# Spectrum of collective excitations of a quantum fluid of polaritons

F. Claude  Laboratoire Kastler Brossel, Sorbonne Université, ENS-Université
PSL, Collège de France, CNRS, 4 place Jussieu, 75252 Paris Cedex 05, France
M. J. Jacquet  Laboratoire Kastler Brossel, Sorbonne Université, ENS-
Université PSL, Collège de France, CNRS, 4 place Jussieu, 75252 Paris Cedex
05, France  I. Carusotto  INO-CNR BEC Center and Dipartimento di Fisica,
Università di Trento, via Sommarive 14, I-38123 Trento, Italy  Q. Glorieux
Laboratoire Kastler Brossel, Sorbonne Université, ENS-Université PSL, Collège
de France, CNRS, 4 place Jussieu, 75252 Paris Cedex 05, France  E. Giacobino
Laboratoire Kastler Brossel, Sorbonne Université, ENS-Université PSL, Collège
de France, CNRS, 4 place Jussieu, 75252 Paris Cedex 05, France  A. Bramati
Laboratoire Kastler Brossel, Sorbonne Université, ENS-Université PSL, Collège
de France, CNRS, 4 place Jussieu, 75252 Paris Cedex 05, France

###### Abstract

We use a recently developed high-resolution coherent probe spectroscopy method
to investigate the dispersion of collective excitations of a polaritonic
quantum fluid. We measure the dispersion relation with high energy and
wavenumber resolution, which allows to determine the speed of sound in the
fluid, and to evidence the contribution of an excitonic reservoir. We report
on the generation of collective excitations at negative energies, on the ghost
branch of the dispersion curve. Precursors of dynamical instabilities are also
identified. Our methods open the way to the precise study of quantum
hydrodynamics of quantum fluids of light.

.

##  I Introduction

Quantum fluids of light, made of photons propagating in a nonlinear medium
have attracted a lot of interest in the past decade, since their evolution is
quite similar to that of atomic Bose-Einstein condensates (BEC) or superfluid
liquids [1, 2]. Both BEC and fluid of light systems can be described by
analogous equations – a generalized Gross-Pitaevski equation. Their quantum
hydrodynamic properties come from the properties of the excitation modes in
the fluid, which are described by the Bogoliubov theory [3]. Interestingly,
fluids of light are not in equilibrium, and corresponding terms have to be
added in the Gross-Pitaevski equation to account for this property. As a
result they show both standard quantum fluid properties, like superfluidity
and sonic dispersion and a rich ensemble of new behaviors revealed by their
collective excitation spectra that we study here.

The quantum fluid of light investigated in this article is a polariton fluid
that results from the strong coupling of photons and excitons in a
semiconductor microcavity.

Under the condition for cavity resonance, the component of the photon
wavevector perpendicular to the cavity $k_{\bot}$ is quantized. The frequency
dependence of photonic modes with wavevector $k$ in the cavity plane is then
$\omega=c/n\sqrt{k_{\bot}^{2}+k^{2}}\approx c/n~{}(k_{\bot}+k^{2}/2k_{\bot})$,
where $n$ is the refractive index of the cavity. Thus, the photon frequency
acquires a quadratic dependence on $k$ for the motion in the cavity plane (see
Fig. 1 (a)) – cavity photons are endowed with an effective mass $m=\hbar
k_{\bot}n/c$ . Meanwhile, excitons are bound electron-hole states created by
laser illumination, i.e. massive interacting excitations.

The strong coupling of photons and excitons generates two new eigenstates of
the system: two polariton branches with different dispersive properties, the
upper and the lower polaritons, which are shown in Fig. 1 (a).

Polaritons inherit the properties of cavity photons and excitons and thus
behave collectively (they interact via the exciton-exciton coupling) as a flow
of massive particles (their mass is a combination of the effective photon mass
with the exciton mass), that is as a quantum fluid [4]. Similarly to ensembles
of cold atoms [5, 6, 7, 8], polariton fluids exhibit Bose-Einstein
condensation [9, 10] and superfluidity [11, 12, 13, 14]. The latter
phenomenology is connected to the dispersion of collective excitations of
quantum fluids (eg Bogoliubov excitations on top of a coherent polariton
fluid) that goes from a parabolic dependence of the energy on the wavenumber
$k$ at large $k$ to a linear dependence at low $k$. While superfluidity
phenomenology has been observed over ten years ago in semiconductor
microcavities illuminated by resonant laser light, the precise measurement of
the spectrum of collective excitations has only been addressed systematically
more recently [15, 16, 17].

Due to the cavity decay rate and to the exciton finite lifetime, the
polaritons generate photoluminescence, which has been used in Refs. [15, 16]
to measure the spectrum of Bogolioubov excitations in polariton fluids. In
[16] the pump laser is far off-resonance and generates a high-energy
incoherent exciton reservoir. In turn, its relaxation toward lower energies
leads to the spontaneous generation of a polariton BEC. In this case,
additional exciton-polariton interaction channels are excited [18] and, as a
result, the shape of the spectra may differ nontrivially from that of the pure
polariton condensate in the low-energy and low-wavenumber region, yielding
additional effects. In order to concentrate on the pure polariton quantum
fluid properties, the excitation of these additional channels can be decreased
by pumping the polaritons on- or near-resonance, thus generating a polariton
fluid whose hydrodynamics are directly set by the properties of the pump, as
in [15, 17].

In these configurations, the presence of the high-density pump at low wave
number hinders the observation of the luminescence in the low-energy region of
the Bogoliubov dispersion. This can be mended by filtering out the pump
component by means of heterodyne four-wave mixing [19] or spectral and
polarisation filtering techniques [15]. This allows to access the sonic
behavior, obtain the speed of sound in the fluid and highlight the indirect
generation of a dark excitonic reservoir. Yet, the spectral resolution of the
experiments [15, 16, 19] is ultimately limited by the bandwidth of the
spectrometer, which is typically larger than the polariton linewidth [20, 21].
Therefore, the shape of the dispersion, which strongly depends on the energy
and intensity of the pump [22], may not be well resolved, potentially hiding
narrow features.

