Nature Physics  Letter
Collective nonperturbative coupling of 2D electrons with highqualityfactor terahertz cavity photons
 Qi Zhang^{1}^{, }
 Minhan Lou^{1}^{, }
 Xinwei Li^{1}^{, }
 John L. Reno^{2}^{, }
 Wei Pan^{3}^{, }
 John D. Watson^{4}^{, }
 Michael J. Manfra^{4, 5}^{, }
 Junichiro Kono^{1, 6, 7}^{, }
 Journal name:
 Nature Physics
 Volume:
 12,
 Pages:
 1005–1011
 Year published:
 DOI:
 doi:10.1038/nphys3850
 Received
 Accepted
 Published online
The collective interaction of electrons with light in a highqualityfactor cavity is expected to reveal new quantum phenomena^{1, 2, 3, 4, 5, 6, 7} and find applications in quantumenabled technologies^{8, 9}. However, combining a long electronic coherence time, a large dipole moment, and a high qualityfactor has proved difficult^{10, 11, 12, 13}. Here, we achieved these conditions simultaneously in a twodimensional electron gas in a highqualityfactor terahertz cavity in a magnetic field. The vacuum Rabi splitting of cyclotron resonance exhibited a squareroot dependence on the electron density, evidencing collective interaction. This splitting extended even where the detuning is larger than the resonance frequency. Furthermore, we observed a peak shift due to the normally negligible diamagnetic term in the Hamiltonian. Finally, the highqualityfactor cavity suppressed superradiant cyclotron resonance decay, revealing a narrow intrinsic linewidth of 5.6 GHz. Highqualityfactor terahertz cavities will enable new experiments bridging the traditional disciplines of condensedmatter physics and cavitybased quantum optics.
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At a glance
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Main
Strong resonant light–matter coupling in a cavity setting is an essential ingredient in fundamental cavity quantum electrodynamics (QED) studies^{14} as well as in cavityQEDbased quantum information processing^{8, 9}. In particular, a variety of solidstate cavity QED systems have recently been examined^{15, 16, 17, 18}, not only for the purpose of developing scalable quantum technologies, but also for exploring novel manybody effects inherent to condensed matter. For example, collective fold enhancement of light–matter coupling in an Nbody system^{19}, combined with colossal dipole moments available in solids, compared to traditional atomic systems, is promising for entering uncharted regimes of ultrastrong light–matter coupling. Nonintuitive quantum phenomena can occur in such regimes, including a ‘squeezed’ vacuum state^{1}, the Dicke superradiant phase transition^{2, 3}, the breakdown of the Purcell effect^{4}, and quantum vacuum radiation^{5} induced by the dynamic Casimir effect^{6, 7}.
Specifically, in a cavity QED system, there are three rates that jointly characterize different light–matter coupling regimes: g, κ and γ. The parameter g is the coupling constant, with 2g being the vacuum Rabi splitting between the two normal modes, the lower polariton (LP) and upper polariton (UP), of the coupled system; see Supplementary Equation (6). The parameter κ is the photon decay rate of the cavity; τ_{cav} = κ^{−1} is the photon lifetime of the cavity, and the cavity Q = ω_{0}τ_{cav} at mode frequency ω_{0}. The parameter γ is the nonresonant matter decay rate, which is usually the decoherence rate in the case of solids. Strong coupling is achieved when the splitting, 2g, is much larger than the linewidth, (κ + γ)/2, and ultrastrong coupling is achieved when g becomes a considerable fraction of ω_{0}. The two standard figures of merit to measure the coupling strength are C ≡ 4g^{2}/(κγ) and g/ω_{0}; here, C is called the cooperativity parameter^{18}, which is also the determining factor for the onset of optical bistability through resonant absorption saturation^{20}. To maximize C and g/ω_{0}, one should construct a cavity QED setup that combines a large dipole moment (that is, large g), a small decoherence rate (that is, small γ), a large cavity Q factor (that is, small κ), and a small resonance frequency ω_{0}.
