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Plasmon-exciton polariton condensation evidenced through spatial coherence and interactions

Polaritons are quasiparticles arising from the strong coupling of electromagnetic waves and elementary excitations in semiconductors. In this frame, localized surface plasmons in metallic nanoparticles have attracted increasing interests in the last decade, since their sub-diffraction-limited mode volume offers the access to extremely high photonic densities and corresponding huge nonlinearities. However, high absorption losses in metals have always hindered the observation of collective coherent phenomena, such as condensation. In this work we demonstrate the formation of a non-equilibrium room temperature plasmon-exciton-polariton condensate with a long range spatial coherence extending well over the excitation area, by coupling Frenkel excitons in organic molecules to a multipolar mode in a lattice of plasmonic nanoparticles. Time-resolved experiments evidence the sub-ps formation dynamics of the condensate and a sizeable blueshift, thus measuring, for the first time, the effect of polariton interactions in plasmonic cavities. Our results pave the way to the observation of room temperature condensation and novel nonlinear phenomena in plasmonic systems, challenging the common belief that absorption losses in metals prevent the realization of macroscopic quantum states.

breakthrough within this emerging field.
A promising route for the exploration of exciton-polariton physics at the nanoscale is offered by localized surface plasmons (LSPs), arising from the coherent oscillation of electrons at the boundary between a metal nanoparticle and a dielectric material. In fact, compared to photonic modes in microcavities, LSPs have unique properties, including the confinement of light in mode volumes far below the diffraction limit (V λ 3 ) and strong electromagnetic (EM) nearfield enhancement. Recently, plasmon-exciton-polariton (PEP) quasiparticles have been reported in several hybrid metal-organic systems [12][13][14][15][16] . Among these, particular attention is devoted to the study of two-dimensional (2D) lattices of plasmonic nanoparticles, where peculiar PEP features can arise, including the ability to propagate in the plane of the array 17 that leads to spatially extended polaritons 18 . Interestingly, lasing action has been demonstrated in these systems interacting with organic media 19,20 . Unlike surface plasmon polaritons and LSPs, plasmonic lattices can support low mass polaritons with relatively long lifetimes, and have thus been predicted to offer the possibility to observe condensation 21 . Very recently a preprint claimed the observation of a Bose-Einstein condensate (BEC) in similar plasmonic structures 22 . While it is well known that in the context of 2D geometries a Berezinsky-Kosterliz-Touless (BKT) transition rather than a BEC should be expected, it is more interesting to note here that for strongly out-of-equilibrium systems, such as PEPs, the phase transition cannot be described in the framework of equilibrium thermodynamics. In this sense, the attempt to fit a Maxwell-Boltzmann or a Bose-Einstein distribution in an open-dissipative system is hardly significant, since it does not imply thermalisation 23 , as also demonstrated in a microcavity under weak coupling regime 24 .
On the other hand, for driven-dissipative systems, it is essential to establish the importance of particle-particle interactions with respect to fluctuations induced by pumping and dissipation.
The spontaneous formation of spatial and temporal correlations in these many body systems can be described as photon lasing when the interactions are absent 22

Non-equilibrium condensation
The optical mode in our system is provided by a 2D array of silver NRs. In periodic geometries, LSPs from individual nanoparticles can coherently couple with each other by means of diffraction orders propagating in the plane of the array, known as Rayleigh anomalies. As a result, hybrid photon-plasmon lattice modes arise, known as Surface Lattice Resonances (SLRs) 26 . These modes are characterized by a strong suppression of losses (higher quality factor) with respect to individual LSPs, at the price of a more spatially delocalized EM mode (higher mode volume) [27][28][29] , while maintaining the strong EM field enhancement typical of plasmonic nanoparticles 30,31 . Moreover, unlike confined optical modes in microcavities, SLRs are able to propagate in the plane of the 4 array.
Our NRs array, whose scheme and scanning electron micrograph are shown in Figure 1a    i.e. k || = (ω/c) sin(θ) where ω is the PEP frequency, c the speed of light and θ the angle with respect to the sample normal when the sample is rotated about the y-axis. At pump fluences below a certain threshold (P < P th , Fig. 1c), the PL is mainly dominated by a flat emission band from the LSP around 2.12 eV, while two dispersive SLR branches appear at high k || . The SLR emission vanishes at k || = 0 due to its antisymmetric EM field distribution 20 , resulting in the reduction of the net dipole moment by the array 32 . As the pump power is raised above P th , stimulated scattering leads to a large carrier accumulation in the long living polariton state associated to the multipolar SLR, resulting, as we will show, to PEP condensation. The emission dispersion collapses into an individual sharp peak centered at E=2.04 eV and k || = 0, as shown in Fig. 1d (P = 1.1P th ).
The real space emission pattern, measured by imaging the emission of the sample as described in the Methods section, also changes radically below and above threshold, as shown in Figs. 1e-f. Indeed, while below threshold (Fig. 1e) the PL is homogeneously distributed in the ex-6 cited area, at P > P th (Fig. 1f) a structured stripe-like pattern arise, extended along the y-direction.
This peculiar spatial distribution can be attributed to the anisotropy in the nanoparticle dimensions and the array, which leads to condensation on the SLR associated to the (0,±1) diffraction order, predominantly oriented along the y-direction 20 . The random allocation of the stripes across the emission pattern can be explained by sample imperfections and inhomogeneities as can be inferred by the different emission patterns (not shown) obtained on different regions of the sample.

