Unstable and stable regimes of polariton condensation

Unstable and stable regimes of polariton condensation

F. Baboux Centre de Nanosciences et de Nanotechnologies, CNRS, Univ. Paris-Sud, Université Paris-Saclay, C2N–Marcoussis, F-91460 Marcoussis, France Laboratoire Matériaux et Phéenomènes Quantiques, Université Paris Diderot, CNRS-UMR 7162, Paris 75013, France    D. De Bernardis TQC, Universiteit Antwerpen, Universiteitsplein 1, B-2610 Antwerpen, Belgium INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, I-38123 Povo, Italy Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1040 Vienna, Austria    V. Goblot Centre de Nanosciences et de Nanotechnologies, CNRS, Univ. Paris-Sud, Université Paris-Saclay, C2N–Marcoussis, F-91460 Marcoussis, France    V.N. Gladilin TQC, Universiteit Antwerpen, Universiteitsplein 1, B-2610 Antwerpen, Belgium    C. Gomez Centre de Nanosciences et de Nanotechnologies, CNRS, Univ. Paris-Sud, Université Paris-Saclay, C2N–Marcoussis, F-91460 Marcoussis, France    E. Galopin Centre de Nanosciences et de Nanotechnologies, CNRS, Univ. Paris-Sud, Université Paris-Saclay, C2N–Marcoussis, F-91460 Marcoussis, France    L. Le Gratiet Centre de Nanosciences et de Nanotechnologies, CNRS, Univ. Paris-Sud, Université Paris-Saclay, C2N–Marcoussis, F-91460 Marcoussis, France    A. Lemaître Centre de Nanosciences et de Nanotechnologies, CNRS, Univ. Paris-Sud, Université Paris-Saclay, C2N–Marcoussis, F-91460 Marcoussis, France    I. Sagnes Centre de Nanosciences et de Nanotechnologies, CNRS, Univ. Paris-Sud, Université Paris-Saclay, C2N–Marcoussis, F-91460 Marcoussis, France    I. Carusotto INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, I-38123 Povo, Italy    M. Wouters TQC, Universiteit Antwerpen, Universiteitsplein 1, B-2610 Antwerpen, Belgium    A. Amo Centre de Nanosciences et de Nanotechnologies, CNRS, Univ. Paris-Sud, Université Paris-Saclay, C2N–Marcoussis, F-91460 Marcoussis, France    J. Bloch Centre de Nanosciences et de Nanotechnologies, CNRS, Univ. Paris-Sud, Université Paris-Saclay, C2N–Marcoussis, F-91460 Marcoussis, France
Abstract

We experimentally investigate the spatiotemporal coherence properties of polariton condensates in GaAs-based microcavities under continuous-wave non-resonant pumping. We provide evidence for a modulational dynamical instability of polariton condensates due to their coupling with the reservoir excitons. This instability results in a strongly reduced spatial and temporal coherence and a significantly inhomogeneous density. We demonstrate how the instability can be tamed by using polariton lattices where condensation occurs into negative effective mass states, leading to largely improved coherence and homogeneity. All results are theoretically reproduced within a generalized Gross-Pitaevskii approach.

thanks: florent.baboux@c2n.upsaclay.fr

Many-body physics predicts that an homogeneous Bose-Einstein condensate at equilibrium cannot form in the presence of attractive interactions: they make the condensate to collapse so as to minimize its (negative) interaction energy Pitaevskii and Stringari (2016); Stoof (1994). In presence of a confining potential however, the zero-point kinetic energy competes with interactions. In 3D, a confined condensate is thus stabilized up to a critical particle number Pitaevskii and Stringari (2016); Bradley et al. (1995); Gerton et al. (2000), and the collapse can be triggered by tuning the interaction strength using Feshbach resonances Roberts et al. (2001); Donley et al. (2001); Chin et al. (2003); Lahaye et al. (2008). In 1D, a condensate with attractive interactions acquires a stable soliton-like shape for any particle number Strecker et al. (2002); Khaykovich et al. (2002). Inspired by well-known dispersion management techniques used in fiber optics to generate solitons and gap solitons Boyd (1992); Butcher and Cotter (1993); de Sterke and Sipe (1994), self-bound condensate droplets in 1D have also been observed for repulsively interacting atoms when the atomic effective mass is turned to negative by means of an optical lattice Anker et al. (2005) or of a spin-orbit coupling Khamehchi et al. (2017). Related instability and pattern formation phenomena have also been studied in optical systems Tai et al. (1986); Hegarty et al. (1999); Kip et al. (2000). However, for driven-dissipative non-equilibrium condensates all this phenomenology has been little explored experimentally so far.

