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Interplay between the Hawking effect and the quasi-normal modes of a one-dimensional sonic horizon in a polariton fluid

Analogue gravity enables the laboratory study of the Hawking effect, correlated emission at the horizon. Here, we use a quantum fluid of polaritons as a setup to study the statistics of correlated emission. Dissipation in the system may quench quasi-normal modes of the horizon, thus modifying the horizon structure. We numerically compute the spectrum of spatial correlations and find a regime in which the emission is strongly enhanced while being modulated by the quasi-normal modes. The high signal-to-noise ratio we obtain makes the experimental observation of these effects possible, thus enabling the quantitative study of the influence of dissipation and of higher order corrections to the curvature on quantum emission.

Quantum fluctuations at the event horizon of black holes cause the emission of correlated waves by the Hawking effect (HE) [1]: while some waves (Hawking radiation) propagate away from the horizon to outer space, others (the partner radiation) fall inside the horizon. Since signaling from inside the horizon is impossible, only Hawking radiation may be detected and correlations between paired waves cannot be measured in astrophysics. The HE may also be observed in the laboratory thanks to analogue gravity setups [2,3]. These are media whose properties may be engineered such that waves within propagate on effectively curved spacetimes [4,5], as has been experimentally demonstrated in a variety of platforms [6][7][8][9][10][11][12][13][14][15][16][17]. For example, there is a horizon for sound waves in a one-dimensional transsonic fluid where the flow velocity of the fluid equals the speed of sound. The HE at the sonic horizon yields the emission of correlated waves just like in astrophysics [4,5], with the notable difference that observation on both sides of the horizon is possible.
Experimental evidence for correlated emission at horizons was recently reported in analogue gravity setups based on classical [18] and quantum fluids [19]. While the thermal fluctuations of classical fluids overpower quantum fluctuations at the horizon such that spontaneous emission cannot be observed there, this can be done with quantum fluids. Spontaneous emission would yield a non-separable state at the output [20][21][22][23][24][25][26][27][28][29], whose degree of entanglement could be quantified from the density and correlation spectra [30,31].
Although most work on spontaneous emission has been dedicated to atomic Bose-Einstein condensate (BEC) analogues [32][33][34][35][36][37][38][39][40], correlated emission with comparable properties may also be observed in quantum fluids of microcavity exciton-polaritons (polaritons) where a sonic horizon has already been experimentally realised in one-and twodimensional microcavities [11,41]. In both quantum fluids, the Hawking effect manifests itself as the emission of socalled Bogoliubov excitations (elementary excitations of the fluid whose kinematics are ruled by its dispersion) that propagate in opposite directions on either side of the horizon.
In this paper, we explore the parameter space of quantum fluids of polaritons to find a regime favourable to the formation of correlations by the HE. The hydrodynamics of the fluid are controlled by its density and phase, which are in turn connected with the optical bistability of the system (the hysteresis cycle of its polariton-density-to-optical-power relationship), and so we investigate spontaneous emission from this perspective. We study the influence of the regime of bistability on either side of the horizon on the properties of emission by the Hawking effect. In doing so, we also find that, besides yielding correlated emission by the HE, quantum vacuum fluctuations also perturb the horizon that rings down. This linear response of the perturbed horizon is understood in terms of damped resonances called quasinormal modes (QNMs) [42][43][44]. These are complex frequency modes whose real and imaginary parts correspond to their oscillation frequency and lifetime, respectively. To date, QNMs have only been observed in two-dimensional rotating flows [45]. They have also been studied in the context of large amplitude perturbations destabilising the horizon and the so-called black-hole laser in one-dimensional conservative quantum fluids [46][47][48][49], but it is the first time they are observed in a driven-dissipative fluid.
