Dissecting Superradiant Phase Transition in the Quantum Rabi Model

TL;DR

This paper investigates the microscopic mechanism of the superradiant phase transition in the quantum Rabi model. It introduces an operator-space diagonalization that yields three fundamental patterns with eigenenergies $\lambda_n$ and pattern operators $\hat{A}_n$, enabling a detailed energy-flow analysis as the coupling $g$ crosses the critical point. The main finding is that pattern $\lambda_1$ actively drives the transition, pattern $\lambda_2$ acts as an inspector and triggers the activation of pattern $\lambda_3$, which then stabilizes the new superradiant phase. The approach reproduces exact diagonalization results and offers a general framework to analyze phase transitions in other light-matter models such as Dicke and spin-boson models.

Abstract

The phase transition is both thermodynamically and quantum-mechanically ubiquitous in nature or laboratory and its understanding is still one of most active issues in modern physics and related disciplines. The Landau's theory provides a general framework to describe \textit{phenomenologically} the phase transition by the introduction of order parameters and the associated symmetry breakings; and is also taken as starting point to explore the critical phenomena in connection with phase transitions in renormalization group, which provides a complete theoretical description of the behavior close to the critical points. In this sense the microscopic mechanism of the phase transition remains still to be uncovered. Here we make a first attempt to explore the microscopic mechanism of the superradiant phase transition in the quantum Rabi model (QRM). We firstly perform a diagonalization in an operator space to obtain three fundamental patterns involved in the QRM and then analyze explicitly their energy evolutions with increasing coupling strengths. The characteristic behaviors found uncover the microscipic mechanism of the superradiant phase transition: one is active to drive the happening of phase transition, the second responses rapidly to the change of the active pattern and wakes up the third pattern to stablize the new phase. This kind of dissecting mechanism explains for the first time why and how happens the superradiant phase transition in the QRM and paves a way to explore the microscopic mechanism of the phase transitions happening popularly in nature.

Figures (5)

• Figure 1: The marks of the obtained patterns and comparisons of the physical quantities with the results obtained by numerical exact diagnoalization. (a1)-(a3) Patterns marked by $\lambda_1, \lambda_2$ and $\lambda_3$ and their wavefunctions with relative phase denoted by $\pm$. Note that there is a total free phase factor $e^{i\pi}$, which is omitted here for simplicity. (b1)-(b3) The first four pattern energy levels as functions of the coupling strengths rescaled by $g_c = \sqrt{1+\sqrt{1+\frac{\Delta^2}{16}}}$Ying2015 and $\Delta = 50$ is taken here and hereafter. (c) The summations of the pattern energy levels (lines) and their comparison with the results obtained directly by numerical exact diagonalization (symbols). (d) Comparison of the summations of the pattern's photon numbers $\langle a^\dagger a \rangle_{\lambda_n}(n=1,2,3)$ (not shown) to those obtained by numerical exact diagonalization (symbols). (e) Comparison of the summations of the pattern's $\langle \sigma^x\rangle_{\lambda_n}(n=1,2,3)$(not shown) to those obtained by numerical exact diagonalization (symbols). The comparisons are made for the first four energy levels.
• Figure 2: (a1) $\&$ (b1) The energy levels of the ground state and the first excited state (heavy black solid ines) and corresponding pattern components (thin red, green, and blue solid lines) as functions of the coupling strength. (a2) $\&$ (b2) The second-order derivatives of the corresponding energy levels (heavy black solid lines) and their pattern components (thin red, green, and blue solid lines).
• Figure 3: Patterns marked by their eigenenergies and eigenfunctions as functions of the coupling strengths. The first column: (a1) the eigenenergy of the pattern $\lambda_1$ and its first- and second-order derivatives with respect to the coupling strength; (b1) the eigenfunctions $u_{1,m}$($m=1,2,3$), (c1) their first-order derivative, and (d1) their second-order derivative for the pattern $\lambda_1$ (red). The second column: (a2) the eigenenergy of the pattern $\lambda_2$ and its first- and second-order derivatives with respect to the coupling strength; (b2) the eigenfunctions $u_{2,m}$($m=1,2,3$), (c2) their first-order derivative, and (d2) their second-order derivative for the pattern $\lambda_2$ (green). The third column: (a3) the eigenenergy of the pattern $\lambda_3$ and its first- and second-order derivatives with respect to the coupling strength; (b3) the eigenfunctions $u_{3,m}$($m=1,2,3$), (c3) their first-order derivative, and (d3) their second-order derivative for the pattern $\lambda_3$ (blue). For each pattern, there are three components corresponding to the operators $i\hat{\sigma}_y$(solid lines), $\hat{\sigma}_z$(dash lines) and $\hat{a}$(dotted-dash lines).
• Figure 4: The wavefunctions of the ground state (a1)-(a3) and the first excited state (b1)-(b3) for three typical coupling strengths $g/g_c = 0.5$ (a1) $\&$ (b1); $1.0$ (a2) $\&$ (b2); $1.5$ (a3) $\&$ (b3), corresponding to the normal, critical and superradiant phase, respectively, in the Fock basis. The heavy black solid lines denote the total wavefunctions and the thin colored lines (red, green and blue) correspond to the three components in patterns $\lambda_1$, $\lambda_2$ and $\lambda_3$, respectively,
• Figure 5: The total photon number (a) and the spin-flip (b), and the corresponding components in patterns for the ground state as functions of the coupling strength. The black solid lines denote total one and the red, green, and blue thin solid lines correspond to the components in patterns $\lambda_1$, $\lambda_2$, and $\lambda_3$, respectively.