Polaritons, arising from the strong coupling of quantum well excitons and cavity photons Carusotto and Ciuti (2013), are an appealing candidate to address this physics. These quasi-particles indeed combine non-equilibrium properties with substantial interactions. Direct interactions between polaritons are of repulsive nature Ciuti et al. (1998), which should preclude the aforementioned instability scenario. However, under the widely used non-resonant pumping scheme, interactions between polaritons and the reservoir cloud of uncondensed excitons need also to be considered. This polariton-reservoir coupling can be recast as effective attractive interactions between condensed polaritons Liew et al. (2015). When these effective interactions overcome the direct repulsive interactions between polaritons, the condensate is expected to enter a modulationally unstable regime Wouters and Carusotto (2007) characterized by a turbulent steady state Bobrovska et al. (2014); Bobrovska and Matuszewski (2015); Liew et al. (2015). This behavior strongly contrasts the usual collapse scenario of attractively interacting cold atom gases and is another intriguing example of the rich non-equilibrium physics of driven-dissipative polariton condensates.

While predicted a decade ago Wouters and Carusotto (2007) this modulational instability of polariton condensates has not been addressed in previous experimental reports, except for a recent study using organic cavities under pulsed excitation Daskalakis et al. (2015); Bobrovska et al. (2016). In the present letter, we experimentally observe the modulational instability of polariton condensates in inorganic cavities and under continuous-wave (CW) pumping. For this purpose, we investigate the first-order spatiotemporal coherence properties of polariton condensates in GaAs-based 1D microcavities in which the sign of the polariton effective mass can be changed. When condensation occurs in a positive mass state, we highlight an unstable steady-state regime: we observe strongly reduced spatial and temporal coherence and a sizable density inhomogeneity. When the cavities are spatially patterned into lattices so that condensation occurs in negative mass states, the modulational instability is suppressed and all previous signatures of instability disappear: the condensates reach a stable steady-state with high homogeneity and coherence. Similar observations are also obtained in 2D geometries 111See Supplemental Material at http://… for details on the modeling and additional theoretical and experimental data., highlighting the generality of the shown phenomena. Suppressing instabilities is of high interest for investigating the peculiar coherence properties recently predicted for driven-dissipative bosonic condensates Sieberer et al. (2013); Chiocchetta and Carusotto (2013); He et al. (2015); Altman et al. (2015); Dagvadorj et al. (2015); Gladilin et al. (2014); Ji et al. (2015), such as KPZ universal scalings.

We start with a theoretical description of polariton condensation showing the physical origin of the reservoir-induced modulational instability. Ignoring the spin degree of freedom, the condensate wavefunction can be described by a generalized Gross-Pitaevskii equation coupled to a rate equation for the exciton reservoir density Wouters and Carusotto (2007):

(1)
(2)

where is the pumping rate, is the effective polariton mass at the condensate energy, and are the polariton and exciton loss rates, is the relaxation rate of the reservoir into the condensate, and and are positive and describe the repulsive polariton-polariton and polariton-reservoir interaction constants.

