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Dissipative Jaynes-Cummings Dynamics: A Systematic Study of Excitation Decay and Coherence Loss in the Strong-Coupling Regime

Miguel Percudani

Published Oct 4, 2026• 0 citations
Publisher

Abstract

This work presents a systematic numerical study of the dissipative Jaynes-Cummings model, consisting of a two-level system (qubit) coupled to a truncated harmonic oscillator, both subject to Markovian dissipation. The model is standard in quantum optics and open quantum systems. The master equation is solved using the Lindblad formalism with BDF integration. The study focuses on two standard observables: the qubit excited-state population P_exc(t) and the qubit Bloch vector length |r(t)|. A three-dimensional parameter scan is performed over the coupling g, the qubit dephasing rate gamma_phi, and the oscillator relaxation rate gamma_a, giving 216 cells in total. The parameters used are omega_0 = 6 Hz, omega_1 = 40 Hz, giving a detuning delta = 34 Hz, and thermal occupation n_th = 0.05. MAIN FINDINGS: The single-exponential description of the excited-state population is valid only at intermediate coupling (g approximately 4 to 8 Hz). Outside this region, the dynamics exhibits a multi-timescale structure. The decay time T is non-monotonic in g, with a minimum at g approximately 7 Hz. The location of the minimum depends only weakly on gamma_a, following a power law with exponent 0.127. At strong coupling (g greater than 15 Hz), a two-exponential model is strongly preferred by the Akaike Information Criterion in 43 out of 45 cells. The two timescales are separated by a factor of approximately 150. The fast timescale T_1 is best approximated by the Purcell formula (median ratio of numerical to theoretical value 0.68). The Fermi golden rule overestimates the decay rate by a factor of approximately 4.5. Neither the dressed-state correction nor the Bloch-Redfield calculation improves on the Purcell formula. The Bloch vector length does not follow any simple functional form (single exponential, stretched exponential, power law, or double exponential). All models have mean R-squared in the range 0.31 to 0.36. Its time derivative is, however, almost perfectly correlated with that of the population (mean correlation 0.91). The results are converged with respect to the Fock-space truncation at N_0 = 12, with identical observables for N_0 between 12 and 20. SCOPE: This is a purely numerical study of a well-defined quantum-optical model. It contains no empirical data, no claims about biological or cognitive systems, and no speculative ontology. The model, the numerical method, and the observables are standard. The results include several negative findings (the failure of perturbative formulas, the lack of a simple functional form for the Bloch vector) which are reported honestly and which identify the regime where non-perturbative methods are required. The purpose of this release is to provide reproducible numerical data and a benchmark for future analytical work on the dissipative Jaynes-Cummings model in the strong-coupling regime. CONTENTS: The LaTeX technical note (approximately 20 pages), the Python scripts for the numerical simulations, and the CSV files with the fit results, regime classification, and theoretical comparison. REFERENCES: E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 (1963).G. Lindblad, Commun. Math. Phys. 48, 119 (1976).H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002).C. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed., Springer (2004).E. M. Purcell, Phys. Rev. 69, 681 (1946).C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions, Wiley (1992).W. P. Schleich, Quantum Optics in Phase Space, Wiley-VCH (2001).

DOI
10.5281/zenodo.23132205

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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.

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The study of phase transitions in dissipative quantum systems based on the Liouvillian is often hindered by the difficulty of constructing a time-local master equation when the system-environment coupling is strong. To address this issue, the complex discretization approximation for the environment is proposed to study the quantum phase transition in the extended Jaynes-Cumming model with an infinite number of boson modes. This approach yields a non-Hermitian effective Hamiltonian that can be used to simulate the dynamics of the spin. It is found that the ground state of this effective Hamiltonian determines the spin dynamics in the single-excitation subspace. Depending on the opening of the energy gap and the maximum population of excitations on the spin degree of freedom, three distinct phases can be identified: fast decaying, localized, and stretched dynamics of the spin. This approach can be extended to multiple excitations, and similar dynamics were found in the double-excitation subspace, indicating the robustness of the single-excitation phase.

