Analogous black-holes in Bose-Einstein condensates - How ......

Analogous black-holes in Bose-Einstein condensatesHow to create them and how to detect the associated (sonic) Hawking

radiation ?

Nicolas Pavloff LPTMS, Université Paris-Saclay

École de Gif, LAL, october 2017

1

Analogous Hawking radiation Unruh, Phys. Rev. Lett. (1981)

horizon

upstream :subsonic

downstream :supersonic

direction of the flow

Vu < cu Vd > cd

“δ-peak”

0

x

ξu

n(x)

nu

U(x) = Λδ(x)

1

Vu < cu Vd > cd

0

x

ξu

n(x)

nu

U(x)

U0= −Θ(x)

−1

1

Vu < cu Vd > cd

“Waterfall”

gravitational black hole

horizon

Hawking radiation 75’

horizon

upstream :subsonic

downstream :supersonic

Vu upstream Vd downstream

Hawking

(Vu − cu)

partner(Vd − cd )

Vu < cu Vd > cd

2

quasi-1D Bose-Einstein condensates

quasi-1D condensatelongitudinal size ∼ 102µmtransverse size ∼ 1µm

1.6 mm

optical guidemagnetic trap

x

yz

BEC g

(a)

(b)

rfknife

Guerin et al., Phys. Rev. Lett. (2006)

tight harmonic radial confinement :

xradial confinementpulsation ω⊥

V⊥(~r⊥ ) =12 mω2

⊥r 2⊥ .

→ 1D model : Ψ(x , t)

3

1D mean field regime Menotti and Stringari, PRA (2002)

domain of validity

~ω⊥~2/ma2 n1a ∼

µ

~ω⊥ 1

a = 3D scattering lengthn1 = 1D linear density

• The first inequality allows to avoid the Tonks-Girardeau regime and impliesEint Ekin. Also Lφ≫ ξ Lφ = ξ exp

[π√

~ n12m aω⊥

]• the second inequality allows to avoid the 3D-like transverse Thomas-Fermiregime and implies that transverse motion is “frozen” (more precisely:Born-Oppenheimer approximation)

0 1 2 3p0

5

hω

cp

Bogoliubov spectrum

p2/2m+µ

0 1 2 3 4 5 60

5

10

15

20

25

30

η=9.4

(a)

k [aρ

−1]

ωnk [

ωρ]

C. Tozzo & F. Dalfovo, PRA (2002)

← η =µ

~ω⊥= 9.4

onlyaxi-symmetricexcitationsincluded (m = 0)

4

A possible dumb hole configuration

P. Lebœuf & N. Pavloff, Phys. Rev. A (2001)

credit: Yan Nascimbene

stationary: Ψ(x , t) = ψ(x) exp−iµt

classical mean field:

− 12ψxx +

(U(x) + g |ψ|2

)ψ = µψ

wherea g = 2ω⊥a

n(x) = |ψ|2 n(x)v(x) = Im(ψ∗·ψx )

aOlshanii, PRL (1998)

U(x) = κ δ(x) (κ > 0)

0

x [a.u.]0

1 n(x)/n1

U(x)

flow

region

supersonic

−1 −0.5 0 0.5 1 κ0

0.5

1

1.5

2

2.5

v /c

0

1

2

0

1

2

0

1

2

5

A possible dumb hole configuration P. Lebœuf & N. Pavloff, Phys. Rev. A (2001)

credit: Yan Nascimbene

stationary: Ψ(x , t) = ψ(x) exp−iµt

classical mean field:

− 12ψxx +

(U(x) + g |ψ|2

)ψ = µψ

wherea g = 2ω⊥a

n(x) = |ψ|2 n(x)v(x) = Im(ψ∗·ψx )

aOlshanii, PRL (1998)

U(x) = κ δ(x) (κ > 0)

0

x [a.u.]0

1 n(x)/n1

U(x)

flow

region

supersonic

−1 −0.5 0 0.5 1 κ0

0.5

1

1.5

2

2.5

v /c

0

1

2

0

1

2

0

1

2

5

BEC analogue of a Laval nozzle A. M. Leszczyszyn et al., Phys. Rev. A (2009)

Nozzle of a V2 rocket

F = m (vout − vin)

For a thick barrier

U(x) of width ξ ∼ (gn)−1/2 :− (n1/2)xx

2n1/2+ 1

2 v2(x) + g n(x) + U(x) = C st ,

n(x)v(x) = C st .

