# Bichromatic state-dependent disordered potential for Anderson localization
of ultracold atoms

Baptiste Lecoutre  Laboratoire Charles Fabry, Institut d’Optique, CNRS,
Université Paris-Saclay, 91127 Palaiseau Cedex, France  Yukun Guo  Laboratoire
Charles Fabry, Institut d’Optique, CNRS, Université Paris-Saclay, 91127
Palaiseau Cedex, France  Xudong Yu  Laboratoire Charles Fabry, Institut
d’Optique, CNRS, Université Paris-Saclay, 91127 Palaiseau Cedex, France  M.
Niranjan  Laboratoire Charles Fabry, Institut d’Optique, CNRS, Université
Paris-Saclay, 91127 Palaiseau Cedex, France  Musawwadah Mukhtar  Laboratoire
Charles Fabry, Institut d’Optique, CNRS, Université Paris-Saclay, 91127
Palaiseau Cedex, France  Valentin V. Volchkov  Laboratoire Charles Fabry,
Institut d’Optique, CNRS, Université Paris-Saclay, 91127 Palaiseau Cedex,
France  Alain Aspect  Laboratoire Charles Fabry, Institut d’Optique, CNRS,
Université Paris-Saclay, 91127 Palaiseau Cedex, France  Vincent Josse
<EMAIL_ADDRESS>Laboratoire Charles Fabry, Institut
d’Optique, CNRS, Université Paris-Saclay, 91127 Palaiseau Cedex, France

###### Abstract

The ability to load ultracold atoms at a well-defined energy in a disordered
potential is a crucial tool to study quantum transport, and in particular
Anderson localization. In this paper, we present a new method for achieving
that goal by rf transfer of atoms of an atomic Bose-Einstein condensate from a
disorder insensitive state to a disorder sensitive state. It is based on a
bichromatic laser speckle pattern, produced by two lasers whose frequencies
are chosen so that their light-shifts cancel each other in the first state and
add-up in the second state. Moreover, the spontaneous scattering rate in the
disorder-sensitive state is low enough to allow for long observation times of
quantum transport in that state. We theoretically and experimentally study the
characteristics of the resulting potential.

##  I Introduction

Ultra-cold atoms offer remarkable quantum simulators to experimentally study
difficult condensed-matter problems [1]. Quantum transport and Anderson
localization have been directly observed by launching atoms in disordered
potentials produced by far off-resonance laser speckle [2, 3, 4, 5, 6, 7, 8].
Convincing results with quantitative comparison to calculations have been
obtained on Anderson localization in 1D speckle disordered potentials [9, 2,
3], and on direct signature of weak localization phenomena [10, 4, 5]. When it
comes to 3D Anderson localization, several observations have been reported [6,
7, 8], but precise quantitative measurements are still lacking. The reason is
that in experiments performed so far, atoms launched in the disorder have a
large energy dispersion so that evaluating the mobility edge demands a
deconvolution leading to large uncertainties.

It would thus be extremely interesting to have a method to launch atoms in the
disorder at a precisely defined energy. By scanning the energy around the
mobility edge, it would allow one to determine precisely the mobility edge.
One might even evaluate critical exponents. Note that there is no exact theory
yielding the value of both quantities [11, 12], and it is thus highly
desirable to compare the results of approximate theoretical treatments [13,
14, 15, 16] or numerical results [17, 18, 19] with experimental results. Among
other experiments that would benefit from such an improved control of the
energy of the atoms launched in the disorder, one can cite measurements of
spectral functions [20], tests of the landscape theory of localization [21],
search for sophisticated signatures of localization [22, 23, 24, 25, 26], or
even the study of 2D localization [27, 28].

Our strategy for launching atoms with a precisely defined energy (see Fig. 1)
consists in performing a rf transition from a state insensitive to disorder
(state $\lvert 1\rangle$) to a state sensitive to disorder (state $\lvert
2\rangle$). The energy of the populated eigenstates in the disorder can be
adjusted by the control of the frequency of the rf. In the disorder, the atom
energy levels form a continuum. Since the initial state is discrete, a priori
one has a one way transition characterized by a rate $\Gamma$ given by the
Fermi Golden Rule, which also sets the minimum energy dispersion of the
transferred atoms. For an interaction time $t_{\mathrm{rf}}$ shorter than
$\Gamma^{-1}$, as in the experiment reported below, the energy dispersion of
the arrival states, i.e. of the atoms transferred to the continuum, is
Fourier-limited and given by $\Delta E=\hbar/t_{\mathrm{rf}}$.

Figure 1: Launching atoms at a well defined energy in a disordered potential.
It consists of transferring atoms of a Bose-Einstein condensate from the
discrete, disorder insensitive state $\lvert 1\rangle$ with a well defined
energy, to a state $\lvert 2\rangle$ sensitive to disorder, thus belonging to
a continuum. By tuning the rf transfer frequency $\omega_{\mathrm{rf}}$ one
can select the mean energy of the atoms transferred in $\lvert 2\rangle$,
while adjusting the rf power and duration of the transfer allows one to
control the spread in energy $\Delta E$ of the transferred atoms.

A first implementation of that scheme was demonstrated in the work of Volchkov
et al. [20], where we used a rf atom transfer between states with different
sensitivities to a monochromatic laser speckle disorder. These different
sensitivities stemmed from the very different detunings of the laser used to
produce the disorder, for the initial state $\lvert 1\rangle$ and the final
state $\lvert 2\rangle$ (see Fig. 2). The narrow Fourier-limited energy
distribution we obtained –two orders of magnitude lower than in previous
experiments [18, 20]– allowed us to make a quantitative study of the spectral
function of the atoms in that disorder, exploring different regimes of quantum
transport from the quantum low disorder regime to the classical strong
disorder regime [29, 30, 31]. Striking differences between the cases of
attractive (red detuned) and repulsive (blue detuned) disorder were observed
and interpreted. The method of [20] is, however, strongly limited by a serious
problem. It relies on a laser tuned between the two resonances associated with
the two atomic ground levels, and the detuning for the upper state $\lvert
2\rangle$ cannot be large enough to avoid resonant scattering of photons in
that state. This entails a rapid destruction of the coherence of the spatial
wave function describing the atomic motion and thus of Anderson localization.

Figure 2: State dependent disorder in a monochromatic speckle. The laser at
frequency $\omega$ is closer to resonance for the $\lvert
2\rangle\rightarrow\lvert F^{\prime}\rangle$ transition than for the $\lvert
1\rangle\rightarrow\lvert F^{\prime}\rangle$ transition. The light shift
induced in $\lvert 2\rangle$ is thus larger than the light shift induced in
$\lvert 1\rangle$, and for a speckle induced by that laser the resulting
disorder is stronger in $\lvert 2\rangle$. On the figure, corresponding to a
blue detuned laser, the disorder in $\lvert 2\rangle$ is repulsive, and is
limited below. For a detuning $\delta$ of opposite sign (red detuned laser),
the disorder would be attractive in $\lvert 2\rangle$, and limited above. The
drawback of that method is the large spontaneous scattering of laser photons
in state $\lvert 2\rangle$, resulting in a loss of coherence for the quantum
transport of atoms in state $\lvert 2\rangle$.

The method reported in the present paper overcomes this problem by the use of
a bichromatic speckle potential. It consists of two speckles due to lasers of
almost identical frequencies for which the potentials are of opposite signs
for the initial state of the rf transfer, and of same signs for the final
state of the rf transfer (Fig. 3). It yields a strong suppression of the
sensitivity to the disorder in the initial state, together with a strong
suppression of resonant scattering in the final – disorder sensitive – state.
This scheme will allow one to operate with observation times around one second
or more, which is required to study 3D localization phenomena [7, 8].

This manuscript is organized as follows. In section II, we recall the most
important properties of the state-dependent disordered potential based on a
monochromatic laser speckle as used in Volchkov et al. [20], and discuss the
limitations of that scheme. In section III, we describe the new method where
we introduce a second laser to realize a bichromatic speckle. Firstly we study
the influence of the small difference between the two laser frequencies
creating the two speckles and estimate the fundamental potential decorrelation
in the disorder sensitive state, an important result of this paper. Secondly
we show how it is possible, by a suitable adjustment of the frequencies and
intensities of the two lasers, to minimize the photon-scattering rate in the
disorder sensitive state and to suppress the disorder experienced by the atoms
in the initial state. In section IV, we present an experimental evaluation of
the bichromatic speckle disorder scheme . The reported measurements support
our analysis and are promising for future quantum transport experiments.

Figure 3: State dependent disorder in a bichromatic speckle. The two lasers at
frequencies $\omega_{\mathrm{p}}$ (principal) and $\omega_{\mathrm{c}}$
(compensation) create disordered potentials that add-up in $\lvert 2\rangle$
but cancel each other in $\lvert 1\rangle$. The detunings
$\delta_{\mathrm{p}}$ and $\delta_{\mathrm{c}}$ are chosen so that the
spontaneous scattering rate in $\lvert 2\rangle$ is small enough to have long
coherence times in $\lvert 2\rangle$, i.e., on the order of one second or
more.

##  II State-dependent disordered potential based on a monochromatic speckle

The creation of state dependent potential for alkali atoms has been widely
investigated in the context of optical lattices using circularly polarized
light tuned between the $D_{1}$ and $D_{2}$ lines, see e.g. [32, 33, 34].
However such scheme is efficient only if the two considered state have
different magnetic susceptibilities [35]. Since the study of 3D Anderson
localization requires a magnetic levitation to suspend the atoms against
gravity during their expansion in the disorder [6, 7, 8], both disorder
sensitive and insensitive states must have similar magnetic susceptibility.
Alternative methods to create state dependent disorder must then be developed,
such as the monochromatic speckle state-dependent potential realized in Ref.
[20]. This section recalls the main properties of that scheme and points out
its limits.

