Enhanced Hawking radiation in an out-of-equilibrium quantum fluid
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Enhanced Hawking radiation in an out-of-equilibrium quantum fluid
M. J. Jacquet1∗† , M. Joly1∗ , F. Claude1 , L. Giacomelli2 , E. Giacobino1 , Q. Glorieux1 , I. Carusotto2 , and A. Bramati1
1
Laboratoire Kastler Brossel,
Sorbonne Université, CNRS, ENS-Université PSL,
Collège de France, Paris 75005, France
2
INO-CNR BEC Center and Dipartimento di Fisica,
Università di Trento, via Sommarive 14,
I-38050 Povo, Trento, Italy
∗
These two authors contributed equally to the work.
†
correspondance to maxime.jacquet@lkb.upmc.fr
Spontaneous emission by the Hawking effect may be observed in quantum fluids. However, its detection is
usually rendered challenging by the weak signal strength and the short propagation length of correlated waves
on either side of the horizon. In quantum fluids of polaritons, out-of-equilibrium physics affects the dispersion
arXiv:2201.02038v1 [quant-ph] 6 Jan 2022
relation, and hence the emission and propagation of waves. Here we explore the influence of the fluid properties
on the Hawking effect and find that it may be strongly enhanced by supporting the phase and density of the fluid
upstream of the horizon in the bistable regime. This brings spontaneous emission in out-of-equilibrium systems
within experimental reach.
I. INTRODUCTION the emission of collective Bogoliubov excitations that propa-
gate in opposite directions on either side of the horizon. How-
Quantum fluctuations at the event horizon of black holes ever, to date, theoretical works on polariton analogues have
cause the emission of correlated waves by the Hawking ef- predicted a signal that appears hardly measurable because of
fect [1]: while some waves (Hawking radiation) propagate its low strength (correlations a 10−5 fraction of the fluid den-
away from the horizon to outer space, others (the partner sity) [43–45] and short propagation length (about 12 µm from
radiation) fall inside the horizon. Since signalling from in- the horizon for the experimental configuration of [11]). These
side the horizon is impossible, only Hawking radiation may limitations stem from the combination of various factors that
be detected and correlations between paired waves cannot ultimately influence the hydrodynamics of the fluid.
be measured in astrophysics. The Hawking effect may also The main difference between quantum fluids of atoms and
be observed in the laboratory thanks to analogue gravity se- of polaritons is that the latter are intrinsically out of equi-
tups [2, 3]. These are media whose properties may be engi- librium. Radiative and nonradiative dissipative processes in
neered such that waves within propagate on effectively curved microcavities must be compensated for by optical pumping,
spacetimes [4, 5], as has been experimentally demonstrated in and so the non-equilibrium state is not determined by ther-
a variety of platforms ranging from optical waves in fibres to modynamic equation conditions [46]. The radiative decay of
capillary waves in water tanks and sound waves in quantum polaritons does not solely render real-time diagnosis of the
fluids [6–18]. For example, there is a horizon for sound waves fluid properties possible (a notable experimental simplifica-
in a one-dimensional transsonic fluid (a fluid whose flow ve- tion compared with Bose-Einstein condensates of atoms), it
locity goes from being sub- to super-sonic) where the flow also is at the origin of a unique phenomenology in the col-
velocity of the fluid equals the speed of sound. The Hawking lective dynamics. Specifically, in the regime of nonlinear in-
effect at the sonic horizon yields the emission of correlated teractions of interest to horizon physics, a gap opens between
waves just like in astrophysics [4, 5], with the notable differ- the branches of the dispersion relation [47, 48]. Here we show
ence that observation on both sides of the horizon is possible. how this affects the strength of the Hawking effect and how,
Experimental evidence for correlated emission at horizons in turn, two-point correlations can be used as a diagnostic for
was recently reported in analogue gravity setups based on out-of-equilibrium effects.
classical [12, 19] and quantum fluids [20]. While the ther- In this paper, we explore the parameter space of quantum
mal fluctuations of classical fluids overpower quantum fluc- fluids of polaritons and find a regime favourable to the forma-
tuations at the horizon such that spontaneous emission can- tion of correlations by the Hawking effect. The hydrodynam-
not be observed there, this can be done with quantum fluids. ics of the fluid are controlled by its density and phase, which
Spontaneous emission would yield a non-separable state at the are in turn connected with the optical bistability of the sys-
output [21–30], whose degree of entanglement could be quan- tem (the hysteresis cycle of its polariton-density-to-optical-
tified from the density and correlation spectra [31, 32]. power relationship) [49], and so we investigate spontaneous
Although most work on spontaneous emission has been emission from this perspective. We study the influence of the
dedicated to atomic Bose-Einstein condensate (BEC) ana- regime of density on either side of the horizon on the proper-
logues [33–41], correlated emission with comparable prop- ties of emission. In doing so, we explain how effects of out-
erties may also be observed in quantum fluids of microcavity of-equilibrium in the configurations considered in [11, 44, 45]
polaritons where a sonic horizon has already been experimen- limit the emission of Bogoliubov excitations, and we show
tally realised in one- and two-dimensional flows [11, 42]. In how to engineer the fluid such that the strength and spatial
both quantum fluids, the Hawking effect manifests itself as extension of the correlation signal become amenable to exper-2
imental detection. We find that setting the density and phase modes of the homogeneous medium and construct the “global
of the fluid as close as possible to the turning point of the modes” (GMs) of the inhomogeneous system (including the
bistability aids the emission of correlated Bogoliubov excita- waterfall). These are the modes in which scattering occurs.
