casimir-polder shifts on quantum levitation states=1p-p...
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Casimir-Polder shiftson quantum levitation states1
Pierre-Philippe Crépin
Laboratoire Kastler Brossel
supervised by S. Reynaud, R. Guérout and N. Cherroret
With discussions with V. V. Nesvizhevsky and A. Yu. Voronin
1P-P. Crépin, G. Dufour, R. Guérout, A. Lambrecht and S.Reynaud,Physical Review A 95 (2017) 032501
GBAR Experiment
Gravitational Behavior of Antihydrogen at Rest2
Test the equivalence principle for antimatter by timing the free fall ofantihydrogen H released from trapExperiment under construction at CERNCurrent experimental bound3 : −65g ≤ g ≤ 110g
START : the extra e+
is photodetached
STOP : annihilation ofH on the detector afterits free fall
Expected accuracy for g : 10−2
2P. Indelicato et al. Hyperfine Interact (2014) 228:141-1503ALPHA collab. Nature Communications 4 (2013) 1785
P-P. Crépin - Rencontres de Moriond - March 2017 Casimir-Polder shifts on quantum levitation states
Quantum levitation states
At small distances (<1 µm), H is sensitive to the Casimir-Polderpotential : quantum reflection (QR) occurs4
Potential landscape Reflectivity
H is trapped between Gravity (↓) and Quantum Reflection (↑)→ quantum levitation state
4G. Dufour et al. J. Mod. Phys. Conf. Ser. 30 (2014) 1460265P-P. Crépin - Rencontres de Moriond - March 2017 Casimir-Polder shifts on quantum levitation states
Outlooks
Using QR for spectroscopic measurements5 : in futuregenerations of GBAR, spectroscopic measurements onantihydrogen atoms in quantum levitation statesAnalogy with the GRANIT experiment6 to measure resonancetransitions between the gravitationally quantum states ofneutrons
GOAL : determine precisely quantum levitation states5A. Yu. Voronin, V. V. Nesvizhevsky et al. J. Mod. Phys. Conf. Ser. , 30
(2014) 14602666M. Kreuz, V. V. Nesvizhevsky et al. Nucl. Instr. Meth. A 611 (2009) 326
P-P. Crépin - Rencontres de Moriond - March 2017 Casimir-Polder shifts on quantum levitation states
Scattering length approximation
Scattering length approximation7 :
E1n = λnεg +mga (1)
λnεg : energy of quantum bouncers,
εg =(
~2mg2
2
)1/3(0.6 peV for g = g),
λn are zeros of the Airy function Ai
mga : CP shift due to QR, a is the scattering length
Transition frequencies :
ωmn =E1n − E1m
~= (λn − λm)εg (2)
Measure of ωmn would give a direct access to the value of g !Perform a full quantum treatment of free fall and QR → improve (1)
7A.Y. Voronin, P. Froelich and V. V. Nesvizhevsky P. R. A 83 (2011) 032903P-P. Crépin - Rencontres de Moriond - March 2017 Casimir-Polder shifts on quantum levitation states
Schrödinger equation
Schrödinger equation :
ψ′′(z) + F (z)ψ(z) = 0
F (z) = 2m~ (E − V (z)){
V (z) = mgz + VCP (z) if z > 0
V (z) =∞ if z ≤ 0 0 1 2 3 4 5 6 7 8z (`g)
−2
0
2
4
6
8
V,E
(εg)
0.00 0.05 0.10
−20
−10
0
Liouville transformation :
z(z), ψψψ(z) =√z′(z)ψ(z)
ψψψ′′(z) + F (z)ψψψ(z) = 0
F (z) = E − V (z)
V (z) = z − VCP (z)
E = Eεg
: preserves energy shifts−2 0 2 4 6 8
z
−2
0
2
4
6
8
V,E
−0.1 0.0 0.1 0.2
0
200
400
600
P-P. Crépin - Rencontres de Moriond - March 2017 Casimir-Polder shifts on quantum levitation states
Cavity resonances
New picture : Fabry-Perot cavity
TOP mirror : perfectlyreflecting due to gravity
BOTTOM mirror : partiallyreflecting due to QR
Above and below the bottom mirror, quasi-stationary states :
ψψψm(z) = am2 (Ci+(z − zt) + Ci−(z − zt)), Ci±(z) = Ai(z)± iBi(z)
m : number of bounces, ρ : round-trip factor, am+1 = ρ am
Resonances En (n labels energy levels) correspond to :
ρ ∈ R, ρ ' 1
P-P. Crépin - Rencontres de Moriond - March 2017 Casimir-Polder shifts on quantum levitation states
Casimir-Polder shifts
We solve numerically Schrödinger equation and find ρ(E) and also En.Comparison with scattering approximation En − λnεg (= mga in s.l.a.) :
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mgRe(a) ● En-λnϵg_
2 4 6 8 10
-9
-8
-7
-6
-5
-4
-3
n
En-λnϵ g_,mgRe(a)
(10-4ϵ g)
Approximation works until a fraction of 10−4εg.We need a more precise description.
