# solutions

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```Dominique Delande &amp; Christophe Texier
Master 2 iCFP
Correction de l’examen du 8 juin 2016
Part 1: speckle
A. Diffraction by a square aperture.– Because the plane wave is perpendicular to the
aperture, the amplitude is uniform at z = 0, so that :
ky L
kx L
2
E(kx , ky ) = A L sinc
sinc
(14)
2
2
and
2
4
2
I(kx , ky ) = A L sinc
kx L
2
sinc
2
ky L
2
.
(15)
B. Diffraction by a diffusive plate.
R 2π dφ iφ
1/ The phase
φ(x, y)
of
the
field
is
uniformly
distributed,
hence
hA(~
r
)i
=
A
= 0.
0 2π e
Similarly for A(~r)2 = 0. The squared modulus is of course independent of the phase, so that
only hA(~r)A∗ (~r)i = |A(~r)|2 = A2 survives.
P
~
rn ) eik&middot;~rn makes clear that the field is the sum of a
Discretizing (1) as E(~k) = η 2 N
n=1 A(~
large number of uncorrelated random variables. Hence we can apply the central limit theorem,
which implies that the distribution of E is Gaussian. We have hE(~k)i = 0, hE(~k)2 i = 0 and
X
~
hA(~rn )A∗ (~rm )i eik&middot;(~rn −~rm ) = η 4 N A2 = η 2 L2 A2 = I0
h|E(~k)|2 i = η 4
|
{z
}
n,m
A2 δn,m
is
If we decompose the electric field as E = X + i Y, we deduce from
the
averaged
intensity.
E 2 = X 2 −
Y 2 + 2i hX
Yi = 0 that the two components are uncorrelated, hX Yi = 0, with
2
2
same variance X = Y (isotropy). As
a result the distribution of the electric field
has the
form PE (E, E ∗ ) ∝ exp − (1/2c) X 2 + Y 2 . Using that the averaged intensity is hIi = |E|2 =
2 2
def
X + Y = 2 X 2 = 2c = I0 we finally deduce the form
PE (E, E ∗ ) =
1 −|E|2 /I0
e
.
πI0
(16)
2/ The distribution Rof the intensity is given by writing
the field in polar coordinates E = R eiφ
R
2π
∗
2
and P (I)dI = RdR 0 dφ PE (E, E ) = d(R )/2 dφ PE (E, E ∗ ), hence the Rayleigh law
P (I) = π PE (E, E ∗ ) =
1 −I/I0
e
.
I0
The most probable value is thus I = 0. The first moments are hIi = I0 and I 2 = 2I02 , hence
Var(I) = I02 = hIi2 .
3/ a) Correlations are conveniently computed with the discrete formulation
X
X ~ ~0
~
~0
hE(~k)E ∗ (~k 0 )i = η 4
hA(~rn )A∗ (~rm )i eik&middot;~rn −ik &middot;~rm = η 4 A2
ei(k−k )&middot;~rn
n,m
=
I0
L2
Z
L/2
−L/2
dx
Z
n
L/2
~ ~ 0 )&middot;~
r
dy ei(k−k
−L/2
7
so that :
so that :
✓
◆
✓
◆
ky L kx L hE(~k)E ⇤∗(~k 00)i = I0 sinc ∆kx L sinc ∆ky L
(17)
2
2
sinc
(17)
hE(~k)E (~k )i = I0 sinc
2
2
b) E is Gaussian, therefore we can use Wick’s theorem :
b) E is Gaussian, therefore we can use Wick’s theorem :
~k 0 )E ⇤ (~k 0 )i + hE(~k)E ⇤ (~k 0 )ihE(~k 0 )E ⇤ (~k)i
hI(~k) I(~k∗ 0 )i = hE(0~k)E∗⇤ (~k)ihE(
0
0 {z
~
~
~
~
~
~
|
| ~k 0 )E ∗ (~k)i{z
} ∗ (~k)E ∗ (~k 0 )i
hI(k) I(k )i = hE(k)E (k)ihE(k )E (k )i + hE(~k)E ∗ (~k }0 )ihE(
+ hE(~k)E(~k 0 )ihE
|
{z
}=I02 |
{z
} ⇤ (|~k 0 )i|2
{z
}
=|hE(~k)E
=I02
=|hE(~k)E ∗ (~k 0 )i|2
=0
The third contraction would be hEEihE ⇤ E ⇤ i = 0 (recall that there are (2n 1)!! manners to make
(recall
are (2n − 1)!! manners to make n pairs from 2n variables).
n
pairs that
fromthere
2n variables).