###  II.1 Monochromatic speckle potential

Figure 4: a: Sketch of the generation of the speckle in an effective paraxial
geometry. A laser beam of waist $w$ is focused using a lens of focal length
$f=$15.2(5)\text{\,}\mathrm{mm}$$. A rough plate of diameter $D$ scatters the
laser light within an angle $\theta_{\mathrm{diff}}\approx 5^{\circ}$ which is
fixed by the diffuser. The atomic cloud is centered on the optical axis in the
Fourier plane $\\{x=0,y=0,z=0\\}$. Inset: 3D view of a numerical realization
of a speckle pattern. We define the size of the speckle grains by the half-
widths $\sigma_{\perp}$ and $\sigma_{\parallel}$ of its autocorrelation
function. b: Measured transverse $c_{3D}(\Delta\mathbf{r}_{\perp},\Delta z=0)$
and longitudinal $c_{3D}(\Delta\mathbf{r}_{\perp}=\mathbf{0},\Delta z)$
autocorrelation functions of the speckle pattern, fitted by an effective
paraxial theoretical model with effective parameters
$D=$17.8(1)\text{\,}\mathrm{mm}$$ and $w=$8(1)\text{\,}\mathrm{mm}$$ [36] (see
text). This effective paraxial model successfully reproduces the features of
the measured correlation functions.

We consider the disordered potential $V(\mathbf{r})$ experienced by an atom in
a specific level in the presence of a monochromatic laser speckle pattern of
intensity $I(\mathbf{r})$. For a non saturating laser strongly detuned from
the nearest relevant resonance, it scales as $I(\mathbf{r})/\delta$, where
$\delta=\omega-\omega_{0}$ is the detuning of the laser with respect to the
nearest resonant transition involving the considered atomic level. The speckle
is generated by focusing an expanded laser beam onto the atoms located in the
focal plane and diffracting it through a rough plate of random thickness, as
illustrated in figure 4. The simplest model of such a speckle, as presented in
[37], results from the hypothesis that the rough plate imposes onto the laser
beam a random phase $\phi(\mathbf{r}_{\mathrm{0}})$ with a probability
distribution constant over $2\pi$, and $\delta$-correlated –meaning that the
spatial autocorrelation function of the transmission
$t_{\mathrm{diff}}(\mathbf{r}_{\mathrm{0}})=\exp\\{i\phi(\mathbf{r}_{\mathrm{0}})\\}$
is a $\delta$ Dirac function.

The complex amplitude $\mathcal{A}(\mathbf{r})$ of the field at each point of
the fully developed speckle pattern –close to the focus of the laser– can then
be considered a sum of a very large number of independent random variables
with the same statistical properties. According to the central limit theorem,
the complex amplitude is thus a Gaussian random process, whose properties
allow one to calculate the statistical properties of the intensity
$I(\mathbf{r})$, which is proportional to the squared modulus
$\lvert\mathcal{A}(\mathbf{r})\rvert^{2}$ of the complex amplitude. Since the
complex amplitude at each point has a 2-dimensional Gaussian probability
distribution, the intensity $I(\mathbf{r})$ has an exponential probability
distribution
$\mathcal{P}(I)=\overline{I}^{-1}\exp[-I/\overline{I}]\Theta(I/\overline{I})$
[37] with $\Theta$ the unit step function, and where the symbol
$\overline{\>\cdots\>}$ stands for ensemble averaging, i.e., averaging over
different realizations of the rough plate. The standard deviation of the
intensity fluctuations is then equal to the mean intensity value, i.e.,
$\sigma_{I}=\overline{I}$. A similar property
$\sigma_{V}=\lvert\overline{V}\rvert$ holds for the disordered potential which
can be positive or negative. It is experimentally controlled by adjusting the
laser power and detuning, and can be varied over several orders of magnitude
[20]. Both $\overline{I}$ and $\overline{V}$ are independent of $\mathbf{r}$
because the process is spatially invariant and ergodic for this simple model
of uncorrelated dephasing.

A key property of the fully developed speckle pattern is the size of the
speckle grains, which is equal, within a factor, to the half-width of the
normalized autocorrelation function defined around the focal point where the
atoms are located

$c_{3D}(\Delta\mathbf{r}_{\perp},\Delta z)=\frac{\overline{\delta
I(\mathbf{0},0)\,\delta I(\Delta\mathbf{r}_{\perp},\Delta
z)}}{\overline{\delta I^{2}}}$ (1)

Here $\delta I(\mathbf{r})=I(\mathbf{r})-\overline{I}$ are the intensity
fluctuations, and we note $\Delta\mathbf{r}_{\perp}=\\{\Delta x,\Delta y\\}$
the transverse displacement in the Fourier plane. The function $c_{3D}$ is a
fourth order moment of the complex amplitude $\mathcal{A}(\mathbf{r})$, and
for the Gaussian process considered here, one can use the Wick’s theorem for
classical moments of a Gaussian random process, to express any moment of
$I(\mathbf{r})$ as a function of the second order moments of the amplitude
[37]. The function $c_{3D}$ can then be expressed as a function of the
autocorrelation function of the complex amplitude
$\Gamma_{\mathcal{A}}(\Delta\mathbf{r}_{\perp},\Delta z)$ (calculated in
section A.2 of Appendix A) as

$\displaystyle c_{3D}(\Delta\mathbf{r}_{\perp},\Delta z)$ $\displaystyle=$
$\displaystyle\frac{\left\lvert\Gamma_{\mathcal{A}}(\Delta\mathbf{r}_{\perp},\Delta
z)\right\rvert^{2}}{\left\lvert\Gamma_{\mathcal{A}}(\mathbf{0},0)\right\rvert^{2}}$
(2) $\displaystyle=$
$\displaystyle\frac{\left\lvert\mathrm{FT}\left[I(\mathbf{r}_{0})e^{-i\frac{\mathbf{r}_{\mathrm{0}}^{2}k\Delta
z}{2f^{2}}}\right]_{\frac{k\Delta\mathbf{r}_{\perp}}{f}}\right\rvert^{2}}{\left\lvert\displaystyle\int\mathrm{d}\mathbf{r}_{\mathrm{0}}\,I(\mathbf{r}_{\mathrm{0}})\right\rvert^{2}}$

This correlation function is central to quantum transport studies as it
directly translates into the spatial frequency distribution of the potential
that governs the scattering properties of the atoms [9, 38, 39, 36, 40]. A
full calculation is presented in section A.3 of Appendix A, and here we only
discuss some of its most important features in the transverse and longitudinal
directions.

Within the paraxial approximation, the transverse amplitude correlation
function $\Gamma_{\mathcal{A}}(\Delta\mathbf{r}_{\perp},\Delta z=0)$ is the
Fourier transform of the intensity distribution $I(\mathbf{r}_{\mathrm{0}})$
just before the diffusing plate [37]. This can be interpreted as an example of
the Van Cittert-Zernike theorem [41]. The intensity fluctuations correlation
function is thus nothing else than the squared modulus of the diffraction
pattern corresponding to the intensity distribution at the diffusing plate in
the diffuser

$c_{3D}(\Delta\mathbf{r}_{\perp},\Delta
z=0)\propto\left\lvert\mathrm{FT}\left[I(\mathbf{r}_{\mathrm{0}})\right]_{\frac{k\Delta\mathbf{r}_{\perp}}{f}}\right\rvert^{2}\,.$
(3)

For instance, in the ideal case of a Gaussian intensity profile of standard
waist radius $w$ (following the usual convention for laser beams, $w$ is the
radius at $\mathrm{e}^{-2}$), it is

$c_{3D}(\Delta\mathbf{r}_{\perp},\Delta
z=0)=\exp\left\\{-\frac{\Delta\mathbf{r}_{\perp}^{2}}{\sigma^{2}_{\perp}}\right\\}\,,$
(4)

with $\sigma_{\perp}=\lambda f/\pi w$. We call $\sigma_{\perp}$ the size of a
speckle grain, for consistency with previous papers. Within the same paraxial
hypothesis, the longitudinal autocorrelation function of the intensity
fluctuations is, on the $z$ axis and close to the focusing point of the lens
[37]

$c_{3D}(\mathbf{0},\Delta
z)\propto\left\lvert\int{\mathrm{d}\mathbf{r}_{\mathrm{0}}I(\mathbf{r}_{\mathrm{0}})\exp\left\\{-ik\frac{\mathbf{r}_{\mathrm{0}}^{2}\Delta
z}{2f^{2}}\right\\}}\right\rvert^{2}.$ (5)

In general, the evaluation of the integral in (5) can be done numerically
only. In the ideal case of a Gaussian intensity profile, it yields a
Lorentzian profile of the longitudinal correlation function [37] and we define
the size of a speckle grain by the half-width at half-maximum
$\sigma_{\parallel}$.

A more realistic description of the situation considers a correlated diffuser
with a spatial autocorrelation function of the phase factor – more precisely
of $\exp\\{i\phi(\mathbf{r}_{\mathrm{0}})\\}$ – of finite width
$r_{\mathrm{e}}$ (see section A.1 of the appendix or Ref. [37]). The speckle
pattern has then a mean intensity profile that is no longer uniform, in
contrast to the $\delta$-correlated model above. This profile is given by the
Fourier transform of the autocorrelation function of the phase factor. For our
diffuser, it has a Gaussian shape with an angular width of about
$\theta_{\mathrm{diff}}\approx 5^{\circ}$ ($1/e^{2}$ radius) and yields a
speckle pattern extending on a zone much wider
($\sim$1.3\text{\,}\mathrm{mm}$$) than the atomic cloud
($\sim$15\text{\,}\mathrm{\SIUnitSymbolMicro m}$$). The average intensity on
the atoms is then almost constant and the speckle can still be considered
uniform over the atoms sample. Moreover, the correlation length of the phase
factor of the rough plate is small compared to the width $w$ of the laser beam
on the plate, so that the central limit theorem still applies to the speckle
complex amplitude which is well represented by a Gaussian statistics. We can
thus use the formulae (3) and (5).

