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Atomic Clocks as Detectors of Gravitational Waves

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

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Atomic Clocks as Detectors of Gravitational Waves

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  1. Atomic Clocks as Detectors of Gravitational Waves Sergei Kopeikin University of Missouri, USA

  2. 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

  3. Supernova explosions precision timing of binary pulsars spacecraft Doppler tracking precision global astrometry of quasar’s proper motions atom interferometry optical clocks 2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

  4. Clock’s characteristics: accuracy, stability and precision 2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

  5. Accuracy records for microwave and optical clocks 2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

  6. Frequency stability for microwave and optical clocks The variance of frequency The average fractional frequency: Phase of the oscillator: the number of frequency samples time between each frequency sample the time length of each frequency estimate 2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

  7. Pulsar Clock as GW Detector

  8. 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 ~ 8 hr mp mc · 2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

  9. Detection of periodic sources of GWSazhin M.V., “Opportunities for detecting ultralong gravitational waves”, Sov. Astron., 22, 36, 1978 time Detection of stochastic background of GW 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 pulsar clock Cosmological GW background noise Measuring two-point cross-correlations between pulsars 2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

  10. 2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

  11. Computation of Gravitational Wave Signal

  12. Plane gravitational wave Linear approximation: New variable: Gauge freedom: Harmonic gauge: Einstein’s equations: Plane monochromatic wave: Transverse-traceless (harmonic) gauge: 2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

  13. Regions of space around an emitter of GW Clock A Clock B Clock A Clock B 2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

  14. Coordinates and metrics of an isolated gravitating system (Kopeikin et al., PRD 59, 084023,1999) 2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

  15. TT metric tensor perturbation and GW strength - transverse-traceless quadrupole moment of the system emitting GW For a binary system with reduced mass - the luminosity distance, Distance to Virgo Cluster = 0.02 Gpc - the “chirp” mass of the binary - the observed (redshifted) GW frequency 2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

  16. Detection of gravitational wave and its interpretation

  17. Laser interferometer as detector of GWs Michelson-type Fabry-Perot interferometer (LIGO, VIRGO) free mass gravitational wave free test masses (mirrors) suspended on strings 2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

  18. How to interpret the measurement of gravitational wave signal • 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. • TT coordinates: Test masses are in free fall. They do not change positions with respect to TT coordinates. The coordinate distance between the masses does not change. However, gravitational wave changes refractive properties of space as shown by J.L. Synge (1960). The permittivity tensor of space in the presence of gravity field is (F. de Felici, GRG, 2, 347-357, 1971) • Light in the interferometer moves through space with a variable index of refraction. It does change wavelength of light. • Local inertial coordinates of detector: Free test masses are in free fall. They move along geodesics which 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 wave passes on. Refractive tensor of space is identical to vacuum one. Wavelength of light does not change. • 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 2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

  19. Binary systems – the most promising sources of gravitational waves

  20. GW from a coalescing BBH system and GW150914 signal Ringdown oscillations Chirp signal 2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

  21. Binaries in the gravitational wave universe accuracy of atomic clocks today 2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

  22. 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) 2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

  23. Binary System Clock B Gravitational Wave Earth – the comoving distance TWTFT Link Clock A Gravitational Wave Clock A Doppler GW Detector Clock B Sun LISA Atom Interferometry GW Detector Clock B Clock A Optical Laser Link Drag-free satellite 2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

  24. What type of gravitational field can atomic clocks measure? The Hafele-Keating Experiment of 1971: Science, 177, No. 4044, 166-168, 1972 (nanosecond) + Eastbound flight Clock E Rotation of Earth USNO Clock general relativity special relativity Clock W longitude colatitude the angular velocity of Earth’s rotation Earth’s radius acceleration of gravity gravitational potential on Earth’s surface for eastbound flight for westbound flight Westbound flight gravitational time delay Sagnac effect 2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

  25. Would the Hafele-Keating type experiment measure gravitational wave? (Loeb & Maoz, arXiv:1501.00996 “Using atomic clocks to detect gravitational waves”) Clocks A and B are at the same point Clock A Clock B Clock B has been (adiabatically) transported to a different point Clock B Clock A Gravitational wave package passes through the clocks A and B Clock B Clock A Clock B has been (adiabatically) transported back to the point with clock A. Clock B Clock A Clock’ readings are compared. Are they the same? Clock A Clock B 2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

  26. Is atomic clock sensitive to the field of gravitational wave? A clock measure proper time along the clock’s worldline The metric tensor of gravitational wave has There is no the “Newtonian gauge” with , that is used in relativistic geodesy and 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 for stationary clocks Resume: The Hafele-Keating type of experiment does not allow to measure gravitational waves with clocks Clocks must be connected by EM signal (RF/optical/fiber link) which phase is locked to the clocks EM phase propagation is found by solving the equations of light geodesics (equation of eikonal) in the field of gravitational wave 2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

