#  Anisotropy effects on the quantum transport of atomic matter waves

Afifa Yedjour1 and Abdelâali Boudjemâa2,3 1Faculté de Physique, USTO MB,
B.P.1505 El M’Naour, 31000, Oran, Algérie
2Department of Physics, Faculty of Exact Sciences and Informatics, Hassiba
Benbouali University of Chlef, P.O. Box 78, 02000, Chlef, Algeria.
3Laboratory of Mechanics and Energy, Hassiba Benbouali University of Chlef,
P.O. Box 78, 02000, Chlef, Algeria.

###### Abstract

Abstract

We discuss effects of anisotropic scattering in transport properties of
ultracold atoms in three-dimensional optical potentials. Within the realm of
the first Born approximation, we calculate the self energy, the scattering
mean free time, the scattering mean free path, and the anisotropy factor. The
behavior of the diffusion constant as a function of the wavenumber is also
examined in diffusive and weak localization regimes. We show that these
quantities are affected by quantum corrections due to the interference caused
by disorder. The dimensionless conductance is also evaluated using the scaling
theory of localization. Our results are compared with previous theoretical and
the experimental results.

##  I Introduction

It has long been proven that a disorder is a crucial ingredient to understand
transport properties of waves 1 ; 2 . Cold atoms in a disordered medium have
attracted a considerable attention due to their ability of controlling
disorder and interactions 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 . One of the most famous
phenomena occurring in such disordered systems is the Anderson localization
which is characterized by a phase transition between a localized regime and a
diffusive regime 2 . The localization originates essentially from the
interference of multiple-scattering path in the disordered system 11 ; 12 .
Such an interference effect known as weak localization leads to increase the
probability to return back to the origin and hence slows down the transport
giving rise to reduce the diffusion constant and the conductivity compared to
the Boltzmann or Drude contribution. On the other hand, when the localization
is strong enough, the wave remains localized around its initial position
guiding to a total elimination of the transport, which means that the
diffusion is stopped and the conductivity is strictly zero.

Recently, the Anderson transition has been extensively studied both
theoretically 6 ; 113 ; 133 ; 13 ; 14 ; 15 ; 16 ; 17 and experimentally 18 ;
19 ; 20 ; 21 ; 22 . It has been found that the transport of atomic matter
waves in three-dimensional (3D) media is significantly affected by three
parameters: the disorder amplitude, the initial wave energy and the
correlation energy induced by disorder. Quite recently, the dynamic evolution
of expanding Bose-Einstein condensates (BECs) in 3D random potential has been
deeply examined in space and time 23 .

Furthermore, coherent transport and Anderson localization in disordered media
can be strongly affected by anisotropy effects (see e.gBish ; Pine ; Nick ;
Kao ; Wier ; John ; Gur ; Wolf ; Bart ; Stark ; 24 ; 233 ; 234 ). Recent works
133 ; 13 ; 14 ; 15 ; 233 reveal that the transport anisotropy is reduced near
by the mobility edge. The anisotropy of the coherent diffusion can be
controled by the properties of the disorder-averaged effective scattering
medium.

In this paper, we report on the systematic study of the weak localization
effects in the anisotropic transport properties of matter wave in 3D optical
potentials employing the first Born approximation (FBA). This latter has been
recently proven effective in analyzing the elastic scattering time of matter
waves 22 , the dynamic evolution of expanding BECs 23 , diffusion and
localization in optical disordered potentials (see e.g.13 ; 24 ), and good
agreement is obtained with experimental and numerical data 22 . We calculate
analytically and numerically the self-energy, the scattering mean free time,
the scattering mean free path, and the anisotropy factor. Our results
demonstrate that at low momenta, the scattering mean free time is constant
indicating that the scattering is isotropic (i.e in the Boltzmann classical
regime). Whereas at high momenta, it increases linearly with momentum and then
diverges pointing out that the scattering reaches the anisotropic regime in
agreement with the results of Ref.14 . The anisotropy factor increases with
momentum and becomes more pronounced at large momenta revealing that the
scattering is anisotropic for all momenta due to the long-range correlations
between waves. This is in concordance with the results of 13 . Moreover, the
behavior of the scattering mean free path emphasizes that the anisotropic
effects exists for various momenta at lowest energy $(\varepsilon=0)$. A
crossover from a diffusive regime to a weak localization occurs when the
energy becomes comparable with the disorder correlation energy due to the
long-range correlations between waves. Correction to the diffusion constant
relative to the Boltzmann diffusion constant is also determined in different
regimes. In weak localization regime, we find that the diffusion constant
strongly depends on the energy of the system. For instance, in the limit of
high energy the diffusion constant is reduced. This important manifestation
could be explained by the fact that the wave undergoes multiple scattering
events which evoke a reduced diffusion constant. In the opposite situation
where $\varepsilon<0$, the diffusion constant becomes close to the Boltzmann
diffusion constant. Additionally, we perform a numerical analysis based on the
FBA of the scaling function $\beta$ which is related to the dimensionless
conductance in order to understand the universality of the flow of bosonic
states with the scale transformation.

