In Sec. 5.1 we studied the coherent backscattering effect for wave functions which are solutions to the linear Schr¨odinger equation or equivalently to the Helmholtz equation
5.3. NONLINEAR COHERENT BACKSCATTERING 71
CBS Disorder
Figure 5.6: Experimental setup suggested by G. Labeyrie, where an condensate which is initially confined to the center expands into the disorder region and then gets (coherently) scattered back to the origin, similar to the coherent backscattering of acoustic seismological waves in the near field around a source [103].
in the optical context. Now we apply this mechanism to the nonlinear Gross-Pitaevskii equation, where the nonlinearity arises due to the atom-atom interaction in the s-wave approximation. The Gross-Pitaevskii equation with source term is given by (Eq. 3.2):
i~∂
∂tΨ(r, t) =
−~2
2m∆ +V(r) + ~2g(x)
2m |Ψ(r, t)|2
Ψ(r, t) +S0(t)φS(y)δ(x−x0) e−iµt/~
. (5.12)
The dimensionless nonlinearity strengthg(r) = 4√
2πas/a⊥(r) is determined by the s-wave scattering lengthas and the transverse confinement a⊥(r) =p
~/[mω⊥(r)]. The source term is already included, and we use a constant profileφS(y)≡1 (if not explicitly specified otherwise), which corresponds to an incoming plane wave perpendicular to the disorder region as shown in Fig. 5.7. In the following calculations we fix the incoming current density to jin =~k|Ψ0|2/m, where k =√
2mµ/~ is the wavenumber of the incident beam. This is no restriction since the Gross-Pitaevskii equation can be rescaled by keeping the productg|Ψ|2 constant, and a higher current just renormalizes the nonlinearity strength g to lower values otherwise.
In order to study coherent backscattering numerically, we have to extract the angular resolved current from the simulations. One possibility is to take the disorder average over the wave function and to apply a two-dimensional Fourier transformation in the region between the source and the disorder where the nonlinearity is still negligible small. The nonlinearity has to be negligible, such that the superposition principle is valid, and in order to associate the Fourier modes with the outgoing current in certain
source
y
0 W
x1
x1
Figure 5.7: Scattering geometry and stationary scattering state associated with a randomly generated disorder potential. The left hand side displays the potentialV(x, y) in a gray scale plot and the spatial variation of the nonlinearity g(x). x0 is the position of the source and x1 denotes the position where the backscattered current is evaluated. The upper right panel shows the density plot of the corresponding scattering state. The lower right panel shows the decay of the coherent mode|Ψ|2 and the density |Ψ|2 averaged over y and ≈ 1000 disorder configurations.
directions. Thereby the relevant k-vectors of the transport modes are extracted. A density plot pointing out the intensity distribution as a functionkx and ky is shown in Fig. 5.8 b. With this method we can separate incoming waves (kx>0) from reflected waves (kx < 0). The strong peak in forward direction corresponds to the incoming plane wave, while in backward direction we can clearly see the coherent backscattering signal at ky = 0 and kx<0 and for larger angles we see also the diffusive background.
The drawback of this method is that we need an extra simulation region where the nonlinearity is zero and which is large enough to allow a good resolution. Already in Fig. 5.8(b) the size in x-direction is almost to small resulting in artificial oscillations for large angles due to the finite spacing.
On the other hand the Fourier decomposition in x-direction is only necessary to subtract the incoming wave component. However, we know the value of this amplitude analytically and can also calculate it numerically in a simulation without any disorder potential. We have chosen the second method, because the discretization of the source can introduce errors in the order of 0.2% for typical input parameters, and those deviations from the analytical prediction are eliminated with the numerical calculation of the propagating plane wave Φ0(x, y, t). To avoid these numerical errors we subtract
5.3. NONLINEAR COHERENT BACKSCATTERING 73
a) b)
Figure 5.8: a) The angular dependent current of the backscattered condensate is plotted.
The clear coherent backscattering cone is observed. The analytical formula Eq. 5.10 is fitted and shows good agreement. The current is extracted along the line x = x1. b) Extraction of the coherent backscattering by a two-dimensional Fourier transformation of this part of the averaged wave function right to the source position. The strong peak (height is cut) represents the incoming wave (kx >0). For kx<0 we see the coherent backscattering cone.
The oscillations arise due to the finite spacing of the grid.
the wave function that is numerically obtained in the absence of a scattering potential, namely Φ0(x, y, t)∝ |Ψ0|eikx, from the wave function Φ(x, y, t) to get the reflected part Φref(x, y, t):
Φref(x, y, t) = Φ(x, y, t)−Φ0(x, y, t). (5.13) Note that we do not have to care about a time-dependent phase-factor since the trivial time-dependence has been already separated with the ansatz Ψ(r, t) = Φ(r, t)e−iµt/~in the numerical calculation. In order to calculate the backscattered current we apply a Fourier transformation in y-direction to the disorder averaged reflected wave function Φref(x1, y) at the position x1 close to the source, where the nonlinearity g(x1) is still negligible small (see Fig. 5.7). This results in a decomposition into transverse eigen-modes Φ(x1, ky,n)∼exp(inπy/W), which supports outgoing waves into directions with anglesθn = arcsin[2πnkW].