  27. Clock Comparison – Spacetime Diagram Clock B Clock A time Optical/RF downlink space Optical/RF uplink 2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

  28. Clock Comparison – Relativity at Work TT gauge eliminates , and makes =. Therefore, the overall effect of gravitational wave is enclosed to the optical link is connected to by the solution of the light ray propagation equation in the field of gravitational wave: 2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

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

  30. Effect of a plane gravitational wave on light frequency(Kaufmann W.J., Nature, 227, 157, 1970) Particles with constant spatial coordinates move along geodesics (free fall). Proper time of clocks that are in free fall: Proper distance between the two clocks: GW train clock clock light Doppler shift of EM signal: 2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

  31. Doppler tracking as detector of GW Armstrong, J.W., Living Rev. Relativity (2006) 9: 1. https://doi.org/10.12942/lrr-2006-1 gravitational wave signal Earth • GW signals are observed in the Doppler tracking time series through the three pulse response. • The response depends on the two-way light time  , the cosine of the angle between the GW wavevector and a unit vector from the Earth to the spacecraft, and GW properties (Fourier spectrum and polarization state). • The GW response is not time-shift invariant; if T2 or  change during the time of observation the GW response changes. • The GW response is a high-pass filter: in the low-frequencylimit , the response is attenuated due to GW signal overlap and cancellation.  2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

  32. Atom interferometry and optical clocks as gravitational wave detectors

  33. The effect of laser pulse on atom Two level atom: excited state laser pulse emission absorption ground state laser pulse area pulse duration – the Rabi frequencyHz – phase of laser pulse The Bloch matrix: The Bloch sphere: + 2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

  34. Optical versus atom interferometer correspondence Excited and ground states of atom interferometer are equivalent to two arms of optical interferometer. pulse is equivalent to beam splitter pulse is equivalent to mirror beam splitter/combiner laser pulse: beam splitter beam combiner pulse: mirror 2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

  35. Atomic Clock GW Detector Measurement Protocols

  36. Ramsey sequence protocol: optical clocks GW-induced effective path length clock B clock A laser Laser short noise is subtracted in the differential signal. Spacecraft distance noise caused by the recoil of the laser pulses does not cancel but the requirement to the level of noise is less restrictive than in eLISA: ; from the shot noise of a 1 Watt laser detected by telescope of 30 cm diameter across 1-2 AU baseline. 2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

  37. Raman sequence protocol: atom Mach-Zehnder interferometer 2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

  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 . It also allows to reduce the baseline between the satellites. LMT is useful when both the evolution time and the period of gravitational waves where is the laser pulse transit time, as it yields the enhancement in signal by a factor . 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. 2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

  39. Ground state phase memory protocol: optical clocks from to superposition from to from to beamsplitting superposition and reading out the differential phase Atoms with two ground states like provide additional quantum labels that allow for the independent manipulation of only one arm of interferometer. Transitions may be driven between either and or and by applying laser light with different polarizations or frequencies. The relative phase is stored between two ground states during the evolution period between the interrogation by pulses which can be, thus, extended beyond the spontaneous-decay limited coherence time of the atom. 2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

  40. Summary • Optical clock and atomic interferometer detectors bridges the detection gap between space-based and terrestrial optical interferometers through tunable, narrow band GW detection with sensitivity over a broad frequency range: • LISA and atomic clock detectors require drag-free satellites. Atom interferometer does not require drag-free satellite since the atoms themselves are in free fall, but it requires the atoms to be cooled to pico-Kelvin temperature. • The noise floor of optical interferometers is fundamentally limited by white phase noise arising from photon shot noise, while the noise floor of the clock detector is dominated by white frequency noise arising from atom projection noise. • Two clocks shares a single laser, laser noise is common mode and the Ramsey free precession time can be extended considerably beyond the laser coherence time, which can be pushed out to the radiative lifetime of the clock transition, which can be further extended if 3-level atoms are employed. • The smallest detectable fractional frequency difference between the two clocks (and the smallest measurable GW-induced strain) is limited by the atom projection noise and is given by • where THz is the frequency of the optical clock transition, is a total • measurement time, is the number of atoms in each clock. 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) Atomic Clock Interferometer 2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

  42. Thank you! 2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

  43. Dynamical Decoupling (DD) protocol: optical clocks DD is useful when the evolution time as it yields the enhancement in signal by a factor. 2-nd ISSI Workshop on Clocks, Spacetime Metrology and Geodesy (Bern, Switzerland)

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