The remainder of the paper is organized as follows. In section II we calculate
the quantities of interest employing the FBA and introduce a brief overview
for the speckle potential. We write down also the basic equations governing
the diffusion of an expanding BEC. Section III is devoted to the numerical
simulation. We compute the self-energy and plot its imaginary part. We
evaluate also the scattering mean time, the anisotropy factor and the
diffusion constant. Our results are compared with previous theoretical and
experimental findings. In section IV we discuss the dimensionless conductance
and the $\beta$-function in the diffusive regime. Finally, we conclude the
paper with a summary in section V.

##  II Model

We consider a weakly interacting BEC of $N\gg 1$ atoms of mass $m$ expanding
in a 3D random optical potential $U(\textbf{R})$. The wavefunction
$\psi\left({\bf R},t\right)$ of such a system obeys the time-dependent Gross-
Pitaevskii equation:

$i\hbar\frac{\partial\psi}{\partial
t}=\left[-\frac{\hbar^{2}}{2m}\nabla^{2}+V+U+g|\psi|^{2}\right]\psi,$ (1)

where $g$ is the strength of repulsive interactions between atoms, $V({\bf
R})$ is the external trapping potential with trap frequency $\omega_{0}$, and
$|\psi({\bf r},t)|^{2}=n({\bf r},t)$ is the BEC density in a random potential.
At time $t>\omega_{0}^{-1}$ when the disorder is switched on, the condensate
becomes very dilue and thus, the interactions can be neglected ($g\rightarrow
0$) 6 . In such a case Eq.(1) reduces to the standard Schrödinger equation.
We use the FBA in which the complex self-energy $\Sigma(\varepsilon,k)$ of the
atom is given by 113 ; 25 ; 26 :

$\Sigma(\varepsilon,k)=\sum_{\mathbf{k}^{\prime}}\
C(k,\textbf{k}^{\prime})[\varepsilon-\varepsilon_{\textbf{k}^{\prime}}-\Sigma(\
\varepsilon,\textbf{k}^{\prime})]^{-1},$ (2)

where $\sum_{\mathbf{k^{\prime}}}\equiv\int d^{3}k^{\prime}/(2\pi)^{3}$,
$\varepsilon_{k^{\prime}}=\hbar^{2}k^{\prime 2}/2m$, and
$C(k,\textbf{k}^{\prime})$ represents the vertex associated with incoherent
(Boltzmann) transport defined in Fourier space as 13 ; 14 :

$C(\textbf{q})=\int\frac{d\bf R}{(2\pi)^{3}}\ C({\bf R})\
e^{i\textbf{q}\cdot\textbf{R}},$ (3)

where ${\bf q}={\bf k^{\prime}}-{\bf k}$. The two-point correlation function
$C(\textbf{R})$ for a 3D laser speckle potential is given by
$C(\textbf{R})=\langle
U(\textbf{0})U(\textbf{R})\rangle=U_{0}^{2}\mathrm{sinc}^{2}(R/\xi)$. It is
characterized by a typical width $\xi$ called the correlation length and a
disorder amplitude $U_{0}$.

Inserting Eq. (3) into (2), the self-energy can be rewritten as:

$\Sigma(\varepsilon,k)=\frac{2\
U_{0}^{2}\xi^{2}}{\pi}\int^{\infty}_{0}dk^{\prime}k^{\prime
2}\dfrac{I(k,\textbf{k}^{\prime})}{\varepsilon-\varepsilon_{\textbf{k}^{\prime}}-\
\Sigma(\ \varepsilon,\textbf{k}^{\prime})},$ (4)

where

$I(\ k,\
\textbf{k}^{\prime})=\int_{0}^{\infty}dR\sin^{2}\left(R/\xi\right)\text{sinc}(kR)\,\text{sinc}(k^{\prime}R).$
(5)

