Testing General Relativity with Atom Interferometry Savas - PowerPoint PPT Presentation

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

Testing Large Distance GR Cosmological Constant Problem suggests Our understanding of GR is incomplete (unless there are ~10 500 universes!) CCP+DM inspired proposals for IR modifications: Damour-Polyakov MOND DGP Beckenstein ADDG (non-locality) ... Ghost condensation Brans-Dicke ... Bimetric ...

Precision long distance tests GR: Principle of Equivalence tested to 3 × 10 -13 most other tests ~ 10 -3 to 10 -5 time delay (Cassini tracking) 10 -5 light deflection (VLBI) 10 -3 perihelion shift 10 -3 Nordtvedt effect 10 -3 Lense-Thirring (GPB) QED: 10 digit accuracy g-2, EDMs, etc

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

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

An old idea Atom Interferometry can measure minute forces Galileo ∼ g ∼ 10 − 11 g Current ∼ 10 − 17 g Future A.Peters

An old idea Atom Interferometry can measure minute forces Galileo ∼ g ∼ 10 − 11 g Current ∼ 10 − 17 g Future dv dt = −∇ φ + GR M e φ = G N R e A.Peters

Outline • Post Newtonian General Relativity • Atom Interferometry • Preliminary estimates

Post-Newtonian Approximation Expansion in potential and velocity Small Numbers v atoms ∼ 10 m sec ∼ 3 × 10 − 8 Atom velocity: φ = G N M earth ∼ 1 2 × 10 − 9 Earth’s potential: R earth height 6 × 10 6 m ∼ 1 10 m 6 × 10 − 5 Gradient: ∼ R earth

Particle equation of motion φ Newtonian Gravitational Potential ψ Kinetic Energy Gravitational Potential ζ Rotational Energy Gravitational Potential d� v dt = −∇ ( φ + 2 φ 2 + ψ ) “scalar potential” − ∂� ζ v × ( ∇ × � “vector potential” ∂t + � ζ ) v ∂φ v 2 ∇ φ +3 � ∂t + 4 � v ( � v · ∇ ) φ − �

Non-abelian gravity In empty space ∇ 2 φ = 4 π G N ρ = 0 ∇ · � Newton g = ∇ 2 δφ = ( ∇ φ ) 2 ∼ ∇ 2 φ 2 Einstein ⇒ δφ ∼ φ 2 ⇒ “ ∇ · � g � = 0” = dt = −∇ ( φ + 2 φ 2 + ψ ) − ∂� ζ d� v v × ( ∇ × � ∂t + � ζ ) v ∂φ v 2 ∇ φ +3 � ∂t + 4 � v ( � v · ∇ ) φ − �

Non-abelian gravity In empty space ∇ 2 φ = 4 π G N ρ = 0 ∇ · � Newton g = ∇ 2 δφ = ( ∇ φ ) 2 ∼ ∇ 2 φ 2 Einstein ⇒ δφ ∼ φ 2 ⇒ “ ∇ · � g � = 0” = dt = −∇ ( φ + 2 φ 2 + ψ ) − ∂� Effect ∼ 10 − 9 g ζ d� v v × ( ∇ × � ∂t + � ζ ) only gradient measurable → 10 − 15 g v ∂φ v 2 ∇ φ +3 � ∂t + 4 � v ( � v · ∇ ) φ − �

“Kinetic Energy Gravitates” v 2 ∇ φ + 4 � v ( � v · ∇ ) φ − � E ff ect ∼ v 2 ∼ 10 − 15 g atoms g dt = −∇ ( φ + 2 φ 2 + ψ ) − ∂� ζ d� v v × ( ∇ × � ∂t + � ζ ) v ∂φ v 2 ∇ φ +3 � ∂t + 4 � v ( � v · ∇ ) φ − �

General Relativity effects on equation of motion dt = −∇ ( φ + 2 φ 2 + ψ ) − ∂� ζ d� v v × ( ∇ × � ∂t + � ζ ) v ∂φ v 2 ∇ φ +3 � ∂t + 4 � v ( � v · ∇ ) φ − �

General Relativity effects on equation of motion dt = −∇ ( φ + 2 φ 2 + ψ ) − ∂� ζ d� v v × ( ∇ × � ∂t + � ζ ) 10 − 9 1 v ∂φ v 2 ∇ φ +3 � ∂t + 4 � v ( � v · ∇ ) φ − �

General Relativity effects on equation of motion dt = −∇ ( φ + 2 φ 2 + ψ ) − ∂� ζ d� v v × ( ∇ × � ∂t + � ζ ) 10 − 9 1 10 − 15 v ∂φ v 2 ∇ φ +3 � ∂t + 4 � v ( � v · ∇ ) φ − �

General Relativity effects on equation of motion dt = −∇ ( φ + 2 φ 2 + ψ ) − ∂� ζ d� v v × ( ∇ × � ∂t + � ζ ) 10 − 9 10 − 13 ∼ 0 1 10 − 15 v ∂φ v 2 ∇ φ +3 � ∂t + 4 � v ( � v · ∇ ) φ − �

General Relativity effects on equation of motion dt = −∇ ( φ + 2 φ 2 + ψ ) − ∂� ζ d� v v × ( ∇ × � ∂t + � ζ ) 10 − 9 10 − 13 ∼ 0 1 10 − 15 v ∂φ v 2 ∇ φ +3 � ∂t + 4 � v ( � v · ∇ ) φ − � 10 − 15 ∼ 0

General Relativity effects on equation of motion dt = −∇ ( φ + 2 φ 2 + ψ ) − ∂� ζ d� v v × ( ∇ × � ∂t + � ζ ) 10 − 9 10 − 13 ∼ 0 1 10 − 15 v ∂φ v 2 ∇ φ +3 � ∂t + 4 � v ( � v · ∇ ) φ − � 10 − 15 ∼ 0 Can these terms be measured in the lab?

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

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

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

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

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

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

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

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!

Raman Transition ω 1 ω 2 Ω 1 Ω 2 k ef f ω 2 � 2,p � k � � 1,p � k e f f = ω 1 + ω 2 ∼ 1 eV ω ef f = ω 1 − ω 2 ∼ 10 − 5 eV

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

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

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

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

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

Measuring Gravity a constant gravitational field produces a phase shift: t 2T � � 1 � 2 mv 2 − mgh φ propagation = dt T 0 r ∆ φ propagation = mg × (area) = mg × k mT × T ∆ φ tot = kgT 2 ∼ 10 8 radians

Gravity Phases  GkMT 2 1 . × 10 8 R 2 e − 2 GkMT 3 v L − 2 . × 10 3   R 3 e  − GMT 2 ω − 1 . × 10 3  R 2  e GMT 2 ω A  1 . × 10 3  R 2 e  7 G 2 kM 2 T 4 1 . 16667 × 10 2  6 R 5  e  3 GkMT 2 v L 3 . × 10 1  R 2  e − 3 G 2 kM 2 T 3  − 3 .  R 4 e  − Gk 2 MT 3 − 1 .  mR 3  e 7 GkMT 4 v 2  3 . 5 × 10 -2 L  2 R 4  e 2 GMT 3 ω v L  2 . × 10 -2  R 3 e  − 2 GMT 3 v L ω A − 2 . × 10 -2  R 3  e  3 Gk 2 MT 2 1 . 5 × 10 -2  2 mR 2  e G 2 kM 2 T 2 1 . × 10 -2   R 3 e  − 11 G 2 kM 2 T 5 v L − 5 . 5 × 10 -3  2 R 6  e − 7 G 2 M 2 T 4 ω  − 1 . 16667 × 10 -3  6 R 5 e  7 G 2 M 2 T 4 ω A 1 . 16667 × 10 -3  6 R 5  e  − 8 GkMT 3 v 2 − 8 . × 10 -4 L  R 3  e − 3 GMT 2 ω v L  − 3 . × 10 -4  R 2 e  35 G 2 kM 2 T 4 v L 1 . 75 × 10 -4  2 R 5  e  5 GkMT 2 v 2 5 . × 10 -6 L  R 2  e − 11 G 2 k 2 M 2 T 5  − 2 . 75 × 10 -6  4 mR 6 e  − 15 G 2 kM 2 T 3 v L − 1 . 5 × 10 -6 R 4 e