Ultrastrong coupling between THz phonons and photons caused by an enhanced vacuum electric field

Ultrastrong coupling (USC) between phonons and vacuum photons can result in fascinating quantum phenomena, though it is difficult to achieve due to the small dipole moments of phonons in solids. Here, we investigate the vacuum Rabi splitting by coupling phonons in perovskite $\mathrm{C}{\mathrm{H}}_3\mathrm{N}{\mathrm{H}}_3\mathrm{Pb}{\mathrm{I}}_3$ films with photons in split ring resonators. As the gap size of the resonator decreases, the coupling strength η increases due to the enhanced vacuum field in the gap, reaching the USC regime $(\eta \sim 0.24)$ at a gap size of 100 nm. Our results show that nanoresonators are an excellent platform for studies of vacuum-dressed phonon properties.

Figure 1

(a) Left panel: Illustration of SRRs covered with an $\mathrm{MAPb}{\mathrm{I}}_3$ thin film. The normally incident THz pulse is linearly polarized along the $x$ axis. Right panel: Optical microscope images of SRRs with and without $\mathrm{MAPb}{\mathrm{I}}_3$. (b) Transmission spectra of an 850-nm-thick $\mathrm{MAPb}{\mathrm{I}}_3$ film (red solid curve) and bare SRRs with $G=1\mu \mathrm{m}$ (blue dashed curves). (c) Resonance frequency of the 1-μm-gap bare SRRs as a function of $L$ (blue circles). The blue dashed line is the fit to a linear function. The $\mathrm{T}{\mathrm{O}}_1$ phonon frequency of $\mathrm{MAPb}{\mathrm{I}}_3$ is indicated by the black dashed line.

Figure 2

(a) Two-dimensional map of the transmission spectra for the samples with a gap size of 1 μm. The position of the $\mathrm{T}{\mathrm{O}}_1$ phonon is indicated by the black dashed line. (b) Transmission spectra that correspond to the on-resonance case. The polariton peaks are indicated by triangles. (c) Experimental polariton peak positions (circles) and the theoretical polariton dispersion predicted by the two-oscillator model (dashed curves) for different gap sizes as a function of the cavity resonance frequency. The $x$-axis positions of the black arrows are about 0.95 THz.

Figure 3

(a) Upper panel: The region around the gap of the SRR model used in the FDTD simulations. The volume of $\mathrm{MAPb}{\mathrm{I}}_3$ inside the gap is represented by $V_0$. Lower panel: The distribution of the enhancement factor ${\alpha }_{\mathrm{vac}}$ in the plane $z=–50\mathrm{nm}$ ($z=0$ defines the quartz/SRR interface) for samples with $G=0.1$ and 1 μm. (b) Theoretical gap-size dependence of the enhancement factor ${\alpha }_{\mathrm{vac}}$ averaged over $V_0$ (red circles) and its fitting result (red dashed curve). The blue dashed curve describes the gap-size dependence of the contribution of the oscillators in Eq. (5). (c) Experimental gap-size dependence of ${\mathrm{\Omega }}_{\mathrm{R}}$ (black circles), the simulation result (gray squares), and the predicted curve proportional to $G^{-0.43}$ (black dashed curve). The error bars on gray squares refer to the deviation of simulation results when considering the thickness of $\mathrm{MAPb}{\mathrm{I}}_3$: $t_{MA}\left(t_{\mathrm{MAgap}}\right)\pm 20\mathrm{nm}$. The vertical axis on the right side of the figure represents the corresponding values of η. (d) Calculated transmission spectra of the on-resonance case. The polariton peaks are indicated by triangles.

Figure 4

(a) The susceptibility of $\mathrm{MAPb}{\mathrm{I}}_3$ obtained from the transmission spectrum (solid curves) and the corresponding fitting result (dashed curves). Red and blue data represent the real and imaginary parts of the susceptibility, respectively. (b) The calculated gap-size dependence of the vacuum field in the gaps (black circles) together with the fitting result (black dashed curve). The calculated gap-size dependence of the vacuum field in the gaps (black circles) together with the fitting result (black dashed curve). The error bars come from the measurement error of thickness of ${\mathrm{MAPbI}}_3$ in gap (200 $\pm$ 20 nm) and the fitting error of the parameter $\sqrt{N}{d}_{{\mathrm{TO}}_1}$ [($2.8\pm 0.2$)$\times {10}^{-16}{\mathrm{C}/\mathrm{m}}^{0.5}$].