The widely used adiabatic approximation assumes that the reservoir follows instantaneously the condensate dynamics and is expected to be accurate if the reservoir decay time is the fastest time scale 222More formally, three independent analytical conditions need to be fulfilled so as to apply the adiabatic approximation, as shown in Ref. Bobrovska and Matuszewski (2015). Under this condition, Eq. (2) reduces to . Reinjecting into Eq. (1) yields a modified Ginzburg-Landau equation Carusotto and Ciuti (2013) in which is replaced by an effective interaction constant Liew et al. (2015):

(3)

This means that, as far as linear stability is concerned, the coupling to the reservoir can be recast as an effective interaction between polaritons. The condensate is then stable for repulsive interactions (, defocusing effective nonlinearity), and unstable for attractive interactions (, self-focusing effective nonlinearity).

This polariton instability can be interpreted as a reservoir-induced modulational instability as follows. A local increase of polariton density (due to a quantum or thermal fluctuation of the condensate, or to pump noise) induces a local depletion of the reservoir density via a spatial hole burning Wouters and Carusotto (2007); Estrecho et al. (2017). Such a depletion creates a potential well which further attracts the condensate polaritons, making the initial fluctuation to exponentially grow in time. This positive feedback loop is eventually broken by gain saturation and by polariton propagation, so that the density fluctuation is ejected from its initial position and starts moving through the condensate. The turbulent behavior results from the chaotic evolution of several of such density fluctuations Bobrovska et al. (2014); Bobrovska and Matuszewski (2015); Liew et al. (2015).

In typical polariton experiments Kasprzak et al. (2006); Deng et al. (2007); Manni et al. (2012); Roumpos et al. (2012); Nitsche et al. (2014); Fischer et al. (2014), and , so that the stability condition requires high pump powers difficult to achieve in practice, in particular for large pump spots in a CW regime. This stability condition can be relaxed by using a relatively small pumping spot (m) Bobrovska et al. (2014); Daskalakis et al. (2015); Bobrovska et al. (2016), as was done in most previous experimental studies Kasprzak et al. (2006); Deng et al. (2007); Manni et al. (2012); Roumpos et al. (2012); Nitsche et al. (2014); Fischer et al. (2014). In that case, the confinement-induced kinetic energy competes with attractive interactions so that stable condensates can be achieved even for negative .

The fact that dynamical stability has so far restricted experiments to small condensate sizes may appear as a serious limitation of polaritons as a platform for simulating novel driven-dissipative phenomena Sieberer et al. (2013); Chiocchetta and Carusotto (2013); He et al. (2015); Altman et al. (2015); Dagvadorj et al. (2015); Gladilin et al. (2014); Ji et al. (2015). In contrast to previous works, we experimentally evidence the turbulent behavior of condensates under CW pumping and large pumping spots (m). Building on this understanding, we then demonstrate a method to tame the instability using polariton lattices and obtain stable condensates of large size.

Figure 1: (a) Scanning electron micrograph of 1D wire cavities. (b)-(c) Far field emission (TM polarization) of a moderate wire cavity at pump power (b) below and (c) above the condensation threshold (). (d) Real space image of the condensate. (e) Interferogram obtained by superposing two mirror-symmetric images of the condensate in a Michelson interferometer. (f) Measured and (g) calculated spatial coherence at (red), compared to the spatial profile of the condensate (black) and of the pump spot (blue). (h)-(k) Same quantities as (d)-(g) obtained in a high wire cavity. (l) Measured and (m) calculated evolution of the temporal coherence, for the moderate (green line) and high wire cavity (blue line).
Figure 2: (a) Scanning electron micrograph of a 1D Lieb lattice of micropillars. (b)-(c) Far field emission (TM polarization) of a moderate Lieb lattice at pump power (b) below and (c) above the condensation threshold (). (d) Real space image and (e) interferogram of the condensate. (f) Measured and (g) calculated spatial coherence at (red), compared to the spatial profile of the condensate (black) and of the pump spot (blue). (h) Measured and (j) calculated evolution of the temporal coherence.

Our microcavities, grown by molecular beam epitaxy, consist of a GaAlAs layer surrounded by two GaAlAs/GaAlAs Bragg mirrors. To vary the quality factor we fabricate two sets of cavities: (1) Moderate factor with 26 and 30 pairs in the top/bottom mirrors, respectively, yielding a nominal quality factor ; (2) High factor with 28 and 40 pairs in the top/bottom mirrors, yielding a nominal . For both types of cavities twelve GaAs quantum wells of width 7 nm are inserted in the structure, resulting in a 15 meV Rabi splitting. The planar cavities are patterned into 1D wire cavities or 1D lattices of coupled micropillars, by e-beam lithography followed with dry etching (down to the GaAs substrate).

