Testing General Relativity with Atom Interferometry

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2 Testing General lativity with Atom Interferometry Savas Dimopoulos with Peter Graham Jason Hogan Mark Kasevich

3 Testing Large Distance GR Cosmological Constant Problem suggests Our understanding of GR is incomplete (unless there are ~ universes!) CCP+DM inspired proposals for IR modifications: Damour-Polyakov DGP ADDG (non-locality) Ghost condensation... MOND Beckenstein... Brans-Dicke Bimetric...

4 Precision long distance tests GR: Principle of Equivalence tested to most other tests ~ 10-3 to 10-5 time delay (Cassini tracking) light deflection (VLBI) perihelion shift Nordtvedt effect Lense-Thirring (GPB) QED: 10 digit accuracy g-2, EDMs, etc

5 Precision GR tests mostly use: Planets and photons over astronomical distances Can we study GR using atoms over short distances (meters)?

6 Precision GR tests mostly use: Planets and photons over astronomical distances Can we study GR using atoms over short distances (meters)? Yes, thanks to the tremendous advances in Atom Interferometry Unprecedented Precision (see Nobel Lectures 97, 01, 05 ) Several control variables (v, t, ω, h) We are at crossroads where atoms may compete with astrophysical tests of GR

7 An old idea Atom Interferometry can measure minute forces Galileo Current Future g g g A.Peters

8 An old idea Atom Interferometry can measure minute forces Galileo Current Future g g g dv dt = φ + GR A.Peters φ = G N M e R e

9 Outline Post Newtonian General lativity Atom Interferometry Preliminary estimates

10 Post-Newtonian Approximation Expansion in potential and velocity Small Numbers Atom velocity: v atoms 10 m sec Earth s potential: φ = G NM earth R earth Gradient: height R earth 10 m m

11 Particle equation of motion φ ψ ζ Newtonian Gravitational Potential Kinetic Energy Gravitational Potential Rotational Energy Gravitational Potential d v dt = (φ + 2φ2 + ψ) ζ t + v ( ζ) +3 v φ t + 4 v( v )φ v2 φ scalar potential vector potential

12 Non-abelian gravity In empty space Newton g = 2 φ = 4πG N ρ = 0 d v dt = (φ + 2φ2 + ψ) ζ t + v ( ζ) Einstein 2 δφ = ( φ) 2 2 φ 2 δφ φ 2 = g 0 +3 v φ t + 4 v( v )φ v2 φ

13 Non-abelian gravity In empty space Newton g = 2 φ = 4πG N ρ = 0 Einstein 2 δφ = ( φ) 2 2 φ 2 δφ φ 2 = g 0 d v dt = (φ + 2φ2 + ψ) ζ t + v ( ζ) +3 v φ t + 4 v( v )φ v2 φ Effect 10 9 g only gradient measurable g

14 Kinetic Energy Gravitates v 2 φ + 4 v( v )φ Effect v 2 atomsg g d v dt = (φ + 2φ2 + ψ) ζ t + v ( ζ) +3 v φ t + 4 v( v )φ v2 φ

15 General lativity effects on equation of motion d v dt = (φ + 2φ2 + ψ) ζ t + v ( ζ) +3 v φ t + 4 v( v )φ v2 φ

16 General lativity effects on equation of motion d v dt = (φ + 2φ2 + ψ) ζ t + v ( ζ) v φ t + 4 v( v )φ v2 φ

17 d v General lativity effects on equation of motion dt = (φ + 2φ2 + ψ) ζ t + v ( ζ) +3 v φ t + 4 v( v )φ v2 φ

18 d v General lativity effects on equation of motion dt = (φ + 2φ2 + ψ) ζ t + v ( ζ) v φ t + 4 v( v )φ v2 φ

19 d v General lativity effects on equation of motion dt = (φ + 2φ2 + ψ) ζ t + v ( ζ) v φ t + 4 v( v )φ v2 φ

20 d v General lativity effects on equation of motion dt = (φ + 2φ2 + ψ) ζ t + v ( ζ) v φ t + 4 v( v )φ v2 φ Can these terms be measured in the lab?

21 Light Interferometry beamsplitter output ports mirror accurate measurement of y beamsplitter mirror L L λ L (phase resolution) x

22 Atom Interferometry 2T t beamsplitter output ports T mirror mirror 0 beamsplitter similar to light interferometer but arms are separated in space-time instead of space-space r

23 Atom Interferometry 2T t beamsplitter output ports T mirror mirror 0 beamsplitter similar to light interferometer but arms are separated in space-time instead of space-space r

24 Atom Interferometry 2T t beamsplitter output ports T mirror mirror 0 beamsplitter similar to light interferometer but arms are separated in space-time instead of space-space r

25 Atom Interferometry 2T t beamsplitter output ports T mirror mirror 0 beamsplitter similar to light interferometer but arms are separated in space-time instead of space-space r

26 Atom Interferometry 2T t beamsplitter output ports T mirror mirror 0 beamsplitter similar to light interferometer but arms are separated in space-time instead of space-space r

27 Atom Interferometry t 2T T 0 use lasers as beamsplitters and mirrors r

28 Atom Interferometry t 2T T 0 r slow atoms fall more under gravity and the interferometer can be as long as 1 sec ~ earth-moon distance!

