The collective interaction of electrons with light in a high-quality-factor cavity is expected to reveal new quantum phenomena1, 2, 3, 4, 5, 6, 7 and find applications in quantum-enabled technologies8, 9. However, combining a long electronic coherence time, a large dipole moment, and a high quality-factor has proved difficult10, 11, 12, 13. Here, we achieved these conditions simultaneously in a two-dimensional electron gas in a high-quality-factor terahertz cavity in a magnetic field. The vacuum Rabi splitting of cyclotron resonance exhibited a square-root 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 high-quality-factor cavity suppressed superradiant cyclotron resonance decay, revealing a narrow intrinsic linewidth of 5.6 GHz. High-quality-factor terahertz cavities will enable new experiments bridging the traditional disciplines of condensed-matter physics and cavity-based quantum optics.
At a glance
Strong resonant light–matter coupling in a cavity setting is an essential ingredient in fundamental cavity quantum electrodynamics (QED) studies14 as well as in cavity-QED-based quantum information processing8, 9. In particular, a variety of solid-state cavity QED systems have recently been examined15, 16, 17, 18, not only for the purpose of developing scalable quantum technologies, but also for exploring novel many-body effects inherent to condensed matter. For example, collective -fold enhancement of light–matter coupling in an N-body system19, 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 state1, the Dicke superradiant phase transition2, 3, the breakdown of the Purcell effect4, and quantum vacuum radiation5 induced by the dynamic Casimir effect6, 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; τcav = κ−1 is the photon lifetime of the cavity, and the cavity Q = ω0τcav at mode frequency ω0. 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 ω0. The two standard figures of merit to measure the coupling strength are C ≡ 4g2/(κγ) and g/ω0; here, C is called the cooperativity parameter18, which is also the determining factor for the onset of optical bistability through resonant absorption saturation20. To maximize C and g/ω0, one should construct a cavity QED set-up 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 ω0.
Group III–V semiconductor quantum wells (QWs) provide one of the cleanest and most tunable solid-state environments with quantum-designable optical properties. Microcavity QW-exciton-polaritons represent a landmark realization of a strongly coupled light–condensed-matter system that exhibits a rich variety of coherent many-body phenomena21. However, the large values of ω0 and relatively small dipole moments for interband transitions make it impractical to achieve large values of g/ω0 using exciton-polaritons. 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 ω0, typically in the mid-infrared and terahertz range, and their enormous dipole moments (tens of e-Å). Experimentally, ultrastrong coupling has indeed been achieved in GaAs QWs using ISBTs10, 11 and CR12, 13. In the latter case, a record high value of g/ω0 = 0.87 has been reported13. 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 strong-coupling criterion (C 1) was not satisfied.
Here, we simultaneously achieved small γ and small κ in ultrahigh-mobility two-dimensional electron gases (2DEGs) in GaAs QWs placed in a high-Q one-dimensional (1D) terahertz photonic-crystal cavity (PCC) in a perpendicular magnetic field. We achieved C > 300 and g/ω0 ~ 0.1, observing vacuum Rabi splitting (Rabi oscillations) in the frequency (time) domain. We demonstrated that the influence of this non-perturbative coupling extends even to the region where Δ > ω0. This can occur only when g2/(ω0κ) > 1, which we satisfied through a unique combination of strong light–matter coupling, a small resonance frequency, and a high-Q cavity. Furthermore, we observed a -dependence of 2g on the electron density (ne), signifying the collective nature of light–matter coupling19, 23, 24, 25, 26, 27. A value of g/ω0 = 0.12 was obtained with just a single QW with a moderate ne. Finally, the previously identified superradiant decay of CR in high-mobility 2DEGs28 was significantly suppressed by the presence of the high-Q 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.
High-mobility GaAs 2DEG samples were studied using terahertz time-domain 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 50-μm-thick 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 substrate-removed 4.5-μm-thick GaAs 2DEG sample was placed on the central ‘defect’ layer of the PCC, which was a 100-μm-thick 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 full-width at half-maximum (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 Q-factors are one to two orders of magnitude higher than those reported for the terahertz metamaterial resonators employed in previous untrastrong-coupling studies using 2DEG CR12, 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 (ωc = eB/m∗, where m∗ = 0.07me is the electron effective mass of GaAs and me = 9.11 × 10−11 kg) and the cavity mode frequency (ω0): Δ ≡ ωc − ω0. 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 CR-inactive circular-polarization 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 (Br = 1.00 T for this mode), we determined (2g, κ, γ, ω0)/2π = (74,2.6,5.6,403) GHz, yielding C = 360 and g/ω0 = 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 time-domain oscillations23, 26, 29. Experimentally, for an incident terahertz beam linearly polarized in the x direction, we measured the y-polarization component, Ey, of the transmitted terahertz wave, in both positive (+B) and negative (−B) fields, and took the difference, ΔEy = Ey(+B) −Ey(−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 ΔEy 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 time-domain oscillations. The beating lasts for dozens of picoseconds, indicating a long intrinsic CR coherence time.
