The control of light–matter coupling by embedding matter into a photonic resonator is an essential ingredient for the development of single photon devices and cavity quantum electrodynamic studies and applications. The two parameters of photonic resonators that play a key role in light–matter coupling are the quality factor Q, which is inversely proportional to the photon decay rate of the cavity, and the mode volume V, which quantifies the confinement of the electric field. In many cases, photonic resonators associating strong electric field confinement and a high Q are highly desired. For instance, in the weak-coupling regime, the intensity of light–matter interaction is enhanced by the ratio Q/V, as described by Purcell effect [1]. Also, as the light–matter coupling constant g scales with Ne/V, with Ne the number of emitters in the matter part collectively involved in the interaction [2–4], small V are needed to reach the ultrastrong and deep-strong coupling regimes at the single electron level [5–9]. Moreover, the in-depth study of the ultrastrong coupling regime requires a high degree of coherence of the light–matter coupling [10–14], quantified by the cooperativity C=4g2Q/(ω0γ), with γ being the matter non-radiative decay rate and ω0 being the photonic mode angular frequency, which implies a high Q. Therefore, a current challenge in the development of the light–matter coupling platform is to build photonic resonators that conciliate a small V with a high Q.

The ultrastrong coupling regime in high cooperativity systems has been recently studied using cyclotron resonances in a high-mobility two-dimensional electron gas coupled to various types of terahertz (THz) photonic resonators [15]. For example, using a high-Q THz Fabry–Perot cavity made of distributed Bragg reflectors (DBRs; Q>180), Zhang et al. reported g/ω0∼0.1 with C>300 and showed that these unique conditions increase cyclotron resonance lifetime via the suppression of superradiant decay [16]. More recently, in a similar THz Fabry–Perot cavity, Li et al. reported g/ω0∼0.36 with a record-high cooperativity C=3513, allowing the observation of a vacuum Bloch–Siegert shift in ultra-narrow Landau polaritons [17]. Also, Mavrona et al. demonstrated a cooperativity of C=57 and g/ω0∼0.175 using a Fabry–Perot cavity based on weakly transmitting hole patterned metal layer [18].

However, in these works based on high-Q Fabry–Perot cavities, the light–matter coupling involves a very large number of electrons Ne due to the large V, limited by diffraction to the order of (λ2)3. A current challenge to explore advanced quantum phenomena such as the fermionic Rabi model for coupled systems is to reduce the number of electrons involved in the collective interaction while keeping a high coherence in the system. To this end, using a single complementary LC circuit THz resonator, Rajabali et al. recently achieved g/ω0∼0.33 with C=94 implying an estimated optically active electron number of only 30,000, and even of 2000 with C=14 [19]. Such a small number of electrons are obtained thanks to the confinement in truly all space dimensions of the electromagnetic mode, which leads to a deep sub-wavelength mode volume V, the cooperativity remaining nevertheless moderate due to the low Q factor typical of LC circuit resonators (Q=11).

Potential candidates that have recently emerged to simultaneously achieve a high Q and a strong electric field confinement are hybrid THz resonators based on Fabry–Perot cavities strongly coupled to LC circuit metamaterials. In these hybrid resonators, the strength of the coupling is given by the resonator-to-resonator coupling constant G defined by analogy to the light–matter coupling constant g. The ultrastrong coupling between a THz LC circuit resonator and a THz Fabry–Perot cavity was first reported by Meng et al. [20]. The authors demonstrated nonlocal collective interaction of spatially separated metamaterial layers mediated by the cavity photons [20] and also the potential of these hybrid THz resonators for phase-sensitive THz detection of chemical and biological materials [21]. More recently, they have demonstrated ultrastrong coupling between the effective magnetic dipole moments of metallic complementary metasurfaces and the magnetic field of a cavity mode [22]. Also, Jeanin et al. reported an hybrid THz cavity based on the coupling between LC circuit resonators and a dipolar antenna to engineer the radiative coupling properties and reach unity absorption [23].