Higher resolution can be achieved by probing the Bogoliubov excitations with
an additional laser field with adjustable energy and wavenumber. This
versatile spectroscopic method, developed in Ref. [17] allows a very precise
determination of the dispersion curve in various conditions. In this paper, we
make use of this technique to perform a detailed study of the spectrum of
collective excitations on both the normal and ghost branches, and we study the
evolution of the Bogolioubov spectrum along the bistability loop of the fluid,
as the pump parameters are scanned. We also study the effects of a dark
excitonic reservoir [15] on the speed of sound in the fluid and we compare it
with the theoretical predictions. Finally, at low pump power, we observe
plateaus at the crossing of the normal and ghost branches, highlighting the
onset of dynamical instability precursors in the fluid.

##  II Theoretical background

###  II.1 Bogoliubov method

The evolution equation of the macroscopic wavefunction $\psi(\mathbf{r},t)$ of
a dilute Bose-Einstein condensate of ultracold atoms is given by the Gross-
Pitaevski equation (GPE) [4]:

$i\hbar\frac{\partial\psi}{\partial
t}=-\frac{\hbar^{2}}{2m}\nabla^{2}\psi+\hbar
g|\psi|^{2}\psi+V_{ext}(\mathbf{r})\psi,$ (1)

with $g$ the interaction constant between the particles and
$V_{ext}(\mathbf{r})$ the external trapping potential. The macroscopic
fraction of particles occupying the condensate mode behaves collectively: its
behaviour can be accurately described within the mean-field approximation,
whereby the quantum operator of the atomic matter field
$\hat{\psi}(\mathbf{r},t)$ is replaced by the classical wave
$\psi(\mathbf{r},t)=\psi_{0}(\mathbf{r})e^{-i\mu t/\hbar}$. Its phase
oscillates at the frequency set by the chemical potential $\mu$, equal to the
energy of the condensate, related in free space to the spatially uniform
density $\left\lvert\psi_{0}(\mathbf{r})\right\rvert^{2}$ via the equation of
state $\mu=\hbar gn$.

The evolution of small perturbations $\delta\psi(r,t)$ on top of the
condensate can be calculated with the Bogoliubov theory [3], which consists in
looking for the linearized response of the system by setting
$\psi_{(}\mathbf{r},t)=(\psi_{0}(\mathbf{r})+\delta\psi(\mathbf{r},t))e^{-i\mu
t/\hbar}$ in Eq.(1). This leads to the Bogoliubov dispersion relation

$\displaystyle\omega_{B}(k)$ $\displaystyle=$
$\displaystyle\pm\sqrt{\left(\frac{\hbar
k^{2}}{2m}+gn\right)^{2}-\left(gn\right)^{2}}$ (2) $\displaystyle=$
$\displaystyle\pm\sqrt{\frac{\hbar k^{2}}{2m}\left(\frac{\hbar
k^{2}}{2m}+2gn\right)},$ (3)

associating the energy $\hbar\omega_{B}(k)$ of the weak perturbation (measured
from the pump frequency $\omega_{0}$) to its wavenumber $k$. The + (-)
solution is called the positive (negative) Bogoliubov branch, or normal
(ghost) branch.

Depending on the value of $k$ compared to the inverse of the healing length
$\xi=\sqrt{\hbar/mgn}$, the collective excitations of the fluid behave in
fundamentally different ways. When $k\xi\gg 1$, the dispersion relation has
the standard single-particle parabolic shape,

$\omega_{B}(k)=\pm\left(\frac{\hbar k^{2}}{2m}+gn\right),$ (4)

while when $k\xi\ll 1$, it is linear, giving rise to a sonic-like behaviour,

$\omega_{B}(k)=\pm\ c_{s}k,$ (5)

where the sound velocity is defined as $c_{s}=\sqrt{\hbar gn/m}$.

###  II.2 Out-of-equilibrium case

The Bogoliubov approach can be extended to polariton fluids, provided that one
accounts for the driven-dissipative dynamics in semiconductor microcavities.
Indeed, in contrast to atomic condensates, polaritons are intrinsically out-
of-equilibrium systems, whose decay rate $\gamma$ has to be compensated by
continuous and quasi-resonant pumping
$\mathcal{F}(\mathbf{r},t)=F_{0}(\mathbf{r})e^{-i\omega_{0}t}$. The time-
dependent evolution of the mean-field wave function is provided by a
generalized form of the GPE written as [4]

$\displaystyle i\hbar\frac{\partial\psi_{LP}}{\partial t}$ $\displaystyle=$
$\displaystyle\left(\hbar\omega_{LP}-\frac{\hbar^{2}}{2m_{LP}}\nabla^{2}+\hbar
g|\psi_{LP}|^{2}-i\hbar\frac{\gamma}{2}\right)\psi_{LP}$ (6) $\displaystyle+$
$\displaystyle i\hbar\eta(k)F_{0},$

where $\omega_{LP}$ is the frequency of polaritons,
$\left(\hbar^{2}/2m_{LP}\right)\nabla^{2}$ gives their kinetic energy
(referred as their flow) along the cavity plane, $m_{LP}$ their effective
mass, and $\hbar g$ their interaction strength, which is positive (repulsive
interaction) in the case where the excitons are generated in a single circular
spin state $\sigma$. The coefficient $\eta(k)$, proportional to the
transmission of the optical cavity input mirror, quantifies the coupling
efficiency of the driving field to polaritons.

In our experiments, the laser excites the cavity in the vicinity of the lower
polariton branch $\omega_{0}\simeq\omega_{LP}$ (Fig. 1). Since the Rabi energy
$\hbar\Omega_{R}$ is much larger than the characteristic energies of the
system, in particular the interaction energy $\hbar gn$, it can be assumed
that the fluid contains only lower polaritons, allowing to truncate its wave
function to that of the lower branch,
$\psi(\mathbf{r},t)=\psi_{LP}(\mathbf{r},t)$, without loss of generality.

