Universit di Urbino Italy Atom interferometers for gravitational

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

Università di Urbino Performance and Sensitivity Italy Frequency response: phase difference at the output when the input is a “unity amplitude” GW output input Frequency Response 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) F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 2

The Ingredients of Sensitivity - 1 Università di Urbino Italy 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) F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 3

The Ingredients of Sensitivity - 2 Università di Urbino Italy 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 Power Spectral Density Correlation fluctuations at the output (η 1 is a kind of “reading” efficiency) (Shot Noise is a white noise) 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 F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 4

Why we hope in Atom Interferometry ? Università di Urbino Italy Shot Noise limited sensitivity - Matter Waves versus Optical Waves: a naive approach Probe Term: Shot Noise max gain for fast – not relativistic atoms min loss for 100 W laser and the max value found in literature for Atom flow 18 (~ 10 ) 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)? F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 5

Università di Urbino Towards the evaluation of the S. N. limited sensitivity g, 0 e, k g, 0 Italy Detection e, k g, 0 Source 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). F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 6

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

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

Evolution of a gaussian wave packet under ABCD description Università di Urbino Italy 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. ) F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 9

The Beam Splitter influence Università di Urbino Standard 1 st order perturbation approach for weak dipole interaction Italy 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(t. A), qcl 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). F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 10

Phase shift for a sequence of pairs of homologous paths - 1 Università di Urbino Italy q kβ 1 kβ 3 kβ 2 kβN kβi β 3 βN β 2 Mβ 1 Mβ 2 Mβ 3 M βi MβN βi Mα 1 Mα 2 Mα 3 Mαi MαN kα 1 kα 2 α 3 kαi αi kαN αN βD α D t t 1 t 2 t 3 ti t. N t. D F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 11

Phase shift for a sequence of pairs of homologous paths - 2 From previous results: Splitting at the exit of the interferometer Università di Urbino Italy w. p. propagation Phases imprinted by the B. S. on the atom waves 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 qj are evaluated by using ABCD matrices. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 12

The Machine 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 qj in Δφ expression • Fully simplify • Print ΔΦ • End Note : the job should be worked in the frequency space (Fourier transform) F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 13

How about coordinates ? - 1 Università di Urbino Italy 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. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 14

How about coordinates ? - 2 Università di Urbino Italy 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). F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 15

Fermi Coordinates - 1 Università di Urbino Italy 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. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 16

Fermi Coordinates - 2 Università di Urbino Italy 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) F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 17

Fermi Coordinates - 3 Università di Urbino Italy 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. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 18

Einstein Coordinates - 1 Università di Urbino Italy 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: F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 19

Einstein Coordinates - 2 Università di Urbino Italy 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). F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 20

Descriptions and Result 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. A. S. L. B. 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. A. S. Two different descriptions; same (physical) result, obviously. F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 21

The main contributions - 1 Università di Urbino Italy 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 / 2 M* (in next few slides we put G(Ω) = [Ω T …… ]/2) F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 22

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

Shot Noise Limited Sensitivity Università di Urbino Italy 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). F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 24

The Configuration Term Università di Urbino Italy To. F 50 s To. F 0. 1 s l. G(Ω)l To. F 0. 01 s To. F 1 ms Frequency [Hz] F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 25

The Scale Factor Σ Università di Urbino Italy Σ = Σ 1 Σ 2 We need to have Σ 2 as larger as we can, but: • T is not free (the bandwidth behaves as 1/T) • v. T 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) F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 26

Some sensitivity curves Università di Urbino Italy We represent the first branch only of the sensitivity curves Let us consider in some detail a specific interesting example (see next slide) F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 27

Università di Urbino A rough picture of Sources & Detectors Italy h [1/sqrt Hz] SN core collapse -18 1 2 3 1 LISA LIGO – Virgo A. I. Intermediate BH-BH Coalescence 2 3 -20 of ce en sc BH ale ve Co assi m -22 Galactic binaries ms Pulsars Slow Pulsars NS NS C Co oa Bi les na ale Bin a ce ry sc ry l e r hr n s nce ce s LMXRBs & Perturbed “newborn”NS -24 -4 10 -2 10 F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 0 10 2 10 4 10 f [Hz] 28

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

optimistic 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 F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 30