Few-Electron Ultrastrong Light-Matter Coupling at 300 GHz with

Letter Cite This: Nano Lett. XXXX, XXX, XXX−XXX

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Few-Electron Ultrastrong Light-Matter Coupling at 300 GHz with Nanogap Hybrid LC Microcavities Janine Keller,† Giacomo Scalari,*,† Sara Cibella,‡ Curdin Maissen,† Felice Appugliese,† Ennio Giovine,‡ Roberto Leoni,‡ Mattias Beck,† and Jérôme Faist† †

ETH Zürich, Institute of Quantum Electronics, Auguste-Piccard-Hof 1, Zürich 8093, Switzerland Istituto di Fotonica e Nanotecnologie (IFN), CNR, via Cineto Romano 42, 00156 Rome, Italy

‡

S Supporting Information *

ABSTRACT: Ultrastrong light-matter coupling allows the exploration of new states of matter through the interaction of strong vacuum fields with huge electronic dipoles. By using hybrid dipole antenna-split ring resonator-based cavities with extremely small effective mode volumes Veff/λ30 ≃ 6 × 10−10 and surfaces Seff/λ20 ≃ 3.5 × 10−7, we probe the ultrastrong light-matter coupling at 300 GHz to less than 100 electrons located in the last occupied Landau level of a high mobility two-dimensional electron gas, measuring a normalized coupling ratio of ΩR/ωc = 0.36. Effects of the extremely reduced cavity dimensions are observed as the light-matter coupled system is better described by an effective mass heavier than the uncoupled one. These results open the way to ultrastrong coupling at the single-electron level in two-dimensional electron systems. KEYWORDS: Ultrastrong coupling, few electrons, THz, hybrid LC resonator, Landau levels, two-dimensional electron gas

S

ubwavelength metallic resonators offer interesting possibilities in basic and applied science due to their capability of greatly increasing the interaction of electromagnetic waves with small nano objects with very different dimensions. Optical antennas,1 plasmonic resonators, metamaterials, and patch cavities2 were recently used to demonstrate several advancements in the field of strong light-matter coupling. One interesting research direction is the one studying the so-called ultrastrong light-matter coupling regime,3 achieved when the coupling strength ΩR becomes comparable to the unperturbed frequency of the system ω. Here peculiar cavity quantum electrodynamics effects are predicted due to the creation of an exotic ground state where correlated photons are present.4,5 Such ultrastrong coupling regime has been now observed in different physical platforms, as intersubband (ISB) transitions in semiconductor quantum wells6 and plasmons,7 electronic transitions in organic molecules,8 Cooper pair boxes in cavity QED (Quantum Electro Dynamics),9 and by our group with magnetopolaritons, where cyclotron10 and magneto-plasmon transitions11 in two-dimensional electron gases (2DEG) are coupled to terahertz (THz) metamaterials. The scaling of the normalized coupling ratio ΩR toward values larger than

process. Most of the systems cited above rely on the collective enhancement of the interaction strength that scales with √N. Single quantum elements in strong16,17 and eventually ultrastrong coupling18 provide a powerful platform to implement quantum computing protocols. The absolute control on the number of coupled elements and its scaling toward low values would offer a great tool to study ultrastrong coupling physics in a different regime. In previous realizations, the system could be treated as bosonic due to the very diluted number of excitations with respect to the total number N of electronic excitations. A reduction toward a few electron regime would provide a change in the accessible physics, because the system will be better described by the so-called quantum Rabi model.18,19 Theoretical calculations on an intersubband two level system in such a regime predict peculiar spectral signatures.20 In order to reduce the number of excitations coupled to the light field, the most straightforward way is to reduce the cavity effective interaction volume by fully exploiting the role of metallic confinement. This approach recently led to the demonstration of single molecule strong coupling at room temperature at visible frequencies21 with a nano cavity of V normalized volume (λ / n)3 ≃ 4 × 10−7 . In our previous experi-

unity12,13 is expected to significantly increase the number of virtual photons present in the ground state14,15 of the ultra strongly coupled system. Together with the strength of the coupling, another highly relevant parameter is the number N of electronic oscillators effectively involved in the strong coupling

ments the electron number at the anticrossing for a coupling

ωc

© XXXX American Chemical Society

Received: July 28, 2017 Revised: November 22, 2017 Published: November 27, 2017 A

DOI: 10.1021/acs.nanolett.7b03228 Nano Lett. XXXX, XXX, XXX−XXX

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ΩR ωc

= 0.27 is typically about 27000. In other strong

coupling experiments with ISB transitions and arrays of metallic cavities the number of strongly coupled electrons per cavity are 2400 (near-IR22) and 1400 (mid-IR23). Our system operates in the THz/GHz range where the large conductivity of noble metals enables very small interaction volumes combined with low losses.24 For a 2DEG, the number of electrons per unit surface Nel which are optically active in the single particle, cyclotron regime (i.e., where magnetoplasmonic corrections25,26 can be ignored, as in the present case) is given by the ratio between the electron sheet density ρ2DEG and the filling factor ν:27 ρ ρ eB Nel(B) = 2DEG = 2π2DEG = ℏρ2DEG h ν eB