Here, we report on hybrid resonators based on the coupling between a THz Tamm cavity and a THz LC circuit metamaterial. The LC circuit resonators are directly patterned on the metallic mirror of the Tamm cavity and are designed to concentrate the THz electric field. Using transmittance measurements and temporal coupled mode theory, we analyze the properties of the coupled modes and demonstrate the reduced coupling constant G/ω0∼0.1, showing that the two resonators are in the ultrastrong coupling regime. We further describe how the properties of the coupled modes emerge from the properties of the uncoupled metamaterial and the Fabry–Perot resonators. We show high quality factors, Q=37, explained by the quasi-total suppression of the radiative losses provided by the Tamm cavity and small V, in the (2–3)×10−4λ3 range, resulting from the principal quality of LC metamaterials. Our study demonstrates that, by combining high Q and small V, Tamm cavity-LC metamaterial coupled cavities fulfill the challenging requirement to conciliate strong THz electric field confinement with high quality factor, which is promising for the exploration of ultrastrong THz light–matter coupling with a high degree of coherence in the limit of a few electrons and for the realization of single photon THz emitters and detectors.

Our dual-cavity THz resonator is based on a hybrid THz Tamm cavity and an LC metamaterial structure and is presented in Fig. 1(a). The Tamm cavity [24,25], on its own, consists of a DBR composed of two high-resistivity silicon wafers of thickness 66 μm (∼3λ4) separated by a vacuum layer of thickness 75 μm (∼λ4) and a 100-nm-thick metallic gold mirror deposited on the top silicon wafer of the DBR [26]. The calculated reflection spectrum of this Tamm cavity, obtained by the finite element method (FEM, COMSOL Multiphysics), exhibits a Tamm mode at 0.98 THz and a resonance linewidth of ΓTamm=0.098THz, which gives a quality factor Q=100 [see Fig. 1(b)]. The isolated LC metamaterial, whose base unit cell is presented in Fig. 1(a), is basically a gold layer including patterned holes giving its resonance properties. This pattern should not be mistaken for a complementary split ring resonator that is efficient for concentrating the magnetic field. The model we present concentrates the electric field at its center, similar to conventional split ring resonator, which is crucial for coupling it to matter systems involving the electric dipole interaction, such as in the common two-dimensional electron gas. For this reason, we call this pattern grid split ring resonators (GSRRs). The calculated transmission spectrum of an isolated GSRR metamaterial of dimensions w=4.4μm, l=4.4μm, and L=16.5μm, shows an LC metamaterial resonant mode at 1 THz with a linewidth ΓLC=0.222THz, resulting in a quality factor Q=4.5 [see Fig. 1(b)]. The hybrid THz resonator structures were fabricated by manually stacking commercially available thin silicon wafers and precision machined thin metal strips opening vacuum gaps, creating the THz DBR. The stacking is realized in a custom-made sample holder dedicated to maintaining the metal spacers positions and holding the stack together by applying pressure. Prior to stacking, the LC metamaterials were patterned on the top silicon wafer by performing a simple optical lithography and the deposition of a 3/100 nm chromium/gold metallic mirror via thermal evaporation under vacuum, followed by a lift-off.

We fabricated nine different Tamm cavity-LC metamaterial coupled resonators, tuning the uncoupled resonance frequencies of the LC metamaterial fLC by applying an homothetic transformation to the w, l, and L dimensions of the LC circuit resonator at a fixed metamaterial period p=65μm. We probe their optical properties in a transmission experiment using a Bruker Vertex 70v FTIR under vacuum, a Globar blackbody source, and a helium-cooled bolometer detector. Figure 2(a) reports the measured transmission spectra of the Tamm cavity-LC metamaterial coupled resonators for different fLC (represented by the colour dots). We observe for each transmission curve two well-separated resonance peaks that form two distinct lower and upper frequency coupled modes. These modes are the result of the coupling between the Tamm cavity and the LC metamaterial. Around the frequency matching between the Tamm cavity mode and the LC metamaterial mode (dark green curve), the two peaks are approximately symmetric and separated by a splitting of 0.20 THz. Figure 2(b) shows the frequencies of the two measured resonance peaks corresponding to the frequency of the upper (red squares) and lower (blue squares) coupled modes as a function of fLC. We clearly observe the two distinct branches forming the anti-crossing pattern, similar to the behavior observed in the strong light–matter coupling regime, demonstrating the strong coupling between the Tamm cavity and the LC metamaterial.