Moreover, the pump term explicitly breaks the rotational invariance of the
global phase of the condensate that prevailed in the equilibrium case.
Accordingly, the polariton fluid is not a manifestation of a spontaneous
symmetry breaking and therefore it is a very peculiar type of condensate.
Crucially, rather than being governed by the chemical potential $\mu$, the
oscillation frequency of the polariton wave function is fixed by the pump
frequency
$\psi_{LP}(\mathbf{r},t)=\psi_{LP}^{0}(\mathbf{r})e^{-i\omega_{0}t}$.
Therefore, a polariton analogue of the equation of state can be derived in
terms of the steady-state generalised GPE. In the case of a pump laser at
normal incidence, i.e. with zero wave number, the equation is

$n(\mathbf{r})\left[\left(\frac{\gamma}{2}\right)^{2}+\left(gn-\delta\right)^{2}\right]=\eta^{2}|F_{0}(\mathbf{r})|^{2},$
(7)

which relates the polariton density $n=\left\lvert\psi_{LP}\right\rvert^{2}$
to the laser intensity $I=\left\lvert F_{0}(\mathbf{r})\right\rvert^{2}$ and
the frequency detuning $\delta=\omega_{0}-\omega_{LP}$, expressed with respect
to the bare lower polariton frequency. Remarkably, when
$\delta>\sqrt{3}\gamma/2$, the system exhibits a bistable behaviour [23],
similarly to Kerr non-linear media in an optical cavities. The plot of
$n=f\left(I\right)$ in Fig. 1(b) features a hysteresis loop, where the fluid
can either be in a high or low density regime, corresponding to a high or low
interaction regime respectively. The intermediate negative slope density
branch is an unstable solution of the generalized GPE that cannot be observed
in the standard experimental steady state regime (however, see Ref [24] for
phenomenology beyond the standard case).

Figure 1: Properties of microcavity polaritons (a) Energy dispersion of the
upper (UP) and lower (LP) polaritons. (b) Bistability curve at a fixed pump
frequency, indicating the working points A, B, C. The red dot highlights the
turning point (TP) of bistability

This dependence of the polariton density on the pump parameters induces a
radical transformation of the collective excitations of polaritons. The
evolution of small perturbations on top of a stationary state fluid can be
studied by linearizing the generalized GPE with
$\psi_{LP}(\mathbf{r},t)=\left[\psi_{LP}^{0}(\mathbf{r})+\delta\psi_{LP}(\mathbf{r},t)\right]$.
In a spatially homogeneous geometry, with a pump at normal incidence with
respect to the cavity plane, this leads to the Bogoliubov dispersion relation
[4, 22]:

$\displaystyle\omega_{B}(k)$ $\displaystyle=$
$\displaystyle\pm\sqrt{\left(\frac{\hbar
k^{2}}{2m_{LP}}+2gn-\delta\right)^{2}-(gn)^{2}}$ (8) $\displaystyle-$
$\displaystyle i\gamma/2.$

where $k$ is the wave number of the perturbation. This equation is similar to
Eq. (3) for the equilibrium case, except for the additions of an extra
frequency shift term $gn-\delta$, coming from the quasi-resonant driving, and
of an imaginary term $i\gamma/2$, coming from the polariton finite lifetime.
It again exhibits two branches with opposite sign, here called the normal and
ghost branches.

When the fluid is injected at the bistability turning point (TP), $\delta=gn$
and thus the real part of the Bogoliubov dispersion relation is exactly the
same as in Eq. (3). Consequently, we recover the phenomenology of the
equilibrium condensates: when $k\xi\ll 1$, the collective excitations behave
like phonons, with a speed of sound given by

$c_{s}=\sqrt{\dfrac{\hbar gn}{m_{LP}}}=\sqrt{\dfrac{\hbar\delta}{m_{LP}}}$ (9)

and the polariton fluid may exhibit superfluidity according to the Landau
criterion, as was experimentally observed in Refs [13, 14]. On the other hand,
when $k\xi\gg 1$ the collective excitations behave like single-particles.

###  II.3 Interplay with the intensity

Away from the bistability turning point (TP), the solutions of the Bogoliubov
equation are modified as compared to the equilibrium case.

On the bistability higher branch, the excess of interaction energy over the
pump energy $gn>\delta$ results in a splitting of the normal and ghost
branches, due to an opening gap whose size increases as the intensity rises.
In addition, the dispersion curve recovers a parabolic shape: the regime of
collective excitations is lost.

On the bistability lower branch, the positive and negative branches cross each
other as the interaction energy drops to $gn<\delta/3$. This causes the energy
of Bogoliubov excitations to become purely imaginary at wavenumbers for which
the argument of the square root in Eq. (8) becomes negative:

$\delta-3gn<\dfrac{\hbar k^{2}}{2m_{LP}}<\delta-gn.$ (10)

If the imaginary part $\mathfrak{Im}(\omega_{B}(k))$ turns out to be positive,
the solutions are unstable. However, the losses $\gamma$ can stabilize the
solutions by keeping their imaginary part negative over a large intensity
range along the bistability lower branch. In this case, the collective
excitations are related to precursors of instabilities [22] and can be studied
in a steady state regime.

In the following we present an experimental investigation of the dispersion
curves and the properties of polariton fluids for these specific situations.

###  II.4 Effect of the reservoir on sound propagation

In order to properly model the behavior of the collective excitations, a dark
exciton reservoir has to be considered, as explained in Refs. [15, 25]. This
provides an additional two-body interaction channel of energy $\hbar
g_{r}n_{r}$, fed by the decay of a fraction of polaritons into non-radiative
excitons, at energies well below that of the radiative excitons contributing
to the strong coupling. Therefore, as demonstrated in [15], its density
$n_{r}$ scales proportionally to the polariton density $n$. This is the model
we will use here, and we will check this behaviour in our experiments.

