On the Origin of the Hidden Symmetry in the Asymmetric Quantum Rabi Model

TL;DR

This paper investigates the origin of the hidden symmetry in the asymmetric quantum Rabi model (AQRM) by applying two successive diagonalizations in the strong-coupling regime where the two-level splitting Δ is large relative to the mode frequency ω. It shows that degeneracies reappear at specific bias values η corresponding to multiples of the mode frequency, and identifies energy-level matching within an asymmetric double-well as the microscopic origin, a conclusion supported by the effective potential and detailed wavefunction analyses. The work also discusses an excited-state quantum phase transition linked to the hidden symmetry, suggesting symmetry breaking in strong coupling and restoration in weaker coupling. Overall, the study provides a coherent physical picture of the AQRM in a largely unexplored parameter regime and has implications for circuit QED experiments where the bias term can be tuned externally.

Abstract

The introduction of an asymmetric term into the quantum Rabi model generally lifts energy-level degeneracies. However, when the asymmetry parameter takes specific multiples of the bosonic mode frequency, level degeneracies reappear$-$a phenomenon referred to as the hidden symmetry in the asymmetric quantum Rabi model. Identifying the origin of this hidden symmetry and its explicit operator form constitutes two central tasks in studying this system. Here, we investigate the origin of this hidden symmetry using the method of two successive diagonalizations, with a focus on physics in the regime where the ratio between the two-level splitting $Δ$ and the mode frequency $ω$ satisfies $Δ/ω\gg 1$. We find that the hidden symmetry stems from energy-level matching within the asymmetric double-well potential, a picture strongly supported by the wavefunctions of both the ground and excited states. Moreover, the emergence of an excited-state quantum phase transition is identified and qualitatively discussed, which arises from the breaking and restoration of this hidden symmetry across different coupling regimes. Our results provide deeper insight into the physics of the asymmetric quantum Rabi model, particularly in the previously less-explored strong-coupling regime where $Δ/ω\gg 1$.

Figures (6)

• Figure 1: Energy spectrum of the AQRM as functions of the coupling strength scaled by $g_c = \sqrt{1+\sqrt{1+\frac{\Delta^2}{16}}}$Ying2015 with various asymmetric parameters (a) $\eta=0$, (b) $\eta=0.2$, (c) $\eta=0.5$, (d) $\eta=0.8$, (e) $\eta=1.0$, and (f) $\eta=1.5$. The lines are the results computed using our method and the solid dots are those obtained by numerical ED with the same parameter $\Delta = 10$,
• Figure 2: Schematic diagram of the effective potential. The orange dashed curve represents the standard QRM ($\eta=0$), while the solid black curve corresponds to the AQRM ($\eta \neq 0$).
• Figure 3: Schematic diagram of energy-level matching in an asymmetric double-well potential. (a) $2\eta = n\hbar\omega$, the ground state in the higher well becomes degenerate with a certain excited state in the lower well; (b) $2\eta \neq n\hbar\omega$, all energy levels are non-degenerate and arranged in order of increasing energy.
• Figure 4: Wavefunctions of the AQRM at $\eta=0.5$. The first, second, and third rows correspond to the ground state, first excited state, and second excited state, respectively. Each column represents a different coupling strength scaled by $g_c = \sqrt{1+\sqrt{1+\Delta^2/16}}$. Solid lines denote the results obtained by our method and circles represent those obtained from ED. Red and blue curves indicate the spin-up and spin-down components of the wavefunctions.
• Figure 5: Wavefunctions of the AQRM at $\eta=0.8$. Other information is the same as Fig. \ref{['fig4']}.