C011 - Drive (A) realizes the Hamiltonian H_A = (1/2) hbar eta Omega S_{x,e} (a e^{i delta t} +...
Verdict: verified
Location: Experiment
Type / expected artifact: math / math
Claim: Drive (A) realizes the Hamiltonian H_A = (1/2) hbar eta Omega S_{x,e} (a e^{i delta t} + a^dag e^{-i delta t}), a Molmer-Sorensen-type force whose phase depends on the eigenstate of S_{x,e}.
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/main.tex:115 (Experiment).
Conclusion
H_A = (1/2) hbar eta Omega S_{x,e}(a e^{i delta t}+a^dag e^{-i delta t}) is term-by-term the standard bichromatic Lamb-Dicke MS interaction restricted to {down,e} and promoted to the two-ion collective S_{x,e}. State-dependent displacement force confirmed by diagonalizing in the S_{x,e} eigenbasis. Definition/identification confirmed; C012 corroborates quantitatively.
Verification details
Derivation excerpt: 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}...