Atomic Clocks as Detectors of Gravitational Waves · Resume: 1) The Hafele-Keating type of experiment does not allow to measure gravitational waves with clocks 2) Clocks must be connected

Atomic Clocks as Detectors of

Gravitational Waves

Sergei Kopeikin

University of Missouri, USA

Plan of lecture:

• Gravitational wave spectrum and detectors • Accuracy and stability of clock oscillators • Pulsar clock as GW detector• Computation of GW signal• Binary systems as sources of GW signal• Are atomic clocks sensitive to gravitational

wave?• Interaction of gravitational wave with light• Measurement protocols• Summary

Image credit: Chris Henze, NASA Advanced Supercomputing Division, NASA Ames Research Center

2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

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precision timing of binary pulsars

spacecraft Doppler tracking

Supernovaexplosions

𝟏𝟎𝟒

precision global astrometry of quasar’s proper motions atom interferometry

optical clocks

Clock’s characteristics: accuracy, stability and precision

Image credit to J.R VigSLCET-TR-92-1, October 1992


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Figure credit: P. Delva et al., In: Relativistic Geodesy, D. Puetzfeld & C. Lämmerzahl (eds.) https://link.springer.com/chapter/10.1007%2F978-3-030-11500-5_2


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Accuracy records for microwave and optical clocks

Figure credit: Hartnett & Luiten,Rev. Mod. Phys., 83, 1-9, 2011


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Frequency stability for microwave and optical clocks

The 𝐴𝑙𝑙𝑎𝑛 variance of frequency:

𝜎𝑦2 𝑀,𝑇, 𝜏 =

1

2ത𝑦𝑛+1 𝜏 − ത𝑦𝑛 𝜏 2

𝑀

The average fractional frequency:

Phase of the oscillator: 𝜙 𝑡

𝑀 − the number of frequency samples𝑇 − time between each frequency sample𝜏 − the time length of each frequency

estimate

ത𝑦𝑛 𝜏 =1

2𝜋ν0

𝜙 𝑡𝑛+1 + 𝜏 − 𝜙 𝑡𝑛𝜏

𝑡𝑛 = 𝑛 𝑇 (𝑛 = 0,1,2, … ,𝑀)

Pulsar Clock as GW Detector

Gravitational Wavesthe indirect evidence through pulsar timing

Hulse & Taylor binary pulsar

PSR 1913 + 16 -- discovered in 1974

Rotational period P = 1/17 sec

Orbital period 𝑷𝒃~ 8 hr

mp

mc

8March 25-28, 2019


•

•

time

Detection of periodic sources of GWSazhin M.V., “Opportunities for detecting ultralong gravitational waves”, Sov. Astron., 22, 36, 1978

March 25-28, 2019 2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

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pulsar clock

Detection of stochastic background of GW

𝑓−5

Cosmological GW background noise

Detweiler S., “Pulsar timing measurements and the search for gravitational waves”, ApJ., 234, 1100, 1979

Foster R.S. & Backer D.C., “Constructing a pulsar timing array”, ApJ., 361, 300, 1990

Measuring two-point cross-correlations between pulsars

March 25-28, 20192-nd ISSI Workshop on Clocks, Spacetime

Metrology and Geodesy (Bern, Switzerland)10

Computation of Gravitational Wave Signal

Plane gravitational wave



Linear approximation: 𝑔𝜇𝜈 = 𝜂𝜇𝜈 + ℎ𝜇𝜈 𝑡, Ԧ𝑥

New variable: ℎ𝜇𝜈 = ℎ𝜇𝜈 −1

2𝜂𝜇𝜈ℎ

Gauge freedom: ℎ𝜇𝜈 = ℎ𝜇𝜈 + 𝜕𝜇𝑤𝜈 + 𝜕𝜈𝑤𝜇

Harmonic gauge: 𝜕𝜈 ℎ𝜈𝜇 = 0

Einstein’s equations: □ℎ𝜇𝜈 = 0 □𝑤𝜇 = 0

Plane monochromatic wave: ℎ𝜇𝜈 = Re 𝐵𝜇𝜈 cos(𝑘𝛼𝑥𝛼 + 𝜙0)

