quantum slit di raction in space and time and semi ... · from (dit)-(dis) theory to the...

IntroductionFrom (DIT)-(DIS) theory to the semi-classical approximation

Corrections to the truncation approximationCold atomsConclusion

Quantum slit diffraction in space and timeand semi-classical approximation

Dr. Mathieu Beau, Postdoctoral researcher, STP-DIAS

Collaborator : Prof. Tony Dorlas, STP-DIAS

Dr. Mathieu Beau, Postdoctoral researcher, STP-DIAS 1 / 52



I Summary1. Introduction : quantum diffraction and slit experiments2. From (DIT)-(DIS) theory to the semi-classical approximation3. Corrections to the truncation approximation4. Application to the cold atoms experiments5. Conclusion : remarks and discussions




Young slit experiment : from light to massive particlesFeynman Slit experiment : the wave-particle dualityTruncation approximation - Feynman model

1. Introduction : quantum diffraction and slit experiments

I R.Feynman, A.Hibbs, Quantum Mechanics and Path Integrals3rd ed (New York :McGraw-Hill), 1965.

I R. P. Feynman, R. B. Leighton, M. L. Sands : The FeynmanLectures on Physics (Addison-Wesley, Reading, MA, 1965).

I A. Zecca, Int. J. Theo. Phys. 38(3), 911-918 (1999).

I A. O. Barut, S. Basri, Am. J. Phys. 60(10), 896 (1992).

I M.Beau, Eur.J.Phys 33 1023 (2012).





History : outline

I 1801 : T.Young ⇒ double-slit Light interference

I 1897 : J.Thomson, experiment on cathode rays⇒ discovery of the electron

I 1924 : L.de Broglie, wave particle duality λ = hp

⇒ massive particles ∼ wave for m and v small enough

I 1925 : E.Schrodinger equation : Hψ = i~∂tψI 1927 : Davisson-Germer experiment : diffraction of electrons

by a crystalline nickel target, proof of de Broglie hypothesis

I 1927 : W.Heisenberg, uncertainty principle : ∆x∆p ≥ ~limit on the measurement accuracy

I 1961 : C. Jonsson ⇒ double-slit Electrons interference





Figure: Feynman experiment - Roger Bach et al. 2013 New J. Phys.15, 033018 and Feynman Lecture 1965





Figure: Mask movement - Roger Bach et al. 2013 New J. Phys. 15.





Figure: Buildup of electron diffraction - A. Tonomura et al. Am. J.Phys. 1989 57





Slit experiments :

I 1976 : P.G.Merli, G.F.Missiroli, G.Pozzi⇒ Single-electrons interference (biprism)

I 1988 : A. Zeilinger, R. Gahler, C. G. Shull, W. Treimer, W.Mampe ⇒ Cold neutrons slit experiment

I 1992 : F. Shimizu, K. Shimizu, H. Takuma⇒ Cold atoms slit experiment under gravity

I 2003 : O. Nairz, M. Arndt, A. Zeilinger⇒ Slit experiment for large molecules

I 2007 : S. Frabboni, C. Frigeri, G. C. Gazzadi, G. Pozzi⇒ Electrons (nano-slit)

I 2013 : R. Bach, D. Pope, S. -H. Liou, H. Batelaan⇒ Realization of the Feynman’s experiment (at last !)





2d

2a

x

(x,z)

x-x1x1

SourceSource

SlitsSlits ScreenScreen

z

yTruncation model: x>>d

Motion 1 Motion 2

Schematic representation of the apparatus.Dr. Mathieu Beau, Postdoctoral researcher, STP-DIAS 9 / 52




I Truncation assumption :Motion separated in two parts and x � d ,⇒ classical motion along x and v ∼ vx

⇒ tc ∼ x1x t

I Propagator

K(a)Trunc(z , t; 0, 0|tc) =

∫ a

−adz1

eim|z−z1|

2

2~(t−tc )

(2iπ~(t − tc)/m)1/2e i

mz212~tc

(2iπ~tc/m)1/2

I Intensity on the screen : (hypothesis : thin wave packet)