To quantitatively describe the experimentally observed trends, we use a quantum formalism similar to the Rabi Hamiltonian, the Hopfield model [27], which involves bosonic operators only. In a general picture involving two resonators, creation and annihilation operators a^†, a^ and b^†, b^ are associated with the photonic modes A and B. Here, A is the Tamm cavity mode, and B is the LC metamaterial mode. The Hamiltonian of the coupled resonators can then be expressed as H^=ℏωA(a^†a^+12)+ℏωB(b^†b^+12)+ℏG(a^†b^+a^b^†−a^b^−a^†b^†),(1)where G is defined here as an energy exchange rate between the two harmonic oscillators. The uncoupled eigenstates are written as |NA,NB′⟩ with N, N′ the numbers of photons in modes A and B, respectively. Under the rotating wave approximation and considering for simplicity the subspace with one photon in the system, the eigenstates are two coupled photonic modes |+1⟩ and |−1⟩ that are linear combinations of the uncoupled eigenstates |1A,0B⟩ and |0A,1B⟩, $$\begin{gathered} |+1⟩=cosθ|1A,0B⟩+sinθ|0A,1B⟩, \\ |−1⟩=sinθ|1A,0B⟩−cosθ|0A,1B⟩, \end{gathered}$$(2)where tan(2θ)=−2Gδ. θ varies from 0 to π2 with the resonator detuning δ=ωB−ωA describing the continuous transition from |1A,0B⟩ to |0A,1B⟩, θ=π4 at zero detuning. The lower (−) and upper (+) resonance angular frequencies read ω±=ωA+ωB2±12(ωA−ωB)2+4G2.(3)

Note that the significant reduction of the resonator coupling constant G when the LC pattern size is reduced (i.e., fLC increases) is accounted for using an empirical linear model G=G0(1−β(ωB−ωA)); see Appendix B for details and arguments for this description. We find a good quantitative agreement between experiment and theory [Eq. (3)] for G≈0.10THz at frequency matching [solid black lines in Fig. 2(b)]. From this analysis, we extract a reduced coupling constant G/ω0∼0.1 demonstrating the ultrastrong coupling regime between the Tamm cavity and the LC metamaterial.

The experimental transmission spectra also provide a measure of the quality factors of the hybrid Tamm cavity-LC metamaterial resonators that we investigate in this section. We report on Fig. 3(a) the evolution of the linewidth of the two resonance peaks Γ± as a function of fLC. The two curves exhibit a crossing at zero detuning (corresponding to fLC=0.98THz) resulting from the exchange of properties between the Tamm cavity and the LC metamaterial in the coupled modes. At the crossing, we measure Γ+=0.061THz and Γ−=0.064THz yielding the quality factors Q+=18 and Q−=14 for the upper and lower coupled modes. Such values are surprisingly high, considering standard theory of coupled modes [28], which predicts linewidths Γ±=ΓTamm+ΓLC2∼0.11THz resulting in Q±,CM≈9 (i.e., Q±,CM≈2QLC since ΓTamm≪ΓLC).

To understand the physical mechanism underlying these higher than expected Q values of the coupled modes, we use the temporal coupled mode theory [23,29] and analyze the different decay and coupling rates involved in the system. The complete study, which includes the full derivation, the expression of the S-parameters, and the expression of the resonator coupling constant G, is presented in Appendix A. We consider a set of two coupled planar resonators A and B in a series, exchanging energy radiatively with an input channel s1 (excited by a wave s+,1) and an output channel s2, respectively, as sketched in Fig. 3(b). The coupled equations of motion are {dAdt=(iωA−ΓA2)A+iCBCAGB+Γrad1,ACAs+,1dBdt=(iωB−ΓB2)B+iCACBGA,(4)where A and B are the complex field amplitude of mode A and B, respectively [30], G is the resonator coupling constant, CA(CB) defines the relation between the complex field amplitudes A(B) and the mode energy in the resonator WA=CA|A|2 (WB=CB|B|2), and ΓA (ΓB) is the amplitude decay rate of mode A (B) with ΓA=Γrad1,A+Γloss,A (ΓB=Γrad2,B+Γloss,B).

