Mixing of positive and negative frequency waves at the optical horizon

Maxime Jacquet, Vyome Singh and Friedrich König Quantum Optics Group, University of St Andrews ICNFP’17 – 22/08/17

1 Curvature of spacetime and event horizon

Nasa website Luminet, 1979

General Relativity identifies gravity with curvature of spacetime

Point of no return: event horizon Penrose, 1972 General Relativity: nothing can escape a black hole

2 Spontaneous emission from the vacuum out

in

3 Spontaneous emission from the vacuum out

† ∗ in: 휙 = d휔(푎휔푓휔 + 푎휔푓휔 ) 푎 0 = 0

† ∗ out: 휙 = d휔(푎 휔퐹휔 + 푎 휔퐹휔 ) 푎 0 = 0

′ ∗ Express out modes in terms of in modes: 퐹휔 = d휔 (훼휔휔′푓휔′ + 훽휔휔′푓휔′)  0 ≠ 0

푎 0 = 훽휔휔′ 1 > 0 휔′ Spontaneous emission from the vacuum! in Black hole  Hawking radiation 휅푡 휔푖푛 ∼ 푒 휔표푢푡 4 surface gravity of the black hole Negative frequency waves positive negative frequency frequency out wave wave

† ∗ in: 휙 = d휔(푎휔푓휔 + 푎휔푓휔 ) 푎 0 = 0 positive negative frequency frequency 휙 = d휔(푎 퐹 + 푎 † 퐹∗ ) 푎 0 = 0 out: 휔 휔 휔 휔 wave wave

′ ∗ Express out modes in terms of in modes: 퐹휔 = d휔 (훼휔휔′푓휔′ + 훽휔휔′푓휔′) positive frequency wave  0 ≠ 0 mixing of positive and negative frequency waves  Spontaneous emission from the vacuum!  mixing of creation and annihilation operator Black hole  Hawking radiation

in 휅푡 휔푖푛 ∼ 푒 휔표푢푡 5 surface gravity of the black hole Negative frequency waves positive negative frequency frequency out wave wave

† ∗ in: 휙 = d휔(푎휔푓휔 + 푎휔푓휔 ) 푎 0 = 0 positive negative frequency frequency 휙 = d휔(푎 퐹 + 푎 † 퐹∗ ) 푎 0 = 0 out: 휔 휔 휔 휔 wave wave

′ ∗ Express out modes in terms of in modes: 퐹휔 = d휔 (훼휔휔′푓휔′ + 훽휔휔′푓휔′) positive frequency wave  0 ≠ 0 mixing of positive and negative frequency waves  Spontaneous emission from the vacuum!  mixing of creation and annihilation operator Black hole  Hawking radiation Does this effect occur only in 휅푡 in a gravitational setup? 휔푖푛 ∼ 푒 휔표푢푡 6 surface gravity of the black hole Spacetime flow at the horizon

Subluminal flow of spacetime Superluminal flow of spacetime

푑푠2 = −푑푡2 + 푑푟 + 훽푑푡 2 + 푟2 푑휃 2 + 푠푖푛2휃 푑휙 2

Space flows radially inwards at velocity 훽 = 2푀/푟. At the horizon, 훽 = 푐: nothing can escape from the inside of a black hole. t ever increases towards the horizon -> light from infalling objects redshifts

7 Spacetime flow at the horizon

Subluminal flow of spacetime Superluminal flow of spacetime

Event horizon separates region of sub from superluminal flow

Unruh (1974): spacetime as a moving fluid – the waterfall model 8 Spacetime flow at the horizon – in the lab Unruh, 1981: Flow gradient can be realized in laboratories + analogue systems can produce HR • Superfluid helium, superconducting circuits, waveguides, slow light… • Water waves • Rousseaux et al. NJP 2008. Transfer of energy to negative frequency wave at the white hole horizon. • Weinfurtner et al. PRL 2011. Frequency shift of water waves at the white hole horizon. • Rousseaux et al. PRL 2016. Noise correlations across the water-wave horizon. • Phonons in a BEC • Steinhauer Nat Phys 2016. Hawking radiation at the black hole horizon?

