PHYSICAL REVIEW B 87, 075324 (2013)

Role of geometry for strong coupling in active terahertz metamaterials

Daniel Dietze,* Karl Unterrainer, and Juraj DarmoPhotonics Institute, Vienna University of Technology, 1040 Vienna, Austria

(Received 21 December 2012; published 28 February 2013)

We discuss the influence of geometry and type of meta-atom resonance on important parameters for strongand ultrastrong light-matter coupling experiments based on localized plasmons in planar metamaterials andintersubband plasmons in quantum wells. An analytic model based on the density matrix formalism is used toextract the radiative linewidth, the achievable coupling rate, and its dependence on distance between a singlequantum well and a metasurface for five different types of meta-atoms. Depending on these values, the opticalresponse of the coupled system gives rise to different regimes ranging from the weak-coupling regime over thehybridization-induced transparency regime to the strong-coupling regime. As a result of our investigation, weshow that the achievable coupling rates exhibit only a weak dependence on the actual geometry and thus thephysical nature of the metamaterial resonance. In all cases, the coupling rate can be made large enough to reachthe strong-coupling regime by using multiple quantum wells.

DOI: 10.1103/PhysRevB.87.075324 PACS number(s): 42.50.Nn, 78.67.De, 78.67.Pt

I. INTRODUCTION

Metamaterials, artificial structures consisting of a regulararray of subwavelength resonators (meta-atoms),1 constitutean exciting possibility for the efficient coupling of free-spaceradiation to a wide variety of quantum systems. The two-dimensional version of such a metamaterial, the metasurface,is thereby especially appealing due to its easy fabrication andthe wide design freedom. In recent years, the coupling of suchmetasurfaces to a range of systems has been demonstrated,including phonons,2 intersubband transitions in quantumwells,3,4 vibrational modes of molecules,5 and cyclotronresonances in a two-dimensional electron gas.6 Especially,the combination of such metasurfaces with intersubbandtransitions in quantum wells opens up a whole new rangeof applications. Examples include large-area surface-emittingquantum-cascade lasers and photodetectors, coherent terahertz(THz) amplifiers, and active THz modulators.7

An important aspect is the achievable coupling strengthbetween the quantum wells and the metamaterial. The couplingis mediated by the strong evanescent field of the localizedmetamaterial plasmons, which are excited by the normallyincident radiation used to probe the system. These localizedplasmons take the role of the photon field in classic cavityquantum electrodynamics. Energy can be exchanged peri-odically via the near field between the localized plasmonsof the metamaterial and the quantum system. In the THzfrequency range, the interaction can be strong enough to reachthe strong and ultrastrong-coupling regimes, respectively.2,4,6

The choice of meta-atom geometry thereby plays an importantrole for setting the coupling strength and the radiative decayrates. Depending on the type of resonance, the decay lengthof the evanescent field into the substrate can reach severalmicrons. The field confinement along the growth axis isdirectly related to the achievable field enhancement and thecoupling strength between the localized plasmons and thequantum well intersubband plasmons.

Reports so far have only been concerned with proof-of-principle demonstrations and have been restricted to a singlemeta-atom geometry. In most cases, a split-ring-resonator-based metasurface has been used. In this paper, we use finite-

difference time-domain simulations to characterize severaldifferent meta-atom geometries with respect to the radiativelifetime of the resonances and the extent of the localizedplasmon field. A single parabolic quantum well is used asa probe to investigate the decay characteristics of the couplingas a function of distance between the quantum well and themetasurface. The parabolic quantum well represents the idealsystem to approach the ultrastrong-coupling regime as highcarrier densities can be used,8 and thus a single quantum wellis already enough to achieve strong coupling.4 Based on theseresults, we present a modified expression for the couplingrate that is used to estimate the limiting coupling strengththat can be achieved for a given meta-atom geometry. It isfurther shown that, depending on the radiative linewidth ofthe metamaterial, the coupled system of the metasurface andquantum well gives rise to three different regimes, namely, theweak- and strong-coupling regimes, as well as the regime ofhybridization-induced transparency.

This paper is organized as follows. Section II presents theprinciples underlying the coupled system of the metasurfaceand quantum well and introduces an analytic model thatcan be used to determine the coupling strength over a widerange of values. In Sec. III, the finite-difference time-domaincode is briefly presented, which has been used to calculatethe transmission of several different meta-atom geometriescoupled to a single parabolic quantum well. Finally, the resultsare discussed in Sec. IV.

