S. Bar-Ad,

O. Farberovich,

I. Fouxon

M. Elazar

Y. Vinish

Tel Aviv University

R. Schilling, Mainz U.

EPL, 92, 14002 (2010) Phys.Rev.A, 86, 063821 (2012) Phys.Rev. A 85, 045602 (2012) Phys.Rev. A 60, 013802 (2013) Chapter in: Analogue Gravity Phenomenology, Series: Lecture Notes in Physics, Vol. 870, Springer Int.J.Mod.Phys.B 30, 1650 (2016)

11/6/2017 QFC2017 - Pisa

Analogue Hawking radiation in a "flow of light"

Analogue gravity by an optical vortex. Resonance enhancement

of Hawking radiation Marco Ornigotti Shimshon Bar-Ad Alexander Szameit Victor Fleurov

11/6/2017 QFC2017 - Pisa

arXiv:1704.07609 PRA - submitted

• Black hole • Analogue gravity-Transonic flow. • Maxwell equations and hydrodynamics. • Mimicking horizon and curved space, Hawking

radiation • Fluctuations in transonic regime – linear geometry • Vortex as a background • Superradiance and resonant enhancement

11/6/2017 QFC2017 - Pisa

Black hole – naïve approach

11/6/2017 QFC2017 - Pisa

22 mcmvR

MmG <= Escape velocity cannot exceed speed of light

2cMGRR s => Radius of event horizon

24 MS π=

Temperature M

MS

ES

Tπ81

=∂∂

=∂∂

=

Entropy = surface area

Black hole Hawking radiation

KM

M

KM

kgGMk

cT

s

BH

8

232

106

10227.18

−⋅=

⋅==

π

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Black hole radiates according to the Planck law for black body radiation

Modeling black hole in a flow – Unruh, 1981

Pt

t

∇−=∇⋅+∂∂

=⋅∇+∂∂

ρ

ρρ

1)(

0)(

vvv

v

ϕϕϕψψψ

+→+→

00

00

Continuity equation

Euler equation

Laminar solution for a transonic flow

plus fluctuations near this solution

00

00

ϕρρ ψ

∇==

ve

W. G. Unruh, Phys. Rev. Lett. 46, 1351 (1981)

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Equation for fluctuations

( ) ( ) 00000 =⋅∇+∇⋅∇+∂∂ ψρϕρψρ v

t

020 =+∇⋅+

∂∂ ψϕϕ s

tv

( ) ( )[ ] [ ] 0ttt

200002

2

=∇⋅∇−∇⋅⋅∇+

∂∂

⋅∇+∇⋅∂∂

+∂∂ ϕϕϕϕϕ svvvv

Excluding ψ

11/6/2017 QFC2017 - Pisa

01=

∂∂

−∂∂

−ϕj

iji x

ggxg

−−

−=

2200000

0022

000

000022

0

0001

1

svvvvvvvvsvvvvvvvvsvv

vvv

sg

oxzyzxx

zyoxyxy

zxyxoxx

zyx

ij

( ) 2det/1 −−== sgg ij

)ln(21)(

20

20

ss

rra

tvsdrrvt −+≈

−+= ∫ ⊥

⊥

τ

[ ]( )rrrvvv ddddtdtss

dxdxgd jiij ⋅−⋅+⋅−== 0

200

22 21σ

Contravariant metric in a “curvilinear space”

ji

ljil gg δ=

Connection between contravariant and covariant metrics

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1 + 1 Klein-Gordon equation

01=

∂∂

−∂∂

−ψ

xgg

xg jij

i

−

−=

22

22

1 0

0 1

sv

sv

sg

x

x

ij

−

= 22

11svv

vs

gxx

xij

( )22222 2)(1 dxdtdxvdtvss

d xx ++−−=σ

222

222

2 dxvs

sds

vsdx

x

−+

−−= τσ

dxsv

vddtx

x22 −

−= τ

xavs sx =− 22Schwarzschild metric

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Hawking radiation

[ ]

⋅−

−⋅−==00

2

22

0022 1

vvvv

ssdrds

sdxdxgd ji

ij τσ

The corresponding equation for fluctuation is equivalent to that considered by Hawking.

Hawking radiation in a laboratory

“Schwarzschild metric”

ra

tvsdrrvt

s

ln21)(

20

20 +≈−

+= ∫ ⊥

⊥

τ

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s > v, subsonic

s < v, supersonic

“Mach horizon”

sonar

Submarine can be located by the sonar

Submarine can still be located by the sonar but with a very long waiting time. Time slowing.