I (z , t) ∝ |K (a)Trunc(z , t)|

2

I Double-slit :

K (dble) = K (1) + K (2) ⇒ I (dble) ∝ I (1) + I (2) + I (12)





Result :

K(a)Trunc(z , t) =

e iz2

2t

(2iπ~t/m)1/2Ft,tc (z , a)

where :

Ft,tc (z , a) ≡ C [αt,tc (z , a)] + C [αt,tc (z ,−a)]

+ iS [αt,tc (z , a)] + iS [αt,tc (z ,−a)]

αt,tc (z , a) ≡√γNF

(1− z

aγ

),

where γ = |x − x0|/|x1 − x0| and where the Fresnel number isdefined as

NF ≡ 2a2

λL

where L = |x − x1| and λ ≈ 2π~m(x/t) when x � a.





Regimes :

I NF � 1 : Fraunhofer regime ⇒ fringes ∆z ∼ λL2a

I NF � 1 : Fresnel regime ⇒ shape of the aperture and fastoscillations

I NF ∼ 1 : Intermediate regime ⇒ transition between bothprevious regimes

Rem. For Nf � 1, since p = hλ ,

∆pp ∼ a

L , we obtain :

∆z∆p ∼ h

2





-300 -200 -100 100 200 300x�a

1��

2

1

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-7.5 -5 -2.5 2.5 5 7.5x�a

1��

2

1

H2bL

-3 -2 -1 1 2 3x�a

0.5

1.3

H2cL

Figure: Interference patterns for single-slit. We have NF = 0.01 for (a),NF = 0.5 for (b) and NF = 100 for (c)





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1

H3aL

-150 -100 -50 50 100 150x�a

1

H3bL

-60 -40 -20 20 40 60x�a

1��

2

1

H3cL

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1��

2

1

H3dL

Figure: Interference patterns for double-slit. We have d/a = 13 andNF = 0.001 for (a), 0.015 for (b) and 0.12 for (c) and 6 for (d).





I Question :semi-classical regime x � λ (classical motion along x-axis)but a is not too small compare to x , is the truncation modelstill valid ?

I Problem :tc 7→ tc +∆tc , since we have to take into account theuncertainty along the axis y , z .⇒ diffraction in time model




(DIT) : a brief introBrukner-Zeilinger (BZ) model : (DIT)-(DIS)Semi-classical approximation : new formulasPatterns

2. From diffraction in space (DIS) and in time (DIT) theoryto the semi-classical approximation

I M. Moshinsky, Phys. Rev. 88(3), 625-631 (1952).

I P. Szriftgiser, D. Guery-Odelin, M. Arndt, J. Dalibard, (1996).

I C. Brukner, A. Zeilinger, Phys. Rev. A. 56(5), 3804 (1997).

I G. Kalbermann, J. Phys. A : Math. Gen. 34, 6465 (2002).

I A. del Campo, G. Garcıa-Calderon, J.G. Mugad, Phys. Rep.476(1)-(3), 1-50 (2009).

I A. Goussev, Phys. Rev. A. 85(1), 013626 (2012) ; Phys. Rev.A. 87(5), 053621 (2013).





x1=0

Shutter closed for t<0

�=exp(ikx) �=0





x1=0

Shutter open at t>0

>� 0

x

Detector

ttc

QuantumClassical

Den

sity

pro

file

1





Moshinsky model (1D) for perfectly absorbing shutter{ ~22m∂

2xψ(x , t) + i~ ∂

∂tψ(x , t) = 0 t > 0ψ(x , 0) = e ikxΘ(−x)

(1)

Solution :

ψ(x , t) =∫ +∞−∞ dx0

e im(x−x0)

2

2~t√2iπ~t/m

ψ(x0, 0) =∫ 0−∞ dx0

e im(x−x0)

2

2~t√2iπ~t/m

e ikx

|ψ(x , t)|2 = 1

2

{(1

2+ C [α]

)2

+

(1

2+ S [α]

)2}

(2)

where α =√

~tmπ (k − mx

~t ).