II. THEORY

The coupled system of the metasurface and quantumwell is shown schematically in Fig. 1(a). In this picture,the metasurface takes the role of the optical cavity thatexchanges energy with the quantum system at a rate g.This energy exchange takes place via the evanescent field ofthe metamaterial resonant mode polarized along the growthdirection of the quantum well. The loss rate of the cavity isdenoted by κ and takes into account both radiative and Ohmiclosses. It is related to the full width at half maximum (FWHM)linewidth of the metamaterial resonance �f via κ = π�f .The polarization in the quantum well decays at a rate γ ,

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DANIEL DIETZE, KARL UNTERRAINER, AND JURAJ DARMO PHYSICAL REVIEW B 87, 075324 (2013)

FIG. 1. (Color online) (a) Scheme of the metasurface (MM)cavity coupled to a quantum well (QW) below. (b) Equivalent levelscheme in the strong-coupling regime, g � κ,γ . UP and LP denotethe upper and lower polariton branches. (c) Equivalent level schemein the hybridization-induced transparency regime, κ > g > γ . |1〉represents the ground state, and |2〉 and |3〉 denote the metasurfaceand the quantum well, respectively.

which is given by the inverse of the effective dephasing timeT ∗

2 , γ = 1/T ∗2 . In the experimental configuration used, the

metasurface is additionally coupled to an external probe fieldETHz, whose transmission through the structure gives insightinto the internal state of the coupled system.4 This additionalprobe field is not taken into account in the usual cavity QEDformalism but gives rise to an additional effect termed cavity-or hybridization-induced transparency.9

Depending on the value of the coupling rate g, the systemcan be represented in one of the following two equivalentpictures. In the so-called strong-coupling regime shown inFig. 1(b), in which g � κ,γ , the metasurface acts as an opticalcavity which modifies the energy levels of the quantum welland leads to the formation of upper (UP) and lower (LP)polariton branches.10 If the cavity is resonant to the quantumwell transition, the two polariton branches are separated bytwice the vacuum Rabi energy hg. Note that strong coupling isa necessary prerequisite for the construction of a lasing spaserbased on metasurfaces.11,12 In the special case κ > g > γ ,the combined system of the external probe, metasurface, andquantum well can be represented as a three-level system,where states |1〉, |2〉, and |3〉 represent the ground state,the metasurface, and the quantum well, respectively. Thisequivalent description is similar to the one given in Ref. 5 fora metasurface coupled to an atomic medium. However, in thecase at hand, the external field couples only to the metamaterialand not to the quantum well. Thus, it shows more similarities tothe situation discussed in Ref. 9. The transmission coefficientin this configuration shows a transparency feature due to theinterference of the two pathways, |1〉 → |2〉 and |1〉 → |2〉 →|3〉 → |2〉. This is a general behavior that can be observed in alarge variety of physical systems and is, for example, termedelectromagnetically induced transparency (EIT),13 cavity-

induced transparency,9 plasmon-induced transparency,14 orhybridization-induced transparency.5

The coupled system of the metamaterial and quantum wellcan be modeled using two coupled two-level systems describedby the usual density matrix formalism.12 In the limit of weakelectric fields, the level occupations of the quantum well canbe assumed to be constant over time, i.e., ρ

QW11 = ρ

QW22 = 0.

In addition, all electrons are assumed to be in the ground state,ρ

QW11 = 1 and ρ

QW22 = 0. This leads to the system of coupled

equations for the off-diagonal elements of the density matrices:

ρQW + (iωQW + γ )ρQW = iμQW

ε0hLPMM, (1)

ρMM + (iωMM + κ)ρMM = iμMM

ε0hLPQW + i

μextMM

hETHz,

(2)

where ωQW and ωMM are the resonant frequencies, ε0 isthe vacuum permittivity, μQW and μMM are the transitiondipole elements for excitation with the electric field polarizedalong the z axis, μext

MM is the transition dipole moment ofthe metasurface for excitation with an electric field polarizedalong the x axis, and L is some geometrical factor describingthe coupling strength between the two systems via the inducedpolarizations along the z axis. Note that L has units of inverselength. The macroscopic two-dimensional sheet polarizationdensities are further given by

PMM = nMMμMMρMM, (3)

PQW = nQWμQWρQW, (4)

where nMM and nQW are the (sheet) densities of the two-levelsystems associated with the metasurface and the quantum well,respectively. Using the usual rotating wave approximation,ρ ∝ exp(−iωt), Eqs. (1) to (4) can be rewritten as

(ω − ωQW + iγ )ρQW = −VMMρMM, (5)

(ω − ωMM + iκ)ρMM = −VQWρQW − μextMM

hETHz, (6)

where we have introduced effective coupling rates

VMM = nMMμMMμQW

ε0hL, (7)

VQW = nQWμMMμQW

ε0hL. (8)

Equations (5) and (6) now serve as the starting point for thefollowing discussion.

