The study of the interaction of resonant electromagnetic fields with quantum systems of different nature is an important part of contemporary quantum optics and quantum information technologies [1]. One of the fundamental effects of this interaction is Rabi oscillations which are the periodic energy exchange between the field and the quantum medium. In recent years, Rabi oscillations have been extensively investigated in a wide range of quantum systems, including ultracold gases, quantum dots, individual spins, superconducting qubits, exciton-polaritons in microcavities, etc. [2–5]. The emergence of Rabi flopping is a feature of the interaction of the electromagnetic field with both single quantum emitters and ensembles of oscillators, including collective spontaneous decay (superradiance) [6].

It has been generally accepted that Rabi oscillations require strong coupling between the electromagnetic field and the quantum emitter [1, 7, 8]. The figure of merit of this coupling is the parameter g0 = \(\mu \sqrt {\hbar \omega {\text{/}}2\varepsilon V} \) where μ is the transition matrix element, \(\hbar \omega \) is the quantum energy, ε is the dielectric constant of the medium, and V is the volume of the optical mode. It is obvious that microcavities of different types with small values of V should be used for increasing the coupling parameter. It has been shown both experimentally and theoretically that an increase in the number N of e-mitters interacting with the field results in the rise of the coupling parameter according to the formula \(g(N) = {{g}_{0}}\sqrt N \) [6, 9, 10]. A necessary condition of the Rabi oscillations generation has the form

$$2g > (\kappa + \gamma ){\text{/}}2,$$
(1)

where κ is the reciprocal of the photon lifetime in the cavity and γ is the spontaneous recombination rate [1]. This relation demonstrates the advantages of exploiting high-Q resonators with small values of κ [5, 11‒13].

Quantum dots and exciton-polaritons in quantum wells placed in microcavities are the main semiconductor objects for research in cavity quantum electrodynamics [14–16]. They allow for the observation of Bose–Einstein condensation of exciton-polaritons at appropriate conditions at cryogenic temperatures [17]. However, widespread laser heterostructures with typical cavity lengths of 100–500 μm have never been considered as candidates for the investigation of strong coupling and Rabi oscillations. Indeed, the value of the coupling strength g0/2π ~ 108 Hz or 0.4 μeV at characteristic parameters of μ, ε, and V. This value is smaller by a few orders of magnitude than typical values of (κ + γ) [18], which makes impossible the observation of the effects under study in those devices.

However, as was noted above, the effective method of enhancing light-matter coupling is increasing the number of emitters interacting with the same optical mode. It has been experimentally found in our previous research the fact of establishing the superradiant phase transition in laser heterostructures at room temperature [19–22]. It turned out that the electron–hole ensemble exhibited an off-diagonal long-range order [23], the coherence of superradiance exceeded that of lasing observed in the same samples, and the pulse propagation in the medium showed a superluminal nature [23, 24]. Furthermore, superradiance has super-Poisson statistics and its Wigner functions have broad areas of negative values. This implies that superradiant pulses have a quantum nature [25]. In this work, it is experimentally demonstrated that the condensation of electrons and holes towards the bottoms of the bands, which occurs with the mediation of resonant photons during the superradiant phase transition, enables the realization of strong coupling and the observation of Rabi flopping.

Modified GaAs/AlGaAs laser heterostructures were used in the experiment. The active layer was 0.2‑μm-thick intrinsic GaAs. The length of samples varied from 100 to 450 μm. The width of the emitting area was in the range of 6–7 μm. The reflection coefficient of the facets of the samples was 0.32. Three sections were formed along the cavity axis for the realization of the superradiant regime [20, 22]. The photograph of the two samples is presented in Fig. 1. Two areas at the facets of the structure were pumped by current pulses with an amplitude exceeding the lasing threshold by a few times.

Fig. 1.
figure 1

Photographs of two heterostructures (top view) with a length of (top) 350 and (bottom) 100 μm. The horizontal line corresponds to the 5–6-μm-wide active area. The arrows show the directions of the generated radiation.

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A dc reverse bias in the range of –(1–10) V was applied to the central section of the samples for preventing laser generation. This ensured the achievement of the e–h pair concentration as high as 6 × 1018 cm–3, fulfilled the quantum degeneracy criterion [26], and brought about the non-equilibrium condensation in phase space and superradiant phase transition [21]. All samples generated the standard laser radiation at small driving currents and without the reverse bias. The lasing spectrum consisted of a single longitudinal mode or a few modes of the cavity. The samples emitted femtosecond pulses with an energy of 10–30 pJ under superradiance. Here, the optical spectrum contained doublets at certain pumping ranges.

Figure 2 presents typical optical doublets of the superradiance generated from different samples. All spectra are shifted towards longer wavelengths by more than 10 nm as compared to the lasing spectra. The spectral splitting between the components was from 0.49 to 2.61 nm depending on the parameters of the structures and parameters of pumping.

Fig. 2.
figure 2

Optical spectra of six samples under superradiance.