⇒ analogous to the Fresnel (near field) light diffraction and also toa stationary plane wave diffraction by a half plane.⇒ “diffraction in time” by an edge in time.





-|x0| x1=0 x

(y,z)

Emission at t0=0

Shutter closed for t < t1 , (t1 � 0)

Detection at t2=t

=� �=0�





-|x0| x1=0 x

(y,z)

Emission at t0=0 Shutter open

for t > t1 � 0

Detection at t2=t

0��





I Schrodinger equation~22m∇2ψ(r, t) + i~ ∂

∂tψ(r, t) = 0 for x ≥ x1 and t ≥ t1ψ(r, t) = 0 for x > x1 and t < t1,ψ(r1, t) = φ(r1, t) for x = x1 and t ≥ t1.

(3)

I Comments :

1. (3) is a Schrodinger equation on a half spaceV = [0,+∞)× R2 i.e., on the r.h.s of the plane x1 = 0.

2. The surface term of the solution will be restricted to theaperture of the slit : ∂V = {x1 = 0} × [−b, b]× [−a, a].

3. Boundary conditions : (BZ) take a plane waveφ(r1, t) = e−iω0t . Here we take a Gaussian wave packet(details later on).





I Solution of (3) via the point source method :

~2

2m∇2G (r, t, r′, τ) + i~

∂G (r, t, r′, τ)

∂t= i~δ3(r − r′)δ(t − τ) (4)

I Causality condition :

G (r, t < τ, r′, τ) = 0 ,

I General homogeneous boundary conditions (far fielddiffraction) :

G (r, t, r1, τ) = λ1G0(x−x1, r⊥−r⊥,1; t−τ)+λ2G0(x+x1, r⊥−r⊥,1; t−τ) .

I Free Green function (for infinite volume) :

G0(r − r′; t − τ) =

(m

2iπ~(t − τ)

)3/2

eim|r−r′|22~(t−τ) θ(t − τ)





I General solution : sum over point sources solutions

ψ(r, t) =

∫Vd3r′G (r, t, r

′, t1)ψ(r

′, t1)+

i~2m

∫ t

t1

dτ

∫∂V

dS1 [G (r, t, r1, τ)∇r1ψ(r1, τ)]x1=0

− i~2m

∫ t

t1

dτ

∫∂V

dS1 [ψ(r1, τ)∇r1G (r, t, r1, τ)]x1=0 (5)

I Particular cases :1. λ1 = 1λ2 = −1 : Dirichlet conditions, G (r, t, r1, τ)|x1=0 = 0

⇒ perfectly reflective2. λ1 = 1λ2 = 0 : free Green’s function G0(r − r1; t − τ)

⇒ perfectly absorbing3. λ1 = 1λ2 = 1 : Neumann conditions, ∂x1G (r, t, r1, τ)|x1=0 = 0

⇒ partially absorbing





I Boundary conditions (wave function)

Gaussian wave packet :

ψ(r1, τ) =

∫R3

dR G0(r1 − R; τ)φ(R)θ(τ − t1) , (6)

where the normalized Gaussian wave packet φ is given by :

φ(R) =1

(2πσ2)3/4e−

|R−r0|2

4σ2 , (7)

such that |φ(R, 0)|2 → δ3(R− r′) when σ → 0n.b. Validity of this assumption : σ � |r0|





I Solution for the slit model (Now we take t1 = t0 = 0)

By (5) and (6), we get the following formula :

ψ(r, t) =

∫R3

dR K (a,b)(r, t,R, 0)φ(R) (8)

where the single-slit propagator is defined by :

K (a,b)(r, t,R, 0) =∫ t

0dτ

∫ a

−adz1

∫ b

−bdy1

[−X

τη1 +

x

t − τη2

]G0(r−r1, t−τ)G0(r1−R, τ)

(9)





I Explicit integral formulan.b. Since σ � |r0|, ψ(r, t) ∝ K (a,b)(r, t)

We rewrite (9) :

K (a,b)(r, t, r0, 0) =

∫ a

−adz1

∫ b

−bdy1K (r, t; r0, 0|r1) , (10)

where the three-dimensional one-point source propagator is :