Here $I(\ k,\ \textbf{k}^{\prime})$ gives the main contribution to the
correlation function, it vanishes for $k^{\prime}>k+2/\xi$ and for
$k^{\prime}<k-2/\xi$. Keeping in mind that the FBA implies that $k$ is a
positive quantity, one has

$\text{Im}\Sigma^{0}(k,\varepsilon)=\frac{2\
U_{0}^{2}\xi^{2}}{\pi}\int^{k+2/\xi}_{0}\ dk^{\prime}k^{\prime
2}\dfrac{I(k,\textbf{k}^{\prime})}{\varepsilon-\varepsilon_{\textbf{k}^{\prime}}+i\
0}.$ (6)

The self-energy $\Sigma(\varepsilon,k)$ is a complex function. Its real part
exhibits a logarithmic divergence near
$\varepsilon=\frac{\hbar^{2}}{2m}\big{(}k+\frac{2}{\xi}\big{)}^{2}$. For
$\varepsilon<\frac{\hbar^{2}}{2m}\big{(}k+\frac{2}{\xi}\big{)}^{2}$, we obtain
from Eq. (6) the imaginary part of the self-energy:

$\text{Im}\Sigma^{0}(k,\varepsilon)=-U_{0}^{2}\xi^{2}\int^{k+2/\xi}_{0}\
dk^{\prime}k^{\prime 2}\ I(k,\textbf{k}^{\prime})\delta(\varepsilon\ -\
\varepsilon_{\textbf{k}^{\prime}}).$ (7)

At small momenta $k\ll 2/\xi$, one has
$\text{Im}\Sigma^{0}(k,\varepsilon)=-\pi U_{0}^{2}/(4\varepsilon_{\xi})$,
where $\varepsilon_{\xi}=\hbar^{2}/(2\xi^{2}m)$ accounts for the quantum
correlation energy. We see that $\text{Im}\Sigma^{0}(k,\varepsilon)$ is
constant, and it depends only on the characteristics of the random potential.
In the limit $k=0$, the imaginary part of the self-energy can be approximated
by a Heaviside step function as: $\text{Im}\Sigma^{0}(0,\varepsilon)=-(\pi
U_{0}^{2}/4\varepsilon_{\xi})\Theta(\varepsilon-\varepsilon_{\xi})$ 22 , which
indicates that some atoms can diffuse from the optical disordered potential.

In the frame of the FBA which requires the inequality $\varepsilon_{k}\gg
U_{0}^{2}/\varepsilon_{\xi}$, the scattering mean-free time is defined as 13 ;
27 :

$\tau_{s}(\varepsilon,k)=\frac{\hbar}{2\text{Im}\Sigma^{0}(\varepsilon,k)}.$
(8)

For an isotropic diffusion,
$\tau_{s}(\varepsilon,k)=2\hbar\,\varepsilon_{\xi}/(\pi U_{0}^{2})$. The
scattering mean free path $\ell_{s}(\varepsilon,k)$ and the scattering mean
free time are connected with each other via:
$\ell_{s}(\varepsilon,k)=v\tau_{s}(\varepsilon,k)$ where $v=\hbar\ k/m$.

Then, in the diffusive regime and for weak disorder, the diffusion constant is
given by 6 ; 27 :

$D_{B}=\frac{v\ell_{B}}{3},$ (9)

In the weak-scattering limit $\varepsilon_{k}\backsimeq
U_{0}^{2}/\varepsilon_{\xi}$, the diffusion constant is reduced due to the
dephasing processes. In such a situation, the transport (Boltzmann) mean-free
path $\ell_{B}$ is the average distance required to completely erase the
memory of the initial direction of propagation. For the above 3D disorder
speckle, the transport mean-free path is given by $\ell_{B}=(2\ell_{s}/3)\
[(k\xi)^{2}\Theta(k\xi\ -\ 1)+\Theta(1\ -\ k\xi)]$ 13 . It is related to
$\ell_{s}$ through 13 :

