Real-space collapse of a polariton condensate

Microcavity polaritons are two-dimensional bosonic fluids with strong nonlinearities, composed of coupled photonic and electronic excitations. In their condensed form, they display quantum hydrodynamic features similar to atomic Bose–Einstein condensates, such as long-range coherence, superfluidity and quantized vorticity. Here we report the unique phenomenology that is observed when a pulse of light impacts the polariton vacuum: the fluid which is suddenly created does not splash but instead coheres into a very bright spot. The real-space collapse into a sharp peak is at odd with the repulsive interactions of polaritons and their positive mass, suggesting that an unconventional mechanism is at play. Our modelling devises a possible explanation in the self-trapping due to a local heating of the crystal lattice, that can be described as a collective polaron formed by a polariton condensate. These observations hint at the polariton fluid dynamics in conditions of extreme intensities and ultrafast times.


Supplementary
. Femtosecond experiment with a 16 µm FWHM gaussian spot and linear polarization in a position with a microcavity-exciton negative detuning (-0.8 meV). Each row is relative to a dierent initial density. The rst column represents the time-integrated images of the bare emission in space directly aquired on the camera. The second and third columns represent the density and phase proles, respectively, along a central diameter and versus time, retrieved by means of the ultrafast imaging. The powers are increasing from top to bottom rows and correspond to initial total populations of P1P6 = 20 × 10 3 , 40 × 10 3 , 70 × 10 3 , 110 × 10 3 , 200 × 10 3 and 420 × 10 3 polaritons. The associated initial top densities are: 70 µm −2 , 140 µm −2 , 240 µm −2 , 380 µm −2 , 690 µm −2 and 1440 µm −2 , respectively. The maxima on the colour bars of the density charts are expressed in terms of the initial top densities and represent the achieved enhancement factors. We report in the y-axis the blueshifted energy of the brightest point in the far-eld measurements and in the x-axis the mean density values of the real-space 2D gaussian. The solid line is a linear t to the data which gives a nonlinearity of g ∼ 1.5 µeVµm 2 . We assume that the total population is equally distributed between the three QWs and that each results in the same blueshift. Hence the overall nonlinearity should be renormalized to g ∼ 4.5 µeVµm 2 when referring to the single QW.

Supplementary
Supplementary Figure 6. Propagating bright peak activated under resonant fs pulse injection with (a) 0.7 µm −1 and (b) 1.6 µm −1 in-plane wavevectors ky. The cuts of the spatial polariton density as a function of time are taken along the propagation direction. The colour bars represent the density relative to the top density at the initial time (in the present case t0 = 0). Figure 7. Picosecond experiment. The excitation pulse is a 3.5 ps width laser pulse quasi-resonant (0.5 nm blueshifted) on the LPB. The three panels show the time-space graphs of the amplitude (a), phase (b) and k-space (c) cross cuts with a time step of 0.5 ps. The Look-Up Table (LUT) colour scale in the case of the amplitude chart in (a) is relative to the initial top amplitude, and represents an enhancement factor of 4 (in amplitude) at around 10 ps after the pulse arrival, which corresponds to a factor of 16 in density. The time of the pulse arrival in these time-space charts is t0 = 6.5 ps.

Supplementary
Supplementary Figure 8. Interferometric scenario from a ring source. The ring corresponds to a double gaussian along a diameter cross-cut. The ring radius and width are let to expand in time, which correspond to the two gaussians moving and expanding, respectively. The phase is modulated by a radial in-plane wavevector which slightly increases in time. The part of the ring which reaches the centre emerges on the other side and interferes with the opposing wavevector uid, causing a structure similar to that observed in the experiment.
Supplementary Figure 10. Evolution of the photon eld intensity in the model with k-dependent nonlocal reduction of the Rabi coupling. Parameters are the same as in Supplementary Figure 9 (for ps pulse excitation at the LP branch) with a = 4 and kc = 0.2 µm −1 .