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This work presents a systematic numerical study of the dissipative Jaynes-Cummings model, consisting of a two-level system (qubit) coupled to a truncated harmonic oscillator, both subject to Markovian dissipation. The model is standard in quantum optics and open quantum systems. The master equation is solved using the Lindblad formalism with BDF integration. The study focuses on two standard observables: the qubit excited-state population P_exc(t) and the qubit Bloch vector length |r(t)|. A three-dimensional parameter scan is performed over the coupling g, the qubit dephasing rate gamma_phi, and the oscillator relaxation rate gamma_a, giving 216 cells in total. The parameters used are omega_0 = 6 Hz, omega_1 = 40 Hz, giving a detuning delta = 34 Hz, and thermal occupation n_th = 0.05. MAIN FINDINGS: The single-exponential description of the excited-state population is valid only at intermediate coupling (g approximately 4 to 8 Hz). Outside this region, the dynamics exhibits a multi-timescale structure. The decay time T is non-monotonic in g, with a minimum at g approximately 7 Hz. The location of the minimum depends only weakly on gamma_a, following a power law with exponent 0.127. At strong coupling (g greater than 15 Hz), a two-exponential model is strongly preferred by the Akaike Information Criterion in 43 out of 45 cells. The two timescales are separated by a factor of approximately 150. The fast timescale T_1 is best approximated by the Purcell formula (median ratio of numerical to theoretical value 0.68). The Fermi golden rule overestimates the decay rate by a factor of approximately 4.5. Neither the dressed-state correction nor the Bloch-Redfield calculation improves on the Purcell formula. The Bloch vector length does not follow any simple functional form (single exponential, stretched exponential, power law, or double exponential). All models have mean R-squared in the range 0.31 to 0.36. Its time derivative is, however, almost perfectly correlated with that of the population (mean correlation 0.91). The results are converged with respect to the Fock-space truncation at N_0 = 12, with identical observables for N_0 between 12 and 20. SCOPE: This is a purely numerical study of a well-defined quantum-optical model. It contains no empirical data, no claims about biological or cognitive systems, and no speculative ontology. The model, the numerical method, and the observables are standard. The results include several negative findings (the failure of perturbative formulas, the lack of a simple functional form for the Bloch vector) which are reported honestly and which identify the regime where non-perturbative methods are required. The purpose of this release is to provide reproducible numerical data and a benchmark for future analytical work on the dissipative Jaynes-Cummings model in the strong-coupling regime. CONTENTS: The LaTeX technical note (approximately 20 pages), the Python scripts for the numerical simulations, and the CSV files with the fit results, regime classification, and theoretical comparison. REFERENCES: E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 (1963).G. Lindblad, Commun. Math. Phys. 48, 119 (1976).H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002).C. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed., Springer (2004).E. M. Purcell, Phys. Rev. 69, 681 (1946).C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions, Wiley (1992).W. P. Schleich, Quantum Optics in Phase Space, Wiley-VCH (2001).
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: - Rmath(t)=∣r(t)∣R_{\text{math}}(t) = |\mathbf{r}(t)|Rmath​(t)=∣r(t)∣, the Bloch vector length of the reduced qubit (coherence proxy) - Rbio(t)=Tr(ρ(t) ∣1⟩⟨1∣)R_{\text{bio}}(t) = \text{Tr}(\rho(t)\,|1\rangle\langle 1|)Rbio​(t)=Tr(ρ(t)∣1⟩⟨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 Rbio(t)R_{\text{bio}}(t)Rbio​(t) is valid only at intermediate coupling (g≈4g \approx 4g≈4–888 Hz). Outside this region, the fit fails or is a lower bound. 2. **Two-exponential structure**: In the failed regime (g≳15g \gtrsim 15g≳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 ∼150\sim 150∼150. 3. **Negative result on naive scaling**: The predictions T1∼1/ΩRT_1 \sim 1/\Omega_RT1​∼1/ΩR​ and T2∼1/γtotalT_2 \sim 1/\gamma_{\text{total}}T2​∼1/γtotal​ are **not** confirmed. The two timescales depend on the full parameter triplet (g,γϕ,γa)(g, \gamma_\phi, \gamma_a)(g,γϕ​,γa​). 4. **Purcell formula**: Among four theoretical models (Fermi golden rule, Purcell, dressed-state, Bloch-Redfield), the **Purcell formula** is the best approximation for T1T_1T1​ (median ratio 0.68). 5. **No simple form for coherence**: No simple functional form (single exponential, stretched exponential, power law, double exponential) fits Rmath(t)R_{\text{math}}(t)Rmath​(t) well. All have mean R2R^2R2 in the range 0.31–0.36. 6. **Proxies are complementary**: The two proxies are uncorrelated in value (mean corr≈+0.09\text{corr} \approx +0.09corr≈+0.09) but almost perfectly correlated in their time derivatives (mean corr≈+0.91\text{corr} \approx +0.91corr≈+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 N0∈[12,20]N_0 \in [12, 20]N0​∈[12,20] (relative change 0.00%). The results are fully converged at N0=12N_0 = 12N0​=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 RmathR_{\text{math}}Rmath​ are reported honestly as negative results, which motivate further theoretical analysis.
