Second Order Glauber Corre-lation of Gravitational Wavesusing the LIGO observatoriesas Hanbury Brown and Twissdetectors

January 31, 2021

Speaker: Alexander BarrettPreston JonesEmbry-Riddle AeronauticalUniversity - Prescott

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Classical interferometers

Figure 1: Classical amplitudeinterferometer [1].

Figure 2: Classically connectedintensity interferometer [2].

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The LIGO-Virgo HBT interferometer

Figure 3: An HBT interferometer isnot classically connected. Theinterference is in the non-classicalwave functions [3].

2nd order correlation function

g(2) (τ) =〈h2a(t) h2b(t+τ) 〉〈h2a(t)〉〈h2b(t+τ)〉

Figure 4: Time lag [4] ∆t→ τfor LIGO-Virgo network.

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2nd order correlations

The section on Glauber corre-lations is a demonstration thatsignals detected by LIGO-Virgoare not coherent.

Coherent or “steady” signal〈I (t)〉 = 〈I (t+ τ)〉

Second order Glauber correlationfunction

g(2) (0) = 1 +〈[∆h2(t)]2〉〈h2〉2

g(2) for AM signals are similarto chaotic signals [5].

Figure 5: Two chaotic and onecoherent signal’s second orderGlauber correlation [6]

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Time Weighted AverageUsing the conventional method the correlation function takes the

form: g2(τ) =

1

2T

∫ T−T I(t)I(t+ τ)dt

1

4T 2

∫ T−T I(t)dt

∫ T−T I(t+ τ)dt

An already known

correlation was investigated, which was generated from theoscillatory intensity described by:I(t) = I0(1 +A0 sin(ωt))

Figure 6: Glauber correlation of the oscillatory intensity using a timeweighted average [7]

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Intensity Weighted Average

g2(τ) =G(2)(t, t+ τ)

G(1)(t)G(1)(t+ τ)

G(2)(t, t+ τ) =

∫∞−∞ I(t)I(t+ τ)I(t+ 0)dt∫∞

−∞ I(t+ 0)dt

G(1)(t+ τ) =

∫∞−∞ I(t+ τ)I(t+ 0)dt∫∞

−∞ I(t+ 0)dt

(1)

Figure 7: Glauber correlation of the oscillatory intensity using an intensityweighted average [7]

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Comparison of Glauber Correlation Functions

Figure 8: Standard time average [7]

g2(τ) =1

2(2 +A2

0 cos(ωτ)) (2)

Figure 9: Intensity weighting [7]

g2(τ) =4(1 +

1

2A4

0 cos(ωτ)

(2 +A20)(2 +A2

0 cos(ωτ)(3)

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Sine-Gaussian Approximation of a Black Hole MergerUsing a function of the form: h(t) = Ar

d e−( t+tm

b)2 cos(2πωt)

Figure 10: Discovery response [8]

Figure 11: Sine-Gaussian Model[7]

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Sine-Gaussian Correlation With a Time Weighted Average

The correlation was calculated using the time weighted average:

Figure 12: Sine-Gaussian correlation with a time weighted average [7]

g2(τ) =2e−

τ2

b2 T ((T−2tm−τb ) + (T+2tm+τb ))

b√π(((T−2tm√

2b) + (T+2tm√

2b))((T−2(tm+τ)√

2b) + (T+2(tm+τ)√

2b)))

(4)

limτ→∞

g2(τ) = 0

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Sine-Gaussian Correlation With an Intensity WeightedAverage

The correlation was calculated using an intensity weighted average:

Figure 13: Calculated Correlation [7]

g2(τ) =2√3e−τ 2

3b2 (5)

limτ→∞

g2(τ) = 0

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Some Characteristics of 2nd Order Glauber CorrelationFunctions

I Coherent light of a single frequency [9, 10] is defined asg2(τ) = 1.

I For a laser g2(τ = 0) = 2 for chaotic light [11].I g2(τ = 0) > 1 for most signals except e.g. steady and

bunched [9, 10].I For chaotic light g2(τ) = 0 as τ → ∞, [6].

Figure 14: Two chaotic and one coherent signal’s second order Glaubercorrelation [6]

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Questions

Figure 15: Discovery response [8]

Figure 16: Sine-Gaussian Model[7]

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References I

[1] Lumen Learning, ”Wave Optics, Young’s Double SlitExperiment”.

[2] Frank Sebastian Rotondo, “Copy Imaging with Amplitude andIntensity Interferometers.” (2004).

[3] Preston Jones, Andri Gretarsson, Ellie Gretarsson, BrennanHughey, Darrel Smith, Michele Zanolin, and DouglasSingleton, (2019), arXiv:1907.00100 [gr-qc].

[4] Yoshinori Fujii, “Study of the fast localization of coalescingbinaries with a hierarchical network of gravitational wavedetectors April 10, 2017”.

[5] A. Lebreton, I. Abram, R. Braive, I. Sagnes, I. Robert-Philip,and A. Beveratos, Phys. Rev. A 88, 013801 (2013).

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References II

[6] Wikipedia contributors, ”Degree of coherence,” Wikipedia,The Free Encyclopedia, ”Degree of coherence” (accessedJanuary 30, 2021).

[7] Alexander Barrett, Preston Jones, ”Second Order GlauberCorrelation of Gravitational Waves using the LIGOobservatories as Hanbury Brown and Twiss detectors”

[8] B.P. Abbott, et al. ”Observation of gravitational waves from abinary black hole merger.” Physical Review Letters, 116(6),Feb 2016

[9] Mark Fox, Quantum optics - an introduction (OxfordUniversity Press, 2006).

[10] R. Louden, The Quantum Theory of Light (Oxford UniversityPress, third edition, 2000).

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References III

[11] A. Lebreton, I. Abram, R. Braive, I. Sagnes, I. Robert-Philip,and A. Beveratos, Phys. Rev. A 88, 013801 (2013).

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