Camino 1 — Extended Analysis: Regime Classification, Two-Exponential Structure, and Proxy Comparison in the Two-Mode Jaynes-Cummings Model

This internal working note documents the results of the Camino 1 program, an extension of the v8.1-3D analysis of the two-mode Jaynes-Cummings model with Lindblad dissipation. It is a **methodological and phenomenological** contribution: it uses the Jaynes-Cummings formalism as an effective language for two coexisting dynamical modes, and it does not claim that the underlying system is quantum. ## What is formal and verifiable The model is the same two-mode Jaynes-Cummings system with Lindblad dissipation used in v8.1-3D, integrated with BDF. Two coexistence proxies are analyzed: - $R_{\text{math}}(t) = |\mathbf{r}(t)|$, the Bloch vector length of the reduced qubit (coherence proxy)- $R_{\text{bio}}(t) = \text{Tr}(\rho(t)\,|1\rangle\langle 1|)$, the qubit excited population (population proxy) ## What is phenomenological The proxies are phenomenological simplifications of coherence and population. No claim is made that they correspond to specific physical quantities in a real system. ## Main findings 1. **Single-exponential validity**: The single-exponential description of $R_{\text{bio}}(t)$ is valid only at intermediate coupling ($g \approx 4$–$8$ Hz). Outside this region, the fit fails or is a lower bound. 2. **Two-exponential structure**: In the failed regime ($g \gtrsim 15$ Hz), a two-exponential model is strongly preferred by AIC in 43 out of 45 cells. The two timescales are separated by a factor of $\sim 150$. 3. **Negative result on naive scaling**: The predictions $T_1 \sim 1/\Omega_R$ and $T_2 \sim 1/\gamma_{\text{total}}$ are **not** confirmed. The two timescales depend on the full parameter triplet $(g, \gamma_\phi, \gamma_a)$. 4. **Purcell formula**: Among four theoretical models (Fermi golden rule, Purcell, dressed-state, Bloch-Redfield), the **Purcell formula** is the best approximation for $T_1$ (median ratio 0.68). 5. **No simple form for coherence**: No simple functional form (single exponential, stretched exponential, power law, double exponential) fits $R_{\text{math}}(t)$ well. All have mean $R^2$ in the range 0.31–0.36. 6. **Proxies are complementary**: The two proxies are uncorrelated in value (mean $\text{corr} \approx +0.09$) but almost perfectly correlated in their time derivatives (mean $\text{corr} \approx +0.91$). The value correlation changes sign with the regime. Both proxies are needed for a complete characterization. 7. **Fock-space convergence**: The observables are identical for $N_0 \in [12, 20]$ (relative change 0.00%). The results are fully converged at $N_0 = 12$. ## Contents of this release The LaTeX technical note and the associated CSV files with the fit results, regime classification, and proxy comparison. The generating scripts are provided in the companion release of v8.1-3D. ## Scope disclaimer No empirical dataset is analyzed or claimed in this note. The purpose is methodological: to document a reproducible numerical procedure and its regime of validity. The failure of the naive scaling and the lack of a simple functional form for $R_{\text{math}}$ are reported honestly as negative results, which motivate further theoretical analysis.