Next, we explain the key experimental features in Figs. 1 and 2: (a) Rabi oscillations, (b) coherent gain, (c) spectral shifts and splitting and (d) bound biexciton formation for different values of ℏΩ/E and different excitation geometries, i.e. for dominating Coulomb many-body effects in co-linearly excited TMDCs as well as for moderate Coulomb interaction and comparably more pronounced light-matter coupling in co-circularly excited (Ga,In)As QWs.
(a) Rabi oscillations. The microscopic model traces the temporal Rabi oscillations back to oscillations of the total coherently excited 1s exciton density ∣P∣ and incoherent 1s exciton density N.
In the (Ga,In)As MQW, we focus the discussion on N, since it provides the strongest contribution. N can be generated by exciton-phonon, exciton-light and exciton-exciton interaction. In the MQW case, incoherent exciton formation via exciton-phonon scattering is negligible, cf. Eq. (18). Rabi oscillations are solely determined by Pauli-blocking effects of incoherent excitonic occupations N in fourth order of the optical field, cf. first and second line in Eq. (14) in the SM, since exciton-exciton interaction, cf. third and fourth line in Eq. (14) in the SM, is also of minor importance.
Compared to the MQW, in the monolayer MoSe, exciton-phonon, Pauli-blocking and exciton-exciton interactions scale differently. Here, in contrast to the (Ga,In)As MQW, optical blocking in the incoherent occupations N is of minor importance, so that this formation mechanism is even outcompeted by exciton-phonon interaction at the applied pump powers: The stronger Coulomb interaction reduces the Pauli-blocking contribution in the incoherent excitonic occupations N, Eq. (14) first line, cf. also Tab. SI and SII in the SM, since the excitonic wave functions are more spread out in q-space, and enhances the exciton-exciton interaction in the excitonic transitions P, cf. third and fourth line in Eq. (6) in the SM. Thus, the incoherent occupations N do not contribute to the Rabi-flopping dynamics. Similarly, the coherently excited exciton density ∣P∣ is especially decreased by the excitation-induced dephasing via the biexciton and exciton-biexciton continuum. The different scaling of these mechanisms result as a direct consequence of the stronger confinement in atomically thin TMDC, which increases exciton-exciton and exciton-phonon interaction compared to exciton-light interaction. All in all, in the Coulomb-dominated monolayer MoSe, no Rabi oscillations are observed.
(b) Coherent gain. In the (Ga,In)As QW, the full spectro-temporal dynamics from our calculations, shown in Fig. 2a on the right, also capture the emergence of gain. Its attribution of a coherent nature originates from the two-pulse superposition in the blocking contribution (third term in the first line of Eq. (6) in the SM), which transfers parts of the pump-induced dynamics in the direction of the probe pulse, i.e. it emerges only during the presence of the pump pulse. This "wave-mixing-like" process is significantly different from the more common, incoherent gain due to inversion of an incoherent charge-carrier population. The low excitation density of 16 μJ/cm and the resulting absence of population inversion conclusively rule out any possibility of incoherent gain. Occupation gratings due to Coulomb interaction (second line in Eq. (6) in the SM) and unbound spin-like exciton-biexciton transitions (second term in the last line in Eq. (6) in the SM) play a minor role, but do contribute to an overall enhancement of the coherent gain. In contrast, in the MoSe monolayer, the strong Coulomb interaction, coupled with the presence of spin-unlike biexcitons and exciton-biexciton transitions due to co-linear excitation, effectively suppresses the coherent gain signatures at the applied pump powers.
(c) Energy shifts and splitting. The narrow linewidth of the MQW sample allows for a detailed quantitative analysis of the light-driven spectral shifts and Rabi splitting using our many-body model. To this end, we examine the individual spectral features induced by the optical excitation. Figure 3a illustrates the measured nonlinear absorption for co-circularly polarized pump and probe pulses at a time delay of 300 fs for increasing photon densities. The exciton resonance begins to split into two distinct absorption peaks at energy densities of 1.4 μJ/cm. In particular, the low-energy branch, positioned at 1.4666 eV, maintains its spectral position as the energy density increases, while the high-energy branch shifts from 1.4677 eV to 1.4727 eV, which it reaches at the highest energy density of 99.0 μJ/cm. Both branches have narrower linewidths than the linear 1s exciton resonance at low driving photon fluences. Moreover, additional absorption features emerge as the excitation strength increases. A weak resonance appears at lower energies for pump energy densities of 2.8 μJ/cm, subsequently shifting from 1.465 eV to 1.460 eV as the excitation is further increased. Another weak absorption peak arises between the two initial branches at 1.468 eV for energy densities of 17.0 μJ/cm, maintaining its spectral position even under stronger driving conditions.
Figure 3b traces the spectral positions of all observed absorption peaks relative to the 1s exciton resonance in the linear absorption against the square root of the excitation power, i.e. the field amplitude. In this regime, an approximate analytical formula of the Rabi splitting for vanishing exciton-exciton interaction V = 0 and vanishing pump saturation is derived (cf. Eq. (64) in the SM):
where the Coulomb-enhanced blocking parameter reads: with the 1s excitonic wave function φ at relative momentum q. Eq. (1) is formally equivalent to the splitting observed in a classical 2-LS and therefore shows a linear splitting in the Rabi energy. This linear splitting behavior is observed in the experiment as well as in the full simulations, cf. Fig. 3b. We conclude that the (Ga,In)As MQW sample displays an effective, modified 2-LS response due to its relatively weak Coulomb interaction and the co-circular polarization conditions (no spin-unlike biexcitons or exciton-biexciton transitions). Note that in the simulations, a weak saturation onset at higher fluences can be observed, which is traced back to the onset of optical blocking of the 1s exciton density.
