{'fraction_non_alphanumeric': 0.024284023668639052, 'fraction_numerical': 0.007715976331360947, 'mean_word_length': 4.886319308999721, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Polariton condensation in photonic molecules

We report on polariton condensation in photonic molecules formed by two coupled micropillars. We show that the condensation process is strongly affected by the interaction with the cloud of uncondensed excitons. Depending on the spatial position of these excitons within the molecule, condensation can be triggered on both binding and anti-binding polariton states of the molecule, on a metastable state or a total transfer of the condensate into one of the micropillars can be obtained. Our results highlight the crucial role played by relaxation kinetics in the condensation process.

We report on polariton condensation in photonic molecules formed by two coupled micropillars. We show that the condensation process is strongly affected by the interaction with the cloud of uncondensed excitons. Depending on the spatial position of these excitons within the molecule, condensation can be triggered on both binding and anti-binding polariton states of the molecule, on a metastable state or a total transfer of the condensate into one of the micropillars can be obtained. Our results highlight the crucial role played by relaxation kinetics in the condensation process. Most of the experimental studies in atomic Bose condensates have explored conditions of thermodynamic equilibrium since typical condensate lifetimes are much longer than interaction times. Recent theoretical proposals have shown that out of equilibrium bosonic systems present qualitatively new behaviors [1]. One proposed way to reach this regime is the use of photonic systems with effective photon-photon interactions and dissipation provided by inherent optical losses [2]. Localized to delocalized phase transitions [3,4], highly entangled states [5], or fermionisation effects in a ring of coupled sites [6] are predicted in such systems.
Microcavity polaritons are a model system for the investigation of the physics of driven-dissipative boson condensates [7][8][9][10][11][12][13]. They are the quasi-particules arising from the strong coupling between excitons confined in quantum wells and the optical mode of a microcavity. Because of their light-matter nature, polaritons present peculiar properties: they interact efficiently with their environment through their excitonic part [14,15] while their photonic part enables efficient coupling with the free space optical modes. Polariton condensates can be generated in zero dimensional micropillars [11] or in arrays of pillars with fully controlled coupling [16,17]. In this configuration, the non-equilibrium nature of polariton condensates should allow the realization of metastable collective states, such as the self-trapped states in a bosonic Josephson junction [18][19][20].
In the present paper we investigate polariton condensation in photonic molecules obtained by coupling two micropillars. We demonstrate that polariton interactions strongly affect the way condensation occurs in such coupled system, not only modifying the wavefunction of the polariton condensate, but also the relaxation dynamics. This effect, specific to an out-of-equilibrium bosonic system, is illustrated by considering different positions of the non resonant excitation within the molecule. When the excitation spot is placed at the center of the molecule, polariton condensation is observed on both binding and anti-binding states. Interactions induce strong changes in the condensate wavefunction, the most important one being the change in its spatial anisotropy.
When the excitation spot is positioned on one of the two coupled micropillars, condensation occurs in a very different way. As the excitation power is increased, the polariton condensate is first created in a metastable state localized in the excited pillar by an effect similar to self trapping [18]. Further increasing the pumping power, the condensation tends to progressively evolve from a kinetic regime to a regime closer to thermodynamical equilibrium: massive occupation is observed on the lowest energy state of the system, mainly localized in the non-excited pillar. These results are simulated using a relaxation model including semi-classical Boltzmann equations and nonlinear Schrödinger equation in a selfconsistent way. The key features of our experiments can only be reproduced if one properly includes the changes of the relaxation rates due to interaction-induced modifications of the overlaps between the different polariton states.
The microcavity sample, described in Ref. [11], consists in a λ/2 cavity with a quality factor exceeding 16 000 and containing 12 GaAs quantum wells. Coupled micropillars were fabricated using electron beam lithography and Inductively Coupled Plasma dry etching. The diameter d of the micropillars ranges from 3 to 4 µm and their center to center distance d CC is varied from 2.3 to 3.7 µm, corresponding to an expected coupling constant between 0.1 and 1 meV . Note that d CC is always kept smaller than twice the radius of the pillars, ensuring the direct coupling of the polariton modes of the two micropillars. A scanning electron micrography of an array of such photonic molecules is shown in Fig.1.a. Microphotoluminescence experiments are performed on single molecules using a single mode cw Ti:Saph laser focused onto a 2 µm diameter spot with a microscope objective. The sample is maintained at 10 K and the excitation laser energy is tuned typically 100 meV above the lower polariton reso- nance, thus providing non-resonant optical excitation of the system. The emission is collected through the same objective and imaged on the entrance slit of a monochromator. The spectrally dispersed emission is detected with a nitrogen-cooled CCD camera. We define the detuning δ = E c − E x as the energy difference between the lowest energy photonic mode and the exciton resonance.
