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# Metamaterials and Ultra-strong Coupling

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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 *E _{vac}*

^{8 }of the same cavity mode.

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}:

Here,

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).

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.

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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