; 1n

dndx

[v 2 − c2

]= dU

dx where c2(x) = g n(x)

v(x) ≶ c(x) ↔ sign(

dndx

)= ∓ sign

(dUdx

)

supersonic

subsonicn(x)

transonicU(x)

transonic :Ux (x0) = 0v(x0) = c(x0)

/

6

How to form a sonic horizon ? A. Kamchatnov & N. Pavloff, Phys. Rev. A (2012)

∆

U(x)

0− +XX (t) (t)

n(x,t)

x

DSW

−1000 −500 0

x

0

1

2

n(x) analytical

−1000 −500 00

1

2

n(x) numerical ∆=5

−1000 −500 00

1

2


−1000 −500 00

1

2


The profile in the region of the horizon only depends on vu/cu, but not of theinitial profile. No hair theorema ? not quite, see below.

aF. Michel, R. Parentani, R. Zegers, Phys. Rev. D (2016)

7

How to form a sonic horizon ? A. Kamchatnov & N. Pavloff, Phys. Rev. A (2012)

−590 −580 −570

x

0

1

2

n(x)

−4 0

x

0

1

2

red=numericsblack=Whitham+transonic

-250 -200 -150 -100 -50 0

x0

0.1

0.2

nmin

−1000 −500 0

x

0

1

2

n(x) analytical

−1000 −500 00

1

2


−1000 −500 00

1

2


−1000 −500 00

1

2


Soliton train in the upstream plateau... experimental problem ?

7

Stationary solutions of the NLS equation

− 12Axx +

[g n +

J2

2n2 − µ]

A = 0 , where J = n(x)v(x) and A =√

n

first integral:

12A2

x + W (n) = Ecl , where W (n) = −g2 n2 + µn +

J2

2n .

n

W(n

)gn

+J2/2n

2

−10 0 10 20 x / ξ0

0.5

1

n(x

)/n

2−10 0 10 20

0

0.5

1

n(x

)/n

2

−10 0 10 200

0.5

1

n(x

)/n

2

soliton Ecl = W (n2)

cnoidal wave W (n1) < Ecl <W (n2)

harmonic wave Ecl →W (n1)

8

Stationary solutions of the NLS equation

n(x , t) = 14 (λ4 − λ3 − λ2 + λ1)2 + (λ4 − λ3)(λ2 − λ1)sn2 (k (x − Vt),m) ,

k =√

(λ4 − λ2)(λ3 − λ1), V = 14 (λ1 + λ2 + λ3 + λ4), m = mλi ∈ [0, 1],

v(x , t) = V − Cλin(x,t)

.

−10 0 10 20 x / ξ0

0.5

1

n(x

)/n

2

−10 0 10 200

0.5

1

n(x

)/n

2

−10 0 10 200

0.5

1

n(x

)/n

2

soliton limit: λ2 → λ36?∼ λ4 − λ3

cnoidal wave: λ1 < λ2 < λ3 < λ4KdV 1895

6?∼ λ2 − λ1sinusoidal limit: λ2 → λ1

9

Whitham modulation equations Forest & Lee (1986), Pavlov (1987)

slow modulations λi → λi(x , t) with ∂λi∂t + Vi(λj) ∂λi∂x = 0

V1(λj) = 12∑4

i=1 λi −(λ4−λ1)(λ3−λ1)K(m)

(λ4−λ1)K(m)−(λ3−λ1)E(m)m = (λ2−λ1)(λ4−λ3)

(λ4−λ2)(λ3−λ1)

Gurevich-Pitaevskii problem Gurevich & Pitaevskii (1973)

simple case: decay of a initial discontinuity → dispersive shock wave

no characteristic length : self-similar solution depending on ζ = x/t andmatching to the right and left boundaries with a non dispersive flow.

u(x , t = 0) =

1 if x < 0 ,0 if x > 0 .

ut + uux + uxxx = 0

-30 -20 -10 10 20

0.5

1

1.5

2

x

u(x)t = 4

xx +-

(Vi − ζ) dλidζ = 0

λ3

λ2(ζ)

λ1

x− x+

u = λ3 = 1*

u = λ1 = 0

10

Dispersive shock + soliton train A. Kamchatnov & N. Pavloff, Phys. Rev. A (2012)

−3 −2 −1

x/t

−2

−1

0

1

2

λ2(x/t)

λ3

λ4

λ1

λ+

Aλ

+

C

λ−

C

λ−

A

region Bregion regionA C

0V−− V

+

n(x,t)

dispersiveshock

flowx

U(x) soliton train:

F λ1, λ4: fixedλ2( x

t ) = λ3( xt )