Another more realistic feature of our experiment is the fact that the phase
distribution at the diffuser may not be strictly uniform over $2\pi$. In fact,
as shown in [42] and in section A.1 of the Appendix A, a model with a Gaussian
phase distribution of standard deviation $\sigma_{\phi}$ allows us to show
that the formulae (3) and (5) can still be used as long as $\sigma_{\phi}\gg
2\pi$, for a monochromatic speckle. We will see in section III that a
bichromatic speckle requires a more elaborated description, for which we will
use the model of a plate with a Gaussian distribution of thickness.

The experimental determination of the correlation function (1) was done with
an optical microscope [43] and the results are shown in Figure 4.b. Because of
the large numerical aperture $\mathrm{NA}=0.55(2)$, the theoretical
description requires a beyond-paraxial model [20]. In fact, the measured
transverse and longitudinal correlation functions were found well reproduced
by equations (3) and (5), provided that we introduce a geometrical scaling
factor [36, 44]. More precisely, we can match the measured correlation
functions using formulaes (3) and (5) with a geometrical factor of $0.875$,
resulting in the effective numerical aperture
$\mathrm{NA}_{\mathrm{eff}}=0.50$. In particular, the transverse profile of
the three-dimensional correlation function is very well described by a
gaussian function (4) of $1/e$ radius
$\sigma_{\perp}=$0.50(1)\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. The
longitudinal profile is also well reproduced by the result of a numerical
evaluation of (5) based on a truncated gaussian illumination 111The case of a
pure gaussian intensity profile on the diffuser leads to a Lor?entzian
longitudinal correlation function while it is described by a
$\mathrm{sinc}^{2}$ function in the case of a circularly truncated homogeneous
illumination. In both cases, we characterize its width by the HWHM. [36]. A
longitudinal correlation length can still be defined by the half-width at
half-maximum, yielding
$\sigma_{\parallel}=$2.05(5)\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ [20].
Altogether, this indicates that an effective paraxial approximation is well-
suited to describe the spatial correlations of our speckle.

###  II.2 Energy-resolved transfer scheme in a monochromatic state-dependant
optical potential for ${}^{87}\mathrm{Rb}$

In this subsection, we address the two problems that influence the energy
spread of ${}^{87}\mathrm{Rb}$ atoms transfered into a disordered potential:
the fluctuations of the energy difference between the magnetic levels due to
magnetic field fluctuations, and the energy dispersion of the initial state.

####  II.2.1 Suppression of the effect of the magnetic field fluctuations

The atomic sample consists of a Bose-Einstein Condensate (BEC) prepared in the
disorder insensitive state $\lvert 1\rangle$ with energy $E_{1}$ and then
coupled by a rf field to the disorder sensitive state $\lvert 2\rangle$ with
energy $E_{2}$. The final kinetic energy $E_{f}$ relevant to the transport is
$E_{f}=E_{i}+\hbar\delta_{\mathrm{rf}}$ where $E_{i}$ is the initial kinetic
energy, which is null for atoms initially at rest, and
$\delta_{\mathrm{rf}}=\omega_{\mathrm{rf}}-\Delta_{\mathrm{hf}}$ is the
detuning of the rf field with respect to the bare $\lvert
1\rangle\rightarrow\lvert 2\rangle$ transition. The energy $E_{f}$ can be
chosen by tuning the rf frequency, allowing us to address precise energy
values of the atoms in the disorder-sensitive state.

If the bare resonance frequency $\Delta_{\mathrm{hf}}$ between the two
magnetic sublevels fluctuates because of magnetic field fluctuations, the
energy $E_{f}$ will also fluctuate. In order to cancel these fluctuations of
$E_{f}$, we use the so-called “clock states” $\lvert 1\rangle\equiv\lvert
F=1,m_{\mathrm{F}}=-1\rangle$ and $\lvert 2\rangle\equiv\lvert
F=2,m_{\mathrm{F}}=+1\rangle$, which are separated at zero field by a
splitting of $\Delta_{\mathrm{hf}}/2\pi=$6.835\text{\,}\mathrm{GHz}$$, and we
impose a bias magnetic field at the so-called magic value of
$B_{0}^{*}=3.23\,\mathrm{G}$. The magnetic susceptibilities of states $\lvert
1\rangle$ and $\lvert 2\rangle$ of ${}^{87}\mathrm{Rb}$ are then identical,
meaning that the energy separation between these states is insensitive to
magnetic field fluctuations [46]. Moreover, this choice of the clock states is
also crucial regarding the use of the magnetic levitation to study the
propagation of the atoms in 3D.

The rf coupling consists in fact of a two-photon transition, involving a
microwave and a rf field, to match the angular momentum difference $\Delta
m_{F}=2$ [20]. It can be considered a direct transition induced by a field
with an effective frequency $\omega_{\mathrm{rf}}$ equal to the sum of the two
frequencies of the two fields, and an effective Rabi frequency proportional to
the product of the Rabi frequencies of the two fields.

####  II.2.2 Suppression of the sensitivity of the initial state to the
disordered potential

In order to have a speckle acting strongly on the atomic state $\lvert
2\rangle$ but very little on the atomic state $\lvert 1\rangle$, we use a
laser of frequency $\omega$ close to resonance for the transition $\lvert
2\rangle\rightarrow\lvert F^{\prime}\rangle$ and far from resonance for the
transition $\lvert 1\rangle\rightarrow\lvert F^{\prime}\rangle$ (see Fig. 2).
More precisely, since the natural linewidth of the transition
($\Gamma_{\mathrm{Rb}}/2\pi\simeq$6.07\text{\,}\mathrm{MHz}$$) is small
compared to the splitting
$\Delta_{\mathrm{hf}}/2\pi\simeq$6.8\text{\,}\mathrm{GHz}$$ [47], it is
possible to operate in the regime of
$\Gamma_{\mathrm{Rb}}\ll\delta\ll\Delta_{\mathrm{hf}}$, with
$\delta=\omega-\omega_{2,F^{\prime}}$ the detuning of the speckle laser from
resonance for state $\lvert 2\rangle$. Then the detuning from resonance for
state $\lvert 1\rangle$ is almost equal to $\Delta_{\mathrm{hf}}$, so that the
average speckle potentials for level $\lvert 1\rangle$ and level $\lvert
2\rangle$ are in a ratio

$\frac{V_{1}}{V_{2}}\sim\frac{\delta}{\Delta_{\mathrm{hf}}}\ll 1\,.$ (6)

With well chosen laser intensity $I$ and detuning $\delta$, one can then
obtain an almost disorder-insensitive state $\lvert 1\rangle$, i.e., a
disordered potential $V_{1}$ small compared to the chemical potential $\mu$,
so that the screening effect in the Bose-Einstein condensate prepared in state
$\lvert 1\rangle$ absorbs the residual potential $V_{1}$ and suppresses
further its eventual perturbation [48]. In contrast, state $\lvert 2\rangle$
is sensitive to the disorder, as expected. For instance, the choice of
$\delta/2\pi\approx\pm$80\text{\,}\mathrm{MHz}$$ in Volchkov et al. leads to a
disordered potential ratio of $\lvert V_{2}/V_{1}\rvert\sim 100$ with
$V_{1}\leq\mu$.

This simple implementation of a state-dependent disordered potential allowed
us to determine the spectral functions of ultra-cold atoms in a speckle
potential at various energies. The results were found in remarkable agreement
with a numerical theoretical treatment in all regimes of disorder [20]. This
result shows the interest of a transfer between a weakly disorder sensitive
state and a strongly disorder sensitive state, in order to make energy-
resolved measurements on transport of atoms in disorder.

However, equation (6) with the requirement $V_{1}<\mu$ prevents one to have a
large value of $\delta$, so that the spontaneous photon scattering rate in
state $\lvert 2\rangle$, which scales as $\Gamma\sim I/\delta^{2}$, is large
enough to make it impossible to study quantum transport of the atoms in state
$\lvert 2\rangle$ for a long time. This is because the spontaneous scattering
of a photon by an atom in state $\lvert 2\rangle$ destroys the motional wave-
function coherence, which is at the heart of quantum transport and Anderson
localization. For instance, in the configuration of Volchkov et al., the
photon scattering rate was as large as a few $10^{3}\,\mathrm{s}^{-1}$,
forbidding measurement times beyond a few milliseconds, see Table 1. This
inadequacy of the monochromatic speckle potential to allow for long-lasting
energy-resolved quantum transport experiments can be overcome by the use of a
bichromatic speckle potential, as explained in next section.

##  III State-dependent disordered potential based on a bichromatic speckle

In subsection III.1, we present the improved scheme to experimentally study
quantum transport of atoms, based on a disordered potential created by two
lasers at two different frequencies. In subsection III.2, we give a
quantitative evaluation of the residual disorder due to a fundamental
potential decorrelation resulting from the difference in frequencies between
the two lasers. It is characterized by a normalized bichromatic correlation
function, whose form is remarkable. It allows us to show that the residual
disorder in state $\lvert 1\rangle$ has the same order of magnitude as in the
monochromatic scheme. In subsection III.3, we evaluate the residual
spontaneous scattering rate of laser photons in state $\lvert 2\rangle$, and
we show that is reduced by several orders of magnitude, which is the main goal
of the new scheme.