Transverse-traceless (harmonic) gauge: 𝐵00 = 0, 𝐵0𝑖= 0,𝐵11 + 𝐵22 + 𝐵33 = 0

ℎ𝑖𝑗 ≡ ℎ𝑖𝑗𝑇𝑇 = Re 𝐴𝑖𝑗 cos(𝑘𝛼𝑥

𝛼 + 𝜙0)

Regions of space around an emitter of GW

March 25-28, 20192-nd ISSI Workshop on Clocks, Spacetime Metrology and

Geodesy (Bern, Switzerland) 13

Clock A Clock B

Clock A Clock B



Coordinates and metrics of an isolated gravitating system(Kopeikin et al., PRD 59, 084023,1999)



ℎ𝑇𝑇 = 9 × 10−18Gpc

𝐷𝐿

𝑀𝑧

106𝑀⊙

53 𝑓

mHz

23

𝑀𝑧 = 1 + 𝑧𝑀1𝑀2

35

𝑀1+𝑀2

15

- the “chirp” mass of the binary

𝐷𝐿 = 1 + 𝑧 𝑟 - the luminosity distance,

𝑓 =𝜔

2𝜋

1

1+𝑧- the observed (redshifted) GW frequency

ℎ𝑖𝑗𝑇𝑇 =

2𝐺

𝑐4𝑟

𝑑2𝑰𝑖𝑗𝑇𝑇

𝑑𝑡2

𝑰𝑖𝑗𝑇𝑇 - transverse-traceless quadrupole moment

of the system emitting GW

TT metric tensor perturbation and GW strength

ℎ𝑖𝑗𝑇𝑇 =

2

𝜋

𝐺𝑀53𝜔

23

𝑐41

𝑟

For a binary system with reduced mass

1 Gpc ≃ 0.2 𝑧

𝑀 =𝑀1𝑀2

𝑀1 +𝑀2

Distance to Virgo Cluster = 0.02 Gpc

Detection of gravitational wave and its interpretation

Michelson-type Fabry-Perot

interferometer (LIGO, VIRGO)

Laser interferometer as detector of GWs

free test masses (mirrors)

suspended on strings

free mass

17March 25-28, 20192-nd ISSI Workshop on Clocks, Spacetime

Metrology and Geodesy (Bern, Switzerland)

gravitational wave



Question: If a gravitational wave stretches the distance between free test masses it must also stretch the wavelength of the laser light. How can we detect the gravitational wave if it stretches both the space distance and the light wavelength?

Answer: crucially depends on the choice of coordinates and their physical interpretation.

a) TT coordinates: Test masses are in free fall. They do not change positions with respect to TTcoordinates. The coordinate distance between the masses does not change. However, gravitationalwave changes refractive properties of space as shown by J.L. Synge (1960). The permittivity tensor ofspace in the presence of gravity field is (F. de Felici, GRG, 2, 347-357, 1971)

𝜖𝑖𝑗 =−𝑔𝑔𝑖𝑗

𝑔00= 𝛿𝑖𝑗 + ℎ𝑖𝑗

𝑇𝑇

Light in the interferometer moves through space with a variable index of refraction. It does changewavelength of light.

b) Local inertial coordinates of detector: Free test masses are in free fall. They move along geodesicswhich deviate due to the presence of time-dependent curvature of space caused by gravitational wave.Thus, the coordinate distance between the test masses changes periodically as the gravitational wavepasses on. Refractive tensor of space is identical to vacuum one. Wavelength of light does not change.

c) An intermediate choice of coordinates is possible attributing the effect of gravitational waves partially to th change in coordinate distance between the test particles and partially to the change of the refractive properties of space. The overall physical effect of gravitational wave always remains invariant (Kopeikin et al., PRD, 59, 084023, 1999).How to