tions and enhances their propagation length upstream of the The waterfall separates two regions of fluid density n and
horizon. Thus, we obtain an order of magnitude increase in phase θ. To simplify the discussion, we consider regions of
both the correlation strength and length when the fluid is kept homogeneous properties whose laboratory-frame dispersion
at the sonic point of the bistability upstream of the horizon, is modelled by
and left free to evolve downstream. Fine control upon the s
working point provides us with a better understanding of the ~δk 2 ~δk 2
influence of the properties of the quantum fluid of polaritons ω± (k) = ± ∆p − − 3gn ∆ p − − gn
2m∗ 2m∗
on the propagation of Bogoliubov excitations therein as well
as on emission by the Hawking effect in systems out of equi- +vδk − iγ/2,
librium (see companion paper [50]. Our results open the way (1)
to the experimental observation of spontaneous emission from
the vacuum in polaritonic systems. ~k2
with ∆p = ωp − ω0 − 2mp∗ the effective detuning between
the pump energy ~ωp and that of polaritons ~ω0 , where kp is
~
the wave-number of the pump field. v = m ∂x θ is the flow
II. SONIC HORIZON IN A POLARITON FLUID velocity of the polariton fluid.
The laboratory frame dispersion relation (1) depends on
Our study is based on the consideration of the so-called wa- both the flow velocity and density of the fluid. In Appendix A,
terfall geometry of the density of the quantum fluid in the lab- we show that there is a hysteresis relationship between gn
oratory frame. This waterfall is illustrated in Fig. 1 in lab- and the pump intensity. We will henceforth refer to this re-
oratory frame coordinates x and t. This flow profile is re- lationship as the ‘bistability loop’. In the special case where
alised in a device called a wire, an elongated microcavity in ∆p = gn, the dispersion curve (in the rest frame of the
which the polariton dynamics are effectively one-dimensional. ∆ =gn
fluid) has a linear slope at low k: ω± p −−−→ cs k, with
We pump the microcavity with a continuous wave, coherent k→0
pump laser incident at a given angle with respect to the nor- cs := ∂ω/∂k|ω=0 the ‘speed of sound’ in the fluid. In this case,
mal to form a stationary flow along the wire. We structure cs = cB . At large k, the dispersion is that of free massive
. par-
∆p =gn 2 ∆p =gn
the light field with a step-like intensity profile (black line in ticles, ω± −−−−→ ~k /2m. There, ∂ω± ∂k >
k→∞
Fig. 1 (c)). In the region where the pump lies, the density and cs — the gradient of the dispersion curve is larger than the
phase properties of the polariton fluid are set by those of the speed of sound, so the dispersion is said to be ‘superluminal’
pump, while in the region where the pump intensity is zero, (in analogy with superluminal corrections to the dispersion
polaritons propagate ballistically [51]. As in [11, 45], we in eg [52, 53]). Because of the linear, sound-like, dispersion
consider a cavity with an attractive defect (a localised 1 µm of excitations at short k, the case ∆p = gn is referred to as
long broadening of the wire) placed after the region where the “sonic point” of the bistability loop [54]. Operation at
the pump lies. The defect at x = 0 will create a dip in the ∆p 6= gn is also possible, in which case the linear behaviour
fluid density and a spike in the flow velocity (in red in Fig. 1 at short k disappears and the dispersion is always quadratic.
(c)) because of approximate conservation of the flow current. As we will show in Section II B, this bears consequences on
We show p the speed of collective (or Bogoliubov) excitations the emission and propagation of Bogoliubov excitations in the
cB = ~gn/m∗ (~ the reduced Planck constant, g the ef- fluid. For now, we consider that ∆p = gn.
fective nonlinearity, n the fluid density, m∗ the effective mass In the configuration of Fig. 1 (a), the fluid flow is
of polaritons) in blue in Fig. 1 (c): we see that the speed is transsonic: it goes from being sub- to supersonic with a sonic
almost flat before the defect and decreases afterwards. The horizon (v = cs ) at x = −2 µm [55]. The region where
polariton dynamics are driven-dissipative: upon de-excitation the flow is subsonic is upstream of, or outside, the horizon.
(after their lifetime 1/γ), polaritons release photons that leak The region where the flow is supersonic is downstream of,
out of the cavity, enabling the direct monitoring of the density or inside, the horizon. We plot the dispersion curve in the
and phase of the fluid. laboratory frame (the real part of Eq. (1)) of the subsonic
fluid flow in Fig. 1 (d), and of the supersonic fluid flow in
Fig. 1 (f): blue (orange) curves correspond to ω+ (ω− ) solu-
A. Modes of the system tions of Eq. (1). Note that in the rest frame of the fluid, these
modes have strictly positive (negative) energies. Modes of the
In Appendix A, we review the theory for polariton hydrody- field are normalised with respect to a (Klein-Gordon) scalar
namics in a homogeneous system as described by a modified product that is not positive definite [56]: modes with positive
Gross-Pitaevskii equation. There we show how the polari- (negative) energies in the rest frame of the fluid have posi-
tons behave as a fluid and describe the dispersive properties tive (negative) norm. However, in the laboratory frame, the
of Bogoliubov excitations therein. Here we review the disper- Doppler effect modifies the shape of the dispersion relation.
sive properties of the fluid in the waterfall configuration and For subsonic fluid flows, the negative norm branch (blue) is at
the kinematics of Bogoliubov excitations therein: We find the negative laboratory frame energies. For supersonic flows, part3
(a)
Pump profile modes of positive norm and two modes of complex ω and k,
Bragg
which are exponentially growing and decaying modes. For
Horizon
mirrors ω < ωmax in the downstream region, there are four propagat-
ing modes, two of which have positive norm while the other
Subsonic flow two have negative norm. For ω > ωmax , there are two propa-
Supersonic flow
Quantum gating modes of positive norm and two exponentially growing
Upstream region
wells and decaying modes.