Group III–V semiconductor quantum wells (QWs) provide one of the cleanest and most tunable solidstate environments with quantumdesignable optical properties. Microcavity QWexcitonpolaritons represent a landmark realization of a strongly coupled light–condensedmatter system that exhibits a rich variety of coherent manybody phenomena^{21}. However, the large values of ω_{0} and relatively small dipole moments for interband transitions make it impractical to achieve large values of g/ω_{0} using excitonpolaritons. Intraband transitions, such as intersubband transitions (ISBTs)^{1} and cyclotron resonance (CR)^{22}, are much better candidates for accomplishing ultrastrong coupling because of their small ω_{0}, typically in the midinfrared and terahertz range, and their enormous dipole moments (tens of eÅ). Experimentally, ultrastrong coupling has indeed been achieved in GaAs QWs using ISBTs^{10, 11} and CR^{12, 13}. In the latter case, a record high value of g/ω_{0} = 0.87 has been reported^{13}. In all these previous intraband studies of ultrastrong light–matter coupling, however, due to ultrafast decoherence (large γ) and/or lossy cavities (large κ), the value of C remained small; that is, the standard strongcoupling criterion (C 1) was not satisfied.
Here, we simultaneously achieved small γ and small κ in ultrahighmobility twodimensional electron gases (2DEGs) in GaAs QWs placed in a highQ onedimensional (1D) terahertz photoniccrystal cavity (PCC) in a perpendicular magnetic field. We achieved C > 300 and g/ω_{0} ~ 0.1, observing vacuum Rabi splitting (Rabi oscillations) in the frequency (time) domain. We demonstrated that the influence of this nonperturbative coupling extends even to the region where Δ > ω_{0}. This can occur only when g^{2}/(ω_{0}κ) > 1, which we satisfied through a unique combination of strong light–matter coupling, a small resonance frequency, and a highQ cavity. Furthermore, we observed a dependence of 2g on the electron density (n_{e}), signifying the collective nature of light–matter coupling^{19, 23, 24, 25, 26, 27}. A value of g/ω_{0} = 0.12 was obtained with just a single QW with a moderate n_{e}. Finally, the previously identified superradiant decay of CR in highmobility 2DEGs^{28} was significantly suppressed by the presence of the highQ terahertz cavity. As a result, we observed ultranarrow polariton lines, yielding an intrinsic CR linewidth as small as 5.6 GHz (or a CR decay time of 57 ps) at 2 K.
Highmobility GaAs 2DEG samples were studied using terahertz timedomain magnetospectroscopy (see Supplementary Section 2). The magnetic field quantized the density of states of the 2DEG into Landau levels. As schematically shown in Fig. 1a, terahertz cavity photons are coupled with the transition between adjacent Landau levels, that is, CR. Figure 1b shows our 1D terahertz PCC design, consisting of two layers of 50μmthick undoped Si wafers on each side as a Bragg mirror. Thanks to the large contrast of refractive index between Si (3.42 in the terahertz range) and vacuum, only a few layers of Si were required to achieve sufficient cavity confinement of terahertz radiation with high Q values. A substrateremoved 4.5μmthick GaAs 2DEG sample was placed on the central ‘defect’ layer of the PCC, which was a 100μmthick Si (sapphire) wafer in Cavity 1 (Cavity 2). Calculated electric field distributions inside Cavity 1 are shown in Fig. 1c–e, for the first, second and third cavity modes, respectively. The spatial overlap of the 2DEG and the electric field maximum ensured the strongest light–matter coupling.
Figure 1f shows a terahertz transmission spectrum for Cavity 1, containing a 2DEG, at 4 K. Three photonic band gaps are seen as transmission stopbands. At the centre of each stopband, a sharp cavity mode is observed. As shown in Fig. 1g–i, the fullwidth at halfmaximum (FWHM) values, or κ, of these cavity modes were 2.6 GHz, 5.0 GHz and 3.8 GHz, corresponding to Q factors of 150, 243 and 532, respectively; note that these numbers are slightly lower than those for an empty cavity without including the 2DEG, which were 183, 450 and 810, respectively. These Qfactors are one to two orders of magnitude higher than those reported for the terahertz metamaterial resonators employed in previous untrastrongcoupling studies using 2DEG CR^{12, 13}. In the following, experimental data recorded with Cavity 1 are shown. Results obtained with Cavity 2 are described in Supplementary Section 4.
