Casimir-Polder shifts on quantum levitation states=1P-P...

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


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


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


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


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 !


THANK YOU FOR YOU ATTENTION !


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)


Casimir-Polder shifts on quantum levitation states=1P-P...

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