In the experiments described below, polaritons are excited non-resonantly with a CW monomode laser tuned to 740 nm. The sample temperature is K and the cavity-exciton detuning is meV (defined with respect to the lowest energy cavity mode). The polariton emission is collected with a numerical aperture objective and focused on the entrance slit of a spectrometer coupled to a CCD camera. Imaging of the sample surface (resp. the Fourier plane) allows for studying polariton properties in real (resp. reciprocal) space.

We first consider polariton condensation in positive mass states. A wire cavity (width m and length m, see Fig. 1a) of moderate factor, is excited with an elliptical spot of length m (intensity FWHM). Fig. 1b shows the far field emission at very low pump power (for the TM polarization, i.e. parallel to the wire axis), evidencing a parabolic-like dispersion (near ) with an effective mass ( is the free electron mass). When increasing the pump power, stimulated scattering causes the emission to collapse into a narrow spectral line Kasprzak et al. (2006) centered at , as seen in the spectrum of Fig. 1c obtained at . The real space image of the resulting polariton condensate (Fig. 1d) and the corresponding spatial profile (Fig. 1f, black line) reveal inhomogeneities in the condensate density, with a typical contrast (ratio between maximum and minimum density) at the center of the pump spot.

To investigate the coherence properties of the condensate we employ Michelson interferometry Kasprzak et al. (2006). We superpose the condensate image with its mirror symmetric, obtained by reflection on a retroreflector, so that each point of the condensate interferes with the point located at . The corresponding interferogram is shown in Fig. 1e, for zero temporal delay () between the two arms of the interferometer. By extracting the fringe visibility through Fourier analysis, we obtain the first order spatial coherence plotted in Fig. 1f (red line). The measured coherence extends over a much shorter length scale than the condensate density, with a coherence length (at ) of m. This short spatial coherence is a first hint of the presence of an instability and of the consequent turbulent behavior. To gain further insight into this phenomenon, we investigate the temporal coherence. Figure 1l (green line) shows the evolution of when scanning the temporal delay of the interferometer. The decay is non-monotonic (a revival is seen near ps) Note1 () and the envelope decays within a coherence time ps.

The density inhomogeneity (Fig. 1f) suggests that disorder is playing a role in the experiment, leading to a slight localization of the condensate. If we use a wire cavity with a high factor and similar disorder strength (Fig. 1h-j) we observe an increased density inhomogeneity of the condensate, with a typical contrast at (Fig. 1h and j, black line). The higher ratio between the disorder amplitude and the polariton linewidth results in a condensate fragmented into distinct lobes with no mutual coherence (see interferogram in Fig. 2i and extracted spatial in Fig. 2j, red). Hence, the coherence length is limited to the lobe size and is of the order of m, comparable to the one of the moderate condensate of Fig. 1f. The coherence time, on the other hand, is here twice longer with ps for each lobe (Fig. 1l, blue line).

Let us now investigate condensation in negative mass states, as it can be achieved by patterning the cavity into a lattice Jacqmin et al. (2014); Baboux et al. (2016). Our intuition is here guided by the adiabatic approximation, that led to the derivation of Eq. (3) for the effective interaction between polaritons. In this approximation one can easily show that, all other parameters kept the same, inverting the sign of the mass reverses the effect of interactions Boyd (1992); Butcher and Cotter (1993); de Sterke and Sipe (1994), suppressing the feedback loop at the heart of the modulational instability.

