Atom Interferometry for Gravitational Waves. Flavio Vetrano Università di Urbino and INFN Firenze

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Paula Ramsey
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1 Am Inefemey f Gaviainal Waves Flavi Vean Univesià di Ubin and INFN Fienze Flavi Vean Rma May 3, 00

2 Ramsey Inefeence Tw level ams (gund, excied Tw successive elecmagneic ineacins N exenal mmenum exchange (f he mmen Only inenal degees f feedm ae invlved in he cesses e.m. e.m. [w..] g g, e Ovela (g, e decay e g 0 N ssibiliyknwwhichae g and which ae e alng he (ime ah: in he finale sae hee is inefeence beween he w saes: his is he Ramsey Inefeence. Flavi Vean Rma May 3, 00

3 M.W. Evluin - Sandad uanum aach (in aing sysem: fis de eubain hey f a dile e.m./w.. ineacin. If f simliciy we assume ze deuning δ, he ulain f he e sae afe he ineacin ime τ wih he e.m. field (by a lase f feuency ωl is ce( τ (- cs Ωegτ ( whee Ω eg is he Rabi feuency h Ω - e d E g f, d being he dile mmenum eg f he am and E he elecic scillaing field. Puing ωeg ωe ωg, he deuning is defined by δ ωl ωeg. Assumiming δ 0, he euain ( descibes he s called esnan Rabi scillains. Assume: 0 cg(0 ² ; τ such ha Ω egτ π/ (π/ ulse. Fm ( we have : ce ( τ cg ( τ Flavi Vean Rma May 3, 00 3

4 M.W. Evluin - 0 i π/ ulse This is a efec Beam Slie f inenal saes / 0 g e τ g e Flavi Vean Rma May 3, 00 4

5 M.W. Evluin - 3 Use w π/ ulses in de have Ωegτ π (π ulse: we have 0 0 ce (τ π ulse This is a efec Mi f inenal saes g e e g 0 τ Ne: If δ 0, wih w π/ ulses same cnsideains as befe wih c (τ T ( csδ T Flavi Vean Rma May 3, 00 e 5

6 M.W. Evluin - 4 Take in accun exenal D.O.F. cnsideing he mmenum exchange : e, e > e > e > g,g > g V - d E > g > E E cs ( k - ωl φ kineic em in he Hamilnian (and in he hase ecil em in he deuning δ: δ ω L ( ω eg k M h k M Even if he inefeence is n inenal d..f., he ecil ens he ah: if we lk f inefeence, we mus clse i saially. Flavi Vean Rma May 3, 00 6

7 The simles clsed am inefemee g e g e Cune Suce g π/ π π/ c e, hk (T τ ΔΦ δτ c (T e, hk ( - cs - he absin (emissin f mmena mdifies bh inenal and exenal saes simulaneusly ΔΦ akes in accun all ha haens he ams (GW Flavi Vean Rma May 3, 00 7

8 The Ingediens f Sensiiviy As an examle lk a he efmance f an ical inefemee (a Michelsn wih suiable echnical sluins when a lane GW wih laizain is iminging n i alng a diecin eendicula is ams: Feuency Resnse: uu (hase diffeence Δ Δϕ L c h Ω L sinc cs Ω c L c L 4 π Ω L sinc cs Ω λ c L c ; Δ ϕ h π c Δ λ Inu (GW Feuency Resnse Gemeical Tem Pbe Tem Cnfiguain Tem Gemeical Tem: Scale fac elaed he dimensin f he deec (he lengh f Michelsn ams, and hei angula elain Pbe Tem: he Physics f deecin (inefeence f ical beam Cnfiguain Tem: he gemeical aangemen f cmnens f he deec (efacin, eflecin and ecmbinain f he same beam n susended mis in an hgnal ams Michelsn Flavi Vean Rma May 3, 00 8

9 Why we he in Am Inefemey? Sh Nise limied sensiiviy - Mae Waves vesus Oical Waves: a naive aach ~ h ( Ω L N& λ G ( Ω λ λ MW π f hω c 0 OW mpv c mpc v Pbe Tem: L 8 h max gain f fas n elaivisic ams N& N& Sh Nise Phn 0 Am min lss f 00 W lase and he max value fund in lieaue f Am flw 8 (~ 0 Six de f magniude a u dissal assuming he same de f magniude f gemeical em. Ae we able use his esuce? And wha abu he cnfiguain em G(Ω? Flavi Vean Rma May 3, 00 9

10 Twads he evaluain f he S.N. limied sensiiviy g, 0 e, k g, 0 e, k Deecin Suce g, 0 T T Single inefemee wih M.Z. gemey and ligh-field beam-slies The absin (emissin f mmena mdifies bh inenal and exenal saes We use he ABCD fmalism, alied a wave acke eesened in a gaussian basis (e.g. Hémie-Gauss basis. Flavi Vean Rma May 3, 00 0