Using this model, we derive the expressions of the reflection and transmission coefficients of the coupled resonator system (see Appendix A) and determine the analytical expression of the resonance linewidth of the upper and lower coupled modes, Γ±, in the ultrastrong coupling regime. Considering the Tamm cavity as A and the LC metamaterial as B, Γ± reads at zero detuning, Γ±=Γrad1,Tamm+Γrad2,LC+Γloss,Tamm+Γloss,LC2.(5)

In this equation, Γrad1,LC and Γrad2,Tamm vanish as the energy emitted by the LC metamaterial in the left direction is absorbed by the Tamm cavity and reciprocally [see Fig. 3(b)]. This coupling mechanism is responsible for the narrow resonance linewidths of the coupled modes observed experimentally in Fig. 3(a) compared to those predicted by the standard theory of coupled modes, which explains the large values of Q− and Q+. It also highlights the strong link between the coupling strength and the radiative coupling properties of the LC metamaterial, which is the origin of the large value of G as Γrad1,LC is the dominant loss term of the uncoupled LC metamaterial, consistently with the expression of G reported in the Appendix C, which scales like Γrad1,LC.

The resonance linewidth of the coupled modes can be further simplified. Since there is no way to distinguish Γloss,Tamm from Γloss,LC as they both result from ohmic losses in the metal layer, we note Γloss,metal=Γloss,Tamm+Γloss,LC. Moreover, the DBR mirror is highly reflective, leading to Γrad1,Tamm≪Γrad2,LC. The resonance linewidth of the coupled modes reduces then to Γ±≈Γrad2,LC+Γloss,metal2. This simplified expression highlights that Γ± can be ultimately reduced down to its fundamental limit Γloss,metal2 by suppressing the radiative losses Γrad2,LC. To this aim, we fabricate Tamm cavity-LC metamaterial coupled resonators and add a supplementary mirror at a distance of 75 μm over the LC metamaterial, which suppresses radiative losses in s2 at the expense of preventing the use of these hybrid cavities in transmission.

Figure 4 shows the experimental reflection spectra of two Tamm cavity-LC metamaterial coupled resonators based on DBRs made of two and three silicon layers and, respectively, one and two vacuum layers, including the additional gold mirror blocking transmission. We observe two high contrast peaks at resonance frequencies 0.84 and 1.06 THz, with Q=25.2±1.6 and Q=32.6±1.2 for the two-silicon-layer-based structure and Q=35±6 and Q=37±5 for the three-silicon-layer-based structure. The uncertainty regarding the three layers’ cavity quality factor is attributed to the resolution of our setup (6 GHz) combined with limited peak contrast and non-negligible parasitic features in the spectrum. Increasing the number of silicon layers in the DBR increases its reflectivity and accordingly reduces the value of Γrad1,Tamm so that the quality factor increases. However, adding more than three layers reduces the peak contrast dramatically because of critical coupling arguments [26].

These results show that our approach improves the quality factor Q by a factor 8 over the uncoupled LC metamaterial. Also, our analysis reveals that the Q enhancement results from the suppression of the dominant decay rate that is the radiation loss rate of the LC metamaterial so that the quality factor is almost ultimately limited by the losses in the metallic layer.

We now examine the mode volume V of the Tamm cavity-LC metamaterial coupled resonators. The unit cell pattern of the LC metamaterial is designed to concentrate the electric field in the central capacitive gap between the metallic tips, and we are interested in how this electric field confinement is modified by the coupling to the Tamm cavity. To this purpose, using the eigenfrequency solver of COMSOL Multiphysics, we compute the electromagnetic field distributions of the lower and upper frequency resonant modes of the coupled resonators. Simulations were run in a single period of the metamaterial bounded by orthogonal perfect electric and perfect magnetic conductor boundary conditions to create the array periodicity, and they are bounded by perfectly matched layers in the propagation direction. The gold layer is modeled using the refractive index of gold interpolated from data from Ref. [31]. Figure 5(a) displays the in-plane electric field amplitude enhancement of the upper coupled mode in the LC metamaterial plane for fLC=1.01THz compared to an input plane wave of unity amplitude (see Appendix C for the electric field profile along the optical axis). We observe a large increase of the electric field between the metallic tips and around their edges, reaching a factor up to 20. From the computed electric field distributions, we calculate the effective mode volumes according to the definition used for dissipative resonators [32], V=∫(ϵE2−μH2)d3r2ϵ(r0)E2(r0),(6)where r0 is taken at the center of the LC metamaterial pattern in vacuum above the layer surface [ϵ(r0)=ϵ0]. This equation is appropriate in our case as the quality factors of the coupled resonators remain in the range of a few tens. For the following, we consider only the real part ℜ(V) of this complex expression. We also compute the mode volume of the uncoupled LC metamaterial on an infinite silicon substrate and obtain VLC≈1.1×10−4λ3.