29 Raman Transition ω 1 ω 2 Ω 1 Ω 2 k e f f ω 2 1,p 2,p k k e f f = ω 1 + ω 2 1 ev ω e f f = ω 1 ω ev

30 Raman Transition ψ = c 1 1, p + c 2 2, p + k c 1 2, c Ω 1 Ω ,p 2,p k 1,p 2,p k Π 2 Π 3 Π 2 2 Π t Rabi 1 π/2 pulse is a beamsplitter π pulse is a mirror

31 AI Phase Shifts Total phase difference comes from three sources: φ tot = φ propagation + φ laser + φ separation

32 Propagation Phase φ tot = φ propagation + φ laser + φ separation φ propagation = mdτ = Ldt = p µ dx µ integral taken over each arm of interferometer t 2T T 0 r

33 Laser Phase φ tot = φ propagation + φ laser + φ separation φ laser = vertices (phase of laser) out H int in = out µ E 0 e i k x in the laser imparts a phase to the atom just as a mirror or beamsplitter imparts a phase to light

34 Separation Phase 2T t x µ φ separation = x µ p ν dx ν T 0 r

35 Measuring Gravity a constant gravitational field produces a phase shift: t 2T T φ propagation = ( ) 1 2 mv2 mgh dt 0 r φ propagation = mg (area) = mg k m T T φ tot = kgt radians

36 Gravity Phases GkMT 2 2 2GkMT 3 v L 3 GMT 2 ω R 2 e GMT 2 ω A R 2 e 7G 2 km 2 T 4 6R 5 e 3GkMT 2 v L 2 3G2 km 2 T 3 4 Gk2 MT 3 mr 3 e 7GkMT 4 v 2 L 2R 4 e 2GMT 3 ωv L 3 2GMT 3 v L ω A R 3 e 3Gk 2 MT 2 2m 2 G 2 km 2 T G2 km 2 T 5 v L 2 6 7G2 M 2 T 4 ω 6R 5 e 7G 2 M 2 T 4 ω A 6 5 8GkMT 3 v 2 L 3 3GMT 2 ωv L R 2 e 35G 2 km 2 T 4 v L 2R 5 e 5GkMT 2 v 2 L R 2 e 11G2 k 2 M 2 T 5 4mR 6 e 15G2 km 2 T 3 v L R 4 e

37 Gravity Phases GkMT 2 2 2GkMT 3 v L 3 GMT 2 ω R 2 e GMT 2 ω A R 2 e 7G 2 km 2 T 4 6R 5 e 3GkMT 2 v L 2 3G2 km 2 T 3 4 Gk2 MT 3 mr 3 e 7GkMT 4 v 2 L 2R 4 e 2GMT 3 ωv L 3 2GMT 3 v L ω A R 3 e 3Gk 2 MT 2 2m 2 G 2 km 2 T G2 km 2 T 5 v L 2 6 7G2 M 2 T 4 ω 6R 5 e 7G 2 M 2 T 4 ω A 6 5 8GkMT 3 v 2 L 3 3GMT 2 ωv L R 2 e 35G 2 km 2 T 4 v L 2R 5 e 5GkMT 2 v 2 L R 2 e 11G2 k 2 M 2 T 5 4mR 6 e 15G2 km 2 T 3 v L R 4 e non-relativistic constant g NR gravity gradient

38 Gravity Phases GkMT 2 2 2GkMT 3 v L 3 GMT 2 ω R 2 e GMT 2 ω A R 2 e 7G 2 km 2 T 4 6R 5 e 3GkMT 2 v L 2 3G2 km 2 T 3 4 Gk2 MT 3 mr 3 e 7GkMT 4 v 2 L 2R 4 e 2GMT 3 ωv L 3 2GMT 3 v L ω A R 3 e 3Gk 2 MT 2 2m 2 G 2 km 2 T G2 km 2 T 5 v L 2 6 7G2 M 2 T 4 ω 6R 5 e 7G 2 M 2 T 4 ω A 6 5 8GkMT 3 v 2 L 3 3GMT 2 ωv L R 2 e 35G 2 km 2 T 4 v L 2R 5 e 5GkMT 2 v 2 L R 2 e 11G2 k 2 M 2 T 5 4mR 6 e 15G2 km 2 T 3 v L R 4 e non-relativistic constant g NR gravity gradient Doppler shift

39 Gravity Phases GkMT 2 2 2GkMT 3 v L 3 GMT 2 ω R 2 e GMT 2 ω A R 2 e 7G 2 km 2 T 4 6R 5 e 3GkMT 2 v L 2 3G2 km 2 T 3 4 Gk2 MT 3 mr 3 e 7GkMT 4 v 2 L 2R 4 e 2GMT 3 ωv L 3 2GMT 3 v L ω A R 3 e 3Gk 2 MT 2 2m 2 G 2 km 2 T G2 km 2 T 5 v L 2 6 7G2 M 2 T 4 ω 6R 5 e 7G 2 M 2 T 4 ω A 6 5 8GkMT 3 v 2 L 3 3GMT 2 ωv L R 2 e 35G 2 km 2 T 4 v L 2R 5 e 5GkMT 2 v 2 L R 2 e 11G2 k 2 M 2 T 5 4mR 6 e 15G2 km 2 T 3 v L R 4 e non-relativistic constant g NR gravity gradient Doppler shift GR φ 2

