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Mon & Wed is for theory, Tues & Thur is for experiments, Fri is for drinkin` and thinkin - Close enough! Spontaneous parametric down conversion with a depleted pump as an analogue for gravitational particle production RQI 2017

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signal idler

Paul M. Alsing & Michael L. Fanto

Air Force Research Laboratory, Rome, NY USA Collaborator: Perry Rice,

  • Univ. of Miami, Oxford, OH

RQI 2017 Kyoto, Japan ４－７July2017

Spontaneous parametric down conversion with a depleted pump as an analogue for gravitational particle production

Mon & Wed is for theory,

  • Close enough!

Fri is for “drinkin` and thinkin Tues & Thur is for experiments,

$$ - AFOSR LRIR: “Relativistic Quantum Information” Approved for public release 88ABW-2015-3227, 88ABW-2016-1701; distribution unlimited. PM: Dr. Tatjana Curcic

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Black Hole Information Problem

29 April 2011

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Outline

  • Classical information transmission capacity of quantum black holes;

Adami & Ver Steeg, Class. Q. Grav. 31 (2014) 075015; arXiv:gr-qc/0407090v8

– Classical information is not lost in black hole dynamics; re-emitted in stimulated emission – Hawking radiation is spontaneous emission

  • Analogy to SPDC (spontaneous parametric down conversion)

– Hawking radiation is a two-mode squeezed state; observed state is thermal

  • Depleted BH `pump’ model (PDC) (Alsing: CQG 32, 075010, (2015); arXiv:1408.4491)

– Quantized the BH `pump’ source – Short time behavior, Long time behavior – Page Information Curves

  • One Shot Decoupling Model (Bradler & Adami: arXiv:1505.02840;

Alsing & Fanto: CQG 33, 015005 (2016), arXiv:1507.00429)

– Suggested by Alsing: CQG:2015 Future Work; closer analogy to SPDC – Page Information Curves redux

  • Summary and Conclusion

S( ) τ ( ) I τ

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4

29 April 2011

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29 April 2011

Simple Derivation of Unruh Effect: zero vs. constant acceleration

´ ´

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29 April 2011

Simple Derivation of Unruh Effect: Bosons

Frequency Transformations in SR: a = 0 (constant velocity)

Alsing & Milonni, Am.J.Phys. 72 1524 (2004); T. Padmanabhan, “Gravitation: Foundations & Frontiers,” Cambridge (2010).

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29 April 2011

Simple Derivation of Unruh Effect: zero vs. constant acceleration

´ ´

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8

29 April 2011

Simple Derivation of Unruh Effect: Bosons

Frequency Transformations in SR: a = constant; (uniform acceleration)

( , ) i t z

e φ ⇒

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9

29 April 2011

Simple Derivation of Unruh Effect: Bosons

1 ln

( ) Re 0, Re

s by s b

dy y e e s b s

∞ − − −

  = Γ     > >  

∫

2

( ) ,

i

s i c a i a c b i c a i e

π

ω

−

= Ω = Ω     = − − =  

/

1 1 2

Unruh

Unruh kT

a c kT e π

Ω

≡ ⇒ = −

Alsing & Milonni, Am.J.Phys. 72 1524 (2004)

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10

29 April 2011

Simple Derivation of Unruh Effect: Fermions

Alsing & Milonni, Am.J.Phys. 72 1524 (2004)

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29 April 2011

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12

29 April 2011

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29 April 2011 Sean Carroll, Spacetime and Geometry, Chap 9, (2004)

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29 April 2011

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15

29 April 2011

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29 April 2011

( / ) ( / ) 2 2

U H B B

a c c T T k k κ π π = ⇒ =  

4 2

, 4

s

GM c r GM κ = =

2

2

s

GM r c =

surface gravity Schwarzschild radius

2 2 2 2

( ),

z

e dt dz

κ

κ ≈ −

( )

z

z eκ ρ =

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29 April 2011

The Hawking Effect: Modes

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29 April 2011

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Channel (Holevo) Capacity

2 2 /( /c)

tanh z r e

πω κ −

= =

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Black Hole Information Problem