$\frac{\ell_{s}}{\ell_{B}}=1\ -\ \langle\cos\theta\rangle,$ (10)

where the anisotropy factor $\langle\cos\theta\rangle$ is given by 13

$\langle\cos\theta\rangle=\int d\Omega\cos\theta\ {\cal C}(k\xi,\theta),$ (11)

where $\cos\theta$ is averaged over the phase function ${\cal
C}{(k\xi,\theta)}=\sin\theta\,C(k\xi,\theta)/\big{[}\int
d\Omega\,C(k\xi,\theta)\big{]}$ with $C(k\xi,\theta)$ being the angular
correlation function versus the scattering angle $\theta$ between $k$ and
$k^{\prime}$ at fixed on-shell momenta $k=k^{\prime}$, and
$d\Omega=2\pi\sin\theta d\theta$. For $\langle\cos\theta\rangle=0$, the
scattering is fully isotropic and hence, $\ell_{B}=\ell_{s}$ 13 . Isotropic
scattering exists only for $\delta$-correlated disorder (uncorrelated
disorder). Contrary, when $\langle\cos\theta\rangle\neq 0$, the diffusion is
anisotropic and the corrections due to weak localization are pronounced. In
this case, a large number of scattering events is necessary to deviate the
particle completely, thus $\ell_{B}\gg\ell_{s}$ 13 .

The diffusion constant $D_{\varepsilon}$ must be written as 24 ; 25 ; 26 :

$\frac{1}{D_{\varepsilon}}-\frac{1}{D_{B}}=\frac{2\hbar}{mkD_{B}}\int_{0}^{\infty}\frac{q\
dq}{q^{2}-i0},$ (12)

In 3D, the integral over momentum $q$ in Eq. (12) requires only a cutoff at
large distances (see e.g. 13 ; 28 ) yielding:

$D_{\varepsilon}=D_{B}\left[1-\frac{3}{\pi(k\ell_{B})^{2}}\right].$ (13)

This equation shows that the weak localization reduces the diffusion constant
due to the corrections induced by the fluctuations caused by the random
potential (weak localization effects). Such interference corrections are small
in the weak disorder regime $k\ell_{s}\gg 1$ 13 .

Figure 1: Imaginary part of the self-energy $\text{Im}\Sigma^{0}$ for
$U/\varepsilon_{\xi}=1$.

##  III Numerical results

We solve numerically Eq. (6) at $k=0$ for different energies. Our calculation
is based on the FBA which offers more scattering channels to particles
travelling along the diffusion.
Figure 1 shows that $\text{Im}\Sigma(\varepsilon)$ presents a step function
behavior with $\varepsilon$ and strongly depends on the energy in agreement
with the above analytical results.

Figure 2: The mean free time $\tau_{s}$ normalized by
$h/(2\pi\varepsilon_{\xi})$ as a function to momentum for various disorder
amplitude on a logarithmic scale.

The scattering mean free time is obtained from Eq.(8). Figure 2 depicts that
the curves have the same behavior for various disorder amplitudes and diverge
at large momentum in similar fashion with the numerical and experimental
findings of Ref.22 . Within Boltzmann approximation for $k\xi<1$, the
contribution of the scatterers in the direction of incident propagation
vanishes by the ensemble average over all realizations of the random potential
giving rise to an isotropic $\tau_{s}$. Our results show that for small
momentum, the scattering mean free time converges to a constant values
$\tau_{s}=0.96\hbar/\varepsilon_{\xi}$ until $k\xi=0.2$ in agreement with the
theoretical results of Ref.14 . For $k\xi\gtrsim 0.2$, the scatterers start to
deviate for all values of disorder amplitude. Another important remark is that
as the disorder amplitude increases, $\tau_{s}$ increases linearly with $k\xi$
in the high-momenta limit. It is also shorter for large amplitudes in the
whole range of $k\xi$.

Figure 3: Anisotropic factor as function of momentum. At lowest momentum
$\langle\cos\theta\rangle=0.45$ emphasizing the anisotropic effects.

Actually according to Boltzmann approximation for $k\xi\gg 1$, the scattering
is strongly deviated forward in a scattering cone of an angle $\theta$. The
deviation which regards only the faster atoms may modify the trajectory of the
waves during the diffusion. The scattering angle $\theta$ which is defined as
the angle between the initial wave vector $k$ and the final wave vector
$k^{\prime}$, is averaged over the phase function $\cos\theta$. For $k\xi\gg
1$, one has $\theta\ll 1$ yielding $\cos\theta=1-\theta^{2}/2$. Therefore,
equation (10) gives $\tau_{B}=2\tau_{s}/\langle\theta^{2}\rangle$ marking that
when forward scattering is enhanced, the Boltzmann time largely exceeds the
Drude time. For $k\xi=1$, $\tau_{s}=1.3\hbar/\varepsilon_{\xi}$ which is
bigger than the initial momentum ($\tau_{s}=0.96\hbar/\varepsilon_{\xi}$)
caused by the dephasing process. The deviation can be attributted to the
velocity of the atom which causes fluctuations in its trajectory showing that
$\tau_{s}$ exhibits an anisotropy behavior.