9 Creating analogue horizons with moving Refractive Index Front (RIF)

Philbin et al., Science (2008)

Pulse creates RIF  modifies speed of light by Kerr effect Subluminal Superluminal Subluminal

Moving analogue horizon in medium at rest

Resulting HR temperature can be very high (up to 1000K)

10 Optical analogue and RIF

휔 휕휔 Dispersion: 휔 = 푓 휆 → 푣푝 ≠ 푣푔 → 푘 ≠ 휕푘 휔 = 푓 푟푒푓푟푎푐푡푖푣푒 푖푛푑푒푥

Philbin et al., Science (2008) Back of pulse blueshifts incoming light Front of pulse redshifts incoming light

Moving frame: incoming wave slows down, turns around frequency shifted ≡ event horizon 11 Theoretical advances

Recent efforts in optical event horizons Rubino et al. PRL (2012), Robertson PRE 90 (2014), Belgiorno et al. PRD 91 (2015), Linder et al. PRD 93 (2016)

Finazzi and model for spontaneous emission Carusotto, at a moving RIF in a nonlinear dielectric PRA 89 (2013) Light emitted from the vacuum in the presence and absence of a horizon

Jacquet and RIF can act as black hole, white hole and König, PRA 92 (2015) horizonless emitter First complete spectrum of light from vacuum mixing of positive and negative frequencies 12 Dispersion relation of the dielectric in the frame of the pulse

n (ζ)-nR

Moving frame dispersion relation δn

0 0 ζ

Conserved co-moving frequency 휔′ 휕휔′ ω' Group velocity 휕푘′

k'

Negative lab-frequency waves Positive lab-frequency waves 13 Dispersion relation of the dielectric in the frame of the pulse

n (ζ)-nR

Moving frame dispersion relation δn

0 0 ζ

ω' ' ω

k'

14 Simultaneous black- and white-hole horizons!

n (ζ)-nR

Moving frame dispersion relation δn

0 0 ζ

Analogue black hole horizon ω'

Analogue white hole horizon 휔′ k' Frequency is conserved Horizonless waves can scatter into other frequency intervals waves that have same 휔′ 15 Spontaneous emission of light

Incoming modes scatter into outgoing modes Analogue black-hole horizon  mixing positive and negative norm waves Hawking Partner Quasi pair-wise emission radiation wave

1

0 ω'

Mode-mode correlation map in the moving frame

16 Spontaneous emission of light

Lab Frame spectrum of spontaneous emission in bulk fused silica Incoming modes scatter into outgoing modes  mixing

) positive and negative norm 1

- Black-hole emission

waves

m 1

- in the partner wave (s Hawking Partner radiation

wave density Black-hole emission

in Hawking radiation Spectral Spectral Spectral correlations?  on the arxiv at the Wavelength (nm) end of the month 17 Scattering of a positive norm CW at the horizon

• Scattering of coherent state: Spontaneous emission 푙푎푠푒푟, 2 2 훼푙푎푠푒푟 푛 훼푙푎푠푒푟 = a + 훽 훼푙푎푠푒푟 Parametric amplification

18 Scattering of a positive norm CW at the horizon

• Scattering of coherent state: Laboratory frame dispersion relation ω ω'

k' ω’=cst contour line k

19 20 Scattering of positive norm CW at horizon

Frequency shifting of a 532nm CW at a horizon in PCF NL 1.5-670

21 Scattering of positive norm CW at horizon

• Scattering of coherent state: Laboratory frame dispersion relation ω ω'

훼푙푎푠푒푟 푛 훼푙푎푠푒푟 푙푎푠푒푟, 2 2 = a + 훽 훼 k' 푙푎푠푒푟 ω’=cst contour line Spontaneous emission Parametric amplification k

Expect about 0.1 photon/second at 222nm

22 Negative frequency at the horizon Scattering of light at a refractive index front

• Pulse in nonlinear dielectric creates an analogue system to black and white holes • The front of the pulse can act as a black hole, a white hole, or a horizonless emitter  Towards the experimental • Pairs emitted via the Hawking effect can measurement of pairs be observed in the optical frequency range, partner in the UV and Hawking radiation in the visible.