Finally we consider the TMDC monolayer case where the spectral shifts observed in Fig. 1 appear at first glance to be similar to those observed in the QW sample (cf. Fig. 2). The low-energy resonance remains fixed at ~1.633 eV, while the high-energy peak continuously shifts to higher energies with increasing excitation, as shown in Fig. 3b on the right, displaying a highly asymmetric splitting. Here, the initial symmetric Rabi splitting into a repulsive (upper) and an attractive (lower) branch is significantly altered by the strong Coulomb interaction, which dominates over the optical interation (ℏΩ/E ~10): The upper branch exhibits a strong Coulomb-dominated density-dependent blue shift, which is a common behavior of TMDCs, while the lower branch remains relatively stable as it is more light-dominated (see SM for more details). Notably, a closer analysis reveals a slight sublinear increase of the splitting with the Rabi energy, even though the optically excited exciton density is still far from optical saturation, which deviates from the effective 2-LS behavior, cf. Equation (1). In the MoSe monolayer case, where Coulomb interactions completely overshadow optical interactions, a regime of vanishing Pauli-blocking emerges as long as the exciton density is well below the Mott transition. Under such conditions, an approximate analytical expression for the Rabi splitting E at zero detuning can be derived (cf. Eq. (62) in the SM):
Here, E is the exciton energy, V represents the effective exciton-exciton interaction, and ℏΩ = ∑φℏΩ is the excitonic Rabi energy. This Hartree-Fock description already predicts a weak sublinear splitting with respect to the Rabi energy ℏΩ. The analysis of the microscopic model identifies the origin of this weak sublinear behavior in a regime far from optical saturation: First, we note that the splitting itself originates from an occupation grating rather than a polarization grating, since, at zero pump-probe delay, i.e. maximal splitting, the overall excitonic density is already dominated by incoherent occupations. At that time, four-particle biexcitons (Eq. (15) in the SM) are already decayed and six-particle exciton-biexciton transitions (Eq. (16) and Eq. (17) in the SM) are formed, since they possess a combined coherent (probed transitions) and incoherent (pump-induced excitonic occupation) source. Second, in co-linear excitation, next to spin-like exciton-biexciton transitions, additional spin-unlike exciton-biexciton transitions are induced, which are absent in co-circular excitation. Due to their Coulomb-correlated intervalley nature, they cause a dynamic coupling between both K and K' valleys, which causes an overall attenuation of the Rabi splitting.
The qualitative validity of Eq. (2) results from the appearance of intervalley exciton-biexciton transitions which overcompensate the excitation-induced dephasing by amplifying coherent Coulomb renormalization effects in the probe-induced transition (cf. the second line in Eq. (6) in the SM). The experimentally observed linewidth narrowing of the splitting peaks corroborates this interpretation: Compared to the 6.7 meV FWHM of the linear absorption resonance, the lower-energy peak narrows to between 3.32 and 6.18 meV, while the higher-energy peak exhibits a FWHM in the range of 3.11 to 4.74 meV as the photon density increases. The simulations also display a linewidth narrowing at small to moderate photon densities.
(d) Biexciton formation. An additional resonance on the low-energy side is clearly observed in co-linear measurements in the MoSe monolayer, cf. Figures 1 and 4. Its spectral position is ~30 meV below the excitonic resonance similar to other works, which report biexcitonic binding energies in the range of 20-30 meV. Its spectral position experiences a fluence-dependent red shift, which mirrors a density-dependent repulsive interaction between the biexcitonic and the excitonic resonance, as well as an increase in oscillator strength. In co-circular excitation in a (Ga,In)As MQW, no biexcitonic resonance appears. Here, only biexcitonic continua occur, since only one electron-heavy-hole spin configuration is optically addressed. The observed features in the experiments are well reproduced by the microscopic theory, confirming the contributions of bound biexcitons and the biexcitonic continuum.
In summary, we present experimental evidence and theoretical confirmation for Rabi splitting of the 1s exciton resonances in (Ga,In)As QWs and MoSe monolayers under close-to-resonant excitation conditions. The fundamentally different nature of both 2-LS, particularly regarding the interplay of Coulomb interactions and light-matter coupling, in conjunction with specific excitation conditions, dictates the splitting behavior, occurrence of Rabi oscillations, and coherent gain: In (Ga,In)As MQWs with co-circular excitation and weaker Coulomb interaction, the Rabi splitting scales nearly linearly with the Rabi energy and is accompanied by pronounced Rabi oscillations and coherent optical gain. Conversely, in MoSe monolayers, with co-linear excitation and stronger Coulomb interaction, only sublinear Rabi splitting is observed, with no evidence of Rabi oscillations or gain. Our findings are corroborated by a microscopic theory that elucidates the physical mechanisms underlying these effects. These insights are pivotal for leveraging exciton-based coherent phenomena in next-generation ultrafast optoelectronic and switching devices.