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 idler Kyoto, Japan ４－７ July2017 signal Paul M. Alsing & Michael L. Fanto Air Force Research Laboratory, Rome, NY USA Collaborator: Perry Rice, $$ - AFOSR LRIR: “Relativistic Quantum Information” Approved for public release 88ABW-2015-3227, 88ABW-2016-1701; distribution unlimited. Univ. of Miami, Oxford, OH PM: Dr. Tatjana Curcic
Black Hole Information Problem 29 April 2011 2
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 I τ ( ) – Short time behavior, Long time behavior τ S( ) – 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 3
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Simple Derivation of Unruh Effect: zero vs. constant acceleration ´ ´ 29 April 2011 5
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). 29 April 2011 6
Simple Derivation of Unruh Effect: zero vs. constant acceleration ´ ´ 29 April 2011 7
Simple Derivation of Unruh Effect: Bosons Frequency Transformations in SR: a = constant; (uniform acceleration) e φ ⇒ i ( , ) t z 29 April 2011 8
Simple Derivation of Unruh Effect: Bosons = Ω = Ω s i c a i ( a c ) − π = − ω − = i 2 b i c a , i e ∞ ∫ − − − = Γ s 1 by s ln b dy y e e ( ) s 0 > > Re 0, Re 0 b s 1 a c ≡ ⇒ = kT Ω − π Unruh / kT e 1 2 Unruh Alsing & Milonni, 29 April 2011 Am.J.Phys. 72 1524 (2004) 9
Simple Derivation of Unruh Effect: Fermions 29 April 2011 Alsing & Milonni, Am.J.Phys. 72 1524 (2004) 10
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29 April 2011 Sean Carroll, Spacetime and Geometry , Chap 9, (2004) 13
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≈ κ κ − ρ = e κ 2 2 z 2 2 z e ( dt dz ), ( ) z κ ( / ) a c ( / ) c 4 GM c = ⇒ = 2 GM κ = = = T T , r π π U H s 2 2 2 k 2 k r 4 GM c s B B 29 April 2011 surface gravity Schwarzschild radius 16
The Hawking Effect: Modes 29 April 2011 17
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Channel (Holevo) Capacity = = − πω κ 2 2 /( /c) z tanh r e 19
Black Hole Information Problem 29 April 2011 20
BH as PDC with depleted pump P.M. Alsing, Classical & Quant. Grav. 32 , 075010 (2015); arXiv:1408.4491 29 April 2011 21
Justification for Model 29 April 2011 22
BH as PDC with depleted pump see Heisenberg approach: P. Nation and M. Blencowe: New J. Phys. 12 095013 (2010), arXiv: 1004.0522 23
BH as PDC with depleted pump n n 0 , n p 0 s 29 April 2011 24
BH as PDC with depleted pump 29 April 2011 25
BH as PDC with depleted pump ≈ n n 0 , n p 0 s 29 April 2011 p336 p446 26
Channel (Holevo) Capacity 29 April 2011 27
Page Information Curves τ τ = d n ( ) d 0 p I τ I τ I τ ( ) ( ) ( ) τ τ S( ) S( ) Page, PRL 71 , 1291 (1993); gr-qc/9305007 Page, PRL 71 , 3743 (1993); gr-qc/9306083 29 April 2011 28
Page Information Curves τ τ = d n ( ) d 0 p I τ I τ I τ ( ) ( ) ( ) τ τ S( ) S( ) Page, PRL 71 , 1291 (1993); gr-qc/9305007 Page, PRL 71 , 3743 (1993); gr-qc/9306083 τ τ = τ τ = d n ( ) d 0 d n ( ) d 0 p p 29 April 2011 29
Page Information Curves τ τ = d n ( ) d 0 p I τ I τ I τ ( ) ( ) ( ) τ τ S( ) S( ) Page, PRL 71 , 1291 (1993); gr-qc/9305007 Page, PRL 71 , 3743 (1993); gr-qc/9306083 I τ thermal τ ( ) S ( ) τ S( ) 29 April 2011 30
Relative Entropy of BH ’pump’ to emitted HawkRad signal τ = 0.42 τ = Signal: initial vacuum 0 BH `pump’ Signal Initial BH `pump’ CS Final BH ’pump’: τ = 0 Single-mode squeezed state τ = 0.55 Signal: final τ = 0.55 31
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 32
Justification for Model 29 April 2011 33
Spontaneous parametric down conversion as an analogue for gravitational particle production One Shot Decoupling Model Bradler and Adami, arXiv:1505.0284 BH `pump’ empty Hawking mode radiation modes U k U N U 2 U 1 29 April 2011 34
Spontaneous parametric down conversion as an analogue for gravitational particle production One Shot Decoupling Model 29 April 2011 35
Spontaneous parametric down conversion as an analogue for gravitational particle production Reduced Density Matrices ‘ ′ Φ = N j N ≡ (notation: j k ) 36 N
Spontaneous parametric down conversion as an analogue for gravitational particle production Probabilities Entropy τ S( ) τ I( ) τ S( ) Page (1993) τ I( ) τ S( ) 29 April 2011 Page (2013) 37
Spontaneous parametric down conversion as an analogue for gravitational particle production Original Probabilities Refinement of Probabilities ≡ (notation: j k ) N = = n 10 n 25 p p 0 0 29 April 2011 38
Spontaneous parametric down conversion as an analogue for gravitational particle production Page Information Curves = n 25 p 0 n n n n s i , p s i , p 0 0 = n 100 p 0 n n s i , p 0 39
Analogy of BH evaporation to SPDC process 29 April 2011 40
Consideration of coherence length of BH `pump’ source particles 29 April 2011 41
Conclusion thermal τ S ( ) I τ I τ ( ) ( ) τ S( ) τ S( ) Alsing: CQG 32, 075010, (2015) Page (1993) τ I( ) τ S( ) Alsing and Fanto, CQG 33, 015005 (2016) Page (2013) 42
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