The current density in direction θn, normalized with respect to the total incoming currentW jin, is then calculated by:
j(θn) = 2π W jin
~
m kx,n|Φ(x1, ky,n)|2 W cos(θn). (5.14) The wave vector kx,n is related to the transverse mode by kx,n = q
|k|2−k2y,n (see Fig. 5.9). The last factorWcos(θn) arises due to the geometrical fact that the outgoing
ky
∆ky = 2πW
kx ky,4
kx,4
Θ4
Figure 5.9: The angular resolution of the backscattered current is limited by the widthW of the disorder region. This implies a spacing of the wave vector ∆ky = 2πW in y-direction.
The angle corresponding to the transversal Fourier modenis given by: θn= arcsin[2πnkW].
current in direction θn passes through a perpendicular line of length Wcos(θn). A typical angular resolved backscattered current is shown in Fig. 5.8 a. The current is then normalized in the following way:
Z 2π 0
j(θ)dθ = 2π . (5.15)
Note that this integration (or summation over discrete angles) also takes angles in forward direction into account.
In the following we want to summarize the numerical implementation and the param-eters we have chosen. Afterwards we present the results we yield in this way: We con-sider a Gauss correlated disorder potential characterized byV(r)V(r′) =V02exp(−|r−r2σ2′|2).
Furthermore we specify an average height of the potential of V0 = 0.614µ and a cor-relation length of kσ = 0.5 with the wave vector k = √
2mµ/~. This corresponds to almost isotropic scattering as already seen in the previous chapter. Using the results of Eq. 4.27, Eq. 4.53 and Eq. 4.63 we can calculate the scattering and transport mean free path. Additionally we have extracted the length scales numerically and have found a scattering mean free pathkℓs ≃9.61 and a transport mean free path ofkℓtr ≃9.75, which is in good agreement with the analytical result. The result ℓs ≈ ℓtr confirms that we have almost isotropic scattering. For the optical thickness b of the disorder medium we have chosen b = L/ℓs = 4.1 (kL = 40) and the width kW = 120. The width of the system results in an angular resolution of ∆θ = 2π/kW ≃ 0.05[rad]. In order to perform the disorder average we repeat the time-dependent integration of the Gross-Pitaevskii equation approximately 103 times with randomly generated disorder realizations for each set of parameters. The error bars in the following graphs represent
5.3. NONLINEAR COHERENT BACKSCATTERING 75
a) b)
c) d)
Figure 5.10: The coherent backscattering peak for the nonlinear Gross-Pitaevskii equation is calculated for several nonlinearity values g. The error bars correspond to the statistical deviations arising from the disorder average (∼1000 realizations). a) In the linear regimeg= 0 the well known behavior is observed with a cone height by a factor of two larger compared with the diffusive background. b) For a nonlinearity g = 0.01 the coherent backscattering peak vanishes. c) and d) The cone reverts into a pronounced dip (g = 0.02), implying destructive instead of constructive interference. The underlying interference phenomenon is still active. This is observed already at a small interaction energyg|Ψ(r)|2∼10−2.
the standard deviation arising from the disorder average.
In the linear case g = 0 we encounter the well known coherent backscattering peak as explained in the previous section. We observe a peak height of roughly a factor two compared to the diffusive background as shown in Fig. 5.10(a). This is expected since our potential has a vanishing mean V(r) = 0 and furthermore the real part of the refractive index in the effective medium is modified only marginal as we have seen in Fig. 4.7. This implies a small single scattering contribution. For a small nonlinear-ity the cone height is reduced (Fig. 5.10(a)) and vanishes eventually if the interaction
Figure 5.11: We compare the coherent backscattering, where the incident wave enters the disorder region perpendicular (left graph) and where the incoming wave is tilted by an angle φ−6 ≃ −0.32 (right graph). In the latter case coherent backscattering is observed in retro-reflection, in contrast to ordinary specular reflection, where the peak would appear at φ+6 ≃+0.32. This is also observed in the nonlinear case, which confirms that in both cases the feature arises due to interference between time reversed paths.
strength increases (Fig. 5.10(b)). But it is very interesting that the underlying inter-ference effect is not washed out but reverts the cone (Fig. 5.10(c) and Fig. 5.10(d)) for intermediate strength of the nonlinearity. We find the novel phenomenon that the atom-atom interaction reverts the interference from constructive to destructive, imply-ing that the interference phenomenon is still effective. The shape of the cone and the dip are quite similar, especially the width is comparable. The intermediate interaction strength g = 0.02, where the dip is quite pronounced, is still weak, corresponding to an interaction energy ofg|Ψ(r)|2 ∼10−2µ.
In the case of a perpendicular incident wave we cannot distinguish between coherent backscattering and ordinary specular reflection. In the case of coherent backscattering one expects the peak to appear in exactly backward direction in contrast to specular reflection, where the reflected wave forms with the incoming wave an angle of 2φn. In order to prove that the dip is still related to coherent backscattering we investigated the angular resolved current in the case where the incident wave enters the disorder medium with an angle φ, as depicted in Fig. 5.11 in the right side. This can actually be achieved in our numerical simulation by changing the transverse function of the source. To this end we choose the source amplitude φS(y) of Eq. 5.12 to be an excited transverse eigenmode:
φS,n(y) =ei2π nW y. (5.16)
The incident wave is tilted by the angle φn= arcsin[2πnkW] in this case.
We specifically have chosen n = −6 and φ−6 ≃ −0.32. In the case of coherent
5.4. DIAGRAMMATIC NONLINEAR COHERENT BACKSCATTERING 77