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Generation of a maximally entangled state using collective optical pumping

verification/C079/attempts/R002/claim_report.md

C079 - The repump (B) is modelled by Markovian jump operators L_{e->down}^{(k)} = sqrt(p_{e->dow...

Verdict: verified Location: Supp. Mat. S6, Eq. (S10) Type / expected artifact: math / math Claim: The repump (B) is modelled by Markovian jump operators L_{e->down}^{(k)} = sqrt(p_{e->down} kappa)|down>up}^{(k)} = sqrt(p_{e->up} kappa)|up>down}/p_{e->up} = tan^2(gamma). 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:174-183 (Eq. (S10)).

Conclusion

Jump operators L_{e->dn}=sqrt(p_dn kappa)|dn>up}=sqrt(p_up kappa)|up> (single-ion decay), total rate out of |e> = (p_dn+p_up)kappa = kappa (since sin^2+cos^2=1), and branching p_dn/p_up = sin^2/cos^2 = tan^2(gamma). All three properties of Eq S10 hold exactly. Conclusion: verified.

Verification details

Derivation excerpt: Per-ion basis $0=\ket{\downarrow},1=\ket{\uparrow},2=\ket{e}$. The two single-ion jump operators (Eq. S10) are $$ L_{e\to\downarrow} = \sqrt{p_{e\to\downarrow}\,\kappa}\,\ket{\downarrow}\bra{e}, \qquad L_{e\to\uparrow} = \sqrt{p_{e\to\uparrow}\,\kappa}\,\ket{\uparrow}\bra{e}, $$ with $p_{e\to\downarrow}=\sin^2\gamma$, $p_{e\to\uparrow}=\cos^2\gamma$ (same branching parametrisation as the discrete model, P.branching).

Executable rerun: sympy_check.py exited 0 in 0.624s; log verification/C079/attempts/R002/sympy_check.log.

Output excerpt:

L_dn|e> = Matrix([[sqrt(kappa)*Abs(sin(gamma)), 0, 0]])  (should be sqrt(p_dn kappa)|down>)
L_up|e> = Matrix([[0, sqrt(kappa)*Abs(cos(gamma)), 0]])  (should be sqrt(p_up kappa)|up>)
jumps annihilate |down>,|up>: OK (decay only from |e>)
total rate out of |e> = kappa (should be kappa)
p_dn/p_up = tan(gamma)**2 = tan^2(gamma)? True
ALL CHECKS PASS: single-ion e-decay, total rate kappa, branching tan^2(gamma)

Supporting files