Submarine cannot be located by the sonar

axsxvxx

aaxsaxs

a

axdx

axsdx

xvsdx

xvsdxt

x

x

x

x

x

x

x

x

+=+++

≈−

++

≈−

++

= ∫∫∫∫

)( ;ln122ln1

2)()(

1

0

0

1

0

1

1

0

0

1

1

0

0xx=0 x

1x

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Madelung transformation:

Gross-Pitaevskii Eq.:

=>

Bose-Einstein Condensate

E. Madelung, Quantum theory in hydrodynamic form, Z. Phys. 40, 322 (1927).

Euler flow equations for light

- ψψψψ 22 ||

21 gm

i xt +∂−=∂

)exp( ϕρψ i−=

+

∇−∇−=∇+∂

=∇+∂

ρρρ

ρρ

gmm

vv

v

t

t

22

211

21

0)(

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QFC2017 - Pisa

Nonlinear Schrödinger equation. I

0. B ,BE

0, D ,DH

=•∇∂∂

−=×∇

=•∇∂∂

=×∇

t

t

HB

dtrEtrEtrD

µ

τττεε

=

−== ∫∞

,),()(),(ˆ),(0

0)ˆ(),( 2

22 =

∂∂

+∇− Et

trE εµ

Maxwell equations

Helmholtz equation

11/6/2017

Non-Linear Schrödinger (NLS) equation

)exp(),,,(2

);,,( 0 tizizyxAdtzyxE ωβωπω

−= ∫

0ω

0ˆ 2

22

2

2

2

2

2

2

=∂∂

−+∂∂

+∂∂

+∂∂

tEEE

yE

xE

zE εα

Helmholtz equation deduced from the Maxwell equations

paraxial approximation

Wave packet around the frequency and wave vector 0β

11/6/2017 QFC2017 - Pisa I.Carusotto, C. Ciuti RMP, 2013

Non-Linear Schrödinger (NLS) equation

Refraction coefficient plays the role of potential

Spatial coordinate ‘z’ is now “time”

Time ‘t’ is now a “spatial coordinate”

2

2

02

2

2

2

~2t

Dyx ∂

∂+

∂∂

+∂∂

=∆ πβ

AAgAyxnAAz

i 2220

0

)],([2

1+−+∆−=

∂∂ β

β

Kerr nonlinearity

11/6/2017 QFC2017 - Pisa

z

x

Hydrodynamic representation

zAiAAgAUext ∂∂

=+

+∇− 22

021β

ϕifeA −=

2

2

0

2

0

2

21

21

0)(

sg

gUz

z

ext

=

++

∇−∇−=∇+

∂∂

=∇+∂∂

ρ

ρρρ

ββ

ρρ

vv

v

The density field

The velocity field

Continuity equation

Euler equation

Quantum potential

Substituted into

ϕβ

ρ

−∇=

==

vfA

0

22

sound velocity

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Behavior of fluctuation modes near the “horizon”. QP neglected

)ln(32

1

xs

ii ee−

= αν

ντξ

νανα

ννξ ξνντ

231

1

2)( ;)(

2

i

i

ske iki

−

−== −

HTs eeν

απν

⇒32

The first mode oscillates like crazy near the horizon (x=0) and acquires the factor when crossing it. This factor defines the “Hawking temperature”.

πα2

3 sTH =

“Left” mover Right mover

se απν

32

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QFC2017 - Pisa

Straddled fluctuations

∫ −= )()();( xegdzxf ziω

ω ξωω

( )2

2

2

21)( Γ

−−

Γ=

g

egωω

πω

Zero at the horizon

11/6/2017

Derivation of Hawking temperature s

rl e απν

ϕϕ 32

+

134

32

−=−=+ ls

rs

rl ee ϕϕϕϕ απν

απν

1

1

34

−se απν

The fluctuation is cut in two parts

KG normalization requires that

The relative weight of the left part is

It is identified as Planck distribution απλ

πα

ν ssT

eH

TH38 ;

43 ;

1

1 2

H ==

−

Assuming a symmetry lr ϕϕ =

Hawking “temperature” corresponds to the wave length

11/6/2017 QFC2017 - Pisa

QFC2017 - Pisa

Close to the horizon. Role of QP ( )

( ) 0)]()([)1(41

0)(

22

2

=∂−+−+∂−−+−

=++∂∂−+−

kkkkkk

kkkkkk

ksksiisikki

kkiksksii

ξαξνχαβχαβ

ξβ

ξβ

αχαχν

222223

2

||21

21 ;...

18),(

Agslkilk n

n

ββαν ==

+−=Λ

),(exp)33

2(),( 21 ikxkikkdkezxC

zi +Λ−−= ∫− νανχ γγν

322221 81

48114

641 ;

241 ν

αν

ναν

γαν

γ nn lilii−+−−=−=

11/6/2017

QFC2017 - Pisa

Close to the horizon 3

23

2

22

211

814

272

32 ,

sli

sl

six nn

s ανν

αν

γγγχ γ +−=−−=≈ −

( ) 3/11,min

ααν n

nr l

llxs=>>

This solution holds in the window

For smaller x fluctuations are regular and do not zero on the horizon

3

22

1

818

34

)(1

;1)(

sl

sT

eN

n

H

TH

ανπ

απ

ν

ων

+=

−=

−

Frequency distribution of the HR deviates from the Planck function at high frequencies.

