The superradiant phase is a finite size effect in two-photon processes

TL;DR

The paper addresses whether the two‑photon Dicke model supports a genuine thermodynamic superradiant phase. By contrasting two classical limits—a squeezed‑vacuum mean‑field and a Glauber coherent‑state mean‑field—alongside numerical results, it demonstrates that the superradiant phase is a finite‑size crossover, disappearing as $j\to\infty$, while spectral collapse persists at $\gamma_{\text{sc}} = \omega/2$. The squeezed‑state analysis yields SP solutions only when $j^2\omega_0^2 < \gamma^2$, which become unphysical in the thermodynamic limit; the coherent‑state limit shows SP is not physical and reveals spectral collapse as a phase‑space phenomenon. These findings resolve prior ambiguities and place constraints on realizing superradiant behavior in two‑photon platforms; the role of dissipation in stabilizing a true SP remains an open question.

Abstract

Two-photon light-matter interactions exhibit distinctive features such as spectral collapse. The two-photon Dicke model has been reported to exhibit a superradiant phase which could be useful in quantum applications. Here we show that this superradiant phase is not a genuine thermodynamic phase but a finite-size effect. Combining analytical and numerical analyses, we demonstrate that the superradiant region shrinks with increasing system size and disappears in the thermodynamic limit, while spectral collapse remains. Our results clarify the nature of superradiant conditions in two-photon systems and constrain its realization in quantum platforms.

Figures (6)

• Figure 1: (a1)-(a3) Phase diagrams of the normalized atomic excitation $1 + \langle \hat{J}_{z} \rangle/j$ [Eq. \ref{['eq:JzVariable']}] as a function of the scaled atomic frequency $\omega_{0}/\omega$ and the scaled coupling parameter $\gamma/\omega$. The color scale represents the numerical value of the last observable. The green solid line represents the scaled critical coupling $\gamma_{\text{c}}/\omega=\sqrt{\omega_{0}j/(2\omega)}$. (b1)-(b3) Numerical implementations of panels (a1)-(a3). Each column identifies a different system size: (a1)-(b1) $j=10$, (a2)-(b2) $j=25$, and (a3)-(b3) $j=100$.
• Figure 2: Projections of (a1) displaced ground-state energy $E_{0} + j|\omega_{0}|$ [Eq. \ref{['eq:GroundStateEnergy']}], (b1) number of photons $\langle \hat{a}^{\dagger}\hat{a} \rangle$ [Eq. \ref{['eq:PhotonNumber']}], and (c1) atomic operator $\langle \hat{J}_{z} \rangle/j$ [Eq. \ref{['eq:JzVariable']}], as a function of the scaled atomic frequency $\omega_{0}/\omega$. Each color identifies a different system size $j=20,40,100,200$. The red dashed line represents the thermodynamic limit $j\to\infty$. (a2)-(c2) Numerical implementations of panels (a1)-(c1). The coupling parameter is $\gamma/\omega=0.45$.
• Figure 3: (a1)-(a4) Energy surface $h_{\text{D}}(q,p)$ in Eq. \ref{['eq:E_surface_bosonic']} as a function of the bosonic coordinates $(q,p)$ for increasing coupling parameter $\gamma/\omega=0.1,0.2,0.4,0.5$, scaled by the field frequency $\omega$. (b1)-(b4) Orthogonal projections of panels (a1)-(a4) over the bosonic plane $(q,p)$. The color scale shows values of the scaled classical energy $E/j$ that begins at the scaled ground-state energy $E_{0}/j=-|\omega_{0}|$. System parameters: $\omega=1$ and $\omega_{0}=2\omega$.
• Figure S1: (a1)-(a3) Phase diagrams of the number of photons $\langle \hat{a}^{\dagger}\hat{a} \rangle$ [Eq. (6) in the main text] as a function of the scaled atomic frequency $\omega_{0}/\omega$ and the scaled coupling parameter $\gamma/\omega$. The color scheme represents the value of the number of photons in a logarithmic scale. The orange solid line represents the scaled critical coupling $\gamma_{\text{c}}/\omega=\sqrt{\omega_{0}j/(2\omega)}$. (b1)-(b3) Numerical implementations of panels (a1)-(a3). Each column identifies a different system size: (a1)-(b1) $j=10$, (a2)-(b2) $j=25$, and (a3)-(b3) $j=100$.
• Figure S2: (a1)-(a3) Phase diagrams of the displaced ground-state energy $E_{0} + j|\omega_{0}|$ [Eq. (5) in the main text] as a function of the scaled atomic frequency $\omega_{0}/\omega$ and the scaled coupling parameter $\gamma/\omega$. The color scheme represents the value of the displaced ground-state energy. The black solid line represents the scaled critical coupling $\gamma_{\text{c}}/\omega=\sqrt{\omega_{0}j/(2\omega)}$. (b1)-(b3) Numerical implementations of panels (a1)-(a3). Each column identifies a different system size: (a1)-(b1) $j=10$, (a2)-(b2) $j=25$, and (a3)-(b3) $j=100$.