First, we discuss the normal mode frequencies of thecoupled system in the strong-coupling limit. After setting theexternal probe field to zero, Eqs. (5) and (6) can be writtenas a homogeneous matrix equation, which has only nontrivialsolutions if the determinant of the coefficient matrix is identicalto zero. This leads to the characteristic equation

(ω − ωQW + iγ )(ω − ωMM + iκ) − VMMVQW = 0, (9)

with the complex solutions

ω± = ωMM + ωQW

2− i

κ + γ

2

±√

1

4[(ωMM − ωQW ) − i(κ − γ )]2 + VMMVQW . (10)

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In Eq. (10), the real part gives the coupled mode frequenciesand the imaginary part gives the respective linewidths.

Of special interest is the case when the metamaterial isresonant to the quantum well transition, ωMM = ωQW = ω0,and the coupling rates VMM and VQW are much faster than thedecay rates κ and γ . Then, Eq. (10) simplifies to

ω± = ω0 ± √VMMVQW − i(κ + γ )/2. (11)

The two normal modes are thus separated by a frequency2√

VMMVQW and have an equal linewidth of (κ + γ )/2. Therate

√VMMVQW can thereby be associated with the vacuum

Rabi frequency g as it is usually used in cavity QED.To get access to the internal dynamics of the coupled

system, we have to rely on experimentally accessible param-eters, such as the optical response of the combined systemto an external probe field ETHz. The parameter of interest isthereby the transmission coefficient. For a metasurface witha thickness much smaller than the relevant wavelength, thetransmission coefficient is most conveniently calculated as thatof an infinitely thin conductive sheet on a bulk substrate:15

t = 2

1 + ns + σ/(c0ε0), (12)

where c0 is the vacuum speed of light, ns is the refractiveindex of the bulk substrate, and σ is the sheet conductivity ofthe metasurface. To calculate σ , we make use of the surfacecurrent density j = σETHz = (∂)/(∂t)P ext

MM , with P extMM =

nMMμextMMρMM . Using again the rotating-wave approximation

and solving the set of Eqs. (5) to (6) for ρMM yields the finalresult for σ :

σ = iωA(ω − ωQW + iγ )

(ω − ωMM + iκ)(ω − ωQW + iγ ) − g2, (13)

with the amplitude

A = nMM

(μext

MM

)2

h. (14)

As has been discussed in Ref. 14, this result is mathematicallysimilar to the susceptibility obtained for EIT in three-levelatomic systems16 and thus supports similar transparencyfeatures, provided the ratio of the decay and coupling ratesare chosen conveniently.9,16

Figure 2 shows the transmission coefficient of a metasurfacecoupled to a quantum well for different values of the coupling

FIG. 2. (Color online) Line shape of the transmission coefficientfor different values of the coupling rate g: (a) κ,γ > g (weak-couplingregime), (b) κ > g > γ (hybridization-induced transparency regime),and (c) g > κ,γ (strong-coupling regime). For the calculations,κ = 3γ . The black dotted curve is the bare metasurface withouta QW.

strength g. For the calculations, we have chosen the parametersA = 0.1,ns = 3.6, and κ = 3γ . The transmission coefficienthas been normalized to the transmission coefficient of thesubstrate alone. Depending on the choice of g, we candistinguish three different regimes. In the case of weakcoupling [Fig. 2(a)], the introduction of the quantum well leadsto a slight increase of the linewidth of the bare metasurface(shown as a black dotted line). When the coupling rate isthe fastest rate of the problem, as plotted in Fig. 2(c), theresulting transmission coefficient shows clear signatures ofvacuum Rabi splitting. The frequency splitting of the twopolariton branches is given by 2g. A special case is obtainedwhen the coupling rate is intermediate, i.e., κ > g > γ , andthe system coupled to the external probe field is the one withthe larger dissipation.9 The resulting transmission coefficient isshown in Fig. 2(b) and looks fundamentally different from thestrong-coupling case. The line shape is given by the differenceof two Lorentzians rather than by the sum, as is the case inFig. 2(c).