The polariton modes in these photonic molecules are investigated by photoluminescence measurements at low excitation power. An example of an emission spectrum measured on a single molecule is presented in Fig. 1.a. Discrete emission peaks are observed corresponding to polariton quantum states fully confined in the microstructure. The broader line at higher energy is due to emission of the excitonic reservoir. The two lowest energy modes (labeled B and A) are attributed to the binding and anti-binding states arising from the hybridization of the lowest energy mode of each micropillar. The splitting between these states is proportional to the coupling between the two micropillars. It can be contin- uously tuned by changing the center to center distance d CC , as illustrated in Fig. 1.b. The left column of Fig. 1.c shows the spatial distribution (near field) of the four lowest energy modes in a molecule made of two 4 µm pillars with d CC = 3.73 µm (the first two panels correspond to the B and A modes). The right column shows the calculated polariton wavefunctions considering a confinement potential taken as infinite outside the photonic molecule and equal to zero inside. As schematically illustrated in Fig. 1.d, all these states result from the hybridization of the optical modes of each individual micropillar, showing the strong analogy between our system and the orbitals of a diatomic molecule.
We now discuss polariton condensation in these molecules under non resonant optical excitation. As we have shown in our recent work on single micropillars [21], such an excitation scheme not only populates the confined polariton states, but also creates a population of uncondensed excitons called the excitonic reservoir. Because of the limited diffusion length of excitons, this reservoir remains localized in the excitation area. Repulsive interactions between polaritons and the excitonic reservoir strongly influence the precise quantum state in which polaritons accumulate. In the following we will consider two different locations of the excitonic reservoir within the molecule, which can be selected via the position of the excitation laser spot. First we will consider excitation conditions where the reservoir of uncondensed excitons is at the center of the molecule. Then we will address the case of asymmetric excitation, in which the excitonic reservoir is injected only in one of the two micropillars. Specific spatial behavior of the polariton condensates is observed in each case, driven by the polariton interaction with the excitonic reservoir. Figure 2.a-c present the measured emission distribution along the axis of a photonic molecule excited at its The inset shows the calculated spatial shape of the binding state center. As summarized in Fig. 3.a, above a well defined excitation threshold, a strong non-linear increase of the emission intensity is observed, one of the signatures of polariton condensation [11]. Under this excitation condition, a massive accumulation of polaritons occurs both in the binding and anti-binding lowest energy polariton states. Interestingly, as the excitation power is increased, a progressive change of the spatial shape of the polariton wavefunction is observed, with vanishing probability density at the center of the molecule. This behavior is due to repulsive interactions with the excitonic reservoir injected in that region.
To describe our experiments, we simulate polariton relaxation using coupled semi-classical Boltzman equations and nonlinear Schrödinger equations in a self-consistent way [22]. As described in the supplementary information, on one hand, a nonlinear Schrödinger equation is solved at each time iteration, in the presence of the potential induced by the excitonic reservoir and by the occupation of the polariton states, to find the eigenstates of the system. On the other hand, the time evolution of these occupation numbers are calculated using the semiclassical Boltzmann equations in which the scattering rates are obtained using the exact shape of the polariton states obtained from the non-linear Schrödinger equation. Both exciton-phonon and exciton-exciton scattering mechanisms are considered. The reservoir is assumed to be thermalized at the lattice temperature. The key ingredient to describe our experiments is to consider scat-tering rates proportional to the spatial overlap between initial and final states. For instance, the scattering rate of two excitons from the reservoir resulting in one polariton in state i and one exciton in the reservoir is proportional to |ψ i (x, y)| 2 (ρ R (x, y)) 3 dxdy, where ψ i (x, y) is the wavefunction of the polariton state i and ρ R (x, y) is the spatial distribution of the excitonic reservoir, described by a gaussian of width w R . To fit the data we use an exciton-exciton scattering rate W XX = 2 10 3 s −1 and an exciton-phonon scattering rate W XP = 10 9 s −1 . The lifetime of the reservoir is equal to 400 ps, and that of the polariton states to 30 ps, a value extracted from coherence measurements in ref. [12].