The reservoir contribution is included in the generalized GPE (6) by adding
its interaction energy with polaritons. It leads to the following Bogoliubov
dispersion relation:

$\displaystyle\omega_{B}(k)$ $\displaystyle=$ (11) $\displaystyle\pm$
$\displaystyle\sqrt{\left(\frac{\hbar
k^{2}}{2m_{LP}}-\delta+g_{r}n_{r}+2gn\right)^{2}-(gn)^{2}}$ $\displaystyle-$
$\displaystyle i\dfrac{\gamma}{2}.$

Note that, in this expression, the reservoir energy is only added to the
average energy of polaritons, fixed by the laser detuning $\delta$, and not to
the intrinsic dynamics of the fluid, which is governed by the interaction
energy $gn$ between polaritons. Therefore the reservoir is assumed not to
contribute to the dynamics of collective excitations. This simplified picture
holds for excitations far from the pump frequency, as disscussed in Ref. [25].
In particular, at the bistability TP, the polariton density is now associated
to both the laser and reservoir energy via $\delta=gn+g_{r}n_{r}$.
Correspondingly, one calculates the following Bogoliubov equation, same as
before:

$\omega_{B}(k)=\pm\sqrt{\left(\frac{\hbar
k^{2}}{2m_{LP}}\right)\left(\frac{\hbar
k^{2}}{2m_{LP}}+2gn\right)}-i\gamma/2.$ (12)

At low-k one recovers two linear branches, corresponding to the onset of
phonon-like excitations, of speed of sound $c_{s}^{r}=\sqrt{\hbar gn/m_{LP}}$
which does not depend on the reservoir energy.

The only significant effect of the reservoir is to lower the renormalization
of the system energy due to the interactions between polaritons, with respect
to the laser detuning, i.e. $gn=\delta-g_{r}n_{r}$ at the TP. Accordingly, the
speed of sound expressed as a function of $\delta$ is also changed by a factor
of proportionality $\alpha$ with respect to the speed of sound of the pure
polariton fluid:

$c_{s}^{r}=\alpha\hskip
1.42271ptc_{s}=\alpha\sqrt{\frac{\hbar\delta}{m_{LP}}},$ (13)

corresponding to a blueshift contribution

$g_{r}n_{r}=(1-\alpha^{2})\hskip 1.42271pt\delta.$ (14)

We will experimentally verify this model.

##  III Experimental set-up

Our semiconductor microcavity is made of two highly reflecting planar
GaAs/AlGaAs Bragg mirrors, spaced from each other by a 2$\lambda$ thick GaAs
substrate to form a high finesse ($\mathcal{F}=3000$) optical cavity, which
embeds three InGaAs quantum wells (QW) at the antinodes of its field. It is
placed in a open-flow helium cryostat at the temperature of 4K in order to be
able to support robust QW excitons. The strong coupling results in a Rabi
energy equal to $\hbar\Omega_{R}=5.07$ meV and provides lower polaritons with
lifetime of about 15 ps, dominated by the mirror losses.

A small wedge between the two mirrors allows to continuously change the
detuning between the photons and excitons, from +8 to -4 meV, and consequently
the fraction of photons and excitons contributing to the polariton states. It
also leads to an energy gradient of the bottom of the lower polariton branch,
of about 0.7 $\mu$eV$\cdot\mu$m-1 along its axis.

Figure 2: Experimental set-up. The microcavity is placed in a cryostat at 4K.
It is illuminated by a pump laser orthogonal to the cavity. The probe laser,
modulated at 5 MHz, with an adjustable frequency and in-plane wavevector +k,
is detected either in transmission or in reflection (not shown here). When the
probe resonates with the pump polariton fluid, it is transmitted and generates
an additional 4WM signal at -k. The polariton fluid is imaged in the
reciprocal space on a DMD displaying a pinhole of tunable position, switched
between +k and -k. By selecting the +k (-k) angle, a direct (4WM) detection of
the probe transmission is operated. The pinhole output is detected by a
photodiode, connected to a spectrum analyser demodulating the signal at 5MHz.

In Fig. 2 we show the core of the experimental set-up. A cw Ti:Sapphire pump
laser with a linewidth less than $1\text{\,}\mathrm{MHz}$ illuminates the
cavity at normal incidence in order to create a 2D polariton fluid at rest in
an area of $100\mu m$, corresponding to the waist of the pump beam. Its
frequency is slightly blue detuned from the polariton resonance
($\delta>\sqrt{3}\gamma/2$) in order to operate in the bistable regime, where
different densities of the upper and lower branches of the hysteresis loop can
be excited by tuning the intensity. The phase and intensity profiles of the
pump beam are controlled with a spatial light modulator (SLM).

In order to examine the collective excitations of the fluid, a second very
weak cw probe laser beam is applied on top of the fluid with a spot of $75\mu
m$ in diameter. Its angle of incidence with respect to the cavity plane is
controlled with another SLM, displaying a blazed grating of tunable step size
to change the in-plane wavevector $k_{pr}$ of the first diffraction order sent
to the sample.

For each selected $k_{pr}$, the probe frequency $\omega_{pr}$ is scanned
around the pump frequency $\omega_{p}$ over a range of 200 GHz ($0.8$ meV),
such that only the lower branch of polaritons is excited. When the probe
resonates with one of the excitation modes of the fluid, it goes through the
sample, resulting in a peak (drop) in the cavity transmission (reflectivity)
at the corresponding wavevector and frequency ($k_{pr}$, $\omega_{pr}$),
giving direct access to the dispersion curve of collective excitations of the
fluid.