How to interpret the measurement of gravitational wave signal

Binary systems – the most promising sources of gravitational waves

20March 25-28, 2019

Chirp signal

Ringdownoscillations


GW from a coalescing BBH system and GW150914 signal

Δ𝑓

𝑓= 0.3

𝑀𝑧

106𝑀⊙

53 𝑓

mHz

83 Δ𝑡obs

1 hour

Image credit to M. Colpi & A. SesanaarXiv:1610.05309http://inspirehep.net/record/1492489/plots

2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland) 21

Binaries in the gravitational wave universe

accuracy of atomic clocks today

Recent works on atom interferometry and optical clocks as gravitational wave detectors



➢ Dimopoulos S. et al., “Gravitational wave detection with atom interferometry”, Phys. Lett. B, 678, 37 (2009)

➢ Yu N. & Tinto M. “Gravitational wave detection with single-laser atom interferometers”, Gen. Rel. Grav., 43, 1943 (2011)

➢ Baker J.G. & Thorpe J.I. “Comparison of atom interferometers and light interferometers as space-based gravitational wave detectors”, PRL, 108, 211101 (2012)

➢ Graham P. W. et al., “New method for gravitational wave detection with atomic sensors”, PRL, 110, 171102 (2013)

➢ Loeb A. & Maoz D., “Using atomic clocks to detect gravitational waves”, arXiv: 1501.009996 unpublished (2015)

➢ Vutha A., “Optical frequency standards for gravitational wave detection using satellite Doppler velocimetry”, New J. Phys., 17, 063030 (2015)

➢ Kolkowitz S. et all, “Gravitational wave detection with optical lattice atomic clocks”, PRD 94, 124043 (2016)

➢ Graham P. W. et al., “A resonant mode for gravitational wave detectors based on atom interferometry”, PRD, 94, 104022 (2016)

➢ Hogan J.M. & Kasevich M.A. “Atom-interferometric gravitational-wave detection using heterodyne laser links, PRA 94, 033632 (2016)

➢ Chaibi W. et al., “Low frequency gravitational wave detection with ground-based atom interferometer arrays”, PRD, 93, 021101 (2016)

➢ Norcia M.A. et al., “Role of atoms in atomic gravitational-wave detectors”, PRA 96, 042118 (2017)

➢ Su J. et al., “Low-frequency gravitational wave detection via double optical clocks in space”, Class. Quantum Grav., 35, 085010 and 249501 (2018)



Clock AOptical Laser Link

𝑀1

𝑀2

Binary System

Atom Interferometry GW Detector

Sun

EarthClock A

LISA

TWTFT Link

Clock B

Clock A

Drag-free satellite

What type of gravitational field can atomic clocks measure?



The Hafele-Keating Experiment of 1971:Science, 177, No. 4044, 166-168, 1972

Rotationof Earth

Clock E

Clock W

USNO Clock

𝜏 − 𝑡 =1

𝑐2−𝑈⊕ + 𝑔⊕ℎ −

1

𝑐21

2𝑣⊕2 + Ԧ𝑣⊕ ⋅ Ԧ𝑣 +

1

2𝑣2 𝑡

Δ𝜏 =𝑔⊕ℎ

𝑐2−

Ω⊕𝑐2

𝑅⊕2 sin2 𝜃 ሶ𝜙

ሶ𝜙 > 0 for eastbound flight ሶ𝜙 < 0 for westbound flight

+𝜙

𝜙 − longitude𝜃 − colatitudeΩ⊕ − the angular velocity

of Earth’s rotation𝑅⊕ − Earth’s radius

𝑔⊕ − acceleration of gravity

𝑈⊕ − gravitational potential onEarth’s surface

Sagnac effectgravitational time delay

general relativity special relativity

Δ𝜏 ≡ 𝜏 − 𝜏USNO (nanosecond)

Would the Hafele-Keating type experiment measure gravitational wave?