The results of the model for central pumping are shown in Fig. 2d-f. The potential of the reservoir increases with the optical pumping and has a maximum at the center of the molecule, inducing the spatial separation of the states of each micropillar. As a result, both the coupling and the associated splitting between the two lowest energy states decrease. Indeed, this feature is shown in Fig. 3.b: simultaneous to the spatial separation of the polariton distributions, the measured splitting goes down to a minimum at about P = 4 P th . At higher powers, the measured splitting tends to increase again. In the following, we will show that at such high density the observed splitting is no longer related to the binding anti-binding splitting but is an anisotropic splitting induced by the lateral spatial shrinking of the condensate wavefunction in each micropillar. This feature is not reproduced by our model which does not include the polarization degree of freedom.
Polarization resolved measurements were performed for different excitation densities. Figure 3.c and 3.d display emission spectra linearly polarized parallel and perpendicular to the molecule axis, below and above threshold. In the low density regime (Fig. 3.c), the presence of uncondensed excitons has a negligible effect, and we observe a polarization splitting in both the binding and anti-binding states. As previously reported theoretically and experimentally [23], this splitting is larger for the binding state and amounts to 70 µeV . The inset of Fig.3.c shows that the overall shape of the polariton binding state is elongated along the molecule. This is why the lowest energy line of the binding state doublet is polarized parallel to the molecule axis. The situation is different at high density (P > 4 P th ), where a stronger potential barrier is induced by the excitonic reservoir at the center of the molecule. In this case, the coupling between the states of each micropillar becomes too small to be resolved in our experiment. The remaining splitting is then purely a polarization splitting. In each micropillar, the polariton wavefunction is shrunk by the excitonic repulsive potential and thus it is more elongated perpendicularly to the molecule axis. This change in anisotropy is evidenced in the polarization of the emission.The lowest energy emission line is now strongly polarized perpendic- Different dynamics is observed when the laser spot is positioned on top of one of the micropillars forming the molecule. Emission distributions measured under these conditions are reported in Fig. 4.a-c. Close to threshold, condensation occurs in the pillar that is optically pumped. Interparticle interactions blue shift the energy of the ground state in the pumped pillar, decoupling it from the states of the other pillar , and severely limiting the Josephson transfer of particles from one pillar to the other. This localisation is very analogous to the original self-trapping effect, except that here the blue shift is mainly induced by the interaction between the exciton reservoir and the condensate and not by interactions within the condensate itself. As the excitation power is increased, the better relaxation kinetic and the presence of excited states in the un-pumped pillar destroy the metastable state [24] and allow the system reaching its ground state. Condensation then occurs in the nonpumped micropillar (Fig. 4.c).
We can reproduce this overall behavior using the selfconsistent polariton relaxation model presented above. As shown in Fig. 4.d, at low power the reservoir potential creates an asymmetry in the two bottom states of the molecule, whereas the upper states are not affected. The populations of the excitonic states in the reservoir are small and their relative distribution is given by the lattice temperature. When pumping increases (Fig. 4.e), condensation starts at the eigenstate possessing the largest overlap integral with the reservoir. This corresponds to the third polariton state (population n 3 ). The reservoir potential also significantly perturbs the two bottom states, which become both confined in the right pillar. This blocks the direct relaxation from the reservoir to-wards these states and further enforce condensation in the third metastable state localized in the pumped pillar. At high pumping (Fig. 4.f) a new effect is observed: condensation is triggered onto the ground state (of population n 1 ). This does not arise from direct scattering from the reservoir, since the overlap between the ground state and the reservoir remains very small. This condensation is the result of efficient scattering from the intermediate third state toward the ground state via acoustic phonons. This regime occurs when the phonon assisted scattering rate W XP n 3 (1 + n 1 ) from the intermediate state into the ground state overcomes the ground state radiative losses n 1 τ 1 . This happens when n 3 ≈ 100 and further increase of the pumping does not change n 3 , but only increases n 1 . The second quantized state remains weakly populated: it has a smaller overlap with the reservoir than the third state and a higher energy than the ground state. Overall the complete condensation dynamics can only be understood considering the modifications of the relaxation rates induced by interactions (as shown in the supplementary [22] when constant scattering rates are considered, condensation in the metastable states can not be reproduced).