Figure 4: The scattering mean free path $\ell_{s}$ as a function of the
$\varepsilon/\varepsilon_{\xi}$ for $U/\varepsilon_{\xi}$ =1.

The anisotropy factor $\langle\cos\theta\rangle$ as a function of the
wavenumber $k\xi$ is captured in figure 3. We see that for very small wave
$k\xi\sim 0$, $\cos\theta=0.45$ confirming that the anisotropic effects
persist even at small momenta. The numerical calculation gives
$\tau_{s}=0.55\tau_{B}$ at $k\sim 0$ indicating that some atoms are sensitive
to dephasing processes and therefore correspond to diffusive propagation. The
anisotropy factor increases with $k\xi$ and becomes noticeable
($\langle\cos\theta\rangle\sim 1$) at large momenta. We infer that the
scattering is anisotropic at lowest energy $(\varepsilon=0)$ for all momenta
due to the fact that the disorder is not pointlike. It is thus related to the
finite disorder correlation length. These results are comparable with those
found in the literature 23 , where $\langle\cos\theta\rangle=1/3$ for all
$k\xi\ll 1$.

Figure 5:  (a) Boltzmann diffusion constant normalized by $(2\pi m)/h$ as a
function of the energy for three different values of momenta. At
$\varepsilon=0.1\varepsilon_{\xi}$ deviations in $k\xi=1$ and $k\xi=1.2$ are
due to week localzation effects. (b) Weak localization correction relative to
the Boltzmann diffusion constant versus $\varepsilon/\varepsilon_{\sigma}$ at
$\varepsilon=0$ for for three different values of $k\xi$.

Now let us analyze the scattering mean free path $\ell_{s}$ in terms of the
pertinent parameter $\varepsilon/\varepsilon_{\xi}$. Figure 4 shows that
$\ell_{s}$ first decreases at lower energies ($\varepsilon<\varepsilon_{\xi}$)
until it reaches $\ell_{s}=0.66\xi$ at $\varepsilon=\varepsilon_{\xi}$
signaling the presence of the residual anisotropy. In the opposite, at higher
energies ($\varepsilon>\varepsilon_{\xi}$), $\ell_{s}$ increases displaying a
strong anisotropy for fast atoms as predicted in Refs.13 ; 14 . This
nonmonotonic behavior would indicate that the long-range correlations in the
optical potential may substantially affect the scattering anisotropy.
The situation is quite different at the initial state ($k\xi\simeq 0$), where
the scattering mean free path $\ell_{s}$ decreases rapidly at $\varepsilon\sim
0$ towards the same value $\ell_{s}=0.66\xi$ and it remains constant
regardless of the particle energy. Our analysis based on the anisotropy factor
shows that $\ell_{s}=0.55\ell_{B}$ which is close to the results of 13
($\ell_{s}=2\ell_{B}/3$). We conclude that the anisotropic character in 3D
case is always present during the diffusion process.

The Boltzmann diffusion constant in the diffusive regime can be computed
through Eq. (9). Its behavior as a function of the energy for three different
momenta is displayed in figure 5 (a). It is clearly visible that at very low
momenta $k\simeq 0$, the diffusion constant is a linear function of the
energy, $D_{B}\propto\varepsilon$, for $0<\varepsilon/\varepsilon_{\xi}<4$
indicating the persistence of the anisotropy of $D_{B}$ even at very low
momenta. This emphasizes the absence of a 3D white-noise limit (i.e.
nonexistence of isotropic diffusion) in good agreement with the results of 14
. However, for $k\xi=1$ and $k\xi=1.2$, $D_{B}$ increases as a power-law
$\propto\varepsilon^{\alpha}$ ($\alpha=0.05$ for $k\xi=1$ and $\alpha=0.06$
for $k\xi=1.2$) in the region $\varepsilon/\varepsilon_{\xi}\lesssim 3.5$. For
$\varepsilon\gtrsim 3.5\varepsilon_{\xi}$, the diffusion constant begins to
saturate at $2\pi mD_{B}/h\approx 0.46$. The two curves for $k\xi=1$ and
$k\xi=1.2$ match at $\varepsilon/\varepsilon_{\xi}\lesssim 3.8$. This can be
understood by the fact that the anisotropy factor has a non-negligeable effect
for $k\xi\neq 0$ as already indicated in figure 3 giving rise to a significant
anisotropic diffusion constant.

