Metamaterials and Ultra-strong Coupling

Main content

Enhancement and tunability of light-matter interaction is crucial for fundamental studies of cavity quantum electrodynamics (QED) and for applications in classical and quantum devices 1.  The coupling between one cavity photon and one elementary electronic excitation is quantified by the vacuum Rabi frequency W  1. The non-perturbative strong light-matter coupling regime is achieved when OMEGA is larger than the loss rates of the cavity and electronic excitations. Recently, growing interest has been generated by the so-called ultrastrong coupling regime 2,3 which is obtained when the vacuum Rabi frequency OMEGA becomes an appreciable fraction of the unperturbed frequency omega. In such a regime the system under consideration has to be treated beyond the rotating wave approximation, analogously to what happens in spin resonance for high irradiation powers leading to effects such as the Bloch-Siegert shift. The consequence of these additional terms is the modification of the ground and excited state properties of the light-matter coupled system. The ultrastrong coupling regime of cavity QED has been predicted to display intriguing and peculiar quantum electrodynamics features:  Casimir-like squeezed vacuum photons upon either non-adiabatic change or periodic modulation in the coupling energy 3, non-classical radiation from chaotic sources 4 and others.

Semiconductor-based systems operating in the Mid-IR and THz are especially attractive for the study of this peculiar regime as very large intersubband dipole moments d can be achieved 5,6,7. The system can also benefit from the  enhancement of the light-matter coupling by deriving from the simultaneous coupling of N electronic excitations with dipole d to the vacuum fluctuations Evac 8 of the same cavity mode.

MM and 2DEG  
Fig. 1 Schematic of the light-matter coupling experiments with metallic metasurfaces and 2DEG. The magnetic field applied perpendicularly to the surface of the 2DEG gives rise to a parabolic potential where equidistantly spaced Landau levels are formed. The complementary metasurface is based on split-ring resonators which are LC circuits where the current J and the electric field E are separated spatially. The in-plane electric field E that couples with the TE polarized Inter-Landau-level transition (calculated with 3D finite element solver) is plot in color scale and is mainly concentrated in the gap between the metals. Oscillating currents J are flowing in the metallic apertures.

The matter part of our system is still constituted by a semiconductor quantum well but we do not exploit intersubband transitions coming from size quantization. The THz and microwave-active transition we consider is the one arising between consecutive Landau levels which are created when the 2-dimensional electron gas (2DEG) is immersed in a DC magnetic field parallel to the growth axis of the heterostructure.  As discussed in  9, it can be shown that the normalized light-matter coupling ratio for this cyclotron-based system  scales in the following way with the relevant physical parameters 10,11:

Relative coupling strength



is the 2DEG fill factor, rho is the 2DEG sheet carrier density, alpha is the fine structure constant, e the elementary charge and A is a quantity related to the vector potential of the THz mode.

We experimentally implement this scheme using a strongly subwavelength resonant cavity constituted by split-ring resonators which are the building blocks of metamaterials and  metasurfaces 13 14 . One of the salient features of the split-ring resonators is their extreme sub-wavelength dimensions which allow the concentration of the electromagnetic fields in extremely reduced volumes. For our purposes this greatly enhances the vacuum field fluctuations allowing us to achieve extremely high normalized coupling ratios.

A schematic of the typical sample employed in our experiments is presented in Fig. 1. We leveraged on this experimental arrangement and we demonstrated record-high coupling ratios up to 0.87. (Fig 2(a)), ultrastrong coupling with superconducting cavities (NJP) and electrically tunable ultrastrong coupling. We were able to identify peculiar behavior of ultrastrongly coupled systems as illustrated in Fig. 2(b,c,d).

Fig. 2 (a): Transmission through sample with Nb resonator with the LC-mode at 310 GHz. (b):  The evolution of the normalized polariton frequencies at resonance 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). (c): Calculated polariton branches as a function of the coupling strength for a resonator frequency of 350 GHz.  (d): Normalized polaritonic gap as function of the normalized coupling ratio.(Adapted with permission from 11).

The possibility to modulate the ultrastrong coupling regime by changing the characteristics of the cavity can be implemented by using superconducting resonators with high switching capabilities.  To maximize the switching effect allowed by the presence of the superconductor, we adopted the following design strategy: we increased the radiative Q factor in order to have a structure whose resonance line width is loss limited. The radiative quality factor can be engineered by acting on the capacitor gap dimension in order to reduce the efficiency of the dipolar coupling as well as the inter-meta-atom spacing . An SEM picture of the fabricated metasurface is reported in Fig. 3 (a) together with 3D simulations of the surface currents. Measurements of the metasurface reported in Fig.3c yield a quality factor Q=54 increasing by a factor of 10 the usual quality factor of split-ring resonators.

Fig. 3 (a): SEM picture of one meta-atom constituting the complementary Nb metasurrface (100 nm thick film ). (b): finite element 3D simulation of the currents flowing in the meta-atom for the Nb in the superconducting state. The highest current density is in the long and narrow wires.(c): Metasurface intensity transmission at T=2.6 K normalised to the transmission value at T=10 K for a long scan of 40 mm. Inset (high): color plot of the metasurface amplitude transmission as a function of the temperature, normalised to the transmission at 10 K. Inset (low): Metasurface transmission at 10 K referenced to empty sample holder: no resonance is observed (d): Experimental transmission at 2.6 K normalized to transmission at 10 K for the Nb metasurface (blue line, left y-axis) together with simulated curve for the same quantity (red line, left y-axis). Simulated transmission for Perfect Electric Conductor (light blue line, right y-axis) and Gold (yellow line, right y-axis).

The ultrastrong coupling with 2DEGs is at the heart of our ERC project MUSiC and the next developments will included millikelvin measurements of ultrastrongly coupled 2DEGs in the fractional quantum Hall regime, superconducting cavities and few electron ultrastrong coupling.

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