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Generation of a maximally entangled state using collective optical pumping

verification/C003/attempts/R002/claim_report.md

C003 - The collective excitation unitary U_A(Phi) = exp(-i Phi S_{x,e}^2), with S_{x,e}=sigma_{x...

Verdict: verified Location: Dissipation scheme, Eq. (1) Type / expected artifact: math / math Claim: The collective excitation unitary U_A(Phi) = exp(-i Phi S_{x,e}^2), with S_{x,e}=sigma_{x,down e} tensor 1 + 1 tensor sigma_{x,down e}, provides full transfer from |down,down> to |ee> for Phi = pi/4. 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:102-106 (Eq. (1)).

Conclusion

U_A(Phi)=exp(-i Phi S_xe^2). Computed ||^2 = 1.0 (paper: full transfer =1). Closed form sin^2(2 Phi) confirmed over Phi scan; U_A unitary. Exact match. -> verified.

Verification details

Derivation excerpt: On the two-level subspace $\{|{\downarrow}\rangle,|e\rangle\}$ of each ion, $\sigma_{x,de}=|{\downarrow}\rangle\langle e|+|e\rangle\langle{\downarrow}|$. The collective operator is $$S_{x,e}=\sigma_{x,de}\otimes\mathbb1+\mathbb1\otimes\sigma_{x,de}.$$

Executable rerun: sympy_check.py exited 0 in 0.524s; log verification/C003/attempts/R002/sympy_check.log.

Output excerpt:

|<ee|U_A(pi/4)|dd>|^2 = 1.0
U_A unitary : True
Phi/pi, prob, sin^2(2Phi):
  0.000  0.000000  0.000000
  0.125  0.500000  0.500000
  0.250  1.000000  1.000000
  0.375  0.500000  0.500000
  0.500  0.000000  0.000000
ALL_CHECKS_PASS = True

Supporting files