(1)

Only electrons on the topmost Landau level participate to the optical transition as the lower ones are Pauli blocked.27 Then the number of electrons nel for a cavity of effective surface Seff is equal to the number of independent electron orbits which eB corresponds to the number of flux quanta Φ(B) = h within the given surface nel(B) =

⎡ 1 ⎤ eB × Seff ≈ 2.42 × 1014 × ⎢ 2 ⎥ ·B ·Seff ⎣m T ⎦ h

(2)

It is worth noting that the electron number depends only on the effective surface, linearly on the magnetic field B and on the fundamental constants e,h. Since our target is to operate in the ultrastrong coupling regime, we have to ensure that the number of electrons coupled to the resonator in such a surface still yields a normalized coupling ratio larger than 10%. It has been already demonstrated theoretically27 and experimentally10,12 that the normalized coupling ratio of the magnetopolaritons scales with √ν. In this case, our experimental platform is particularly suited for pursuing few-electron ultrastrong coupling experiments, since the low magnetic field regime, where less and less electrons are contained in the surface Seff, corresponds to a high filling factor and hence to a high value of the normalized coupling ratio ΩR .

Figure 1. (a) Overall scanning electron microscopy micrograph of the resonator array with unit cell dimensions. (b) Resonator with dimensions. (c) Detail of the gap at the center of the two antenna lobes with a separation Δ = 172 nm.

ωc

The clean observation of magnetopolaritons in the case of our transmission experiments12 has been obtained with complementary metasurfaces,28 which naturally filter out the weakly coupled cyclotron peak obtaining a signal which is solely due to the (ultra)strongly coupled metasurface. We use in the present experiment a resonant cavity which is a hybrid between a dipole antenna, widely used for light-matter coupling at the nanoscale from visible29 to the THz range30,31 and a complementary LC (inductance-capacitance) metasurface (or nanoslit32). Our design features an electric field distribution highly concentrated in regions of the order of ≈200 × 200 nm2 keeping the LC nature of the resonator by employing long inductive elements. The layout of the sample and the details of the resonators together with the dimensions are visible in Figure 1. The cavities are defined by electron beam lithography and successive lift off of the metallic surface (Ti/Au, 5/180 nm) deposited on top of a GaAs/AlGaAs triangular well heterostructure. We have a total of Nres = 102 resonators, all probed by the THz beam of our THz-time domain spectroscopy (TDS) system. In Figure 2 we report finite element electromagnetic simulations (CST microwave STUDIO) of the

first three cavity modes together with the simulated S parameter and the surface current distribution for the low frequency mode at νr = 300 GHz (λr ≃ 1 mm). From the current distribution we can identify that the resonant mode is of the LC kind, making this cavity partly behaving as “plasmonic” because part of the energy is stored in the currents which are circulating in the structure33 all along its perimeter (see Figure 2c), allowed by the connected nature of the complementary metasurface. The overall length of both tapered antenna lobes (200 μm) will set the resonance frequency of the cavity: we then define the small cavity volume by interrupting the triangular structures and creating a capacitor with an extremely narrow gap Δ = 170 nm (this value is the result of the mean of the gap of 10 cavities picked on different rows on the sample, the statistics gives Δ = 170 nm with σsdev = 12 nm). The control over the narrow gap will finally set the extension of the effective mode surface and then the number of electrons B

DOI: 10.1021/acs.nanolett.7b03228 Nano Lett. XXXX, XXX, XXX−XXX

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Figure 2. (a) In-plane (Eplane = Ex2 + Ey2 at z = −90 nm) electric field distribution for the first three modes of the cavity. (b) Simulated transmission (red line) and measured sample transmission at 300 K (blue). (c) Simulated surface current distribution for Mode 1.

actually coupled to the resonator as described by eq 2. The extension of the optical mode between the gaps in the direction perpendicular to the plane of the resonator is of the order of the gap (200−300 nm, as visible in Figure 2a). In order to enhance the coupling we employed an optimized shallow 2DEG with the channel located 90 nm below the surface (fringing field of the capacitor, Figure 2a) with an electron density of 3.1 × 1011 cm−2 and a Hall mobility μ = 2.5 × 106 cm2/(V s) at 4 K. We calculate the cavity volume and the effective mode surface by adopting the definitions usually employed in microcavity QED and quantum optics:34 Veff =

∫V ϵE2 dV Max(ϵE2)V

Seff =

∫z = z

2DEG

Figure 3. (a) Transmission of the hybrid cavity deposited on top of the 2DEG at a temperature of 2.7 K as a function of the magnetic field. (b) Sections of the previous color plot for B = 0.9 T, B = 2.5 T, and B= 4T with the different resonant peaks of lower polariton (LP) and upper polariton (UP) highlighted. (c) Zoom of the colorplot of panel a with enhanced contrast to highlight the two anticrossing with modes M2 and M3.

