The sensitivity of atom interferometers to gravitational waves

Transcription

1 The sensitivity of atom interferometers to gravitational waves The Galileo Galilei Institute for Theoretical Physics Arcetri, Florence February 24, 2009 Pacôme DELVA ESA DG-PI Advanced Concepts Team

2 Gravitational Wave Detection Context Actual laser interferometers: first detection soon? Very few events expected (<1 detection/year). Amelioration of terrestrial antennas (2013) ~1det./day to 1det./week. Exploration of a new frequency range (low frequency): LISA (ESA/NASA 2018). New type of detectors: atom interferometers. Applications: inertial sensors, gyrometer and absolute gravimeter (see the review by Miffre et al. Phys. Scr. 74, 2006) 2

3 Outline 1. Interest 2. The phase difference 1. Operational coordinates 2. Active and passive change of coordinates 3. MWI vs. LWI 4. Sensitivity curves 5. Another configuration 6. Conclusion 3

4 Black hole binary coalescence Compact binaries MWI interest Compact binaries coalescence Stellar collapse 10 7 km 100 km Λ 10-2 Hz 1 Hz 10 3 Hz F Space based interferometer LISA : L tot ~ km Ground based interferometer with Fabry-Perot cavities VIRGO (3 km) + Fabry-Pérot (Finesse = 50) : L tot ~ 150 km The interferometer frequency domain depends only on the flight time T of the particle in the interferometer arm F ~ 1 / T ~ V / L tot For the same frequency domain, reducing the particle velocity reduce the arm length Particle = atoms Reducing the dimension of the interferometer helps to fight the different noises, and especially thermal noise 4

5 Calculation of the phase difference Weak-field approximation g μν = η μν + H μν, H μν 1 Calculation of the phase difference within the eikonal and the weak-field approximation (Linet & Tourrenc 1976). [φ o ] B A = k μx μ B k μx μ A [φ] B A =[φ o] B A +[δφ]b A [δφ] B A = ~c2 2 R tb t A H μν k μ k ν dt E In the Einstein frame : ds 2 = η μν dx μ dx ν + h rs dx r dx s ; r, s =1, 2 Z ' z ' 0 Xî ζ Λ x a = f a + 1 4ḣjkXĵX ˆk + O(ζ 4 ) x r = f r + X ˆr 1 2 h r sxŝ + O(ζ 4 ) In the Fermi frame : ds 2 = ηˆα ˆβdX ˆα dx ˆβ + 1 2ḧrsX ˆr XŝdT 2 ; r, s =1, 2 5

6 Operational coordinates x Einstein Frame X Fermi Frame L L y Y O φ = 4π L λ ψ = ΩT/2 sin 2ψ 2ψ h + φ =4π L λ O 1 sin 2ψ 2ψ h + WHY? -> TWO DIFFERENT EXPERIMENTS 6

7 Operational coordinates By defining our atom interferometer in a non covariant way (ie. its definition depends on the coordinate system we use), we assume that we can experimentally realize this coordinate system with a certain protocol -> we give a physical meaning to the coordinate system -> operational coordinates Free experiment -> the different part of the interferometer do not move in the Einstein frame Rigid experiment -> the different part of the interferometer do not move in a Fermi frame Free Michelson in the Fermi Frame X Rigid Michelson in the Fermi Frame x r = X ˆr 1 2 h r sxŝ + O(ζ 4 ) φ o L φ =4π L λ 1 sin 2ψ 2ψ h + O Y Delva et al. 06 7

8 The rigid Ramsey-Bordé interferometer Y L Ω = 2πc Λ, Λ À L = v 0T θ X We assume that the center of mass of the interferometer (= origin of the frame) is located at the center of symmetry of the atom trajectory φ(ω) =4π L λ F 0(Ω)tanθ Ψ 3 at low frequency µ h (Ω) tan θ 2 h +(Ω) F 0 (Ω) =i sin Ψ Ψ = ΩT 2 = ΩL 2v 0 µ cos Ψ sin Ψ Ψ D Ambrosio et al. 07 8

9 Change of the origin of the frame Y 0 O θ X 0 φ = φ o + δφ + δφ + φ o The center of mass follows a geodesic (doesn t move in the Einstein frame) Same result as before X r = 1 2 h r s Xs 0 As should be, the phase difference does not depend on the origin of the frame -> passive change of coordinates D Ambrosio et al. 07 9

