Laboratoire Kastler Brossel (LKB)

Master 2 Lumiere, Matiere, Interactions, Sorbonne Universite

Simulation of a 1D black hole in a superfluidof polaritons

Author :M. Yuhao LIU

Supervisor :M. Alberto BRAMATIM. Maxime JACQUET

Version ofJuly, 2020

Contents

1 Introduction 1

2 Polariton in 1D cavity 12.1 Excitons in semiconductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Photons in cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Polariton in microcavity (dressed state) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3.1 Nonlinear regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3.2 Bistability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3.3 Superfluidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Truncated Wigner Approximation 8

4 Analogue gravity with 1D superfluid of polaritons 94.1 Hawking radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2 Similarity of polariton flow and curved spacetime . . . . . . . . . . . . . . . . . . . . . . . 104.3 Density and velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.4 The defect configuration of the polariton flow . . . . . . . . . . . . . . . . . . . . . . . . . 124.5 Bogoliubov excitation modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.6 Correlation detection in real space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.7 Correlation detection in momentum space . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 Conclusion 18

Appendix A: Position of the defect in the polariton flow 19

Appendix B: * 20

1 Introduction

In General Relativity, a well-known idea is the black hole, neither particles nor light can escape from itand its non returnable boundary is called the event horizon. As demonstrated by Stephen Hawking in1974 [1], the black hole also has a black-body radiation: quantum fluctuations of the vacuum scatteringat the horizon yield pairs of phonons by the Hawking effect. One phonon propagates away from thehorizon (it is called Hawking radiation) whilst its partner falls inside the horizon. The temperature ofHawking radiation is inversely proportional to the mass of the black hole: for a black hole of one solarmass has a temperature of only 60 nK which is much lower than the temperature of the cosmic mi-crowave background. Thus, Hawking radiation from black holes cannot be measured. Fortunately, BillUnruh showed in 1981 that it is possible to create effective horizons for waves in media and to observespontaneous emission by the Hawking effect in such setups [2]. This idea gave the birth of a new line ofresearch: analogue gravity, whereby laboratory systems are used to study phenomena of quantum fieldson effectively curved spacetimes [3] [4] [5].To understand the analogy between these systems, we can first imagine the motion of fish in a riverflowing towards a waterfall. The flow velocity of the river increases towards the waterfall (from up- todownstream). In this river, there exist a point where the flow velocity equals the maximal swimmingspeed of fish, that is a point of no-return for the fish. This point will separate the water flow into down-stream regime and upstream regime. Once the fish have passed the no-return point and go from upstreamto downstream, they could never come back to upstream regime which is similar as the light near to theevent horizon. Now we can replace fish by phonons in the transsonic flow. In a transsonic flow, the flowvelocity from up- to downstream and the velocity of sound wave in the flow can equal the flow velocity ata point called transsonic point. The transsonic point is a the effective horizon of phonons. The Hawkingeffect will rule the scattering of sound waves at the sonic horizon, and so incoming vacuum fluctuationswill yield pairs of phonons.In our laboratory, we want to use a transsonic polariton flow, quasi-particle from exciton-photon cou-pling, to observe the Hawking effect around the horizon. Pairs of phonons created by Hawking effect atthe horizon will cause a small modulation of polariton flow’s density. Using a density-density correlationdetection, my numerical simulation shows that it’s possible to observe this effect in this system. The ex-perimental realization of this analogue gravity can help us study the hawking effect in the laboratory. Inthis report, we follow the literature [6] [4] and study pairs of phonons in real-space. We also numericallycalculate the correlation signature of Hawking effect in momentum space. Unlike the Hawking effect in ablack hole, in which we can’t detect the inside of black hole, we can measure the Hawking radiation andits partner from the density of the polariton flow.The report is structured as follows. Firstly, I will talk about polariton physics in semiconductor micro-cavities: I will explain how photons may couple to excitons in semiconductors enclosed in cavities toform polaritons. I will then introduce the non-linear Schrodinger equation that rules the polariton flowfrom which we have a Bogoliubov dispersion of the polariton flow. In the second part, I will show howto simulate this system numerically by using the Truncated Wigner Approximation. In the last part, Iwill review you the mathematical relationship between phonons in a polariton flow and light on curvedspacetime and show how to detect the Hawking effect signal from the density-density correlation functionin real space and in momentum space.

2 Polariton in 1D cavity

To create a polariton flow, we use a semiconductor microcavity and a coherent laser beam. Boundelectron-hole particles, known as excitons, are created in semiconductor. In a quantum well with athickness of the order of a few nanometers, the momentum of the excitons will be quantized in thedirection perpendicular to the well (note the direction k‖). Set quantum wells in a pair of Bragg mirrorsforms a microcavity with strong photon-exciton coupling.

