## Abstract

Analogue gravity enables the study of fields on curved space–times in the laboratory. There are numerous experimental platforms in which amplification at the event horizon or the ergoregion has been observed. Here, we demonstrate how optically generating a defect in a polariton microcavity enables the creation of one- and two-dimensional, transsonic fluid flows. We show that this highly tuneable method permits the creation of horizons. Furthermore, we present a rotating geometry akin to the water-wave bathtub vortex. These experiments usher in the possibility of observing stimulated as well as spontaneous amplification by the Hawking, Penrose and Zeld’ovich effects in fluids of light.

This article is part of a discussion meeting issue ‘The next generation of analogue gravity experiments’.

### 1. Introduction

Analogue gravity experiments use well-controlled laboratory systems to study phenomena that are either currently eluding our comprehension or simply unreachable by their very nature, such as the behaviour of small-amplitude waves near black holes [1]. For example, it is possible to create static black- [2–7] or white-hole [2,4,6,8–12] event horizons for waves in media and rotating geometries with horizons as well as ergoregions [13,14]. Thus, analogue gravity enables the study of classical and semi-classical effects of fields on curved space–times, such as the Hawking effect at static horizons [15,16] or superradiance [17–20] and the quasi-normal modes of rotating geometries [21]. These effects were recently observed in water experiments [5,13,14]. In these seminal experiments, excitations were seeded by a classical state to stimulate the emission. Unfortunately, because of its low temperature, spontaneous emission (stemming from quantum fluctuations at the horizon or the ergosurface) cannot be observed in these systems. This can be done with fluids at lower temperatures, such as atomic Bose–Einstein condensates [7]. Here, we discuss another experimental platform for analogue gravity: polariton fluids in microcavities.

Fluids of light can display properties similar to those of matter fluids when the environment, such as confinement in an optical cavity, provides the photon with an effective mass and when the optical nonlinearity of the medium provides effective photon–photon interactions. With polaritons in microcavities, two-dimensional^{1} fluids in the plane of the cavity have been demonstrated to exhibit Bose–Einstein condensation [22,23] and superfluidity [24–27]. These hydrodynamic properties, together with the wide range of available control and engineered tunability of the optical set-ups, enable the simulation of systems described by Hamiltonians including various kinds of interactions. For example, elementary excitations of a polariton fluid can experience an effective curved space–time determined by the physical properties of the flow [28].

A flow featuring a horizon was first realized with a polariton fluid by means of an engineered defect in the cavity [11]. Here, by contrast, we show how an optical defect can be used to realize a transsonic flow. This technique has the advantage of being widely adaptable to various cavity geometries without requiring the mechanical manufacturing of a new geometry for each gravity scenario to be investigated. We demonstrate this advantage by realizing three scenarios: one-dimensional and two-dimensional static horizons and draining, rotating vortex flow.

The paper is structured as follows: we first introduce the quantum fluid properties of microcavity polaritons. We show how sub- and supersonic flows may be created by propagation across an optical defect and discuss the dispersion relation of Bogoliubov modes in these flows. We then present an experiment demonstrating the advent of superfluidity in a two-dimensional flow. In the third section, we revisit the canonical derivation of the wave equation for Bogoliubov modes and discuss the analogy with a curved space–time metric. In the fourth section, we present experimental data demonstrating the creation of one- and two-dimensional static horizons for the Bogoliubov modes before moving to a rotating geometry in the final section.

### 2. Polariton fluids in microcavities

The system used for analogue gravity is a semiconductor microcavity, made of quantum wells of InGaAs located between two Bragg mirrors forming a thin cavity of high finesse, as shown in figure 1*a*. In the quantum wells, electrons and holes form bound states, named excitons, that couple efficiently to photons. Under these conditions, the strong coupling regime can be reached between excitons and photons and the vacuum Rabi splitting takes place [29], resulting in mixed photon–exciton states, known as cavity polaritons. The two eigenstates, the upper and the lower polariton, are shown in figure 1*b*. They are separated by 5 meV, and in the following we will work with the lower polariton branch, excited close to resonance by a continuous wave, monochromatic laser. To avoid thermal noise, the system is operated at 5 K in a helium cryostat.

