Quantum-classical correspondence in resonant and nonresonant Rabi-Stark model

TL;DR

This work analyzes quantum–classical correspondence in the Rabi–Stark model using mean-field semiclassical dynamics, showing that the nonlinear Stark coupling U reshapes the semiclassical phase space and that correspondence emerges for large atom–field frequency ratios. The authors link phase-space structure to quantum entanglement via time-averaged linear entanglement entropy, demonstrating clear Q–C signals in nonresonant regimes. In the resonant regime, they find that Q–C can be recovered in the resonant RSM when U → ±1, with the U → −1 case acting as a new few-body thermodynamic-like limit where truncated photon numbers remain manageable. These results highlight how Stark-like nonlinearities enable quantum chaos and entanglement signatures to reflect classical dynamics, offering pathways for quantum simulations and deeper insights into few-body quantum systems.

Abstract

Testing the correspondence principle in nonlinear quantum systems is a fundamental pursuit in quantum physics. In this paper, we employed mean field approximation theory to study the semiclassical dynamics in the Rabi-Stark model (RSM) and showed that the nonlinear Stark coupling significantly modulates the semiclassical phase space structure. By analyzing the linear entanglement entropy of coherent states prepared in the classical chaotic and regular regions of the semiclassical phase space, we demonstrate that quantum-classical correspondence can be achieved in the RSM with large atom-light frequency ratios. While this correspondence fails in the resonant Rabi model because its truncated photon number is insufficient to approach the large quantum number limit, we discovered that in the resonant RSM when the nonlinear Stark coupling $U \to \pm 1$, the time-averaged linear entanglement entropy correlates strongly with the semiclassical phase space. In particular, when $U \to -1$, the truncated photon number in the resonant RSM is very close to that in the resonant Rabi model, but the time-averaged linear entanglement entropy still corresponds well with the semiclassical phase space. This result demonstrates that quantum-classical correspondence can be realized in the few-body resonant RSM.

Figures (5)

• Figure 1: The Poincaré sections for the RSM in the case: $q_{2}=0, p_{2}>0$, with $\omega=14$, $\omega_{0} =1$, $g=5$ and the system energy $E = 7$. The Stark coupling parameters from left to right are U = -0.3, 0, and 0.3, respectively.
• Figure 2: The distribution of the time-averaged linear entanglement entropy of Poincaré sections in Fig. \ref{['NonRSM-ps']}. The Stark coupling parameters from left to right are U = -0.3, 0, and 0.3, respectively. Here, the time interval is $t\in[0, 500]$ and the system photon number is truncated at $180$, $160$ and $150$, respectively.
• Figure 3: The correspondence between classical dynamics and quantum linear entanglement entropy for the resonant Rabi model with system energy $E = 1.3$. Part (a) is the classical Poincaré section in the case: $q_{2}=0, p_{2}>0$, with $\omega=\omega_{0} =1$, $g=0.4$ and the coordinates of point $C_1$ are $q_{1} = 0,p_{1} =-0.9,q_{2} =0,p_{2} = 1.67033$. Part (b) shows the time evolution of the linear entanglement entropy $S(t)$ with different initial system photon numbers $N_p$ for the initial state centred at point $C_{1}$. Part (c) shows the distribution of the time-averaged entanglement entropy corresponding to the Poincaré section in part (a). Here, the photon number was truncated at 18.
• Figure 4: The correspondence between classical dynamics and quantum linear entanglement entropy for the resonant RSM with system energy $E = 0.15$. Part (a) is the classical Poincaré section in the case: $q_{2}=0, p_{2}>0$, with $\omega=\omega_{0} =1$, $g=0.6$, $U=0.999$ and the coordinates of point $C_2$ are $q_{1} = 0,p_{1} =-0.9,q_{2} =0,p_{2} = 0.77769$. Part (b) is the time evolution of linear entanglement entropy $S(t)$ with different initial system photon numbers $N_p$ for the initial state centred at point $C_{2}$. Part (c) shows the distribution of time-averaged entanglement entropy corresponding to the Poincaré section in part (a). Here, the photon number was truncated at 100.
• Figure 5: The correspondence between classical dynamics and quantum linear entanglement entropy for the resonant RSM with system energy $E = 0.6$. Part (a) is the classical Poincaré section in the case: $q_{2}=0, p_{2}>0$, with $\omega=\omega_{0} =1$, $g=0.2$, $U=-0.999$ and the coordinates of point $C_3$ are $q_{1} = 0,p_{1} =-0.9,q_{2} =0,p_{2} = 1.08086$. Part (b) is the time evolution of linear entanglement entropy $S(t)$ with different the initial system photon number $N_p$ for the initial state centred at point $C_{3}$. Part (c) shows the distribution of time-averaged entanglement entropy corresponding to the Poincaré section in part (a).Here, the photon number was truncated at 16.