2.1 Excitons in semiconductor

The energy of electrons in typical semiconductors can be describe by the materiel’s electronic bandstructure Fig 1 shows the band structure of GaAs used in our laboratory: the valence band (red curve)is separated from the conduction band (green curve) by a bandgap of a few eV. When the semiconductorabsorbs a photon of higher energy than the band gap. This excitation will create a positive electronhole, and the corresponding electron will be excited to the conduction band. The Coulomb force fromother electrons in the semiconductor will combine the wave functions of the hole and the excited electron

1

yielding a quasi-particle called as exciton. The Hamiltonian of the exciton can be describe as [7]:

He = − ~2

2m∗c∇2c −

~2

2m∗v∇2v + Eg −

e2

ε|rc − rv|(2.1)

m∗cand m∗v present the effective mass of electron and hole(In GaAsm∗cme

=m∗vme

= 0.066 where me is theelectron mass), respectively, rc and rv the spatial coordinate of the electron and the hole, respectively,∇2i is the Laplace operator for ri(i = v, c) and Eg the energy of the gap. In the relative coordinate

r = rc − rv, He can be separated in two part, a part present plane waves of wave number k and anotherpresent the relative energy:

He = Hk +Hint, (2.2)

with

Hk = Eg +~2k2

2M,

where M =m∗c +m∗v and k = kc + kv

(2.3)

Hint = − ~2

2µ∇2r −

e2

εr, (2.4)

where µ = m∗cm2v/(m

∗c + m2

v) is the reduced mass of the system and ε is the dielectric constant of themedium.The dielectric constant of semiconductors is generally large (ε/ε0 is order of 10). The electric fieldscreening will reduce the Coulomb interaction between electron and hole, yielding a Wannier-Mott exciton[8]. And we can solve the Hint in the same way as for the Hamiltonian of hydrogen atom. So the energyand radius are

Eint(n) = −µ

meε2rRy

n2= −Rx

n2(2.5)

and

ax =e2

2Rx, (2.6)

where n is the quantum number of the state, Ry is the Rydberg constant of energy, εr = ε/ε0 is therelative permittivity, Rx is the effective Rydberg constant and ax is the effective Bohr radius.

Figure 1: Energy diagram of a GaAs gap(figure from https://en.wikipedia.org/wiki/Direct and indirect band gaps]/media/File:Direct.svg)

Two-dimensional confinement of excitons in a quantum wellIn order to steady the exciton in two dimension, we used a hetero-structures semiconductor. In oursample, we use the structure of quantum wall present in Figure 2(a). Thin layer of typical thickness

2

(a) Structure of A InGaAs quantum wall (b) The energy band of a InGaAs quantum wall

Figure 2: The structure of the quantum wall (figure from [9])

100 A is inserted between two layers with a larger gap. The breaking of translation invariant along thegrowth of the axis of the wall (axis z in fig 2(a)) causes the confinement of the electron in this direction.The movement of electron in the plan of the layer (plan perpendicular to z) remains free.This confinement change the coupling between exciton and photon. In the case of a semiconductor,translation invariant causes the conservation of the total momentum and energy of the exciton-photonsystem. An optical exciton with a given wavevector k and energy ω will couple with one mode of electronicfield only. This dependence of coupling gives rise to the degeneracy between the energy of the exciton andphoton. The result of this coupled photon-exciton, the polariton, is a stationary state in a semiconductor.In the case of a quantum well, the symmetry is broken along axis z and only the transverse componentk‖ of momentum is conserved in the interaction. The momentum of the exciton (kexc

‖ ,kexcz ) radiatively

recombine by emitting a photon with transverse momentum kph‖ = kexc

‖ , the longitudinal component kphzremains free. The exciton will couple with a continuum of modes of electromagnetic field, with a lifetimecalculated by Fermi’s golden rule—typically a exciton in a 10 nm quantum well of GaAs has a lifetimearound 10 ps [10].

2.2 Photons in cavity

In our group, the Fabry-Perot cavity is formed by a pair of Bragg mirrors made of multiple layers,alternating two materials of different refractive index nh=3.48 and nb=2.95. The thickness of each layeris λ0/4nc where λ0 is the wavelength of light for which the reflectivity is maximal and nc is the reflectiveindex of the medium in the cavity. The reflectivity R is approximately given by [11]:

R =

[nc(nh)2N − nc(nb)2N

nc(nh)2N + nc(nb)2N

]2(2.7)

where N is the number of repeated pairs of low/high refractive index material. This structure createa high reflectivity mirror over a bandwidth (the stop band) of 100 nm centered on λ0/4nc [12]. R≈1over the stop band and oscillates very quickly outside this band. The reflectivity as a function of thewavelength is shown in figure 4, a narrow hollow appears in the center of stop band, which correspondsto the resonant wavelength λ0/nc of the microcavity when it is excited at normal incidence. Outside thestop band, the reflectivity oscillates as with one Bragg mirror only.The dispersion relation of A Fabry-Perot cavity is given by:

kphz = 2πnc/λ0, (2.8)

With nc(GaAs) = 3.84. Inside the cavity, the photon propagates with an energy given by:

Ecav(kph‖ ) = ~ckphcav, (2.9)

where kphcav =∥∥kph

cav

∥∥ is

kphcav =√

(kphz )2 + (kph‖ )2, (2.10)

3

Figure 3: Reflectivity of intensity in function of the wavelength for a Bragg mirror composed by 20 pairsof Ga0.9Al0.1As/AlAs

Figure 4: Reflectivety of intensity in function of the wavelength for the cavity

Here kphz is the mode given by the quantification condition of the cavity, combine equation (2.9) andequation (2.10), the dispersion relation of the cavity can be written by:

Ecav(kph‖ ) = ~c

√2πncλ0

2

+ (kph‖ )2, (2.11)

For small kph‖ ,

Ecav(kph‖ ) =

hcncλ0

+hcλ0(kph‖ )2

2nc(2.12)

From equation (2.12) the effective mass of cavity photons near the bottom of the dispersion relation (lowk and low ω) is defined as:

1

m∗ph=

1

~2d2Ecav(k

ph‖ )

d(kph‖ )2

∣∣∣∣∣kph‖ =0

(2.13)

So we have:

m∗ph =nch

λ0c(2.14)

Thereby a photon in the cavity have a non zero effective mass. This mass is 104 times less than theeffective mass of exciton( the dispersion of an exciton is flat at small k‖ ).

