Walid CHAIBI

ARTEMIS, Université Côte d’Azur, NICE-France

AtomQT Workshop, Hanover 25-27 Feb 2019

03/04/2019

Newtonian and Shapiro noises in Atom

Interferometry based Gravitational Wave Detectors

1

2

Summary

Mitigation of Newtonian noise

Shapiro noise…What is it? Does it really exist?

B. Chauvineau & O. Minazzoli

03/04/2019

Measuring a GW Isolated system in free fall

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Equivalence principle We need at least two systems in free fall

Laser

synchronization comparison

Gravitational Wave

Changes the propagation time

Free falling Free falling

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Principle of detection

𝑑2𝑥

𝑑𝑡2= −𝑐2𝜕𝑥ℎ

00 ≅ 0

metric 𝑔𝛼𝛽 = 𝜂𝛼𝛽 + ℎ𝛼𝛽

ℎ𝛼𝛽 =

ℎ00 0 0 00 ℎ+ 0 00 0 −ℎ+ 00 0 0 0

𝜂𝛼𝛽 =

−1 0 0 00 1 0 00 0 1 00 0 0 1

ℎ00 =4𝐺𝑀

𝑐2𝑅M : source mass

R : distance to the sourceMinkowski

Transverse Traceless (TT) gauge

𝑑𝑠2 = 0 = 𝑔𝛼𝛽 𝑑𝑥𝛼𝑑𝑥𝛽Light : Null geodesic

−𝑐2𝑑𝑡2 + 1 + ℎ+ 𝑑𝑥2 = 0

𝑑𝑥 ≅ ±𝑐 𝑑𝑡 1 +1

2ℎ+

1st order

𝐿 = 𝑐 Δ𝑡 + 𝑐 Δ𝑡

ℎ+ 𝑡′ 𝑑𝑡′Shapiro time delay

Spatial coordinates remain unchanged

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Effect of the Gravitational wave

Motion equation

𝑇ℎ ≫ 𝐿/𝑐𝑑2𝑥𝑚𝑖𝑟𝑟𝑜𝑟

𝑑𝑡2=1

2𝐿 ℎ+

Fermi coordinates Reduce the GW effect to a mirror

motion

5

Detectorphf ~

Sensitivity increased by the Finesse F

Light VS Atoms

Δ𝜑𝐷𝑒𝑡𝑒𝑐𝑡𝑜𝑟~𝐹 × 𝐻𝑜𝑝 × ℎ

laser

L

@ high frequency

x1x2

Ultimate limit at low frequency : Newtonian Noise = Gravitational motion of test masses

ℎ = ℎ+ + 𝜒𝑁𝑁 + 𝜀𝑠ℎ𝑜𝑡03/04/2019

Detectoratf ~

Sensitivity increased by the number of recoil n

Δ𝜑𝐷𝑒𝑡𝑒𝑐𝑡𝑜𝑟~𝑛 × 𝐻𝑎𝑡 × ℎ

2-photon atom interferometry : Phase difference of interrogation lasers

Laser1 Laser2

12int LL

laser

L

@ high frequency

x1x2

Dimopoulos et al. , PRD 78 122002 (2008)

Monte-Carlo approximation

Estimation of the integral 𝐼 𝑓 = 01𝑓 𝑥 𝑑𝑥 = 𝑓 𝑥𝒇 𝒙 : deterministic function

Monte- Carlo approximation : 𝒉+,𝑵 =𝟏

𝑵 𝒏=𝟎𝑵 𝒉𝒏

Noise : 𝝈𝑵 =𝑪𝑵𝑵×𝝈𝑵𝑵

𝑵

𝟐+

𝑪𝒔𝒉𝒐𝒕×𝝈𝒔𝒉𝒐𝒕

𝑵

𝟐

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We consider 𝑥 as a random variable 𝑓 𝑥 is a random variable, 𝜎𝑓2 its variance