23 24 Light-matter interaction in a dispersive medium Laboratory frame dispersion relation ω ω2

Lagrangian for light matter Optical interaction: 휕 퐴 2 휕 퐴 2 ℒ = 푡 − 푥 frequencies 8휋푐2 8휋 3 2 2 휕푡푃푖 푃푖 퐴 + 2 − + 휕푡푃푖 2휅푖휔푖 2휅푖 푐 푖=1 ω1 k Sellmeier dispersion relation:

2 2 3 2 2 c k =ω[1+∑i=1 (4πκi)/(1-ω /ωi )]

25 Light-matter interaction in a dispersive medium Laboratory frame dispersion relation ω ω2

Lagrangian for light matter optical interaction:

2 2 Positive

휕 퐴 휕 퐴 frequencies ℒ = 푡 − 푥 8휋푐2 8휋 3 2 2 휕푡푃푖 푃푖 퐴 + 2 − + 휕푡푃푖 2휅푖휔푖 2휅푖 푐 푖=1 ω1 -ω1 k optical

Sellmeier dispersion relation: negative frequencies

2 2 3 2 2 c k =ω[1+∑i=1 (4πκi)/(1-ω /ωi )]

26 -ω2 Light-matter interaction in a dispersive medium Laboratory frame dispersion relation Lagrangian for light matter ω

interaction: Positive

2 2 waves 휕푡퐴 휕푥퐴 ℒ = 2 − 푎

8휋푐 8휋 norm 3 2 2 휕푡푃푖 푃푖 퐴 + 2 − + 휕푡푃푖 2휅푖휔푖 2휅푖 푐 푖=1

k Negative

Sellmeier dispersion relation: waves

2 2 3 2 2 † norm c k =ω[1+∑i=1 (4πκi)/(1-ω /ωi )] 푎

27 Light-matter interaction in a dispersive medium Lorentz boost to frame moving at speed u: Laboratory frame dispersion relation 휔′ = 훾(휔 − 푢푘) ω ω' 푘′ = 훾(푘 − 휔 푢/푐2)

Stationary RIF in the moving frame: n (ζ)-nR

δn

0 0 ζ k' Moving frame dispersion relation

k

ω'

28 k' 5 1 2 3 4 5 4 ω' 3 2 1 k'

1 ' 2 ' 3 ' 4 ' 5 ℏ ω minLℏ ω minR ℏ ω maxL ℏ ω maxR 10- 1

'

ω

' - 2 I 10

y

t

i

s

n

e

d

x

u - 3 l 10

F

10- 4 0.03 0.06 0.09 0.12 29 ℏ휔′ℏω'(eV(eV)) 30 Scattering of positive norm CW at horizon Spectral density

Long IR Short IR wavelength: wavelength: Green CW wave Green CW wave interacts with CW @532 nm InfraRed Pulse interacts with back of pulse front of pulse  blue shift  red shift

532 850 Wavelength31 (nm) 32 Negative frequency waves – nonlinear optics

Coupling to waves that have negative laboratory frequency is possible  creation of photon pairs  linear scattering process Fourier analysis: all light waves oscillate with both positive and negative frequencies +∞ 퐴 = 퐴 휔 푒−푖휔푡푑휔 −∞

Electric field is real  퐴 −휔 = 퐴 ∗ 휔

33 Negative frequency waves – nonlinear optics Momentum conservation in the lab frame ≡ energy conservation in the moving frame

Couple 7-fs pulse into few mm-long PCF Rubino, PRL (2012), McLenaghan, NJP (2014)

퐴 −휔 = 퐴 ∗ 휔

Negative frequency waves recently observed in water experiments Weinfurtner, PRL (2011), Rousseaux, NJP (2008), 34