The sensitivity of atom interferometers to gravitational waves The Galileo Galilei Institute for Pacôme DELVA Theoretical Physics ESA DG-PI Advanced Concepts Team Arcetri, Florence http:// www.esa.int/ act February 24, 2009
Gravitational Wave Det ection 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): LIS A (ES A/ NAS A 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) http://www.esa.int/act 2 Pacôme DELVA - GGI, Arcetri, Florence - February 24, 2009
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 http://www.esa.int/act 3 Pacôme DELVA - GGI, Arcetri, Florence - February 24, 2009
MWI int erest Compact binaries coalescence Black hole binary coalescence S tellar collapse Compact binaries 10 7 km Λ 100 km 10 -2 Hz 10 3 Hz 1 Hz F S pace based interferometer Ground based interferometer with Fabry-Perot cavities LISA : L tot ~ 5.10 6 km 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 http://www.esa.int/act 4 Pacôme DELVA - GGI, Arcetri, Florence - February 24, 2009
Calculation of the phase difference g μν = η μν + H μν , H μν ¿ 1 • Weak-field approximation • Calculation of the phase difference within the eikonal and the weak-field approximation (Linet & Tourrenc 1976). [ φ o ] B A = k μ x μ B − k μ x μ A [ φ ] B A = [ φ o ] B A + [ δφ ] B R t B A = ~ c 2 [ δφ ] B t A H μν k μ k ν d t A 2 E • In the Einstein frame : d s 2 = η μν d x μ d x ν + h rs d x r d x s ; r, s = 1 , 2 f a + 1 j X ˆ k + O ( ζ 4 ) ˙ x a h jk X ˆ = Z ' z ' 0 4 r − 1 ı ≤ ζ ¿ Λ f r + X ˆ s + O ( ζ 4 ) ¯ X ˆ x r h r s X ˆ = 2 d s 2 = η ˆ α d X ˆ β + 1 s d T 2 ; r, s = 1 , 2 2 ¨ β d X ˆ h rs X ˆ r X ˆ • In the Fermi frame : α ˆ http://www.esa.int/act 5 Pacôme DELVA - GGI, Arcetri, Florence - February 24, 2009
Operational coordinates X x Fermi Frame Einstein Frame Fermi Frame Einstein Frame L L y Y O O ψ = Ω T/ 2 ∙ ¸ ∆ φ = 4 π L 1 − sin 2 ψ ∆ φ = − 4 π L sin 2 ψ ˜ ˜ h + h + λ 2 ψ λ 2 ψ WHY ? -> TWO DIFFERENT EXPERIMENTS WHY ? -> TWO DIFFERENT EXPERIMENTS http://www.esa.int/act 6 Pacôme DELVA - GGI, Arcetri, Florence - February 24, 2009
Operational coordinates • By defining our atom interferometer in a non covariant way (ie. its definition • 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 depends on the coordinate system we use), we assume that we can experimentally realize this coordinate system with a certain protocol -> we experimentally realize this coordinate system with a certain protocol -> we give a physical meaning to the coordinate system -> operational coordinates give a physical meaning to the coordinate system -> operational coordinates • Free experiment -> the different part of the interferometer do not move in • Free experiment -> the different part of the interferometer do not move in the Einstein frame the Einstein frame • “Rigid” experiment -> the different part of the interferometer do not move • “Rigid” experiment -> the different part of the interferometer do not move in a Fermi frame in a Fermi frame Free Michelson in the Fermi Frame Free Michelson in the Fermi Frame X Rigid Michelson in the Fermi Frame Rigid Michelson in the Fermi Frame r − 1 x r = X ˆ s + O ( ζ 4 ) ¯ h r s X ˆ ∆ φ o 2 ∙ ¸ ∆ φ = 4 π L 1 − sin 2 ψ L ˜ h + λ 2 ψ Y Delva et al. 06 Delva et al. 06 O http://www.esa.int/act 7 Pacôme DELVA - GGI, Arcetri, Florence - February 24, 2009
The rigid Ramsey-Bordé interferometer Y Ω = 2 π c L , Λ À L = v 0 T Λ X θ • We assume that the center of mass of the interferometer (= origin of the • We assume that the center of mass of the interferometer (= origin of the frame) is located at the center of symmetry of the atom traj ectory frame) is located at the center of symmetry of the atom traj ectory µ ¶ cos Ψ − sin Ψ F 0 ( Ω ) = i sin Ψ µ ¶ Ψ ∆ φ ( Ω ) = 4 π L h × ( Ω ) − tan θ λ F 0 ( Ω ) tan θ h + ( Ω ) Ψ = Ω T = Ω L 2 2 2 v 0 ∝ Ψ 3 at low frequency D’ Ambrosio et al. 07 D’ Ambrosio et al. 07 http://www.esa.int/act 8 Pacôme DELVA - GGI, Arcetri, Florence - February 24, 2009
