Superradiant Phase Transition and Statistical Properties in Dicke-Stark Model

TL;DR

The study tackles superradiant phase transitions (SPT) and quantum statistics in the finite-size Dicke-Stark (DS) model under strong light–matter coupling and thermal environments. It combines an extended coherent state space diagonalization with a quantum dressed master equation to obtain accurate spectra and thermal states, and it analytically derives the SPT critical point for the infinite DS model via mean-field/Holstein-Primakoff methods, including a finite-temperature generalization $\lambda_c(T) = \sqrt{\frac{\Delta}{4}\left[ \frac{\omega}{\tanh(\Delta/(2k_B T))} - \frac{U}{2} \right]}$, with numerical confirmation for finite N showing Stark-field tunability of the transition. The open-system analysis reveals rich photon-statistics dynamics, with $G^{(2)}(0)$ evolving from bunching to anti-bunching and back, and entanglement $N(\rho)$ and spin squeezing $\xi^2$ remaining robust at low $T$ but diminishing with temperature, modulated by the Stark strength $U$. These results elucidate how Stark-type nonlinearity and strong coupling shape quantum correlations and phase behavior, offering tunable pathways for quantum devices and thermally assisted quantum engines.

Abstract

In this study, the energy spectrum and thermal equilibrium states of the finite-size Dicke-Stark model were numerically obtained within the extended coherent state space by solving the dressed master equation for strongly coupled light-atom systems. The critical point of the superradiant phase transition in the infinite-size Dicke-Stark model was analytically derived using the mean-field approach and confirmed with numerical calculation. Under thermal equilibrium conditions, analyses of the negativity, zero-time-delay two-photon correlation function, and atom-spin squeezing parameters in the finite-size Dicke-Stark model reveal that as the coupling strength increases, the light field undergoes a transition from photon bunching to anti-bunching and then back to bunching. The Stark field can modulate both the maximum and minimum values of the two-photon correlation function and their corresponding coupling strengths. At low temperatures, the system exhibits entanglement and spin squeezing. As temperature rises, entanglement gradually diminishes, while strong coupling facilitates the preservation of entanglement in the system state. Atom-spin squeezing spin squeezing is highly sensitive to temperature and vanishes rapidly with increasing temperature. This work contributes to the fundamental understanding of quantum phenomena in Dicke-Stark systems.

Figures (7)

• Figure 1: Ground-state energy spectrum of DS model as a function increasing coupling strength $\lambda$. (a). A comparison of ground-state spectrum calculated by DCS method with photon truncation number $K_{tr}=50$ and by the DFS method with photon truncation numbers $N_{tr}= 8, 16, 32, 64, 128$ at fixed Stark parameter $U=1.0$. (b). Demonstrating the effects of the stark strength $U = 1.5,0.5,0,-0.5. -1.5$ on the ground-state energy spectrum, simulated using the DCS method with $K_{tr}=50$, $N=32$, $\Delta=1$.
• Figure 2: Comparison of the ground-state calculation accuracy between the DCS (red lines) and DFS (black lines) method. (a). Energy error analysis: $K_{tr}$ and $N_{tr}$ represent the minimum truncation number required to achieve a calculation accuracy of $10^{-6}$ for atom number $1\le N\le 128$. (b). Wavefunction error analysis: Effects of the photon number truncations $N_{tr}$ and $K_{tr}$ on calculation the error $-\log_{10}(\Delta P)$ of the ground-state wavefunction with atom number $N=128$. The vertical axis represents calculation accuracy, with higher values of $-\log_{10}(\Delta P)$, indicating smaller wavefunction errors. Other parameters: $U = 1.0$, $\Delta = 1$, $\lambda = 0.5$.
• Figure 3: Comparison of the accuracy between DCS and DFS method in the excited-state calculation of the Dicke model. (a) Excited-state energy error $\log_{10}(\Delta E)$ as a function of the number of excited state $k$. (b) Excited-state wavefunction error $\log_{10}(\Delta P)$ as a function of the excited state $k$. Red circles represent the results of the DCS method, while black squares represent the results of DFS method. Blue dashed lines serve as a reference of calculation accuracy $10^{-6}$. Other parameters: $N = 32$, $U = 1.0$, $\lambda = 0.5$, $\Delta = 1$, and $N_{{tr}}= K_{{tr}} = 50$.
• Figure 4: Variations in the average photon number of the ground state in the isolated finite-size Dicke-Stark model: (a) The variation of the average photon number with coupling strength for different atomic numbers $N=8, 32,128$ at $U=1$. (b) Phase diagram of the average photon number in the plane of coupling strengths and Stark field strengths ($\lambda-U$ with atomic number $N=128$. (c) The variation of the average photon number with coupling strength for different Stark field strengths $U=-1.5, 0, 1.5$ at $N=128$. (d)-(f) Evolution of average photon number as a function of coupling strength for different Stark field strengths $U=-1.5, 0, 1.5$ with atomic number $N=16$. Other parameters: $K_{tr}=50, \Delta=1$.
• Figure 5: The zero-time delay two-photon correlation function ${G^{(2)}}(0)$ as a function of the coupling strength $\lambda$. ${G^{(2)}}(0)$. Panel (a) demonstrates the variation for positive Stark field strengths, while (b) illustrates the case for negative Stark field strengths. In both panels, the black line corresponds to $U = 0$, while lines of different colors distinguish different Stark field strengths. Other parameters, $N = 8$, $T = 0.1$,$\Delta=1$,$K_{tr}=50$.