Here we show that dissipation quenches these modes which form atmospheres (higher order corrections to the curvature of the effective spacetime) on either side of the horizon, thus modulating correlated emission by the HE. Fine control upon the regime of bistability provides us with a better understanding of the influence of the properties of the quantum fluid of polaritons on the propagation of Bogoliubov excitations as well as on emission by the HE therein. Our results open the way to the experimental observation of spontaneous emission from the vacuum in polaritonic systems and to further study of fluctuation-driven instabilities of sonic horizons. Sonic horizon in a polariton fluid Our study is based on the experiment [11]: the physical device is a GaAs microcavity sandwiched between a pair of planar Bragg reflectors with alternating λ/4 layers of GaAs/AlGaAs. An InGaAs quantum well is inserted in the microcavity layer, whose thickness is chosen so that the cavity mode is resonant with the quantum well excitonic transition to obtain polaritons. The micro-cavity is elongated over 500 µm in one direction and kept to a constant transverse width of 3 µm to form a wire in which the polariton dynamics are effectively one-dimensional. As in [11,50,51], we pump the microcavity with a continuous wave, coherent pump laser incident at a given angle with respect to the normal to form a stationary flow along the wire. However, instead of using a Gaussian spatial mode, we structure the light field to pump with a step-like intensity profile. In the region where the pump lies, the density and phase properties of the polariton fluid are set by those of the pump, while in the region where the pump intensity is zero, polaritons propagate ballisticaly. As in [11,51,52], we consider a cavity with an attractive defect (a localised 1 µm long broadening of the wire to a width of 5.6 µm) placed downstream of the region where the pump lies. The defect at x = 0 will create a dip in the fluid density and a spike in the flow velocity because of the conservation of the flow current. The density profile of the polariton fluid gn/m ( g the interaction energy, n the mean-field density of polaritons, m their effective mass) is shown in blue in Fig. 1 (c) in laboratory frame coordinates x and t. We see that the polariton density is almost flat before the defect and decreases afterwards. The polariton dynamics are driven-dissipative: polaritons have a lifetime γ after which they de-excite, releasing a photon that leaks out of the cavity, enabling the direct monitoring of the density and phase of the fluid.
In Appendix A, we review the theory for polariton hydrodynamics in a homogeneous system as described by a modified Gross-Pitaevskii equation (GPE). There we show how the polaritons behave as a fluid and describe the dispersive properties of elementary (or Bogoliubov) excitations therein. These depend on the effective detuning between the pump energy ω p and that of polaritons ω 0 , ∆ p = ω p − ω 0 − k 2 p 2m , where k p is the wavenumber of the pump field. In the case ω p > ω 0 (∆ p > γ √ 3/2), the steady-state GPE is with F p the laser field. The equation of state (1) describes the bistable behaviour of the fluid [53]: as shown in Fig. 1 (a), there is a hysteresis relationship between gn and |F p | 2 , which we will henceforth refer to as the 'bistability loop'.
The kinematics of Bogoliubov excitations of the fluid are governed by the dispersion relation. In Appendix A, we calculate and study the properties of the dispersion relation as a function of ∆ p in the frame co-moving with the fluid at velocity v = m ∂ x θ. Here we write the dispersion relation ω L ± in the laboratory frame (where measurements are made) by means of a Galilean transform (ω L → ωvδk, with δk = k − k p ): The dispersion relation (2) depends on both the density n and the velocity of flow v of the fluid [54].
The case of interest is that of a transsonic fluid flow, that is a flow which goes from being sub-to supersonic with a sonic horizon (v = c s ) at x = 0 [55]. This is the canonical 'waterfall configuration' of the transsonic flow studied in the analogue gravity literature with quantum fluids. The waterfall is composed of two homogeneous regions of different density (and hence speed of sound) and phase (and hence flow velocity). The region where the flow is subsonic is upstream of, or outside, the horizon. The region where the flow is supersonic is downstream of, or inside, the horizon.