Supplementary Figure 11. Evolution of the photon eld intensity and of the reservoir density in the reservoir-ring model. Parameters are A = 0, δ = 0, mC = 5 × 10 −5 me, ΩR = 6 meV, τC = 2.5 ps, τX = 300 ps, g = 12 × 10 −3 meV µm 2 ,g = 2g, R = 2.6 × 10 −3 meVµm 2 , W = 12.5 µm, Tp = 50 fs, ∆ω = 0. The model commonly used to describe the dynamics of polaritons is based on the mean-eld approach. The time evolution of the photonic (excitonic) wavefunctions Ψ C (r) (Ψ X (r)) is given by coupled Gross-Pitaevskii equations [1,2]: where m C (m X ) is the cavity photon (exciton) eective mass, g is the exciton-exciton interaction constant, Ω R is the Rabi frequency determining the exciton-photon coupling, describing the pulsed excitation of the cavity with a Gaussian spot of diameter 2W and pulse-duration of 2T p and F 0 is a parameter determining the density of photo-created carriers. While this model has been successful to describe much of the uid dynamics of polaritons, including ballistic motion [3,4], superuidity [5], solitons [6], etc. [7], in presence of repulsive interactions, g > 0, it cannot account for real-space localization. Many additional ingredients to these equations can produce qualitatively the most important feature of our experiment, the formation of a high density peak in the center. However, these extended models, based on dierent physical assumptions, have some implications, present other qualitative features and/or require certain ranges or parameters that rule them out as an explanation of the phenomenon. Some hypotheses that work reasonably well demand assumptions that are dicult to justify, such as explicit attractive interactions, g < 0. We found that one model only is suciently consistent with all the observations and provides a reasonably close agreement with the data to be retained as a possible mechanism for our experiment. It is presented last in a series of alternative descriptions, below: the loss of strong coupling in Supplementary Note 2, a localisation due to the exciton reservoir or dark excitons in Supplementary Note 3, polaritons with attractive interactions in Supplementary Note 4all of these models failing in some fatal way to account for the experimentand, ultimately, the collective polaron eect, in Supplementary Note 5, which, on the contrary, reproduces adequately the ndings, based on likely assumptions and parameters corresponding to our sample.
Supplementary Note 2. Loss of strong-coupling.
The measured emission of total emitted photons from a single experiment suggests that the exciton density is close to the saturation value of n sat ≈ 10 3 µm −2 . At such a high density, the reduction of the Rabi splitting should occur due to phase space lling [1,811] and lead to a modication of the polariton dispersion. At the rst order of perturbation in the exciton density, the saturation is momentum-independent, or local in space (precisely, the range of the interaction is of the order of the exciton radius) [911]. The linear regime Rabi coupling Ω R0 is then renormalized to: where a is the coecient dependent on the quantum well geometry, g the polariton-polariton interaction strength and Ψ X the exciton wavefunction (as already dened before). This leads to eective attractive interactions but only for the UP branch, while the interactions remain repulsive for the LP branch [1]. This is because the UP branch is redshifted whilst the Rabi coupling decreases, while the LP branch is blueshifted, just as it would be from repulsive interactions. A numerical simulation for this mechanism is shown in Supplementary Figure 9. This model correctly reproduces the experimental Rabi bending and allows to describe a central density peak with both a femtosecond pulse and a picosecond pulse tuned to the UP branch. Critically, however, the ps pulse excitation tuned to the LP branch leads to strong defocusing in this model, in contradiction with the experimental data.
To overcome the defocusing in the case of LP branch excitation, we can consider a momentum-dependent loss of the Rabi coupling. Assuming a stronger reduction of Ω R in the vicinity of k = 0, where lies the condensate whose density is responsible for the phase-space lling, allows to create a negative eective mass for the lower polariton.