###  III.1 Bichromatic speckle scheme

As shown in figure 3, we use two lasers sufficiently detuned from resonance
with state $\lvert 2\rangle$. The principal laser (purple in Fig. 3) is
largely detuned and we fix its detuning
$\delta_{\mathrm{p}}/2\pi=$105\text{\,}\mathrm{GHz}$$ with respect to the
transition $\lvert 1\rangle\rightarrow\lvert F^{\prime}\rangle$. The potential
on state $\lvert 1\rangle$ is suppressed by the use of a less-detuned
compensation laser (orange in Figure 3), with an opposite detuning sign. We
can add the two potentials without considering possible interferences between
the lasers because the beat-note frequency ($\sim$100\text{\,}\mathrm{GHz}$$
in the implementations presented below) is high enough that it is averaged out
by the inertia of the atoms and has no effect on the atomic motion. In the
linear regime, the total disordered potential for atoms in state $\lvert
F,m_{F}\rangle$ can thus be written as

$\displaystyle V_{F,m_{F}}(\mathbf{r})$ $\displaystyle=$
$\displaystyle-\frac{1}{2\epsilon_{0}c}\big{(}\;\mathrm{Re}[\alpha_{F,m_{F}}(\delta_{\mathrm{p}})]I_{\mathrm{p}}(\mathbf{r})$
(7)
$\displaystyle+\mathrm{Re}[\alpha_{F,m_{F}}(\delta_{\mathrm{c}})]I_{\mathrm{c}}(\mathbf{r})\;\big{)}\;,$

where the complex atomic polarizability $\alpha_{F,m_{F}}(\delta)$ describes
the atomic dipolar response of the internal state $\lvert F,m_{F}\rangle$ to
an external electric field of detuning $\delta=\omega-\omega_{1,F^{\prime}}$.
The subscript $\mathrm{p}$ stands for principal and $\mathrm{c}$ stands for
compensation.

Following the scheme sketched in Fig. 3, we want to adjust the two potentials
such that they cancels each other in state $\lvert 1\rangle$, while summing up
in state $\lvert 2\rangle$. We denote $V_{\mathrm{R}}$ the amplitude of the
total laser speckle field in this disorder sensitive state, which is the
relevant quantity when considering the study of quantum transport phenomena in
such disordered potential.These two conditions write

$\left\\{\begin{array}[]{ll}V_{1}(\mathbf{r})=V_{\mathrm{p,1}}(\mathbf{r})+V_{\mathrm{c,1}}(\mathbf{r})\\\
V_{2}(\mathbf{r})=V_{\mathrm{p,2}}(\mathbf{r})+V_{\mathrm{c,2}}(\mathbf{r})\end{array}\right.\;\mathrm{with}\;\left\\{\begin{array}[]{ll}\overline{V_{1}}=0\\\
\overline{V_{2}}=V_{\mathrm{R}}\end{array}\right.$ (8)

Cancelling the total potential $V_{1}(\mathbf{r})$ on state $\lvert 1\rangle$
for all $\\{\mathbf{r}\\}$ requires the two speckle patterns to be as
identical as possible. To do so, a first mandatory condition is to shine two
identical laser modes on the diffuser (as shown in figure 4 for a single laser
beam). This is experimentally done by injecting the two lasers into the same
optical fiber before shining them onto the diffuser.

A second condition lies in the stability of the laser intensities. Power
fluctuations of one of the beams of the order of a few percents can lead to
strong potential fluctuations on state $\lvert 1\rangle$ – up to a few
percents of $V_{\mathrm{R}}$ – which can be limiting, see table 1. Each of the
lasers are therefore power-stabilized.

However, strict equality of the two identical speckle patterns cannot be
achieved with different wavelengths, because the speckle patterns depend on
diffraction and therefore on the laser wavelengths. The use of different laser
wavelengths then leads to fundamental potential decorrelation preventing the
complete cancelling of the disorder for state $\lvert 1\rangle$. That is the
subject of the next section.

###  III.2 Fundamental potential decorrelation in the disorder insensitive
state

Recalling that the detunings and intensities of the two lasers are adjusted
such that $\overline{V_{\mathrm{c,1}}}=-\overline{V_{\mathrm{p,1}}}$, we want
to characterize the residual fluctutations $\delta V_{1}(\mathbf{r})=\delta
V_{\mathrm{p,1}}(\mathbf{r})+\delta V_{\mathrm{c,1}}(\mathbf{r})$ due to the
frequency difference between the two lasers. We thus evaluate the variance of
$V_{1}(\mathbf{r})$ at each point. For a speckle pattern, one has
$\sigma_{V}=\overline{V}$, and the variance can be expressed as:

$\sigma_{V_{1}}^{2}(\mathbf{r})=2\lvert\overline{V_{\mathrm{p,1}}}(\mathbf{r})\,\overline{V_{\mathrm{c,1}}}(\mathbf{r})\rvert\left(1-c_{2\lambda}(\mathbf{r},\lambda_{\mathrm{p}},\lambda_{\mathrm{c}})\right)\mathrm{,}$
(9)

where the normalized bichromatic correlation function $c_{2\lambda}$ is
defined as

$c_{2\lambda}(\mathbf{r},\lambda_{\mathrm{p}},\lambda_{\mathrm{c}})\equiv\frac{\overline{\delta
V_{\mathrm{p,1}}(\mathbf{r})\delta
V_{\mathrm{c,1}}(\mathbf{r})}}{\overline{V_{\mathrm{p,1}}(\mathbf{r})}\>\overline{V_{\mathrm{c,1}}(\mathbf{r})}}\,.$
(10)

The correlation function (11) quantifies the correlation of the two speckle
fields at position $\mathbf{r}$ in space as a function of their wavelengths.
In the case of totally decorrelated speckle fields ($c_{2\lambda}=0$), the
variance of the total potential is consistently given by the sum of the
variances of each field. In contrast, for exactly identical speckle patterns,
the variance of the total potential would be null. Studying the amplitude of
the residual potential then comes down to investigating the behavior of the
normalized bichromatic correlation function
$c_{2\lambda}(\mathbf{r},\lambda_{\mathrm{p}},\lambda_{\mathrm{c}})$.

Figure 5: a: Illustration of the correlation between two speckle potentials
generated by the same diffuser with two slightly different wavelengths. Close
to the Fourier plane, the two patterns are identical up to a spatial scaling
factor, yielding an almost perfect overlap of the potential that decreases
with the distance to the optical axis. The correlation length $l_{c}$
describes the typical distance for which the speckle patterns become out of
phase. b: Plot of the variance of the total potential as a function of the
position in the Fourier plane. Close to the optical axis, the two speckle
patterns are similar and the only decorrelation term arises from the phase
difference due to the propagation through the diffuser. Far away from the
optical axis, the two speckle patterns do not superimpose and the variance of
the total potential is the sum of the two individual variances. Plot obtained
for a realistic inverse relative frequency difference $\mathcal{F}=4000$, a
well-exaggerated diffuser’s roughness
$\sigma_{\mathrm{e}}^{2}=10^{5}\lambda_{\mathrm{p}}^{2}$ and potentials of
opposite average values $V_{0}$.

Relying on the same paraxial assumption as for the calculation of the spatial
correlation function of a single speckle pattern, we find (see Appendix A)
that the correlation function between the two speckles generated by the same
diffuser, for the same spatial mode but for different frequencies, can be
expressed as:

$c_{2\lambda}(\mathbf{r}_{\perp},z,\lambda_{\mathrm{p}},\lambda_{\mathrm{c}})=e^{-\sigma_{\Delta\phi}^{2}}\;c_{3D}\Big{(}\frac{\mathbf{r}_{\perp}}{\mathcal{F}},\frac{z}{\mathcal{F}}\Big{)}\,,$
(11)

where $\sigma_{\Delta\phi}$ is a term discussed below and the parameter
$\mathcal{F}=\omega_{\mathrm{p}}/\delta\omega=\lambda_{\mathrm{p}}/\delta\lambda\approx
4000$ characterizes the frequency difference of the two speckle fields.
Expression (11) constitutes a major result of this letter. It is the product
of two terms that have a simple interpretation.

The first term $\exp{\\{-\sigma_{\Delta\phi}^{2}\\}}$ is associated to the
fluctuations of the phase difference for the two different wavelengths
$\lambda_{p}$ and $\lambda_{c}$ propagating inside the rough plate at the same
point. More precisely, the phases at each point of the diffuser differ for the
two lasers according to
$\Delta\phi(\mathbf{r}_{0})=\phi_{\mathrm{p}}-\phi_{\mathrm{c}}=2\pi(n-1)(\lambda_{\mathrm{p}}^{-1}-\lambda_{\mathrm{c}}^{-1})\delta
e(\mathbf{r}_{0})$, where $\delta e$ corresponds to the thickness
fluctuations. When averaged over the diffuser, i.e. over
$\mathbf{r}_{\mathrm{0}}$, and considering a Gaussian distribution for the
phase as discussed in section II.1, one obtains an average phase factor
$\overline{\exp{\\{i\Delta\phi\\}}}=\exp{\\{-\sigma_{\Delta\phi}^{2}/2\\}}$,
with $\sigma_{\Delta\phi}^{2}$ the variance of $\Delta\phi$. This fluctuating
phase difference term therefore results into a decorrelation factor constant
in the speckle pattern. It is solely due to the roughness of the diffusing
plate, and it can be rewritten as
$\exp{\\{-\sigma_{\Delta\phi}^{2}\\}}\approx\exp{\\{-4\pi^{2}(n-1)^{2}\sigma_{\mathrm{e}}^{2}/\lambda_{\mathrm{p}}^{2}\mathcal{F}^{2}\\}}$
222The factor two compared to the first factor in Eq. (11) is due to the fact
that we consider the correlation of the potential that is proportional to the
light intensity and not to the field amplitude, see appendix A.. Here
$\sigma_{\mathrm{e}}$ is the thickness fluctuations r.m.s. value, i.e. the
diffuser’s roughness. It means that the rougher the diffuser and the bigger
the frequency difference, the less the speckle patterns will be correlated,
even at the center of the patterns. This term has been visually exaggerated in
figure 5.b and explains why the minimum of the variance does not reach
perfectly 0.

