=(eta^2 Omega^2/4 delta^2)(delta t - sin delta t) has secular rate dPhi/dt -> eta^2 Omega^2/(4 delta) = (eta Omega/2)^2/delta = J, matching the claimed J exactly. Drive-(C) term hbar(Omega_C/2)(sx x 1 + 1 x sx) and detuning hbar beta(|up>
Verification details
Derivation excerpt: From the main text (Eq. near line 115), the bichromatic drive (A) realises a Molmer-Sorensen interaction coupling the collective operator $S_{x,e} = \sigma_{x,\downarrow e}\otimes\mathbf{1} + \mathbf{1}\otimes\sigma_{x,\downarrow e}$ to a single motional mode $\hat a$: $$ H_A = \tfrac{1}{2}\,\hbar\,\eta\Omega\,S_{x,e}\,\big(\hat a\,e^{i\delta t} + \hat a^{\dagger} e^{-i\delta t}\big), $$ with $\eta$ the Lamb-Dicke parameter, $\Omega$ the carrier Rabi frequency and $\delta$ the detuning from the sidebands.