Downstream region
(b) (c)
Local modes in a homogeneous region may be sorted by
C their respective group velocity vg = ∂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 transsonic fluid, the global
modes (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
(d) (e) (f) 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 uin , while the
unique local mode with negative group velocity defines an out
GM uout . In the downstream region, modes with negative
group velocity define in GMs d1in and d2out and modes with
positive group velocity define out GMs d1out and d2out .[58]
GMs uin , uout , d1in and d1out are positive-norm modes while
Figure 1. Properties of the transsonic polariton fluid flow. (a) GMs d2in and d2out are negative-norm modes. Each in GM
Sketch of the system: a step-like laser field pumps polaritons, creat- describes the scattering of a plane wave to various outgoing
ing a transsonic fluid flow across an attractive obstacle. (b) Optical plane waves. Conversely, each out GM describes a single
bistability of the polariton fluid. (c) Spatial properties of the fluid plane wave resulting from the scattering of various incoming
when setting the fluid density and phase in the upstream region as waves. The scattering can be described in the in as well as
close as possible to point C of the bistability loop. Black, pump in-
the out basis, and the transformation between the two bases
tensity; red, fluid velocity; blue, speed of excitations. (d) Subsonic
dispersion exactly at point C. (e) Subsonic dispersion away from defines the scattering matrix (see [36, 56, 57, 59] for an ana-
point C on the upper branch of the bistability loop. (f) Supersonic lytical derivation of the scattering matrix). Because the vac-
dispersion of a ballistic fluid. Blue, positive-norm branch; orange, uum is basis dependent, spontaneous emission at the horizon
negative-norm branch. Circles, local modes of positive group ve- will occur in correlated pairs uout − d1out , uout − d2out and
locity; filled dots, local modes of negative group velocity. ωmin , d1out − d2out on top of the classical background formed by
lower frequency of the gaped positive-norm branch; ωmax , upper the polariton fluid (the mean-field).
frequency of the negative-norm branch.
B. Effects of out-of-equilibrium physics
of the negative norm branch (orange) is pulled up to positive
laboratory frame energies (up to a maximum energy which we So far we have discussed the dispersive properties of the
denote by ωmax ) while part of the positive norm branch (blue) fluid when ∆p = gn, when operating at the sonic point of the
is pulled down to negative laboratory frame energies. bistability loop. In the regime ∆p < gn, the microcavity acts
Now that we have described the dispersive properties of as an optical limiter [47]: as can be seen on Fig. 1 (a), the
the transsonic fluid, we consider the kinematics of Bogoli- growth of the speed of excitations cB with the pump strength
ubov excitations therein. Because of the time invariance of is sub-linear. While the fluid is stable in this regime, a gap
the system, these are plane wave modes. Eq. (1) is a fourth- opens between the positive- and negative-norm branches of
order polynomial, so there are four (positive laboratory-frame the dispersion curve, see Fig. 1 (e). We mark the bottom of
frequency) solutions to the equations of motion in each spa- the ω+ curve as ωmin .
tial region on either side of the interface. These solutions are This behaviour is markedly different from that observed
found at the intersection point of an ω = cst line with the dis- in systems close to thermal equilibrium like quantum fluids
persion branches at positive energies in Fig. 1 (d), (e) and (f). of atoms. There, the oscillation frequency of the condensate
Although these solutions share the same ω (which manifests wavefunction corresponds to the chemical potential. Instead
energy conservation in the laboratory frame), they have dis- here it corresponds to ωp — the opening of the gap illustrates
tinct k, i.e. they are local modes of the homogeneous system. how tuning ∆p gives access to a unique phenomenology of
For ω > 0 in the upstream region there are two propagating collective dynamics. Specifically, the linear behaviour at short4
k disappears as soon as the gap opens and the dispersion is al- of the horizon may be tuned by controlling the wave-number
ways quadratic (in this case cB 6= cs as there is no meaning to of the fluid in either region by means of the pump (kp,u or
cs ) so excitations are massive and elastic scattering is possi- kp,d in the up- or downstream region, respectively), see Ap-
ble, meaning that the polariton ensemble cannot be superfluid pendix A 2. On the other hand, the fluid density may be sup-
even though its flow velocity is subsonic. As we will see in ported on the higher branch of the bistability loop by means
Section III A, this departure from superfluid propagation mod- of the pump intensity. There are, roughly speaking, 6 differ-
ifies the density of the fluid in the region x < 0 as a function ent working points along the bistability loop, meaning that in
of the pump strength and profile. order to explore the full parameter space we have computed
In Section II A, we have seen that the Hawking effect con- 36 correlation spectra. Not all combinations are interesting,
sists in the mixing of in GMs of opposite sign of norm at the though, so in Fig. 2 we present four configurations that give
horizon, uin from x < 0 and d1in and d2in from x > 0. typical behaviours: row (a), the fluid density is set near (but
The downstream modes only exist over the limited interval not at) the sonic point in both regions (kp,u = 0.25 µm−1 ,
0 < ω < ωmax . When the gap between ω− and ω+ opens, kp,d = 0.55 µm−1 ), the pump strength ramps down towards
uin only exists for ω > ωmin > 0 so the frequency interval the horizon; row (b) the fluid density is set away from the
for scattering is reduced to ωmin < ω < ωmax . The curva- sonic point on the upper branch of the bistability loop in
ture of the ω+ curve also affects the condition of momentum both regions (kp,u = 0.25 µm−1 , kp,d = 0.58 µm−1 ); row
conservation between in and out modes. We will show in the (c), the fluid density is set away from the sonic point on the
simulations (cf Section III B) that the efficiency of the Hawk- upper branch of the bistability loop in the upstream region
ing effect is thus decreased. (kp,u = 0.25 µm−1 ) and left free to evolve in the downstream
In brief, when ∆p < gn, out-of-equilibrium physics mani- region; row (d), the fluid density is set near (but not at) the
fests itself in the opening of a gap between the branches of the sonic point in the upstream region (kp,u = 0.25 µm−1 ), the
dispersion relation and a modification of the shape of the dis- pump strength is set abruptly to zero at x = −7 µm and so the
persion to a purely quadratic form. This affects the generation fluid is left free to evolve from that point on (across the defect
of Bogoliubov excitations as well as their propagation in the into the downstream region).