Monte Carlo estimation

Limit : lim𝑁→∞

𝐼𝑁 𝑓 = 𝐼 𝑓

Error : 𝜀𝑁 = 𝐼 𝑓 − 𝐼𝑁 𝑓𝑰𝑵 𝒇 =

𝟏

𝑵

𝒏=𝟎

𝑵

𝒇 𝒙𝒏

ℎ 𝜂 = ℎ+ + 𝜒𝑁𝑁 + 𝜀𝑠ℎ𝑜𝑡

𝜂 = 𝜒𝑁𝑁 , 𝜀𝑠ℎ𝑜𝑡 is centered, stationary, random variable

Gravitational signal : 𝒉+ = 𝒉 𝜼

Noise : 𝜎ℎ = 𝜎𝑁𝑁2 + 𝜎𝑠ℎ𝑜𝑡

2

Extraction of the GW signal

𝝈𝜺𝑵 ≈𝝈𝒇

𝑵If we change the statistics of 𝑥𝑛 𝝈𝜺𝑵 ≈

𝑪 × 𝝈𝒇

𝑵

𝑪 < 𝟏 : Variance reduction

We repeat the same experiment at different position with the same

interrogating beam

Multiple atom interferometers

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Shot noise

𝐶𝑠ℎ𝑜𝑡 = 1Shot noise contribution (unless atoms clouds are entangled with each other)

𝝈𝒔𝒉𝒐𝒕

𝑵

Newtonian noise

(upgraded) Saulson model: for afrequency f, space is subdivided into

cubic cells.

ℒ 𝑓 =𝜆

2=𝑐𝑠𝑜𝑢𝑛𝑑2𝑓

Within a single cube: The mass density fluctuates according to noise variance (seismic, infrasound)

Between cubes : residual correlation between nearby cubes to satisfy mass conservation

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Homogeneous medium : Rayleigh Waves

Mykkeltveit et al. Bulletin of the SeismologicalSociety of America, 73 173-186 (1983)

km/s 3~ Hz, 4Hz 6.1 vf

Saulson model

Braun et al. Bulletin of the Seismological Society of America, 98 1876-1886 (2008)

Inhomogeneous medium

Geophysical model

Newtonian noise contribution

𝑪𝑵𝑵 𝒇 𝝈𝑵𝑵

𝑵

Correlation 𝐶 > 1

anti-correlation 𝐶 < 1

decorrelation 𝐶 = 1

𝜹

ℒ(𝒇)

Gravitational acceleration correlation function

03/04/2019

Infrasound NN noise (air)

9~N

Seismic NN noise (ground)

Sensitivity curve

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𝑳 = 𝟏𝟔 km, 𝜹 = 𝟐𝟎𝟎 m, 𝑵 = 𝟖𝟎, multiple diffraction 𝒏𝒅 = 𝟏𝟎𝟎𝟎, 𝝈𝒔𝒉𝒐𝒕 = 𝟏𝟎−𝟕𝐫𝐝/ 𝐇𝐳

03/04/2019

AdV,AdL

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Summary

Mitigation of Newtonian noise

Shapiro noise…What is it? Does it really exist?

B. Chauvineau & O. Minazzoli

03/04/2019

11

GR description of Newtonian Noise

03/04/2019

𝜈𝛼𝛽 ≅

2𝑈

𝑐20 0 0

02𝑈

𝑐20 0

0 02𝑈

𝑐20

0 0 02𝑈

𝑐2

𝑑2𝑥

𝑑𝑡2= −𝜕𝑥𝑈

metric 𝑔𝛼𝛽 = 𝜂𝛼𝛽 + 𝜈𝛼𝛽

Metric chosen so the applied force is Newtonian (post Newtonian development)

Δ𝑈 = −4𝜋𝐺𝜚

Newtonian potential : Poisson equation

Newtonian noise

𝑑𝑠2 = 0 = 𝑔𝛼𝛽 𝑑𝑥𝛼𝑑𝑥𝛽Light : Null geodesic −𝑐2 1 −

2 𝑈

𝑐2𝑑𝑡2 + 1 +

2 𝑈

𝑐2𝑑𝑥2 = 0

𝒄𝜟𝒕 = 𝑳 + 𝜹𝒙𝟐 − 𝜹𝒙𝟏 + 𝑳

𝟐𝑼

𝒄𝟐𝒅𝒙

Shapiro noise

Newtonian noise

Is this a real effect, or is it a representation artefact?