Higher
moments
hInnii =
= hEE
hEE⇤∗ &middot;&middot; &middot;&middot; &middot;&middot; EE
EE⇤∗i.
i. There
There are
are n!
n! manners
manners to
to pair
pair the
the n
n amplitudes
amplitudes to
to
Higher moments :: hI
n
2
n
n
n
2
n
n
the n
n complex
complex amplitudes,
amplitudes, thus
thus hI
hI ii =
= n!
n! h|E|
h|E| ii =
= n!
n! II0 ,, which
which are
are indeed
indeed the
the moments
moments of
of the
the
the
0
exponential
law
(2).
exponential law (2).
4/ a)
a) To
To transform
transform the
the angularly
angularly correlated
correlated intensity
intensity distribution
distribution into
into aa spatially
spatially correlated
correlated
4/
one, the
the easiest
easiest way
way is
is to
converging lens
lens and
and look
look in
in the
the focal
focal plane.
plane. A
A ray
ray with
with direction
direction
one,
~k in the plane (xOz) characterized by the angle θ such that tan θ = kx /kz ≈ kx /k0 is thus at
~k
in the plane (xOz) characterized by the angle ✓ such that tan ✓ = kx /kz ⇡ kx /k0 is thus at
position x/f
x/f =
= tan
tan ✓θ in
in the
the focal
focal plane,
plane, hence
hence
position
∆~~k
∆~
' k ff
(18)
~rr '
(18)
kk00
x
F’
F
θ
f
Figure
Figure 4:
4: The
The atoms
atoms in
in the
the focal
focal plane
plane feel
feel the
the disordered
disordered potential
potential characterized
characterized by
by the
the correlacorrelations
(4).
tions (4).
b)
b) The
The quasi-resonant
quasi-resonant laser
laser field
field modifies
modifies the
the internal
internal energy
energy levels
levels of
of the
the atom
atom (light-shift).
(light-shift). By
By
energy
conservation,
this
creates
for
the
atom
center
of
mass
an
e↵ective
potential
proportional
energy conservation, this creates for the atom center of mass an effective potential proportional
to
rr)) and
to the
the laser
laser intensity
intensity I(~
I(~
and inversely
inversely proportional
proportional to
to the
the detuning
detuning δ.. The
The atom
atom “feels”
“feels” aa
potential
V
(~
r
)
/
I(~
r
),
thus
potential V (~r) ∝ I(~r), thus

✓
✓
x◆
y◆
0
0
2
2
2
~
~
~
∆x
hV
(
k)
V
(~
r
I(
k
sinc
0)i / hI(k)
0)i = I02 1 + sinc 2
2 ∆y
~
~
~
hV (k) V (~r )i ∝ hI(k) I(k )i = I0 1 + sinc
sinc
σ
σ
with = 2f /(k0 L) = f 0 /(⇡L), where we made use of (18). Because of the paraxial approxiwith σ = 2f /(k0 L) = f λ0 /(πL), where we made use of (18). Because of the paraxial approximation L/f must be small. The shortest achievable correlation length is of the order of 0 .
mation L/f must be small. The shortest achievable correlation length is of the order of λ0 .
C. Gaussian aperture.– A Gaussian beam arrives on the di↵usive plate, A(~r) = A exp[ ~r 22/(2r022)]
C. Gaussian aperture.– A Gaussian beam arrives on the diffusive plate, A(~r) = A exp[−~r /(2r )]
(just before the plate). Using the Fraunhofer formula, we obtain that the correlation function 0
(just before the plate). Using the Fraunhofer formula, we obtain that the correlation function
of the field after the di↵usive plate is
of the field after the diffusive plate isZ
⇣
⌘2
Z
~2 2
⇤ ~0
2 2
i(~k ~k 00)&middot;~
r ~
r 22/(2r022) 2
~
~
hE(~k)E
(
k
)i
=
⌘
A
d~
r
e
e
= I˜˜0 e −∆k~k 2rr002/4
∗ ~0
2 2
i(k−k )&middot;~
r
−~
r /(2r0 )
/4
~
hE(k)E (k )i = η A
d~r e
e
= I0 e
with I˜˜0 = ⇡r022 ⌘ 22A22. The intensity correlations are
with I0 = πr0 η A . The intensity correlations are
h
i
~k 0 )i = I˜2 h1 + e ~k~ 22r022/2 i .
hI(~k)
I(
hI(~k) I(~k 0 )i = I˜002 1 + e−∆k r0 /2 .