Indeed, the LP polariton branch blueshifts more at k = 0, creating a negative curvature of the dispersion around the ground state, allowing for self-focusing even with repulsive interactions. Technically, this can be achieved by ltering Eq. (2) in k space. In the simplest formulation, this is done by transforming the nonlinear elds Ψ X = |Ψ X | 2 Ψ X and Ψ C = |Ψ X | 2 Ψ C to k-space, applying a Gaussian lter Ψ X,C (k) = Ψ X,C (k) exp(−k 2 /2k 2 c ), transforming them back, and subtracting from the ( Ω R /2)Ψ X,C terms in Eqs. (1) after multiplication by ag/2. This nonlocal and nonlinear reduction of the Rabi coupling is equivalent to introducing interactions that are nonlocal in real space. The range of the nonlocal eective interaction is given by k −1 c . Supplementary Figure 10 shows an example of self-focusing in this model in the case of picosecond LP branch excitation. In the case of femtosecond excitation, the results are similar to the ones obtained with the previous model.
While this model provides a good agreement with experiments performed at dierent settings, its justication is not straightforward. While the k-dependent reduction of Rabi coupling can be predicted by the simple model of phase space lling or exchange eects [8], it occurs at k vectors comparable to the inverse exciton Bohr radius a −1 B . The dependence of the Ω R on momentum for k comparable to µm −1 requires an existence of correlations over distances much larger than the exciton radius, for which we see no rationale in the mean-eld framework. An explanation along these lines is thus thwarted in a straightforward model and would require a drastic reconsideration of the physics of polaritons based on the mean-eld approximation and perturbation theory in the exciton basis. In the close vicinity of the Mott transition, the description in terms of the dilute exciton gas may become questionable, and collective eects of strongly-correlated quantum gases could be required instead. Indeed, BCS-related physics is expected to occur close to the phase space lling threshold [12].
where A is the energy relaxation constant, F p (r, t) = F 0 e −r 2 /2W 2 −t 2 /2T 2 p −i∆ωt is the pumping eld and with an initial reservoir density n R (r, t = 0) = n 0 e −r 2 /2W 2 . In this simple model, the backscattering from the condensate to the reservoir is not taken into account, but the initial reservoir density is assumed to be excited by the incoming pulse.
A numerical simulation of this model is shown in Supplementary Figure 11. The depletion in the center of the reservoir is created due to the stimulated scattering to the condensate, which is faster in the center, where the condensate density is larger. The bright density peak is then created because the polariton waves emitted form the remaining ring-shaped reservoir focus in the center. While the main feature of the experiment is thus reproduced, there are several problems with quantitative features: (a) the central peak appears much later than is observed in the experiment and (b) the bending of the Rabi oscillations is initially in the incorrect direction (backwards in the center).
There are even more serious conceptual diculties: (c) in such a model, the peak cannot move ballistically, as is observed in the experiment if the excitation pulse is injected at an angle, since the reservoir would stay essentially in the position where it is created. (d) The peak persists when resonantly exciting the lower branch, which is expected to produce little or no reservoir excitons. (e) Finally, the eect does not show any strong polarization dependence.
In the regime of quasiresonant pulsed excitation we expect the reservoir to be polarised as the typical exciton spinrelaxation time in our system is supposed to be long compared to the characteristic time scale of the Rabi-oscillations and collapse dynamics Variations on this theme are possible. For instance, one can assume the intermission of dark excitons. Since the coupling is incoherent and assuming the exciton mass as innite, which allows to treat dark excitons as a density eld, their eect can be modeled by the following equations: A numerical simulation of this model is shown in Supplementary Figure 12. The scattering to polaritons creates a hole in the Gaussian distribution of excitons. The ow towards the hole can create a localization, however, the eect is not as clear as in the experiment: the localization is rather broad and has a low intensity.

Supplementary Note 4. Attractive interactions between excitons.
The interaction constant in the Gross-Pitaevskii equations, and variants, that describe the polariton dynamics is positive, as indeed same-spin excitons are largely believed to be repulsive, but several eects could potentially give rise to attractive interactions, as we discuss shortly. Since we actually observe a real-space collapse, it is instructive to simply assume attractive interactions between polaritons. In this case self-focusing is indeed possible for a positive eective mass. The model is identical to the one above, Eqs. (1), but with g < 0. We do not include the reservoir excitons that we have seen can be ruled out on several grounds. A result of numerical simulation is shown in Supplementary Figure 13. As expected, the high intensity peak appears due to self-focusing.