In practice, the diffuser’s roughness $\sigma_{\mathrm{e}}$ is fixed by
manufacturing. It was measured at the optics workshop of Institut d’Optique,
using a profilometer. Several measurements of the surface’s profile have been
performed along $1\text{\,}\mathrm{mm}$ long straight lines onto several areas
of our diffuser, leading to a r.m.s. roughness of
$\sigma_{\mathrm{e}}=$1.3\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. For
$\mathcal{F}\approx 4000$, the term $e^{-\sigma_{\Delta\phi}^{2}}$ (associated
to the first factor of expression 11) is $\sim 1-10^{-6}$, indicating almost
perfect correlation between the two speckles and thus a negligible residual
disorder.

The second contribution in equation (11) describes the loss of correlation as
the position in the speckle is shifted away from the optical axis. It is due
to the different geometrical scaling factors of the two speckle patterns,
proportional to the wavelengths, as illustrated in figure 5. Remarkably, the
resulting term is expressed with the spatial correlation function (1) of a
monochromatic speckle, with a magnifying factor $\mathcal{F}$. The bichromatic
correlation function then has a width $l_{c}=\mathcal{F}\sigma_{\perp}$ which
defines a “bichromatic correlation length”, corresponding to the distance from
the center at which the two patterns are shifted by one speckle grain size.

This correlation length must be compared to the maximum size of the atomic
sample. Taking again $\mathcal{F}\approx 4000$ corresponding to a frequency
difference $\delta\omega/2\pi\approx$100\text{\,}\mathrm{GHz}$$ (see section
III.1), the bichromatic correlation length is of the order of
$l_{c}\sim$2\text{\,}\mathrm{mm}$$. It is much larger than the largest size of
our atomic sample about
$R_{\mathrm{TF}}\sim$45\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, see section
IV. We can then estimate this decorrelation factor to be at most
$1-c_{3D}(R_{\mathrm{TF}}/\mathcal{F})\sim 5\times 10^{-4}$. Altogether, we
find using Eq. (9) that the fundamental potential decorrelation leads
typically to a residual disorder in state $\lvert 1\rangle$ with a r.m.s.
value of the order of

$\sigma_{V_{1}}\sim 0.02\times V_{\mathrm{R}}\,,$ (12)

i.e. of the same order as in Volchkov et al. [20], see discussion in II.2.2.
For instance, we will see below that Table 1 predicts a residual disorder
$\sigma_{V_{1}}/h=$9.6\text{\,}\mathrm{Hz}$$ for the specific case of
$V_{\mathrm{R}}/h=$416\text{\,}\mathrm{Hz}$$, in agreement with the coarse
estimation of expression (12).

###  III.3 Reduction of the photon-scattering rate for the disorder-sensitive
state

Let us recall that a strong reduction of $\Gamma_{2}$, the photon scattering
rate of the disorder-sensitive state $\lvert 2\rangle$, is the main goal of
our present work. In order to study quantum transport phenomena such as
Anderson localization, a lifetime on the order of one second time or more is
needed [7, 8]. Note that the condition is much less stringent in the initial
state, being only on the order of tens of milliseconds,. This state $\lvert
1\rangle$ is indeed only used as a “source” of atoms of very well defined
energy during the rf transfer 333Once the rf transfer is switched off, the
atoms in the state $\lvert 1\rangle$ can be removed to avoid any detrimental
interactions with the atoms transferred in the state $\lvert 2\rangle$.,
yielding the simpler condition $\Gamma_{1}^{-1}\geq t_{\mathrm{rf}}$ (see
section II.2).

In presence of the bichromatic potential, the total photon scattering rate is
given by the sum of the individual rates, proportional to the imaginary parts
of the atomic polarizability, in the linear regime:

$\displaystyle\Gamma_{F,m_{F}}$ $\displaystyle=$
$\displaystyle\frac{1}{\hbar\epsilon_{0}c}\Big{(}\mathrm{Im}[\alpha_{F,m_{F}}(\delta_{\mathrm{P}})]I_{\mathrm{p}}$
(13)
$\displaystyle\quad\quad\quad+\;\mathrm{Im}[\alpha_{F,m_{F}}(\delta_{\mathrm{c}})]I_{\mathrm{c}}\Big{)}\mathrm{.}$

Table 1 shows the results of the numerical determination of the experimental
parameters for a bichromatic speckle potential corresponding to
$V_{\mathrm{R}}/h=$416\text{\,}\mathrm{Hz}$$. This value is typical for the
experimental study of the Anderson localization or the spectral functions (see
Refs. [7, 20]). For consistency, this value will also corresponds to the
experiments described in the next section.

Table 1: Comparison of the state-dependent disordered potential parameter between the monochromatic configuration and the bichromatic one, the crucial quantity being the photon scattering lifetime in state $\lvert 2\rangle$ ($\Gamma_{2}^{-1}$, bolded line). The determination of these quantities has been performed for a total disorder amplitude of $V_{\mathrm{R}}/h=$416\text{\,}\mathrm{Hz}$$. The quantities $\delta_{\mathrm{p}}$ and $\delta_{\mathrm{c}}$ are defined in figure 3 and $\delta$ is defined in figure 2. $P$, $P_{\mathrm{p}}$, and $P_{\mathrm{c}}$ correspond to the laser powers. Quantity | Monochromatic case | Bichromatic case
---|---|---
$\delta/2\pi$ | $81\text{\,}\mathrm{MHz}$ | —
$\delta_{\mathrm{p}}/2\pi$ | — | $95\text{\,}\mathrm{GHz}$
$\delta_{\mathrm{c}}/2\pi$ | — | $-1.40\text{\,}\mathrm{GHz}$
$\Gamma_{1}^{-1}$ | $26.6\text{\,}\mathrm{s}$ | $73\text{\,}\mathrm{ms}$
$\mathbf{\Gamma_{2}^{-1}}$ | $\mathbf{5.3\>ms}$ | $\mathbf{1.66\>s}$
$\sigma_{V_{1}}/h$ | $6.3\text{\,}\mathrm{Hz}$ | $11.4\text{\,}\mathrm{Hz}$
$P$ | $0.49\text{\,}\mathrm{\SIUnitSymbolMicro W}$ | —
$P_{\mathrm{p}}$ | — | $430\text{\,}\mathrm{\SIUnitSymbolMicro W}$
$P_{\mathrm{c}}$ | — | $4.6\text{\,}\mathrm{\SIUnitSymbolMicro W}$

Here, the detuning of the principal laser is set to
$\delta_{\mathrm{p}}/2\pi=$95\text{\,}\mathrm{GHz}$$. This value is chosen as
a compromise to get a sufficiently large detuning while the two speckle
patterns can be considered identical enough to permit the cancellation of the
total potential onto the disorder insensitive state. Then, the detuning
$\delta_{\mathrm{c}}$ of the compensation laser is determined so that the
lifetime of state $\lvert 1\rangle$ is larger than the duration of rf transfer
( for instance $t_{\mathrm{rf}}=$20\text{\,}\mathrm{ms}$$ for the experiment
shown in section IV). Here we have
$\delta_{\mathrm{c}}/2\pi=-$1.40\text{\,}\mathrm{GHz}$$ and
$\Gamma_{1}^{-1}=$73\text{\,}\mathrm{ms}$$. With these parameters, one can in
particular deduce the potential amplitude generated by both the laser speckle
fields on the state $\lvert 1\rangle$ and $\lvert 2\rangle$:
$V_{\mathrm{p,1}}/h=-V_{\mathrm{c,1}}/h=$366\text{\,}\mathrm{Hz}$$,
$V_{\mathrm{p,2}}/h=$348\text{\,}\mathrm{Hz}$$ and
$V_{\mathrm{c,2}}/h=$68\text{\,}\mathrm{Hz}$$.

The most notable result of Table 1 is the improvement of the scattering
lifetime $\Gamma_{2}^{-1}$ of the disorder-sensitive state $\lvert 2\rangle$
by more than two orders of magnitude, going from a few milliseconds for the
monochromatic configuration of [20] up to more than one second in the
bichromatic configuration. Additionally, the residual disordered potential
$\sigma_{V_{1}}$ applied to the disorder-insensitive state is of the same
order as in the monochromatic configuration, yielding the same state-
selectivity.

The analysis described here for the specific case of
$V_{\mathrm{R}}/h=$416\text{\,}\mathrm{Hz}$$ can be reproduced for a wide
range of disorder amplitudes, both for a globally attractive
($V_{\mathrm{R}}<0$) and repulsive ($V_{\mathrm{R}}>0$) disorder in state
$\lvert 2\rangle$, leading to the same conclusion. For instance, we checked
that we obtain similar improvements for the range of disorder amplitudes
$\lvert
V_{\mathrm{R}}/h\rvert\in[$40\text{\,}\mathrm{Hz}$,\,$4\text{\,}\mathrm{kHz}$]$
used in Volchkov et al. [20].