We are interested in the propagation of light in the cavity plane, with a small incidence angle. In an optical cavity, the resonance condition of an incoming laser with a cavity mode is given by *p*(*λ*/2) = *l*cos*θ*′, with *p* an integer (here *p* = 4), *λ* the mode wavelength, *l* the cavity thickness and *θ*′ the optical angle of incidence (inside the cavity). This implies that the component of the wavevector *k* perpendicular to the cavity plane, *k*_{z}, is fixed to *pπ*/*l*. In the case of a small angle of incidence (or a small component of the wavevector parallel to the cavity plane *k*_{x}) the dispersion can then be written as

*m*is the effective photon mass for motion in the cavity plane, $m=\hslash {k}_{z}/c$. This geometry gives a parabolic dispersion curve for the photons inside the cavity. The strong coupling modifies the photon and the exciton dispersion curves, introducing a curvature on the lower polariton branch, with an effective mass similar to that of the photon. With such a dispersion curve the group velocity of the polaritons can now be written as $v=\hslash {k}_{x}/m$. Moreover, excitons interact together via electromagnetic interaction, resulting in an interaction between polaritons.

As a result, cavity polaritons are interacting composite bosons with a small effective mass (∼10^{−5}m_{e}). They have a large coherence length of the order of 1–2 μm at the operating temperature (5 K). This enables the building of many-body coherent quantum effects, such as condensation and superfluidity at these temperatures. The quantum fluid properties of photons in a cavity and polaritons were predicted [30–32] and later observed by several groups [22–27]. The main interest of this system for quantum simulation is the evolution of the polariton wave function, *ψ*, which is described by an equation similar to the Gross–Pitaevskii equation

*V*

_{ext}, the fourth and fifth terms are the losses (

*γ*) owing to the cavity and the resonant pumping by a laser. These last two terms are additions to the usual Gross–Pitaevskii equation: they account for the driven–dissipative nature of the system, with losses of light out of the cavity, compensated by pump photons.

The onset of superfluidity can be studied using the Boguliubov method, where the effect of a small perturbation is studied by linearizing the Gross–Pitaevskii equation in the vicinity of the operating point. The resulting Bogoliubov dispersion is given by

**δk**=

**−**

*k*

*k*_{p}is the mean difference between the wavevector of the perturbation and the wavevector of the pump and

*n*= |

*ψ*

_{0}|

^{2}. Equation (2.3) was established from the Gross–Pitaevskii equation for matter superfluids. For laser-driven systems like polaritons, it is only valid if the detuning between the laser energy and the lower polariton energy (in the absence of interaction) is equal to the nonlinear interaction energy, which corresponds to the upper turning point of the bistability curve [32,33]. This is the point where the superfluidity demonstrations in polaritons were performed [27]. In the rest frame of the fluid, the full dispersion relation is centro-symmetric around the {

*k*= 0,

*ω*= 0} point—there are two branches: a positive (fluid frame) frequency branch and a negative (fluid frame) frequency branch (the so-called ghost branch). In figure 2, we show the dispersion relation in the laboratory frame when the polariton flow is subsonic (

*a*) and supersonic (

*b*). The positive (negative) fluid-frame frequency branch is shown in black (red dashed). Depending on the value of

*δk*compared with the inverse of the healing length $\xi =\sqrt{\hslash /mgn}$ the dispersion curve can be changed considerably. If

*δk*is large,

*δkξ*≫ 1, the above equation gives $\omega (\delta k)=\mathit{\delta}\mathit{k}\mathbf{\cdot}{\mathit{v}}_{\mathbf{0}}\pm \hslash \delta {k}^{2}/2m+gn$—the usual parabolic dispersion for a massive system, shifted by the interaction energy 2

*gn*. If

*δk*is small,

*δkξ*≪ 1, we have

*ω*(

*δk*) =

**δk ·**

*v*_{0}±

*c*

_{s}|

*δk*| with the speed of sound ${c}_{s}=\sqrt{(\hslash gn/m)}$. Figure 2 shows a sonic dispersion with a discontinuity in the slope at

*δk*= 0. This point corresponds to the minimum (maximum) of the positive (negative) frequency branch, as can be seen in figure 2

*a*. Moreover, in the subsonic case (

*v*

_{0}<

*c*

_{s}), the slopes on each side of

*δk*= 0 have opposite signs. This gives rise to superfluidity because no elastic scattering (for example, against an obstacle) is possible in this case, thus fulfilling the Landau criterion. The critical velocity below which superfluidity occurs has been shown here to be the sound velocity. However, in some experiments, the presence of an incoherent reservoir of excitons modifies the interaction energy and should be taken into account for the definition of the critical velocity [34]. In the following

*c*

_{s}will denote this critical velocity.