The developed model is thus capable of reproducingthe resulting transmission coefficient over a wide range ofcoupling rates. In the following, we use this model to extractthe coupling strength g from finite-difference time-domainsimulations for various geometries of meta-atoms, as well asfor different distances of the quantum well to the metasurface.

III. SIMULATIONS

We have developed a finite-difference time-domain (FDTD)code to simulate the interaction of single-cycle THz pulseswith meta-atoms on a dielectric substrate coupled to aquantum well.4 The three-dimensional FDTD code is basedon the standard Yee grid and the leapfrog method for spatialand temporal staggering of the electric and magnetic fieldcomponents.17,18 The temporal step size �t is related to thespatial grid size �r by c0�t = C�r , where C is the Courantnumber. To ensure stable computation, we use a value ofC = 0.5.18 The spatial step size has to be chosen with respect tothe smallest feature size of the coupled system, i.e., meta-atomgeometry and/or distance between the quantum well and themetasurface. We use a value of �r = 500 nm throughout thepaper.

The simulation volume is shown in Fig. 3(a). The growthdirection of the quantum well is taken as the z axis. Thecomputational space is divided in two parts, air or vacuum onone side and substrate on the other side. The substrate is takeninto account as frequency independent and lossless dielectricwith permittivity ε = 12.96, which is a suitable approximationin the lower THz region up to 5 THz. The simulation spacecontains a single meta-atom, which is located in the x-y planeat the interface between air and substrate. The unit cell sizehas been chosen as 50 × 50 μm to reduce the effects of mutualcoupling between neighboring elements. We simulate themetalized parts of the meta-atom as a perfect electric conductorby fixing the tangential components of the electric field tozero. This is a very good approximation in the THz frequencyrange, where Ohmic losses in the metal can be neglected. Asboth the substrate and the metal are taken as lossless, the finitelinewidth of the MM resonances can be fully attributed toradiative losses in the following. Apart from influencing the

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FIG. 3. (Color online) (a) Sketch of the simulation volumeshowing a linear dipole meta-atom. The THz pulses propagate fromport 1 to port 2 with the electric field vector aligned parallel tothe x axis. All meta-atoms are tuned for a resonance frequency of2.3 THz. The wire and gap widths are, in all cases, 2 μm. (b) Rect-angular ring. (c) Split-ring resonator tuned for the hybrid resonance.(d) Split-ring resonator tuned for the LC resonance. (e) Doubly splitring resonator.

resonance amplitude, the unit cell size can also have an impacton the radiative linewidth of the metasurface.19–21 Thus, thecell size and the resonance frequency have to be kept constantwhen different meta-atom geometries are to be compared.We assume the THz pulses to be normally incident on thesubstrate with the electric field vector pointing along the x axis.Therefore, we can use perfect periodic boundary conditions atthe four side walls to simulate an extended metasurface. Atthe top and bottom planes [ports 1 and 2 in Fig. 3(a)], we usefirst-order Engquist-Majda absorbing boundary conditions andthe total-field/scattered-field approach.22–24 We have verifiedthat the residual scattering that occurs on the edges and cornersis negligible by choosing different sizes of the simulationspace and appropriate time windows. The source is modeledas a modulated Gaussian, cos(ω0t) × exp(−4 ln 2t2/T 2), tosimulate a broadband, single-cycle THz pulse with FWHM T

and center frequency ω0.To investigate the suitability of a chosen meta-atom

geometry for strong- and ultrastrong-coupling experiments,we use a single parabolic quantum well located at a variabledistance d below the meta-atom. The electronic response ofthe quantum well to the external electric field is simulatedusing an anisotropic Lorentz model, which is an admissibleapproximation for low-excitation field strengths. In this model,the quantum well polarization acts only on the z component

of the electric field and is given by18

Pz(ω) = f12n0e2/meff

ω2QW − ω2 + 2iγ ω

Ez(ω), (15)

where ωQW is the transition frequency, n0 = 2n2d/�r isthe number density of electrons per unit cell, e is theelectron charge, meff is the effective mass, γ = 1/T ∗