In Fig. 1, we plot the dispersion relation (2) of the inhomogeneous fluid in the laboratory frame for two different pumping strengths when pumping at point C of the bistability loop. Blue (orange) curves correspond to positive (negative) energy solutions of Eq. (A10) in the frame co-moving with the fluid. As usual in field theories in analogue gravity, the modes with positive (negative) energies in the rest frame of the fluid have positive (negative) Klein-Gordon norm [56]. For subsonic fluid flows, the negative norm branch is at negative laboratory frame energies, while for supersonic flows, part of the negative norm branch is pulled up to positive laboratory frame energies by the Doppler effect, up to a maximum energy which we denote by ω max . Now that we have described the dispersive properties of the inhomogeneous fluid, we consider the kinematics of Bogoliubov excitations therein. Because of the time invariance of the system, these are plane wave modes. Eq. (2) is a fourthorder polynomial, so there are four (positive laboratory-frame frequency) solutions to the equations of motion in each spatial region on either side of the interface. These solutions are found at the intersection point of an ω 0 = cst line with the dispersion branches at positive energies in Fig. 1. Although these solutions share the same ω (which manifests energy conservation in the laboratory frame), they have distinct k, i.e. they are local modes of the homogeneous system. For ω > 0 in the upstream region there are two propagating modes of positive norm and two modes of complex ω and k, which are exponentially growing and decaying modes. For ω < ω max in the downstream region, there are four propagating modes, two of which have positive norm while the other two have negative norm. For ω > ω max , there are two propagating modes of positive norm and two exponentially growing and decaying modes. The interface at x = 0 is a sonic horizon only at frequencies for which there are two propagating local modes in the upstream region and four propagating local modes (including negative-norm modes) in the downstream region [35,56,57], i.e., for ω ∈ [0, ω max ]. In Appendix A 4 we explain how to construct the "global modes" (GMs) of the inhomogeneous system (including the waterfall).
Before moving on to the calculation of spontaneous emission at the horizon, we remark that, in reality, the media on either side of the interface are not strictly homogeneous. Instead their density and phase vary in space. Nevertheless, if the amplitude of these variations is small, the description of the system as two media remains valid, only one has to calculate the dispersion and bistability at all points. However, this renders an analytical calculation overly lengthy (if at all possible), unlike in [35,56,57]. So, instead of calculating the scattering matrix analytically, we use the Truncated Wigner Approximation (see Appendix B) to evolve the wave function and obtain the properties of the fluid at all points in the cavity as well as the dynamics of the Bogoliubov excitations therein.
Enhanced Hawking effect We now perform simulations with the cavity parameters of [11]: γ = 0.047 meV, g = 0.005 meVµm, m = 3 · 10 −5 m e . In Appendix C, we study a variety of configurations of pump spatial profile, wavenumber and intensity and we observe the influence of the regime of density of the fluid (optical bistability) on spontaneous emission at the horizon and propagation in either region thereafter. In doing so, we can explain that the configurations [11,51,52] limit the emission of Bogoliubov excitations because of operation with inhomogeneous profiles far from the sonic point of the bistability loop in the upstream region. In Fig. 2, we show the two-point correlations g (2) (x, x ) − 1 for the optimal configuration, which is when the density and phase of the fluid in the upstream region are homogeneously supported in the bistable regime, specifically at point C in Fig. 1 (a). In all configurations, correlations may be sorted by the region in which the involved modes propagate, which correspond to four quadrants in the plots. The South West quadrant (x < 0, x < 0) corresponds to correlations in the upstream region; the South East and North West quadrants correspond to correlations across the horizon in the up-and downstream regions; the North East quadrant corresponds to correlations in the downstream region. All configurations have some common features, which are evident in Fig , where the pump was also structured to obtain a flat fluid density in the nonlinear-bistable density regime. In [52], just like in all other configurations considered in Appendix C, there is no effective cavity between the edge of the pump and the horizon (unlike in [11,51]). Yet, unlike in Fig. 2, the Hawking moustache is short, with a total length of about 25 µm whereas we obtain a total length of over 110 µm. We note that we obtain similar downstream-downstream correlation patterns to [52], so the difference does not lie in the physics in that region. In fact, the fluid accumulates before the horizon: as can be seen in Fig. 2, there is a small but non-negligible bump of width about 7 µm in the fluid density just before the horizon. If, as in [52] the fluid density is already at or above the sonic point before that bump, then the density rises such that it is far in the non-linear regime just before the horizon. In contrast, in Fig. 2 the density is supported at the sonic point in the upstream region, and the pump stops 10 µm before the horizon such that the accumulation of fluid brings the fluid density on the bump at the sonic point as well. Given the enhanced trace length compared with [52], we postulate that operating at the sonic point upstream of the horizon enhances the propagation of Bogoliubov excitations in that region. We confirm this in Appendix D, where we study the influence of the fluid properties near the horizon on spontaneous emission: we find that Bogoliubov excitations are emitted in the upstream region as long as the fluid density is supported on the upper branch of the bistability loop in the vicinity of the horizon. We also demonstrate that operating at exactly the sonic point there strongly enhances spontaneous emission and, in turn, propagation in the upstream region.