The experiments demonstrating this effect^{2} were performed by shining a laser quasi-resonant with the lower polariton branch on the microcavity with a small angle of incidence [27]. This configuration gives a small velocity to the fluid of light in the cavity plane, so that it is smaller than the speed of sound at high intensity. The dispersion for Bogoliubov excitations is then that of figure 2*a*. On the other hand, when the fluid flow is supersonic, the dispersion relation is as in figure 2*b*.

In figure 3, we show what happens when the fluid of light hits a defect. There is a drastic change when, by increasing the polariton density, one goes from the linear regime (where the particle–particle interactions are negligible (*a*,*b*)) to the nonlinear regime (*c*,*d*): in figure 3*c*, we see interference fringes between the incoming flux and the scattered light disappear in the nonlinear regime. As can be seen by comparing figure 3*b*,*d*, the scattering ring in momentum space also disappears. This is due to superfluidity. Importantly, the flow velocity goes from subsonic (on the right-hand side of figure 3*c*) to supersonic (on the left-hand side of figure 3*c*) over the optical defect.

### 3. Analogue gravity with polaritons

The ability to generate a transsonic flow in one and two spatial dimensions is at the basis of the use of polariton quantum fluids for analogue gravity. Here, we revisit the derivation establishing the analogy and explain how polariton flows can be tailored to generate the corresponding experimental configuration.

We begin by converting the Gross–Pitaevskii wave equation for the polaritons into hydrodynamic equations using the Madelung transformation. At this point it should be noted that the mathematical analogy [1] between the wave equation for Bogoliubov excitations in a fluid flow and on a curved space–time is valid at low *k* only [35], where the dispersion relation (2.3) is linear—the excitations behave as genuine sound waves with a constant group velocity *c*_{s}. In this ‘hydrodynamic regime’, the influence of the pump, losses and external potential on the kinematics of sound waves is negligible, and thus the last three terms of (2.2) may be dropped:^{3} the wave equation reduces to the nonlinear Schrödinger equation [37]

*ψ*|

^{2}the polariton intensity. We convert equation (3.1) into the hydrodynamic continuity and Euler equations by writing the complex scalar field in terms of its amplitude and phase—$\psi =\sqrt{\rho}\hspace{0.17em}{\text{e}}^{\text{i}\varphi}$, with

*ρ*the polariton density, corresponding to the fluid density:

**is a two-dimensional vector (**

*v***= {**

*v**v*

^{x},

*v*

^{y}} with

*x*and

*y*the spatial coordinates) and

*v*is its modulus.

We obtain the wave equation for the Bogoliubov excitations (sound waves at low *k*) by linearizing equations (3.2) and (3.3) around a background state: we set *ρ* = *ρ*_{0} + *ϵρ*_{1} + *O*(*ϵ*^{2}) and *ψ* = *δρ*/*ρ* and we neglect the quantum pressure (last term in (3.3)) to arrive at

Equation (3.4) is strictly isomorphic to the wave equation of a massless scalar field propagating on a 2+1D curved space–time,^{4}

*η*is written in terms of a 3 × 3 symmetric matrix with

*η*= det(

*η*

_{μν}), and

The various components of this ‘acoustic metric’ [1] are given by the fluid velocity. Depending on the configuration of the flow, i.e. on ${v}_{0}^{x}$ and ${v}_{0}^{y}$, one can set up different fluid flows and study the propagation of fields on different space–times, such as: (i) an accelerating flow along one spatial dimension only, (ii) a fluid flowing at increasing radial velocity in two spatial dimensions, and (iii) a rotating flow like a vortex. These various flow configurations may feature different regions characteristic of a black hole space–time, which may all be read from the metric (3.6). Notably, there is an event horizon in all three configurations—where the flow velocity equals the speed of sound in (i)^{5} and where the radial component of the fluid velocity equals the speed of sound (*v*_{r} = *c*_{s}) in (ii) and (iii). Configuration (iii) also has an ergoregion, which is bounded by the surface where the flow velocity exceeds the speed of sound (${v}_{0}^{2}>{c}_{s}^{2}$) [35].