2 is thedamping coefficient, and f12 is the oscillator strength. For allsimulations, we used the values ωQW = 2π × 2.3 THz, n2d =5 × 1011 cm−2, meff = 0.074m0, f12 = 1.08, and T ∗

2 = 2.5 ps,corresponding to a single 140-nm-wide Al0.3Ga0.7As/GaAsparabolic QW. These values have been adapted from Ref. 4,where it has been demonstrated that a single parabolic QWis sufficient to reach the strong-coupling regime. Thus, thesingle QW can act as a probe of the light-matter interactionmediated by a specific meta-atom geometry. The finite-difference implementation of Eq. (15) is based on the auxiliarydifferential equation method.18

Figure 3 shows the meta-atom geometries which areinvestigated in the following: the linear dipole [Fig. 3(a)], therectangular ring [Fig. 3(b)], the split-ring resonator [Figs. 3(c)and 3(d)], and the doubly split ring resonator [Fig. 3(e)]. Thesplit-ring resonator exhibits two lowest-order resonances forthe used polarization, namely, the LC resonance [Fig. 3(c)]and the dipole resonance of the closed ring modified bythe presence of the slit [hybrid resonance, Fig. 3(d)]. Allmeta-atoms have been scaled to be resonant with the transitionfrequency of the quantum well at 2.3 THz under illuminationwith x-polarized light. Note that both the split-ring resonator[Figs. 3(c) and 3(d)] and the doubly split ring resonator[Fig. 3(e)] exhibit different resonance frequencies when theincident field is polarized along the y axis. In this case,however, the metasurface is not resonant with the quantumwell, and thus no coupling effects are observable.

IV. RESULTS

Figure 4 shows the simulated transmission coefficientsobtained for the five different meta-atom geometries. Thetransmission coefficients have been normalized to the trans-mission of the bare GaAs substrate. The linewidth of the baremetasurface (shown as a black dotted line) ranges from about100 GHz in the case of the LC resonance in the split-ringresonator [Fig. 4(c)] to over 650 GHz in the case of the lineardipole [Fig. 4(a)]. As has been mentioned above, the only lossmechanism for the bare metamaterial taken into account inthe simulations is radiative damping. Thus, the differencesin linewidth are directly related to the physical nature ofthe resonance. The insets in Fig. 4 show the normalizedenergy density associated with the electric field parallel tothe growth direction calculated at the resonance frequency of2.3 THz. As can be seen, the resonant mode corresponds toa pure electric dipole mode for the linear dipole [Fig. 4(a)],the rectangular ring [Fig. 4(b)], and also, to a large degree,the hybrid mode of the split-ring resonator [Fig. 4(d)]. Thestored energy can be efficiently radiated to free space, whichexplains the large linewidth of the resonances. In the caseof the LC resonance of the split-ring resonator [Fig. 4(c)],the stored energy is confined to a small region around the

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FIG. 4. (Color online) Transmission coefficients of bare meta-surface (black dashed line) and metasurface coupled to a quantumwell 250 nm below the surface (red solid line). The insets showthe orientation of the incident field (arrows) and the calculatedenergy density contained in the electric field polarized along thegrowth direction in the plane of the quantum well. (a) Lineardipole. (b) Rectangular ring. (c) Split-ring resonator (LC resonance).(d) Split-ring resonator (hybrid resonance). (e) Doubly split ringresonator.

slit, corresponding to the fundamental magnetic resonance.As this resonance couples to free space only via the crossterms of the polarizability tensor, radiative losses are ratherinefficient, leading to a narrow transmission resonance. Aspecial case is the doubly split ring resonator [Fig. 4(e)],which shows both an LC and an electric-dipole-like modepattern. The linewidth in this case is, however, dominated bythe LC part of the resonance. The different geometries alsoshow different scattering cross sections, as can be seen fromthe different depths of the transmission minimum. Table Ishows a comparison of different parameters extracted fromthe simulated data.

When the coupling between the metasurface and the singleparabolic quantum well is switched on (red solid line inFig. 4), the transmission coefficient is altered drasticallycompared to the uncoupled case. The single resonance dipof the bare metasurface is now split in two distinct minima.Note that similar hybridization effects can be observed inspecially designed metamaterials, e.g., plasmonic molecules14

or stereometamaterials.25 There, the line splitting is based onthe coupling between different elements of the metasurface

TABLE I. Extracted parameters for the five investigated meta-atom geometries: dipole (DIP), rectangular ring (RR), split-ringresonator (hybrid resonance, SRR1), split-ring resonator (LCresonance, SRR2), and doubly split ring resonator (dSRR). Q

denotes the quality factor of the resonance, g250 nm is the couplingrate at a distance d = 250 nm, and gmax = limNQW →∞ g.

