Ultrastrong coupling in the near field of complementary split-ring resonators

The ultrastrong light-matter interaction regime was investigated in metallic and superconducting complementary split-ring resonators coupled to the cyclotron transition of two-dimensional electron gases. The subwavelength light confinement and the large optical dipole moment of the cyclotron transition yield record high normalized coupling rates of up to $\frac{{\Omega }_R}{{\omega }_c}=0.87$. We observed a blue-shift of both polaritons due to the diamagnetic term of the interaction Hamiltonian.

Figure 1

(a) Schematic drawing of the experiment depicts the polarization and wave vector of the probing THz pulse and the sample with the complementary split-ring resonator surface and a 2DEG. The static magnetic field is applied perpendicular to the sample surface/growth direction (Faraday geometry). (b) The ratio of the transmission coefficients $t\left(B_{\mathrm{res}}\right)/t(B=0\text{T})$ for the heterostructures without resonator show a minimum at the bare cyclotron resonance frequency ${\omega }_c$ [$B_{\mathrm{res}}$ is the resonant magnetic field: ${\omega }_c\left(B_{\mathrm{res}}\right)={\omega }_{\text{LC}}$]. In panels (c) and (d), FEM simulations of the direct and complementary split-ring resonator show the in-plane electric field distribution $E_{\mathrm{plane}}=\sqrt{|E_x{|}^2+{\left|E_y\right|}^2}$ 100 nm below the semiconductor surface(color scale) and the current distribution in the gold structure (red arrows). (e) Complementary split-ring resonators show a complementary transmission spectrum compared to their direct counterpart. The insets clarify the polarization of the electric field for both cases (yellow: Au, black: GaAs). The simulation with the conductivity of gold $\left({\epsilon }_{\text{Au}}\right)$ fits well the measured transmittance. FEM simulations predict a Nb-resonator frequency ${\omega }_{\text{LC}}=2\pi 340$ GHz. Panel (f) shows a cut along the dashed white line in panel (d).

Figure 2

(a) From FEM simulation: the electric field in the polarization localized in the gap ($E_x$ for complementary and $E_y$ for direct split-ring resonators) integrated over the $x\text{−}y$ plane is plotted as function of the $z$ position. (b) Simulations show a modification of the field confinement in $z$ direction due to the coupling with the cyclotron transition. The gray (black) lines indicate the positions of the 2DEGs in the D111118 (EV1452). The effective volume $V$ is increased as more 2DEGs are added.

Figure 3

(a) Contour plot of the transmittance through the direct split-ring resonator on EV1452. The anticrossing of the LC mode with the cyclotron transition takes place at $B_{\mathrm{res}}=1.2$ T. Uncoupled areas of the 2DEG give rise to the minimum evolving linearly with $B$ (transmission measurements were performed at an interval of $\Delta B=0.2$ T; this periodicity appears as an apparent modulation of the transmission). (b) The same measurement for the complementary version of of the split-ring resonator on EV1452 (color scale is inverted). The uncoupled parts of the 2DEG do not contribute to the signal. (c) Comparison of the transmission at three characteristic values of the magnetic field for the direct and complementary split-ring resonators.

Figure 4

Transmittance through samples $\mathbf{C}$ (4 QWs) and $\mathbf{D}$ (20 QWs) are shown in (a) and (b), respectively. The normalized coupling rates are ${({\Omega }_R/{\omega }_c)}_C=0.57$ and $({\Omega }_R/{\omega }_c{)}_D=0.72$. Solid lines are fits of the polariton dispersion [see Eq. (17) in Ref. [26]) to the transmittance maxima (${\Omega }_R/{\omega }_c$ is the fitting parameter).

Figure 5

(a) The transmission measurement through sample $\mathbf{E}$ (InAs quantum well) with a normalized coupling rate ${\left(\frac{{\Omega }_R}{{\omega }_c}\right)}_E=0.69$. Panel (b) shows the transmission through sample $\mathbf{F}$ (niobium split-ring resonator on D091113) with the LC-mode frequency at 310 GHz. In this sample, we attain a record high splitting $2{\left(\frac{{\Omega }_R}{{\omega }_c}\right)}_F=2\times 0.87$ at the resonant field $B_{{\mathrm{res}}_F}=0.75$ T.

Figure 6

(a) Normalized polaritonic gap as function of the normalized coupling ratio. Data points include samples with direct and complementary resonators [see (b) for the legend]. The inset depicts the extent of the polaritonic gap for sample $\mathbf{E}$. (b) The evolution of the normalized polariton frequencies at resonance $\left({\omega }_c={\omega }_{\text{LC}}\right)$ follows clearly the blue-shifted prediction (in blue). The gray lines show the linear behavior expected in the strong coupling regime (without diamagnetic and counter-rotating terms).

Figure 7

In (a) and (b) is the electric field transmitted through sample D111118 and E110913 measured for finite magnetic fields, normalized to the electric field transmitted at zero magnetic field.