F. Vetrano Università di Urbino & INFN Firenze, Italy

1. Atom interferometers for gravitational wave detection: a look at a “simple” configuration F. Vetrano Università di Urbino & INFN Firenze, Italy

2. Performance and Sensitivity Frequency response: phase difference at the output when the input is a “unity amplitude” GW Noise spectrum: power spectral density of phase fluctuations read at the output Sensitivity: the smallest amplitude wave that can be detected at a fixed S.N.R. (usually 1) output input Frequency Response

3. The Ingredients of Sensitivity - 1 As an example look at the performance of an optical interferometer (a Michelson with suitable technical solutions when a plane GW with “+” polarization is impinging on it along a direction perpendicular to its arms): Frequency Response: output (phase difference) Input (GW) Frequency Response Geometrical Term Probe Term Configuration Term Geometrical Term: Scale factor related to the dimension of the detector (the length of Michelson arms, and their angular relation) Probe Term: the Physics for detection (interference of optical beam) Configuration Term: the geometrical arrangement of components of the detector (refraction, reflection and recombination of the same beam on suspended mirrors in an orthogonal – arms Michelson)

4. The Ingredients of Sensitivity - 2 Because of the discrete nature of light and/or atomic beams, we have a unavoidable limit in reading the interferometer output: the Shot Noise. We adopt the “Shot Noise limited Sensitivity” as a first criterium for comparing performances. Noise spectrum (Shot Noise only): Assuming poissonian distribution we have: Standard Deviation fluctuations at the output Power Spectral Density (η1 is a kind of “reading” efficiency) (Shot Noise is a white noise) Correlation The minimal detectable signal amplitude at S.N.R. = 1 is supplied by (η2 is a “efficiency of the process”) where η=η2/η1 is a “efficiency number” (we put η=1 from now on) and for a Michelson interferometer

5. Why we hope in Atom Interferometry ? Shot Noise limited sensitivity - Matter Waves versus Optical Waves: a naive approach Probe Term: max gain for fast – not relativistic atoms min loss for 100 W laser and the max value found in literature for Atom flow (~ 10 ) Shot Noise 18 Six order of magnitude at our disposal assuming the same order of magnitude for geometrical term. Are we able to use this resource? And what about the configuration term G(Ω)?

6. Towards the evaluation of the S.N. limited sensitivity Detection g, 0 e, k e, k g, 0 g, 0 Source T T Single interferometer with M.Z. geometry and light-field beam-splitters The absorption (emission) of momenta modifies both internal and external states We use the ABCD formalism, applied to a wave packet represented in a gaussian basis (e.g. Hérmite-Gauss basis).

7. Determine the ABCD Matrices - 1 Suppose the Hamiltonian quadratic at most: Evolution (via the Ehrenfest theorem) through Hamilton’s equations:

8. Determine the ABCD Matrices - 2 The integral of Hamilton’s equations is: A perturbative expansion leads to: time ordering operator

9. Evolution of a gaussian wave packet under ABCD description Under paraxial approximation, the evolution of the gaussian wave packet is determined by the classical action Scl and by the use of the ABCD matrices: where: (X/Y is the complex radius of curvature for the gaussian w.p.)

10. The Beam Splitter influence Standard 1st order perturbation approach for weak dipole interaction ttt theorem The B.S. (neglecting possible dispersive properties) introduces a multiplicative amplitude Qbs and a phase factor simply related to the laser beam quantities ω*, k*, Φ* where q* = qcl(tA), qcl being the central position of the incoming atomic w.p., with respect to the laser source, and tA = central time of e.m. pulse (used as an atom beam splitter).

11. Phase shift for a sequence of pairs of homologous paths - 1 q kβN kβ1 kβ3 kβi kβ2 β1 β3 βD βN β2 Mβ1 Mβ2 Mβ3 Mβi MβN βi Mα1 Mα2 Mα3 Mαi MαN αD αN α1 α2 α3 αi kα1 kα2 kαN kαi kα3 t t1 t2 t3 ti tN tD

12. Phase shift for a sequence of pairs of homologous paths - 2 From previous results: w.p. propagation Phases imprinted by the B.S. on the atom waves Splitting at the exit of the interferometer Space integration around the mid (exit) point, equal masses on both the paths and identical starting points q1α= q1β lead to simplified expression where all qj are evaluated by using ABCD matrices.

13. The Machine • Choose a system of coordinates • Calculate ABCD matrices in presence of GW at the 1st order in the strain • amplitude h • Apply ΔΦ expression (previous slide) to the settled interferometer • Use ABCD law to substitute all qj in Δφ expression • Fully simplify • Print ΔΦ • End Note : the job should be worked in the frequency space (Fourier transform)

14. How about coordinates ? - 1 Coordinates (and GW) are in the Hamiltonian: Starting from usual Lagrangian function (signature +,-,-,-) where gμν is the metric tensor, in the weak field approximation the first order expansion leads to the Hamiltonian function : To be compared with previous general expression.