The potential correlations for the atoms are Gaussian as well.
The potential correlations for the atoms are Gaussian as well.
8
8
Part 2 : localization of an atom
A. Structure of the Green functions
e 0 (k, k 0 ; E) = h φk |(E − H0 + i0+ )−1 | φk0 i = δk,k0 (E − εk + i0+ )−1
1/ a) Free Green function : G
2
2
with εk = ~ k /(2m).
b) The corresponding expression in x-space is
G0 (x, x0 ; E) =
Z
0
eik(x−x )
m
dk
0
= 2
eikE |x−x |
+
(2π) E − εk + i0
i ~ kE
def
where kE =
√
2mE/~ .
(19)
Integral is easily computed with residue’s theorem by closing the contour by a semi-circle of
radius R → ∞ in the upper (lower) plane for (x − x0 ) &gt; 0 (&lt; 0, resp.).
2/ The Green function is defined by G(E) = 1/(E − H + i0+ ) and satisfies the Dyson equation G = G0 + G0 V G.
The Green function is different for every realization of the disorder. Its average describes the
spatio-temporal evolution of the average field (NOT the average intensity). The Dyson equation
for the averaged Green function is
G = G0 + G 0 Σ G ,
(20)
and corresponds to a reorganisation of the perturbative expansion in terms of irreducible diagrams. We deduce Eq. (8).
Translational invariance is restored after disorder averaging, which implies that G and Σ are
diagonal in the k-representation : h φk |Σ(E)| φk0 i = Σ(k; E) δk,k0 .
3/ In real space we have
G(x, x0 ; E) =
Z
0
dk
eik(x−x )
.
(2π) E − εk − Σ(k; E)
(21)
In the weak disorder limit, |Σ(k; E)| |E|, the poles k&plusmn; of the integrand, i.e. zeros of E − εk −
Σ(k; E), are only slightly shifted and remain close to &plusmn;kE . Thus, assuming that Σ is a smooth
function we can simply perform the substitution Σ(k; E) → Σ(kE ; E). Expanding the position
of the poles for small Σ gives
r
2m
i Im Σ
k&plusmn; ' &plusmn;
(E − Re Σ) 1 −
2
2E
}
| ~ {z
def
= k̃E 'kE
we see that the calculation of G is the same as the one of G0 , provided the substitution
kE −→ k̃E −
im
Im [Σ(kE ; E)] .
~2 kE
Making this substitution in Eq. (19), we obtain the structure
|x − x0 |
0
0
G(x, x ; E) ≈ G0 (x, x ; E) exp −
2`(E)
with
1 def
2m
=− 2
Im [Σ(kE ; E)]
`(E)
~ kE
9
Re Σ appears as a (negative) correction to the energy due to the average disorder. It is a
simple shift of the energy axis. `(E) is of course the mean free path for atom with energy E.
It corresponds to the typical distance between two collisions on the disorder (`(E)/vE is the
B. Born approximation.
1/ Cf. lecture’s notes. The perturbative expansion of G reads
G = G0 + G0 V G 0 + G0 V G 0 V G 0 + &middot; &middot; &middot;
2/ We deduce Σ = V0 +&middot; &middot; &middot; at lowest order. This is a (trivial) shift in energy due to hV i = V0 6= 0.