Abstract Using a dissipative quantum Rabi model, we study the dynamics of a slow qubit coupled to a fast quantum harmonic oscillator interacting with a bosonic bath from weak to strong and ultra-strong coupling regimes. Solving the quantum Heisenberg equations of motion, perturbative in the internal coupling between qubit and oscillator, we derive functional relationships directly linking the qubit coordinates in the Bloch sphere to oscillator observables. We then perform accurate time-dependent Matrix Product State simulations and compare our results both with the analytical solutions of the Heisenberg equations of motion, and with numerical solutions of a Lindblad master equation, perturbative in the external coupling between oscillator and environment. Indeed, we show that, up to the strong coupling regime, the qubit state accurately fulfils the derived functional relationships. We analyse in detail the case of a qubit starting with generic coordinates on the Bloch sphere of which we evaluate the three components of the Bloch vector through the averages of oscillator observables. Interestingly, a weak to intermediate oscillator coupling to the bath is able to simplify the Bloch vector evaluation since qubit-oscillator relationships are more immediate. Moreover, by monitoring the qubit fidelity with respect to free limit, we find the parameter regime where the combined effect of internal and external couplings is able to hinder the reliable evaluation of the qubit Bloch vector. Finally, in the ultra-strong coupling regime, non-Markovian effects become robust and the dynamics of qubit and oscillator are inextricably entangled making the qubit Bloch vector evaluation difficult. Graphical Abstract
Environmental influences on the dynamics of a coupled qubit-oscillator system are studied analytically in a regime where deviations from the exactly solvable Jaynes–Cummings model are important. We investigate the case of a quasi-degenerate qubit within the ultra-strong coupling regime for which the qubit frequency is much smaller than the frequency of the oscillator, and the coupling between the qubit and the oscillator is large, both of which invalidate the usually employed rotating wave approximation. In contrast to the standard quantum optics master equation, we explicitly take the qubit-oscillator coupling into account while microscopically deriving a dressed state master equation. Using the derived master equation, we discuss a spectroscopic technique which can be used to probe the dressed energy-level structure of the qubit-oscillator system.
We study the dissipative dynamics of a biased two-level system (TLS) coupled to a harmonic oscillator (HO), the latter interacting with an Ohmic environment. Using the Van Vleck perturbation theory and going to second order in the coupling between the TLS and the HO, we show how the Hamiltonian of the TLS–HO system can be diagonalized analytically. Our model represents an improvement on the usually used Jaynes–Cummings Hamiltonian as an initial rotating wave approximation is avoided. By assuming a weak coupling to the thermal bath, analytical expressions for the time evolution of the populations of the TLS are found: the population is characterized by a multiplicity of damped oscillations together with a complex relaxation dynamics towards thermal equilibrium. The long time evolution is characterized by a single relaxation rate, which is largest at resonance and whose expression can be given in a closed analytic form.
We study the dissipative dynamics of a biased two-level system (TLS) coupled to a harmonic oscillator (HO), the latter interacting with an Ohmic environment. Using Van-Vleck perturbation theory and going to second order in the coupling between TLS and HO, we show how the Hamiltonian of the TLS-HO system can be diagonalized analytically. Our model represents an improvement to the usually used Jaynes-Cummings Hamiltonian as an initial rotating wave approximation is avoided. By assuming a weak coupling to the thermal bath, analytical expressions for the time evolution of the populations of the TLS are\nfound: the population is characterized by a multiplicity of damped oscillations together with a complex relaxation dynamics towards thermal equilibrium. The long time evolution is characterized by a single relaxation rate, which is largest at resonance and whose expression can be given in closed analytic form.
The study of phase transitions in dissipative quantum systems based on the Liouvillian is often hindered by the difficulty of constructing a time-local master equation when the system-environment coupling is strong. To address this issue, the complex discretization approximation for the environment is proposed to study the quantum phase transition in the extended Jaynes-Cumming model with an infinite number of boson modes. This approach yields a non-Hermitian effective Hamiltonian that can be used to simulate the dynamics of the spin. It is found that the ground state of this effective Hamiltonian determines the spin dynamics in the single-excitation subspace. Depending on the opening of the energy gap and the maximum population of excitations on the spin degree of freedom, three distinct phases can be identified: fast decaying, localized, and stretched dynamics of the spin. This approach can be extended to multiple excitations, and similar dynamics were found in the double-excitation subspace, indicating the robustness of the single-excitation phase.