To test this prediction, we fabricate a 1D Lieb lattice of coupled micropillars Baboux et al. (2016), as shown in Fig. 2a. We here present data for a moderate cavity but we obtained similar results at high . Each pillar has a diameter of m and the lattice period is m. The fundamental (s-symmetry) states of the pillars hybridize to form three bands. For our lattice parameters the two lowest are superimposed within the linewidth, but the upper band is well separated, as seen in the far field emission of Fig. 2b (TM polarization). This upper band shows a negative curvature at the center of the second Brillouin zones (, see vertical red arrows), yielding a negative effective mass nearly equal to the opposite of the mass of the wire cavities studied above. Fig. 2c shows the condensate far field emission at , which is concentrated at the top of the upper band. The real space image (Fig. 2d) and corresponding spatial profile (Fig. 2f, black) show that the condensate possesses a regular Gaussian-like shape, despite the presence of disorder.

The spatial coherence at , extracted from the interferogram of Fig. 2e is shown in Fig. 2f (red circles). As the antisymmetric character of the upper band makes the condensate density to vanish in between neighboring pillars, we restricted our spatial sampling of the coherence function to the center of the pillars. We observe that the spatial coherence extends over the whole condensate, yielding a coherence length m about times higher than for positive mass condensates (Figs. 1f and 1j). The temporal coherence , shown in Fig. 2h, reveals a slow and monotonic decay with a long coherence time  ps, four times higher than the one of the wire cavity of same factor (Fig. 1l, green). This strongly suggests that the condensate is here dynamically stable.

We obtained similar results in 2D cavities in bands with positive and negative mass Note1 (), highlighting that the physical mechanism of the polariton instability (spatial hole-burning effect) is independent of the dimensionality Wouters and Carusotto (2007).

To get further physical insight on these behaviors, let us now compare these experimental results to the predictions of the theory. We start from linear stability analysis of the time-independent steady-state, assuming a spatially homogeneous system. Using parameters taken from the experiment, we calculate the spectrum of the elementary (Bogoliubov) excitations Note1 (). We first consider the adiabatic approximation. Figure 3a shows the imaginary part of the spectrum for the positive mass condensate in the moderate wire cavity. In the low wavevector region, the upper Goldstone branch takes positive imaginary values: perturbations at these wavevectors are exponentially amplified by the system, corresponding to a modulationally unstable regime of condensation. Figure 3b shows the spectrum calculated for the negative mass condensate in the moderate lattice: here, all excitation modes have negative imaginary part and are thus exponentially damped, corresponding to a stable regime of condensation.

Figure 3: (a)-(b) Imaginary part of the Bogoliubov excitation spectrum calculated under the adiabatic approximation, for a positive (a) and negative (b) mass condensate of moderate factor, assuming a constant polariton linewidth . (c)-(f) Same quantities calculated with the full model of coupled Eqs. (1)-(2), with a constant linewidth (c,d) and with a momentum-dependent linewidth (e,f). Formulas and parameters are given in the Supplemental Material.

However, the adiabatic approximation does not hold in the majority of GaAs- of CdTe-based microcavities in which condensation has been reported. In these materials and, in particular, in the present experiments, the reservoir lifetime is significantly longer than the polariton lifetime. Therefore, we need to consider the full model formed by the coupled Eqs. (1)-(2) Bobrovska et al. (2014). As seen in Fig. 3c, the qualitative shape of the spectrum for the positive mass condensate remains essentially unchanged, although precise values of the strength and the position of the modulational instability are modified. For the negative mass condensate however, see Fig. 3d, the spectrum is qualitatively altered compared to the adiabatic approximation, and a novel instability appears. Such specifically non-adiabatic instability, not discussed previously in the literature, is a priori relevant given our experimental parameters, but is not observed in our measurements.

Several mechanisms can be invoked to explain its suppression. In particular, in the antisymmetric band of a lattice, the radiative linewidth is expected to increase monotonically away from the maximum of the band Aleiner et al. (2012); Stepnicki and Matuszewski (2013). In our experiments we can quantify this phenomenon by extracting the linewidth of the low power photoluminescence as a function of the wavevector Note1 (). We find an approximate linear increase of the linewidth, , with eV for high (moderate) lattices, and m. When introducing such momentum-dependent broadening in the simulations, the instability is indeed suppressed, as shown in the spectrum of Fig. 3f. For the 1D wire cavities, photoluminescence measurements also reveal a slight -dependency of the linewidth, m, which could be linked to energy relaxation effects or position-dependent losses. For completeness, we include this effect in the simulations of the positive mass condensate as well: as shown in Fig. 3e, the condensate remains in the unstable regime.