29 April 2011

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29 April 2011

BH as PDC with depleted pump

P.M. Alsing, Classical & Quant. Grav. 32, 075010 (2015); arXiv:1408.4491

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29 April 2011

Justification for Model

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BH as PDC with depleted pump

see Heisenberg approach: P. Nation and M. Blencowe: New J. Phys. 12 095013 (2010), arXiv: 1004.0522

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BH as PDC with depleted pump

0, p s

n n n ฀

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29 April 2011

BH as PDC with depleted pump

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29 April 2011

BH as PDC with depleted pump

p336 p446

0, p s

n n n ≈

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29 April 2011

Channel (Holevo) Capacity

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29 April 2011

( ) I τ S( ) τ ( ) I τ

Page, PRL 71, 1291 (1993); gr-qc/9305007 Page, PRL 71, 3743 (1993); gr-qc/9306083

( )

p

d n d τ τ =

Page Information Curves

( ) I τ S( ) τ

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29 April 2011

( ) I τ S( ) τ ( ) I τ

Page, PRL 71, 1291 (1993); gr-qc/9305007 Page, PRL 71, 3743 (1993); gr-qc/9306083

( )

p

d n d τ τ =

Page Information Curves

( )

p

d n d τ τ =

S( ) τ ( ) I τ

( )

p

d n d τ τ =

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29 April 2011

( ) I τ S( ) τ ( ) I τ ( ) I τ

Page, PRL 71, 1291 (1993); gr-qc/9305007 Page, PRL 71, 3743 (1993); gr-qc/9306083

S ( )

thermal τ

S( ) τ

( )

p

d n d τ τ =

Page Information Curves

S( ) τ ( ) I τ

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Initial BH `pump’ CS Signal: initial vacuum

Relative Entropy

  • f BH ’pump’ to emitted HawkRad signal

τ =

Final BH ’pump’: Single-mode squeezed state Signal: final

0.55 τ = 0.55 τ = 0.42 τ = τ =

Signal BH `pump’

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Outline One Shot Decoupling Model

  • Justification for use of trilinear Hamiltonian for BH evaporation/particle

production – Semi-classical Hamiltonian for a collapsing spherical shell

  • One Shot Decoupling Model of Bradler and Adami, arXiv:1505.02840

– Simplified version of Master Equation suggested by Alsing: CQG 32, 075010, (2015); arXiv:1408.4491

  • Analytic formulation by Alsing and Fanto, CQG 33, 015005 (2016),

arXiv:1507.00429 – Extension of models by Alsing and by Nation and Blencowe – Page Information Curves

  • Summary and Conclusion

29 April 2011

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29 April 2011

Justification for Model

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29 April 2011

Spontaneous parametric down conversion as an analogue for gravitational particle production

1

U

2

U

N

U

One Shot Decoupling Model

Bradler and Adami, arXiv:1505.0284

k

U

BH `pump’ mode empty Hawking radiation modes

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29 April 2011

Spontaneous parametric down conversion as an analogue for gravitational particle production

One Shot Decoupling Model

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Spontaneous parametric down conversion as an analogue for gravitational particle production

Reduced Density Matrices

‘

N

N j

′ Φ =

(notation: )

N

j k ≡

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29 April 2011

Spontaneous parametric down conversion as an analogue for gravitational particle production

Probabilities Entropy

S( ) τ

S( ) τ I( ) τ S( ) τ I( ) τ

Page (1993) Page (2013)

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29 April 2011

Spontaneous parametric down conversion as an analogue for gravitational particle production

Original Probabilities Refinement of Probabilities

10

p

n = 25

p

n =

(notation: )

N

j k ≡

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Spontaneous parametric down conversion as an analogue for gravitational particle production

Page Information Curves

25

p

n =

, p s i

n n

100

p

n =

, p s i

n n

, p s i

n n

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29 April 2011

Analogy of BH evaporation to SPDC process

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29 April 2011

Consideration of coherence length

  • f BH `pump’ source particles
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Conclusion

S ( )

thermal τ

S( ) τ ( ) I τ

Alsing: CQG 32, 075010, (2015) S( ) τ I( ) τ Alsing and Fanto, CQG 33, 015005 (2016) Page (2013)

S( ) τ ( ) I τ

Page (1993)