To measure such resonances, it is crucial to dissociate the probe signal from
the intense background of pump photons transmitted (reflected) by the cavity
mirrors. This is achieved by modulating the probe amplitude with an acousto-
optic modulator (AOM) before entering the cavity, at the frequency of
$f_{\mathrm{mod}}=$5\text{\,}\mathrm{MHz}$$, chosen to be significantly higher
than the spectral width of the pump laser and its low-frequency noise
bandwidth. Then, with a spectrum analyser we perform a demodulation at
$f_{\mathrm{mod}}$ of the transmitted (reflected) cavity signals focused on
photodiodes, which isolates the probe signal from the pump signal. In this
way, the spectral resolution of the measurements is only limited by the
spectral width of the probe laser, here lower than 1 MHz
($4\text{\,}\mathrm{neV}$).

Meanwhile, the angular resolution of the detection system is precisely
controlled by means of a digital micromirror device (DMD) positioned in the
Fourier plane of the cavity. This device displays a tunable radius and
position pinhole that acts as a spatial frequency filter before the detection
photodiode, providing a high $k$-resolution of $\delta
k=\pm$0.0005\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$.

##  IV Bogoliubov dispersion curves

###  IV.1 Shape of the dispersion curves

Figure 3: Bogoliubov dispersion curves (a) Energy scan of the cavity
reflectivity at fixed k = 0.40$\mu$m-1 (corresponding to the vertical red
dashed line in (b). (b) Dispersion in the linear regime (no pump laser). The
red points show the minima of the reflectivity dip for each energy scan. The
red arrow indicates the energy $\hbar\omega_{0}$ of the pump used in panels
(b, c). (c) Dispersion in the bistable nonlinear regime (point B in the
bistability curve of Fig 1.). (d) Dispersion measured close to the bistability
turning point A. The shape of the curve is changed as compared to (c) and the
gap is decreased.

We now study the dispersion relation as a function of the pump intensity
$I=|F_{p}|^{2}$, for a fixed detuning $\delta=$0.2\text{\,}\mathrm{meV}$$
between the pump and the lower polariton energy at $k=0$.

Fig. 3 (b-d) shows typical spectra obtained in reflection with the pump probe
technique, in the linear and nonlinear regimes. Each successive vertical slice
corresponds to a scan of the probe frequency at a given k, separated from each
other by $\Delta$k = 0.0189 $\pm$ 0.0005 $\mu$m-1. Fig. 3 (a) gives an example
of measured probe signal, with a dip associated to the detection of resonance.
For each slice, the energy of the reflectivity minimum gives the real part of
the energy of the Bogoliubov dispersion, highlighted by the red dots in Fig. 3
(b-d).

In the linear regime (Fig. 3 (b)), the probe scans the cavity without any
fluid injected by the pump. We recover the standard dispersion relation of
lower polaritons, parabolic over the range of wavenumbers scanned in [-0.95,
+0.95] $\mu$m-1. From this dispersion we extract the cavity parameters at the
considered working point, including the polaritons decay rate given by the
full width at half maximun (FWHM) of the reflectivity dips $\hbar\gamma=80\mu
eV$, and the polariton mass $m_{LP}$ = 5.5$\cdot$10-35 kg. The reliability of
such a measurement is confirmed by comparison with spectra measured in an out-
of-resonance excitation scheme, by detecting the photoluminescence signal of
the lower branch with a spectrometer.

To study the Bogoliubov dispersion of the polariton fluid in non linear
regime, we add the pump beam and increase its intensity. As mentioned above,
the pump laser frequency is blue detuned from the polariton resonance in order
to work in the bistable regime ($\hbar\delta=0.2$ meV). To operate on the
higher branch of the associated hysteresis loop, the laser intensity must
first be ramped up until it is above the lower turning point of the
bistability (near C in Fig. 1), where the fluid switches from the low to the
high density regime. Then, it is ramped down to address the density region
near the TP (points B and A).

The dispersion curve shown in Fig. 3 (c), corresponding to point B in Fig. 1
exhibits a blue shift that brings the bottom of the dispersion curve at $k=0$
from 1.4825 to 1.4828 meV, above the pump energy.

In Fig. 3 (d), although the laser intensity at its center is close to the
bistability turning point, corresponding to point A in Fig. 1, where the
linear dispersion should appear at low k, starting from zero detuning as
predicted by Eq. (8), the dispersion curve keeps displaying a small gap with
respect to the laser frequency and the dispersion is steeper than parabolic,
but not fully linear. These effects are investigated in the next section.

###  IV.2 Spatial dependence of the polariton density

Figure 4: Analysis of the shape of the spectrum linewidth. (a) Vertical slices
taken along the blue dashed lines at $k=0.16\mu m^{-1}$ and orange dashed
lines at $k=0.64\mu m^{-1}$ of the spectrum shown in (b). (b) Reflection map
of the probe with respect to its wavevector $k$ and energy $\hbar\omega$, at a
detuning $\hbar\delta=0.2$ meV. The horizontal red dotted line indicates the
energy of the pump $\hbar\omega_{0}$. The red points highlight the minima of
reflectivity $R_{min}$; the green points the correspond to the reflectivity
dips at $R=0.11\times\left(R_{max}-R_{min}\right)$. The red and green curves
are the corresponding theoretical fits, obtained for $\hbar gn=122\mu eV$ and
$\hbar gn=114\mu eV$ respectively. (c) Analytical dispersion curves
corresponding to concentric rings of equal width in the Gaussian pump, of
reflectivity calculated assuming a Gaussian probe. The red curve corresponds
to $\hbar gn=122\mu eV$; the linear green curve to $\hbar gn=114\mu eV$.

In Fig. 3(d) phonon-like Bogoliubov excitations were not observed because of
the Gaussian intensity profile of the laser, which creates a non-uniform
polariton density [15]. As a result, over the spatial extent excited by the
probe beam, collective excitations with different dispersions are scanned and
therefore the observed shape of the dispersion curve differs from that
expected when considering a uniform pump intensity profile. Specifically, in
the present case, gapped dispersions are excited in addition to the gapless
one, preventing the observation of a purely sonic dispersion.