In analogy to the physics of many-atom light–matter interactions19, 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 (ne), 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 square-root dependence on ne (Fig. 2e), which is strong evidence for collective vacuum Rabi splitting, as observed in atomic gases23, spin ensembles24, 25, 26, and intersubband transitions27. 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 Landau-quantized 2D electrons and terahertz cavity photons can be described by the following Hamiltonian22: , where , , , and . The first two terms, and , represent, respectively, the energy of the cavity mode at ω0 and the energy of the 2DEG in a magnetic field B with frequency ωc. 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 counter-rotating terms, , which are usually neglected under the rotating-wave approximation. Also included in the total Hamiltonian is the diamagnetic term, , also known as the A2 term because, mathematically, it is proportional to the square of the vector potential A of the light field. The pre-factor ℏg2/ωc of the A2 term suggests that this term is negligible in the weak-coupling regime, but can have measurable effects when g is a significant fraction of ωc.
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/ω0 was determined to be 0.09 (for Cavity 1). The full CR-cavity Hamiltonian provided the best fit, while the other two failed to show the non-negligible 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/ω0 = 0.1, the contribution of the A2 term is expected to be of the order of 0.01ω0. This 1% contribution from the A2 term is indeed responsible for the observed blueshift (see also Supplementary Fig. 7).
Furthermore, we have evidence that the influence of this non-perturbative coupling extends even to the region where Δ > ω0, 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, ω0, independent of B. Here, instead, the UP mode is already slightly blueshifted from ω0 even at B = 0, and then redshifts with increasing |B|, asymptotically approaching ω0 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 counter-rotating 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 = ω0), g2/ω0 > κ must be satisfied. This condition, g2/(ω0κ) > 1, can be met only through a unique combination of strong light–matter coupling, a small resonance frequency, and a high-Q 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 ultrahigh-mobility 2DEGs, showing a decay rate that is proportional to ne (ref. 28). This radiative decay mechanism is very strong and dominant at low temperatures, faster than any other phase-breaking 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 high-Q cavity and thus re-excites coherent CR multiple times. Hence, this reversible emission and absorption in a strongly coupled cavity-2DEG system strongly suppresses the superradiant decay, revealing the intrinsic CR decoherence rate, ΓCR (s−1) = (πτCR)−1. This dramatic suppression of radiative decay is opposite to the Purcell effect in a high-Q cavity expected in the weak-coupling regime, and can only be understood within the framework of strong coupling30, where superradiant decay is suppressed by the reversible absorption and emission processes.
Figure 4a presents temperature-dependent 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, τCR, measured in the cavity (blue solid circles) is much longer than the superradiance-limited value in free space (red solid circles); at 2 K, the former is 57 ± 4 ps, while the latter is 10 ps. Therefore, the τCR value measured in our high-Q 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, τd.c., obtained from the electron mobility, μe = eτd.c./m∗. Above 20 K, where piezoelectric scattering and polar optical phonon scattering dominate, τCR approaches τd.c.. At 2 K, τCR is still lower than τd.c.. How the CR linewidth in a high-mobility 2DEG changes with the magnetic field and temperature is a long-standing question31, and a systematic study of ‘superradiance-free’ CR widths should provide significant new insight.
We have demonstrated collective, coherent, and non-perturbative light–matter coupling between the CR of a 2DEG and terahertz cavity photons in a high-Q 1D photonic-crystal cavity, with a cooperativity up to 360. A high-Q cavity is particularly important for cavity-2DEG systems, since it increases not only the cavity photon lifetime but also the matter lifetime via the suppression of superradiant decay. Furthermore, unlike the near-field 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 many-body excitations in the terahertz range; for example, magnetically ordered systems, high-temperature superconductors, and heavy-fermion systems. Hence, our high-Q terahertz cavity-based techniques open a door to a plethora of new possibilities to combine the traditional disciplines of many-body condensed-matter physics and quantum optics of cavity QED.
We used polarization-resolved time-domain terahertz magnetospectroscopy to study the 2DEG samples in terahertz cavities placed inside a split-coil superconducting magnet. Our laser source was a Ti:sapphire regenerative amplifier (1 kHz, 775 nm, 200 fs, Clark-MXR, Inc., CPA-2001). The nonlinear crystals used for both terahertz generation and detection were 1-mm-thick 110-oriented zinc telluride (ZnTe). We studied two samples of modulation-doped GaAs 2DEG with a single quantum well. Sample 1 had density and mobility values of 1.9 × 1011 cm−2 and 2.2 × 106 cm2 V−1 s−1, respectively, in the dark, while after illumination at 4 K they changed to 3.1 × 1011 cm−2 and 3.9 × 106 cm2 V−1 s−1; Sample 2 had 5 × 1010 cm−2 and 4.4 × 106 cm2 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.
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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We thank A. Chabanov, H. Pu and A. Belyanin for useful discussions. J.K. acknowledges support from the National Science Foundation (Grant No. DMR-1310138). 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. DE-AC04-94AL85000. 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. DE-SC0006671. M.J.M. acknowledges additional support from the W. M. Keck Foundation and Microsoft Research.
- Supplementary information (3,947 KB)