Loss of integrability in a system with two-photon interactions

TL;DR

This work investigates the loss of integrability in the isolated two-photon Dicke model, focusing on the normal phase and leveraging a mean-field classical limit together with quantum diagnostics such as Peres lattices and spectral-ratio statistics. In the integrable limit $\omega_{0}=0$, a Bogoliubov transformation yields analytic spectra $E_{n_{c},m_{x}} = \Omega_{m_{x}}(n_{c}+\tfrac{1}{2}) - \tfrac{\omega}{2}$, while turning on $\omega_{0}$ breaks integrability and induces observable deviations in both classical trajectories and quantum observables. Across resonant $\omega_{0}=2\omega$ and nonresonant $\omega_{0}=\tfrac{\pi}{5}\omega$ cases (with $\gamma<\gamma_{\textsc{sc}}$), the average spectral ratio $\langle r \rangle$ transitions from Poisson to GOE as coupling grows, and Peres lattices become progressively disordered, signaling chaos and mixed dynamics. A classical analysis via the mean-field Hamiltonian confirms a progression from regular to chaotic motion at high energies, consistent with the quantum signatures, and reveals a regime of mixed dynamics near the ground state. These results advance understanding of how integrability breaks down in two-photon light–matter systems and point to rich avenues for exploring quantum chaos, localization, and experimental verification in related platforms.

Abstract

Light-matter systems that exhibit two-photon interactions have emerged as powerful platforms for exploring quantum applications. In this work, we focus on the two-photon Dicke model, a system of significant experimental relevance that displays spectral collapse and undergoes a phase transition from a normal to a superradiant phase. We analyze the normal phase, where a classical limit with two degrees of freedom can be derived using a mean-field approximation. Our study presents a detailed investigation of the loss of integrability in the two-photon Dicke model, employing both quantum and classical diagnostics. These results allow us to explore various dynamical features of the system, including the onset of chaos and the existence of mixed phase-space behavior.

Figures (8)