F xt = 1

2∑4

i=1 λi

yields → λ2,3( xt

)F nmin = f (λi

′s)= f ( x

t )

−250 −200 −150 −100 −50 0

x

0

0.1

0.2

0.3

nmin

11

Technion experiment Lahav et al. PRL (2010)

The arrow indicates the direction ofthe harmonic potential relative tothe stationary step like potential(v ∼ 0.3 mm/s).

left plot:

v(x) = −1n

∫ x

nt dx ′ ,

c(x) =√

g n(x) .

velocity

green dashed line: black hole horizonyellow dash-dot: white hole horizon

green andblue curves:density

red curve:−v(x)

black curve:c(x)

12

Steinhauer, Nature Physics 2016 :

profile near the horizon ' waterfallnu/nd = 5.55 5.55 cu/cd = 2.4 2.36Vu/cu = 0.375 0.4245 Vd/cd = 3.25 5.55

theoretical “waterfall”

0

x

ξu

n(x)

nu

U(x)

U0= −Θ(x)

−1

1

Vu < cu Vd > cd

Vd

Vu=

nu

nd=( cu

Vu

)2=

Vd

cd=( cu

cd

)2

Beware offluctuations !cf. case of δ-peakconfiguration:U(x) = κ δ(x)

0 1 2 3 4 5

κ

0

0.5

1

1.5

2

u0

13

Sonic black holes : “dumb holes” Unruh (1981)

horizon

flow

sub-sonicregion

super-sonicregion

v < c v > c

v = c

stimulated Hawking radiation:in the laboratory :

E(p) = c |p|+ v p ↓

comoving Doppler

p p

EE

supersonic regionsubsonic region

ref

inc

trans

trans

14

the position of the horizon is energy-dependent

model configuration :

supersonic

velocity

subsonic x

v(x)

cE − v(x).p = ± c p

Ep

E

p

±c p

E

p

±EB(p)

6

BBBBM

E − v(x).pQQk

v(x) < c

region x < 0

v(x) > c

region x > 0

phase space :

inc

trans

p

x

Hawking

partner

p

x

ref

inc

trans

inc

15

the position of the horizon is energy-dependent

model configuration :

supersonic

velocity

subsonic x

v(x)

c

E − v(x).p = ±EB(p)

with

EB(p) = c p√

1 + ξ2p2/4

Ep

E

p

±c p

E

p

±EB(p)

6

BBBBM

E − v(x).pQQk

v(x) < c

region x < 0

v(x) > c

region x > 0

phase space :

inc

trans

p

x

Hawking

partner

p

x

ref

inc

trans

inc

15

Numerical test, model configuration: Recati, Pavloff & Carusotto, Phys. Rev. A (2009)

U(x) and g(x) step like withU(x) + g(x)n0 = C st such thatψ0(x) =

√n0 expik0 x, verifies ∀x

− 12ψ′′0 +

[U(x) + g(x)|ψ0|2

]ψ0 = µψ0 ,

C st = µ− k202 .

p

x

uin

uout

d1|out

d1|in

d2|out

d2|in

16

Partial conclusion :

Up to now, only classical description (= wave mechanics).

Realistic “dumb hole” configuration: i.e., formation of a sonic event horizon

“δ-peak “: dynamical process well described by Whitham approach

“waterfall” configuration: co-validated by experiment

“stimulated Hawking radiation”: quantum reflection + mode conversion

17

Spectrum of Hawking radiation

i ∂tΨ = − 12∂

2x Ψ +

[U(x) + g Ψ†Ψ

]Ψ

Energy current associated to emission ofelementary excitations

Π = − 12∂tΨ

†∂x Ψ + h.c.

at T = 0, 〈Π〉 = −∫ ∞0

dω2π ~ω |Su,d2|2

Hawking radiation in the u|outchannel. Equivalent to a black bodyradiation of temperature TH ∼ 10% µ

|Su,d2|2ω→0∼ Γ

exp~ω/kBTH − 1

→ kBTH

gnu∼ 0.1→ TH ∼ 5− 10 nK

00

ω in units of cu/ξu

|Su,d2|2

andΓ×n

TH

0.1 0.2 0.3 0.4

0.5

1

1.5Γ× nTH

|Su,d2|2

Ω

configuration withVucu = 0.5

19

Hand waving mathematics Volovik : The universe in a helium droplet

long wave length limit : E − v(x) · p = ±c(x) · p → p = Ev(x)± c(x)

phase space :

inc

trans

p

x

Hawking

partner

Tunnel probability

P ∝ e−2S/~ where S =∣∣Im ∫ p(x) dx

∣∣near the horizon

Ev(x)−c(x)

