⟨σ_112⟩(t) exceed 1 when setting `order=3` about quantumcumulants.jl

[Papageno2]

For collective spin calculations, if either $Re(\sigma_1^{12})$ or $Im(\sigma_1^{12})$ exceeds 0.5, it implies that the projection of the collective spin exceeds $N/2$ for a system of $N$ spin-1/2 atoms.
$$
\begin{align}
\langle \hat{J}_x\rangle &= (N/2)(\langle \hat{\sigma}_1^{12}\rangle+\langle \hat{\sigma}_1^{21}\rangle),\
\langle \hat{J}_y\rangle &= (iN/2)(\langle \hat{\sigma}_1^{12}\rangle - \langle \hat{\sigma}_1^{21}\rangle),\
\langle \hat{J}_z\rangle &= (N/2)(2\langle \hat{\sigma}_1^{22}\rangle - 1).
\end{align}$$

[ChristophHotter]

Hi @Papageno2,
You used the ClusterSpace which is outdated. Please use the Symbolic Sums and Indices instead. This should resolve your problem.
The superradiant laser example might be useful for you: https://qojulia.github.io/QuantumCumulants.jl/stable/examples/superradiant_laser_indexed/
Here's some documentation about the symbolic sums: https://qojulia.github.io/QuantumCumulants.jl/stable/symbolic_sums/

[Papageno2]

Hi @Papageno2, You used the ClusterSpace which is outdated. Please use the Symbolic Sums and Indices instead. This should resolve your problem. The superradiant laser example might be useful for you: https://qojulia.github.io/QuantumCumulants.jl/stable/examples/superradiant_laser_indexed/ Here's some documentation about the symbolic sums: https://qojulia.github.io/QuantumCumulants.jl/stable/symbolic_sums/

Thank you for reminding me about the new API: Sumbolic Sums and Indices, now I get the same results for order=2 and order=3. The inconsistency in results that I previously encountered may be attributed to the instability of the ODE due to the sizable detuned parameter $\Delta$.