fluid. In all configurations, the fluid builds up in the region
−10 µm < x < xd : a relatively small amplitude bump in
the density forms before the defect. This is yet another in-
III. EMISSION BY THE HAWKING EFFECT dication that the hydrodynamics in the upstream region are
not superfluid, even when forcing operation near (but not at)
We now perform calculations with the cavity parameters the sonic point. On the other hand, while the density of the
of [11]: ~γ = 0.047 meV, ~g = 0.005 meVµm, m∗ = 3 · fluid is mostly flat in the configuration of Fig. 2 (a), in that of
10−5 me . Importantly, ωp −ω0 = 0.49 meV was kept constant Fig. 2 (b) its amplitude undulates widely over 100 µm down-
throughout. stream of the horizon before flattening down. This illustrates
We study spontaneous emission via non-local correlations how attempting to force the fluid properties to a working point
in the fluid density [33], which we quantify with the nor- away from the sonic point after it has propagated across an ob-
malised spatial correlation function stacle destabilises it. Meanwhile, the fluid density smoothly
decreases when there is no pump in the downstream region.
G(2) (x, x0 )
g (2) (x, x0 ) = . (2) Given the variety of fluid properties and the possible fast
G(1) (x)G(1) (x0 ) variations within, the description of the system as two homo-
G(2) (x, x0 ) and G(1) (x) are the diagonal four-points and two- geneous media adopted in Section II and amenable to analyt-
points correlation function of the field, respectively (cf Ap- ical solutions is not always valid. Instead we must calculate
pendix B). the bistability and dispersion at all points. To this end, we
In Fig. 2 we plot the operation point on the bistability loop use the Truncated Wigner Approximation (see Appendix B)
of the fluid on either side of the horizon (solid line, upstream, to evolve the wave function and obtain the properties of the
dashed line, downstream), the pump profile (black line) and fluid at all points in the cavity as well as the dynamics of the
ensuing properties of the inhomogeneous fluid – characterised Bogoliubov excitations therein [60]. This numerical method
by its velocity (red line) and the speed of excitations cB (blue was first used in the context of analogue gravity with atomic
line) – as well as the resulting density-density correlations (2). quantum fluids [33] and was adapted to polaritonic quantum
We are interested in the fluid properties and their influence fluids in [44]. Here, it enables the study of spontaneous emis-
on correlated emission, so although we plot cB , since this is sion on highly varying backgrounds. All maps result from
proportional to square root of the fluid density we will refer to 100 000 Monte-Carlo realisations.
the density in the discussion.
B. Correlation spectra
A. Fluid configurations
In all configurations, correlations may be sorted by the spa-
We consider various flow profiles on either side of the hori- tial region in which the involved modes propagate, which cor-
zon. On the one hand, the bistability of the fluid on either side respond to four quadrants in the plots. The South East quad-5
(a)
(b)
(c)
(d)
Figure 2. Correlated emission at the horizon. Left column, bistability loop: solid black, upstream region; dashed grey, downstream region.
Middle column, solid black, pump strength upstream; dashed black, pump strength downstream; blue, speed of excitations; red, fluid flow
velocity kp,u . Right column, spatial correlation function g (2) (x, x0 ) (Eq. (2)), colour scale from −10−3 to 10−3 .
rant (x < 0, x0 < 0) corresponds to correlations in the up- (x = 0) fringes. While traces (i) and (ii) are generic fea-
stream region; the South Wast and North East quadrants corre- tures of the Hawking effect in dispersive quantum fluids, see
spond to correlations across the horizon in the up- and down- eg [33, 41, 44, 45, 56, 61], the strictly horizontal and verti-
stream regions; the North West quadrant corresponds to cor- cal correlation lines (iii) are novel features that indicate cor-
relations in the downstream region. All configurations have relations between the propagating modes uout and d2out and
some common traces, which are most visible in Fig. 2 (d): (i) a mode bound to the horizon. In the companion paper [50],
anti-correlations along the x = x0 diagonal that indicate anti- we explain that these are quasi-normal modes of the effective
bunching under repulsive polariton interactions; (ii) a nega- spacetime whose emission stems from vacuum-driven pertur-
tive moustache-shaped trace in the upstream-downstream re- bations.