While some features of the experiment are well reproduced in this model, the existence of attractive polariton interactions is highly questionable. Potentially the strongest eect in this sense is the indirect exchange through the biexciton state, but this occurs only for opposite spins, while we did not observe any strong spin dependence of the focusing eect in the experiment. The other possible sources of attractive interaction, such as the indirect exchange [1416], or Van der Waals interactions appear to be negligible for the low-momentum excitons created by resonant pumping [16]. It was also recently observed that the sign of the interactions may change to negative at large polariton momenta [15], and other eects could similarly give rise to polariton attraction [15,16], but the proposed congurations are dicult to t in with our experiment. The change of sign of the polariton interactions with no further justication is thus not tenable and would be in conict with a large literature that successfully describes various experimental regimes on a physically well motivated assumption of repulsive interactions.

Supplementary Note 5. Collective polaron eect.
We now conclude this series of possible models to reproduce the real-space collapse of the polariton condensate with the one model that we found could stand as an explanation for the observed eect.
When the exciton density is large enough, as in the case of pulsed excitation, the macroscopic occupation of the ground state with excitons may lead to enhancement of the exciton-phonon interaction that produces a cooling of the excitonic reservoir against its blueshift due to their attraction [17,18]. The corresponding released energy is transfered to the crystal lattice via acoustic phonons. Another plausible mechanism of heating is through the polariton-polariton Auger process followed by emission of the cascade of acoustic phonons as discussed by Klembt et al. [19]. In any case, a local heating may occur which leads to a band-gap narrowing, a variation of the exciton-photon detuning and a redshift of the lower polariton branch. The latter eect may lead to self-focusing of the polariton condensate. Such an eect can be referred to as a collective polaron eect. Formally, it can be described by the modulation of the exciton energy through a retarded function: where β characterises the magnitude of band-gap renormalization and τ H is the characteristic heat relaxation time in the crystal lattice. Both β and τ H are tting parameters. The fourth power of the exciton wavefunction Ψ X (r, t ) under the integral accounts for the quadratic dependence of the polariton Auger process on the concentration of polaritons.
The dynamics of the system is governed by the initial pumping strength. The Rabi oscillations generally have an anharmonic behavior due to the changes of the bare exciton energy and the interplay between the blueshift g|Ψ X | 2 and the redshift E X (t). The eective frequency of these oscillations is then determined by Ω eff = Ω 2 R + (∆ − g|Ψ X | 2 ) 2 [20]. In addition to the change in detuning, at larger population densities close to the Mott transition, the coupling Ω R itself would be reduced, resulting from the change of the exciton oscillator strength due to the exciton phase space lling [8].
The result of the numerical simulation of Eqs. (1) with an Eq. (5) dependence is shown in Supplementary Figure   14. At the initial stage, the Rabi oscillations between photon and exciton states are seen. The dependence of the oscillations frequency on the local detuning results in appearance of the radial waves spreading from the center of the spot. The modied density-dependent value of the Rabi frequency allows to reproduce some retardation of their motion in the spot center. When the density decreases due to the photonic decay, the direction of the wave-spreading changes. In addition, the heating reservoir forms a potential well, in which the polaritons are trapped through their excitonic fraction, that results in a stabilization of both excitonic and photonic components of the polariton ow.
Further dynamics exibits the decay of both components of the polariton many-body wavefunction. One can see the typical behavior of the sharply localised peak appearance extracted at dierent pumping power, which is modeled by setting increasing amplitude of F p in Eq. With a threshold of polariton lasing of the order of 10 mW in GaAs based microcavities, this seems to be a correct order of magnitude. The uid redistribution dynamics and the heat relaxation dynamics appear to be of the similar timescales. A detailed information on cooling/heating of the crystal lattice in the presence of exciton-polaritons is given in Klembt et al. [19]. From the comparison with the paper by Luo, it follows that about 5% of the energy optically injected to the microcavity is likely to be thermally dissipated, the rest being radiatively emitted (with radiative lifetime, say, 10 ps). If the thermal dissipation rate is 20 times lower, we end up with the eective heat wave which would propagate over a distance comparable with the spot size (20 µm) in 20·10 ps = 200 ps. This yields the speed of 0.1 µm ps −1 . For comparison, in another work, the actual propagation rate of the heat wave is determined by the acoustic sound velocity, which is of the order of 0.4 µm ps −1 for GaAs in Matsuda et al. [22]. Note that in reality there is no single heat wave but rather the heat propagates diusively.