To go beyond the stability analysis and simulate the spatial and temporal coherence of the condensate, we consider the full nonlinear model of Eqs. (1)-(2), including the -dependent linewidth and the pump profile. We add a gaussian noise term to Eq. (1) so as to effectively account for all quantum, thermal or pump laser fluctuations, as well as a disorder potential with standard deviation of eV, corresponding to the typical disorder strength of our cavities Baboux et al. (2016). The shape of the disorder is adjusted to fit the experimentally observed condensate density profiles (e.g. Fig. 1f and j). Figures 1g and 1k show the simulated time-averaged condensate density (black) and spatial coherence at (red) for the positive mass condensate, for moderate and high wire cavities. Figure 1m shows the corresponding temporal coherence, all in good agreement with the experiment.

This difference between moderate and high cavities can be intuitively understood from the interplay between instability and disorder: At moderate , density fluctuations chaotically propagate along the condensate, resulting in a strongly reduced spatial and temporal coherence; the non-monotonicity in the temporal coherence (Fig. 1l and m, green line) arises from the scattering of density fluctuations on the disorder. At high on the contrary, due to the higher disorder/linewidth ratio, the condensate is pinned into localized high density areas with higher temporal coherence (Fig. 1l and m, blue line).

To calculate the spatio-temporal coherence of negative mass condensates, we simulate the negative mass band of the Lieb lattice by a linear chain of pillars with negative tunnel coupling between them, yielding a single band of negative curvature, and we introduce disorder with the same amplitude than for positive mass. Figures 2g and 2i show the spatial coherence at zero delay () and the time decay of the coherence (). The simulations, in good agreement with the experimental results (Fig. 2f,h), show a smooth density profile and a high spatial and temporal coherence.

In summary, we compared the spatio-temporal coherence of polariton condensates of positive and negative effective mass, in GaAs cavities under continuous-wave pumping. Due to effective attractive interactions mediated by the exciton reservoir, positive mass condensates are dynamically unstable as evidenced by a strongly reduced spatial and temporal coherence and a spatially inhomogeneous density. Using a lattice to invert the sign of the polariton mass allows to suppress this instability and prepare extended condensates with high spatial and temporal coherence. This method, along with the use of cavities with very long lifetimes and reduced wedge Ballarini et al. (2017), opens exciting possibilities in view of investigating novel driven-dissipative phenomena Sieberer et al. (2013); Chiocchetta and Carusotto (2013); He et al. (2015); Altman et al. (2015); Dagvadorj et al. (2015); Gladilin et al. (2014); Ji et al. (2015) with bosonic condensates.

This work was supported by the French National Research Agency (ANR) project ”Quantum Fluids of Light” (ANR-16-CE30-0021) and program Labex NanoSaclay via the project ICQOQS (Grant No. ANR-10-LABX-0035), the French RENATECH network, the ERC grant Honeypol and the EU-FET Proactive grant AQUS (Project No. 640800).