• Figure 1: Peres lattices of the operators (a1)-(c1) $\hat{c}^{\dagger}\hat{c}$, (a2)-(c2) $\hat{J}_{x}$, and (a3)-(c3) $\hat{J}_{x}^{2}$, as a function of the scaled energy $\epsilon=E/j$. Each column identifies a different coupling strength: (a1)-(a3) $\gamma=0.2$, (b1)-(b3) $\gamma=0.3$, and (c1)-(c3) $\gamma=0.4$. In each panel, the black dots represent the numerical expectation values using the eigenstates of the two-photon Dicke Hamiltonian in Eq. \ref{['eq:DickeHamiltonian']} and the red lines describe the analytical values using Eq. \ref{['eq:AnalyticalSpectrum']}. In panels (a1)-(c1) and (a2)-(c2), each red line represents $m_{x}$ as a continuous variable for discrete values of $n_{c}$. In panels (a3)-(c3), the continuous variable is $m_{x}^{2}$. System parameters: $\omega=1$, $\omega_{0} = 0$, and $j = 15$. We use a perturbation parameter $\varepsilon=0.001$. The truncation value is $n_{\text{max}} = 200$.
• Figure 2: Peres lattices of the operator $\hat{J}_{x}^{2}$ as a function of the scaled energy $\epsilon=E/j$. In each panel, the black dots represent the numerical expectation values using the eigenstates of the two-photon Dicke Hamiltonian in Eq. \ref{['eq:DickeHamiltonian']} and the red lines describe the analytical values using Eq. \ref{['eq:AnalyticalSpectrum']}. In panel (a), we set $\omega_0 = 0$ with $\varepsilon=0.001$ and plot $m_{x}^2$ as a continuous variable for discrete values of $n_{c}$. In panels (b) and (c), we set $\omega_{0} = 0.05$ to compute the numerical expectation values (black dots), but we use the analytical results (red lines) from panel (a). In panel (b), the continuous variable is $m_{x}^{2}$ as described in panel (a), while in panel (c), the continuous variable is $n_{c}$ for discrete values of $m_{x}^{2}$. System parameters: $\omega=1$, $\gamma=0.3$, and $j = 15$. The truncation value is $n_{\text{max}} = 200$.
• Figure 3: Peres lattices of the operators (a) $\hat{a}^{\dagger}\hat{a}$, (b) $\hat{J}_{z}$, and (c) $\hat{J}_{x}^{2}$, as a function of the scaled energy $\epsilon=E/j$. In each panel, the green and blue dots represent the numerical expectation values using the eigenstates of the two-photon Dicke Hamiltonian in Eq. \ref{['eq:DickeHamiltonian']}. The blue dots correspond to eigenstates that tend to behave as eigentates of the operator $\hat{J}_x$. The green dots correspond to eigenstates that tend to behave as eigenstates of the operator $\hat{J}_{z}$. System parameters: $\omega=1$, $\omega_0=0.05$, $\gamma=0.3$, and $j = 25$. The truncation value is $n_{\text{max}} = 200$.
• Figure 4: (a1)-(e1) Average spectral ratio $\langle r \rangle$ and Peres lattices of the operators (a2)-(e2) $\hat{a}^{\dagger}\hat{a}$, (a3)-(e3) $\hat{J}_{z}$, and (a4)-(e4) $\hat{J}_{x}^{2}$ covering a wide energy spectrum, where $\epsilon=E/j$. Each column identifies a different coupling strength: (a1)-(a4) $\gamma=0.01$, (b1)-(b4) $\gamma=0.05$, (c1)-(c4) $\gamma=0.1$, (d1)-(d4) $\gamma=0.2$, and (e1)-(e4) $\gamma=0.3$. In panels (a1)-(e1), the black solid line represents the average spectral ratio in Eq. \ref{['eq:SpectralRatio']} using the converged eigenvalues of the two-photon Dicke Hamiltonian in Eq. \ref{['eq:DickeHamiltonian']}. We average the spectral ratio over the four parity subspaces to smooth spectral fluctuations. For each panel from (a1) to (e1), we get around $2\times10^4$ converged eigenvalues for each subspace. The red (blue) dashed line represents the analytical average spectral ratio of the GOE (Poisson) distribution. In panels (a2)-(e2), (a3)-(e3), and (a4)-(e4), the green dots represent the numerical expectation values using the eigenstates of the two-photon Dicke Hamiltonian for the parity subspace with eigenvalue $p=1$. System parameters: $\omega=1$, $\omega_0=2$, and $j = 25$. The truncation value is $n_{\text{max}} = 2\times10^3$.
• Figure 5: (a1)-(e1) Average spectral ratio $\langle r \rangle$ and Peres lattices of the operators (a2)-(e2) $\hat{a}^{\dagger}\hat{a}$, (a3)-(e3) $\hat{J}_{z}$, and (a4)-(e4) $\hat{J}_{x}^{2}$ covering a wide energy spectrum, where $\epsilon=E/j$. Each column identifies a different coupling strength: (a1)-(a4) $\gamma=0.01$, (b1)-(b4) $\gamma=0.05$, (c1)-(c4) $\gamma=0.1$, (d1)-(d4) $\gamma=0.2$, and (e1)-(e4) $\gamma=0.3$. In panels (a1)-(e1), the black solid line represents the average spectral ratio in Eq. \ref{['eq:SpectralRatio']} using the converged eigenvalues of the two-photon Dicke Hamiltonian in Eq. \ref{['eq:DickeHamiltonian']}. We average the spectral ratio over the four parity subspaces to smooth spectral fluctuations. For each panel from (a1) to (e1), we get around $2\times10^4$ converged eigenvalues for each subspace. The red (blue) dashed line represents the analytical average spectral ratio of the GOE (Poisson) distribution. In panels (a2)-(e2), (a3)-(e3), and (a4)-(e4), the blue dots represent the numerical expectation values using the eigenstates of the two-photon Dicke Hamiltonian for the parity subspace with eigenvalue $p=1$. System parameters: $\omega=1$, $\omega_0=\pi/5$, and $j = 25$. The truncation value is $n_{\text{max}} = 2\times10^3$.