2 ϵExy dS

2 Max(ϵExy )z = z 2DEG

effect of the ohmic losses on the cavity’s quality factor together with the thickness (180 nm) of the gold film. This conductance value is then used to evaluate all the cavity-related quantities, as the field distribution for the three modes reported in Figure 2a and the surface current for the mode M1 plotted in Figure 2c. For the lowest frequency mode of the cavity, we calculate a V270GHz ≃ 4.3 × 10−19 m3 corresponding to a normalized mode eff

(3)

In Figure 2b, we report the computed sample transmission for an array of resonators on top of an undoped GaAs wafer, together with a measurement of our sample at 300 K, where the effect of the 2DEG carriers is highly reduced, due to the much lower mobility and the different carrier distribution. We can see that a good agreement between simulation and experiment is obtained when we consider a gold surface with a conductance of σAu = 7.5 × 105 S/m, which is lower than what usually reported at similar frequencies.35 The presence of long (100 μm) and narrow antenna lobes (see Figure 1) enhances the

volume of 270GHz Seff

270GHz Veff

λ3 −7

≃ 3 × 10−10 and a normalized surface

≃ 3 × 10 . This extremely subwavelength confinement enhances the vacuum field fluctuations and the calculated field enhancement factor is |Eres/Ein| ≃ 460 for the lowest frequency mode M1 = 270 GHz. In Table 1 we summarize the principal λ2

Table 1. Relevant Figures for the Three Considered Cavity Modes

mode 1 mode 2 mode 3

freq (GHz)calc

freq (GHz)4K,HighB

Veff (m3)

Veff/λ30

Seff (m2)

Seff/λ20

270 745 1223

307 770 1210

9.3 × 10−19 2.3 × 10−18 3.9 × 10−18

7 × 10−10 3.5 × 10−8 2.6 × 10−7

4.2 × 10−13 6.7 × 10−13 6.4 × 10−13

3.4 × 10−7 4.1 × 10−6 1 × 10−5

C

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Figure 4. (a) Hybrid nanogap cavity transmission experiment: extracted transmission maxima fitted with a Hopfield-like model (see refs 10 and 27 and Supporting Information) and constant cyclotron mass as measured on a bare 2DEG. (b) Hybrid nanogap cavity transmission experiment: extracted transmission maxima fitted with Hopfield model and an effective mass varying linearly with the magnetic field. (c) RMS values for the three modes normalized coupling ratios in the case of linear varying mass (solid lines) and constant bare 2DEG mass (dashed lines). (d) Residuals from the fits of panels (a,b). (e) Standard 300 GHz split ring cavity transmission experiment: extracted maxima fitted with Hopfield model with bare 2DEG mass and heavy 2DEG mass. (f) RMS values for the 300 GHz split ring cavity coupling ratios in the case of bare 2DEG mass (solid line) and heavy mass (dashed line). (g) Residuals from the fits of panel e.

figures of the three modes of the resonators (calculated and measured resonance frequencies) together with their normalized volumes and surfaces. The measured value of the cold cavity resonance is deduced by assuming that our light-matter coupled system is well modeled by the Hopfield approach detailed in refs 27 and 36. In this model, the empty cavity frequency can be retrieved by measuring the value of the lower branch frequency for high magnetic field values highly detuned from resonance. The asymptotic values measured from the data of Figure 4 are indeed the empty cavity frequencies reported in Table 1. If we compare the numbers of Table 1 with the same quantities evaluated for a dipole antenna30,31 of identical dimensions (see Supporting Information), we see that our cavity has a resonant frequency which is roughly 40% lower, and normalized modal volume and surface roughly a factor of 10 smaller, thanks to the added inductance of the

complementary structure. These values are already remarkable but they are clearly limited by the losses in the metal; future experiments employing superconductors36,37 should bring significantly higher quality factors and field enhancements. In Figure 3a, we report the sample transmission as a function of the applied magnetic field recorded at T = 2.7 K. We use a THz-TDS spectrometer coupled to a He cryostat equipped with a superconducting magnet and described in ref.10 Clear anticrossings of the three cavity modes with the linear cyclotron dispersion are observed at the resonant fields and sections of the colorplots are reported in Figure 3b to highlight the different polariton branches. From the field distributions reported in Figure 2a we expect the different polaritonic branches to be coupled since the photonic modes are not orthogonal. As expected, we observe a high value of the normalized coupling ratio for the low frequency mode and then D