4

2.3 Polariton in microcavity (dressed state)

The cavity photons interact with the active medium, here InGaAs quantum wells placed at the anti-nodesof the GaAs microcavity electromagnetic field. For small value of k‖, the light-matter coupling can bedescribed by the Hamiltonian:

H = ~ω(k‖)a†a + Eexcb

†b +~ΩR

2(a†b + b†a) (2.15)

where a(b) denote the annihilation operator of a photon (an exciton). The third term represent thelight-matter interaction defined by the Rabi frequency ΩR, which is much larger than the photon andexciton decay rate and cause the strong coupling regime. The eigenstates are linear superpositions ofphoton and exciton states, known as cavity polaritons. Their decay rate is the weighted average of thecavity and exciton decay rates. The two eigenstates, called upper and lower polariton branches, areseparated by an energy of the order of the Rabi splitting (~ΩR) shown in figure 5. The dispersion curveof the polariton depends on the dispersion of photon and the exciton and in particular, the effective massof the lower polariton curve is comparable to the photon’s. In the following , we will only discuss lowerpolariton branch.

Figure 5: Dispersion relation of upper and lower polariton branches (taken by Anne Maitre)

2.3.1 Nonlinear regime

The effective Coulomb interaction between excitons is negligible compared to the Rabi splitting, and thisinteraction can be projected on the polariton basis, giving us an effective polariton-polariton interactionrepresented by a Kerr-like nonlinear term. So we can use a nonlinear Schrodinger equation to describethe evolution of the lower polariton wavefunction:

i~∂ψLP∂t

= (~ωLP +~2

2m∗∇2 + ~g|ψLP |2 − i~γ)ψLP + i~ηF0 (2.16)

The first term on the right-hand side represents the ground energy of the lower polariton state ~ωLP ; thesecond term is the kinetic energy in the cavity plane, with m∗ the effective mass for the lower polaritonbranch; the third term is the nonlinear interaction ~g, which is positive, meaning that the interaction isrepulsive; the fourth and fifth terms are, respectively, the losses due to the polariton decay rate and thepolariton source from the pump laser field of amplitude ~F0 which generates polaritons with efficiency η.Using the mean field theory, we can treat the polariton-polariton interaction as a small perturbation ofdensity and linearize the non-linear Schrodinger equation. Because of dissipation and pumping, thereare also quantum fluctuations, which are not included in the equations for the simulation of many-bodyquantum fluids. However these quantum fluctuations can be included in the model and do not preventfrom reaching the single or few particles quantum regime for simulating other systems. We will discussit precisely in the next section. [13]

5

2.3.2 Bistability

In a dynamical system, bistability means the system has two stable equilibrium states. In our case, Whenthe system is pumped by a quasi-resonant laser in a nearly pane wave geometry, the wavefunction of thepolariton has the same frequency as the input laser pump:

ψLP = ψ0eik‖re−iω0t, (2.17)

where ω0 and k‖ are the frequency and wave number of the laser pump in the cavity plane. The solutionof the polariton wavefunction is obtained by relating equation (2.17) and equation (2.16):

(ω0 − ωLP −~

2m∗k2‖ − g|ψ0|2 + iγ)ψ0 = iηF0. (2.18)

The mean number of polaritons is obtained by multiplying equation (2.18) by its complex conjugate:

n[(δ + gn)2 + γ2] = η2I0, (2.19)

where δ = −ω0 +ωLP + ~2m∗ k

2‖, n = |ψ0|2 and I0 = |F0|2. The equation (2.19) shows that there exist two

stable point of the polariton density n as function of the pump power I0. This bistability is combined byhysteresis, the next state of system depends on the previous state. The bistable behavior is obtained forδ < 0 and δ2 > 3γ, in this case we can define the turning points by:

gn = −2

3± 1

3

√δ2 − 3γ2 (2.20)

We can see in figure 6, the top-left turning point is the the point correspond to gn = −δ(if δ γ). Thispoint is the working point of the system (point B in figure 6). In the experiment, we will increase thepump intensity to higher than the intensity of the bottom right turning point, than decrease the intensityto reach the working point in order to obtain a stable relative high polariton density profile.

Figure 6: Bistability of mean polariton density (figure from [9])

2.3.3 Superfluidity

The evolution of the lower polariton wavefunction described by the non-linear Schrodinger equation (2.16)is similar to the Gross-Pitaevskii equation which describe the Bose-Einstein condensates or quantum fluid

6

at zero temperature. This kind of evolution gives polariton the opportunity to simulate these systems.But the last two term in equation (2.16) which represent the losses of polariton density cause this systemto a driven-dissipative system and give a different nature from conservative systems.Similar to their conservative counterpart, polariton fluids were predicted to show collective phenomenasuch as Bose Einstein condensation or superfluidity, and these properties have been observed by severalgroups in the past years. [14–16]. The superfluidity of the polariton flow is studied via the Landau criterionfor a superfluid in conservative system. Using the Bogoliubov method, the properties of superfluid canbe studied by investigating the evolution of small perturbations. First we can linear the Gross-Pitaevskiiequation in the vicinity of the working point:

ψLP (r, t) = ψ0LP (r, t) + δψ(r, t) (2.21)

