Polaron picture of the two-photon quantum Rabi model

We employ a polaron picture to investigate the properties of the two-photon quantum Rabi model (QRM), which describes a two-level or spin-half system coupled with a single bosonic mode by a two-photon process. In the polaron picture, the coupling in the two-photon process leads to spin-related asymmetry so that the original single bosonic mode splits into two separated frequency modes for the opposite spins, which correspond to two bare polarons. Importantly, the tunneling causes these two bare polarons to exchange their components with each other, thus leading to additional induced polarons. According to this picture, the variational ground-state wave function of the two-photon QRM can be correctly constructed, with the ground-state energy and other physical observables in good agreement with the exact numerics in all the coupling regimes. Furthermore, generalization to multiple induced polarons involving higher orders in the tunneling effect provides a systematic way to yield a rapid convergence in accuracy even around the difficult spectral collapse point. In addition, the polaron picture provides a distinctive understanding of an incomplete spectral collapse behavior that is about the existence of discrete energy levels apart from the collapsed spectrum at the spectral collapse point. This work illustrates that the polaron picture is helpful to capture the key physics in this nonlinear light-matter interaction model and indicates that this method can be applicable to more complicated QRM-related models.

Figure 1

(a)–(c) Schematic diagrams for bare potentials and tunneling-induced deformation of effective potentials. (a) In the absence of the tunneling between two spin states, i.e., $\mathrm{\Omega }=0$, the single bosonic mode is nonlinearly coupled with two spin states labeled by $\pm$ to form two bare potentials $v^{\pm }$ which determine two bare polarons (${\mathrm{\Psi }}^{-}$ and ${\mathrm{\Psi }}^{+}$ filled in blue and red). (b) When the tunneling is switched on, namely, $\mathrm{\Omega }\ne 0$, the bare polarons couple with each other and induce an additional potential $\delta v^{+}=-\frac{\mathrm{\Omega }}{(1-2g^{\prime })\omega }\frac{{\mathrm{\Psi }}^{-}}{{\mathrm{\Psi }}^{+}}$ and $\delta v^{-}=-\frac{\mathrm{\Omega }}{(1+2g^{\prime })\omega }\frac{{\mathrm{\Psi }}^{+}}{{\mathrm{\Psi }}^{-}}$. (c) Total effective potential $v_{\mathrm{eff}}^{\pm }=v^{\pm }+\delta v^{\pm }$. The solid lines denote the effective potentials and the dashed lines are the bare potentials. (d)–(f) Tunneling-induced polarons in different regimes of tunneling strength. In the presence of tunneling, each wave function ${\mathrm{\Psi }}^{\pm }$ (sketched by thick solid lines) consists of two separated components, one originating from the bare polaron and the other induced by the tunneling, depicted, respectively, by ${\mathrm{\Psi }}^{+}\left(x\right)={\alpha }_1{\phi }_{{\alpha }_1}+{\alpha }_2{\phi }_{{\alpha }_2}$ and ${\mathrm{\Psi }}^{-}\left(x\right)={\beta }_1{\phi }_{{\beta }_1}+{\beta }_2{\phi }_{{\beta }_2}$ [see Eq. (8)]. The wave function ${\mathrm{\Psi }}^{-}\left(x\right)$ consists of an original polaron with low frequency (filled in red) and an induced polaron with high frequency (filled in blue). Meanwhile, the wave function ${\mathrm{\Psi }}^{+}\left(x\right)$ consists of an original polaron with high frequency (filled in blue) and an induced polaron with low frequency (filled in red). Representative cases in the (d) weak, (e) intermediate, and (f) strong tunneling regimes show successively the increasing weight for the induced polarons.

Figure 2

Ground-state energy as a function of the coupling strength $g$ renormalized by the frequency $\omega$. The red solid line ($N=2$) represents the result obtained by our frequency-shifted two-pair polaron trial state given by Eq. (8). The yellow dash-dotted line is the result obtained by the bare state given by Eq. (9) and the blue dashed line ($N=1$) denotes the result from the simple trial state given by Eq. (10). In addition, the green circles give the exact diagonalization result, which provides a benchmark. Here we set $\omega =1$. For simplicity, we set $\mathrm{\Omega }=1$ hereafter.