Hawking radiation and quasinormal modes We now turn to the new interference patterns observed in Fig. 2. Since these features develop strictly horizontally and vertically, they correspond to correlations between a propagating mode with a mode whose group velocity is zero. Whence these features signal the excitation of a mode localised near the horizon. The parts of the horizontal and vertical features that go from x = 0 towards x < 0 and x < 0 are correlations between the localised mode and the Hawking radiation (u out ) in the upstream region, while the parts that go towards x > 0 and x > 0 are correlations of the same localised mode with the partner wave (d2 out ) in the downstream region. So the localised mode resonates with outgoing Bogoliubov excitations: it is a long-lived trapped mode that couples to a propagating mode, i.e., a quasi-bound state of the horizon [43]. These are predicted to occur in particular for massive fields around Kerr and Schwarzshild black holes [59][60][61].
We have verified that the fluid density barely evolves within 2γ before the steady-state is reached, meaning that we do operate in the linear regime of interactions. This implies that these localised modes may only be populated by quantum fluctuations of the vacuum that perturb the horizon in addition to generating correlated waves by the Hawking effect. The ringdown frequency of the horizon is obtained by reading out the wavenumber of propagating modes that res- onate with the quasi-bound state on either side of the horizon, k u = −0.645 µm −1 and k d2 = 0.985 µm −1 . These share a frequency ω = Re(ω Q i ) = 0.7697 ps −1 . In Fig. 3, we show the result of the excitation of the horizon by an incoming wave packet: instead of letting solely the vacuum impinge on the horizon, we send in an additional, small amplitude wave packet (a classical coherent wave) of frequency Re(ω Q i ) coming from the upstream region. We observe the relaxation of the horizon, that is the evolution of the density at the horizon as a function of time and we see that it oscillates with frequency Re(ω Q i ) while damping down exponentially. This damping is a measure of the negative imaginary part of the complex frequency of the mode, Im(ω Q i ) = −0.042 12 ps −1 . We have also observed the coupling of this mode with an outgoing wave packet on each side of the horizon, confirming the quasi-bound nature of this excited state. We have varied the frequency of the incoming wave packet and observed that the ringdown was not significantly affected. So the oscillation frequency Re(ω Q i ) is the characteristic ringdown frequency of the horizon, and the observed localised mode is a quasinormal mode (QNM) of the horizon. Spatially, the bound part of the QNM is located near the horizon at x = 0 and the transverse width of the horizontal and vertical correlation features indicate that it extends over 10 µm.
The oscillating pattern of the strictly horizontal and vertical features is due to a phase difference between the bound part of the QNM and the outgoing modes, while their diagonal inclination implies that the QNM does not have a purely standing wave shape but oscillates in time. Note that the anomalous modulation of the correlations in both the up-and downstream region stem from these fringe patterns, so the QNM effectively modulates correlated emission by the Hawking effect [62]. Because of the simultaneous spontaneous excitation of correlated waves and QNMs in the two regions, the latter modify the Hawking spectrum: the frequency distribution of the Bogoliubov excitations emitted upstream will not smooth as in conserved quantum fluids [40], but will exhibit peaks at the frequencies Re(ω Q i ) of the QNMs and with widths give by Im(ω Q i ). As a result, the QNMs increase the Hawking emission, which is in turn dominated by Bogoliubov excitations of frequencies Re(ω Q i ). This is similar to the phenomenology observed in eg [48], but the source here is different: the instability of the black hole results from a dissipative quench and not from two-boundary interactions.
Discussion We showed how engineering the density of a quantum fluid of polaritons can enhance the emission and propagation of paired Bogoliubov excitations in a transsonic flow. Our work sheds light on the interplay between optical bistability and parametric amplification in fluids of light. The bistable behaviour of a system can thus be exploited to study field theoretic effects like the Hawking effect in the laboratory.
Here we observed the generation and propagation of paired Bogoliubov excitations of the quantum fluid on either side of a sonic horizon when supporting the density of the fluid at various points in the bistable regime. Support of an inhomogeneous fluid density and velocity may be achieved by changing the wavenumber of the pump. In an experiment, this is easily implemented with high spatial resolution (limited by the diffraction limit) thanks to spatial light modulators [63]. We found that supporting the density of fluid at the turning point of the bistability loop (the sonic point) yields Hawking correlations of the order of 10 −3 of the fluid fraction over more than 100 µm. These are a tenfold and a four-to tenfold enhancement, respectively, compared to previous results and render the observation of the HE realistic.