In the case of a coherent pump with a frequency ω0 and a wave number in the cavity plane k‖ = 0. Sothe steady state of the system is ψ0

LP = ψ0eiω0t and δψ(r, t) is given by:

i~∂

∂t

(δψ(r, t)δψ∗(r, t)

)= ~LBog

(δψ(r, t)δψ∗(r, t)

)(2.22)

The Bogoliubov operator LBog can be written as:

LBog =

(~k22m∗ + ∆ + 2gn− iγ gn

−gn − ~k22m∗ −∆− 2gn− iγ

), (2.23)

where k is the wave number of the perturbation, ∆ = ωLP − ω0 is the detuning of the laser pump, andn = |ψ0|2 is the polariton density. For a detuning ∆ = −gn which correspond the working point of thebistability curve in figure 6, the eigenenergy of the Bogoliubov operator is the Bogoliubov dispersion forBogoliubov excitation:

ωB(k) = ±√

~k22m∗

(~k22m∗

+ 2gn)− iγ. (2.24)

The behaviour of the system depends on the ratio between the wavelength of the perturbation and thehealing length ξ =

√~/m∗gn (the shortest distance over which the wavefunction can change). Now we

focus to the real part. For small wave number k 1/ξ, a sound-like dispersion appear:

ωB(k) = ±csk (2.25)

where the speed of sound is defined by cs =√

~gn/m∗. For k 1/ξ, the dispersion is parabolic:

ωB(k) = ±(~k2

2m∗+ gn) (2.26)

If we have a moving polariton flow (k0 6= 0) with velocity v0, the dispersion relation in the laboratoryreference frame should be rewritten to:

ωB(k) = ∆k · v0 ±√

~∆k2

2m∗(~∆k2

2m∗+ 2gn) (2.27)

where ∆k = k− k0 and the detuning for the working point of bistability should be changed to ∆ =

ωLP +~k202m∗ − ω0 = −gn.

The polariton density n will influence the polariton-polariton interaction. The superfluid propriety isexperimentally studied by the scattering of the polariton flow in the vicinity of the defect. When thepolariton density is low, the interaction is negligible, polaritons are elastically scattered by the defect,generating cylindrical waves propagating away ( figure 7(a) ). When the laser intensity is increased upto the value where gn ∼ ∆, polariton-polariton interactions increase and cause a dramatic change in thepolariton dispersion. For subsonic regime (v0 < cs), elastic scattering is not possible thus fulfilling theLandau criterion. The interference fringes disappear, demonstrating the emergence of a friction-less flowcharacteristic of the superfluid regime ( figure 7(b) ). These effects have been studied in details in theLKB group [17–30].

7

(a) Low density (b) High density

Figure 7: Superfluidity of polariton flow. Left figure shows the flow scatter at a defect in the low densitycase and right figure shows the superfludity of the flow at the defect, fringes disappear, in the high densitycase (Figures from [14])

3 Truncated Wigner Approximation

In order to solve the non-linear Schrodinger equation for the polariton flow numerically, we will use theTruncated Wigner Approximation (TWA).The exact numerical solution of polaritons flow can only be obtained for few particles. For a large many-body system, we need to use the mean-field theory (MFT) to simplify the system. But the quantumfluctuation is not included in MFT. When we want to go beyond the mean-field theory, here we can usethe Truncated Wigner Approximation (TWA) which include the quantum fluctuation of the polaritondensity.In TWA, we describe the polariton flow by a Wigner quasi-probability distribution, introduced in quantumoptic, the non-linear Schrodinger equation can be treated as a Langevin equation, a stochastic differentialequation, which can be solved numerically. In our case for a polariton flow, thanks to the presence of theloss and the pump, the equation can be truncated into the form [31]:

i~dψLP (x, t) =

(~ωLP +

~2k2

2m∗+ ~g(|ψLP (x)|2 − 1

∆x)− i~γ

)ψLP (x)dt+ i~ηF0dt+

~√4∆x

√γdW (x)

(3.1)where ∆x is the spatial grid of the simulation which should satisfy the validity condition g γ∆x.Physically the interaction between polaritons should be weak enough for TWA, so a single polariton ona grid cell should not have significant impact on Wigner distribution of the system. This idea give usthe validity condition of the numerical simulation. dW (x) are zero-mean independent complex stochasticnoises introduced to take account the quantum fluctuation of the field with a form:

dW (x)dW (x′) = 0

dW (x)dW ∗(x′) = 2dtδx,x′ ,(3.2)

Where overline represent the average value and δx,x′ is the Kronecker delta function (equal to 1 only ifx = x′) . In the numerical simulation, the Wigner distribution at a certain time t is the average of alarge number N (105 in my simulation) of independent realizations. First, we evaluate the system usingequation (3.1) with different stochastic noises for a few hundred of realizations (200 realizations represent1ns evolution in my simulation) to reach a steady state. Then we measure the Wigner distributionperiodically with a period Ts (5ps in my case) and calculate the density of the flow by average allrealizations. Inspired from previous numerical studies of acoustic black holes in atomic condensates byCarusotto [32], we shall look for the signature of analog Hawking radiation in the normalized, zero-delaycorrelation of the cavity field intensity defined as:

g(2)(x, x′) =

⟨ψ†(x)ψ†(x′)ψ(x′)ψ(x)

⟩〈ψ†(x)ψ(x)〉〈ψ†(x′)ψ(x′)〉

(3.3)

With the Wigner formalization, different terms of equation (3.3) have the form [31]:

G(1)(x) = 〈ψ†(x)ψ(x)〉 = 〈|ψ(x)|2〉W −1

2∆x(3.4)

8

and

G(2)(x, x′) = 〈ψ†(x)ψ†(x′)ψ(x′)ψ(x)〉

= 〈|ψ(x)|2|ψ(x′)|2〉W +1

4∆x2(1 + δx,x′)−

1

2δx(1 + δx,x′)〈|ψ(x)|2 + |ψ(x′)|2〉W ,

(3.5)

where the 〈...〉W averages indicate the arithmetic mean of wave function from N different configurations.To verify this correlation method, I tested the program with a zero pump input where I set the pumppower to zero, so there is no polariton in the cavity (shown in figure 8). In this case the correlationfunction should be 0 in the entire space. As we said, we use a Monte-Carlo simulation, so the number

Figure 8: Non-normalized correlation function G(2)(x, x′) (eq (3.5)) of a zero pump input, the cavity isnot pumped (xhorizon will be explained in the next section)

of realizations should be large enough to have a physical result. From figure 9 we can see, only when wehave a order of 6× 104, the average data turns stable. That’s why we use N = 105 in the simulation.

1 2 3 4 5 6 7 8 9 10

104

-10

-8

-6

-4

-2

0

10-5

Figure 9: Normalized correlation function g(2)(x, x′) − 1 (eq (3.3)) for a given point (x0, x′0) in the

correlation map

4 Analogue gravity with 1D superfluid of polaritons

Analogue gravity is a research program that studies (quantum) fields on effectively curved spacetime inthe laboratory. Thus, effect connected with the kinematics of fields on a curved spacetime described by

9

general relativity may be studied using other physical system such as sound waves in fluids.The analogy relies on the fact that the wave equation for sound waves (Bogoliubov excitations) in a fluidflow is strict isomorphic to the equation of small amplitude excitations of a massless field on a curvedspacetime.

In this section, we will simulate the hydrodynamics of polaritons in a 1-dimension wire and show howsuch a flow create an effective horizon. We will then study spontaneous emission from the vacuum atthis horizon— the Hawking radiation.

4.1 Hawking radiation

In general relativity, a black hole is a region where the gravitational field is so strong that nothing (neitherparticle nor light) can escape from it. The boundary of the black hole is a surface called the event horizon.The escape velocity from the horizon outwards (away from the black hole) is the speed of the light. In1974, Hawking developed a theory to explain the evaporation of static black holes. Quantum fluctuationof the electromagnetic field at the event horizon will yield a pair of photons, one can escapes from theblack hole called Hawking radiation while its partner will fall inside the horizon [1]. This phenomenongives static black holes a thermal temperature but the temperature of Hawking radiation of a solar massblack hole is only 60nK which is much lower than the cosmic microwave background and is hard todetect. Analogue gravity allows us to study the Hawking effect in situations in which the signal to noiseratio is more favorable to detection.

4.2 Similarity of polariton flow and curved spacetime

To demonstrate the similarity between this two system, we need to rewrite the wavefunction of polaritonas a fluid flow. Using the Madelung transformation, with ψLP =

√ρeiφ (φ being the phase of the field),

the nonlinear Schrodinger equation (2.16) can be transformed into fluid-like equations: the hydrodynamiccontinuity and Euler equations

∂ρ

∂t+∇(ρv) = 0,

∂φ

∂t+m∗

2~v2 + gρ+

~2m∗√ρ∇2(√ρ) = 0,

(4.1)

where the flow velocity of the polaritons is v = ~m∗∇φ. The wave equation of Bogoliubov excitation

(sound wave at low k) in the flow is obtained by linearizing equation (4.1) around a steady state by theBogoliubov method (see equation (2.21)) and neglecting the quantum pressure:

ρ0cs

((c2s − v2)∂x2 − ∂t2 − 2v∂t∂x)δψ = 0 (4.2)

with δψ = δρρ , cs =

√~gρ/m∗ the local speed of sound. Equation (4.2) is isomorphic to the d’Alembertian

∆ψ1 ≡ 1√−η∂(µ)(

√ηηµν∂νδψ) = 0 of scalar waves on a 1+1D spacetime of metric ηµν

ds2 ≡ ηµν∂µ∂ν =ρ0cs

(−(c2s − v2)dt2 + dx2 − 2vdtdx). (4.3)

In equation (4.3), when cs = v, a horizon appears in the spacetime. This condition naturally occurswhen the polariton flow pumped by a Gaussian beam, because the density of polariton will decrease awayfrom the center of the pump. When the density of polariton decrease, cs, proportional to square root ofdensity, will decrease but inversely v will increase due to the interaction energy being converted to kineticenergy. If at the pump center cs > v (subsonic flow), there will exist a point in the cavity where cs = vand, after that point, v > cs (supersonic flow). Spontaneous emission of phonon pairs from the vacuumis expected to occur at sonic horizons in polariton by the Hawking effect. [33]

4.3 Density and velocity

We create a horizon by pumping a 1-dimensional wire with a Gaussian beam. Figure 10 shows the processto create the polariton flow, I use a 17 um width Gaussian beam centred at 87 um pump in a 200 umlength microcavity. We choose a pump frequency detuning from the bottom of upper polariton dispersion(see figure 5) ~∆ = 0.5meV and a small angle of the pump kp = 0.3µm−1 to give a prefers direction topropagation of the flow. Figure 11(a) show the simulation of density profile of the polariton flow in thecavity calculated by TWA, where n is the density of the polariton flow in unit of µm−1. Figure 11(b)