Figure 3

Ground-state wave function obtained by the wave-function ansatz with two pairs ($N=2$) of polaron (solid lines), in comparison with the exact numerics (circles) for (a) spin-down and (b) spin-up components. The dash-dotted lines denote the original polaron and dashed lines show the polaron induced by the tunneling interaction. The results (dotted lines) for a pair of polarons ($N=1$) are also presented for comparison.

Figure 4

Physical observables as a function of the coupling strength $g/\omega$: (a) ground-state energy, (b) spin polarization $\langle {\sigma }_x\rangle$, (c) coupling correlation function $\langle {\sigma }_z({\left(a^{†}\right)}^2+a^2)\rangle$, and (d) mean photon number $\langle a^{†}a\rangle$. The green circles are the exact numerical results taken as benchmarks and the red solid lines are our results obtained by two pairs of polarons ($N=2$) for the wave-function ansatz (8). The results (blue dashed lines) for one pair of polarons for the wave-function ansatz ($N=1$) are also presented for comparison. Here we set $\omega =1$.

Figure 5

Ground-state energy as a function of pair number of polarons $N$ at the spectral collapse point ($g/\omega =0.5$). Here $E_{\mathrm{ED}}=-0.743858$ is the energy from exact diagonalization and $E_{\mathrm{N}}$ is the energy in the multipolaron method. When $N\ge 4$, the error is almost invisible in comparison to the exact numerical result (dashed line). The inset shows the relative error $(E_{\mathrm{N}}-E_{\mathrm{ED}})/\left|E_{\mathrm{ED}}\right|$ for the ground-state energy as a function of $N$.

Figure 6

Schematic comparison of the polaron picture in (a) the one-photon QRM and (b) the two-photon QRM. (a) shows displaced effective harmonic oscillators, while (b) shows frequency-shifted effective harmonic oscillators where the original frequency of the bosonic mode is taken to be 1 as an example. Here ${\mathrm{\Psi }}^{+}$ (${\mathrm{\Psi }}^{-}$) is the spin-up (spin-down) component indicated by the position above (below) the dimension axis. The longer solid (shorter dashed) arrows mark the locations of the original (induced) polarons, with their arrow directions labeling the opposite spins.

Figure 7

(a)–(c) Spectral properties of the two-photon QRM as a function of the coupling strength $g/\omega$ at different tunneling strengths $\mathrm{\Omega }/\omega$. One can see coexistence of the discrete and collapsed energy levels of the spectrum structure at the collapse point for an finite $\mathrm{\Omega }/\omega$. Complete collapse of the spectrum happens as $\mathrm{\Omega }/\omega \to 0$. The black solid arrow points to the location of the spectral collapse energy $E_n/\omega =-1/2$. The spectra are calculated by ED with the cutoff in photon number equal to (a) 1600, (b) 800, and (c) 800. (d)–(f) Schematic diagram of the potential curve at $g/\omega =0.5$, where the effective potential would be the tunneling induced potential $v_{\mathrm{eff}}^{-}=v^{-}+\delta v^{-}=\delta v^{-}$. The depth of the tunneling induced potential well $\delta v^{-}$ is related to the tunneling strength $\mathrm{\Omega }/\omega$.

Figure 8

Overview of the ground-state wave function ${\mathrm{\Psi }}^{\pm }(x,g/\omega )$. Here $x$ is the position and $g/\omega$ is the coupling strength. (a) and (b) Tunneling strength $\mathrm{\Omega }/\omega \to 0$. The ground-state property is determined by the bare potential $v^{\pm }$. Thus, ${\mathrm{\Psi }}^{+}$ will become singular and ${\mathrm{\Psi }}^{-}$ will spread out as $g/\omega$ increases. (c) and (d) Tunneling strength $\mathrm{\Omega }/\omega =10$. A large tunneling makes ${\mathrm{\Psi }}^{+}$ and ${\mathrm{\Psi }}^{-}$ neutralize each other.

Figure 9

Spectral properties of the two-photon QRM as a function of the tunneling strength $\mathrm{\Omega }/\omega$. The inset shows the energy-level interval changes as a function of the tunneling strength $\mathrm{\Omega }/\omega$.

Figure 10

Schematic diagram of the potential curve above the spectral collapse point. The tunneling process also contributes an additional potential well, which may lead to a spurious convergence in ED.