In order to solve this problem we made a more detailed analysis of the data,
as shown in Fig. 4. In panel (b), the red dots highlight the minima of the
reflectivity dips in the trace of the probe energy scans for each value of
wavevector $k$ analysed. They are fitted with Eq. (11) by the red theoretical
curve that corresponds to $\hbar gn=122\mu eV$ (including the reservoir
contribution of the reservoir $\hbar g_{r}n_{r}$). Note that while the FWHM is
only 5% broader than the polariton decay rate at high-$k$, it is much broader
at low-$k$, where the Bogoliubov energy has a stronger dependence on $gn$.

To account for this behaviour, we modelled the intensity of the Gaussian pump
beam as a set of concentric rings of equal width. Each ring is associated with
a discrete $gn$ value, varying from 90 to 156 $\mu eV$, from which we
calculate a dispersion curve by using Eq. (11). These various curves are
presented in Fig. 4 (c), where the red one corresponds to the red plot fitting
the experimental data in Fig. 4 (b) ($\hbar gn=122\mu eV$). From this
analysis, we see that the dispersion curve changes significantly at low $k$:
the fluid switches from gapped parabolic modes at high $gn$ to gapless non-
parabolic modes at low $gn$. The linear dispersion highlighted in green is
obtained when $\hbar gn$ is equal to 114 $\mu$eV.

Now, in order to reproduce the experimental measurement as closely as
possible, it is crucial to weight the contribution of each dispersion of these
pump rings to the overall shape of the spectrum, i.e. to assign them a
reflectivity when the probe resonates with them. In the experiment, the
circular probe spot overlaps with the pump. When the probe has a given
frequency $\omega$ and wavevector $k$, it resonates for a specific value of
$gn$ given by Eq. (11). The corresponding ring defined above is thus
transmitted ($R=0$). In order to simplify the model, we assume that all the
other non-resonant pump rings have a unitary reflectivity ($R=1$). In
addition, we take into account the Gaussian distribution of the probe
intensity and the area of each ring, as they affect the transmitted and
reflected intensity.

Based on these considerations, it is simple to assign an intensity dependent
reflectivity $R_{gn}$ to each ring. By comparison with the experimental
spectrum in Fig 4 (b), we can thus in Fig 4 (c) associate to the red
dispersion curve a minimum reflectivity ($R_{122\mu eV}=R_{min}$), and
extrapolate the reflectivity of the green sonic dispersion curve, equal to
$R_{114\mu eV}=0.11\times\left(R_{max}-R_{min}\right)$, with respect to the
maximum $R_{max}$ and minimum $R_{min}$ of the full reflectivity map.

Remarkably, the experimental curve in Fig. 4(b) linking the points with a
reflectivity of $R=0.11\times\left(R_{max}-R_{min}\right)$ reveals a clear
linear dispersion at low k, in agreement with the model. This confirms that
one can isolate from the experimental spectrum the dispersion corresponding to
the bistability TP, calculate its interaction energy and finally extract a
speed of sound value.

###  IV.3 Speed of sound in the fluid

Thanks to the high resolution in momentum and energy of our technique, we
collected precise data on the shape of the polariton spectra depending on the
density, and we extracted the dispersion relation corresponding exactly to the
turning point of the bistability loop. This was done from the fit of the
reflectivity isoline exhibiting at low-k two linear branches, whose slope are
equal to the speed of sound. The validity of such a model was confirmed
experimentally in Ref. [17], by reshaping the probe into a narrow ring with an
SLM in order to measure the spectrum only in the regions of the fluid
corresponding to the bistability turning point.

Figure 5: Experimental measurements (dots) and fitting (solid line) of (a) the
speed of sound $c_{s}$, (b) the reservoir factor $\alpha$ and (c) the dark
exciton reservoir energy $\hbar g_{r}n_{r}$, as a function of the pump
detuning $\delta$.

We measured the speed of sound $c_{s}$ as a function of the pump detuning
$\delta$, by fitting the low-k linear region of the spectra with Eq. (12). As
shown in Fig. 5(a), we recover the square-root dependence law expected from
Eq. (13). As a result of the effect of the reservoir interactions on the
dynamics of the Bogoliubov excitations, the values of the speed of sound are
smaller than those predicted by the pure polariton interaction case in Eq.
(9). Rather, they are given by $c_{s}^{r}=\alpha\sqrt{\hbar\delta/m_{LP}}$,
with $\alpha$ the reservoir factor.

Figure 6: Dispersion relation of the exciton reservoir (a) Full reflection map
of the probe. (b) Right panel: zoom in the region of the red frame in (a).
Left panel: vertical slice along the red dashed line of the right panel. The
normal (NB) R$<$1 and ghost (GB) R$>$1 branches are observed. The additional
narrow flat branch, locally amplified at the pump energy and indicated by the
red arrow, corresponds to the dispersion relation of the exciton reservoir,
detected at the same energy as the pump. (c) Numerical calculation of the
polariton dispersion relation for the same cavity and pump parameters as the
experiment.

From the curve in Fig. 5(a), we measure $\alpha$ plotted in Fig. 5(b). The
reservoir factor is nearly constant, in agreement with the theory, with a mean
value $\left<\alpha\right>=0.59$. In addition, Fig. 5(c) shows the energy
contribution of the dark exciton reservoir $g_{r}n_{r}$ obtained from the fit
of data, which increases linearly with $\delta$ as expected, following a slope
of $1-\left<\alpha\right>^{2}=0.65$.

Thus, we obtained precise data on the polariton Bogoliubov dispersion
relations, we measured both the speed of sound and the excitonic reservoir
contribution, crucial parameters in a wide variety of experiments. Moreover,
in the next sections, we show that our technique also provides access to the
ghost branches, whose emission is usually several orders of magnitude below
the emission of the positive branches, and to a quantitative characterization
of the stability of Bogoliubov modes, via the measurement of their linewidth.

