1. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 1 Università di Urbino Italy F. Vetrano Università di Urbino & INFN Firenze, Italy Atom interferometers for gravitational wave detection: a look at a simple configuration

2. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 2 Università di Urbino Italy 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) outputinput Frequency Response

3. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 3 Università di Urbino Italy 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: Input (GW) output (phase difference) 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. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 4 Università di Urbino Italy 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 Correlation The minimal detectable signal amplitude at S.N.R. = 1 is supplied by (η 2 is a efficiency of the process) where η=η 2 /η 1 is a efficiency number (we put η=1 from now on) and for a Michelson interferometer (Shot Noise is a white noise) (η 1 is a kind of reading efficiency )

5. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 5 Università di Urbino Italy 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 Shot Noise min loss for 100 W laser and the max value found in literature for Atom flow (~ 10 ) 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. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 6 Università di Urbino Italy Towards the evaluation of the S.N. limited sensitivity Source T T g, 0 e, k g, 0 e, k Detection 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). Single interferometer with M.Z. geometry and light-field beam-splitters

7. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 7 Università di Urbino Italy Suppose the Hamiltonian quadratic at most: Determine the ABCD Matrices - 1 Evolution (via the Ehrenfest theorem) through Hamiltons equations:

8. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 8 Università di Urbino Italy Determine the ABCD Matrices - 2 The integral of Hamiltons equations is: A perturbative expansion leads to: time ordering operator

9. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 9 Università di Urbino Italy where: Under paraxial approximation, the evolution of the gaussian wave packet is determined by the classical action S cl and by the use of the ABCD matrices: Evolution of a gaussian wave packet under ABCD description (X/Y is the complex radius of curvature for the gaussian w.p.)

10. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 10 Università di Urbino Italy 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 Q bs and a phase factor simply related to the laser beam quantities ω*, k*, Φ* where q* = q cl (t A ), q cl being the central position of the incoming atomic w.p., with respect to the laser source, and t A = central time of e.m. pulse (used as an atom beam splitter).

11. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 11 Università di Urbino Italy Phase shift for a sequence of pairs of homologous paths - 1 kβ1kβ1 kβ2kβ2 kβ3kβ3 kβikβi kβNkβN kα1kα1 kα2kα2 kα3kα3 kαikαi kαNkαN t 1 t 2 t 3 t i t N t D M β1 M β2 M β3 M βi M βN M α1 M α2 M α3 M αi M αN β1β1 β2β2 β3β3 βiβi βNβN βDβD α1α1 α2α2 α3α3 αiαi αNαN α Dα D t q

12. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 12 Università di Urbino Italy 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 q 1α = q 1β lead to simplified expression where all q j are evaluated by using ABCD matrices.

13. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 13 Università di Urbino Italy Choose a system of coordinates Calculate ABCD matrices in presence of GW at the 1 st order in the strain amplitude h Apply ΔΦ expression (previous slide) to the settled interferometer Use ABCD law to substitute all q j in Δφ expression Fully simplify Print ΔΦ End Note : the job should be worked in the frequency space (Fourier transform) The Machine

14. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 14 Università di Urbino Italy How about coordinates ? - 1 Coordinates (and GW) are in the Hamiltonian: Starting from usual Lagrangian function ( signature +,-,-,- ) where g μν 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. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 15 Università di Urbino Italy 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. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 16 Università di Urbino Italy 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. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 17 Università di Urbino Italy 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. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 18 Università di Urbino Italy 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. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 19 Università di Urbino Italy 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. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 20 Università di Urbino Italy 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 k j 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. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 21 Università di Urbino Italy 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. Descriptions and Result A.S. L.B. A.S.

22. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 22 Università di Urbino Italy 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 k² ћ / 2M* (in next few slides we put G(Ω) = [Ω T …… ]/2)

23. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 23 Università di Urbino Italy Its easy to rewrite the phase difference as: that is: to be compared with what we wrote in slide 3 (optical Michelson) The main contributions - 2 opening angle Geometrical dimension Geometrical term Probe (matter wave) Configuration term

24. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 24 Università di Urbino Italy 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. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 25 Università di Urbino Italy The Configuration Term lG(Ω)l Frequency [Hz] ToF 50 s ToF 0.1 s ToF 0.01 s ToF 1ms

26. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 26 Università di Urbino Italy The Scale Factor Σ Σ= Σ1Σ1 Σ2Σ2 We need to have Σ 2 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) P tr T/M is the transversal dimension of the A.I. (coherence and handling problems)

27. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 27 Università di Urbino Italy 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. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 28 Università di Urbino Italy NS Binary Coalescence lrs Slow Pulsars LMXRBs & Perturbed newbornNS A rough picture of Sources & Detectors -24 -22 -20 -18 h [1/sqrt Hz] f [Hz] - 4 - 2 0 2 4 10 10 10 10 10 Galactic binaries Coalescence of massive BH Intermediate BH-BH Coalescence SN core collapse ms Pulsars 1 2 3 1 LISA 2 LIGO – Virgo 3 A.I. NS Binary Coalescence hrs

29. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 29 Università di Urbino Italy Numbers F [Hz] h [1/ Hz ] H Virgo S.N.-limited Sensitivity A.I. S.N.-limited Sensitivity

30. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 30 Università di Urbino Italy 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 optimistic