By varying the magnetic field (B), we continuously changed the detuning between the cyclotron frequency (ω_{c} = eB/m^{∗}, where m^{∗} = 0.07m_{e} is the electron effective mass of GaAs and m_{e} = 9.11 × 10^{−11} kg) and the cavity mode frequency (ω_{0}): Δ ≡ ω_{c} − ω_{0}. Clear anticrossing behaviour, expected for strong coupling, is shown in Fig. 2a for the first cavity mode in Cavity 1. Two polariton branches (LP and UP) were formed through the hybridization of CR and the terahertz cavity photons. The central peak originates from the transmission of the CRinactive circularpolarization component of the linearly polarized incident terahertz beam, which does not interact with the 2DEG and whose position is practically independent of B in this field range. The FWHM of the central peak is thus essentially given by κ alone, while that for the LP and UP peaks at Δ = 0 is given by (κ + γ)/2. Therefore, from the Δ = 0 spectrum (B_{r} = 1.00 T for this mode), we determined (2g, κ, γ, ω_{0})/2π = (74,2.6,5.6,403) GHz, yielding C = 360 and g/ω_{0} = 0.09. Parameter values determined in this manner for all modes in both Cavities 1 and 2 are summarized in Table 1, together with cavity parameters and resonance conditions.
As in other cavity QED systems based on atoms and microcavity excitons, vacuum Rabi splitting in the frequency domain can be directly observed as timedomain oscillations^{23, 26, 29}. Experimentally, for an incident terahertz beam linearly polarized in the x direction, we measured the ypolarization component, E_{y}, of the transmitted terahertz wave, in both positive (+B) and negative (−B) fields, and took the difference, ΔE_{y} = E_{y}(+B) −E_{y}(−B), to eliminate any background noise; see Supplementary Section 5 for more details. The CR inactive mode was numerically filtered out. As shown in Fig. 2b, the measured ΔE_{y} signal showed strong beating between the two polariton modes, which can be viewed as coherent repetitive energy exchange between the matter resonance and the cavity photons. At each beating node (indicated by an arrow), energy is stored in the 2DEG CR. The average time separation between two adjacent beating nodes was about 13–15 ps, matching the 2g splitting in the frequency domain; see also Fig. 2c for the Fourier transform of the timedomain oscillations. The beating lasts for dozens of picoseconds, indicating a long intrinsic CR coherence time.
In analogy to the physics of manyatom light–matter interactions^{19}, one crucial question is whether the Rabi splitting observed here is a fully coherent behaviour of a large number of individual electrons in the 2DEG. Figure 2d, e, shows three spectra exhibiting polariton manifestation at Δ = 0 for different electron densities (n_{e}), when CR is in resonance with the first cavity mode in Cavity 1. The vacuum Rabi splitting (2g) between the LP and UP peaks exhibited a squareroot dependence on n_{e} (Fig. 2e), which is strong evidence for collective vacuum Rabi splitting, as observed in atomic gases^{23}, spin ensembles^{24, 25, 26}, and intersubband transitions^{27}. This observation validates the notion that billions of 2D electrons are interacting with a common cavity terahertz photon field in a fully coherent manner. By extrapolation, the vacuum Rabi splitting for CR of a single electron is estimated to be 0.14 MHz. It is also worth noting that we should be able to increase the vacuum Rabi splitting further by using multiple layers of a 2DEG with a higher electron density.
The coupled system of Landauquantized 2D electrons and terahertz cavity photons can be described by the following Hamiltonian^{22}: , where , , , and . The first two terms, and , represent, respectively, the energy of the cavity mode at ω_{0} and the energy of the 2DEG in a magnetic field B with frequency ω_{c}. The operators and ( and ) are the annihilation and creation operators for cavity photons (collective CR excitations), respectively. The light–matter interaction term, , with coupling strength g includes counterrotating terms, , which are usually neglected under the rotatingwave approximation. Also included in the total Hamiltonian is the diamagnetic term, , also known as the A^{2} term because, mathematically, it is proportional to the square of the vector potential A of the light field. The prefactor ℏg^{2}/ω_{c} of the A^{2} term suggests that this term is negligible in the weakcoupling regime, but can have measurable effects when g is a significant fraction of ω_{c}.