' E(x±iε) d

dx (v−c)|0→ ±iδ(x) E

ddx (v−c)|0

S ' π Eddx (v−c)|0

Hawking temperature

P ∝ e−E/(kBTH ) with kBTH = ~2π∣∣∣ ddx (v − c)

∣∣∣0

20

Example of the waterfall configurationlinear relation connecting the out-goingmodes to the in-going ones(

u|outd1|out

(d2|out)†

)= S(ω)

(u|ind1|in

(d2|in)†

) at T = 0 outgoing energy current

〈Π〉 = −∫ ∞0

dω2π ~ω |Su,d2|2

Su,d2 =fu,d2√ω/gnu

+ hu,d2√ω/gnu +O(ω3/2)

|Su,d2|2 = Γ(ω)ntherm(ω)

= (Γ +O(ω2))( kBTHω + 1

2 +O(ω))

in the waterfall configuration

kBTH

gnu=

12

(1−M4u)3/2

(2 + M2u)(1 + 2M2

u)

where Mu = Vu/cu0

01

kBTH

mc2u

δ -peakWaterfall

00

Vu / cu

1

Γ

1

0.2 0.4 0.6 0.8

0.5

0.05

0.1

0.15

0.2

0.25

0.3

0.5

Vu / cu

21

Density correlations Balbinot, Carusotto, Fabbri, Fagnocchi, Recati, Phys. Rev. A & New J. Phys. (2008)

0

q [a.u.]

d2|in

d2|out

d1|out

d1|in

Ω

−Ω

downstream region

0

q [a.u.]

0

ω

[a.u

.]

u|inu|out

upstream region

out inoutin

New theoretical and experimental interest:study of density correlation on each side ofthe horizon

g (2)(x , x ′) =〈:n(x)n(x ′) :〉〈n(x ′)〉〈n(x)〉

− 1

F example of induced correlation:u|out d1|out

x = (vd + cd)t correlates withx ′ = (vu − cu)t

F affects the density correlationpattern

-100 0 100 200

x / ξu

-100

0

100

200

x, /

ξu d1-d2

u-d1u-d2

Larré et al., Phys. Rev. A (2012)

22

Uniform 1D Condensate (no black hole)

g (2)(x , x ′) =1

n0 ξF( |x − x ′|

ξ

)where

F (X) = −

1π X

∫ ∞0

dtsin(2tX)

(1 + t2)3/2

F (0) = −2/π .

(µ = g n0 =

1ξ2

)

-20 0 20 40 60 80 100

x / ξ

-20

0

20

40

60

80

100

x, /

ξ

-0.0015

-0.001

-0.0005

0

0.0005

0.001

-6 -4 -2 0 2 4 6

(x-x’) / ξ

-0.6

-0.4

-0.2

0.0

n0ξ

g(2

) (x,x

’) numerics

x’=+50ξ-x

analytical

-2/π=-0.64result

23

Uniform 1D Condensate

Popov result (T = 0)

1n0ρ(1)(x , x ′) = exp

−F (|x − x ′|/ξ)

2πn0ξ

where

F (X) =

∫ ∞0

dt(

t2 + 2t√

t2 + 4− 1)(

1− cos(X t))

0 1 2 3 4 5-0.5

0

0.5

1

|x − x ′|/ξ

1n0ρ(1)(x , x ′)

naive Bogoliubov

24

Two-body Hawking signal Recati, N. Pavloff & I. Carusotto, Phys. Rev. A (2009)

Comparison of numerical and analytic results (model configuration)

0 20 40 60

(x+x’)/2ξu

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

n0ξ

u G

0

(2) (x

,x+

39

.5ξ

u)

main correlation signal :

u|out d2|out

x = Vd2|out t correlates with x ′ = Vu|out t

where Vd2|out = Vd − cd and Vu|out = Vu − cu

25

Steinhauer, Nature Physics 2016 :

density profile near the horizon 'waterfall nu/nd = 5.55 5.55cu/cd = 2.4 2.36Vu/cu = 0.375 0.4245 Vd/cd = 3.25 5.55