Furthermore, we observed quasi-normal modes of the horizon, which manifest themselves as a quasi-bound mode in the steady state. Differently from [45,64], the physics we observe here in the steady-state takes place after the ringdown phase and the horizon instability is initially driven by vacuum fluctuations and not a classical perturbation. In other words, quantum fluctuation at the horizon not only yield pairs by the Hawking effect but also perturb the horizon under a dissipative quench. This phenomenology confirms the potential of polaritonic quantum fluids for analogue gravity: the tunability that was exploited to identify the optimal configuration to observe the HE also provides novel associated phenomena that emerge naturally. Typically, the present interplay with QNMs was, to our knowledge, never observed in similar configurations in conservative platforms such as atomic condensates, while it is relevant to black holes physics in general. Finally, our methods open the way to the theoretical and experimental study of the quantum statistics of the HE in driven-dissipative systems: for example, one could calculate (and observe) the Hawking correlations in reciprocal space [39], thus gaining frequency-resolved information on them [40] which could in turn be used to measure entanglement [30,31]. Physics  In the case where the pump's intensity is zero in the downstream region, the fluid coming from upstream will propagate balistically there, as in eg [11,51,52]. The fluid density will decrease away from the horizon while the flow velocity will always be supersonic. In this case, the dispersion in the downstream re-gion will be [65] where k0 = mv/ denotes the wavevector of the ballistic fluid. It is the same sonic Bogoliubov dispersion relation as for ∆p = gn.
[55] We remark that the 'speed of sound' is an ill-defined concept in such highly dispersive media as our quantum fluid of polaritons. However, it is generally accepted that the 'local speed of sound' cs is given by the gradient of the tangent of the dispersion relation at ω = 0 in the frame co-moving with the fluid. A sub-(super-)sonic flow is thus a flow for which v is lower (larger) than this gradient. In that sense, it is possible to define a horizon and to consider the region downstream from it as its inner region although there may be waves that propagate in both directions therein. There exist stricter definitions of the curvature of the effective spacetime in analogue gravity, see eg [30], but the related considerations do not impact the conclusions we draw in the present work.
Our system is a one-dimensional [66] quantum fluid of exciton-polaritons whose flow velocity goes from being subto super-sonic, thus forming a sonic horizon where the local flow velocity of the fluid equals the local speed of sound. For simplicity, we may consider that the horizon separates two spatial regions whose properties are independent of spacetwo homogenous regions, although we shall eventually depart from this simplified picture. For now, we begin with the theoretical description of a homogeneous quantum fluid (of its phase and density) and of the propagation of quantum (i.e. small-amplitude density) fluctuations therein.

Polariton fluid and Bogoliubov excitations
Exciton-polaritons are quasi-particles resulting from the interaction of light with matter in a semiconductor microcavity. Photons emitted by a laser will be trapped in a cavity formed by two Bragg mirror, wherein their dispersion is the usual Fabry-Perot dispersion. These trapped photons create excitons -bound electron-hole pairs -in the semiconductor microcavity. Strong coupling between the photons and excitons trapped in quantum wells gives rise to two eigenstates for the total Hamiltonian, known as the lower polariton (LP) and upper polariton (UP) branches. Furthermore, the Coulomb interaction between excitons results in an effective non-linearity for exciton-polaritons (polaritons). The dynamics of the mean-field are governed by a generalised Gross-Piteavskii equation, which leads to Euler and continuity equations describing the system as a quantum fluid. Historically, polaritons have first been described as two-dimensional quasiparticles [67], although the theory may be reduced to onedimensional cavities called 'wires' [11,51,52,68], as in the present case.
In the majority of cases, all energies involved are small compared to the Rabi splitting so the exciton-polariton system can be described by the mean field approximation [65]. At this level the system is described by a single scalar field Ψ, the field of lower polaritons, whose dynamics are governed by the driven-dissipative Gross-Pitaevskii equation (GPE) ω 0 is the frequency of the lower polaritons at the bottom of the branch, m is their effective mass, V is the 'external potential' (that is controlled via the interplay of the density profile of the pump and the cavity's own potential), g is the effective nonlinearity, γ is the loss rate, F p is the field of the pump laser. The field Ψ(x, t) is written in the laboratory frame.
