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Characterizing Superradiant Phase of the Quantum Rabi Model

Yun-Tong Yang, Hong-Gang Luo

TL;DR

This work addresses the superradiant phase transition in the quantum Rabi model by introducing a two-step diagonalization that yields accurate ground- and excited-state wavefunctions across weak to deep-strong couplings. By analyzing these wavefunctions, the authors uncover an evolution from a single-well to a double-well effective potential and study photon populations in Fock space, revealing distinct statistical regimes. Near the transition, photon populations follow a Poissonian-like distribution, which shifts to GUE statistics with stronger coupling, while certain excited states display GOE behavior due to the emergent potential barrier. These findings deepen understanding of the superradiant phase and offer insight into the spectral-statistics structure of the QRM and related models.

Abstract

Recently, a superradiant phase transition first predicted theoretically in the quantum Rabi model (QRM) has been verified experimentally. This further stimulates the interest in the study of the process of phase transition and the nature of the superradiant phase since the fundamental role of the QRM in describing the interaction of light and matter, and more importantly, the QRM contains rich physics deserving further exploration despite its simplicity. Here we propose a scheme consisting of two successive diagonalization to accurately obtain the ground-state and excited states wavefunctions of the QRM in full parameter regime ranging from weak to deep-strong couplings. Thus one is able to see how the phase transition happens and how the photons populate in Fock space of the superradiant phase. We characterize the photon populations by borrowing the distribution concept in random matrix theory and find that the photon population follows a Poissonian-like distribution once the phase transition happens and further exhibits the statistics of Gaussian unitary ensemble as increasing coupling strength. More interestingly, the photons in the excited states behave even like the statistics of Gaussian orthogonal ensemble. Our results not only deepen understanding on the superradiant phase transition but also provide an insight on the nature of the superradiant phase of the QRM and related models.

Characterizing Superradiant Phase of the Quantum Rabi Model

TL;DR

This work addresses the superradiant phase transition in the quantum Rabi model by introducing a two-step diagonalization that yields accurate ground- and excited-state wavefunctions across weak to deep-strong couplings. By analyzing these wavefunctions, the authors uncover an evolution from a single-well to a double-well effective potential and study photon populations in Fock space, revealing distinct statistical regimes. Near the transition, photon populations follow a Poissonian-like distribution, which shifts to GUE statistics with stronger coupling, while certain excited states display GOE behavior due to the emergent potential barrier. These findings deepen understanding of the superradiant phase and offer insight into the spectral-statistics structure of the QRM and related models.

Abstract

Recently, a superradiant phase transition first predicted theoretically in the quantum Rabi model (QRM) has been verified experimentally. This further stimulates the interest in the study of the process of phase transition and the nature of the superradiant phase since the fundamental role of the QRM in describing the interaction of light and matter, and more importantly, the QRM contains rich physics deserving further exploration despite its simplicity. Here we propose a scheme consisting of two successive diagonalization to accurately obtain the ground-state and excited states wavefunctions of the QRM in full parameter regime ranging from weak to deep-strong couplings. Thus one is able to see how the phase transition happens and how the photons populate in Fock space of the superradiant phase. We characterize the photon populations by borrowing the distribution concept in random matrix theory and find that the photon population follows a Poissonian-like distribution once the phase transition happens and further exhibits the statistics of Gaussian unitary ensemble as increasing coupling strength. More interestingly, the photons in the excited states behave even like the statistics of Gaussian orthogonal ensemble. Our results not only deepen understanding on the superradiant phase transition but also provide an insight on the nature of the superradiant phase of the QRM and related models.
Paper Structure (5 sections, 11 equations, 4 figures)

This paper contains 5 sections, 11 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Model and Method
  3. Results and Discussion
  4. Summary and Outlook
  5. Acknowledgments

Figures (4)

  • Figure 1: The energy levels of the ground state and the other $9$ low-lying excited states as functions of the coupling strength scaled by $g_c = \sqrt{1+\sqrt{1+\frac{\Delta^2}{16}}}$Ying2015. The lines (blue and red ones denote different parity) are our results and the symbols those obtained by numerical ED with the same parameter $\Delta = 10$. The inset presents the result of photons as functions of the coupling strength for the ground-state and the first three excited states.
  • Figure 2: The effective potentials (a1-a3) and the wavefunctions (blue and red lines) for the ground state (b1-b3) and the first (c1-c3), the second (d1-d3), and the third (e1-e3) excited states for three coupling strength $g/g_c = 0.5$, $1.0$, and $1.5$. For comparison, we also present the results of numerical ED (symbols). The parameters used are the same as in Fig. \ref{['fig1']}. From the wavefunctions obtained the parity of the states is not obviously broken.
  • Figure 3: The photon population $P(n)$ in Fock space for the ground state in three coupling strengths $g/g_c = 0.5, 1.0$, and $1.5$ for different $\Delta$'s: $5$ (the first row, a1-a3), $10$ (the second row, b1-b3), $20$ (the third row, c1-c3), and $30$ (the fourth row, d1-d3). The red dots and blue triangles denote the photon population in Fock basis with odd and even parity. The dash lines represent the fits of the Poissonian statistics-like and the dashed-dot lines represent the fits of the statistics of GUE-like. The fitting parameters and details are given in SM sm2022.
  • Figure 4: The photon population of (a) the ground state, (b) the first, (c) the second, and (d) the third excited states in the superradiant phase. The parameters used read $\Delta = 10$ and $g/g_c = 1.5$. The insets in (b) and (d) show the details of the tails of GUE and GOE, in which the GUE decays more rapidly than that of the GOE. In the second and third excited states the populations of lower low-lying energy levels are scattered and do not exhibit typical statistical features.