40 Gravity Phases GkMT 2 2 2GkMT 3 v L 3 GMT 2 ω R 2 e GMT 2 ω A R 2 e 7G 2 km 2 T 4 6R 5 e 3GkMT 2 v L 2 3G2 km 2 T 3 4 Gk2 MT 3 mr 3 e 7GkMT 4 v 2 L 2R 4 e 2GMT 3 ωv L 3 2GMT 3 v L ω A R 3 e 3Gk 2 MT 2 2m 2 G 2 km 2 T G2 km 2 T 5 v L 2 6 7G2 M 2 T 4 ω 6R 5 e 7G 2 M 2 T 4 ω A 6 5 8GkMT 3 v 2 L 3 3GMT 2 ωv L R 2 e 35G 2 km 2 T 4 v L 2R 5 e 5GkMT 2 v 2 L R 2 e 11G2 k 2 M 2 T 5 4mR 6 e 15G2 km 2 T 3 v L R 4 e non-relativistic constant g NR gravity gradient Doppler shift GR φ 2 GR v 2 φ + 4 v( v )φ

41 Gravity Phases GkMT 2 2 2GkMT 3 v L 3 GMT 2 ω R 2 e GMT 2 ω A R 2 e 7G 2 km 2 T 4 6R 5 e 3GkMT 2 v L 2 3G2 km 2 T 3 4 Gk2 MT 3 mr 3 e 7GkMT 4 v 2 L 2R 4 e 2GMT 3 ωv L 3 2GMT 3 v L ω A R 3 e 3Gk 2 MT 2 2m 2 G 2 km 2 T G2 km 2 T 5 v L 2 6 7G2 M 2 T 4 ω 6R 5 e 7G 2 M 2 T 4 ω A 6 5 8GkMT 3 v 2 L 3 3GMT 2 ωv L R 2 e 35G 2 km 2 T 4 v L 2R 5 e 5GkMT 2 v 2 L R 2 e 11G2 k 2 M 2 T 5 4mR 6 e 15G2 km 2 T 3 v L R 4 e GR φ non-relativistic constant g NR gravity gradient Doppler shift GR φ 2 GR v 2 φ + 4 v( v )φ

42 Gravity Phases kgt 2 2 kgt 2 v LT R e 3 kgt 2 v L kgt 2 φ 5 kgt 2 v 2 L 15 kgt 2 v LT R e φ doppler shift GR terms experimentally controllable parameters are: k, v L, T

43 Gravity Phases kgt 2 2 kgt 2 v LT R e 3 kgt 2 v L kgt 2 φ 5 kgt 2 v 2 L 15 kgt 2 v LT R e φ same scalings, measure with gradient experimentally controllable parameters are: k, v L, T

44 Gravity Phases kgt 2 2 kgt 2 v LT R e 3 kgt 2 v L kgt 2 φ 5 kgt 2 v 2 L 15 kgt 2 v LT R e φ same scalings g 0 experimentally controllable parameters are: k, v L, T

45 Gravity Phases kgt 2 2 kgt 2 v LT R e 3 kgt 2 v L kgt 2 φ 5 kgt 2 v 2 L 15 kgt 2 v LT R e φ unique scaling experimentally controllable parameters are: k, v L, T

46 Atomic Interferometer 10 m 10 m atom drop tower. currently under construction at Stanford

47 Atomic Equivalence Principle Test 10 m 10 m atom drop tower.

48 Atomic Equivalence Principle Test 10 m Co-falling 85 Rb and 87 Rb ensembles 10 m atom drop tower. Will reach accuracy Compared to Lunar Laser Ranging

49 Atomic Equivalence Principle Test 10 m Co-falling 85 Rb and 87 Rb ensembles 10 m atom drop tower. Will reach accuracy Compared to Lunar Laser Ranging Then will test PN GR

50 Other equivalence principle measurements Atomic Equivalence Principle Test at Stanford g 2008 MICROSCOPE g 2011 Galileo Galilei g launch 2009? STEP g?

51 Can we measure H? Pioneer anomaly? Radio ranging of Pioneer Laser ranging of atoms BUT equivalence principle says only tides measurable and Riemann R H 2 way too small

52 Can we measure H? Pioneer anomaly? Radio ranging of Pioneer Laser ranging of atoms BUT equivalence principle says only tides measurable and Riemann R H 2 way too small Similarly DM is not measurable But, Sun s radiation pressure is measurable at the g level, and causes the earth not to be an inertial frame

53 GR Experimentation Precession of Mercury and light bending Hibernation Now Golden Era, many astronomical tests New epoch? High precision atom interferometry allows for greater control and ability to isolate and study individual effects in GR such as 3-graviton coupling and gravitation of kinetic energy

54 Good to Go!

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