DIP RR SRR1 SRR2 dSRR

κ (1012 rad/s) 2.05 2.00 1.97 0.53 0.37�f (THz) 0.65 0.64 0.63 0.17 0.12Q = ωMM/2π�f 3.49 3.48 3.57 13.4 19.0A 0.18 0.29 0.25 0.03 0.181 − tmin (%) 22 30 27 16 62

g250 nm (1012 rad/s) 1.23 1.09 1.10 1.24 1.10

g1 (1012 rad/s) 0.86 0.78 0.74 0.92 0.76ζ1 (μm) 3.48 5.58 5.84 2.62 3.49g2 (1012 rad/s) 0.82 0.58 0.63 0.75 0.69ζ2 (μm) 0.39 0.47 0.53 0.40 0.43

g1 + g2 (1012 rad/s) 1.68 1.36 1.37 1.67 1.45gmax (1012 rad/s) 3.11 3.45 3.46 2.86 2.77

resonators and thus is an inherent property of the metamaterial.This mechanism is not to be confused with the present case,where the hybridization occurs between the metasurface and aquantum system. In Fig. 4, two different classes of line shapescan be clearly distinguished. By a qualitative comparison withFig. 3, we can attribute the line shapes in Figs. 4(a), 4(b),and 4(d) to the induced transparency regime, while the lineshapes in Figs. 4(c) and 4(e) correspond to the strong-couplingregime.

In order to get a quantitative handle on the coupling strengthand its dependence on the distance d between the quantumwell and the metasurface, we can use the model developedin Sec. II. Figure 5 shows the transmission coefficient of thesplit-ring resonator metasurface tuned for the LC resonancefor various values of d. The red solid line is a least-squares fitof the analytic model as given by Eq. (12) to the simulated data

FIG. 5. (Color online) Transmission coefficient of the split-ringresonator metasurface (LC resonance) coupled to a single quantumwell located at different distances ranging from 250 nm (bottom) to6250 nm (top) in steps of 1 μm. The open circles are the resultsof FDTD simulations, and the red solid lines are the results of theanalytic model. The curves have been offset for clarity.

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FIG. 6. (Color online) Dependence of coupling rate g on thedistance between quantum well and metasurface for different meta-atom geometries: dipole (DIP), rectangular ring (RR), split-ring res-onator (hybrid resonance, SRR1), split-ring resonator (LC resonance,SRR2), doubly split ring resonator (dSRR). Open symbols are theresults of FDTD simulations, and the solid lines show the fit resultsof a biexponential decay.

(open circles). To get a reliable value for the coupling strengthg, we have used the following fitting procedure. First, theparameters of the bare metasurface (ωMM, κ , and A) have beenextracted from the transmission coefficient in the uncoupledcase (see Fig. 4). The refractive index of the substrate ns andthe parameters of the quantum well (ωQW and γ ) are thentaken from the simulation parameters. Thus the only remainingfitting parameter used for the curves in Fig. 5 is the couplingrate g.

We have repeated this procedure for all five meta-atomgeometries. The results are plotted in Fig. 6, where the couplingrate g is shown as a function of distance d between the quantumwell and the metasurface. The simulated data (open symbols)are excellently reproduced by a biexponential decay law,

g(d) = g1e−z/ζ1 + g2e

−z/ζ2 , (16)

shown as solid lines. The reason for the existence of two decaylengths, ζ1 and ζ2, is thereby related to the inhomogeneouselectric field profile across the surface of the meta-atom (seeinsets in Fig. 4). The strongest electric fields and shortestdecay lengths into the substrate are obtained close to the edgesof the meta-atom. Away from the edges, the decay lengths arelonger, with lower field strengths. However, these regions areextended over a wider area in the x-y plane. Therefore, bothcontributions have approximately the same amplitudes. Theextracted parameters are listed in Table I.