In the present model as well, we assume that the incoherent reservoir is either completely empty or does not play any role. This assumption is certainly valid in the case of selective picosecond excitation of the lower polariton (LP) branch. We note also that the weak-coupling saturable optical nonlinearity could arise from instantaneous eects or sample heating [23]. The instantaneous part, however, provides only defocusing nonlinearity for excitation below the band edge, as follows from the nonlinear Kramers-Kronig relations [23,24]. Heating generally leads to redshift which corresponds to focusing nonlinearity but has a longer response time.
It is not excluded that more exotic many-body physics is at play, such as a variation of the BCS mechanism, holding the Bosonic condensate together in a way similar as the superconducting phase molds the Fermion liquid, or a dynamical Casimir eect [25,26] pulling non-zero-momentum particles out of the suddenly hit polariton vacuum.
This last interpretation deserves a special mention as it is so closely related to our experiment. If someone would want to check the dynamical Casimir eect as proposed in the works just cited, they would prepare an experiment close to that described in our work. The theory, based on the Truncated Wigner Approximation (TWA), is however yet not ripe to be confronted to our observations. It has indeed so far be implemented only in 1D (though for both homogeneous [25] and inhomogeneous [26] systems). Dimensionality could be crucial in our case. Nevertheless, a preliminary comparison could be that, as stated in these works, the zero pulse width approximation (see Ref. [25] Eq. 2) holds for a pulse duration τ L < /[gn c (0)], with gn c (0) the initial blueshift of the condensate. With blueshifts of 0.5 meV, 1.0 meV or 1.5 meV, this means τ L < 1.3 ps, 0.65 ps and 0.43 ps, respectively. This holds in the case of the fs (130 fs) experiments we show, while the ps (3.5 ps) experiment is at the border of this range (when imparting a blueshift less than 0.2 meV). If the pulse duration is less than any other time scale of the dynamics, it only sets the initial condition of the polariton gas which then can freely evolve. Another quantity to consider is the amount of nite-k particles generated. In the 1D homogeneous case, this reaches a 10% or 20% fraction of the k=0 condensate, starting from an initial blueshift of 0.5 meV and 1.5 meV, respectively (see [25] Fig. 3), supposedly enough to produce a visible structure. For the inhomogeneous case (Gaussian), the nite-k state is ejected initially with a velocity which then decreases, similarly to the experiments, though it reaches lower values (0.25 µm −1 ) than in the experiment (1.25 µm −1 ). The ring in k-space is however ascribed to the repulsive expansion of the condensate and there is no evidence of a ring in real space. In any case, these magnitudes, as well as their manifestation depend strongly on the dimensionality of the system, and in a 2D polar system the same amounts of particles could give rise to very dierent phenomenology (we recall that just ∼6% of the particles contribute to the bright peak of the experiments).
At this stage any further consideration is just speculation and any quantitative comparison could be misleading, without the implementation of the 2D model. While the collective polaron model provides a better agreement with the observations, we should therefore exert caution at this stage not to discard alternative beyond mean-eld eects, that could provide further insights into the dynamics at play.

SUPPLEMENTARY DISCUSSION
In conclusions, while there is no compelling argument that impose the collective polaron eect as the mechanism causing the collapse of the polariton condensate, our simulations show that it is a reasonable candidate, when other more straightforward scenarios fail to provide even a qualitative agreement or demand assumptions too strong to be justied. We feel that it is left to theorists and further investigations to elucidate which exact physics is causing the peculiar phenomenology that we report. In any case, involving self-localization by the phonon eld as suggested by our model, or through the manifestation of a more complex, as yet unidentied, many-body mechanism, our observations establish a rich, unsuspected and potentially useful dynamics of ultra-fast and ultra-dense bosonic gases, as well as the predispositions of polaritons for the study of such regimes.