Effect on a light beam propagation

2 gravitational effects : GW + local gravitational field

test mass test mass

moving mass

Guedanken experiment

12

Pulsing sphere

Schwarzschild metric

Schwarzschild metric

MinkowskimetricTest

massTest

mass

Test masses rigidly attached

photon path

No Newtonian noise, Shapiro delay independent of the rigidity of the test masses attachment.

If the problem can be brought into a pure motion of test masses, it would depend on the rigidity of the attachment.

Shapiro noise is a “real” physical effect03/04/2019

03/04/2019 13

along LIGO 4 km arm, integration over 1

year

Not a new effect…

Pulsar Time Array (PTA) to measure very low frequency GW

Takes into account both the motion of the earth and the modification of

the surrounding gravitation field

Saulson model ℒ

Δ𝑧𝐿

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ℎ𝑁𝑁 ≅1

2 𝑅0

𝑅𝐸 𝐺 Δ𝑀

𝑟2

2

×2𝜋𝑟 × 𝑑𝑟 × Δ𝑧

ℒ3

1/2

×1

4𝜋𝑓2×1

𝐿

Newtonian Noise

Newtonian acceleration

quadratic summation

𝒉𝑵𝑵 ≅𝑮

𝟒 𝟐𝝅𝟑𝟐

𝜟𝝔

𝑳×

𝜟𝒛 𝓛𝟑/𝟐

𝒇𝟐×

𝟏

𝑹𝟎𝟐−

𝟏

𝑹𝑬𝟐

𝟏/𝟐

Local effect

ℎ𝑆𝑁 ≅1

𝑐2 𝑅0

𝑅𝐸 𝐺 Δ𝑀

𝑟

2

×2𝜋𝑟 × 𝑑𝑟 × Δ𝑧

ℒ3

1/2

×1

𝐿/ℒ

Shapiro Noise

Newtonian potential

integrated over 𝐿

𝒉𝑺𝑵 ≅𝟐 𝟐𝝅𝑮

𝒄𝟐𝜟𝒛

𝑳× 𝜟𝝔 × 𝒍𝒏

𝑹𝑬𝑹𝟎

𝟏/𝟐

Non local effect

Estimation…

𝑅0

10-5 N

/𝐇𝐳

Hz

air 8 km 170 m 4.8 × 10-8 kg m-3/√Hz

ocean 4 km 1000 km 750 m 1.4 kg m-3/√Hz

ground 2 km 2 × 10-6 kg m-3/√Hz

𝑅0

2ℒ

ℒ 1 Δ𝜚 1 Δ𝑧

ℒ/2

ℒ/2

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Estimation…

“Acoustic Ambient Noise in the Ocean: Spectra and Sources” G. M. Wenz, The Journal of the Acoustical Society of America 34, 1936 (1962)

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OceanShapiro noise

03/04/2019

Sensitivity curve : Very rough estimation

OceanShapiro noise

ET sensitivity curves

Effect of the Michelson configuration? The Monte-Carlo mitigation?

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Conslusion : site choice…

Sensitivity law scale : 𝑺 ∝ 𝑳 × 𝑳/ℒ 𝟏/𝟐

baseline Number of AIs

Shapiro noise is a non local effectCorrelated noise between

different detectors

Seismic Newtonian Noise Δ𝜚 = 𝜚0 ×Δ𝑥𝑠𝑒𝑖𝑠𝑚

ℒℒ 𝑓 =

𝑐𝑔

2 𝑓

Lower 𝑐𝑔 → Higher Δ𝑥𝑠𝑒𝑖𝑠𝑚, but lower 𝜚0

Find the compromise