K (r, t; r0, 0|r1) ≡∫ t

0dτ

[−x0τη1 +

x

t − τη2

]G0(r−r1, t−τ)G0(r1−r0, τ)

(11)





By direct calculus we get :

K (r, t; r0, 0|r1) = At(r; r0|r1)e iϕt(r;r0|r1) (12)

With the phase and the amplitude given by :

ϕt(r, r0|r1) ≡m

2~t(|r − r1|+ |r1 − r0|)2 (13)

At(r, r1 − r0) ≡ η1A(N)t (r, r1 − r0) + η2A

(D)t (r, r1 − r0). (14)

Dirichlet part :

A(D)t (r, r0|r1) =

x

(2iπ~t/m)3/2

(m

2iπ~t(|r − r1|+ |r1 − r0|)2

|r − r1|2|r1 − r0|+

1

2π|r − r1|3

).

(15)

Neumann part : similar (with |r1 − r0| ↔ |r − r1|).Dr. Mathieu Beau, Postdoctoral researcher, STP-DIAS 28 / 52




I Semi-classical approximation

Semi-classical condition :

µ ≡ m|r|2

~t� 1 ⇔ |r| � λ0 ≡

√2π~tm

(16)

⇒ Stationary phase approximation in (11) :∫ t

0dτ f (τ)e iµφ(τ) ≈ f (τsc)e

iµφ(τsc )

∫ t

0dτ e

iµ2φ′′(τsc )(τ−τsc )2 , µ� 1

(17)

with :

f (τ) = 1((2iπ~/m)2(t−τ)τ)3/2

(−x0τ η1 +

x(t−τ)η2

)φ(τ) = |r−r1|2

|r|2(1−τ/t)+ |r1−r0|2

|r|2τ/t

(18)

and where the saddle point τsc is the solution of φ′(τ) = 0.





K(a,b)sc (r, t; r0, 0) =

eimx2

2~t

(2iπ~t/m)1/2

∫ a

−adz1

∫ b

−bdy1 σt,τsc (x , x0)

eim[(y−y1)

2+(z−z1)2]

2~(t−τsc )

2iπ~(t − τsc)/m

eim[y21+z21 ]

2~τsc

2iπ~τsc/m(19)

where the boundary condition characteristic function is :

σt,τsc (x , x0) ≡λ20ρ

(−mx02π~τsc

η1 +mx

2π~(t − τsc)η2

)(20)

and where the semi-classical time is :

τsc = |r1−r0||r−r1|+|r1−r0| t (21)





��

��

��

� ��

��

��

��

��

��

��

Schematic representation of the apparatus.Dr. Mathieu Beau, Postdoctoral researcher, STP-DIAS 31 / 52




I Interpretation

Conservation of the classical energy of the particle :

E (r1, τsc) = E (r, t) ⇔ m

2

∣∣∣∣ r1 − r0τsc − t0

∣∣∣∣2 = m

2

∣∣∣∣ r − r1t − τsc

∣∣∣∣2leads to (21).However : the classical momentum is not conserved.Rem. If x � a, b we find τsc ≈ |x1−x0|

|x−x1|+|x1−x0| t = tc .





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Figure: We take x = 1, x1 = 0, x0 = −1, a = 0.01, b = 0.1(~ = m = 1). Dirichlet (Fig.2.1a-2.1c) and Neumann (Fig.2.3a-2.3c)boundary conditions with t = 1 (µ ∼ 1) for the figures at the left (a),t = 0.05 (µ ∼ 20) at the middle (b) and t = 0.005 (µ ∼ 200) at theright (c).





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2.4a

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2.4b

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2.4c

Figure: Free condition (Fig.2.3a-2.3c) and truncation approximation(Fig.2.4a-2.4c), with t = 1 (µ ∼ 1) for the figures at the left (a), t = 0.05(µ ∼ 20) at the middle (b) and t = 0.005 ((µ ∼ 200)) at the right (c).