Polaritons can be optically tailored to generate either of these configurations. Take the event horizon: when the polaritons are excited with a small wavenumber *k* and a large enough intensity, the flow is superfluid inside the excitation area. Outside the excitation area, the polariton density *ρ*_{0} decreases, owing to the dissipative nature of the polaritons, and therefore the interaction energy decreases. This induces a reduction in the speed of sound (${c}_{s}\propto \sqrt{\hslash g{\rho}_{0}}$). Moreover, because of the conservation of the polariton energy in the cavity, the sum of the kinetic energy and of the interaction energy is constant [38]

*v*

_{0}is the polariton flow velocity and Δ is the energy detuning at low power between the pump laser and the polariton branch at zero velocity. This induces an increase in the kinetic energy and then in the flow velocity. These effects result in the change of the flow characteristics from superfluid/subsonic to supersonic, thus creating a horizon. As shown in [38,39], the sharpness of this transition depends significantly on the shape of the pump spot and on the presence of an obstacle.

### 4. One- and two-dimensional horizons

In order to improve the sharpness of the horizon a defect can be placed close to the border of the pumping zone, as proposed in [39] and demonstrated by etching [11] a defect in the cavity. Here, we demonstrate the use of a defect generated optically (figure 4). The defect is generated by a control laser beam with orthogonal relative polarization to avoid interferences. The interaction between polaritons with orthogonal polarizations is weaker than the interaction between polaritons with the same polarization, but this can be compensated by increasing the power of the control beam creating the obstacle. Here, the linear obstacle (not directly visible, represented by a red dotted line) is placed at the border of the pumped region. The defect is at a 45° angle to the polariton flow, which has a velocity of 0.64 μm ps^{−1}.

In the low-density regime, one can observe interference fringes between the incoming and reflected flows. At high density, since the fluid velocity is lower than the speed of sound (*c*_{s} = 0.8 μm ps^{−1}), the system is in the superfluid regime. In this case, there is no reflected flow, as attested by the disappearance of the interference fringes in the density distribution and of the reflected light in the momentum distribution. Because of the driven–dissipative nature of the polariton system linked to the cavity losses, part of the population in the superfluid pumped region goes out of the cavity. The remaining part of the flow goes through the barrier, and, in this unpumped region, the flow has a much lower density. The density decreases by about a factor of 2 and the speed of sound is now 0.56 μm ps^{−1}. Moreover, owing to energy conservation the fluid velocity increases and is now 0.90 μm ps^{−1}, enhancing the supersonic character of the fluid [40]. This experiment demonstrates the presence of an event horizon, with a sharp transition between the upstream (subsonic) and downstream (supersonic) regions. The spatial mode of the laser fixes the derivative of the flow velocity at the horizon. Here, a Gaussian beam was used but it is possible to create steeper (and sharper) horizons by shaping the laser beam with, for example, a spatial light modulator.

This process can be extended to create a closed 2D horizon. By generating a polariton ensemble arranged in a square, one produces a converging flow towards the centre of the square. For this, a pump laser beam at *k* = 0 is focused on the microcavity surface. A square mask is placed in the pump laser beam, and is imaged on the microcavity. The image of the mask generates a square dark region (with a 45 μm side) at the centre of a bright Gaussian spot (with diameter at half maximum of 100 μm). At low intensity (figure 5*a*), polaritons flow from the four sides of the mask towards the centre, generating four plane waves that create an interference pattern clearly visible on the figure. Vortex–antivortex pairs also form because of the local disorder. At high intensity (figure 5*b*), the polaritons are generated resonantly at *k*=0 outside of the masked region and then enter by diffraction in the masked region, where they acquire a momentum in order to conserve energy. The repulsion between polaritons leads to an enlargement of the lattice unit cell owing to an enhancement of the interaction, as confirmed by the numerical model (figure 5*c*), and to the appearance of a superfluid region.