DOI: 10.1021/acs.nanolett.7b03228 Nano Lett. XXXX, XXX, XXX−XXX

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heavier effective mass is a feature of many-body physics which emerges naturally in the Landau−Fermi liquid theory39 and has been observed in a variety of 2DEG in transport experiments (see for example ref 40). A more detailed understanding of the physical mechanism underlying this observed heavier mass value is beyond the scope of this paper and will be carried out elsewhere. The normalized coupling ratios measured in our experiment from Figure 4c are

ΩR | ωc M1

= 0.36,

ΩR | ωc M2

= 0.16 ,

ΩR | ωc M3

= 0.09

all with RMS values of less than 2.5%, placing M1 and M2 clearly in the ultrastrong coupling regime. We now concentrate on the first two modes and evaluate the average number of electrons ultrastrongly coupled to the resonator. The signal-to-noise ratio at low frequencies where the anticrossings with the first two modes of the cavity take place can be increased by employing a low pass optical filter with a cutoff at about 1.2 THz. In Figure 5a, we report this transmission measurement together with an electron number scale relative to mode M1 obtained by employing eq 2 with the effective mode surface from Table 1. We see that at the anticrossing field of 0.9 T we have approximately 90 electrons ultrastrongly coupled per resonator with a normalized ratio of

Figure 5. (a) Number of electrons coupled to the first mode M1 of the resonator as a function of the magnetic field together with the corresponding polaritonic branches experimentally measured. (b) Number of electrons as a function of the magnetic field for the cavity presented in the paper (blue curve) and prediction for a similar cavity with gap 30 nm and width 30 nm.

ΩR ω

lower coupling strengths for the excited ones since they occur for lower filling factors. We can extract the maxima of transmission and then use the theoretical model and the fitting procedure developed in refs 10 and 27 to extract the coupling ratios for the three modes. The extracted maxima and relative fits are reported in Figure 4. The polaritonic branches are first fitted using as an input parameter the cold cavity frequencies and assuming a constant effective mass for the electrons equal to the one measured on the bare 2DEG without any cavity (m*/m0 = 0.06973 ± 0.00005, see Supporting Information for more details). The fitting parameter is the normalized coupling ratio ΩR . From the fitting residuals

= 0.36. The cooperativity for this kind of system yields also Ω2

a very good figure of C = 4 γκR = 13 (assuming γ/2π = 40 GHz and κ/2π = 80 GHz, resolution limited) which makes roughly Cel = 0.15 per single electron. It is worth noting that the total number of electrons yielding the measured signal at the anticrossing is of the order of 90 × Nres ≃ 9000. For the mode M2, we can estimate a number of electrons ultrastrongly coupled of about 400 at the anticrossing field B = 1.9 T. We can now discuss the possible developments of this experimental platform and give an outlook on the limits reachable by this approach. In Figure 5b, we plot the number of electrons as a function of the magnetic field for the cavity dimensions presented in this paper and for a similar design with gap width and antenna width of 30 nm yielding an effective mode surface of Seff = 2.5 × 10−14 evaluated at 150 GHz. Such dimension should be still within the reach of our fabrication technology. For a resonant frequency of 250 GHz or below (corresponding to magnetic fields lower than 0.75 T for GaAs), the system would enter the regime where only 4 or less electrons are coupled, where theoretical calculation on slightly different system20 predicts new spectral features due to the change of regime. The light-matter coupled system would be described by the quantum Rabi model but still taking into account antiresonant and A2 terms typical of the ultrastrong coupling regime, as discussed for a two level system in the case of a superconducting QBIT in ref 18. In conclusion, we demonstrated a new type of THz cavity featuring extremely small normalized volume and surface. When strongly coupled to the cyclotron transition of a highmobility 2DEG the system allows to enter the few electron

ωc

(reported in Figure 4d), it is clear that the data relative to the first anticrossing with the cavity mode at 300 GHz is better described employing a heavier effective mass, whereas the other resonances are well described by a lighter mass but still heavier than the uncoupled one. If we let the effective mass vary linearly with the magnetic field (see Figure 4b), we obtain a better fitting for all the three modes as testified by the root mean square (RMS) value for the deduced couplings reported in Figure 4c for the different values of the effective mass. The value of the mass for the coupling to the first mode M1 is m*/ m0 = 0.0727, significantly heavier than the value for the bare 2DEG m*/m0 = 0.0697 (for the detailed calculation of the RMS value see the Supporting Informations). As a control, we performed another strong coupling experiment with a standard split ring cavity with a micrometric-sized gap (see Supporting Information for details) at the same frequency of 300 GHz deposited on the very same 2DEG. In this case the model fits well with the uncoupled effective mass m* = 0.0697m0, as visible in Figure 4e−g), yielding a Ω normalized coupling ratio ωR |SR = 0.32 . These data suggest

(