Second order approximation : truncationFourth order approximation : correctionsPatternsDouble-slit

3. Corrections to the truncation approximation

I Conditions :1. semi-classical regime : µ� 12. elongated apparatus : |x |, |x0| � a, b, |z |, |y |

I ⇒ Semi-classical time :

τsc =|r1 − r0|t

|r − r1|+ |r1 − r0|≈ |x0|

|x − x0|t ≡ tc . (22)





I Second order : truncation approximation

Keeping the order O(z2/x2), O(y2/x2), O(a2/x2), O(b2/x2) :

K(a,b)t (r, r0) ≈

∫ a

−adz1

∫ b

−bdy1

{

σt,tc (x , x0)e im

(x−x0)2

2~t√2iπ~t/m

eim

(y−y1)2+(z−z1)

2

2~(t−tc )

2iπ~(t − tc)/m

e imy21+z212~tc

2iπ~tc/m

}. (23)

⇒ Truncation approximation + mult. factorRem. Result does not depend on the boundary condition.

Distance between two fringes : ∆z ≈ λL2a





I Fourth order : correction to the truncation approximation

Additional conditions :

NF (a) =2a2

λL� 1, NF (b) =

2b2

λL� 1

where L = |x | and λ = 2π~/(mv), with v ≈ vx = |x − x0|/t.

b � a ⇒ ∆z � ∆y � b

⇒ We keep the orderO(z4/x4), O(z2a2/x4), O(y2/x2), O(b2/x2)⇒ neglect terms of the orderO(y4/x4), O(y41 /x

4).





We get a similar formula to the second order approx. but with :

τ′= τ

(1− z2

|x |2tct

), a

′= a

(1− z2

2|x |2tct

)And so the distance between two fringes is about :

∆z ′ ≈ ∆z(1 + z2

2γL2

)(24)

where ∆z = λL/(2a) and γ = |x1 − x0|/|x − x1|.





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Figure: We take x = 50, x1 = 0, x0 = −50, a = 0.01, b = 0.1(~ = m = 1). Dirichlet (Fig.3.1a-3.1c) and Neumann (Fig.3.3a-3.3c)boundary conditions with t = 1 (µ ∼ 104) for the figures at the left (a),t = 0.05 (µ ∼ 2× 105) at the middle (b) and t = 0.005 (µ ∼ 2× 106) atthe right (c).





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Figure: We take x = 50, x1 = 0, x0 = −50, a = 0.01, b = 0.1(~ = m = 1). Free condition (Fig.3.3a-3.3c) and truncationapproximation (Fig.3.4a-3.4c), with t = 1 (µ ∼ 104) for the figures at theleft (a), t = 0.05 (µ ∼ 2× 105) at the middle (b) and t = 0.005(µ ∼ 2× 106) at the right (c).





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Figure: We take x = 50, x1 = 0, x0 = −50,a = 0.01, b = 0.1, d = 0.1 (~ = m = 1). Dirichlet (left) and Neumann(middle) and free boundary conditions (right) with t = 1 for the figureson the top (a) and t = 0.005 on the bottom (b).




Experiment and modelInterpretation

4. Application to the cold atoms experiments

I F. Shimizu, K. Shimizu, H. Takuma, “Double-slit interferencewith ultracold metastable neon atoms”, Phys. Rev. A. 46(1),R17-R20 (1992).

I M. Gondran, A. Gondran, “Numerical simulation of thedouble slit interference with ultracold atoms”, Am. J. Phys.73(6), 507-515 (2005).





g

z

z1

0

b

a

d

yx





Figure: Buildup of atoms diffraction [Gondran, 2005] fortmin ≤ t ≤ tmax , tmin = 175ms, tmax = 208ms and forz1 = 76mm, z − z1 = 116mm, a = 2µm, b = 2.8mm, d = 6µm.Numerical results (a)-(c) and Shimizu experiment for (d).





I Model [Gondran] :- Truncation approximation- Initial condition : ’narrow’ Gaussian wave packet- Initial velocity along z-axis : v0,z = 0 ⇒ t1 =

√2z1/g

- Integration of the propagator tmin ≤ t ≤ tmax

I Problem :Imagine we perform the same experiment with z1, z ≤ 1mm.Then : t1 =

√2z1/g is then not valid ⇔ uncertainty on v0z

⇒ semi-classical model taking into account (DIT).