Spatial and temporal coherence
Spatial coherence from PEP systems has already been reported in the linear regime 33,34 , and has been addressed to the plasmon-exciton hybridization induced by strong coupling. In our nonlinear system, the 2D emission image of the sample above threshold carries all the information about spatio-temporal correlations of the PEP condensate, that can be extracted by using interferometric techniques. In particular, we have employed a Michelson configuration, schematically shown in Since the sample is excited non-resonantly, the carriers relax incoherently into the PEP band, with a phase that is not imposed by the pump. At sufficiently high PEP densities, resonant phononic relaxation from the exciton reservoir rapidly populate the long-living polariton state at k || = 0 20 , leading to bosonic stimulation and, finally, to condensation 35,36 . Measuring the visibility of interference fringes, we can thus obtain a complete spatial reconstruction of the first-order correlation function g (1) (r 1 , r 2 , ∆t) of the condensate (see supporting informations), at each temporal delay ∆t between the pulses in the two interferometer arms: where Ψ * and Ψ are the creation and annihilation operators for the space-time point (r, t).
A typical interference pattern, measured at a pumping power P = 1.2P th and ∆t = 0, is shown in Figure 2b, with the maximum fringes visibility in the center of the image (i.e., the autocorrelation point, r = r 0 ). The spatial map of the g (1) (−r, r, 0) is displayed in Fig. 2c, as calculated from Fig. 2b, while the profile along the black dashed-line starting from the autocorrelation point is reported in Figure 2d. By fitting the extracted decay with an exponential function (see supporting informations), as shown in Figure 2d, a coherence length of L x c 100µm is obtained.
In addition to the presence of the long-range correlations along the stripes, one could wonder if, despite the disorder, the PEP condensate can still manifest the 2D nature of SLRs. In order to verify this property, the g (1) along the direction perpendicular to the emitting stripes is measured by rotating the sample, as reported in Figs. 2e and 2f. Here, regardless of the spatial fragmentation of the condensate, a high degree of coherence is found to be maintained from stripe to stripe. In particular, by fitting all the g (1) maxima positions along the stripes, an exponential decay is obtained, shown as a red line in Figure 2f, with a coherence length of L y c 120µm. This value, very similar to what obtained for the x-direction, demonstrates the 2D nature of the PEP condensate.