10

is the velocity profile of this polariton flow calculated by equations (4.1) and (4.2). The velocity of thepolariton flow is proportional to the gradient of the phase of the field, but at the edge of the cavity,the density is same order of magnitude as stochastic noises. In this area, the velocity of the flow losesphysical meaning, so in the rest of this report, we will only focus on the regime near to the pump centeras shown in figure 12

Figure 10: Scheme of the purely ballistic flow configuration to create a flowing polariton superfluid withan acoustic horizon (Figure from [6])

0 20 40 60 80 100 120 140 160 180 2000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Density

(a) Density profile of polariton flow

0 20 40 60 80 100 120 140 160 180 200

-30

-25

-20

-15

-10

-5

0

5

10

15

20 v

cs

(b) Velocity profile of polariton flow

Figure 11: Density and velocity profile of the polariton flow pumped by a Gaussian beam in the micro-cavity calculated by eq (4.1). In figure 11(b) blue curve represent the velocity of the flow, orange curverepresent the speed of sound.

From figure 12 we can see that at the right side of the pump when the flow flow away from the pumpcenter, v increases and c decreases. There is a cross of velocities at x = 109µm. This point separatestwo regimes of polariton flow: for x < 109µm, v < cs so the flow is subsonic; for x > 109µm, v > cs sothe flow is supersonic. In analogy with wave motion near a black hole, we identify xhor = 109µm as aneffective black hole horizon.

11

50 60 70 80 90 100 110 120 130 140 150

-5

0

5

10

15

v

cs

Figure 12: Velocity profile of polariton flow near to the pump center where blue and orange curve representthe velocity of the flow and the speed of sound

4.4 The defect configuration of the polariton flow

The Hawking temperature TH is the radiative temperature of the Hawking radiation. It’s determined bythe surface gravity [5] [3]

κ ≡ 1

2cs(x)

d

dx

(v2(x)− c2s(x)

)∣∣∣∣x=xhor

, (4.4)

according to TH = ~κ/kB with kB the Boltzmann constant. The hawking temperature is proportionalto the gradient of the flow density at the horizon and inverse proportional to the decay rate γ. For theconfiguration figure 11(a), TH ≈ 0.1K is limited by the decay rate γ. In figure 12, the significant variationof flow density at two side of the horizon cause a problem for the detection of Hawking radiation signal,which is very low near to the horizon.One way to overcome this difficulty is insert an optical defect at the edge of the pump [6]. This defectwill create a narrow repulsive potential and ’absorb’ the flow density. A simulation of a flow profile witha defect is shown in figure 13, this method can create a sharp edge in the flow density at the horizoncorresponds to sudden increase of the flow velocity. The flow density near to the horizon is flatter thanthe ballistic flow configuration one. And TH ≈ 10K so we have a factor of 100 compared to the ballisticflow. The feasibility of such a flow access a sonic horizon was demonstrated in Bloch’s group in 2015. [4]In their experiment, they measured the flow velocity and dispersion of Bogoliubov wave in both the up-and downstream regime. Their data very well matched the theoretical predictions obtained by TWA.

0 20 40 60 80 100 120 140 160 180 2000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Density

(a) Density profile of polariton flow

50 60 70 80 90 100 110 120 130 140 150

-5

0

5

10

15

v

cs

(b) Velocity profile of polariton flow

Figure 13: Density and velocity profile of polariton flow with a optical defect in the cavity at 120 µmwhere blue and orange curve represent the velocity of the flow and the speed of sound

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4.5 Bogoliubov excitation modes

The dispersion (2.26) of Bogoliubov excitation has two branches shown in figure 14, one has a positivenorm frequency correspond to create operator of field a, and another has a negative norm frequencycorrespond to annihilation operator of field a† in fluid frame. In laboratory frame, the Doppler effectfrom the flow velocity will changed the dispersion (2.26) into equation (2.27). The spatial dependence ofthe velocity change the dispersion in the two sides of horizon (figure 15). In upstream subsonic regime,in figure 15, for a giving detuning, there are two modes satisfy the Bogoliubov dispersion. These modeswith positive frequency correspond to the propagating plane waves. For different direction of propagation(group velocity vg = ∂ω/∂k), we note in-going (’in’) if their group velocity points toward the horizon andout-going (’out’) in the opposite case.In downstream supersonic regime, we use the similar notation. In this case, if the modes of negative normbranch have an energy which allow the energy blow the maximum of negative branch dispersion, we canhave a propagating plane wave. The positive norm branch mode called d1 and the negative norm branchmode called d2. The pairs of phonon in modes uout and d2out propagate outward from the horizon withdifferent sign of norm are the phonons of Hawking effect.

Figure 14: Dispersion relation of Bogoliubov excitations eq (2.26) in fluid frame where blue curves arepositive norm Bogoliubov dispersion, red dash curves are negative norm Bogoliubov dispersion.

13

Figure 15: Dispersion relation eq (2.27) of polariton flow in both up and down stream regime in laboratoryframe, blue curves are positive norm Bogoliubov dispersion, red dash curves are negative norm Bogoliubovdispersion.