TH = 1.0 nK TH/(gnu) = 0.36 ?TH/(gnu)|theo ≤ 0.25

-100 0 100 200

x / ξu

-100

0

100

200

x, /

ξu

-0.01

-0.005

0

0.005

d1-d2

u-d1u-d2

26

correlation profile

-10 0 10

xcut

/ ξ

-0.04

-0.02

0

0.02

0

0

-10 0 10

xcut

/ ξ

-0.04

-0.02

0

0.02

0

0

-10 0 10

xcut

/ ξ

-0.04

-0.02

0

0.02

0

0

-10 0 10

xcut

/ ξ

-0.04

-0.02

0

0.02

0

0

u-d1

u-d2(a) (b)

(c) (d)

correlations g2(x , x ′) along a cut x + x ′ = C st. For all the plots: the abscissa isthe coordinate xcut along the cut in unit of ξ =

√ξuξd and the ordinate is√

nundξuξd g2(x , x ′). Left plot: experimental results of Steinhauer. Right plots:theoretical results in the waterfall configuration with Vu/cu = 0.4245 alongdifferent cuts. Figs. (a), (b), (c) and (d): cases when the cut x + x ′ = C st

intercepts the u − d2 correlation lines at x/ξu = 100, 50, 30 and 15. The (red)shaded zone is the forbidden zone around x ′ = 0.

27

One body momentum distribution in the presence of a horizon

T = 0, adiabatic opening of the trap Boiron et al. PRL (2015)

〈n(p

)〉[arb.un

its]

Pu Pd

p

Hawkinguout channel∝ |Sud2|2

BBN

Partnerd2out channel∝ |Sd2u|2+|Sd2d1|2

AAU

d1out channel∝ |Sd1d2|2

-1 0 1 2 3

p ξu

0

1

PdP

u

28

Two body momentum distribution in the presence of a horizon

p, q : absolute momenta in units of ξ−1u

right plot: g2(p, q)→

where g2(p, q) =〈: n(p)n(q) :〉〈n(p)〉〈n(q)〉

0

k [a.u.]

d2|in

d2|out

d1|out

d1|in

Ω

−Ω

downstream region

0

k [a.u.]

0

ω

[a.u

.]

u|inu|out

upstream region

k : momentum relative to the condensatep = k + P(u/d) where P(u/d) = mV(u/d)

T = 0 adiabatic openingBoiron et al. PRL (2015)

-1 0 1 2

p

-1

0

1

2

q

uout

- d1out

d1out

- d2out

d1out- d1

out

uout

- d2out

d2out- d2

out

u out- u o

ut

Pd

Pu

without horizon: g2 ≡ 1

29

Violation of Cauchy-Schwarz inequality (T 6= 0)

C.-S. violation : g2(p, q)

∣∣∣uout−d2out

>

√g2(p, p)

∣∣∣uout× g2(q, q)

∣∣∣d2out

≡ 2

Boiron et al. PRL (2015)

-1 -0.5 0 0.5

p0

2

4

6

1g2(p

,q)

uo

ut -

d2

ou

t

T=2

Pu

T=0T=0.8T=1.2

T=1.6

T in units of µTH = 0.13

Vu/cu = 0.5

Vd/cd = 4

Vd/Vu = 4

nu/nd = 4

30

Quantum effects within 1D mean field ? Popov, Teor. Mat. Fiz. (1971)

The NLS ↔ Gross-Pitaevskii eq. is a nonlinear quantum field eq. :

− 12∂

2x ψ + g ψ†ψ ψ = i ∂t ψ , with

[ψ(x , t), ψ†(y , t)

]= δ(x − y) .

BEC : macroscopic occupation of the lowestquantum state: ψ(x , t) = ψ(0)(x , t) + φ(x , t)(Bogoliubov 1947)

ψ(0) : solution of the (classical) NLS

φ : solution of a linearized (quantum) eq.

makes it possible to consider vacuumfluctuations. In particular : Hawking radiationin a stationary, non uniform setting.

qq′

!" #" " #" !"

!"$

#"$

"$$

#"

!" %

&

!

"

Mathey, Vishwanath, Altman, PRA (2009)

Bouchoule, Arzamasovs, Kheruntsyan, Gangardt, PRA (2012)

31

Phonon evaporation and adiabatic trap opening

φ(x) = eiP(u/d)x∫ ∞0

dω√2π

∑L∈U,D1

[uL(x , ω)aL(ω) + v∗L (x , ω)a†L(ω)

]+ eiP(u/d)x

∫ Ω

0

dω√2π

[uD2(x , ω)a†D2(ω) + v∗D2(x , ω)aD2(ω)

].