gion that indicates correlations across the horizon between Configuration 2 (a) leads to a wider diagonal in the down-
Hawking radiation and its partner radiation (uout − d2out stream region than in the upstream region, and a Hawking
modes); (iii) strictly horizontal (x0 = 0) and strictly vertical moustache of amplitude 6 · 10−4 about 40 µm- and 50 µm-6
long in the up- and down regions, respectively. The dispersive In configuration (a) and (b), the spatial extension of the
features of the diagonal stick to it in both regions, indicating Hawking moustache in the downstream region is short, mean-
a local behaviour. Configuration 2 (b) also leads to a wider ing that the propagation of Bogoliubov excitations is limited
diagonal in the downstream region than in the upstream re- there. Meanwhile, the spatial extension of the Hawking mous-
gion. The Hawking moustache is of amplitude 2 · 10−4 and tache on either side of the horizon is short when the fluid is
about 10 µm- and 20 µm-long in the up- and down regions, pumped away from the sonic point and attains unprecedented
respectively. A new feature emerges in the downstream re- lengths when pumping close to the sonic point.
gion: an anti-correlation trace that begins at 20 µm, which is In brief, We have established that operating such that the
followed by a positive, local-correlations trace that starts at fluid is at the sonic point of the bistability loop upstream of
50 µm. These features follow the amplitude undulations of the horizon and letting the fluid propagate ballistically down-
the fluid density. In configurations 2 (c) and (d), the diagonal stream enhances the emission and propagation of Bogoliubov
width is constant in the upstream region and broadens in the excitations. In that regard, operating with a flat pump profile
downstream region. The Hawking moustache is of amplitude whose spatial extension is well controlled is better than with
1.3 · 10−4 and is about 20 µm-long in both regions in config- a Gaussian profile.
uration 2 (c), while it is of amplitude 7.5 · 10−4 and is about
35 µm- and 105 µm-long in the up- and downstream regions,
respectively, in configuration 2 (d). D. Influence of the fluid properties near the horizon on
In Refs [11, 45] the spatial mode of the pump was Gaus- spontaneous emission
sian and an effective cavity formed between the edge of the
pump and the attractive defect. Emission into uout would Now that we have established the crucial role of the control
form a standing wave in this cavity, thus effectively modulat- of the fluid density in the vicinity of the horizon in the emis-
ing emission at the horizon and reducing emission into d1out sion and propagation of Bogoliubov excitations, we further
and hiding it. The physics at play here is different. In all four investigate the configuration of Fig. 2 (d): the pump strength
configurations, the set of fringes that surround the diagonal in drops abruptly at a certain distance from the defect and the
the downstream region and the Hawking moustache hides cor- fluid accumulates in the region unpumped region thus created
related emission with the witness mode d1out which is weak. and propagates ballistically across the defect into the down-
In configuration 2 (b), the fringes that surround the diago- stream region.
nal are due to spontaneous emission on the spatially varying
background, an effect that goes beyond this work. Besides
this specific case, the fringes are due to the modulation of the 10 µm
Hawking effect by the quasi-normal modes of the effective 9.5 µm
spacetime [50]. Although the pump profile in [44] (where cor- 9 µm
8.5 µm
related emission into d1out was visible) was flat as well, nei- 8 µm
ther the effective cavity nor the modulation by quasi-normal 7.5 µm
7 µm
modes was observed because the defect was repulsive (see
supplemental information in [50]).
C. Spontaneous emission and propagation of Bogoliubov
excitations
In Fig. 2, we observe the influence of the regime of density x µm
of the fluid properties (working point on the bistability loop
on either side of the horizon) on spontaneous emission at the
Figure 3. Density of Bogoliubov excitations (3) in the vicinity of
horizon and propagation in either region thereafter. Emission
the defect for various pump-horizon distances. kp,u = 0.24 µm−1 .
occurs in all configurations. As the pump extends all the way
to the defect in configurations (a), (b) and (c), the dispersion
The fluid density is supported in the upstream region such
is gaped in the upstream region. On the other hand, the pump
that it dips slightly where the pump strength drops, and then
stops before the defect in configuration (d) so the fluid propa-
bumps back up. The shape of the peak in density of Bogoli-
gates ballistically to the defect and across it and the dispersion
ubov excitations follows the shape of the fluid bump. In Fig. 3,
is sound-like at short k in this case [51, 54]. Likewise, the fluid
we compute the density of Bogoliubov excitations on top of
is pumped in the downstream region in configurations (a) and
the fluid,
(b) so the dispersion is gaped there as well, while it is sound- D E D E
like there in configurations (c) and (d). Because the dispersion 2
δ Ψ̂† (x)δ Ψ̂(x) = Ψ̂† (x)Ψ̂(x) − |Ψ(x)| , (3)
is sound-like in the upstream region in configuration (a), the
frequency interval for emission at the horizon is wider in that for different distances between the pump and the horizon
configuration than in any other. This is why the amplitude of xd − |xedge |, while keeping the pump strength constant. Be-
correlations in that configuration is the highest. cause the amount of polaritons that ballistically flow into the7
gap between the pump and the defect is constant, the height haviour of a system can thus be exploited to study field theo-
of the bump decreases with the distance. As the bump height retic effects like the Hawking effect in the laboratory. Specifi-
decreases, the speed of sound decreases while the fluid ve- cally, we have found that fine control over the fluid properties
locity increases — although the flow remains subsonic there, may be achieved with a step-like pump profile.