A comparison of the extracted coupling rate at the closestdistance, g250 nm, and the linewidth of the metasurface reso-nance confirms the conclusions drawn from the line shapeof the coupled system shown in Fig. 4. In the case of thelinear dipole, the rectangular ring, and the hybrid resonanceof the split-ring resonator, the condition for cavity-inducedtransparency, κ > g > γ , is fulfilled. The LC resonances ofthe split-ring resonator and the doubly split ring resonator bothhave a bare linewidth smaller than the achieved coupling rate.Thus, in these two cases, the condition for strong coupling isfulfilled, g > κ,γ .

Interestingly, the decay length and achievable couplingrate show only a weak dependence on the actual geometryof the meta-atom. For instance, the linear dipole and theLC resonance of the split-ring resonator exhibit the largestvalue for the total coupling rate, g1 + g2, despite the factthat the underlying resonant modes are different in nature.Furthermore, for all investigated geometries the coupling ratesshow a biexponential decay with distance d. The short decaylength has a value of around 500 nm; the second decay lengthis on the order of a few microns. Therefore the meta-atomswith the electric dipole mode show higher values for ζ1. Weattribute the short component to field enhancement at the edgesof the metalization, whereas the long component stems fromthe modal distribution across the entire meta-atom, thusfavoring larger structures.

One way to increase the coupling rate between the cavityand the quantum system is to increase the number of quantumwells NQW . However, due to the evanescent nature of theelectric field responsible for the coupling, an effective numberof quantum wells has to be used.7 Equation (16) thus takes theform

g(NQW ) =2∑

i=1

gie−d/ζi

√1 − e−2�dNQW /ζi

1 − e−2�d/ζi, (17)

where d is the distance between the metasurface and the firstquantum well and �d is the distance between subsequentquantum wells. Taking the limit NQW → ∞ in Eq. (17) yieldsa theoretical limit for the achievable coupling rate using a givenmeta-atom geometry. As an example, Table I lists the expectedcoupling rates gmax obtained for the sample values d = 250 nmand �d = 180 nm, corresponding to 140-nm-wide parabolicquantum wells separated by 40-nm AlGaAs barriers. Dueto the long decay length of the coupling strength, the threemeta-atom geometries featuring the electric dipole modesshow the highest values of gmax. In all cases, the coupling ratewould be high enough to reach the strong-coupling regime.The highest value of gmax = 3.46 × 1012 rad/s is therebyreached using the hybrid mode of the split-ring resonator.This large value would account for a vacuum Rabi splittingof 2g = 2π × 1.1 THz, or 48% of the transition frequencyωQW , and is therefore well within the ultrastrong-couplingregime.26

We conclude from the above discussion that all of theinvestigated meta-atom geometries are suited for strong- andeven ultrastrong-coupling experiments. However, when thenumber of quantum wells is limited to one or a few, thenthe split-ring resonator tuned for the LC resonance or thedoubly split ring resonator would have to be used to reach thestrong-coupling regime, as these show the narrowest resonancelinewidths.

V. CONCLUSIONS

To summarize, we have numerically demonstrated thefeasibility of strong- and even ultrastrong-coupling exper-iments based on simple planar metasurfaces and intersub-band transitions in quantum wells. Based on finite-differencetime-domain simulations, we could extract the importantparameters characterizing the interaction between the two

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systems, namely, the radiative linewidth of the metamaterial,the achievable coupling rate, and its dependence on thedistance between the quantum well and the metasurface. Theradiative linewidth depends strongly on the physical nature ofthe metasurface resonance. Due to the wide range of achievablelinewidths ranging from 100 to over 650 GHz and the factthat the metamaterial can be coupled to an external probefield, the system can also be used to investigate the so-calledhybridization-induced transparency regime.9

By investigating five different meta-atom shapes, we foundthat the achievable coupling rate shows only a weak depen-dence on the actual geometry and thus the physical nature ofthe metamaterial resonance. For all investigated geometries,the coupling rate can, in principle, be made large enough toobserve the strong-coupling regime by using multiple quantum

wells. Especially, the coupling strength for metasurfaces basedon electric dipole modes exhibits a long decay length of severalmicrons. This finding has important implications for therealization of surface-emitting THz quantum-cascade lasers,for example. For experiments limited to one or a few quantumwells, meta-atoms based on gap resonances, such as thefundamental LC resonance of the split-ring resonator, yieldthe best results due to their narrow resonance linewidth.

ACKNOWLEDGMENTS

The authors acknowledge partial financial support by theAustrian Society for Microelectronics (GMe), the EuropeanCommission (project ICT-296500 Teracomb), and the AustrianScience Fund FWF (SFB IR-ON F25).

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