11 Flavi Vean Rma May 3, 00 Suse he Hamilnian uadaic a ms: elaivisic mass ~ ; M g( - M ( f M M H α δ γ δ β α Deemine he ABCD Maices - Evluin (via he Ehenfes heem hugh Hamiln s euains: ( ( ( ( ( g( ( f M ( d dh M d dh M d d δ γ β α Γ Γ

12 Deemine he ABCD Maices - The inegal f Hamiln s euains is: ( A(, ( C(, M B(, D(, ( ( M (, Ξ Ψ (, Ξ (, Ψ (, A(, C(, B(, D(, A( ', C( ', τ B( ', f(' d' D( ', g(' α(' β(' ex d' γ(' δ(' τ ex Γ ( ' d' A eubaive exansin leads : ime deing ea A(, C(, B(, D(, 0 ' 0 τ ( ' d' d' ( ' ( '' Γ Γ Γ d''... Flavi Vean Rma May 3, 00

13 Flavi Vean Rma May 3, 00 3 whee: Y, X,,, w..(, ( is ex Y, X,,,..( w cl h Ψ Ξ Y X D C B A Y X M D C B A M Unde aaxial aximain, he evluin f he gaussian wave acke is deemined by he classical acin Scl and by he use f he ABCD maices: Evluin f a gaussian wave acke unde ABCD desciin (X/Y is he cmlex adius f cuvaue f he gaussian w..

14 The Beam Slie influence Sandad s de eubain aach f weak dile ineacin heem The B.S. (neglecing ssible disesive eies induces a mulilicaive amliude Qbs and a hase fac simly elaed he lase beam uaniies ω*, k*, Φ* Q bs ex ( ω k Φ whee * cl(a, cl being he cenal siin f he incming amic w.., wih esec he lase suce, and A cenal ime f e.m. ulse (used as an am beam slie. Flavi Vean Rma May 3, 00 4

15 Phase shif f a seuence f ais f hmlgus ahs - kβ kβ kβ3 kβi kβn β β3 βn βd Mβ β Mβ Mβ3 βi Mβi MβN Mα Mα Mα3 Mαi MαN α D α α α3 αi αn kα kα kα3 kαi kαn 3 i N D Flavi Vean Rma May 3, 00 5

16 Phase shif f a seuence f ais f hmlgus ahs - Fm evius esuls: w.. again Δϕ h N [ ( ( ] N S, S, [( k k ( ω ω ( ] β j j α j j ϑ ϑ β j β j α j α j β j α j j β j α j [ ( ( ] β D h j β D α D α D j Sliing a he exi f he inefemee Phases imined by he B.S. n he am waves Sace inegain aund he mid (exi in, eual masses n bh he ahs and idenical saing ins α β lead simlified exessin Δϕ N β j α j ( k k ( ω ω ( ϑ ϑ β j α j β j α j j β j α j j whee all j ae evaluaed by using ABCD maices. Flavi Vean Rma May 3, 00 6

17 The Machine Chse a sysem f cdinaes Calculae ABCD maices in esence f GW a he s de in he sain amliude h Aly ΔΦ exessin (evius slide he seled inefemee Use ABCD law subsiue all j Fully simlify Pin ΔΦ End in Δφ exessin Ne : he jb shuld be wked in he feuency sace (Fuie ansfm Asen Wine Cnfeence n GW and hei Deecin h:// Flavi Vean Rma May 3, 00 7

18 Hw abu cdinaes? - Cdinaes (and GW ae in he Hamilnian: H α β M δ M γ f ( - M g( Saing fm usual Lagangian funcin (signaue,-,-,- L mc g μν x& μ x& ν whee gμν is he meic ens, in he weak field aximain g μν ημν lμν ; lμν << he Hamilnian funcin : he fis de exansin leads H M [ ] ij i i l il j il η ij M j T be cmaed wih evius geneal exessin. Flavi Vean Rma May 3, 00 8

19 Hw abu cdinaes? - Finally: l c i γ ij j g i i l i c α ij j f i l ij η ij β ij The maices α,β,γ,δ ae fully deemined by he meic ds μ ν μ ν η μν dx dx l μνdx dx (as usual geek indexes un fm 0 3; lain indexes fm 3 In he fllwing we assume f simliciy f g 0 and GWs wih laizain, againg alng he z axis (j 3. Flavi Vean Rma May 3, 00 9

