We consider the situation where one photon enters the system, in resonance with the zero-phonon exciton, and study the subsequent dynamics. In this work we assume that the decay rates are small enough to be safely neglected in the timescales we examine; the effect of losses will be analyzed in a subsequent publication. It can be anticipated, nevertheless, that the high losses present in today’s room-temperature plasmonic cavities would have to be drastically reduced in order to observe any ultrastrong coupling effects. This can perhaps be achieved by lowering the temperature, considering metallodielectric cavities with high dielectric index, or using quantum circuits [22].

Figure 2 renders, for different values of *λ* and *g*, the time evolution of the photon number *P*(*t*)≡〈*a*^{+}*a*〉(*t*) (the exciton number *E*(*t*)=〈*σ*^{+}*σ*^{−}〉(*t*) is complementary to *P*(*t*), as their sum is 1). Each panel shows the comparison between the calculations using the full HQR and the HJC models. In the *λ*=0 case, the vibrational degrees of freedom decouple and, for the considered initial condition, the system is always in the zero-vibration state. Thus, the molecule behaves as a 2LS and the system maps into traditional CQED, where ultrastrong coupling effects are negligible for *g*=0.05 (Figure 2B) and very small even for *g*=0.2 (Figure 2C). As shown in the figure, the frequency of the Rabi oscillations Ω_{R} strongly decreases with *λ*. This occurs because the oscillations mainly involve the two lowest polaritonic states, whose energy decreases with *λ* (as shown in Figure 1C). But, notably, the influence of the CR terms on the dynamics is strongly enhanced for larger values of *λ*, as shown by the incompleteness of Rabi oscillations in the lower panels of Figure 2. This is highlighted in Figure 3, which renders the comparison between the time-averaged values for *P*(*t*), *E*(*t*), and *V*(*t*)≡〈*b*^{+}*b*〉(*t*) when the CR terms have been either considered or neglected, as a function of *λ*. In the last (“Jaynes-Cummings”) case, *P*_{JC}=*E*_{JC}=1/2 for all *λ*. The presence of CR terms change the occupations in two ways. First, they “dress” the bare energies of the states (“Bloch-Siegert” effect). This can be taken into account considering *H*_{CR} as a perturbation to the HJC Hamiltonian. Within second order, the Bloch-Siegert corrections to the bare eigen energies are

Figure 2: Time evolution of the average number of photons, *P*(*t*), for different values of *λ* and *g*.

The photon and the zero-phonon exciton are considered to be in resonance, *ω*_{c}=Δ, and *ω*_{v}=0.075. Each panel shows the comparison between the full HQR model (red curves) and the one without the CR terms (Holstein-Jaynes-Cummings model, black curves). Top panels (A–C) are for *λ*=0, and bottom ones (D–F) are for *λ*=2.5. The case *g*=0.05 is rendered for both values of *λ*, while the other panels are representative of the values *g* needed for ultrastrong coupling effects to appear for each *λ*.

Figure 3: Time-averaged number of excitons (*E*), photons (*P*), and molecular vibrations (*V*) as a function of *λ*.

Continuous lines are computed using the Holstein-Quantum-Rabi model, while dashed lines were computed by neglecting the counter-rotating terms (HJC model). The inset shows the Bloch-Siegert correction to the photon energy, Δ*E*_{BS}(|↓, 1, 0〉), and the effective value of *g* (i.e. half their energy difference between the two lowest polaritons). The parameters used are *ω*_{c}=Δ=1, *ω*_{v}=0.075, and *g*=0.05.

$$\begin{array}{l}\Delta {E}_{\text{BS}}\mathrm{(}\mathrm{|}\mathrm{\downarrow}\mathrm{,}\text{\hspace{0.17em}}\mathrm{1,}\text{\hspace{0.17em}}0\mathrm{\u3009}\mathrm{)}={g}^{2}{\displaystyle \sum _{\tilde{n}}}\frac{|\u3008\uparrow \mathrm{,}\text{\hspace{0.17em}}\mathrm{2,}\text{\hspace{0.17em}}\tilde{n}\mathrm{|}{\sigma}^{+}{a}^{+}\mathrm{|}\mathrm{\downarrow}\mathrm{,}\text{\hspace{0.17em}}\mathrm{1,}\text{\hspace{0.17em}}0\mathrm{\u3009}{|}^{2}}{{\omega}_{c}-\mathrm{(}\Delta +2{\omega}_{c}+\tilde{n}{\omega}_{v}\mathrm{)}}\\ \text{\hspace{1em}}\approx -\frac{2{g}^{2}}{{\omega}_{c}+\Delta +{\lambda}^{2}{\omega}_{v}},\text{\hspace{1em}}\Delta {E}_{\text{BS}}\mathrm{(}\mathrm{|}\mathrm{\uparrow}\mathrm{,}\text{\hspace{0.17em}}\mathrm{0,}\text{\hspace{0.17em}}\tilde{0}\mathrm{\u3009}\mathrm{)}=0,\end{array}$$(4)

where, in the approximation to Δ*E*_{BS}(|↓, 1, 0〉), we have used the following properties of the Frank-Condon factors: (i) $\u3008\tilde{n}\mathrm{|}0\mathrm{\u3009}$ is peaked at $\tilde{n}={\lambda}^{2}$ and (ii) ${\sum}_{\tilde{n}}}\mathrm{|}\mathrm{\u3008}\tilde{n}\mathrm{|}m\mathrm{\u3009}{\mathrm{|}}^{2}=1.$ The important point is that the Bloch-Siegert corrections dress the exciton and photon states differently. This “ultrastrong” mechanism can be incorporated into an effective Jaynes-Cummings model by “renormalizing” the photon frequency *ω*_{c}→ *ω*_{c}+Δ*E*_{BS}(|↓, 1, 0〉), which clearly affects whether the photon is in resonance with the exciton or not. This shift, combined with the strong renormalization of the effective coupling that occurs at large *λ*, brings the exciton and photon out of resonance. This is illustrated in the inset to Figure 3. Assuming the bare resonant condition *ω*_{c}=Δ, the system essentially remains at resonance for values of *λ* such that |Δ*E*_{BS}(|↓, 1, 0〉)|<*g*_{eff}, thus developing complete Rabi oscillations. But these cease to happen when |Δ*E*_{BS}(|↓, 1, 0〉)| and *g*_{eff} are comparable (for $\lambda \gtrsim 2.5$ in the inset to Figure 3, computed for *g*=0.05). As *g*_{eff}~*g*, while Δ*E*_{BS}~*g*^{2}, the BS corrections are more relevant for larger exciton-photon interactions but, admittedly, this effect plays a role only for large values of *λ*. It is worth noting that the CR parity-breaking terms we mentioned when discussing the effective Hamiltonian (3) cancel when the “renormalized” cavity is in resonance with the 2LS, i.e. when *ω*_{c}+Δ*E*_{BS}=Δ. In that case, Hamiltonian (3) is virtually exact in the polariton subspace, for all values of *g*, *ω*_{v}, and *λ*.

The second way in which the CR terms modify the occupations occurs at smaller values of *λ*. It works via mixing states which would be orthogonal within the HJC Hamiltonian but anti-cross when the CR terms are considered (which, as mentioned before and shown in Figure 1, occurs for the states *P*_{0+} and *P*_{2−} at *λ*≈0.74). This mixing allows *P*_{0+} to couple to *P*_{2−}, thus enhancing the average number of phonons *P* (see the peak in *P* in Figure 3, at *λ*≈0.74).

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