##  IV Experimental check of the bichromatic speckle configuration

In this section we present an evaluation of the new bichromatic disordered
potential scheme with ultracold atoms. More precisely, we focus on two main
aspects. Firstly, we check the efficient subtraction of the two speckle
potentials in the state $\lvert 1\rangle$, as described in Fig. 3, so that it
is merely insensitive to the disordered potential. This is done by studying
the mechanical excitation of the atoms in state $\lvert 1\rangle$ following a
quench of the disorder potential. Secondly, we realize the rf transfer
protocol as demonstrated in Volchkov et al. [20] (see Fig. 1) and measure the
spontaneous scattering lifetime of the atoms transferred in state $\lvert
2\rangle$. As predicted in table 1, we find a large improvement by two orders
of magnitude compared to the monochromatic speckle case, the lifetime being
now on the second time scale.

###  IV.1 Experimental set-up

The starting point of the experiment is the creation of a BEC of 87Rb atoms in
the hyperfine state $\lvert 1\rangle\equiv\lvert F=1,m_{F}=-1\rangle$. The
atoms are confined in a crossed optical dipole trap, formed by two orthogonal
laser beams at a wavelength of 1070 nm, in presence of a magnetic levitation.
As described in Ref. [51], the magnetic levitation is created by adding a
magnetic gradient, whose force acts against gravity, to the magic bias field
of 3.23 G (see section II.2.1).

The magnetic levitation enables us to end up the optical evaporation process
with a very decompressed trap configuration
($\omega_{y}\simeq$5\text{\,}\mathrm{Hz}$,\,\omega_{z}/2\pi\simeq$25\text{\,}\mathrm{Hz}$$
and $\omega_{x}/2\pi\simeq$30\text{\,}\mathrm{Hz}$$). At this stage, we obtain
a BEC with around 2$\times$105 atoms, corresponding to a chemical potential
around $\mu/h\simeq$250\text{\,}\mathrm{Hz}$$ and a Thomas-Fermi radii around
$R_{\mathrm{TF}}\sim$ 45, 10 and 8 $\text{\,}\mathrm{\SIUnitSymbolMicro m}$
along each direction. In the Thomas-Fermi regime, the meanfield interatomic
interactions compensate perfectly the trapping potential [51], resulting in an
overall flat potential for atoms in state $\lvert 1\rangle$ and in state
$\lvert 2\rangle$ in the absence of the disordered potential, as sketched in
Figs 1,2 and 3.

The laser speckle disorder geometry and parameters have been extensively
detailed in sections II.1 and III. Let us note, however, that the precise
disorder amplitude calibration is a well-known issue for experiments. As shown
in [51], an efficient method relies on direct comparison between directly
measured spectral functions and numerical calculations. The same method is
thus applied here using the spectral function obtained for our reference
disorder amplitude value $V_{\mathrm{R}}/h=$416\text{\,}\mathrm{Hz}$$ (see
section IV.3).

###  IV.2 Probing the insensitivity of state $\lvert 1\rangle$ to disorder
using a quench

In order to quantify the effect of the residual disorder in state $\lvert
1\rangle$, we introduce a quench protocol. To do so, we switch on abruptly the
total disordered potential (within $100\text{\,}\mathrm{\SIUnitSymbolMicro
s}$) and keep it on for $4\text{\,}\mathrm{ms}$. Then the disorder is switched
off and we measure the momentum distribution of the atoms in $\lvert 1\rangle$
using a standard time of flight technique of duration
$t_{\mathrm{ToF}}=$200\text{\,}\mathrm{ms}$$.

The evolution of the atomic momentum distribution is shown on figure 6, where
we scan the amplitude of the compensation potential from $\lvert
V_{\mathrm{c,1}}\rvert/h=0$ to $700\text{\,}\mathrm{Hz}$ ($V_{\mathrm{c,1}}$
being attractive), while the principal potential is kept constant
$V_{\mathrm{p,1}}/h=$366\text{\,}\mathrm{Hz}$$ ($V_{\mathrm{p,1}}$ being
repulsive). These parameters are chosen to explore the typical configuration
predicted in table 1, where the state $\lvert 2\rangle$ experiences the total
potential $V_{\mathrm{R}}/h=$416\text{\,}\mathrm{Hz}$$ while the disorder is
ideally suppressed in state $\lvert 1\rangle$ for
$V_{\mathrm{c,1}}=-V_{\mathrm{p,1}}$ (see section III.2).

For very low compensation amplitude, $\lvert V_{\mathrm{c,1}}\rvert\ll
V_{\mathrm{p,1}}$, the total potential in state $\lvert 1\rangle$ is
essentially given by the “principal” disorder potential. This repulsive
potential excites the BEC when the disorder is switched on, a part of the
potential energy being transfer to the kinetic energy. This extra kinetic
energy results in a broadening of the atomic momentum distribution (see dashed
horizontal line corresponding to the absence of disorder as a reference).

As the compensating potential is increased, the total disordered potential in
$\lvert 1\rangle$ decreases and the momentum distribution spread decreases
accordingly, with a minimum at the best compensation. When $\lvert
V_{\mathrm{c,1}}\rvert>V_{\mathrm{p,1}}$, the total potential $V_{1}$ turns to
attractive, and the disorder strength increases again. The momentum
distribution broadens then in this regime. The minimum is reached as expected
around $\lvert V_{\mathrm{c,1}}\rvert$ = $V_{\mathrm{p,1}}$ (vertical thin
dotted line), and, most importantly we observe a momentum spread exactly equal
to the one in the absence of disorder. This observation is a strong evidence
of the efficient compensation of the two disordered potentials, the residual
potential yielding no observable excitation of the atomic cloud in $\lvert
1\rangle$.

Figure 6: Evolution of the momentum distribution of the atoms in state $\lvert
1\rangle$ following a sudden switch on of the bichromatic speckle potential
(quench protocol). The amplitude of the principal potential amplitude is fixed
to $V_{\mathrm{p,1}}/h=$366\text{\,}\mathrm{Hz}$$, while the amplitude of
compensating potential is scanned from $\lvert V_{\mathrm{c,1}}\rvert/h$=0 to
$700\text{\,}\mathrm{Hz}$. The dots correspond to the atomic cloud r.m.s. size
measured by fluorescence imaging after a time of flight of 200 ms. The error
bars correspond to the rms uncertainties estimated over five experimental
realizations. For clarity the dots are colored in blue when the total disorder
amplitude $V_{1}=V_{\mathrm{p,1}}+V_{\mathrm{c,1}}$ is repulsive and in red
when it is attractive – see illustrations. The horizontal line corresponds to
the reference case where no disorder is applied to the atoms. The vertical
dotted line indicates the theoretical condition for an optimal cancellation of
the total disorder potential in state $\lvert 1\rangle$, that is $\lvert
V_{\mathrm{c,1}}\rvert$ = $V_{\mathrm{p,1}}$.

###  IV.3 Improved lifetime in the disorder sensitive state $\lvert 2\rangle$

The expected suppression of the disordered potential in state $\lvert
1\rangle$ being verified, we perform the rf transfer towards the disorder
sensitive state $\lvert 2\rangle$ at the energy defined by
$\delta_{\mathrm{rf}}=\omega_{\mathrm{rf}}-\Delta_{\mathrm{hf}}$ (see Fig. 1
and discussion in section II.2.1). Here we continue to investigate the
configuration of Table 1 and we set the bichromatic disorder parameters at the
minimum point of figure 6, that is for $\lvert V_{\mathrm{c,1}}\rvert/h$ = $\
V_{\mathrm{p,1}}/h=$366\text{\,}\mathrm{Hz}$$. The rf power is chosen low
enough to operate in the weak coupling regime where the transfer rate
$\Gamma(\delta_{\mathrm{rf}})$ is well predicted by the Fermi Golden rule
[51]. Moreover, the rf field is applied in the regime of $\Gamma
t_{\mathrm{rf}}\ll 1$ (with $t_{\mathrm{rf}}=$20\text{\,}\mathrm{ms}$$) so
that only a small fraction of the atoms are transferred to state $\lvert
2\rangle$ (not more than $25\%$ at most). As discussed in section III.3, note
that the rf duration is also chosen to be shorter than the lifetime in state
$\lvert 1\rangle$. In these conditions, the energy resolution is time Fourier-
limited to $\Delta E/h=1/t_{\mathrm{rf}}=$50\text{\,}\mathrm{Hz}$$.

Figure 7a shows the evolution of the transferred atom number as a function of
the rf detuning $\delta_{\mathrm{rf}}$, for a fixed rf power. As explained in
[20], the curve $\Gamma(\delta_{\mathrm{rf}})$ constitutes a direct
measurement of the spectral function for the disordered potential in state
$\lvert 2\rangle$. A comparison is shown with a numerical simulation for a
repulsive disorder of amplitude $V_{\mathrm{R}}/h=$416\text{\,}\mathrm{Hz}$$.
As said above this excellent agreement is used to calibrate precisely the
disorder amplitude in the experiments (with a 5$\%$ uncertainty).