Change of the origin of the frame Y 0 ∆ φ = φ o + δφ + δφ + φ o θ O X 0 • The center of mass follows a geodesic (doesn’ t move in the Einstein frame) ∆ X r = 1 ¯ h r s X s 0 2 • S ame result as before • As should be, the phase difference does not depend on the origin of the frame -> passive change of coordinates D’ Ambrosio et al. 07 D’ Ambrosio et al. 07 http://www.esa.int/act 9 Pacôme DELVA - GGI, Arcetri, Florence - February 24, 2009
Change of the center of mass of the apparat us (1/ 2) Y 0 ∆ φ = φ o + δφ + δφ + φ o θ O X 0 • The center of mass follows a geodesic (doesn’ t move in the Einstein frame) ∆ X r = 1 ¯ h r s X s 0 2 • There is a supplementary term • It can be seen also as an active change of coordinates : we define a DIFFERENT experiment http://www.esa.int/act 10 Pacôme DELVA - GGI, Arcetri, Florence - February 24, 2009
Change of the center of mass of the apparat us (2/ 2) Y 0 θ O X 0 ∙ ¸ ∆ φ ( Ω ) = 4 π L h × − tan θ ( F 0 ( Ω ) + F X ( Ω , X 0 )) ˜ ( F 0 ( Ω ) + F Y ( Ω , Y 0 )) ˜ λ tan θ h + 2 µ ¶ Ψ ¿ 1 • at low frequency : cos Ψ − sin Ψ F 0 ( Ω ) = i sin Ψ Ψ F X ' X 0 F Y ' − Y 0 L Ψ 2 L Ψ 2 Ψ = Ω T = Ω L 2 2 v 0 F 0 ' − i 3 Ψ 3 http://www.esa.int/act 11 Pacôme DELVA - GGI, Arcetri, Florence - February 24, 2009
Matter Wave Interferometer vs. Light Wave Interferometer µ ¶ cos Ψ − sin Ψ F 0 ( Ω ) = i sin Ψ µ ¶ Ψ ∆ φ ( Ω ) = 4 π L h × ( Ω ) − tan θ λ F 0 ( Ω ) tan θ h + ( Ω ) Ψ = Ω T = Ω L 2 2 2 v 0 • The maximum phase difference is obtained for T~1/ Ω . Then, if tan θ ' 1 f ∆ φ ∼ 4 π | h | · L mw λ mw • For a light wave interferometer in a Michelson configuration, the maximum phase difference is obtained for L~c/ Ω f ∆ φ ∼ 4 π | h | · L lw λ lw • The shot noise ultimately limit the sensitivity N mw ∼ 10 11 s − 1 ˙ ( Gustavson et al.) f N lw ∼ 10 23 s − 1 1 ˙ √ ∆ φ ∼ Virgo ˙ 2 Nt LISA N lw ∼ 10 8 s − 1 ˙ http://www.esa.int/act 12 Pacôme DELVA - GGI, Arcetri, Florence - February 24, 2009
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 http://www.esa.int/act 13 Pacôme DELVA - GGI, Arcetri, Florence - February 24, 2009
MWI vs. LWI – The low frequency regime Kilometric interferometer to reach the sensitivity of LIS A with thermal atoms (Matter wave cavity ? ) L ∼ v 0 / Ω http://www.esa.int/act 14 Pacôme DELVA - GGI, Arcetri, Florence - February 24, 2009
15 http://www.esa.int/act Pacôme DELVA - GGI, Arcetri, Florence - February 24, 2009 ensitivity curves S
S ensitivity curve in the high frequency range Terrestrial 10 6 m.s − 1 configuration v = L = 1 km √ h/ Hz 10 18 s − 1 ˙ N = 1 μ 10 - 19 10 − 5 tan θ = T = 1 ms 5 μ 10 - 20 2 μ 10 - 20 1 μ 10 - 20 5 μ 10 - 21 Ω 1000 1500 2000 3000 5000 http://www.esa.int/act 16 Pacôme DELVA - GGI, Arcetri, Florence - February 24, 2009
S ensitivity curve in the low frequency range 10 m.s − 1 S patial v = configuration L = 1 km 10 14 s − 1 √ ˙ N = h/ Hz tan θ = 0 . 5 T = 100 s Ω http://www.esa.int/act 17 Pacôme DELVA - GGI, Arcetri, Florence - February 24, 2009
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 traj ectory ∆ φ ( Ω ) = 4 π h D F 1 ( Ω ) D À L λ r T F 1 ( Ω ) = i sin 2 Ψ Ψ = Ω T 2 X L λ r = 2 π ~ = 2 π • The atom wavelength is fixed by the F 1( Ψ ) • The atom wavelength is fixed by the mv r k eff impulsion of the laser impulsion of the laser F 0( Ψ ) • The distance in the amplitude is the • The distance in the amplitude is the distance between the atom distance between the atom interferometer and the laser interferometer and the laser http://www.esa.int/act 18 Pacôme DELVA - GGI, Arcetri, Florence - February 24, 2009
19 http://www.esa.int/act Pacôme DELVA - GGI, Arcetri, Florence - February 24, 2009 MWI vs. LWI
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