(Loeb & Maoz, arXiv:1501.00996 “Using atomic clocks to detect gravitational waves”)

Clock BClock A

Clock A Clock B

Clock BClock A

Clocks A and B are at the same point

Clock B has been (adiabatically) transported to a different point

Gravitational wave package passes through the clocks A and B

Clock’ readings are compared. Are they the same?

Clock BClock A

Clock A Clock B

Clock B has been (adiabatically) transported back to the point with clock A.



Is atomic clock sensitive to the field of gravitational wave?A clock measure proper time 𝜏 along the clock’s worldline

𝜏 = − න

𝑡0

𝑡

𝑔00 + 2𝑔0𝑖𝑣𝑖 + 𝑔𝑖𝑗𝑣

𝑖𝑣𝑗12𝑑𝑡 =

= 𝑡 − 𝑡0 − න

𝑡0

𝑡1

2ℎ00 + ℎ0𝑖𝑣

𝑖 +1

2ℎ𝑖𝑗𝑣

𝑖𝑣𝑗 𝑑𝑡 + 𝑂(ℎ2)

The metric tensor of gravitational wave has ℎ00 = ℎ0𝑖 = 0, ℎ𝑖𝑗 = ℎ𝑖𝑗𝑇𝑇 ≠ 0.

There is no the “Newtonian gauge” with ℎ00 ≠ 0, ℎ0𝑖 ≠ 0 that is used in relativistic geodesyand which causes the gravitational time delay (“red shift”) measured in the Pound-Rebka, Hafele-Keating, GP-A, and other experiments.

Proper time of clock in the field of gravitational wave is 𝜏 = 𝑡 − 𝑡0 −1

2න

𝑡0

𝑡

ℎ𝑖𝑗𝑇𝑇𝑣𝑖𝑣𝑗𝑑𝑡

= 0 for stationary clocksResume:1) The Hafele-Keating type of experiment does not allow to measure gravitational waves with clocks2) Clocks must be connected by EM signal (RF/optical/fiber link) which phase is locked to the clocks3) EM phase propagation is found by solving the equations of light geodesics (equation of eikonal) in the

field of gravitational wave



space

time

𝑑𝑡𝐵𝑑𝜏𝐵

𝑥𝛼(𝜏𝐵)𝑥𝛼(𝜏𝐴)

Clock BClock A

𝑑𝑡𝐴 𝑑𝜏𝐴

Clock Comparison – Spacetime Diagram

𝑡1

𝑡2

𝑡3

ℓ =1

2c(𝜏𝐴3 − 𝜏𝐴1)

𝜏𝐴2 =1

2(𝜏𝐴3 + 𝜏𝐴1)

𝜏𝐴1

𝜏𝐴2

𝜏𝐴3

𝜏𝐵1

𝜏𝐵2

𝜏𝐵3

ℓ



𝑑𝜏2 = −𝑑𝑠2 = 1 − 𝑣2 − ℎ00 − 2ℎ0𝑖𝑣𝑖 − ℎ𝑖𝑗𝑣

𝑖𝑣𝑗 𝑑𝑡2

𝑑𝜏𝐵𝑑𝜏𝐴

=𝑑𝜏𝐵𝑑𝑡𝐵 clock 𝐵

×𝑑𝑡𝐵𝑑𝑡𝐴 optical link

×𝑑𝑡𝐴𝑑𝜏𝐴 clock 𝐴

Clock Comparison – Relativity at Work


=1 − 𝑣2 − ℎ00 − 2ℎ0𝑖𝑣

𝑖 − ℎ𝑖𝑗𝑣𝑖𝑣𝑗

clock 𝐵

12

1 − 𝑣2 − ℎ00 − 2ℎ0𝑖𝑣𝑖 − ℎ𝑖𝑗𝑣

𝑖𝑣𝑗clock 𝐴

12

×𝑑𝑡𝐵𝑑𝑡𝐴 optical link

𝜏 = න

line integralalong the clock path

1 − 𝑣2 − ℎ00 − 2ℎ0𝑖𝑣𝑖 − ℎ𝑖𝑗𝑣

𝑖𝑣𝑗12𝑑𝑡

TT gauge eliminates ℎ00 = ℎ0𝑖 = 0, and makes ℎ𝑖𝑗= ℎ𝑖𝑗𝑇𝑇. Therefore, the overall effect

