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Dominique Delande & 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·~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·(~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·~rn −ik ·~rm = η 4 A2 ei(k−k )·~rn n,m = I0 L2 Z L/2 −L/2 dx Z n L/2 ~ ~ 0 )·~ 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⇤∗ ·· ·· ·· 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 to add add aa converging 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)·~ 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 )·~ 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 ) > 0 (< 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± of the integrand, i.e. zeros of E − εk − Σ(k; E), are only slightly shifted and remain close to ±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± ' ± (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 lifetime of the plane wave). 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 + · · · 2/ We deduce Σ = V0 +· · · 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 + · · · . We sandwich the expression between two plane waves : X Σ(k; E) = V0 + h φk |V G0 (E)V | φk i + · · · = V0 + |h φk |V | φk0 i|2 G0 (k 0 ; E) + · · · 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| > 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 > 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