Observation of bosonic condensation in a hybrid monolayer MoSe2-GaAs microcavity

Bosonic condensation belongs to the most intriguing phenomena in physics, and was mostly reserved for experiments with ultra-cold quantum gases. More recently, it became accessible in exciton-based solid-state systems at elevated temperatures. Here, we demonstrate bosonic condensation driven by excitons hosted in an atomically thin layer of MoSe2, strongly coupled to light in a solid-state resonator. The structure is operated in the regime of collective strong coupling between a Tamm-plasmon resonance, GaAs quantum well excitons, and two-dimensional excitons confined in the monolayer crystal. Polariton condensation in a monolayer crystal manifests by a superlinear increase of emission intensity from the hybrid polariton mode, its density-dependent blueshift, and a dramatic collapse of the emission linewidth, a hallmark of temporal coherence. Importantly, we observe a significant spin-polarization in the injected polariton condensate, a fingerprint for spin-valley locking in monolayer excitons. Our results pave the way towards highly nonlinear, coherent valleytronic devices and light sources.


Supplementary Note 1
In order to describe the eigenenergies of the hybrid polariton resonances, we apply a coupled oscillator model, which reads in the case of three oscillators: the three Hopfield coefficients quantify the admixture of QW-and monolayer-exciton (| 1 | 2 ; | 2 | 2 ) and cavity photon | | 2 . Solving the eigenvalue problem yields the characteristic dispersion relation of hybrid polaritons, featuring three polariton branches. and are photon and exciton energies, respectively, and the exciton-photon coupling strength for the respective oscillator.

Supplementary Note 2
Here, we provide details on a reference study of the power dependent behavior of the photoluminescence from pure GaAs-exciton polaritons emerging in the periphery of the monolayer device with a detuning of -5.9 meV. Sample conditions are comparable to the main text with a temperature of 5 K and optical excitation with an 82 MHz repetition rate, 2 ps pulsed Ti:Sa laser, tuned to an energy of 1.6732 eV. In Supplementary Fig. 1a This feature is typically attributed to a transition from the strong-to the weak coupling regime, which typically occurs at or slightly above the Mott density in high-quality QW-microcavities.
The input power approximately corresponds to the second threshold observed in the main text, attributed to the Mott-transition of the bare GaAs QW directly in the vicinity to the monolayer.

Supplementary Note 3
This section compares the polarization of the emission from our hybrid monolayer-GaAs device with the emission from pure GaAs polaritons, recorded at a comparable laser-lower polariton detuning in relation to the hybrid polaritons of Fig. 4 (main text). The experiment was carried out under the same experimental conditions as described in the main text for Fig 4. In Supplementary Fig. 2 we plot the resulting degree of circular polarization (DOCP) from the emission of the cavity, both subject to + and − pumping. We note, that the DOCP does not show any distinct power dependency within the error margin, as expected in the linear regime by our Boltzmann model, and is significantly lower than the DOCP which we have record from our hybrid polariton condensate under significantly lower pump powers.

Supplementary Note 4
In this supplementary section, we provide additional details on the blueshift fit in Fig. 3f of the main text. The emission from the polariton condensate is subject to an energy shift, depending on the pumping power, both below and above the condensation threshold. At low pumping powers, the occupation of the ground state is negligible and the energy shift arises from the polariton interaction with the excitonic reservoir which builds up with increased excitation power. In the conventional single reservoir model 1 the homogeneous reservoir density is fixed above the threshold due to the stimulated scattering into the ground state. The repulsive polariton-polariton interactions thus govern the further increase of the emission energy above the threshold, which occurs at a reduced rate compared to the large below-threshold blueshift due to polariton-exciton interactions. This model, despite having the advantage of simplicity, is not sufficient for the description of the hybrid polariton devices, where the emission blueshift depends nonlinearly on the pumping power above the lasing threshold.
To account for the hybrid nature of the microcavity we solve the coupled rate equations for the occupations of the polariton condensate mode and the two exciton reservoirs, neglecting the spin polarization and spontaneous scattering probability: Here, is the condensate occupation, 1 and 2 are the densities of exciton reservoirs in the QW and the TMDC monolayer, respectively, 1 and 2 are the corresponding scattering rates, and are the condensate and reservoir lifetimes, and = are the reservoir pumping rates.
Equating the time derivatives to zero we find the stationary values of and .
Below the condensation threshold threshold ( = 0) we express the reservoir densities = from Supplementary Equation (2). Above the threshold ( > 0) both reservoir densities = /( + −1 ) are depleted by stimulated scattering into the condensate. The occupation of the latter reads: where = − −1 −1 has the physical meaning of the occupation of a condensate, which is only coupled to the -th reservoir. The total blueshift is governed by the polariton condensate interaction with itself and with the two reservoirs as follows: where 1 and 2 are the exciton Hopfield coefficients of the polariton mode, corresponding to the QW and TMDC exciton, respectively.
The fits of the experimental dependence of blueshift on the pumping power, obtained with Supplementary Equations (1, 2) and (4) are shown in Fig. 3f. The initial nonlinear increase in the blueshift above the threshold corresponds to a redistribution of the excitonic densities between the two reservoirs, so that the total condensate gain provided by stimulated scattering remains equal to the radiative losses. Note that the two reservoirs model captures the nonlinear behavior of blueshift more accurately than the conventional single reservoir model 1 , demonstrating the important role of the interplay of the two reservoirs in the behavior of the hybrid polariton laser.
The following realistic parameters were used in the fit: = 0.5 ps, 1 = 500 ps, 2 = 10 ps. We also assume equal excitonic interaction constants 2 1 = 2 and take equal Hopfield coefficients 1 = 2 from the polariton branch fit in Fig. 2 of the main text. Note that, under this assumptions, the reservoir contribution to the condensate blueshift is proportional to the total reservoir density = 1 + 2 . The free parameters used for the fit are the two relations between the pumping efficiencies 1 / 2 = 1 and the scattering rates 1 / 2 = 0.02.
The details of the fit presented in Fig. 3f are shown in Supplementary Fig. 3. Below the threshold, the blueshift is linear in the pumping power. The population of QW excitons builds faster than in the TMDC due to longer exciton lifetime in the QW. Above the threshold, both reservoirs are depleted by stimulated scattering into the macroscopically populated polariton mode. Depletion efficiency is governed by the spontaneous scattering rate. Stronger coupling of the TMDC excitonic mode to the polaritonic mode favors stronger TMDC reservoir depletion and results in a slight decrease of its occupation above the threshold. In contrast, the QW reservoir grows so that the total gain provided by the two reservoirs compensates the radiative losses of the condensate.
This results in a nonlinear dependence of the blueshift on the pumping power above the threshold. At higher powers, however, the combined reservoir density reaches a steady state, and the blueshift becomes linear because it is governed by polariton-polariton interactions as in the single-reservoir case 1 .
It is straight forward to quantify the threshold condition of the hybrid polariton laser, applying Supplementary equations 1 and 2. Assuming, as bove, that both reservoirs are pumped with the same efficiency 1 2 = 1 , the threshold reservoir injection rate for the hybrid condensation obeys the condition: Where and are condensation thresholds for the QW and TMDC polaritons, respectively. This means that the pump threshold in the hybrid structure is always lower than that in a QW or TMDC layer alone. Physically this is due to the fact that both reservoirs scatter into the same polariton state, so the reservoir densities required to overcome polariton decay are much lower.