An ultrastrongly coupled single terahertz meta-atom

Free-space coupling to subwavelength individual optical elements is a central theme in quantum optics, as it allows the control over individual quantum systems. Here we show that, by combining an asymmetric immersion lens setup and a complementary resonating metasurface we are able to perform terahertz time-domain spectroscopy of an individual, strongly subwavelength meta-atom. We unravel the linewidth dependence as a function of the meta-atom number indicating quenching of the superradiant coupling. On these grounds, we investigate ultrastrongly coupled Landau polaritons at the single resonator level, measuring a normalized coupling ratio \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{{{\Omega }}}{\omega }=0.6$$\end{document}Ωω=0.6. Similar measurements on a lower density two dimensional electron gas yield a coupling ratio \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{{{\Omega }}}{\omega }=0.33$$\end{document}Ωω=0.33 with a cooperativity C = 94. Our findings pave the way towards the control of ultrastrong light-matter interaction at the single electron/ resonator level. The proposed technique is way more general and can be useful to characterize the complex conductivity of micron-sized samples in the terahertz domain.

1 Point-by-point reply to the Reviewers' comments Reviewer #1: The manuscript entitled "An ultrastrongly coupled single THz meta-atom" by S. Rajabali et al. investigates an individual metallic resonator coupled to inter-Landau level transitions via a novel setup. Overall, the manuscript is well written, properly detailed and technically sound. The three novelties of this work appear to be: i) technical: an asymmetric immersion lens setup (an extension of a symmetric version already utilized to investigate single quantum dots in the visible range) combined with a complementary metallic resonator design that allows performing THz spectroscopy of strongly subwavelength individual elements on a substrate; ii) the observation of quenching of the superradiance decay by decreasing 1 the number of metallic resonators (something already observed in other studies, but not down to the single resonator limit has done here); iii) the characterization of a single metaatom ultrastrongly-coupled to inter-Landau level transitions, leading to the observation of a coupling strength for the single resonator equivalent to the one measured for the array case, yet with a higher cooperativity (thanks to the superradiance quenching). Even if the study does not lead to transformative/unexpected observations, is of a certain relevance for the scientific community, especially because it reports on a useful tool that can allow direct investigations on subwavelength systems (such as 2D-material flakes) via THz far-field spectroscopy, without requiring complex near-field implementations. For this reason, I recommend its publication in Nature Communications after a minor revision (see comments below). Time delay [ps] With aSIL Without aSIL 110 Figure R1: THz signal in time domain from a single resonator (cold cavity) without and with aSIL configuration. The delay between the two signals corresponds to the optical thickness of the aSIL assembly.
Answer: We thank the Reviewer for this question. We would like to clarify it in more detail to avoid confusion. The signal enhancement, as the Reviewer mentions, refers to the peakto-peak value of the THz waveform for the single resonator. It is calculated by dividing the peak-to-peak value of the THz waveform with the aSIL configuration by the peak-to-peak value of the THz waveform without the aSIL assembly. For more clarity, Fig. R1 is added to the supplementary (Fig. S3).
Question 2: lines 225-228: "The lower polariton (LP) mode of the coupled single resonator at its asymptotic limit, at B = 4T, has a Q-factor of 15.4 (using time trace decay method). The Q-factor of the LP at B = 4T for the sample with the coupled 2D array of 3600 resonator without lenses and with lenses are 3.4 and 6.9, respectively". It is really hard to see this directly from the results shown in Fig.  4 (indeed, the linewidth of the LP modes does not seem to be that different for the two cases in Fig. 4b).
Since it is an important point, the authors should show more details regarding this Q-factor estimation in the supplementary information file. Fit y = y 01 + A 1 *exp(-t/τ 1 )*sin(2πf 1 t + θ 1 ) Fit y = y 02 + A 2 *exp(-t/τ 2 )*sin(2πf 2 t + θ 2 ) (b) Figure R2: (a) Normalized THz signal in time domain from a single cSRR and a 60 × 60 array of resonators with the aSIL configuration. The inset shows the magnified image of the tail of the decayed signal to highlight the oscillations. (b) The waveforms in panel (a) are digitally filtered with an ideal bandpass filter (centered at LP mode with a bandwith ∼ 400 GHz) and then fitted by sine functions with exponentially-decaying amplitudes to extract the oscillators' decay times (τ 1 , τ 2 ) and frequencies (f 1 , f 2 ). The Q-factors of the LP mode at B = 4T are then calculated as Q single = f 1 Linewidth = f 1 1/πτ 1 = πτ 1 f 1 = π × 16.3ps × 300GHz = 15.4 for the single resonator and Q array = πτ 2 f 2 = π × 7.87ps × 280GHz = 6.9 for the 60 × 60 array of resonators with aSIL configuration.