Figure 5(b) reports the calculated volumes of the upper and lower coupled modes V+ and V−, respectively, at their resonance angular frequencies ω+ and ω− normalized by VLC for increasing values of fLC (i.e., increasing values of detuning δ2π) and a fixed Tamm cavity resonance frequency fA=1THz. V+ and V− show opposite evolution with fLC: V+ decreases, whereas V− increases when fLC increases. We also observe that the two tendencies cross at a small non-zero detuning δ2π=−0.02THz. This crossover shows that there is an exchange of mode volume properties between the two resonators similar to the Q properties. This behavior results from the continuous transition of the upper coupled mode from a mode mainly given by the Tamm mode to an LC metamaterial mode for increasing resonator detuning corresponding to the continuous evolution of θ in Eq. (2), and inversely for the lower coupled mode. Thus, V+ (V−) converges toward VLC at large positive (negative) detuning, i.e., fLC much higher (lower) than 1 THz. In addition, we note that the mode volume is increased by only a factor ∼2.3 compared to VLC at the curve crossing and by factors 2.1 and 2.5 for V+ and V−, respectively, at zero detuning. The mode volumes at zero detuning are reduced down to 3.2×10−4λ3 for V+ and 2.0×10−4λ3 for V−, revealing a high degree of subwavelength confinement of the coupled modes (details are available in Appendix A).

These simulation results raise the question of the origin of the factor 2 in the increase of V± compared to VLC, and the robustness of this value to the designs of the metamaterials and of the Fabry–Perot cavity. To provide some insight, we model the evolution of V± by computing ⟨±1|E(r)2|±1⟩, where E=EA(r)+EB(r) is electric field operator, with $$\begin{gathered} EA(r)=iEA(afA(r)−a†fA*(r)), \\ EB(r)=iEB(bfB(r)−b†fB*(r)). \end{gathered}$$(7)

Ej=ℏωj2ϵ0Vj is the RMS electric field amplitudes associated to vacuum fluctuations in modes j=A,B, with Vj defined at the central position r0. fj(r) is the electric field amplitude profile of the mode, normalized such that fj(r0)=1. We then estimate the mode volume of the coupled mode V+ using the usual expression involving the electric field only [33], V+=∫ϵ(r)|⟨+1|E(r)2|+1⟩|d3rϵ(r0)|⟨+1|E(r0)2|+1⟩|.(8)

We recognize in the numerator the expression of the electromagnetic energy in the mode, whose value for one photon is directly 3ℏω+2. In the denominator, we use the expressions of the coupled terms from Eq. (2), ⟨+1|E(r0)2|+1⟩=32cos2θEA2+32sin2θEB2+2cosθsinθEAEB,(9)where tan(2θ)=−2Gδ. Then, the coupled mode volume V+ is expressed as V+=VBsin2θω+ωB11+23ωAωBVBVAcotθ+ωAωBVBVAcot2θ,(10)and equivalently for V−, V−=VBcos2θω−ωB11−23ωAωBVBVAtanθ+ωAωBVBVAtan2θ.(11)

The detailed calculation is reported in Appendix B. Considering resonator A stands for the Tamm cavity and resonator B for the LC metamaterial, we compare the simulation results in Fig. 5(b) with the calculated V± [Eqs. (10) and (11)] as a function of the resonator detuning and find quantitative agreement for the only free parameter α=VBVA=0.22 (dashed lines), validating our theoretical description. Note that we take into account the evolution of G with ωB using β=0.18THz−1 determined from fitting the simulated resonance frequencies.

