bug: effective_input_mode uses target coupling in i>2 cross term

Problem

A theory/documentation audit of the multimode virtual-cavity recursion found a likely correctness bug in effective_input_mode for nonzero chains with i > 2.

The current implementation defines gui_t = gu_i(t) and accumulates

coupling_sum += gui_t' * α[k]

for every preceding cavity k < j.

For the three-input case, the explicit derivation in Kiilerich & Mølmer, Phys. Rev. A 102, 023717 (2020), Appendix, gives

\dot\alpha_{u_3}^{(2)}
= -g_{u_2}\left(u_3-g_{u_1}^*\alpha_{u_3}^{(1)}\right)
+ \frac{|g_{u_2}|^2}{2}\alpha_{u_3}^{(2)},

so the cross term must use the coupling of each preceding cavity, g_{u_k}^*, not the target coupling g_{u_i}^*.

The source already notes two typographical errors in the paper's compact general Eq. (A15)/(49): the final sign and the self-coupling index. The explicit two- and three-mode equations also disambiguate the cross-term index.

Current coverage

test/test_utils.jl exercises:

• nonzero effective_input_mode(..., i=2);
• i=3 only with all couplings set to zero.

That means the i>2 cross-coupling path is not currently regression-tested.

Likely fix

Inside multiple_inputs_α!, the sum should use the preceding coupling corresponding to k, e.g.

coupling_sum += gu_buf[k]' * α[k]

rather than gui_t' * α[k].

Add a nonzero three-input regression against the explicit recursive equations before changing the implementation.

This was discovered while restructuring the manual/theory in PR #69; that PR should remain documentation-focused.

Start with test/test_utils.jl and the multiple_inputs_α! implementation that supports effective_input_mode. Run the existing i=2 tests, then add a nonzero three-input case based on the explicit recursive equations in the issue. Done means the regression passes and the i>2 cross-coupling path uses the preceding cavities' couplings.