Corrections

due to QP

Parentani etal, 2009,2011 (numerically)

11/6/2017

VF, R. Schilling, 2012 (analytically)

QFC2017 - Pisa

J.Macher, R.Parentani PRD79,124008(2009), PRA80, 043601(2009)

222242

2 )]([)(2

),( ln xkvxsksklxkix nkk −=+=Ω⇒−=∂ νχ

Saddle point equation

subsonic supersonic

11/6/2017

Quantization of fluctuations

11/6/2017 QFC2017 - Pisa

ϑϑβ

ϑσϑ

ϑσϑϑσϑ

∇∇−+−

−=

++

++

)(4

1)1(21

])()([4

20

0

40

20

fgf

DDfiL

z

yy

ξ20)( fzpx −= Canonical momentum

),( zxχ Canonical coordinate

)'()],(),,'([20 xxiNzxzxf −= δχξ

=

ξ

χϑ

22

1

Y.Vinish, VF, IJMP, 2016 P. E. Larre and I. Carusotto, PRA 2015

Experimental realizations in progress

11/6/2017 QFC2017 - Pisa

1. U. Leonhardt etal – light reflected by a soliton in a waveguide

2. D. Fazzio etal – light reflected by a soliton in a waveguide

3. J. Steinhauer etal – BEC observation of a horizon and radiation

4. S. Bar-Ad etal – “luminous fluid” in a nonlinear medium

5. S. Weinfurthner – “bathtub” experiments

Optical vortex – Laguerre-Gauss beam

11/6/2017 QFC2017 - Pisa

φβ

ϕβ

πφ

ˆˆ),,(1)(

2)(|!|

2),( )(/2)(||

2

2

2

22

2

rnr

frfzr

eeezw

rzwn

zrP inzwrzRri

n

−=∇−=

=

⊥

−−

v

φβ

inzRri

ee )(

2−

11/6/2017 QFC2017 - Pisa

Optical vortex – Laguerre-Gauss beam

“Kerr” metric

−−

−−

=

=∂−∂−

1000000010

0))det(()det(

1

2

20

2

rrvv

rvvvs

g

ggg

r

r

φ

φ

µν

νµνµν

µµν

ξ

11/6/2017 QFC2017 - Pisa

( )

+−

−−−= φφ φdzdrvdrdr

vssdzvs

sdl

r

21 22222

222

022

0 ,0 2220

2 =−=− rvsvs

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Position of event horizon

2

2

0

)(frrPgI

=β

II– subsonic I, III - supersonic

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Ergoregion

rn

frrPgI

20

2

2

2

0

)(ββ

+=

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Geometry of the flow

F. Marino PPA2008.2009

11/6/2017 QFC2017 - Pisa

Superradiance

−−

−

=

+=

2

nrmni

riri

Te

eRe

βν

νν

ψ

ψScattering of a cylindrical wave

−=− 2

22 ||||

1||1hr

mnTRβ

νν

1|| ,0||1 22

2 ><

− R

rmnT

hβν

ν

The scattered wave is stronger than the incident

11/6/2017 QFC2017 - Pisa

Hawking radiation-resonance

−

−=

−

22

1

3s2

||11e)(hr

mnTNβ

νν

νν

απ

2hr

nmβ

νν −=

φφν

φν

πτνπλ

τλβ

ν

vr

mr

nm

h

h

/2 ,/2

02

==

=⇒=−

2||||

12

3)( TsNνπ

αν =

Spectrum of Hawking radiation

Resonance condition: An integer number of wavelengths coincide with the time (propagation distance ) necessary for one full rotation of the vortex

11/6/2017 QFC2017 - Pisa

Does white horizon play a role?

horizonblack -~2

−=

+

+h

h rmnβ

νν

horizon white-~2

−=

−

−h

h rmnβ

νν

+−< hh rr

+−>

hh rr νν

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2

222

222 1

srvs

rmn

sr−

−

−=∂β

νψ

−<

hrνν

−+<<

hh rr ννν

νν <−hr

regular regime

There is a barrier between the horizons reflecting the radiation. Can it be a BH bomb? Press, Teukolsky, Nature, 1972

11/6/2017 QFC2017 - Pisa

Concluding remarks • Hawking radiation can be studied in laboratory. • A possible way is to create an optical transonic flow.

Currently in progress.

• Studying classical fluctuations may be feasible and presents a special interest, especially superradiance.

• Resonance with a vortex can make observation of Hawking radiation by the vortex feasible.

11/6/2017 QFC2017 - Pisa