Answer: We agree with the Reviewer that the difference in Q-factors is not noticeable in Fig. 4. To acquire the colormaps in Fig. 4, we cut the time domain signal before the reflection signal from the cryostat windows and compute its Fourier transform. For a delay of 30 ps, the linewidth of the LP mode is then resolution limited. In order to calculate the correct Q-factor for the LP at its asymptotic limit (B = 4T), we first cut the signal in the time domain before the echo. Then, the signal is digitally filtered by an ideal band-pass filter centered at the peak of the LP mode (bandwidth ∼ 400 GHz) to remove the effect of the second mode. In the end, the filtered signal is fitted with a sine function with exponentiallydecaying amplitude (y = y 0 + Aexp(t/τ )sin(2πf t + θ)) to extract the oscillators' decay time (τ ) and frequency (f ) [1]. To clarify this point, Fig. R2 is added to the supplementary (Fig. S7).
Question 3: lines 238-242: "It is also evident that there is a renormalization of the loaded cavity frequency that blueshifts the resonance by 30 GHz (∼ 0.1 ω) when we reduce the number of resonators from 3600 to 1 . . . We do not observe such a shift when we investigate the cold cavity at 300 K (see Fig.3(a))." My impression is that a comparable slight blue-shift can also be seen in Fig. 3a, at least between the single resonator case and some of the other curves. Perhaps, this could be seen better if presented on a linear scale plot.
Answer: We agree with the Reviewer that there is also a blue-shift in the cold cavity measurements (Fig. 3a) when we reduce the number of resonators. However, it is much smaller than the blue-shift in the loaded cavity (30 GHz). We attribute the larger shift in the case of the loaded cavity to the substantial dielectric contribution of the 2DEG that effectively produces a slow light effect. In the intermediate cases (the smaller array of resonators with more than one resonator), the resonant frequencies show larger shifts in Fig.  3a (as the Reviewer has mentioned). This larger shift can be because of the existence of other modes as a result of the interaction between the THz beam and the boundary of the resonator array : the description of these cases goes beyond the scope of this paper. Hence, we added Fig. R3 to the supplementary (Fig. S5) and modified the manuscript for more clarity: "It is also evident that there is a renormalization of the loaded cavity frequency that blueshifts the resonance by 30 GHz (∼ 0.1ω) when we reduce the number of resonators from 3600 to 1 . . . We observe a much smaller shift (7 GHz) when we investigate the cold cavity at 300 K (see Fig.3(a) and also Fig. S4 in the Supplementary info). We attribute the larger shift in the case of the loaded cavity to the substantial dielectric contribution of the 2DEG that effectively produces a slow light effect, enhancing the collective Lamb shift of the ensemble of meta atoms as observed in other systems." Question 4: Caption of Fig. 4b: "The distance between peaks at B = 0.8T which is twice of the vacuum Rabi splitting is marked with purple" Shouldn't this distance be the vacuum Rabi splitting directly? Ω should be defined in the text (is it the coupling strength, and if so, how is it different from the quantity g used in the cooperativity formula). Answer: We agree with the Reviewer and apologize for the inconsistency in the text. The distance in Fig. 4b is vacuum Rabi splitting which is twice of the coupling strength. Ω is the coupling strength, noted also as g. We have modified the text accordingly and replaced  Figure R3: Cold cavity resonance measurement in Menlo (linear scale) for the single resonator and 60 × 60 array of resonators. Each curve is normalized to its peak value for a better comparison.
g with Ω.
Question 5: lines 251-252: "The linewidth of the cyclotron resonance is smaller than the one measured without lenses due to the superradiance effect". This is not very clear to me. How superradiance quenching can occur here just by varying the spot size of the illuminated area on the 2DEG? Is the density of emitters varying in this case? Answer: In this case we are discussing the broadening of the cyclotron transition without any cavity. It has been proposed theoretically and proved experimentally that the cyclotron transition in 2D electron gases is superradiantly broadened: such research is reported in the paper by Zhang et al., [2]. Intersubband transitions have also been proven to undergo superradiant broadening [3]: an anomalous broadening is observed as a function of the angle of observation of the emission and attributed to an angular-dependent , superradiant dominated spontaneous emission of the electron plasma. What we believe that is happening in the case of lens-coupled cyclotron measurement is that the focusing of the THz radiation and the successive out-coupling via immersion lenses effectively reduce the coupling of the electrons to the electromagnetic field. The distribution of the focused wave vectors implies an angle-dependent coupling of the THz electric field to the in-plane polarized cyclotron transition. In this respect we would expect a result similar to what obtained in Ref. [3] as a function of the extraction angle of the radiation due to the quantum well emission. We modified the manuscript to clarify this point and we plan to investigate further this aspect in the future.