of gravitational wave is enclosed to the optical link

𝑡𝐵 is connected to 𝑡𝐴 by the solution of the light ray propagation equation in the field of gravitational wave: 𝑡𝐵 = 𝑡𝐴 + Δ𝑡


=𝑑𝑡𝐵𝑑𝑡𝐴 optical link



. It yields

𝑡𝐵 − 𝑡𝐴 = Ԧ𝑥𝐵 − Ԧ𝑥𝐴 +1

2𝑘𝑖𝑘𝑗 𝑡𝐴

𝑡𝐵 ℎ𝑖𝑗𝑇𝑇 𝑢 𝑑𝑢

Effect of a plane gravitational wave on light frequency(Kaufmann W.J., Nature, 227, 157, 1970)

Image credit: S. Kolkowitz et al.PRD 94, 124043 (2016)


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𝑑𝑠2 = −𝑑𝑡2 + 1 + ℎ+𝑇𝑇 𝑑𝑥2 + 2ℎ×

𝑇𝑇𝑑𝑥𝑑𝑦 + 1 − ℎ+𝑇𝑇 𝑑𝑦2 + 𝑑𝑧2

ℎ+𝑇𝑇 = ℎ+ cos 2𝜋𝑓 𝑡 − 𝑧 , ℎ×

𝑇𝑇= ℎ× cos 2𝜋𝑓 𝑡 − 𝑧

Particles with constant spatial coordinates move along geodesics (free fall). Proper time of clocks that are in free fall:

𝜏 = 𝑡Proper distance between the two clocks:

ℓ = d + Δ

Δ =1

2𝑘𝑖𝑘𝑗 න

𝑡

𝑡+𝑑𝑐(1−cos 𝜃)

ℎ𝑖𝑗𝑇𝑇 𝑢 − 𝑧 𝑑𝑢

Doppler shift of EM signal: Δ𝜈

𝜈=

𝑑ℓ

𝑑𝑡=

1

21 + cos 𝜃 ℎ 𝑡 +

𝑑

𝑐−

𝑑

𝑐cos 𝜃 − ℎ 𝑡

clock clock

𝑘𝑖light

ℎ 𝑡 = ℎ+ cos 2𝜓 + ℎ× sin 2𝜓 cos 2𝜋𝑓 𝑡 − 𝑧

Doppler tracking as detector of GW Armstrong, J.W., Living Rev. Relativity (2006) 9: 1. https://doi.org/10.12942/lrr-2006-1



gravitationalwave signal

1. GW signals are observed in the Doppler tracking time series through the three pulse response.2. The response depends on the two-way light time 𝑇2 = 2L/c , the cosine of the angle

𝜃 between the GW wavevector and a unit vector from the Earth to the spacecraft, and GWproperties (Fourier spectrum and polarization state).

3. The GW response is not time-shift invariant; if T2 or 𝜃 change during the time of observation theGW response changes.

4. The GW response is a high-pass filter: in the low-frequency limit ( 𝑓gw ≪ 𝑇2−1 ), the response is

attenuated due to GW signal overlap and cancellation.