Figure 7: Implementation of the rf transfer scheme and lifetime measurement
for $V_{\mathrm{R}}/h=$416\text{\,}\mathrm{H}\mathrm{z}$$. a) Normalized
transfer rate $\Gamma(\delta_{\mathrm{rf}})$ from disorder insensitive state
$\lvert 1\rangle$ to the disorder sensitive state $\lvert 2\rangle$. The
compensating and principal disorder amplitudes are set to $\lvert
V_{\mathrm{c,1}}\rvert/h$ = $\ V_{\mathrm{p,1}}/h=$366\text{\,}\mathrm{Hz}$$,
that is the optimal cancellation condition for the disorder in state $\lvert
1\rangle$ (see Fig. 6). The rf power is fixed and the rf field is applied
during $t_{\mathrm{rf}}=$20\text{\,}\mathrm{ms}$$. The blue dots are the
measured points and the red curve is the numerical calculation in [20], taking
into account the resolution $\Delta
E/h=1/t_{\mathrm{rf}}=$50\text{\,}\mathrm{Hz}$$. The squares corresponds to
the detunings $\delta_{\mathrm{rf}}$ chosen to measure the lifetime. b) Atom
number decays on the state $\lvert 2\rangle$ after the transfer for the
detunings $\delta_{\mathrm{rf}}/2\pi=210$ and $350\text{\,}\mathrm{Hz}$. The
fits with exponential decays yield the lifetime of
$1.85(5)\text{\,}\mathrm{s}$ and $1.67(6)\text{\,}\mathrm{s}$ respectively, in
good agreement with Table 1. For both figures, the error bars correspond to
the rms uncertainties estimated over five experimental realizations.

Last we study the scattering lifetime of the atoms transferred in the disorder
sensitive state $\lvert 2\rangle$ for the same configuration corresponding to
$V_{\mathrm{R}}/h=$416\text{\,}\mathrm{Hz}$$. In order to maximise the signal
to noise ratio we choose two rf detuning,
$\delta_{\mathrm{rf}}/2\pi=$210\text{\,}\mathrm{Hz}$$ and
$350\text{\,}\mathrm{Hz}$, close to the maximum of the transfer curve
$\Gamma(\delta_{\mathrm{rf}})$. The decay of the atoms number (once the rf
transfer field is switched off) are shown on figure 7b). The fits by
exponential curves yield very similar lifetime of respectively
$\tau_{210}=$1.85(5)\text{\,}\mathrm{s}$$ and
$\tau_{350}=$1.67(6)\text{\,}\mathrm{s}$$. These values are in very good
agreement with the prediction in table 1, thus validating our analysis. Most
importantly these values are larger than one second, which is crucial for our
ongoing investigation of the Anderson transition.

##  V Summary and outlook

In this paper, we have studied both theoretically and experimentally a new
scheme to produce a 3D state dependent disordered potential with a low
spontaneous photon scattering rate in the disordered sensitive state. It is
realized using a bichromatic speckle disorder whose properties have been
investigated in details, in particular to quantify the conditions under which
the two disordered potentials, created from two laser speckle field at two
slightly different wavelengths, can cancel efficiently together for the
insensitive state. Using this state dependent disorder potential and a rf
transfer from the insensitive state to the disorder sensitive state, one can
load ultracold atoms at a precise energy level in the disordered potential.
The transferred atoms have a typical photon scattering lifetime on the second
time scale, an improvement of two orders of magnitude compared to the
monochromatic speckle scheme [20]. We therefore expect that scheme to allow us
to effect precise measurements of the mobility edge and the critical exponents
the 3D Anderson transition, whose precise determination remains an utmost
experimental challenge [18].

It is worth noting that this scheme could be extended to other kind of optical
potentials used to study disordered systems, such as those created by spatial
light modulators as well as quasi-periodic lattices 444While the general
arguments exposed in section III.2 on the fundamental decorrelation of the
potentials produced by two different wavelength would be essentially similar,
each configuration would require specific calculations. A general analysis is,
however, beyond the scope of this paper., see e.g. [53, 54]. It opens also
many prospects to address fundamental questions related to the Anderson
transition, such as the observation of multifractalilty [55], comparison with
new theoretical approaches such as the one based on the “hidden landscape”
[21, 31], or general Anderson transition with different universality class
[56], dimensions [57], or even in the presence of non-hermitian disorder [58].

###### Acknowledgements.

The authors thanks Vincent Denechaud and Adrien Signoles for early discussions
and work. This work was supported by grant No. 601937 from the Simons
Foundation and by an “Investissements d’Avenir” grant from LabEx PALM
(ANR-10-LABX-0039-PALM). A.A. acknowledges support through the Augustin
Fresnel chair of the Institut d’Optique Graduate School, sponsored by Institut
d’Optique and supported by Nokia Bell labs. He also acknowledges support from
the iXcore-iXlife-IXblue foundation for research.

##  Appendix A Calculation of the bichromatic correlation function

###  A.1 Bichromatic correlation function of the diffuser

We extend the standard calculation of the diffuser correlation function to the
bichromatic case. The laser’s local phase fluctuations just after the diffuser
is expressed as:

$\phi(\mathbf{r}_{\mathrm{0}})=2\pi(n-1)\frac{\delta
e(\mathbf{r}_{\mathrm{0}})}{\lambda}$ (14)

where the points $\mathbf{r}_{\mathrm{0}}=\\{x_{0},y_{0},z=0\\}$ correspond to
the diffuser’s plane. $\delta e(\mathbf{r}_{\mathrm{0}})$ is the local
fluctuation of the thickness of the diffuser, $\lambda$ the laser wavelength
and $n$ stands for the glass index. Note that within this definition, we don’t
consider the averaged phase transmission, that we set to $\overline{\phi}=0$
for simplicity. Moreover, we assume the diffuser to have a gaussian-
distributed thickness with standard deviation $\sigma_{\mathrm{e}}$, resulting
in a phase standard deviation
$\sigma_{\phi}=2\pi(n-1)\sigma_{\mathrm{e}}/\lambda$. The diffuser
transmission is
$t_{\mathrm{diff}}(\mathbf{r}_{\mathrm{0}},\lambda)=e^{i\phi(\mathbf{r}_{\mathrm{0}})}$,
and its ensemble average writes :

$\overline{t_{\mathrm{diff}}}=\overline{e^{i\phi}}=\int{\mathrm{d}\phi\>e^{i\phi}\>\mathcal{P}(\phi)}=e^{-\sigma_{\phi}^{2}/2}$
(15)

with
$\mathcal{P}(\phi)=1/\sqrt{2\pi}\sigma_{\phi}\times\exp{(-\phi^{2}/2\sigma_{\phi}^{2})}$.

The bichromatic correlation function of the diffuser is defined as:

$C_{\mathrm{diff}}(\mathbf{r}_{\mathrm{0}},\mathbf{r}_{\mathrm{0}}^{\prime},\lambda_{\mathrm{p}},\lambda_{\mathrm{c}})=\overline{t_{\mathrm{diff}}(\mathbf{r}_{\mathrm{0}},\lambda_{\mathrm{p}})t_{\mathrm{diff}}^{*}(\mathbf{r}_{\mathrm{0}}^{\prime},\lambda_{\mathrm{c}})}\>.$
(16)

Note that the correlation function of the diffuser with a monochromatic
illumination is obtained by merely setting
$\lambda_{\mathrm{c}}=\lambda_{\mathrm{p}}$. Since the diffuser’s thickness is
a gaussian random variable, the phase difference
$\phi_{\mathrm{p}}(\mathbf{r}_{\mathrm{0}})-\phi_{\mathrm{c}}(\mathbf{r}_{\mathrm{0}}^{\prime})$
is a gaussian variable as well and we can calculate as in equation (15) to
obtain:

$\displaystyle
C_{\mathrm{diff}}(\mathbf{r}_{\mathrm{0}},\mathbf{r}_{\mathrm{0}}^{\prime},\lambda_{\mathrm{p}},\lambda_{\mathrm{c}})=\overline{e^{i(\phi_{\mathrm{p}}(\mathbf{r}_{\mathrm{0}})-\phi_{\mathrm{c}}(\mathbf{r}_{\mathrm{0}}^{\prime}))}}$
(17)
$\displaystyle=\exp{\left[-2\pi^{2}(n-1)^{2}\sigma_{\mathrm{e}}^{2}\left(\frac{1}{\lambda_{\mathrm{p}}^{2}}+\frac{1}{\lambda_{\mathrm{c}}^{2}}\right)\right]}$
$\displaystyle\times\exp{\left[4\pi^{2}(n-1)^{2}\frac{2}{\lambda_{\mathrm{p}}\lambda_{\mathrm{c}}}\overline{\delta
e(\mathbf{r}_{\mathrm{0}})\delta
e(\mathbf{r}_{\mathrm{0}}^{\prime})}\right]}\>.$ (18)

To characterize the granularity correlation function $\overline{\delta
e(\mathbf{r}_{\mathrm{0}})\delta e(\mathbf{r}_{\mathrm{0}}^{\prime})}$ of the
diffuser, we introduce $r_{\mathrm{e}}$, which is the width of the thickness
correlation function and describes the typical size of the transverse
granularity of the diffuser’s surface. By assuming a wide phase distribution
$\sigma_{\phi}\gg 2\pi$ (equivalently $\sigma_{\mathrm{e}}\gg\lambda$), the
wave oscillates many times in a single granularity and the diffuser’s
thickness correlation function can be approximated by a bell-shaped curve:

$\frac{\overline{\delta e(\mathbf{r}_{\mathrm{0}})\delta
e(\mathbf{r}_{\mathrm{0}}^{\prime})}}{\sigma_{\mathrm{e}}^{2}}\approx
1-\frac{(\mathbf{r}_{\mathrm{0}}-\mathbf{r}_{\mathrm{0}}^{\prime})^{2}}{2r_{\mathrm{e}}^{2}}$
(19)

when
$\left\lvert\mathbf{r}_{\mathrm{0}}-\mathbf{r}_{\mathrm{0}}^{\prime}\right\rvert\ll
r_{\mathrm{e}}$. Finally, the bichromatic diffuser correlation function reads
as:

$\displaystyle
C_{\mathrm{diff}}(\mathbf{r}_{\mathrm{0}},\mathbf{r}_{\mathrm{0}}^{\prime},\lambda_{\mathrm{p}},\lambda_{\mathrm{c}})$
$\displaystyle=\exp{\left(-\frac{\sigma_{\Delta\phi}^{2}}{2}\right)}$
$\displaystyle\times\exp{\left(-\frac{(\mathbf{r}_{\mathrm{0}}-\mathbf{r}_{\mathrm{0}}^{\prime})^{2}}{2r_{\mathrm{diff,p}}r_{\mathrm{diff,c}}}\right)}$
(20)

where
$\sigma_{\Delta\phi}^{2}=(\sigma_{\phi_{\mathrm{p}}}-\sigma_{\phi_{\mathrm{c}}})^{2}$
is the variance of the local phase difference
$\Delta\phi(\mathbf{r}_{\mathrm{0}})=\phi_{\mathrm{p}}(\mathbf{r}_{\mathrm{0}})-\phi_{\mathrm{c}}(\mathbf{r}_{\mathrm{0}})$
and where we introduced the diffuser’s phase correlation length
$r_{\mathrm{diff,p,c}}=r_{\mathrm{e}}/\sigma_{\phi,p,c}$ (respectively for the
principal and compensating lasers).