From this image, the Mach number (*M* = *v*/*c*_{s}) is calculated using the local speed of sound of the polariton ensemble (figure 5*d*). In the region inside the trap that is close to the borders, the system is in a subsonic/superfluid regime because of its high density, which is consistent with the disappearance of interference fringes. As polaritons move towards the centre of the trap, their density decreases (this is due to their finite lifetime) and so does the sound velocity of the fluid—the fluid becomes mainly supersonic (except at the centre, where the four flows merge, creating an inner subsonic region). There is an event horizon between the subsonic and supersonic regimes in this convergent flow near the edge of the square. The velocity and the density of the polariton flow can be adjusted to modify the parameters of this horizon.

It has been theoretically predicted that it is possible to observe the Hawking effect at these sonic horizons [11,41]. We propose to begin by seeding the scattering process with a classical, low-amplitude excitation (similar to [5]). This can be done with a probe beam with an energy higher than that of the polariton flow, propagating in the downstream supersonic region towards the horizon. Such a probe wavepacket can be partly transmitted and reflected at the horizon: we will thus retrieve the coefficient of the scattering matrix that rules the Hawking effect [42–45] and observe the Bogoliubov branches of the dispersion by using a method recently implemented in paraxial fluids of light in hot atomic vapours [46].

The spontaneous Hawking process results from the conversion of vacuum fluctuations into correlated phonons on the two sides of the horizon [47], which can be detected as intensity correlations in the near-field emission of the cavity [48–50]. As shown in [41], these correlations correspond to very specific branches of the Bogoliubov dispersion on each side of the horizon, one of them being the negative fluid-frame frequency branch. Their detection, achieved only in cold atoms [7], is non-trivial and remains a challenge for polariton fluids, as a detailed theory for the correlation signal—which would account for the thermal noise (including stimulation of the Hawking effect at the horizon by the ever-present non-thermal, incoherent bath of phonons in the system [51])—is still missing. Yet, quantum noise in semiconductor microcavities was studied experimentally using homodyne detection [52]. This method offers a high sensitivity and should be very efficient for studying the Hawking effect. Furthermore, a measure of the ‘quantumness of the output state’ would discriminate correlations resulting from spontaneous emission from those resulting from noise (although additional noise may degrade the signal-to-noise ratio in the measurement of Hawking correlations). To this end, one could use various measures such as the Perez–Horodecki criterion [47,51,53–58] or the logarithmic negativity [36] to determine whether the output state is non-separable. Importantly, entanglement monotones such as the logarithmic negativity would provide unprecedented insight into the quantum statistics of Hawking radiation.

### 5. Rotating geometry

In the microcavity driven–dissipative system, the use of a pump laser to generate the polaritons also allows an orbital angular momentum to be injected in the fluid. This can be achieved by using a Laguerre–Gauss (LG) beam produced by a phaseplate or a spatial light modulator [59]. Figure 6 shows the density and phase map obtained with a beam with *l* = 8 for low- and high-density regimes. Polaritons rotate and propagate towards the centre with an azimuthal and a radial velocity. A low-density region is present in the centre. In the low-density regime, the phase map shows some disorder. In the high-intensity regime, the disorder decreases as a result of the interactions and, in the density map, one can observe a concentric ring-like structure at the centre of which a dark zone similar to the low-density case is visible [60].

The flow configuration of figure 6 is not only rotational: there is an influx of polaritons towards the centre of the beam, where they naturally escape from the cavity. (Here, polariton losses are not localized solely in the centre of the beam but occur everywhere in the cavity.) The ability to create a rotating flow with a drain is a key advantage of lossy systems such as polaritons in microcavities over conservative systems such as atomic ensembles for the simulation of a rotating black hole. For example, Solnyshkov *et al*. [20] show that focusing an LG beam (with a maximal angular momentum *l* = 18) with sufficient power on a microcavity creates an analogue rotating black hole with a diameter of the order of 20–30 μm. In this case, there is an ergoregion: a region limited on the outside by a static limit, where the total flow velocity is equal to the velocity of sound, and on the inside by the inner horizon, where the radial flow velocity is equal to the velocity of sound.

