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
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
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
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
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
n(x) numerical ∆=10
−1000 −500 00
1
2
n(x) numerical ∆=20
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
n(x) numerical ∆=5
−1000 −500 00
1
2
n(x) numerical ∆=10
−1000 −500 00
1
2
n(x) numerical ∆=20
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
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
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
One-body Hawking signal at equilibrium
linear relation connecting theout-going modes to the in-going ones(
u|outd1|out
(d2|out)†
)= S(ω)
(u|ind1|in
(d2|in)†
)
ω − Vq = ±EB(q)
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
〈u†out uout〉 = |Suu|2 〈u†inuin〉+ |Sud1 |2 〈d†1in d1in〉+ |Sud2 |
2 〈d2in d†2in〉
at T = 0 : 〈u†out uout〉 = |Sud2(ω)|2 needs
u d2 mode conversion
d2-ingoing mode
Spontaneous Hawking radiation
18
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
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
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
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
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
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