Time-resolved photoluminescence
To gather fundamental insights into the PEP condensate dynamics we performed time-and energyresolved PL measurements, by using an imaging spectrometer coupled with a streak camera (≈1.  Fig. 4a) and on the bare polymer layer doped with dye molecules (green dots in Fig. 4a), showing that the polariton decay is given by the reservoir lifetime. However, as the excitation power is raised above the condensation threshold, the decay time at the energy of the PEP condensate reduces by one order of magnitude, to about t A = 3 ps (black dots, Figure 4a).
The PEP condensate time trace in Fig. 4b can be fitted, at each time, with a Gaussian peak profile, whose spectral position and linewidth are shown in Figs. 4c and 4d, respectively. An instantaneous (resolution limited) blueshift as large as 2.5 meV appears when the system is excited.
This blueshift is mainly due to the interactions of the exciton reservoir. With time, the exciton reservoir depletes while the condensate is forming, resulting in a continuous redshift towards the vacuum state (Fig. 4c). As can be seen from the narrowing of the linewidth (Fig. 4d), the conden-sate is already formed a few ps after the excitation pulse, however the system continuos to redshift due to the further reduction of the total population. This behavior is fast enough to exclude any possible heating related effects and similar to what already observed in exciton-polariton condensates with inorganic semiconductor microcavities 39 .
This temporal dynamics not only demonstrates the presence of interactions in PEP system but also shows that plasmon-exciton coupling is still holding while condensation occurs assuring that photonic lasing processes are not taking place.  Time-and energy-resolved PL above PEP condensate threshold, as measured on the streak camera coupled to an imaging spectrometer, and corresponding emission peak energy (c) and linewidth (d) as a function of time.
In conclusion we have reported, for the first time, the evidences of the formation of a plasmon-exciton-polariton condensate in a lattice of metal nanoparticles supporting low losses multipolar plasmonic modes coupled to Frenkel excitons in an organic dye. A widely extended two-dimensional spatial coherence is observed, showing an high degree of robustness against disorder and inhomogeneities. As a result, our system can be described as a single macroscopic state, whose coherence length is longer than ≈100 µm. Time resolved measurements evidence an ultrafast, sub-ps condensate formation, with a 2.5 meV interactions-induced spectral blueshift. Our findings are very promising for studying properties of quantum fluids at room temperature with ultrafast dynamics, thus opening the way towards future plasmon-exciton-polariton based devices. Optical measurements. To measure the angle-resolved PL, we have used a rotational stage that collects PL at different emission angles through an optical fiber and sent to an Ocean Optics spectrometer (USB2000). The sample is excited non-resonantly at normal incidence with 100 fs amplified pulses at E exc = 2.48 eV excitation energy with 1 KHz repetition rate.

Methods
To study the long-range correlations on the PEP condensate, we use a Michelson interferometer. The schematic representation of the setup is shown in Fig. 2(a). The sample is non-resonantly Supplementary Informations for Plasmon-exciton polariton condensation evidenced through spatial coherence and interactions

Molecular dye
As an organic molecule, we used a derivative of rylene dye with the formula [N,N'-Bis(2,6-diisopropylphenyl)-1,7and -1,6-bis (2,6-diisopropylphenoxy)-perylene-3,4:9,10-tetracarboximide] due to its photostability and possibility of reaching high molecular concentrations within the PMMA matrix without aggregation. The absorption and emission spectra of the dye are shown in Fig. S1. The absorption spectrum, in particular, is characterized by one main peak at E = 2.24 eV and a vibronic replica at E = 2.41 eV, as introduced in the main text. Figure S1: Normalized absorption and emission spectra of the bare rylene dye.

Spatial coherence analysis
The two-dimensional spatial map of the first order correlation function g (1) (r, −r) was evaluated using the fast Fourier transform (FFT) to extract the interference pattern modulation from the experimental interferogram, selecting only the frequencies relative to the fringes modulation. This allows the reconstruction of the fringes visibility. In order to normalize the measured visibility, the continuous background signal is used. The normalized visibility (V ) and the first order correlation function have this relation: where I ideal = (I 1 + I 2 )(2 I 1 I 2 ) −1 , with I 1 and I 2 the light intensities measured on the two separated channels of the interferometer, takes into account possible small asymmetries between the two arms.
The function used for the fitting in Fig. 2d and 2f of the manuscript, is an exponential profile with the form: where A is a renormalization factor taking into account the experimental reduced visibility, e.g. for mechanical vibrations of the setup, and b is the parameter containing the coherence length.

Temporal coherence analysis
The temporal decay of coherence was evaluated by looking at the coherence in the autocorrelation point (r 0 ), when lengthening the optical path in one arm of the interferometer by means of a single axis translation micrometer stage. This, in turn, results in a relative temporal delay between the two arms, ∆t, ranging from 0.07ps to ∼ 3ps. Each measured frame is analysed by calculating the first order correlation function in the autocorrelation point, as discussed in the previous sections.
The extracted temporal decay is fitted by using a stretched exponential function as in: where A is a renormalization factor, while l e and β are the parameters containing the temporal coherence decay length, according to the: where l t is the temporal decay length, l e is the renormalization factor scale of the temporal axis, β the exponent of the stretched exponential and Γ the gamma function. From this fitting the extracted β was ≈ 1.5.

Time resolved measurements
The time resolved experiments have been performed by non-resonantly exciting the sample with a pulsed laser of 35 fs pulse width, and collecting the emission on a streak camera set up. Fig. S2 shows the temporal profile of the laser pulse (in log scale) measured on the streak camera, which defines the time resolution of our set up. By fitting the laser profile with a gaussian function (red line), a FWHM of 1.8 ps is obtained.