3/ Next leading order gives Σ = V0 + V G0 V + &middot; &middot; &middot; . We sandwich the expression between two
plane waves :
X
Σ(k; E) = V0 + h φk |V G0 (E)V | φk i + &middot; &middot; &middot; = V0 +
|h φk |V | φk0 i|2 G0 (k 0 ; E) + &middot; &middot; &middot;
k0
Introducing the correlation function of the disorder we easily get
Z
dk 0
2
G0 (k 0 ; E) C(k − k 0 )
Σ(k; E) ' V0 + V0
2π
The important part is the imaginary part. Using that Im[G0 (k 0 ; E)] = −π δ(E − εk0 ) we finally
obtain
mV 2
Im [Σ(k; E)] ' − 2 0 [C(k − kE ) + C(k + kE )] .
2~ kE
As we have seen the elastic mean free path involves the “on-shell” self energy
Im [Σ(kE ; E)] ' −
mV02
[C(0) + C(2kE )] ,
2~2 kE
where the two terms are interpreted as the contributions of forward scattering ∝ C(0) and
backward scattering ∝ C(2kE ).
This leads to the formula for the elastic mean free path
1
m2 V 2
' 4 20 [C(0) + C(2kE )] .
`(E)
~ kE
(22)
C. Localization length.
R
1/ A Fourier transform.– The Fourier transform of sinc(xa/2) is the “door” dx sinc(xa/2) e−ikx =
(2π/a)Πa (k) where Πa (k) = θH (a/2 − |k|) (the converse is easy to check). The Fourier transform
of sinc2 (xa/2) is therefore the convolution (2π/a2 )(Πa ∗ Πa )(k). In order to avoid an explicit
calculation, we nowR make two remarks :
(i ) (Πa ∗ Πa )(0) = dk Πa (k) = a.
(ii ) the overlap between Πa (k 0 ) and Πa (k − k 0 ) increases linearly with k and vanishes for |k| &gt; a,
hence (Πa ∗ Πa )(k) = a(1 − |k|/a) θH (1 − |k|/a).
We deduce
|k|σ
|k|σ
C(k) = π σ 1 −
θH 1 −
.
2
2
2/ a) In 1D, any disorder with short range correlations leads to strong localization of all eigenstates (no mobility edge) 1 . In the lecture we have noticed that in 1D the mean free path and
1
Note that some specific correlations can produce delocalized states, however only on a set of energy of zero
measure.
10
the localisation length are related by ξloc ' 2` (in the weak disorder regime), which is valid
for C(k) = cste, when scattering is isotropic (forward=backward scattering), i.e. hV (x)V (x0 )i ∝
δ(x−x0 ). Hence, the perturbative calculation of the Green function has provided a (perturbative)
formula for the localization length.
b) When scattering is not isotropic (i.e. forward and backward scattering have different probabilities), we have to take into account that only backward scattering leads to localization. This
remark explains that one should perform the subtitution C(0) → C(2k) in the formula for the
elastic mean free path :
1
m2 V 2
1
' 4 20 C(2kE )
'
ξloc
2`tr
~ kE
where we have used (22).
c) Using the expression of the correlation function computed previously, C(2k) = π σ (1 −
|k|σ) θH (1 − |k|σ), we end with
1
ξloc
'
π m2 V02 σ
(1 − kE σ) θH (1 − kE σ)
2
~4 kE
d ) The figure shows that the theoretical prediction is not too bad. As expected for an expansion
in powers of the disorder strength, it works better at small V0 . It always fail at low-k (small
energy) because the self-energy Σ(E) is no longer much smaller than E.
The perturbative expression of the inverse localization length vanishes when the energy is
above a threshold, i.e. for kE &gt; 1/σ. This should however not be interpreted as a mobility
edge ! In 1D, localization takes place at any energy. A calculation involving higher order
contributions to Σ indeed predicts a finite localization length [1], in agreement with the numerics.
For small V0 , the vanishing of the second order contribution manifests as the “accident” (step
like behaviour). As expected, the step is less pronounced for larger V0 .
D. (Bonus) Experimental results.— The observed localization length decays with increasing
V0 , but the 1/V02 prediction is not very good. This is partly because one should include higherorder contributions in V0 and partly due to experimental problems (difficulty of measuring long
localization lengths at small V0 , residual atom-atom interaction killing localization at large V0 ).
As ξloc /σ is at least 100, the formula shows that k` is at least as large. In 2D, this would result
in a huge localization length, scaling like exp(k`), thus larger than the size of the universe ; in
3D, it would be on the diffusive side of the Anderson metal-insulator transition.
11
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