In order to examine the role of weak localization effects around the region
$\varepsilon=0$ in the diffusion constant, we solve numerically Eq. (13). The
results are shown in figure 5 (b). We observe that the diffusion constant is
compatible with the Boltzmann diffusion constant $(D_{\varepsilon}=D_{B})$ for
$\varepsilon/\varepsilon_{\xi}\lesssim 0$. Then for
$\varepsilon/\varepsilon_{\xi}>0$, the diffusion constant decays as
$D_{\varepsilon}<D_{B}$ for all momenta owing to the weak localization
effects. We see also that for $k\xi\sim 0$, the wave form multiple scattering
paths that are first affected by the correlation leading to the diminution of
the diffusion constant. This latter falls quickly to zero for the three values
of momentum. For instance, at $\varepsilon/\varepsilon_{\xi}=0.5$, one has
$D_{\varepsilon}\simeq 0.4D_{B}$ for $k\xi=0$, while this value rises to
$D_{\varepsilon}\simeq 0.8D_{B}$ for $k\xi=1$ and to $D_{\varepsilon}\simeq
0.85D_{B}$ for $k\xi=1.2$.

Figure 6:  Conductance scaling $\beta$-function. The crossing with the
horizontal axis indicates the transition point between the diffusive and the
localized regimes.

##  IV Dimensionless conductance and $\beta$-function

In order to check the universal transport properties, we use the scaling
theory which is formulated in terms of a dimensionless conductance. In the
diffusive regime, the dimensionless conductance $G$ of a sample of linear size
$L$ is defined as: $G=2(k\ell_{B})(kL)/3$ (see e.g 28 ). Our aim is to look at
how the dimensionless conductance evolves with system size, then it is useful
to introduce the scaling $\beta$-function 2 ; 28

$\beta=\frac{d\ln G}{d\ln L},$ (14)

which shows the flow of bosonic states. For $\beta=0$, the conductance $G$
does not vary with $L$. The results of the numerical analysis of equation (14)
based on the FBA for the function $\beta$ are sketched in figure 6. We see
that $\beta$ is negative for small $G$ while it is positive for large values
of $G$. It crosses zero at $G_{c}\approx 1.2$ which is close to the
theoretically predicted value $G_{c}\approx 1$. This unstable fix-point can be
considered as a critical point marking a possible transition from a diffusive
regime to a localized regime.

##  V Conclusions

We investigated the anisotropy effects on the quantum transport of atomic
matter waves using the FBA. We first calculated analytically and numerically
the imaginary part of self-energy which leads us to examine the scattering
mean time of diffusion. Our results indicate that the scattering time is
inversely proportional to the square of the disorder amplitude regardless of
the values of the energy in good agreement with recent experiments. A
deviation of $\tau_{s}$ from isotropy exhibits an angular dependence (i.e.
depends on the anisotropy angle $\theta$). The anisotropy of the scattering
mean free path is revealed in its nonmonotonic character induced by the long-
range correlations of the optical potential. Strictly speaking, the scattering
is never isotropic ($l_{s}\sim 0.45l_{B}$) for small momenta $k\xi\sim 0$ even
though the anisotropy decreases ($\langle\cos\theta\rangle\rightarrow 0$).
While for high momenta $k\xi\gg 1$, the anisotropy is singificant results in
strong scattering anisotropy.

Moreover, the Boltzmann diffusion constant $D_{B}$ has been explored
numerically. At higher energy, we found that $D_{B}$ increases linearly in
agreement with theoretical findings of Refs.13 ; 14 . However, it follows as
power law for $k\xi=1$ and $k\xi=2$ and is significantly anisotropic due to
the correlations induced by the multiple scattering in the optical disorder.
We then analyzed the effects of weak localization on the deviation of the
diffusion constant $D_{\varepsilon}/D_{B}$. Our predictions show that for
$\varepsilon/\varepsilon_{\xi}\leq 0$, $D_{\varepsilon}=D_{B}$ for all
momenta. At higher momenta and for $\varepsilon/\varepsilon_{\xi}>0$, the weak
localization corrections are strong enough to decrease the ratio
$D_{\varepsilon}/D_{B}$.

Furthermore, we found that the monotonic variation of the conductance with $L$
is prominent confirming that there is a universal quantum transition from a
diffusive regime to a localized regime.

We believe that our findings may have implications for understanding the
properties of coherent transport in 3D anisotropic optical disorders and
provide good reference data for future research.

##  VI References

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