In a simplified picture at small detuning, ω+≈ω−≈ωB, and since VB≪VA, the coupled mode volumes can be reduced to V+≈VB/sin2θ and V−≈VB/cos2θ. These expressions describe the crossing of the curves at frequency matching (δ=0 and θ=π/4) with V±=2VB. Thus, our model reproduces the increase of V± by a factor of 2 compared to VLC observed at zero detuning and, importantly, highlights that the mode volumes of the coupled resonator systems are dominated by the metamaterial mode volume at small detuning. The factor 2 can be interpreted as the dilution of the photon electromagnetic energy in the two resonators. Achieving coupled mode volumes of only 2 times, the mode volume of the uncoupled LC metamaterials is then a general result that can be transposed to other coupled systems based on metamaterials and Fabry–Perot cavity. Additionally, the refined model from Eqs. (10) and (11) captures the asymmetry of the V± curves at large detuning, due to the variation of the resonator coupling constant G and the contribution of the mode A. Also, our model reproduces the 0.02 THz shift of the mode volume curve crossing due to the symmetric or anti-symmetric contribution of the Tamm mode (mode A), which results in E+(r0)≠E−(r0) at frequency matching. The deviation for V− at negative detuning is attributed to the much larger wavelength for f−.

The present analysis demonstrates that the main strength of LC metamaterials, which is their extremely small mode volume, is preserved despite their coupling to the Fabry–Perot cavity with a very large mode volume, which underlines the main interest of coupling these two distinct types of photonic resonators.

The properties of our hybrid THz cavities based on the coupling of LC metamaterials with Tamm cavities are very promising for the development of THz light–matter coupling platforms with a high degree of coherence. Indeed, these hybrid cavities conciliate quality factors of up to a few tens, provided by the Tamm cavity, with sub-wavelength electric field confinement, provided by the LC metamaterial, as summarized in Table 1.Table 1.

Comparison of V and Q (theoretical and experimental) performances for the LC metamaterial alone, coupled with a 2 silicon layers Tamm cavity, and coupled with a 3 silicon layers Tamm cavity + additional mirror^a

The lower values of the Q that we measured compared to those predicted are most likely due to the significantly lower conductivity of the chromium–gold layer we used for fabrication compared to the pure gold layer we consider in the simulation, which opens up opportunities for future improvements.

In conclusion, we have experimentally and theoretically investigated original hybrid THz cavities based on the ultrastrong coupling between a THz Tamm cavity and an LC circuit metamaterial. We have demonstrated Q of up to 37 associated with V as low as 3.2×10−4λ3. Our work shows that the principal qualities of resonators, which are a high Q for Tamm cavity and a low V for LC circuit metamaterials, act beneficially in the presence of the coupling. Compared to the uncoupled LC metamaterial, Q is enhanced by a factor of 8 due to the coupling with a Tamm cavity, while V is increased by only a factor of ∼2 as a result of the energy dilution within the two resonators. Thus, investigating light–matter interaction using these hybrid THz cavities should provide a high degree of coherence due to the enhancement of the cooperativity C of at least 3.8, opening interesting perspectives for the development of single photon THz emitters and detectors [34] and quantum technology applications. Also this hybrid system compares favorably to uncoupled LC metamaterials for the study of light–matter interaction in the ultrastrong coupling regime with quantum material having low non-radiative decay rates γ, as the reduction of g given by 1/V remains small compared to the reduction of the resonator linewidth κ. Furthermore, our analysis revealed the suppression of the radiative losses of the LC metamaterials when embedded in the Tamm cavity so that the quality factor is ultimately limited by the ohmic losses of the metallic layer. This property opens vast possibilities in the development of new LC metamaterial patterns previously disregarded due to large radiative losses and renews the interest of investigating low-loss metamaterials based on superconducting materials [35,36] to reach very high Q values. Additionally, our theoretical description of the variations of the mode volume due to the coupling evidences the generality of this concept: possible metamaterials are not limited to the hole array LC circuit we presented but could also include direct split ring resonators similar to Ref. [20] as well as micro-antennas.

Acknowledgment. The authors thank Jean-Francois Lampin and Jean-Michel Gerard for fruitful discussion on the strong coupling regime and cavity.