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The intensity autocorrelation functions during second-harmonic generation in a scanning Michelson interferometer were simultaneously recorded with the spectral measurements. This technique allows for the measurement of coherence and parameters of oscillations of the optical field with a femtosecond accuracy [18]. Figure 3 shows typical intensity autocorrelation functions of two 100 μm long samples. The functions were recorded with the resolution of individual fringes in a scanning Michelson interferometer. The round trip time of the cavity was 3.1–3.2 ps. It is clearly seen the oscillations of the field with a frequency of about 1 THz. The splitting between the spectral components of the doublets and the oscillation frequency depended on the pumping parameters. Figure 4 illustrates the measured spectral splitting for one of the samples as a function of the reverse bias.

Fig. 3.
figure 3

Second-order autocorrelation functions of superradiance from (top two panels) the S03 sample at two values of the reverse bias and (bottom panel) the H16 sample.

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Fig. 4.
figure 4

Splitting between the doublet components versus the reverse bias.

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Table 1 presents the parameters of six samples and corresponding experimental data. The experimental data shown in Figs. 2–4 and Table 1 can be explained within the frame of strong coupling and the Rabi oscillations approach. Indeed, the observation of Rabi flopping is impossible during lasing in the heterostructures under test. This is due to weak coupling between the electron–hole system and the electromagnetic field in the cavity. The typical bandwidth of the laser mode is less than 0.2 nm. Due to the Pauli principle, electrons and holes are strongly spread on energy. So, only a small part of the total number of the carriers injected in the active area interacts with the laser mode. This part can be readily estimated by knowing the density of states within the bands, the threshold density of lasing, the radiation wavelength and the laser mode bandwidth. The estimation gives the value of N for lasing in the range of 103–104, which is not large enough for meeting the criterion (1). However, the non-equilibrium condensation of electrons and holes towards the bottoms of the bands takes place during the superradiant phase transition. This process has been previously described in detail [21, 27, 28]. As a result of the condensation, the number of oscillators, which interact with the field, grows rapidly and the coupling parameter \(g(N) = {{g}_{0}}\sqrt N \) increases by many orders of magnitude. When the number N becomes so large that the criterion (1) is met, then the regime of strong coupling sets in and the observation of Rabi oscillations becomes feasible. We have previously found out that the number of e–h pairs, which were condensed at the bottoms of the bands and taking part in the superradiant process, grows with an increase in the reverse bias [29]. This explains the observed enhancement of the spectral splitting as a function of voltage presented in Fig. 4.

Table 1. Experimental data
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The measured Rabi oscillations Ω in Table 1 are well described by [1]

$$\Omega = 2\sqrt {g_{0}^{2}N - \frac{{{{{(\kappa - \gamma )}}^{2}}}}{4}} .$$
(2)

Strictly speaking, this expression has restricted applicability in the case of highly excited semiconductors where many-body effects can play an important role [30]. However, this formula provides a qualitatively correct estimation of Rabi frequency in our case. Indeed, the number of e–h pairs can be estimated by the value N = (0.58–1.2) × 108 based on the measurement of energies of superradiant pulses. The values of the interband dipole matrix element μ in GaAs are in the range of 20–29 D according to the data published elsewhere [31]. Using the values of κ and V from Table 1, Eq. (2) gives Rabi frequencies \(\Omega {\text{/}}2\pi {\kern 1pt} \) = 0.6–1.7 THz. This corresponds well to the experimental data.

It is worth paying attention to an important feature. Two bottom spectra in Fig. 2 exhibit a fine structure of the doublets consisting of two subharmonics. The existence of the fine structure can be explained by the interaction of quantum emitters. Indeed, as demonstrated in [8, 32, 33], the inclusion of this interaction leads to a more complicated spectrum as compared to the doublet spectrum described by Eq. (2). Subharmonics emerge due to the interaction of dipoles with each other and the spectrum becomes asymmetric. The author used Eq. (5.14) from [8] and Eq. (10) from [32] and calculated the spectrum of Rabi oscillations for the parameters of the sample H11 from Table 1. The comparison of the calculated and experimental spectra is presented in Fig. 5.

Fig. 5.
figure 5

(Top panel) Optical spectrum of Rabi oscillations calculated taking into account dipole interactions and (bottom panel) the corresponding experimental spectrum.

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The shape of the spectra is qualitatively similar despite the opposite position of the subharmonics. The other reason for the appearance of the fine structure of the spectrum of Rabi oscillations can be a non-uniform spatial distribution of the coherent e–h state within the cavity and its division into two symmetric parts. This issue requires an additional study.

To summarize, doublet optical spectra and corresponding coherent oscillations of the electromagnetic field generated by bulk GaAs/AlGaAs heterostructures during superradiance have been studied. It has been demonstrated that superradiance exhibits strong coupling of the field and medium in contrast to lasing in the same samples when weak coupling exists. It has been shown that strong coupling occurs under conditions of the present experiment when only a large enough number of e–h pairs condenses in phase space. Rabi oscillations have been observed with a spectral splitting in the range of 1.3–4.4 meV at 860‒890 nm wavelengths. The corresponding coherent oscillations of the electromagnetic field with frequencies of up to 1.1 THz have been detected. The experimental results are yet another convincing evidence of e–h condensation in bulk GaAs at room temperature discussed in our publications previously.