the Mach number M := v/cs increases. For distances larger Optical bistability is ubiquitous in nonlinear optics and has
than xd − |xedge | = 10 µm, the fluid density drops from the been observed early on in semiconductor microcavities [49],
upper branch under the combined effects of diffraction and where the density of the polariton fluid may be supported on
dissipation. The density of excitations remains roughly con- the upper branch of the bistability loop. When this is done,
stant in the downstream region because the dispersion is al- it is possible to generate and thereafter enhance the (con-
ways sound-like there. On the one hand, the change in M trolled) propagation of topological excitations of the quantum
at the bump affects the generation of Bogoliubov excitations: fluid [62–64]. Here we observed the generation and propaga-
we observe that, in the upstream region, the density of Bo- tion of paired Bogoliubov excitations of the quantum fluid on
goliubov excitations increases until xd − |xedge | = 10 µm and either side of a sonic horizon when supporting the density of
drops afterwards. The ratio between the density of excitations the fluid at various points in the bistable regime. Support of
up- and downstream increases from less than three-fold for an inhomogeneous fluid density and velocity may be achieved
xd − |xedge | = 6 µm to six-fold for xd − |xedge | = 10 µm. by changing the wave-number of the pump. In an experiment,
We have verified that the shape and spatial extension of the this is easily implemented with high spatial resolution (lim-
Hawking moustache is not significantly modified by changing ited by diffraction) thanks to spatial light modulators [65]. We
xd − |xedge |. found that letting the fluid flow ballistically across a repulsive
We now we compute Eq.(3) as we vary the fluid density defect so as to form a horizon yields Hawking correlations
on top of the bump. As we see in Fig. 4, the slope of the of the order of 10−3 fraction of the fluid mean density over
speed of sound at the horizon decreases for increasing bump more than 100 µm. These are a tenfold and a four- to tenfold
heights. In other words, the horizon becomes shallower and enhancement, respectively, compared to previous results and
so the Hawking effect is less efficient and less Bogoliubov render the observation of the Hawking effect realistic. Fur-
excitations are generated. thermore, we showed how moving away from this optimal
configuration reduces the signal strength and length — two
effects which we showed are directly linked to the kinematics
of Bogoliubov excitations in the out-of-equilibrium fluid. In
this way, our work demonstrates that the correlation traces are
a diagnostic for the influence of out-of-equilibrium physics on
mode conversion in inhomogeneous flows. Specifically, we
have observed novel local correlation traces that stem from a
dissipative quench of a mode bound to the horizon that cou-
ples to propagating modes, i.e. quasi-normal modes charac-
teristic of relaxation of the effective spacetime formed by the
fluid geometry. In the companion paper [50] we study the
quantum-fluctuation-driven spacetime ring-down and its in-
x µm fluence on Hawking radiation.
Finally, our methods open the way to the theoretical and
Figure 4. Density of Bogoliubov excitations in the vicinity of the experimental study of the quantum statistics of the Hawking
defect for varying horizon steepness. kp,u = 0.24 µm−1 . effect in driven-dissipative systems: for example, one could
calculate (and observe) the Hawking correlations in recipro-
In conclusion, Bogoliubov excitations are efficiently emit- cal space [40], thus gaining frequency-resolved information
ted in the upstream region as long as the fluid density is sup- on them [41] which could in turn be used to measure the en-
ported on the upper branch of the bistability loop in the vicin- tanglement content of the correlations [31, 32].
ity of the horizon. We have shown that it is not necessary to
operate at the sonic point of the bistability loop, although this
does provide the largest enhancement of the Hawking effect. ACKNOWLEDGEMENTS
We thank Michiel Wouters for discussions on dispersion in
IV. DISCUSSION bistable fluids, Tangui Aladjidi for help with code speed-ups
as well as computer power, and Yuhao Liu for his work early
We showed how engineering the density of a quantum fluid in the project. We acknowledge financial support from the
of polaritons can enhance the emission and propagation of H2020-FETFLAG-2018-2020 project “PhoQuS” (n.820392).
paired Bogoliubov excitations in a transsonic flow. Our work IC and LG acknowledges financial support from the Provincia
sheds light on the interplay between optical bistability and Autonoma di Trento and from the Q@TN initiative. QG and
parametric amplification in fluids of light. The bistable be- AB are members of the Institut Universitaire de France.8
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[54] In the case where the pump’s intensity is zero in any spa- temporal dimension in addition.
tial region, the fluid will propagate ballistically there, as in
eg [11, 44, 45]. In this case,rthe dispersion in the unpumped
2
bal 0) ~(k−k0 )2
region will be [51] ω± = ± ~(k−k 2m 2m
+ 2gn + Appendix A: The physical system
v(k −k0 )−iγ/2, where k0 = mv/~ denotes the wave-number
of the ballistic fluid. It is the same sonic Bogoliubov dispersion In this appendix we present the field theory of polaritons
relation as for ∆p = gn. and their fundamental excitations.
