Mode-multiplexing deep-strong light-matter coupling

Abstract

Dressing electronic quantum states with virtual photons creates exotic effects ranging from vacuum-field modified transport to polaritonic chemistry, and squeezing or entanglement of modes. The established paradigm of cavity quantum electrodynamics maximizes the light-matter coupling strength ${\Omega }_R/{\omega }_c$ , defined as the ratio of the vacuum Rabi frequency and the frequency of light, by resonant interactions. Yet, the finite oscillator strength of a single electronic excitation sets a natural limit to ${\Omega }_R/{\omega }_c$ . Here, we enter a regime of record-strong light-matter interaction which exploits the cooperative dipole moments of multiple, highly non-resonant magnetoplasmon modes tailored by our metasurface. This creates an ultrabroadband spectrum of 20 polaritons spanning 6 optical octaves, calculated vacuum ground state populations exceeding 1 virtual excitation quantum, and coupling strengths equivalent to ${\Omega }_R/{\omega }_c=3.19$ . The extreme interaction drives strongly subcycle energy exchange between multiple bosonic vacuum modes akin to high-order nonlinearities, and entangles previously orthogonal electronic excitations solely via vacuum fluctuations.

Fig. 1. Multi-mode light-matter coupling and ultracompact metasurface.

a Illustration of resonant ultrastrong coupling of a single cavity mode (upper parabola) to a single matter excitation (bottom parabola) with a vacuum Rabi frequency ${\Omega }_{\mathrm{R}}$. The weak population by virtual excitations in the vacuum ground state is indicated by semi-transparent spheres. b Illustration of deep-strong coupling of one light mode (upper parabola) with a frequency of ${\omega }_{j=1}$ to multiple matter excitations (bottom parabolas) with frequencies ${\omega }_{\alpha }$, by vacuum Rabi frequencies ${\Omega }_{\mathrm{R},j,\alpha }$ under off-resonant conditions. Owing to the extremely large light-matter coupling, a significant number of virtual excitations are present. c Three-dimensional cut-away illustration of a conventional metasurface structure (gold shape) and its electric near-field distribution $\tilde{E}$, for the fundamental LC mode at ν₁ = 0.52 THz. The unit cell (size: $60\mathrm{\mu m}\times 60\mathrm{\mu m}$) is indicated by the dashed line. QW: quantum well stack. d Highly compacted metasurface with a unit cell size of $30\mathrm{\mu m}\times 32.5\mathrm{\mu m}$. e Measured THz transmission of the bare resonator array (blue circles), and calculation (red curve).

Fig. 2. Deep-strong light-matter coupling.

a Plasmon frequency ${\nu }_{\mathrm{p},\alpha }$ and coupling strength ${\Omega }_{\mathrm{R},1,\alpha }/{\omega }_{j=1}$ for each MP (magnetoplasmon) mode $\alpha$ in the case of the sample with 48 QWs (quantum wells). b, c Illustration of multi-mode coupling of one cavity mode to several matter modes, MP₀ = CR, MP₁ and MP₂, as a function of the cyclotron frequency, ${\nu }_{\mathrm{c}}$. b uncoupled modes. c Coupled modes comprising of one lower polariton (LP₁) and three upper polaritons (UP_1,1, UP_1,2 and UP_1,3). d THz magneto-transmission $\mathcal{T}$ as a function of ${\nu }_{\mathrm{c}}$ of the single-QW structure. The continuous white curves trace the polariton modes obtained from the multi-mode Hopfield model for the first resonator mode (coupling strength: ${\eta }_1$ = 0.55). The dotted white curves represent the polaritons linked to the higher mode ν₂ (${\eta }_2$ = 0.13). h Spectrum obtained from time-domain quantum model and identical polariton frequencies, for comparison. e Transmission of the 3-QW structure (${\eta }_1$ = 0.76) and i simulation. f Transmission of the 6-QW structure (${\eta }_1$ = 1.34), and j simulation. g Transmission of the 12-QW structure (${\eta }_1$ = 2.32), and k simulation.

Fig. 3. Extremely strong, multi-octave light-matter coupling.

a THz magneto-transmission $\mathcal{T}$ of the 24-QW sample as a function of the cyclotron frequency, ${\nu }_{\mathrm{c}}$ (see Fig. 2). The extended Hopfield model yields coupling strengths of ${\eta }_1=2.80$ and ${\eta }_2=0.85$ for the first and second resonator mode, respectively. Calculated polariton frequencies (solid & dotted curves) with distinct resonances labelled. b Calculated transmission and identical polariton frequencies, for comparison. c Transmission of the 48-QW structure. Coupling strengths: ${\eta }_1=2.83$, ${\eta }_2=0.88$. d Calculated transmission and identical polariton frequencies.

Fig. 4. Dynamics and squeezing of extremely strong light-matter coupling.

a Transmitted THz field of the 48-QW sample (${\eta }_1$ = 2.83) at ${\nu }_{\mathrm{c}}=0.52$ THz (black curve). Inset: Spectral amplitude $\mathcal{A}$ of the THz field. b Calculated expectation value for the population of the first cavity mode $\mathrm{Re}⟨{\widehat{a}}_{j=1}⟩$ after excitation (black curve) by a broadband pulse (grey curve). The shading marks one oscillation period of the bare cavity mode. Inset: Corresponding spectra. c Calculated expectation value of the polarisation of the first MP mode, $\mathrm{Re}⟨{\widehat{b}}_{\alpha =1}⟩$, and spectrum (inset). d $\mathrm{Re}⟨{\widehat{a}}_{j=1}⟩$ for the same coupling strength as in b, yet only for a single pair of light and matter modes. Inset: Corresponding spectrum. e Energy of the first cavity mode for the full calculation (solid curve) and the single-mode reference (dotted curve) as in panel d. f Energies of the first MP mode (solid curve) and CR (dotted curve) for the two cases. g Corresponding coupling energies between cavity mode and MP mode (solid curve) or CR (dotted curve). h Wigner-function representation for the photonic state of a deep-strongly coupled system with ${\Omega }_{\mathrm{R}}/{\omega }_{\mathrm{c}}$ = 3.0 showing the quasi-probability $\mathcal{A}$ as a function of the field quadratures $\mathrm{Re}\left(\alpha \right)$ and $\mathrm{Im}\left(\alpha \right)$.