The flow velocities can be measured in figure 6 using the gradient of the fluid phase as presented in §4. Although the signal-to-noise ratio does not allow the positions of the event horizon and of the ergosurface to be reliably determined, the radial and total velocities are found to vary from 0.5 μm ps^{−1} outside the beam to over 2 μm ps^{−1} when going towards the centre of the beam, while the maximum sound velocity is of the order of 1 μm ps^{−1}. This shows that the system has adequate parameters to define an ergosphere and a horizon. Polaritons are thus a promising system for generating a geometry in which the physics of rotating black holes can be studied.

Thus, in this system, one can test the Penrose effect [18], which is at the heart of instability-driven phenomena such as the black hole bomb. The Penrose effect involves the creation of a pair of positive and negative energy particles—vortices and antivortices in the present case [20]. A vortex–antivortex pair can be generated by creating a localized density dip in the ergosurface with a laser. The antivortex corresponds to the negative energy particle and is absorbed inside the horizon, annihilating a unit of angular momentum. The vortex can escape from the horizon through the static limit. This system is also well adapted to observe the Zeld’ovich effect [19] and its correspondent effect in rotating geometries: superradiance via the amplification of waves scattered by a rotating obstacle. The observation of stimulated superradiance would also enable the study of other classical effects of rotating geometries, such as the apparition of light rings [61], the generation of quasi-normal modes [14,21,62] (damped oscillations emitted upon kicking the vortex) and even a classical effect of the rotating flow on the fluid akin to back-reaction [63]. Furthermore, observing spontaneous superradiance could lead to the study of both semi-classical and purely quantum effects of fields on rotating geometries, such as the rise of a grey-body factor and spontaneous back-reaction of the emitted quanta on the polariton metric.

### 6. Conclusion

In conclusion, polariton fluids in microcavities are a very promising system for analogue gravity. In particular, we have shown how using an optical defect permits the generation of various flow profiles without engineering of the cavity. We have demonstrated the creation of static one- and two-dimensional acoustic black hole horizons in polariton flows. As for the latter, a more regular system could be created with an annular-shaped pumping beam. Owing to the rather high Hawking temperature predicted in these systems, the detection of Hawking radiation should be possible from one- or two-dimensional horizons, thus enabling the first measure of entanglement in an analogue experiment. We have also created a rotating flow implying an ergoregion and an inner horizon, which opens the door to the future observation of the stimulated and spontaneous Penrose and Zeld’ovich effects as well as of regimes of instabilities in black hole systems.

### Data accessibility

The data of figures 3–5 are available in the PhD theses of T.B., E.C. and C.A. All figures are original contributions of the present paper.

### Authors' contributions

All the authors contributed equally to the manuscript.

### Competing interests

We declare we have no competing interests.

### Funding

We acknowledge the support of ANRprojects C-FLigHT (ANR-16-ACHN-0027) and QFL(ANR-16-CE30-0021). This work has received funding from the European Union’s Horizon 2020 Research and Innovationprogramme under grant agreement no. 820392 (PhoQuS). A.B. and Q.G. are members of the Institut Universitaire de France (IUF).

## Acknowledgements

The authors gratefully acknowledge discussions with I Carusotto and J Bloch.

## Footnotes

1 Unless specified otherwise, we write one- and two-dimensional in reference to the spatial dimensions of the system. There is always one temporal dimension in addition.

3 When considering excitations at all possible *k*s, the analogy with wave motion on curved space–times is more complicated but can still be established on kinematic grounds [36].

4 In the covariant notation, the upper and lower Greek indices are indices of coordinates. These indices are used for the time and space components (in this order, ranging over the indexing set {0, 1, 2}, equivalent to the traditional {*t*, *x*, *y*}). Repeated indices are automatically summed over:$y=\sum _{i=0}^{2}{c}_{i}{x}^{i}={c}_{0}{x}^{0}+{c}_{1}{x}^{1}+{c}_{2}{x}^{2}$ is simplified by convention to *y* = *c*_{i}*x*^{i} (Einstein summation convention). *η*_{μν} is the inverse of *η*^{μν} (contravariant metric).

5 In the experimental configuration of figure 4, this would be where $\sqrt{{{v}_{0}^{x}}^{2}+{{v}_{0}^{y}}^{2}}={c}_{s}$.

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