The description of the system as a fluid is supported by the Madelung transformation: we write the field of lower polaritons as Ψ(x, t) = n(x, t)e iθ(x,t) , insert this expression into (A1) and multiply by e −iθ √ n, thus obtaining We write v = m ∂ x θ and insert (A2) into (A1), so that, by taking the real and imaginary parts, we arrive at the Euler and continuity equations for the polariton fluid [69]: The first equation of (A3) is the Euler equation of atomic Bose-Einstein condensates (BECs) plus a term coming from the coherent pumping. The second equation of (A3) is the continuity of the flow with a loss term and a term coming from the coherent pumping. We see that the properties of the fluid depend on two parameters, namely its' density n and phase θ.
The spatial variations of the phase is encapsulated in v, which we identify from (A3) as the flow velocity of the fluid. Now that we have described the polariton fluid in terms of its' density and phase, we consider the propagation of small amplitude fluctuations (such as quantum fluctuations) of the density in this fluid -the so-called 'Bogoliubov excitations'. Bogoliubov excitations are mathematically obtained by linearising the GPE (A1) around a background: Ψ → Ψ + δΨ, and Ψ * → Ψ * + δΨ * . L is the 'Bogoliubov matrix' that describes the dynamics of the Bogoliubov excitations (δΨ, δΨ * ): i∂ t δΨ δΨ * = L δΨ δΨ * . In the steady state, the GPE (A1) becomes where ω p is the frequency of the pump. We first consider a configuration where the wire is pumped with a spatially homogeneous and monochromatic pump of incident wavevector k p (so there is no potential in Eq. (A1), V (x) = 0). The phase of the fluid is then set by, and equal to, k p while its' density is homogeneous. The steady-state GPE (A4) simplifies to where ∆ p is the 'effective detuning' defined as the difference between the pump energy and that of lower polaritons, We go to the reference frame co-moving with the fluid via a Galilean transform (x → x − vt). In the special case of a homogeneous system where the interaction energy matches the detuning, gn = ∆ p , the Bogoliubov matrix L can be written in this frame as Upon diagonalization, we retrieve the Bogoliubov dispersion relation in this co-moving frame, which relates the wavenumber k of Bogoliubov excitations to their frequency ω there: (A8) Figure 4 shows the real part of Eq. (A8), the 'dispersion curve', which is identical to that of atomic BECs. There are two branches ω ∆p=gn ± of the dispersion, which are symmetrical around the point ω = 0, k = 0. At low k, the dispersion curve has a linear slope: ω ∂k > c s -the gradient of the dispersion curve is larger than the speed of sound, so the dispersion is said to be 'superluminal' (in analogy with superluminal corrections to the dispersion in eg [70,71]).

Optical bistability of the polariton fluid
Unlike the configuration considered in the previous paragraph, in the majority of cases the interaction energy does not match the effective detuning and the dispersion curve is thus modified. Furthermore, writing the density of the fluid as a function of the intensity of the laser yields several solutions. This degeneracy of fluid densities is due to optical bistability, which, as we will show, has tremendous influence on the We begin by describing the relationship between the density of polaritons, n, and the intensity of the pump laser, |F p | 2 in the case where the energy of the laser is above that of the lower polaritons, ∆ p > γ √ 3/2: we square Eq. (A5) and find The physics at play may be investigated equivalently in terms of the relationship between the speed of sound and the strength of the pump, as shown in Figure 5. At first, c s increases slowly with |F p | (arrow (1)), until |F p | = F 1 where it increases abruptly (arrow (2)). For |F p | > F 1 , c s increases slowly again. If the pump's strength is decreased from where it falls abruptly (arrow (4)). Since F 1 > F 2 , the c s to |F p | relationship presents a histeresis cycle with two regimes of speed of sound: the linear regime when |F p | < F 1 and c s is low, and the non-linear regime when |F p | > F 2 and c s is high. This histeresis cycle is the manifestation of optical bistability [53], so we will henceforth refer to it as the 'bistability loop'. Note that the dashed line in Fig. 5 is unstable and the speed of sound will actually follow the hysteresis cycle schematised by arrows (1) − (4). Now, in order to explicitly show the dependence of the Bogoliubov dispersion on the density of the fluid as well as the influence of optical bistability thereon, we generalise Eq. (A8): We diagonalise the Bogoliubov matrix L for a ho-mogeneous system pumped with arbitrary strength and obtain (A10) In Fig. 6, we show the dispersion curve for 5 different fluid densities along the bistability loop. As can be seen in Fig. 6  a) and b), the shape of the dispersion does not change much in the linear regime: the two branches of the dispersion curve cross. When the fluid is