4.6 Correlation detection in real space

The signal of spontaneous phonon emission is broadband and low density, so the direct detection ofthe Hawking effect from the polaritons flow density is very difficult. In section 3 we discussed thecorrelation function g(2)(x, x′) of the Wigner distribution ψ(x). This method can help us detect Hawkingradiation due to the entanglement between pairs of spontaneous phonons. Figure 16 shows the numerically

(a) Correlation function of polariton flow without defect (b) Correlation function of polariton flow with a defect

Figure 16: Second order correlation function of polariton flow in real space g(2)(x, x′)− 1

simulated real-space density-density correlation (3.3) for a transsonic flow. As the coherent pump beamcreate a base g(2)(x, x′) = 1, we plot g(2)(x, x′)− 1. In figure 16(a), the flow is ballistic. In figure 16(a),polariton flow across a defect at x = 120µm. There is a strong anti-correlation line along the diagonal,which thins out over the width of the pump beam. Correlations are much stronger and more featurefulwhen there is a defect (figure 16(b)).Now we focus on features near to the horizon. As original predicted in [34], only pairs of spontaneousphonon resonant with Bogoliubov excitation outgoing modes can be observed in the correlation map(uout − d1out and uout − d2out), while the correlation signal of uout − d2out corresponds to the Hawking

14

radiation has an order of g(2) ≈ 10−5 which is no far from analytical prediction. When pairs of phononsare emitted on either side of horizon, they will propagate away from the horizon in Bogoliubov modes.Their relative position x and x’ across the horizon is fixed by their relative group velocity. For x−xhor > 0and x′ − xhor < 0, the correlation is peaked along the half-line

x− xhorvd − cd

=x′ − xhorvu − cu

, (4.5)

where vu,d and cu,d are the velocity of flow and speed of sound in up/downstream regime , so thatvd − cd and vu − cu represent the group velocity of d2out and uout modes for a detuning near to 0. Aswe said before the spontaneous emission is boardband in frequency, so these correlation feature willhave a bandwidth in correlation pattern. The correlation trace of phonon pairs may be calculated byapproximately integrating their group velocity: [4]∫ x

0

xd − xhorvd − cd

dxd =

∫ x′

0

xu − xhorvu − cu

dxu. (4.6)

The correlation traces of uout − d2out, uout − d1out and d2out − d1out are shown in dashed white (andblue), orange and blue in figure 17. These are not straight lines because the group velocity of all thesemodes depends on frequency due to dispersion. The anti-correlation feature at low x, x′ matches well thed2out − uout trace: this is the Hawking signal. This feature is of short extent compared with, e.g. thed1out − d2out feature. This is because uout propagates against the flow in the upstream region (whereasboth d1out and d2out propagate along the flow in the downstream region).

Figure 17: Correlation map with theoretical predictions. White dashed line is the linear prediction ofuout − d2out, blue dashed curve is the prediction of uout − d2out far from the horizon, orange dash curveis the prediction of uout − d1out and red dash curve is the prediction of d1out − d2out

4.7 Correlation detection in momentum space

The driven dissipative nature of the polariton flow can give this system a non-zero thermal temperaturebase. The thermal noise can stimulate emissions at the horizon too, which will mix with the spontaneousHawking effect. To filter thermal noises, we can detect the correlation signal in momentum space becausethe pattern of thermal noise in momentum space has a different frequency nature from the Hawkingeffect. [35] The formula for second order correlations in momentum space is very similar to equation(3.3); we will use the Wigner distribution in momentum space ψ(k) the Fourier transformation of ψ(x) :

ψ(k) =

∫ +∞

−∞

ψ(x)√Le−ikxdx, (4.7)

where L the length of the 1D wire. The wavefunction divided by√L is the normalized wavefunction

in momentum space and the ∆x in equation (3.3) can be chosen as 1 to simplify the calculation. Iperformed the same stress test as for real space and obtain the G2(k, k′) ≈ 0 everywhere for a zero pump

15

Figure 18: Non-normalized correlation function G(2)(k, k′) of a zero pump input

(a) Correlation function of polariton flow in momentumspace

(b) Correlation function of polariton flow zoom on theregion near to 0 with predictions

Figure 19: Second order correlation function of polariton flow in momentum space g(2)(k, k′)− 1

16

input, see figure 18. The numerical result of polariton flow correlation in momentum space is shown infigure 19. There is a lot of high frequency high amplitude noise at high which we can ignore becausefrom the Bogoliubov dispersion figure 15, the d2out mode (Hawking radiation) has a maximum wavenumber around 1µm−1. So we can focus on the zone near k = k′ = 0 as shown in figure 19(b). Beforeinterpreting the correlation map, I made some prediction for different correlation signals: orange blueand red curves indicate where the uout − d2out ,uout − d2out and d1out − d2out correlation traces shouldbe. These curves comes form the Bogoliubov dispersion (2.27) (figure 15). The spontaneous Hawkingradiation is boardband in frequency, so we scan the detuning throw the dispersion relation which meansall possible frequencies from dispersion. Because of the energy conservation condition, we have pairedemission for k and k’ given by the dispersion relation 2.27 at constant omega. (e.g. red horizontal line infigure 15). But unfortunately, we haven’t observed a correlation following the predicted trace for kandk′

near to 0. Around k = 0.5, k′ = −0.5 we can see a little feature match this trace and sudden to the noiseregion.To better distinguish different features in momentum correlation pattern, we can use a windowed FourierTransformation which separately calculate Fourier Transformation in up and down stream region. Thismethod is used in Bose-Einstein Condensate (BEC) [35] [36] [37] where they choose a specific windowfunction in both up and downstream region

Πi(x) = e− (x−Xi)

2

σ2i , (4.8)

and calculate the wavefunction of up and downstream in momentum space separately:

ψi(k) =

∫ +∞

−∞ψ(x)Πi(x)e−ikxdx, (4.9)

where i=u,d denote up and downstream region, Xi and σi are the position and width of the window andshould obey

Xu

∂ωuout/∂k=

Xd

∂ωd2out/∂k′ , and

σu|∂ωuout/∂k|

=σd

∂ωd2out/∂k′ (4.10)

Then the correlation function in momentum space is:

g(2)(k, k′) =

⟨ψ†u(k)ψ†d(k

′)ψd(k′)ψu(k)

⟩〈ψ†u(k)ψu(k)〉〈ψ†d(k′)ψd(k′)〉

(4.11)

and the same TWA transformation between ψi(k) and ψi(k) as introduced in equation (3.5). We apply thiswindowed Fourier transformation because we want to reduce the self-correlation signal of ψ(k). The signalwe want to observe is the uout−d2out one, these two modes propagate in opposite direction on either sideof the horizon. So the correlation trace correspond to two modes in same region (up/downstream region)won’t appear with this method. But this method can’t directly use in the polariton flow. In equation(4.10) the group velocities of different modes are calculated from the dispersion relation. In BEC system,the density flow is ’flat’ on either side of horizon. For one selected region, up- or downstream region, wealways have the same dispersion relation. In polariton flow system the density flow is not flat, so thedispersion relation depends on position on the wire. This will change the validity condition of equation(4.10) so we can’t use the same window as in BEC. For the first try, I use a Heaviside window functionto separate the up and downstream regions

Hu(x) =

0 x > xhor

1 x ≤ xhor, and Hd(x) =

1 x > xhor

0 x ≤ xhor.(4.12)

An extra noise is needed which come from the new operator of wavefunction, the window function. Forthis Heaviside window we have the windowed Fourier transformation

ψi(k) =

∫ +∞

−∞

(ψ(x)Hi(x) +

√(1−Hi(x)2) ∗Noise(x)

)e−ikxdx, (4.13)

where Noise(x) is the same amplitude zero mean white noise as equation (3.2) Here are the result ofthe stress test for a zero pump input, there is no pumped photon in the cavity, and the polariton flowin figure 20. In figure 20(a), we can see there is a big anti-correlation cross pass the point k = 0, k′ = 0and a base of positive correlation of amplitude on the order of 0.06. In figure 20(b), we can’t observethe same feature as 19(b). There is a vertical band of positive correlation line near to 0 and there isno feature matching the prediction trace. The result shows that this method has some problem. Thereare still things to solve in this method for polariton flow. May be a Heaviside function is a too strongapproximation and we should begin with a large bandwidth Gaussian distribution.

17

(a) Non normalized correlation function G(2)(k, k′) ofvacuum state in momentum space with windowed Fouriertransformation

(b) Normalized correlation function g(2)(k, k′) − 1 of po-lariton flow in momentum space with windowed Fouriertransformation

Figure 20: Second order correlation function of polariton flow in momentum space with windowed Fouriertransformation

5 Conclusion

In this report, I presented an acoustic black hole in superfluids of polariton flow in semiconductor mi-crocavities. Pumped by a resonant monochromatic laser beam, the polariton flow in the 1D wire cavityrealized a sonic horizon configuration which is a effective event horizon in curved spacetime [17] [38].Pairs of phonons created by quantum vacuum fluctuation of polariton flow density at the horizon aresignature of Hawking effect in the polariton flow. The Hawking effect at the horizon can be determinedby Hawking temperature TH which is proportional to the gradient of the flow density at the horizon [3].To increase TH at the horizon, I use a small defect at the horizon and obtain a sharp edge of density flowat the horizon which gave us a larger gradient value of the flow. The flow density modulation of phononsare hard to detect directly, so we use the density-density correlation in real space to study the Hawkingeffect.I calculated numerically the correlation function of Hawking effect in real space, my simulation showedthere exist features for uout − d2out modes, which is the Hawking radiation and its partner, in the corre-lation pattern matching the theory. In order to filter the thermal noise of the driven-dissipative natureof polariton system, I tried to calculate the correlation function in momentum space. The signature ofHawking effect is a upstream-downstream correlation because the pairs of phonons created at the horizonhave opposite direction of propagation, one to upstream and another to downstream. To avoid the self-correlation of wavefunction in up/downstream region, we introduced a window Fourier transformation.But the result of correlation in momentum space is still hard to interpret. In the future, we want to usea better window Fourier transformation and analysis what we obtain in momentum space. Then we willuse these result to guide experiments of analogue gravity in a polariton flow in the laboratory.

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Appendix A: Position of the defect in the polariton flow

To understand how the defect works, I choose a series of point on either sides of the position of horizon inthe ballistic flow to set the defect, the result is shown in figure 21. The label under each figure representrelative position of the defect compared to the position of horizon in the ballistic flow. We can see that,from figure 1µm, a defect at 1µm after the horizon without defect, the sharp edge of velocity profileappears. For a defect set before this point, we can’t obtain this sudden increase and decrease of flowvelocity. Figure 22 shows the variance of gradient at the horizon for different positions of defect. Asshown in equation (4.4), the gradient of flow velocity is proportional to the TH . So in order to increasethe TH , we need to put the defect at least 1µm after the position of horizon in the ballistic flow.

Figure 21: Velocity profile for different defect positions

Figure 22: Gradient of flow velocity at the horizon for different defects

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Appendix B: *

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