Earth

𝑇2

Atom interferometry and optical clocks as gravitational wave detectors

The effect of laser pulse on atom



Two level atom:

𝑅𝜃(𝜑) =cos

𝜃

2𝑒𝑖𝜑sin

𝜃

2

𝑒−𝑖𝜑sin𝜃

2cos

𝜃

2

The Bloch matrix:

𝜽 = 𝛀𝝉 – laser pulse area𝝉 – pulse duration

𝛀 = |𝒅 ⋅ 𝑬|/ℏ – the Rabi frequency ≈ 100 Hz

𝝋 = 𝝋𝟎 +𝝎𝒕 + 𝒌 ⋅ 𝒙– phase of laser pulse

The Bloch sphere:

𝜉

𝜂

𝜁| ۧ𝑔 =10

| ۧ𝑒 =01

| ۧ𝜓 = 𝑅𝜃(𝜙) | ۧ𝑔 = cos𝜃

2| ۧ𝑔 + 𝑒−𝑖𝜙 sin

𝜃

2| ۧ𝑒

𝑅𝜃(𝜙) | ۧ𝑒 = 𝑒𝑖𝜙 sin𝜃

2| ۧ𝑔 + cos

𝜃

2| ۧ𝑒

excited state

ground state

| ۧ𝑒 ⊗Ψ𝑒 𝑥 =01⊗Ψ𝑒 𝑥

| ۧ𝑔 ⊗Ψ𝑔 𝑥 =10⊗Ψ𝑔 𝑥

𝐸𝑒

𝐸𝑔laser pulse

𝜉 = sin 𝜃 cos𝜑𝜂 = sin 𝜃 sin𝜑𝜁 = cos 𝜃

Optical versus atom interferometer correspondence

Figure credit:https://en.wikipedia.org/wiki/Interferometry


34

𝑅𝜋/2 =1

2

1 11 1

⟹ ൞𝑅𝜋/2| ۧ𝑔 =

1

2| ۧ𝑔 + | ۧ𝑒

𝑅𝜋/2| ۧ𝑒 =1

2| ۧ𝑔 + | ۧ𝑒

𝑅𝜋 =0 11 0

⟹ ൝𝑅𝜋| ۧ𝑔 = | ۧ𝑒𝑅𝜋| ۧ𝑒 = | ۧ𝑔

𝜋 pulse:mirror

𝜋/2 pulse:beam splitterbeam combiner

Excited | ۧ𝑒 and ground | ۧ𝑔 states of atom interferometer are equivalent to two arms of optical interferometer.

𝜋/2 pulse is equivalent to beam splitter

𝜋 pulse is equivalent to mirrorlaser

beam splitter/combiner

Atomic Clock GW Detector Measurement Protocols

Figure 2b from Norcia M.A. et al., PRA, 96, 042118, 2017


36

Ramsey sequence protocol: optical clocks

laser

clock Aclock B

GW-induced effectivepath length

𝐿𝑗 =𝑐

2

ℎ

𝜔sin𝜔 𝑡𝑗 + 𝑇𝑑 − sin𝜔𝑡𝑗

Φ𝑠 = Φ𝑠 𝑎 − Φ𝑠 𝑏 =2𝜋𝜈

𝑐𝐿2 − 𝐿1

Laser short noise is subtracted inthe differential signal.

Spacecraft distance noise 𝑆𝑥 causedby the recoil of the laser pulses doesnot cancel but the requirement to thelevel of noise is less restrictive than ineLISA:

𝑆𝑥 eLISA ≃ 7 × 10−10 cm/ Hz ;

𝑆𝑥 ≃ 5 × 10−8 cm/ Hzfrom the shot noise of a 1 Watt laserdetected by telescope of 30 cmdiameter across 1-2 AU baseline.

Figure 2c from Norcia M.A. et al., PRA 96, 042118, 2017


37

Raman sequence protocol: atom Mach-Zehnder interferometer

Figure 3 from Norcia M.A. et al., PRA, 96, 042118, 2017


38

Large Momentum Transfer (LMT)-like protocol: optical clocks

LMT is insensitive to laser phase noise. The same precision can be achieved with a reduction in required resources such as atom number or averaging time by a factor ~ 4𝑀 + 1 2 . It also allows to reduce the baseline between the satellites.