References

  • Pitaevskii and Stringari (2016) L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation and Superfluidity (Oxford University Press, Oxford, 2016).
  • Stoof (1994) H. T. C. Stoof, Phys. Rev. A 49, 3824 (1994).
  • Bradley et al. (1995) C. C. Bradley, C. A. Sackett, J. J. Tollett,  and R. G. Hulet, Phys. Rev. Lett. 75, 1687 (1995).
  • Gerton et al. (2000) J. M. Gerton, D. Strekalov, I. Prodan,  and R. G. Hulet, Nature 408, 692 (2000).
  • Roberts et al. (2001) J. L. Roberts, N. R. Claussen, S. L. Cornish, E. A. Donley, E. A. Cornell,  and C. E. Wieman, Phys. Rev. Lett. 86, 4211 (2001).
  • Donley et al. (2001) E. A. Donley, N. R. Claussen, S. L. Cornish, J. L. Roberts, E. A. Cornell,  and C. E. Wieman, Nature 412, 295 (2001).
  • Chin et al. (2003) J. K. Chin, J. M. Vogels,  and W. Ketterle, Phys. Rev. Lett. 90, 160405 (2003).
  • Lahaye et al. (2008) T. Lahaye, J. Metz, B. Fröhlich, T. Koch, M. Meister, A. Griesmaier, T. Pfau, H. Saito, Y. Kawaguchi,  and M. Ueda, Phys. Rev. Lett. 101, 080401 (2008).
  • Strecker et al. (2002) K. E. Strecker, G. B. Partridge, A. G. Truscott,  and R. G. Hulet, Nature 417, 150 (2002).
  • Khaykovich et al. (2002) L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin,  and C. Salomon, Science 296, 1290 (2002).
  • Boyd (1992) R. Boyd, Nonlinear Optics (Academic Press, London, 1992).
  • Butcher and Cotter (1993) P. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University Press, Cambridge, 1993).
  • de Sterke and Sipe (1994) C. M. de Sterke and J. Sipe, ed. Elsevier, Amsterdam , 203 (1994).
  • Anker et al. (2005) T. Anker, M. Albiez, R. Gati, S. Hunsmann, B. Eiermann, A. Trombettoni,  and M. Oberthaler, Physical Review Letters 94, 020403 (2005).
  • Khamehchi et al. (2017) M. Khamehchi, K. Hossain, M. Mossman, Y. Zhang, T. Busch, M. M. Forbes,  and P. Engels, Physical Review Letters 118, 155301 (2017).
  • Tai et al. (1986) K. Tai, A. Hasegawa,  and A. Tomita, Physical review letters 56, 135 (1986).
  • Hegarty et al. (1999) S. Hegarty, G. Huyet, J. McInerney,  and K. Choquette, Physical Review Letters 82, 1434 (1999).
  • Kip et al. (2000) D. Kip, M. Soljacic, M. Segev, E. Eugenieva,  and D. N. Christodoulides, Science 290, 495 (2000).
  • Carusotto and Ciuti (2013) I. Carusotto and C. Ciuti, Rev. Mod. Phys. 85, 299 (2013).
  • Ciuti et al. (1998) C. Ciuti, V. Savona, C. Piermarocchi, A. Quattropani,  and P. Schwendimann, Phys. Rev. B 58, 7926 (1998).
  • Liew et al. (2015) T. C. H. Liew, O. A. Egorov, M. Matuszewski, O. Kyriienko, X. Ma,  and E. A. Ostrovskaya, Phys. Rev. B 91, 085413 (2015).
  • Wouters and Carusotto (2007) M. Wouters and I. Carusotto, Phys. Rev. Lett. 99, 140402 (2007).
  • Bobrovska et al. (2014) N. Bobrovska, E. A. Ostrovskaya,  and M. Matuszewski, Phys. Rev. B 90, 205304 (2014).
  • Bobrovska and Matuszewski (2015) N. Bobrovska and M. Matuszewski, Phys. Rev. B 92, 035311 (2015).
  • Daskalakis et al. (2015) K. S. Daskalakis, S. A. Maier,  and S. Kéna-Cohen, Phys. Rev. Lett. 115, 035301 (2015).
  • Bobrovska et al. (2016) N. Bobrovska, M. Matuszewski, K. S. Daskalakis, S. A. Maier,  and S. Kéna-Cohen, arXiv preprint arXiv:1603.06897  (2016).
