coupling_matrix conjugates off-diagonal entries incorrectly for complex couplings

[oameye]

Summary

coupling_matrix agrees with the documented interaction-picture generator for real couplings, but conjugates the off-diagonal entries incorrectly when the couplings are complex.

For two cascaded modes with

$$G_i = (1, g_i^* a_i, 0),$$

the SLH Hamiltonian gives

$$A = \frac{1}{2} \begin{pmatrix} 0 & g_1 g_2^* \\\ -g_1^* g_2 & 0 \end{pmatrix}.$$

Minimal reproducer

using QuantumInputOutput

g1 = 1 + 2im
g2 = 3 + 4im
A = coupling_matrix((g1, g2))(0.0)

A_expected = 0.5 * [
  0                 g1 * conj(g2)
  -conj(g1) * g2    0
]

@assert A ≈ A_expected

Current result:

ComplexF64[
  0.0 + 0.0im    5.5 - 1.0im
 -5.5 - 1.0im    0.0 + 0.0im
]

Expected result:

ComplexF64[
  0.0 + 0.0im    5.5 + 1.0im
 -5.5 + 1.0im    0.0 + 0.0im
]

Likely cause

The (i, j) indices constructed with divrem appear to be ordered as row-major entries, while the tuple passed to SMatrix is interpreted in column-major order. This effectively transposes the coefficient formula and conjugates the off-diagonal pair.

Impact

The current interaction-picture tests use real Gaussian pulse couplings, for which the mismatch is hidden. Complex temporal modes, for example chirped or detuned pulses, receive the wrong interaction-picture transformation.

Adding a test with non-real g1 and g2 should expose the regression.