Figure 3a presents the best fits to our data with three different Hamiltonians: (full Hamiltonian), , and (the Jaynes–Cummings Hamiltonian, ). At the optimum fitting, the value of g/ω_{0} was determined to be 0.09 (for Cavity 1). The full CRcavity Hamiltonian provided the best fit, while the other two failed to show the nonnegligible blueshift of both polariton modes. Notably, this overall blueshift leads to asymmetric splitting between the two polariton modes with respect to the resonance frequency; this asymmetry is explainable only with the full Hamiltonian. With g/ω_{0} = 0.1, the contribution of the A^{2} term is expected to be of the order of 0.01ω_{0}. This 1% contribution from the A^{2} term is indeed responsible for the observed blueshift (see also Supplementary Fig. 7).
Furthermore, we have evidence that the influence of this nonperturbative coupling extends even to the region where Δ > ω_{0}, as shown in Fig. 3b, c. This unusually large extent of light–matter hybridization affects the region of negative magnetic fields, where, ordinarily, the UP peak would stay at the cavity mode frequency, ω_{0}, independent of B. Here, instead, the UP mode is already slightly blueshifted from ω_{0} even at B = 0, and then redshifts with increasing B, asymptotically approaching ω_{0} at large negative magnetic fields. This counterintuitive behaviour can also be thought of as coupling between the cavity mode and the CR inactive mode at a negative frequency due to the counterrotating terms in ; see Supplementary Section 7 (Supplementary Fig. 10). The blueshift of the UP peak due to hybridization at finite Δ can be estimated to be around B = 0, where Δ 2g. Also, when B ~ 0, the UP linewidth ≈ κ. Therefore, for the UP peak to be blueshifted from the cavity mode frequency at B = 0 (or Δ = ω_{0}), g^{2}/ω_{0} > κ must be satisfied. This condition, g^{2}/(ω_{0}κ) > 1, can be met only through a unique combination of strong light–matter coupling, a small resonance frequency, and a highQ cavity.
Finally, we studied the extracted value of γ as a function of temperature. It has previously been shown that the decay of CR in free space is dominated by collective radiative decay, or superradiance, in ultrahighmobility 2DEGs, showing a decay rate that is proportional to n_{e} (ref. 28). This radiative decay mechanism is very strong and dominant at low temperatures, faster than any other phasebreaking scattering processes; thus, CR lines are much broader than expected from the sample mobility. In the present case, however, the emitted coherent CR radiation cannot readily escape from the highQ cavity and thus reexcites coherent CR multiple times. Hence, this reversible emission and absorption in a strongly coupled cavity2DEG system strongly suppresses the superradiant decay, revealing the intrinsic CR decoherence rate, Γ_{CR} (s^{−1}) = (πτ_{CR})^{−1}. This dramatic suppression of radiative decay is opposite to the Purcell effect in a highQ cavity expected in the weakcoupling regime, and can only be understood within the framework of strong coupling^{30}, where superradiant decay is suppressed by the reversible absorption and emission processes.
Figure 4a presents temperaturedependent transmission spectra at Δ = 0 for the first cavity mode in Cavity 1. The LP and UP peaks significantly broaden above 20 K, becoming unobservable above 80 K. The central peak, on the other hand, remains essentially unchanged as the temperature increases, serving as an excellent linewidth reference. At each temperature, we determined both κ and γ, using the procedure described earlier (Fig. 2a). As shown in Fig. 4b, the CR decay time, τ_{CR}, measured in the cavity (blue solid circles) is much longer than the superradiancelimited value in free space (red solid circles); at 2 K, the former is 57 ± 4 ps, while the latter is 10 ps. Therefore, the τ_{CR} value measured in our highQ cavity is the intrinsic CR decay time due to nonradiative decay mechanisms (that is, scattering). Also shown in Fig. 4b is the temperature dependence of the d.c. momentum scattering time, τ_{d.c.}, obtained from the electron mobility, μ_{e} = eτ_{d.c.}/m^{∗}. Above 20 K, where piezoelectric scattering and polar optical phonon scattering dominate, τ_{CR} approaches τ_{d.c.}. At 2 K, τ_{CR} is still lower than τ_{d.c.}. How the CR linewidth in a highmobility 2DEG changes with the magnetic field and temperature is a longstanding question^{31}, and a systematic study of ‘superradiancefree’ CR widths should provide significant new insight.