bistable, in Fig. 6 b), we observe the appearance plateaus characteristic of an unstable fluid at the crossing points -Goldstone modes [72]. On the other hand, the shape of the dispersion curve changes significantly in the nonlinear regime depending on the position along the bistability loop: at high pump strength (Fig. 6 e)), the two branches are split in energy by a gap that increases with the pump strength. The sonic dispersion relation (A8) is recovered at point C (Fig. 6 c)), while for slightly lower pump strength (Fig. 6 d)), the plateau at low k is characteristic of an unstable fluid (similarly to the Goldstone modes of Fig. 6  d)) . Note that the dispersion curve has a linear slope at low k (and thus a sonic interpretation) at point C only, which is thus sometimes referred to as the 'sonic point' of the bistability. As Eq. (A10) is of order four in k, the dispersion has four complex roots. The real part of these roots is non-zero in the linear regime (Fig. 6 a) and b)) as well as at points C and C (Fig. 6 c) and d)), but not at point D (Fig. 6 e)).
In this appendix, we have seen that the mean-field of a polariton system behaves as a fluid. We have studied the dispersion relation of Bogoliubov excitations in this fluid and seen that optical bistability of the fluid strongly influences the properties of this dispersion relation. These considerations may be generalised to a fluid whose density is not homogeneous.

Dispersion relation in the nonlinear regime
In Fig. 7, we plot the dispersion relation (2) of the inhomogeneous fluid in the laboratory frame when pumping at point D of the bistability loop (see Fig. 6 e)): Fig. 1 a) and b) show the dispersion of a sub-and super-sonic fluid flow, respectively. Blue (orange) curves correspond to positive (negative) energy solutions of Eq. (A10) in the frame co-moving with the fluid. As usual in field theories in analogue gravity, the modes with positive (negative) energies in the rest frame of the fluid have positive (negative) Klein-Gordon norm [56]. For subsonic fluid flows, the negative norm branch is at negative laboratory frame energies, while for supersonic flows, part of the negative norm branch is pulled up to positive laboratory frame energies by the Doppler effect, up to a maximum energy which we denote by ω max . Note that the dispersion up-and downstream features a gap between the positive-and negative-norm branches. We denote the minimum energy of the positive-norm branch in the upstream region by ω min .

Global Modes of the system
In this appendix we explain how to construct the global modes (GMs) of the system from the local modes (LMs) identified in the body of the article.
LMs in a homogeneous region may be sorted by their respective group velocity v g = ∂ω ± /∂k : those which have positive group velocity propagate rightwards while those which have negative group velocity propagate leftwards. We proceed to construct modes of the inhomogeneous fluid, the GMs [56,57] -solutions to the equation of motion that are valid in both regions on either sides of the interface. GMs correspond to waves scattering at the interface, and they describe the conversion of an incoming field to scattered fields in both regions. The GMs are superpositions of the plane wave solutions in the two homogeneous regions on either side of the interface. We identify GMs via their 'defining' local mode. Specifically, in the upstream region, the unique local mode with positive group velocity defines an in GM u in , while the unique local mode with negative group velocity defines an out GM u out . In the downstream region, modes with negative group velocity define in GMs d 1in and d 2out and modes with positive group velocity define out GMs d 1out and d 2out [73]. GMs u in , u out , d 1in and d 1out are positive-norm modes while GMs d 2in and d 2out are negative-norm modes. Each in GM describes the scattering of a harmonic wave to various outgoing harmonic waves. Conversely, each out GM describes a single harmonic wave resulting from the scattering of various incoming waves. The scattering can be described in the in as well as the out basis, and the transformation between the two bases defines the scattering matrix (see [35,56,57,74] for an analytical derivation of the scattering matrix). Because the vacuum is basis dependent, spontaneous emission at the horizon will occur in correlated pairs u out −d 1out , u out −d 2out and d 1out − d 2out on top of the classical background formed by the polariton fluid (the mean-field). region, respectively), while the fluid density may be supported on the higher branch of the histeresis loop by means of the Pigeon effect [76]. There are, roughly speaking, 6 different points along the bistability loop (see Fig. 5), meaning that in order to explore the full parameter space we have computed 36 correlation spectra. Not all combinations are interesting, though, so here we will only comment on spatial correlations obtained with typical (and relevant) inhomogeneous flow profiles.