• If x −ξu : uU(x) = Uu|ineiqu|inx + Suu Uu|outeiqu|outx ,• If x ξd : uU(x) = Sd1,u Ud1|outeiqd1|outx + Sd2,u Ud2|outeiqd2|outx .

adiabatic opening of the trap:(U(ω)V(ω)

)mode

→(10

)or(01

)for d2|out

see also: de Nova, Sols & Zapata PRA (2014)

• adiabaticity always violated for long wave-lengths, when ω × tchar 1

32

truly 1D ? No : transverse excitations

when ~ω⊥ ≤ µ :

0 1 2 3 4 5 60

5

10

15

20

25

30

η=9.4

(a)

k [aρ

−1]

ωnk [

ωρ]

Zaremba, PRA (1998)Stringari, PRA (1998)Fedichev & Shlyapnikov, PRA (2001)Tozzo & Dalfovo, PRA (2002)

modified dispersion relation :ω20(q) = c2

1d q2 (1− 148 (qR⊥)2 + . . .

)new channels :ω2n≥1(q) = 2n(n + 1)ω2

⊥ + 14 (qR⊥ω⊥)2 + . . .

these new channels will be populatedat T = 0

mass term 6= Klein-Gordon→ new “in” modes

33

Conclusion

BECs offer interesting prospects to observe analogous Hawking radiation[Steinhauer, Nature Physics]

general perspective : quantum effects with nonlinear matter waves

One- and two-body momentum distributions accessible by present dayexperimental techniques provide clear direct evidences

ê of the occurrence of a sonic horizon.

å of the associated acoustic Hawking radiation.

+ of the quantum nature of the Hawking process., The signature of the quantum behavior persists even

at temperatures larger than the chemical potential.

34

Cauchy-Schwarz : a wrong theorem ?... cf. sub-Poissonian fluctuations

〈...〉 def= Tr(ρ...)

〈n2〉 = 〈a†aa†a〉

= 〈a†a†a a〉+ 〈a†1 a〉

= 〈a†a†a a〉+ 〈n〉

0 ≤ 〈δn2〉 def= 〈n2〉 − 〈n〉2

= 〈a†a†a a〉 − 〈n〉2︸︷︷︸sign?

+〈n〉

Cauchy-Schwarz: |〈A〉|2 ≤ 〈A†A 〉

Hence |〈aa〉|2 ≤ 〈a†a†a a〉

But |〈a†a〉|2 6≤ 〈a†a†a a〉

stupid theoretical exampleaverage over a number state: ρ ≡ |n〉〈n|〈aa〉2 = 0〈a†a†a a〉 = n(n − 1)〈a†a〉2 = n2 6≤ 〈a†a†a a〉

a number state is clearly sub-Poissonian !

experimental results

Poissonian limit : 〈δN2〉 = 0.34〈N〉Ideal Bose gasYang-YangQuasi-cond. Jacqmin et al., PRL (2011)

35

Violation of Cauchy-Schwarz inequality (δ peak T 6= 0)

C.-S. violation : g2(p, q)

∣∣∣uout−d2out

>

√g2(p, p)

∣∣∣uout× g2(q, q)

∣∣∣d2out

≡ 2

-0.2 0 0.2 0.4

p0

2

4

6

g2(p

,q)

uo

ut -

d2

ou

t

T=0T=0.6

T=1

T=1.6

T in units of µ

TH = 0.12

Vu/cu = 0.5

Vd/cd = 1.83

Vd/Vu = 2.34

nu/nd = 2.37

36

Gravity wave analogues

Schützhold and Unruh, Phys. Rev. D (2002) Rousseaux et al., New Journal of Physics (2008)

Weinfurtner et al., Phys. Rev. Lett. (2011) Euvé et al., Phys. Rev. D (2016)

in a basin of depth h, the dispersion relation of gravity waves is(ω − Vk)2 = g k tanh(k h) , corresponding to c =

√g h

Experimental test of mode conversion :

horizon

white

incident

reflected

anomalous

flow h(x)

xv(x) > c(x) v(x) < c(x)

37

Vancouver experiment Weinfurtner et al., Phys. Rev. Lett. (2011)

1

2

3 4 5

6

7

horizon

white

incident

reflected

anomalous

flow h(x)

x

38

Gravity waves in a flume Euvé et al., Phys. Rev. D (2016)

ω − Vk = ±√

gk tanh(hk)

Poitiers experiment

V (x)- flow

supersonic : V > c subsonic : V < c

k -

ω

6

ω − Vk

k -

ω − Vk

incident

reflected

not detected

reflectedanomalous

ωm − Vk

•

ωm

ωm

F = V /c

39

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