Reviewer #2:
Rajabali et al. report on ultrastrong coupling of a single resonator and Landau level transitions in GaAs and InSb quantum wells using terahertz time-domain spectroscopy (THz TDS). The quenching of the Dicke superradiance has also been shown with respect to decreasing number of resonators. To achieve measurement sensitivity in THz TDS on a single resonator level, the authors utilized an asymmetric silicon immersion lens configuration. These results are novel and of interest to the broad community. Indeed, the combination of the asymmetric lens with THz TDS is even more general, as pointed out by the authors. The manuscript sounds technically correct and well written. Therefore, I recommend its publications in Nature Communications. Few minor comments: Comment 1: Figure 1. "THz" should be added in front of TDS. Answer: We thank the Reviewer. THz is added to the caption of Fig. 1.

Comment 2:
The font size in Figure 1(e) is too small Answer: The font size is changed in Fig. 1e.

The article information and list of changes
In this section, we provided information about the length of the main text and the supplementary document. We also added a new version of the article and supplementary where the changes are highlighted. The deleted parts are crossed out with red lines and the added parts are indicated with blue. The new references are also highlighted in yellow. The length of the text, excluding "Methods", "References", and figure legends, is 4204 words. The "Abstract" section is 149 words. "Methods" section has two subsections and is 273 words. The article includes 5 figures. The length of their legends, excluding their standalone titles, are: • Fig. 1 legend: 79 words  Free-space coupling to strongly subwavelength individual optical elements is a central theme 7 in quantum optics, as it allows to control and manipulate the properties of ::::::: control :::::: over 8 ::::::::::

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As a further confirmation of our model, we can cite the results obtained with a near-field probe 157 measuring a similar complementary single resonator 22 . In that experiment, the single resonator 158 was excited with a focusing lens, providing the necessary in-plane k's ::::: wave :::::::: vectors. The resonant 159 signal was found to remain trapped in the near field since there was no collecting lens. Now that 160 we clarified the mechanism that underlies the observed transmission spectra, we can discuss the 161 experimental measurements in detail.
All error bars represent one σ confidence interval.

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Asymmetric lens setup and sample fabrication The lenses are hyperhemispherical and hemi-