Casimir-Polder shifts on quantum levitation states 1

Transcription

1 Casimir-Polder shifts o quatum levitatio states 1 Pierre-Philippe Crépi Laboratoire Kastler Brossel supervised by S. Reyaud, R. Guérout ad N. Cherroret With discussios with V. V. Nesvizhevsky ad A. Yu. Voroi 1 P-P. Crépi, G. Dufour, R. Guérout, A. Lambrecht ad S.Reyaud, Physical Review A 95 (17) 351

2 GBAR Experimet Gravitatioal Behavior of Atihydroge at Rest Test the equivalece priciple for atimatter by timig the free fall of atihydroge H released from trap Experimet uder costructio at CERN Curret experimetal boud 3 : 65g g 11g Expected accuracy for g : 1 START : the extra e + is photodetached STOP : aihilatio of H o the detector after its free fall P. Idelicato et al. Hyperfie Iteract (14) 8: ALPHA collab. Nature Commuicatios 4 (13) 1785

3 Quatum levitatio states At small distaces (<1 µm), H is sesitive to the Casimir-Polder potetial : quatum reflectio (QR) occurs 4 Potetial ladscape Reflectivity H is trapped betwee Gravity ( ) ad Quatum Reflectio ( ) quatum levitatio state 4 G. Dufour et al. J. Mod. Phys. Cof. Ser. 3 (14) 14665

4 Outlooks Usig QR for spectroscopic measuremets 5 : i future geeratios of GBAR, spectroscopic measuremets o atihydroge atoms i quatum levitatio states Aalogy with the GRANIT experimet 6 to measure resoace trasitios betwee the gravitatioally quatum states of eutros GOAL : determie precisely quatum levitatio states 5 A. Yu. Voroi, V. V. Nesvizhevsky et al. J. Mod. Phys. Cof. Ser., 3 (14) M. Kreuz, V. V. Nesvizhevsky et al. Nucl. Istr. Meth. A 611 (9) 36

5 Scatterig legth approximatio Scatterig legth approximatio 7 : E 1 = λ ɛ g + mga (1) λ ɛ g : eergy of quatum boucers, ( ) 1/3 ɛ g = mg (.6 pev for g = g), λ are zeros of the Airy fuctio Ai mga : CP shift due to QR, a is the scatterig legth Trasitio frequecies : ω m = E1 E 1 m = (λ λ m )ɛ g () Measure of ω m would give a direct access to the value of g! Perform a full quatum treatmet of free fall ad QR improve (1) 7 A.Y. Voroi, P. Froelich ad V. V. Nesvizhevsky P. R. A 83 (11) 393

6 Schrödiger equatio Schrödiger equatio : 8 ψ (z) + F (z)ψ(z) = F { (z) = m (E V (z)) V (z) = mgz + VCP (z) if z > V (z) = if z z (lg) Liouville trasformatio : z(z), ψ(z) = z (z)ψ(z) F (z) = E V (z) V (z) = z V CP (z) E = E ɛ g ψ (z) + F (z)ψ(z) = : preserves eergy shifts V, E (ɛg) V, E z

7 Cavity resoaces New picture : Fabry-Perot cavity TOP mirror : perfectly reflectig due to gravity BOTTOM mirror : partially reflectig due to QR Above ad below the bottom mirror, quasi-statioary states : ψ m (z) = am (Ci+ (z z t ) + Ci (z z t )), Ci ± (z) = Ai(z) ± ibi(z) m : umber of bouces, ρ : roud-trip factor, a m+1 = ρ a m Resoaces E ( labels eergy levels) correspod to : ρ R, ρ 1

8 Casimir-Polder shifts We solve umerically Schrödiger equatio ad fid ρ(e) ad also E. Compariso with scatterig approximatio E λ ɛ g (= mga i s.l.a.) : -3 E-λϵg _, mgre(a) (1-4 ϵg) mgre(a) E-λϵg _ Approximatio works util a fractio of 1 4 ɛ g. We eed a more precise descriptio. Roud-trip factor = QR amplitude r + propagatio phase factor : Resoace coditio : ρ re iθ( E/ɛg), ta θ(x) = Ai(x) Bi(x) θ( E /ɛ g ) + arg( r) = π

9 Effective rage approximatio arg( r) =? New complex legth A(k), such as A() = a ad r = 1 ika(k) 1+ikA(k), k me Effective rage theory suggests 8 for V 4 = C 4 /z 4 potetial : ka(k) = ikl α(kl), l = mc 4 α(k) = 1 + i π 3 K + ( (γ + l ) 9 π 3 i l K) K For Casimir-Polder potetial (V (z) C 4 /z 4 ) : α(k) = α + i π 3 K + ( α α l K ) K where α ad α are determied by a fit. Resoace coditio becomes : θ( E /ɛ g ) Re(arcta(k l α(k l))) = π 8 I. Spruch, T. O Malley ad I, Roseberg, Phys. Rev. Lett. 5 (196) 375

10 Results Correctio to the scatterig legth approximatio : E λ ɛ g mga E = E aa E um 8 6 E-Re(E 1 ) (1-5 ϵg) 6 4 ΔE (1-6 ϵg) Aalytical method would be sufficiet to calculate quatum levitatio states eergies ad deduce from the spectroscopy measuremets the value of g with a accuracy better tha 1 5 g!

11 THANK YOU FOR YOU ATTENTION!

12 Width of resoaces Exted aalytically ρ to C. Cavity respose fuctio : f(e) = Complex resoaces E : ρ(e ) = 1. Fit f A (E Re E ) +(Im E ) ρ(e) 1 ρ(e) f E (ɛg) Re(ΔE), (1-6 ϵg) Im(E-E 1 ), (1-5 ϵg) Eergies are still kow with a accuracy of a few 1 6 ɛ g Good approximatio of the lifetime i cavity : τ = mgb, b = Im(a)