Round-trip factor = QR amplitude r + propagation phase factor :
ρ ' −re2iθ(−E/εg), tan θ(x) = Ai(x)Bi(x)
Resonance condition :
2θ(−En/εg) + arg(−r) = 2nπ
P-P. Crépin - Rencontres de Moriond - March 2017 Casimir-Polder shifts on quantum levitation states
Effective range approximation
arg(−r) =?
New complex length A(k), such as A(0) = a and
r = − 1−ikA(k)1+ikA(k) , ~k ≡
√2mE
Effective range theory suggests8 for V4 = −C4/z4 potential :
kA(k) = −ikl α(kl), l =√2mC4
~α(K) = 1 + iπ3K +
(83 (γ + ln 2)− 28
9 − 2π3 i+ 4
3 lnK)K2
For Casimir-Polder potential (V (z)→ −C4/z4) :
α(K) = α0 + iπ3K +(α2 + 4
3α0 lnK)K2
where α0 and α2 are determined by a fit.
Resonance condition becomes :
θ(−En/εg)− Re(arctan(knl α(knl))) = nπ
8I. Spruch, T. O’Malley and I, Rosenberg, Phys. Rev. Lett. 5 (1960) 375P-P. Crépin - Rencontres de Moriond - March 2017 Casimir-Polder shifts on quantum levitation states
Results
Correction to the scattering lengthapproximation : En − λnεg −mga ∆En = Eana
n − Enumn
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2 4 6 8 10
2
4
6
8
n
En-Re(ℰn1)(10-5ϵ g)
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2 4 6 8 10
-4
-2
0
2
4
6
n
ΔEn(10-6ϵ g)
Analytical method would be sufficient to calculate quantumlevitation states energies and deduce from the spectroscopymeasurements the value of g with an accuracy better than 10−5g !
P-P. Crépin - Rencontres de Moriond - March 2017 Casimir-Polder shifts on quantum levitation states
THANK YOU FOR YOU ATTENTION !
P-P. Crépin - Rencontres de Moriond - March 2017 Casimir-Polder shifts on quantum levitation states
Width of resonances
Extend analytically ρ to C.Cavity response function : f(E) = ρ(E)
1−ρ(E)
Complex resonances En : ρ(En) = 1.Fit |f |2 ' An
(E−Re En)2+(Im En)2
2 3 4 5 6 7 8E (εg)
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
|f|2
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2 4 6 8 10
-2
0
2
4
6
8
n
Re(Δℰn),(10-6ϵ g)
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2 4 6 8 100
1
2
3
4
5
6
n
Im(ℰn-ℰn1),(10-5ϵ g)
Energies are still known with an accuracy of a few 10−6εg
Good approximation of the lifetime in cavity :τ = ~
2mgb , b = −Im(a)
P-P. Crépin - Rencontres de Moriond - March 2017 Casimir-Polder shifts on quantum levitation states