To conclude, we have demonstrated polariton condensation in photonic molecules, a fully controlled system in which the coupling constant between two condensates can be adjusted. Depending on the precise location of the excitonic reservoir, strong renormalization of the polariton states is induced, resulting in strong modification of the condensation dynamics. In particular condensation can occur in a metastable localized state , or in the ground state of the system. These results open the way towards a detailed investigation of Josephson oscillations [25] or, more generally, the pumped-dissipative physics of arrays of coupled condensates in an engineered environment.
We simulate polariton relaxation in a self-consistent way, using coupled semi-classical Boltzmann equations and nonlinear Schrödinger equations. At t = 0 the system is empty and the pumping is switched on. We calculate the time evolution of the system until the establishment of a steady state. The nonlinear Schrödinger equation written below is used to find the spatial shapes and energies E i,t j of the polariton wave functions ψ i,t j (x, y). For the quantized polariton states i the equation reads: 2π 2 e −E iR /k B T and for the reservoir state R the equation is: P is the non-resonant pumping populating the reservoir, τ R = 400 ps (resp. τ i = 30 ps ) is the excitonic reservoir (resp. polariton) lifetime. W XX describes the exciton-exciton interaction rates. The exciton-phonon scattering rate reads: The wave vector conservation rule is replaced by an overlap integral between the spatial extensions of the initial and final states. The overlap integrals change with time, together with the shapes and the energies of the states according to equation (1). The overlap integrals involved are defined as follows: The overlap integrals obtained by summation of scattering rates with different phonon wavevectors read: These overlap integrals are dimensionless.
Finally, the scattering rates W XX and W XP describing exciton-exciton and excitonphonon interaction were used as the fitting parameters of the model, together with the width of the gaussian spatial distribution of the reservoir which we authorize to slightly vary above the width given by the pumping laser spot size. We used W XX = 2 10 3 s −1 , W XP = 10 9 s −1 , w r = 3µm .
In order to understand the terms in the Boltzmann equation, let us consider for example the case of the scattering from the reservoir towards a quantized state i and backward assisted by exciton-phonon and exciton-exciton interaction. We neglect the final state stimulation within the reservoir which means that we assume a weak occupation (n R (E) ≪ 1) of these states. In all cases, the energy difference between the state i and the bottom of the exciton reservoir is E 0 .
Let us first consider the most complicated case of the forward scattering assisted by exciton-exciton interaction (sketched in figure S-1). In order to calculate the corresponding scattering rate, we have to sum over all possible scattering events. The corresponding scattering rates reads: The rate is proportional to N 2 R and is similar to a scattering rate written between individual states and based on particle-particle interaction. The backward process of one particle going from the quantized states to the reservoir reads: The rate is similar to the one of the forward process except by the term : e − E iR k B T /n R (0). The relaxation of particles toward a low energy state is therefore favored only for a minimum energy difference E 0 > k B T ln(n R (0)).
For the phonon-assisted scattering between the reservoir R and a quantized state i the expressions are quite similar. The scattering rates from the reservoir towards the state i reads: W XP I iR (n i + 1) E n R (E) (1 + n ph (E + E iR )) ≈ W XP I iR (n i + 1) N R The scattering rates from the state i toward the reservoir reads: n ph (E) (1 + n R (E)) ≈ W XP I iR n i e −E iR /k B T Sk B T 2π 2 One can note that the expressions for the scattering processes involving exciton reservoir and phonons are quite similar. This is explained by the fact that excitonic reservoir is assumed to be thermalized at the lattice temperature.
Despite the complexity of the equations used in our simulation, we would like to underline that the number of fitting parameter is really limited.  The crucial ingredient which allows catching the basic physics in our experiments is to take into account the overlap integrals between initial and final states in the scattering rates.
As shown in the paper, these wave functions are strongly distorted by interactions, and this dramatically modifies the way polariton relaxation occurs in the system.