##  V Ghost branch and instabilities in the fluid

The Bogoliubov dispersion relation includes an additional component: the ghost
branch, corresponding to the negative solution $\omega_{B-}$ of Eq. (11).
Unlike the $\omega_{B+}$ branch, its inverted parabolic shape does not match
the resonance of the bare optical cavity, making its direct excitation by the
pump less efficient, and as a consequence its observation very challenging.
With our set-up, it is possible to increase the detection efficiency by
exploiting the four-wave mixing (FWM) process where two pump photons at
$\omega_{0}$ are converted into a pair of photons at $\omega_{B+}$ and
$\omega_{B-}$ via the Kerr nonlinearity of polaritons. Indeed, the probe
shining the microcavity at $\omega_{B-}$ seeds the scattering of Bogoliubov
excitations, from the pump mode toward the ghost branch.

To make this scattering processes as efficient as possible we have to increase
the density of polaritons excited in the same state by the pump laser, in
order to be as close as possible to the regime of single mode FWM. This
excitation condition can be reached by using a structured top-hat pump
profile, instead of the previous Gaussian one, to obtain a homogeneous fluid
density. This is done by shaping the pump with a second SLM, and comes at the
price of a loss in intensity of about 60$\%$ of the incident beam. Due to the
unavoidable intensity, frequency and temperature fluctuations, the top-hat
structure makes the system more sensitive to surrounding noise than the
Gaussian shape and it results in a rather unstable operation in the vicinity
of the sonic point. Therefore, it is not possible to reach exactly the sonic
point in that case, preventing the measurement of the speed of sound. However,
as we will show this provides an efficient generation of excitations in the
ghost branch.

###  V.1 Reflectivity experiments

In Fig. 6 we present the typical spectrum obtained in reflection. In panel
(a), for energies higher than that of the pump $\omega>\omega_{0}$, one can
see the normal branch as before, with a dip in the probe reflectivity. In
panel (b), for $\omega<\omega_{0}$, the ghost branch manifests itself as an
increase in the reflected intensity, above the baseline of the total
reflectivity ($R=1$). It is seeded directly by the probe at wavevector k and
energy equal to $\omega_{B-}(k)$. It is then amplified by the FWM process that
converts photons from the pump toward $\omega_{B-}(k)$ and $\omega_{B+}(-k)$.
Since the ghost branch is not resonant with the cavity for $k\neq 0$, its
intensity decreases rapidly with k, whereas it increases when the pump
intensity is set as close as possible to the turning point of the bistability,
as the phase matching conditions of the FWM become quasi-degenerate, thereby
enhancing the efficiency of the scattering processes.

In Fig. 6 (b), in addition to the aforementioned positive and negative
Bogoliubov solutions, a third spectral branch is detected, centered at the
pump energy. Its properties, namely its near-zero curvature for the chosen
wavevector scale and its very narrow linewidth compared to the standard
polariton branches, are typical of those of quantum-well excitons. The
numerical calculations based on the model proposed in [15], accounting for the
relaxation of a fraction of the polariton fluid into a reservoir of dark
excitons, and shown in Fig. 6 (c) accurately reproduce the dispersion
relations observed experimentally, and in particular the narrow spectral line
at the pump energy, with a full width at half maximum equal to the decay rate
set for the excitons in the computation. To match the experimental spectral
linewidth, the loss rate of the excitons is fixed at $\hbar\gamma_{R}$ = 1.5
$\mu$eV, corresponding to a lifetime $\tau$ = 440 ps, consistent with the
expected value. Therefore, the appearance of such a narrow peak in the
dispersion relation in the experiments is a further evidence for the presence
of a dark exciton reservoir.

Quite interestingly, the reservoir line corresponds to an increase in the
probe reflectivity. On the basis of the model developed in [25, 26], this
amplification can be understood as the result of stimulated scattering of the
pump light on the spatial modulation of the reservoir density due to the
interference between the pump and the probe itself.

###  V.2 Four wave mixing experiments

The probe can also be used as an input mode for a four wave mixing (FWM)
process generating collective excitations resonant with the normal branch at
$\omega_{B+}(k)$, which, by combining with polaritons in the pump mode at
$\omega_{0}$, lead to the population of the ghost branch mode at
$\omega_{B-}(-k)$. The photons emitted from the ghost branch can then be
detected in transmission, on the other side of the cavity, by selecting the
signal at the wavevector $-k$ opposite to the probe, with the pinhole of the
DMD placed in the reciprocal space.

To illustrate this, Fig. 7 shows the outcomes of the two different detection
schemes, direct and indirect, performed under the same experimental
conditions. In Fig. 7(a), the detection is operated in reflection, with the
DMD selecting the probe wavevector (direct detection): we recover the normal
branch in the $R<1$ regions and the seeded ghost branch in the $R>1$ regions.
Outside the low-k limit the ghost signal quickly vanishes in the probe
reflectivity baseline at $R=1$ .

In Fig. 7(b), the detection is operated in transmission, with the DMD
selecting the wavevector opposite to the probe wavevector (indirect
detection). Here, the spectrum is plotted as a function of the wavevector $k$
and energy $\hbar\omega$ of the photons originating from the FWM process: the
branch detected below the pump energy corresponds to the ghost branch,
resulting from the FWM between the pump polaritons and the probe at resonance
with the normal branch; the branch detected above the pump energy corresponds
to the normal branch, resulting from the FWM between the pump polaritons and
the probe at resonance with the ghost branch.

Interestingly, the signal obtained with this indirect detection scheme is
better resolved at large $k$ than that obtained in direct detection. This
method gives excellent access to the properties of the ghost branch. In this
case, the measurements were performed for densities in the upper part of the
bistability curve. Remarkably, as we show in the next subsection, it also
allows to investigate novel properties of the normal and ghost branches in the
lower part of the bistability curve.