[55] We remark that the ‘speed of sound’ is an ill-defined concept
Our system is a one-dimensional [66] quantum fluid of
in such highly dispersive media as our quantum fluid of polari-
tons. However, it is generally accepted that the ‘local speed of
exciton-polaritons whose flow velocity goes from being sub-
sound’ cs is given by the gradient of the tangent of the dis- to super-sonic, thus forming a sonic horizon where the local
persion relation at ω = 0 in the frame co-moving with the flow velocity of the fluid equals the local speed of sound. For
fluid. A sub-(super-)sonic flow is thus a flow for which v is simplicity, we may consider that the horizon separates two
spatial regions whose properties are independent of space —10
two homogenous regions, although we shall eventually depart continuity equations for the polariton fluid [46]:
from this simplified picture. For now, we begin with the the- √
Re Fp e−iθ
oretical description of a homogeneous quantum fluid (of its m∗ v 2 ~ ∂x2 n
∂t θ + + √ + V + gn + √ = 0,
phase and density) and of the propagation of quantum (i.e. ~ 2m∗ n n
−iθ
√
small-amplitude density) fluctuations therein. ∂t n + ∂x (nv) = γn − 2 Im Fp e n.
(A3)
1. Polariton fluid and Bogoliubov excitations The first equation of (A3) is the Euler equation of atomic
Bose-Einstein condensates (BECs) plus a term coming from
Exciton-polaritons are quasi-particles resulting from the in- the coherent pumping. The second equation of (A3) is the
teraction of light with matter in a semiconductor microcav- continuity of the flow with a loss term and a term coming from
ity. Photons emitted by a laser will be trapped in a cavity the coherent pumping. We see that the properties of the fluid
formed by two Bragg mirror, wherein their dispersion is the depend on two parameters, namely its’ density n and phase θ.
usual Fabry-Perot dispersion. These trapped photons create The spatial variations of the phase is encapsulated in v, which
excitons — bound electron-hole pairs — in the semiconduc- we identify from (A3) as the flow velocity of the fluid.
tor microcavity. Strong coupling between the photons and Now that we have described the polariton fluid in terms
excitons trapped in quantum wells gives rise to two eigen- of its’ density and phase, we consider the propagation of
states for the total Hamiltonian, known as the lower polari- small amplitude fluctuations (such as quantum fluctuations)
ton (LP) and upper polariton (UP) branches. Furthermore, of the density in this fluid — the so-called ‘Bogoliubov ex-
the Coulomb interaction between excitons results in an ef- citations’. Bogoliubov excitations are mathematically ob-
fective non-linearity for exciton-polaritons (polaritons). The tained by linearising the GPE (A1) around a background:
dynamics of the mean-field are governed by a generalised Ψ → Ψ + δΨ, and Ψ∗ → Ψ∗ + δΨ∗ . L is the ‘Bogoli-
Gross-Piteavskii equation, which leads to Euler and continuity ubov matrix’ that describes the dynamics
of the Bogoliubov
δΨ δΨ
equations describing the system as a quantum fluid. Histori- excitations (δΨ, δΨ∗ ): i∂t =L .
cally, polaritons have first been described as two-dimensional δΨ∗ δΨ∗
quasi-particles [47], although the theory may be reduced to In the steady state, the GPE (A1) becomes
one-dimensional cavities called ‘wires’ [11, 44, 45], as in the
~ 2 2 γ
present case. ω0 − ωp − ∂ + V (x) + g|Ψ(x)| − i Ψ(x)
2m∗ x 2 (A4)
In the majority of cases, all energies involved are small
compared to the Rabi splitting so the exciton-polariton sys- +Fp (x) = 0,
tem can be described by the mean field approximation [46]. where ωp is the frequency of the pump. We first consider a
At this level the system is described by a single scalar field Ψ, configuration where the wire is pumped with a spatially ho-
the field of lower polaritons, whose dynamics are governed by mogeneous and monochromatic pump of incident wavevector
the driven-dissipative Gross-Pitaevskii equation (GPE) kp (so there is no potential in Eq. (A1), V (x) = 0). The phase
of the fluid is then set by, and equal to, kp while its’ density is
~ 2 2 γ homogeneous. The steady-state GPE (A4) simplifies to
i∂t Ψ(x, t) = ω0 − ∂ + V (x) + g|Ψ(x, t)| − i Ψ
2m∗ x 2 h
2 γi
+Fp (x, t). g|Ψ| − ∆p − i Ψ + Fp = 0, (A5)
2
(A1)
where ∆p is the ‘effective detuning’ defined as the difference
ω0 is the frequency of the lower polaritons at the bottom of the between the pump energy and that of lower polaritons,
branch, m is their effective mass, V is the ‘external potential’ ~kp2
(that is controlled via the interplay of the density profile of the ∆p = ωp − ω0 − . (A6)
2m∗
pump and the cavity’s own potential), g is the effective non-
linearity, γ is the loss rate, Fp is the field of the pump laser. We go to the reference frame co-moving with the fluid via a
The field Ψ(x, t) is written in the laboratory frame. Galilean transform (x → x − vt). In the special case of a ho-
The description of the system as a fluid is supported by mogeneous system where the interaction energy matches the
the Madelung transformation: we write the field of lower po- detuning, gn = ∆p , the Bogoliubov matrix L can be written
p
laritons as Ψ(x, t) = n(x, t)eiθ(x,t) , insert this expression in this frame as
√ !
into (A1) and multiply by e−iθ n, thus obtaining gn + 2m~k2
∗ + iγ/2 gne2ikp x
L= ~k2
. (A7)
√ 1 gne−2ikp x gn + 2m ∗ − iγ/2
e−iθ i∂t neiθ = i∂t n − n∂t θ,
2 Upon diagonalization, we retrieve the Bogoliubov dispersion
√ 2 √ √
e−iθ ∂x2 neiθ = n(∂x θ) + n∂x2 n + i∂x (n∂x θ).
relation in this co-moving frame, which relates the wavenum-
(A2) ber k of Bogoliubov excitations to their frequency ω there:
s
~k 2 ~k 2
We write v = m~∗ ∂x θ and insert (A2) into (A1), so that, by ω±
∆p =gn
=± + 2gn − iγ/2. (A8)
taking the real and imaginary parts, we arrive at the Euler and 2m∗ 2m∗11
2.5
2
1.5
1
0.5
0
0 5 10
Figure 5. Dispersion curve of the polaritonic fluid in the rest
frame of the fluid. Real part of the dispersion (A8) for a pump
∆ =gn
vector kp = 0.25 µm−1 . Blue, ω+ p , positive-norm modes; or- Figure 6. Bistability loop for an homogeneous polaritonic fluid.