15. How about coordinates ? - 2 Finally: The matrices α,β,γ,δ are fully determined by the metric (as usual greek indexes run from 0 to 3; latin indexes from 1 to 3) In the following we assume for simplicity f = g = 0 and GWs with “+” polarization, propagating along the z axis (j = 3).

16. Fermi Coordinates - 1 Metric essentially rectangular (near a line), with connection vanishing along the line, and series expansion: Laboratory Reference Frame: where h is the amplitude of the “+” polarized GW. We assume z = 0 as the plane of the interferometer and we develop our calculations on this plane.

17. Fermi Coordinates - 2 It is easy to obtain: α= δ= 0; β= 1; γ = Ω² h/2, which leads to the following expressions for A,B,C,D matrices: (for a single Fourier component)

18. Fermi Coordinates - 3 And finally we write the I/O relation through the response function: where all the quantities are expressed in the FC system and ћ is the reduced Planck constant. The index 1 refers to the first interaction between atoms and photons beams.

19. Einstein Coordinates - 1 In this system the “mirrors” are free falling in the field of the GW, and the metric is Hence α = δ= γ = 0; β = h ­ η, where η is the minkowskian matrix, and h the amplitude of the “+” polarized GW; we deduce immediately the ABCD matrices:

20. Einstein Coordinates - 2 We cannot use the same k for every atoms/photons interaction; from the metric for a null geodesic we have By inserting these kj values in the general expression for Δφ, we obtain where all the quantities are expressed in the EC system. But the transformation matrix S from FC to EC behaves as S = 1 + 0(h); so the two expressions for Δφ in the two systems of coordinates are identical (as expected from the gauge invariance property of Δφ).

21. Descriptions and Result FC: fiducial observer: the laser device is free falling; a tidal force acts on the atoms; the interaction points move and imprinted phases change accordingly. EC: Atoms are free falling; no forces on them; the space between interaction points shows a variable index of refraction; the imprinted phases change accordingly. Two different descriptions; same (physical) result, obviously. A.S. L.B. L.B. A.S.

22. The main contributions - 1 A kind of “clock term”, related to the travel of the beam from the laser to the first interaction point, viewed through the A.I. as a read-out. For a discussion about this term see: S. Dimopoulos et al, Phys.Rev D, 122002 (2008) We discuss here only the first term, in which we have neglected the smaller contribution k²ћ/ 2M* (in next few slides we put G(Ω) = [Ω T …… ]/2)

23. The main contributions - 2 It’s easy to rewrite the phase difference as: that is: to be compared with what we wrote in slide 3 (optical Michelson) Configuration term Geometrical dimension opening angle Probe (matter wave) Geometrical term

24. Shot Noise Limited Sensitivity Considering only the first term of the slide 23, and supposing the A.I. “shot noise” limited as clarifyed in slide 4 at the level of S.N.R. = 1 we have (with η = 1) which has the expected form (see slide 5).

25. The Configuration Term ToF 0.1 s ToF 50 s lG(Ω)l ToF 1ms ToF 0.01 s Frequency [Hz]

26. The Scale Factor Σ Σ1 Σ = Σ2 • We need to have Σ2 as larger as we can, but: • T is not free (the bandwidth behaves as 1/T) • vT is the longitudinal dimension L of the A.I. (coherence problem) • Ptr T/M is the transversal dimension of the A.I. (coherence and • handling problems)

27. Some sensitivity curves We represent the first branch only of the sensitivity curves Let us consider in some detail a specific interesting example (see next slide)

28. A rough picture of Sources & Detectors h[1/sqrt Hz] SN core collapse 1 LISA 2 LIGO – Virgo 3 A.I. -18 1 Intermediate BH-BH Coalescence 2 3 -20 Slow Pulsars Coalescence of massive BH ms Pulsars -22 LMXRBs & Perturbed “newborn”NS NS Binary Coalescence lrs NS Binary Coalescence hrs Galactic binaries -24 - 4 - 2 0 2 4 10 10 10 10 10 f[Hz]

29. Numbers A.I. S.N.-limited Sensitivity h [1/√Hz] F [Hz] Virgo S.N.-limited Sensitivity H

30. optimistic Some conclusions • Comprehensive approach to the problem with (hopefully) reliable • calculation of Frequency Response function for atom interferometers • L and VL frequency disfavoured from the FR behaviour: move the first • non-zero pole towards very low values (at expenses of reduced bandwidth)? • Different, more complex configurations? (e.g.: asymmetric interferometers; • multiple interferometers) • S.N. very hard limit: balance it with LMT? • Heisenberg limit? • Terrestrial solution: the true noise budget has to be investigated (thermal • noise; seismic wall;….) in the low- and intermediate-frequency range; • Space solution: removing seismic wall is of great advantage but in any • case S.N. limit is hard : balance it with LMT and large dimension (but • divergency problem) ? Or very slow atoms (but decay problem)? • In any case, in my opinion required numbers are leaving the realm of • forbidden dreams and are entering the world of exciting challenges