Model :Schrodinger equation :

(− ~2

2m∇2 +mgz)ψ(r, t) = i~ ∂

∂tψ(r, t) for x ≥ x1 and t ≥ t1

ψ(r, t) = 0 for x > x1 and t < t1,

ψ(r1, t) =∫R3 dR Gg (r1, t;R, 0)

e−|R|2

2σ2

(2πσ2)1/4for x = x1 and t ≥ t1.

Green function (in infinite volume) :

Gg (r, t; r′, t ′) = G0(r, t; r

′, t ′) eim2~

(g(z+z ′)(t−t′)− g2

12(t−t′)3

)

Slit propagator :

K (g)(r, t; 0, 0) =

i~2m

∫ t

0dτ

∫ a

−adz1

∫ b

−bdy1 χt,τ (z , z1)Gg (r, t; r1, τ)Gg (r1, t; 0, τ)





Semi-classical approximation :

Similarly, for µ ≡ mr2/(2~t) � 1, we obtain :

K(g)sc (r, t; 0, 0) ≈

∫ a

−adx1

∫ b

−bdy1 Asc(r, t; 0, 0|r1) e iφsc (r,t;0,0|r1)

The semi-classical time is the (real and ∈ (0, t)) solution of :

|r−r1|2τ2−|r1|2(t−τ)2 = gz(t−τ)2τ2+g2

4τ4(t−τ)2−g2

4τ2(t−τ)4

(25)





Broken parabolic trajectories :(1) parabolic trajectory from r0 = 0 at the time t0 = 0 to r1 at thetime τ :{

Energy : E0 =m2 v

20

Velocity : v0 =x1−x0τ−t0

ex +y1−y0τ−t0

ey + ( z1−z0τ−t0

− g2 (τ − t0))ez

(2) parabolic trajectory from r1 at the time τ to r at the time t.{Energy : E1 =

m2 v

21 −mgz1

Velocity : v1 =x−x1t−τ ex +

y−y1t−τ ey + ( z−z1

t−τ − g2 (t − τ))ez

Conservation of energy :

E0 = E1 ⇔1

2|v0|2 =

1

2|v1|2 − gz1

⇒ equation (25).





gz1

z-z1

(0,0)

(x1,z1)(x'1,z1)

v0

v1

(x,z)




Concluding remarksDiscussion

5. Conclusion

5.1. Concluding remarks

I Summary :

1. We obtained a correction to the semi-classical formula andsome prediction to the shift of the fringes positions

2. We gave an interpretation to this approach mixing thetruncation model and the (DIT) one.

3. This approach works for any homogeneous B.C. and for aninitial Gaussian wave packet

I We mention :

1. Valid for general shape of slit (e.g., circle)2. Two-dimensional case : we got a semi-classical formula3. Initial condition : plane wave along the x-axis ⇒ Fourier

transform of our propagator w.r.t. x0.4. for σ > 0 : numerical computation





5.2. Discussion 1 : realistic experimental situation

I Semi-classical parameter µ in the experiments :∼ 1010 in [Zeilinger, 1988] : Cold Neutrons∼ 1010 in [Bach, 2013] : Electrons∼ 1013 in [Zeilinger, 2003] : Large molecule (C60)∼ 107 in [Takuma, 1992] : Cold Atoms (Neon)

⇒ truncation approximation fits very well.





5.3. Discussion 2 : Future investigationsI Conditions :

1. apparatus : mesoscopic scale ∼ 10−6 − 10−3m⇒ µ ∼ 102 − 105.

2. high statistics⇒ fourth order corrections + variation of the amplitude

⇒ mesoscopic slit experiment is a challenge !I Further investigations :

1. atoms diffractions for mesoscopic slit (analytic and numericalcomputations → new predictions)

2. time-diffraction with cold atom under gravity : correctformulas for the propagator


quantum slit di raction in space and time and semi ... · from (dit)-(dis) theory to the...

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