  • (27) See Supplemental Material at http://… for details on the modeling and additional theoretical and experimental data.
  • Sieberer et al. (2013) L. M. Sieberer, S. D. Huber, E. Altman,  and S. Diehl, Phys. Rev. Lett. 110, 195301 (2013).
  • Chiocchetta and Carusotto (2013) A. Chiocchetta and I. Carusotto, EPL (Europhysics Letters) 102, 67007 (2013).
  • He et al. (2015) L. He, L. M. Sieberer, E. Altman,  and S. Diehl, Physical Review B 92, 155307 (2015).
  • Altman et al. (2015) E. Altman, L. M. Sieberer, L. Chen, S. Diehl,  and J. Toner, Phys. Rev. X 5, 011017 (2015).
  • Dagvadorj et al. (2015) G. Dagvadorj, J. M. Fellows, S. Matyjaśkiewicz, F. M. Marchetti, I. Carusotto,  and M. H. Szymańska, Phys. Rev. X 5, 041028 (2015).
  • Gladilin et al. (2014) V. N. Gladilin, K. Ji,  and M. Wouters, Phys. Rev. A 90, 023615 (2014).
  • Ji et al. (2015) K. Ji, V. N. Gladilin,  and M. Wouters, Phys. Rev. B 91, 045301 (2015).
  • (35) More formally, three independent analytical conditions need to be fulfilled so as to apply the adiabatic approximation, as shown in Ref. Bobrovska and Matuszewski (2015).
  • Estrecho et al. (2017) E. Estrecho, T. Gao, N. Bobrovska, M. Fraser, M. Steger, L. Pfeiffer, K. West, T. Liew, M. Matuszewski, D. Snoke, et al., arXiv preprint arXiv:1705.00469  (2017).
  • Kasprzak et al. (2006) J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J. Keeling, F. Marchetti, M. Szymańska, R. Andre, J. Staehli, et al., Nature 443, 409 (2006).
  • Deng et al. (2007) H. Deng, G. S. Solomon, R. Hey, K. H. Ploog,  and Y. Yamamoto, Phys. Rev. Lett. 99, 126403 (2007).
  • Manni et al. (2012) F. Manni, K. G. Lagoudakis, R. André, M. Wouters,  and B. Deveaud, Phys. Rev. Lett. 109, 150409 (2012).
  • Roumpos et al. (2012) G. Roumpos, M. Lohse, W. H. Nitsche, J. Keeling, M. H. Szymańska, P. B. Littlewood, A. Löffler, S. Höfling, L. Worschech, A. Forchel,  and Y. Yamamoto, Proceedings of the National Academy of Sciences 109, 6467 (2012).
  • Nitsche et al. (2014) W. H. Nitsche, N. Y. Kim, G. Roumpos, C. Schneider, M. Kamp, S. Höfling, A. Forchel,  and Y. Yamamoto, Phys. Rev. B 90, 205430 (2014).
  • Fischer et al. (2014) J. Fischer, I. G. Savenko, M. D. Fraser, S. Holzinger, S. Brodbeck, M. Kamp, I. A. Shelykh, C. Schneider,  and S. Höfling, Phys. Rev. Lett. 113, 203902 (2014).
  • Jacqmin et al. (2014) T. Jacqmin, I. Carusotto, I. Sagnes, M. Abbarchi, D. D. Solnyshkov, G. Malpuech, E. Galopin, A. Lemaître, J. Bloch,  and A. Amo, Phys. Rev. Lett. 112, 116402 (2014).
  • Baboux et al. (2016) F. Baboux, L. Ge, T. Jacqmin, M. Biondi, E. Galopin, A. Lemaître, L. Le Gratiet, I. Sagnes, S. Schmidt, H. E. Türeci, A. Amo,  and J. Bloch, Phys. Rev. Lett. 116, 066402 (2016).
  • Aleiner et al. (2012) I. L. Aleiner, B. L. Altshuler,  and Y. G. Rubo, Phys. Rev. B 85, 121301 (2012).
  • Stepnicki and Matuszewski (2013) P. Stepnicki and M. Matuszewski, Phys. Rev. A 88, 033626 (2013).
  • Ballarini et al. (2017) D. Ballarini, D. Caputo, C. S. Muñoz, M. De Giorgi, L. Dominici, M. H. Szymańska, K. West, L. N. Pfeiffer, G. Gigli, F. P. Laussy,  and D. Sanvitto, Phys. Rev. Lett. 118, 215301 (2017).
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