Figure 7: Spectra in direct and FWM detections, at a pump detuning
$\hbar\delta=0.05$ meV. (a) Right panel: reflection map with the detection set
at the wavevector of the probe (direct detection scheme). The pump energy
$\hbar\omega_{0}$ is indicated by the horizontal red dotted line. Left panel:
slice along the vertical red dashed line of the spectrum. The reflectivity dip
R$<$1 corresponds to the normal Bogoliubov branch; the amplification peak
R$>$1 to the ghost branch. (b) Left panel: transmission map with the detection
set at the opposite wavevector of the probe (indirect detection scheme), under
the same experimental conditions as (a). k and $\hbar\omega$ correspond
respectively to the wavevenumber and energy emission of the FWM process. Right
panel: vertical slice along the vertical red dashed line of the spectrum. Two
FWM peaks are detected: the more intense one corresponds to the ghost branch
originating from the parametric scattering of the probe at resonance with the
normal branch at frequencies above the pump ($\omega>\omega_{0}$); the other,
less intense one to the normal branch, also detected via parametric scattering
but originating from the resonance of the probe with the ghost branch at
frequencies below the pump ($\omega<\omega_{0}$). The $\left\lvert
k\right\rvert<$ 0.15 $\mu$m-1 part is masked because the direct transmission
angle of the probe is close to the emission angle of the FWM, preventing the
filtering of one signal from the other with the DMD.

###  V.3 Instabilities in the fluid

When the excitation of the fluid density is such that $gn<\delta/3$ ,
condition fulfilled if the laser intensity lies on the lower branch of the
bistability, the square root argument in the right part of Eq. (8) becomes
negative at the wavevectors where the normal and the ghost branches cross each
other. Therefore, $\omega_{B}$ becomes purely imaginary. If the overall
imaginary part is positive, modulational instability modes would appear.
However, in the case of our polariton dissipative system, the losses stabilize
the fluid by maintaining $\mathfrak{Im}(\omega_{B})$ negative over a wide
range of driving field intensities. Fig. 8(a) shows the dispersion relation
reconstructed under a pump intensity as close as possible to the onset of
instability. Remarkably, two plateaus appear at $\omega_{0}$, exhibiting an
increase in the probe transmission and even the onset of some gain as a
precursor of the modulational instability.

The observation of the ghost branch in Fig. 8(b), is performed in transmission
using FWM detection, as in Fig. 7(b). Quite interestingly, note that the FWM
in Fig. 8(b) does not fulfil the $\omega\leftrightarrow-\omega$ symmetry as
only one Bogoliubov branch is clearly visible, namely the one where the
incident light drives the normal branch. This unexpected asymmetry of the FWM
intensity can be understood by noting that for a sharp-edged top-hat pump in
the proximilty of the instability threshold the density of the fluid develops
a strong spatial modulation that affects the parametric coupling underlying
the ghost branch.

The dispersion curve for the ghost branch is symmetric of the dispersion curve
for the normal branch with respect to $\omega_{0}$, and the instability
precursors emerge at the intersection of the positive and negative Bogoliubov
solutions. Here, thanks to the FWM detection, only the signal issued from the
scattering of polaritons is captured. As a consequence, the cavity resonance,
in the background of the fluid, is eliminated from the spectrum, resulting in
a better resolution of the very narrow linewidth of the plateaus, without the
wide background pedestal of the usual polariton spectral line visible in Fig.
3(a). This narrowing confirms a drastic modification of
$\mathfrak{Im}(\omega_{B})$ with respect to the standard polariton decay rate
$\gamma$, i.e. the precursor of instabilities in the fluid.

Figure 8: Detection of the precursors of dynamical instabilities for an
intensity near the point C of the bistability curve and a detuning
$\hbar\delta=0.16$ meV. (a) Right panel: transmission map of the probe in
direct detection. The zoom in the red dashed inset highlights a narrow plateau
in the spectrum, signature of the onset of precursor of modulation
instabilities at the pump energy $\hbar\omega_{0}$ indicated by the red arrow.
Left panel: slice along the vertical red dashed line of the spectrum. At the
pump energy, a narrowing and amplification of the probe is visible, related to
the change in the imaginary part of the energy of the collective excitations.
(b) Left panel: transmission map of the ghost branch obtained in indirect
detection, under the same experimental conditions as (a). $k$ and
$\hbar\omega$ are respectively the energy and wavevector of the FWM emission.
The ghost branch is detected. The plateau is clearly visible at the crossing
point with the normal branch, with a better resolution than in (a). The
$\left\lvert k\right\rvert<$ 0.25 $\mu$m-1 part is masked for the same reason
as in Fig. 7. Right panel: slice along the vertical red dashed line of the
spectrum. The sub-$\gamma$ linewidth of the resonance peak gives the imaginary
part of the energy of the modulation instabilitiy precursor.

##  VI Conclusion

We have presented an experimental investigation of several new properties of
collective excitations in a polariton quantum fluid. The high energy and
wavenumber resolution of our experiment enables the detailed study of the
shape of the dispersion curve in the regime of high fluid density and the
determination of the speed of sound when operating near the upper turning
point of the bistability. As in [15], we have observed the influence of the
intensity profile of the pump on the dispersion at low wavenumber and
evidenced the signature of a dark excitonic reservoir that modifies the
effective speed of sound in the fluid. The versatility of our methods is
illustrated by the resolution of narrow linewidth plateaus in the dispersion
in the regime of low fluid density.

We anticipate that the spectral resolution demonstrated here will enable the
investigation of other narrow features such as the excitation of Nambu-
Goldstone modes [26] in this driven-dissipative system. The study of
collective excitations in quantum fluids not only yields insight in their
quantum hydrodynamics, but also provides access to the full field theory,
which is essential for analogue quantum simulations and the observation of
quantum scattering effects such as Hawking radiation or rotational
superradiance [27, 28], which rely on the engineering of the kinematics of
collective excitations of the quantum fluid [29, 30].

We acknowledge financial support from the H2020-FETFLAG-2018-2020 project
“PhoQuS” (n.820392). MJJ and AB acknowledge support from the Île de France DIM
SIRTEQ project “FOLIAGE” (n.237992). IC acknowledges financial support from
the Provincia Autonoma di Trento and from the Q@TN initiative. QG and AB are
members of the Institut Universitaire de France.

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