LMT is useful when both the evolution time 𝑇𝑒 ≫ 𝑇d and the period of gravitational waves 𝑇gw ≫ 𝑇d where 𝑇d is the laser pulse transit time, as it yields the enhancement in signal by a factor 𝑇𝑒/𝑇gw ≫ 1.

LMT is similar to dynamical decoupling (DD) sequences but is more flexible as it allows to switch the sign of the phase shift by alternating lasers instead of waiting for the change of GW phase.

Figure from Norcia M.A. et al., PRA, 96, 042118, 2017


39

Ground state phase memory protocol: optical clocks

Atoms with two ground states like 171Yb provide additional quantum labels that allow for the independent manipulation of only one arm of interferometer.

Transitions may be driven between either | ۧ𝑔 1 and ۧ|𝑒 or | ۧ𝑔 2 and ۧ|𝑒 by applying laser light with different polarizations or frequencies.

The relative phase is stored between two ground states | ۧ𝑔 1 and ۧ|𝑔 2 during the evolution period between the interrogation by 𝜋/2 pulses which can be, thus, extended beyond the spontaneous-decay limited coherence time of the atom.

from | ۧ𝑔 1 to superposition | ۧ𝑔 1 ۧ+|𝑒

from ۧ|𝑒 to | ۧ𝑔 2

from ۧ|𝑔 1 to ۧ|𝑒

beamsplitting superposition | ۧ𝑔 2 ۧ+|𝑒and reading out the differential phaseΦ𝑠 = 2𝑘 𝐿2 − 𝐿1

Summary



• Optical clock and atomic interferometer detectors bridges the detection gap between space-based andterrestrial optical interferometers through tunable, narrow band GW detection with sensitivity over a broadfrequency range: 3 mHz ≤ 𝑓𝑔𝑤 ≤ 10 Hz

• LISA and atomic clock detectors require drag-free satellites. Atom interferometer does not require drag-freesatellite since the atoms themselves are in free fall, but it requires the atoms to be cooled to pico-Kelvintemperature.

• The noise floor of optical interferometers is fundamentally limited by white phase noise arising from photonshot noise, while the noise floor of the clock detector is dominated by white frequency noise arising from atomprojection noise.

• Two clocks shares a single laser, laser noise is common mode and the Ramsey free precession time can beextended considerably beyond the laser coherence time, which can be pushed out to the radiative lifetime ofthe clock transition, 𝑇max = 1/(2𝜋Δ𝐴) which can be further extended if 3-level atoms are employed.

• The smallest detectable fractional frequency difference 𝜎min between the two clocks (and the smallestmeasurable GW-induced strain) is limited by the atom projection noise and is given by

𝜎min = ቚ𝛿𝜈

𝜈 𝜏=

Δ𝐴

𝜈 2𝜋𝜏𝑁=

1.1×10−20

𝜏,

where 𝜈 = 430 THz is the frequency of the optical clock transition, 𝜏 is a totalmeasurement time, 𝑁 = 7 × 106 is the number of atoms in each clock.

Image credit to M. Colpi & A. SesanaarXiv:1610.05309http://inspirehep.net/record/1492489/plots

2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland) 41

Sensitivity of GW atomic clock interferometer

Kolkowitz et al.PRD 94, 124043 (2016)

1/𝑓2𝑓




43

Dynamical Decoupling (DD) protocol: optical clocks

)

March 25-28, 2019

)

DD is useful when the evolution time 𝑇𝑒 ≫ 𝑇gw as it yields the enhancement

in signal by a factor𝑇𝑒/𝑇gw ≫ 1.

Documents

Atomic Clocks as Detectors of Gravitational Waves · Resume: 1) The Hafele-Keating type of experiment does not allow to measure gravitational waves with clocks 2) Clocks must be connected