20 The main cnibuins Δϕ ( Ω Ω h ( sin( ΩΤ Ω T k M ( cs( ΩΤ cs( ΩΤ sin( ΩΤ sin( ΩΤ sin ΩΤ/ ΩΤ/ ΩΤ i ΩΤ ( sin ΩΤ/ cs( ΩΤ ΩΤ/ Ω h ( Ω sin( ΩΤ/ T k ΩΤ/ [ cs( ΩΤ isin( ΩΤ ] A kind f clck em, elaed he avel f he beam fm he lase he fis ineacin in, viewed hugh he A.I. as a ead-u. We discuss hee in deail nly he fis em, in which we have negleced he smalle cnibuin k²ћ / M* (in nex few slides we u G(Ω [Ω T ]/ Tin G.M. and Vean F. Class. Quanum Gav., 4, 67 (007 Flavi Vean Rma May 3, 00 0

21 Sh Nise Limied Sensiiviy Cnsideing nly he fis em f he slide 4, and susing he A.I. sh nise limied: Δ [ Φ( Ω ] Sh.N. η N& A he level f S.N.R. we have (wih η ~ MW h ( Ω N& λ 4 πϑ L G ( Ω Flavi Vean Rma May 3, 00

22 The Cnfiguain Tem G G ( Ω 0( Ω ; G( Ω Disance Ω 0 beween ( Ω ΩΤ adiacen les: Ω T sin Ω T scillaes me Ω π v ΔΩ L sin Ω T - Ω T (his and me Ω T sin Ω T deemines aidly beween 0 and he bandwidh f he banches TF 50 s 0.8 TF 0. s lg(ωl TF 0.0 s TF ms Flavi Vean Rma May 3, 00 Feuency [Hz]

23 The Scale Fac Σ ~ MW h ( Ω N& λ 4πϑ L G ( Ω ~ h ( Ω h vt / N& G ( Ω Σ Σ Σ -35 Σ 5 0 J s We need have Σ as lage as we can, bu: T is n fee (he bandwidh behaves as /T vt is he lngiudinal dimensin L f he A.I. (cheence blem P T/M is he ansvesal dimensin f he A.I. (cheence and handling blems Flavi Vean Rma May 3, 00 3

24 Sme sensiiviy cuves We eesen he fis banch nly f he sensiiviy cuves -5 Σ Js TF 50 s Σ TF 0 - s Js Σ TF s Js Le us cnside in sme deail a secific ineesing examle (see nex slide Flavi Vean Rma May 3, 00 4

25 A ugh icue f Suces & Deecs -8 h [/s Hz] Inemediae BH-BH Calescence SN ce cllase LISA LIGO Vig 3 A.I. 3-0 Calescence f massive BH - Slw Pulsas ms Pulsas Galacic binaies NS Binay Calescence hs NS Binay Calescence ls LMXRBs & Peubed newbn NS Flavi Vean Rma May 3, 00 f [Hz] 5

26 Numbes TF 0.4 s; Σ. 0 3 L 4 0 m -3 Js h [/ Hz] 7. 0 TF 0.4 s; 5. 0 Σ Js A.I. S.N.-limied Sensiiviy v 0 m/s & N.7 0 N& F [Hz] Kg m/s H v 6 m/s UV hns ( λ 0, H Lyman α Flavi Vean Rma May 3, 00 6

27 The main cnibuins Δϕ ( Ω Ω h ( sin( ΩΤ Ω T k M ( cs( ΩΤ cs( ΩΤ sin( ΩΤ sin( ΩΤ sin ΩΤ/ ΩΤ/ ΩΤ i ΩΤ ( sin ΩΤ/ cs( ΩΤ ΩΤ/ Ω h ( Ω sin( ΩΤ/ T k ΩΤ/ [ cs( ΩΤ isin( ΩΤ ] A kind f clck em, elaed he avel f he beam fm he lase he fis ineacin in, viewed hugh he A.I. as a ead-u. Ms iman, i scales wih, n wih T/M ~ L : diffeenial cnfiguain? Dimuls S., Gaham P.W., Hgan J.M. and Kasevich M.A. Phys.Le. B, 678, 37 (009 Mülle H., Chiw S., Hemann S. and Chu S. Phys.Rev.Le., 0, (009 Tin G., Vean F., Am Inefemees f Gaviainal Waves deecin, submied GRG (00 Flavi Vean Rma May 3, 00 7

11. HAFAT İş-Enerji Power of a force: Power in the ability of a force to do work

11. HAFAT İş-Enerji Power of a force: Power in the ability of a force to do work MÜHENDİSLİK MEKNİĞİ. HFT İş-Eneji Pwe f a fce: Pwe in he abiliy f a fce d wk F: The fce applied n paicle Q P = F v = Fv cs( θ ) F Q v θ Pah f Q v: The velciy f Q ÖRNEK: İŞ-ENERJİ ω µ k v Calculae he pwe