10 Change of the center of mass of the apparatus (1/2) Y 0 θ φ = φ o + δφ + φδφ o O X 0 The center of mass follows a geodesic (doesn t move in the Einstein frame) There is a supplementary term X r = 1 2 h r s Xs 0 It can be seen also as an active change of coordinates: we define a DIFFERENT experiment 10

11 Change of the center of mass of the apparatus (2/2) Y 0 θ O X 0 φ(ω) =4π L λ tan θ (F 0 (Ω)+F X (Ω,X 0 )) h tan θ 2 (F 0 (Ω)+F Y (Ω,Y 0 )) h + F 0 (Ω) =i sin Ψ Ψ = ΩT 2 = ΩL 2v 0 µ cos Ψ sin Ψ Ψ at low frequency Ψ 1 : F X ' X 0 L Ψ2 F Y ' Y 0 L Ψ2 F 0 ' i 3 Ψ3 11

12 Matter Wave Interferometer vs. Light Wave Interferometer φ(ω) =4π L λ F 0(Ω)tanθ µ h (Ω) tan θ 2 h +(Ω) F 0 (Ω) =i sin Ψ Ψ = ΩT 2 = ΩL 2v 0 µ cos Ψ sin Ψ Ψ The maximum phase difference is obtained for T~1/Ω. Then, if tan θ ' 1 f φ 4π h Lmw λ mw For a light wave interferometer in a Michelson configuration, the maximum phase difference is obtained for L~c/Ω f φ 4π h Llw λ lw The shot noise ultimately limit the sensitivity f φ 1 2 Ṅt (Gustavson et al.) Virgo LISA N mw s 1 N lw s 1 N lw 10 8 s 1 12

13 MWI vs. LWI The high frequency regime Relativistic velocities needed to reach VIRGO sensitivities (Matter wave acceleration, deviation of atoms, measurement frequency) L v 0 /Ω 13

14 MWI vs. LWI The low frequency regime Kilometric interferometer to reach the sensitivity of LISA with thermal atoms (Matter wave cavity?) L v 0 /Ω 14

15 Sensitivity curves 15

16 Sensitivity curve in the high frequency range h/ Hz 1μ μ v = 10 6 m.s 1 L = 1 km N = s 1 tan θ = 10 5 T = 1 ms Terrestrial configuration 2μ μ μ Ω 16

17 Sensitivity curve in the low frequency range h/ Hz v = 10 m.s 1 L = 1 km Ṅ = s 1 tan θ = 0.5 T = 100 s Spatial configuration Ω 17

18 Another configuration Dimopoulos et al. (2007) proposed a different configuration for the detector that takes advantage of the distance between the center of mass of the interferometer (lasers) and the center of symmetry of the atoms trajectory D À L φ(ω) =4πh D λ r F 1 (Ω) T F 1 (Ω) =i sin 2 Ψ Ψ = ΩT 2 F 1(Ψ) L F 0(Ψ) X The atom wavelength is fixed by the impulsion of the laser λ r = 2π~ mv r = 2π k eff The distance in the amplitude is the distance between the atom interferometer and the laser 18

19 MWI vs. LWI 19

20 Conclusion Atom interferometers have not reach their best sensitivities. Important difficulties remain to reach good sensitivities in order to detect gravitational waves: matter wave cavities, efficient splitting, collisions, flux. Matter wave interferometers could compete with space based interferometers such as LISA (low frequency range), but not with earth based ones (high frequency range). Importance of operational coordinates, difference between passive and active change of coordinates Sensitivity comparison (with same flux) Atom interferometer h min λ L tan θ pm Atom interferometer with far away lasers h min λ r D nm LISA h min λ D μm THANK YOU 20

21 Appendice Métrique : ds 2 =(η μν + K μν )dx μ dx ν,k μν 1 Différence de phase dans un interféromètre : φ c2 ~ R Kμν p μ p ν dt E (Formule de Linet-Tourrenc) Accélération a K 00 al c 2 Rotation Ω K 0i ΩL c Onde Gravitationnelle h TT K ij h TT φ ma ~ A v φ mω ~ A φ mhtt ~ A T ~ k A A = ~kvt 2 m ϕ kat 2 ϕ kωvt 2 ϕ kh TT vt 21