∆p =gn
ange, ω− , negative norm modes. Black dashed lines show the ωp − ω0 > 0 and kp = 0. Black, stable points; dashed, unstable
speed of sound. points. The system is bistable for F2 < |Fp | < F1 and follows the
hysteresis cycle (1)-(4).
Figure 5 shows the real part of Eq. (A8), the ‘dispersion
curve’, which is identical to that of atomic BECs. There are it increases abruptly (arrow (2)). For |Fp | > F1 , cs in-
∆ =gn
two branches ω± p of the dispersion, which are symmetri- creases slowly again. If the pump’s strength is decreased from
cal around the point ω = 0, k = 0. At low k, the dispersion |Fp | ≥ F1 , cs decreases slowly until |Fp | = F2 (arrow (3)),
∆ =gn where it falls abruptly (arrow (4)). Since F1 > F2 , the cs to
curve has a linear slope: ω± p −−−→ cs k, with cs ‘speed
k→0 |Fp | relationship presents a histeresis cycle with two regimes
p sound’ in the fluid. Note that, in this case, cs = cB =
of of speed of sound: the linear regime when |Fp | < F1 and
~gn/m∗ , with cB the ‘speed of excitations’ in the fluid in cs is low, and the non-linear regime when |Fp | > F2 and cs
general (including when ∆p 6= gn). At large k, the dispersion is high. This histeresis cycle is the manifestation of optical
∆ =gn
is that of free massive particles, ω± p −−−−→ ~k 2 /2m∗ . bistability [49], so we will henceforth refer to it as the ‘bista-
k→∞
∆ =gn
. bility loop’. Note that the dashed line in Fig. 6 is unstable and
There, ∂ω± p ∂k > cs — the gradient of the disper- the speed of sound will actually follow the hysteresis cycle
sion curve is larger than the speed of sound. 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
2. Optical bistability of the polariton fluid as the influence of optical bistability thereon, we generalise
Eq. (A8): We diagonalise the Bogoliubov matrix L for a ho-
Unlike the configuration considered in the previous para- mogeneous system pumped with arbitrary strength and obtain
graph, in the majority of cases the interaction energy does not s
match the effective detuning and the dispersion curve is thus ~k 2
2
modified. Furthermore, writing the density of the fluid as a ω± (k) = ± + 2gn − ∆p − (gn)2 − iγ/2.
2m∗
function of the intensity of the laser yields several solutions.
(A11)
This degeneracy of fluid densities is due to optical bistabil-
Eq. (1) is the Doppler shifted version of Eq. (A11). In Fig. 7,
ity, which, as we will show, has tremendous influence on the
we show the dispersion curve for 5 different fluid densities
emission and propagation of excitations of the fluid, including
along the bistability loop. As can be seen in Fig. 7 (a) and
Bogoliubov excitations. Here we investigate the influence of
(b), the shape of the dispersion does not change much in the
the bistability on the Bogoliubov dispersion.
linear regime: the two branches of the dispersion curve cross.
We begin by describing the relationship between the density
2 When the fluid is bistable, in Fig. 7 (b), we observe the appear-
of polaritons, n, and the intensity of the pump laser, |Fp | in ance plateaus characteristic of an unstable fluid at the crossing
the case where the energy
√ of the laser is above that of the lower points. On the other hand, the shape of the dispersion curve
polaritons, ∆p > γ 3/2: we square Eq. (A5) and find changes significantly in the nonlinear regime depending on
the position along the bistability loop: at high pump strength
γ2
2 2 (Fig. 7 (e)), the two branches are split in energy by a gap that
(gn − ∆p ) + n = |Fp | (A9)
4 increases with the pump strength. The sonic dispersion rela-
tion (A8) is recovered at point C (Fig. 7 (c)), while for slightly
or, equivalently, lower pump strength (Fig. 7 (d)), the plateau at low k is char-
" 2 # acteristic of an unstable fluid (similarly to Fig. 7 (d)) . Note
m∗ c2s γ 2 m∗ c2s 2 that the dispersion curve has a linear slope at low k (and thus a
− ∆p + = |Fp | . (A10)
~ 4 g~ sonic interpretation) at point C only, which is thus sometimes
referred to as the ‘sonic point’ of the bistability. As Eq. (A11)
The physics at play may be investigated equivalently in is of order four in k, the dispersion has four complex roots.
terms of the relationship between the speed of sound and the The real part of these roots is non-zero in the linear regime
strength of the pump, as shown in Figure 6. At first, cs in- (Fig. 7 (a) and (b)) as well as at points C and C 0 (Fig. 7 (c)
creases slowly with |Fp | (arrow (1)), until |Fp | = F1 where and d)), but not at point D (Fig. 7 (e)).You can also read