We have demonstrated collective, coherent, and nonperturbative light–matter coupling between the CR of a 2DEG and terahertz cavity photons in a highQ 1D photoniccrystal cavity, with a cooperativity up to 360. A highQ cavity is particularly important for cavity2DEG systems, since it increases not only the cavity photon lifetime but also the matter lifetime via the suppression of superradiant decay. Furthermore, unlike the nearfield coupling of metamaterial resonators, our terahertz cavity scheme is applicable to both 2D and bulk materials, which will allow us to study various strongly correlated systems with collective manybody excitations in the terahertz range; for example, magnetically ordered systems, hightemperature superconductors, and heavyfermion systems. Hence, our highQ terahertz cavitybased techniques open a door to a plethora of new possibilities to combine the traditional disciplines of manybody condensedmatter physics and quantum optics of cavity QED.
Methods
We used polarizationresolved timedomain terahertz magnetospectroscopy to study the 2DEG samples in terahertz cavities placed inside a splitcoil superconducting magnet. Our laser source was a Ti:sapphire regenerative amplifier (1 kHz, 775 nm, 200 fs, ClarkMXR, Inc., CPA2001). The nonlinear crystals used for both terahertz generation and detection were 1mmthick 110oriented zinc telluride (ZnTe). We studied two samples of modulationdoped GaAs 2DEG with a single quantum well. Sample 1 had density and mobility values of 1.9 × 10^{11} cm^{−2} and 2.2 × 10^{6} cm^{2} V^{−1} s^{−1}, respectively, in the dark, while after illumination at 4 K they changed to 3.1 × 10^{11} cm^{−2} and 3.9 × 10^{6} cm^{2} V^{−1} s^{−1}; Sample 2 had 5 × 10^{10} cm^{−2} and 4.4 × 10^{6} cm^{2} V^{−1} s^{−1} as the density and the mobility, respectively. The GaAs substrate was removed by selective etching. See Supplementary Information for more experimental details.
Data availability.
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank A. Chabanov, H. Pu and A. Belyanin for useful discussions. J.K. acknowledges support from the National Science Foundation (Grant No. DMR1310138). This work was performed, in part, at the Center for Integrated Nanotechnologies, a US Department of Energy, Office of Basic Energy Sciences user facility. Sandia National Laboratories is a multiprogram laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the US Department of Energy’s National Nuclear Security Administration under Contract No. DEAC0494AL85000. The work at Sandia was supported by the US Department of Energy, Office of Science, Materials Sciences and Engineering Division. Growth and characterization completed at Purdue by J.D.W. was supported by the Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award No. DESC0006671. M.J.M. acknowledges additional support from the W. M. Keck Foundation and Microsoft Research.
Author information
Affiliations

Department of Electrical and Computer Engineering, Rice University, Houston, Texas 77005, USA
 Qi Zhang,
 Minhan Lou,
 Xinwei Li &
 Junichiro Kono

Sandia National Laboratories, CINT, Albuquerque, New Mexico 87185, USA
 John L. Reno

Sandia National Laboratories, Albuquerque, New Mexico 87185, USA
 Wei Pan

Department of Physics and Astronomy, Station Q Purdue, and Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907, USA
 John D. Watson &
 Michael J. Manfra

School of Materials Engineering and School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47907, USA
 Michael J. Manfra

Department of Materials Science and NanoEngineering, Rice University, Houston, Texas 77005, USA
 Junichiro Kono

Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA
 Junichiro Kono
Contributions
Q.Z. designed and fabricated the terahertz photoniccrystal cavities, performed all measurements, analysed all experimental data, and performed all theoretical simulations, under the supervision and guidance of J.K.; M.L. and X.L. assisted Q.Z. with the simulations and measurements. J.L.R., W.P., J.D.W. and M.J.M. provided the 2DEG samples. Q.Z. and J.K. wrote the manuscript. All authors discussed the results and commented on the manuscript.
Competing financial interests
The authors declare no competing financial interests.
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Qi Zhang
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