Bistability loop
Velocity profile and pump Spatial g (2) correlation function In Table I  • I-4: The fluid density is supported on the upper branch in the bistable regime in both regions, with a phase fixed by k p,u = 0.25 µm −1 and k p,d = 0.535 µm −1 , respectively. The fluid density is homogeneous in the upstream region but not in the downstream region where it undulates. Note that, for reasons explained in appendix F, the strength of the defect potential is 30% lower than in all other configurations, yielding weaker spontaneous emission. The Hawking moustache is of amplitude 1 · 10 −4 and is about 12 µm-and 17 µm-long in the up-and downstream regions, respectively. The u out − d1 out correlations are of amplitude 8 · 10 −5 and the trace is 30 µm and 60 µm long in the up-and downstream regions, respectively. The d1 out −d2 out correlations are of amplitude 3.5 · 10 −4 and the trace is 50 µm long. Furthermore, negative correlations appear 22 µm downstream of the horizon, followed by positive correlations after 50 µm.
• I-5: The fluid density is supported on the upper branch in the bistable regime in the upstream region (k p,u = 0.25 µm −1 ), while it is left to evolve ballistically in the downstream region. The Hawking moustache is of am-plitude 2 · 10 −4 and is about 20 µm-and 20 µm-long in the up-and downstream regions, respectively. The u out − d1 out correlations are of amplitude 1.3 · 10 −4 and the trace is 27 µm and 60 µm long in the up-and downstream regions, respectively. The d1 out − d2 out correlations are of amplitude 3 · 10 −4 and the is 120 µm long.
In configuration I-1, 2, 3 and 4, we see that emission occurs even when the phase of the fluid is fixed by the pump in order to support its density on both sides of the horizon. However, correlation traces in the North West quadrant are short in these configurations, meaning that the propagation of Bogoliubov excitations in the downstream region is limited when the phase is fixed there. Furthermore, in configurations for which the fluid density undulates in the downstream region ( I-3 and 4), other short positive-and negative-correlation traces appear well after the horizon. These correspond to spontaneous emission from the vacuum on an effective spacetime without a horizon but whose properties are not homogeneous [30].
Spontaneous emission also occurs when the density of the fluid evolves ballistically in the downstream region, as in configurations I-5 and in Fig.2, and as in [11,51,52]. In all these configurations, the downstream-downstream correlations d1 out − d2 out as well as the the Hawking moustache are clearly visible. However, in [11,51], the u out − d1 out correlations are not visible (if at all present), while they are in [52] and in Table. I-5 and in Fig.2. In [51], it was established that an effective cavity forms between the edge of the (Gaussian) pump and the defect. The Bogoliubov excitations in the upstream region form a standing wave in this effective cavity, which in turn modulates the emission of negative norm waves in the downstream region and thus that of the positive norm companion wave as well. The resulting modulation of the correlation pattern in the downstream region (North East quadrant) was observed in both [11] and [51]. This modulation might be why the up-downstream u out − d1 out are not visible in these works.
So far we have established that operating with a density at the sonic point of the bistability loop upstream of the horizon enhances the propagation of Bogoliubov excitations in the upstream region. In configuration I-1, we also consider the case where the fluid density is also supported at the sonic point in the downstream region. As observed before, the downstreamdownstream trace corresponding to d1 out − d2 out correlations is only of amplitude 8 · 10 −4 , which is overpowered by a positive-correlation trace along the diagonal that stems from local interactions (unlike the physics under study). Yet, the up-downstream correlations are clearly visible and the two traces are over 60 µm long, which confirms that spontaneous emission by the Hawking effect does occur. In comparison, the Hawking moustache and all other correlation traces are stronger and longer in Fig.2. So letting the fluid density evolve ballistically in the downstream region enhances the emission and propagation of Bogoliubov excitations there.