###  A.2 Correlation function of the amplitude

We consider here the general two-point correlation function of bichromatic
speckle field, at any positions $\mathbf{r}$ and $\mathbf{r}^{\prime}$ and for
different wavelengths $\lambda_{\mathrm{p}}$ and $\lambda_{\mathrm{c}}$.
Within the paraxial approximation, it is expressed as

$\displaystyle\Gamma_{\mathcal{A}}(\mathbf{r},\mathbf{r}^{\prime},\lambda_{\mathrm{p}},\lambda_{\mathrm{c}})=\overline{\mathcal{A}(\mathbf{r},\lambda_{\mathrm{p}})\mathcal{A}^{*}(\mathbf{r}^{\prime},\lambda_{\mathrm{c}})}$
$\displaystyle=\frac{e^{ik_{\mathrm{p}}\left(\overline{z}+\frac{x^{2}+y^{2}}{2\overline{z}}\right)}e^{-ik_{\mathrm{c}}\left(\overline{z}^{\prime}+\frac{x^{\prime
2}+y^{\prime
2}}{2\overline{z}^{\prime}}\right)}}{\lambda_{\mathrm{p}}\lambda_{\mathrm{c}}\overline{z}\overline{z}^{\prime}}$
$\displaystyle\times\displaystyle\int{\mathrm{d}\mathbf{r}_{\mathrm{0}}\mathrm{d}\mathbf{r}_{\mathrm{0}}^{\prime}\>\overline{t_{\mathrm{diff}}(\mathbf{r}_{\mathrm{0}},\lambda_{\mathrm{p}})t_{\mathrm{diff}}^{*}(\mathbf{r}_{\mathrm{0}}^{\prime},\lambda_{\mathrm{c}})}}$
$\displaystyle\times\mathcal{A}(\mathbf{r}_{\mathrm{0}},\lambda_{\mathrm{p}})\mathcal{A}^{*}(\mathbf{r}_{\mathrm{0}}^{\prime},\lambda_{\mathrm{c}})$
$\displaystyle\times
e^{ik_{\mathrm{p}}\frac{\mathbf{r}_{\mathrm{0}}^{2}}{2d_{\mathrm{eff}}}}e^{-ik_{\mathrm{c}}\frac{\mathbf{r}_{\mathrm{0}}^{\prime
2}}{2d_{\mathrm{eff}}^{\prime}}}e^{-ik_{\mathrm{p}}\frac{\mathbf{r}.\mathbf{r}_{\mathrm{0}}}{\overline{z}}}e^{ik_{\mathrm{c}}\frac{\mathbf{r}^{\prime}.\mathbf{r}_{\mathrm{0}}^{\prime}}{\overline{z}^{\prime}}}$
(21)

where $\mathcal{A}(\mathbf{r}_{0})$ is the amplitude just before the diffuser,
and the ensemble average $\overline{\>\cdots\>}$ only acts on the
transmissions $t_{\mathrm{diff}}$. The effective distances $d_{\mathrm{eff}}$
are defined with respect to the focal plane by
$1/d_{\mathrm{eff}}=1/\overline{z}-1/f$. Last $\overline{z}=z+f$ is the
longitudinal distance compared to the diffuser ($z$ being the distance to the
Fourier plane, see Fig. 4).

Using the change of variables
$\\{\mathbf{r}_{\mathrm{0}},\mathbf{r}_{\mathrm{0}}^{\prime}\\}\rightarrow\\{\mathbf{r}_{\mathrm{c,0}}=(\mathbf{r}_{\mathrm{0}}+\mathbf{r}_{\mathrm{0}}^{\prime})/2,\>\Delta\mathbf{r}_{0}=\mathbf{r}_{\mathrm{0}}^{\prime}-\mathbf{r}_{\mathrm{0}}\\}$,
we obtain:

$\displaystyle\Gamma_{\mathcal{A}}(\mathbf{r},\mathbf{r}^{\prime},\lambda_{\mathrm{p}},\lambda_{\mathrm{p}})\propto\frac{e^{ik_{\mathrm{p}}\left(\overline{z}+\frac{x^{2}+y^{2}}{2\overline{z}}\right)}e^{-ik_{\mathrm{c}}\left(\overline{z}^{\prime}+\frac{x^{\prime
2}+y^{\prime
2}}{2\overline{z}^{\prime}}\right)}}{\lambda_{\mathrm{p}}\lambda_{\mathrm{c}}\overline{z}\overline{z}^{\prime}}$
$\displaystyle\times\displaystyle\int{\mathrm{d}\mathbf{r}_{\mathrm{c,0}}\mathrm{d}\Delta\mathbf{r}_{0}\>C_{\mathrm{diff}}(\Delta\mathbf{r}_{0},\lambda_{\mathrm{p}},\lambda_{\mathrm{c}})}$
$\displaystyle\times\mathcal{A}(\mathbf{r}_{\mathrm{c,0}}-\Delta\mathbf{r}_{0}/2,\lambda_{\mathrm{p}})\mathcal{A}^{*}(\mathbf{r}_{\mathrm{c,0}}+\Delta\mathbf{r}_{0}/2,\lambda_{\mathrm{c}})$
$\displaystyle\times
e^{i\mathbf{r}_{\mathrm{c,0}}.(\frac{k_{\mathrm{c}}\mathbf{r}^{\prime}}{\overline{z}^{\prime}}-\frac{k_{\mathrm{p}}\mathbf{r}}{\overline{z}})}e^{i\frac{\Delta\mathbf{r}_{0}}{2}.(\frac{k_{\mathrm{p}}\mathbf{r}}{\overline{z}}+\frac{k_{\mathrm{c}}\mathbf{r}^{\prime}}{\overline{z}^{\prime}})}e^{i\frac{\mathbf{r}_{\mathrm{c,0}}^{2}}{2}(\frac{k_{\mathrm{p}}}{d_{\mathrm{eff}}}-\frac{k_{\mathrm{c}}}{d_{\mathrm{eff}}^{\prime}})}$
$\displaystyle\times
e^{-i\frac{\Delta\mathbf{r}_{0}.\mathbf{r}_{\mathrm{c,0}}}{2}(\frac{k_{\mathrm{p}}}{d_{\mathrm{eff}}}+\frac{k_{\mathrm{c}}}{d_{\mathrm{eff}}^{\prime}})}e^{i\frac{\Delta\mathbf{r}_{0}^{2}}{8}(\frac{k_{\mathrm{p}}}{d_{\mathrm{eff}}}-\frac{k_{\mathrm{c}}}{d_{\mathrm{eff}}^{\prime}})}\>.$
(22)

Let us now suppose that the typical size of the diffuser grains is small
compared to the size of the incoming illumination $I(\mathbf{r}_{0})$, meaning
that on the size of $C_{\mathrm{diff}}$, the incoming illumination is
constant. Let us also suppose that the two incoming beams are in the same
spatial mode:
$\mathcal{A}(\mathbf{r}_{\mathrm{0}}-\Delta\mathbf{r}_{0}/2,\lambda_{\mathrm{p}})\mathcal{A}^{*}(\mathbf{r}_{\mathrm{0}}+\Delta\mathbf{r}_{0}/2,\lambda_{\mathrm{c}})\approx\mathcal{A}(\mathbf{r}_{\mathrm{0}},\lambda_{\mathrm{p}})\mathcal{A}^{*}(\mathbf{r}_{\mathrm{0}},\lambda_{\mathrm{c}})=I(\mathbf{r}_{\mathrm{0}})$
(for simplicity we discard from now the c subscript from
$\mathbf{r}_{\mathrm{c,0}}$). We find:

$\displaystyle\Gamma_{\mathcal{A}}(\mathbf{r},\mathbf{r}^{\prime},\lambda_{\mathrm{p}},\lambda_{\mathrm{c}})=\frac{e^{ik_{\mathrm{p}}\left(\overline{z}+\frac{x^{2}+y^{2}}{2\overline{z}}\right)}e^{-ik_{\mathrm{c}}\left(\overline{z}^{\prime}+\frac{x^{\prime
2}+y^{\prime
2}}{2\overline{z}^{\prime}}\right)}}{\lambda_{\mathrm{p}}\lambda_{\mathrm{c}}\overline{z}\overline{z}^{\prime}}$
$\displaystyle\times\displaystyle\int{\mathrm{d}\mathbf{r}_{\mathrm{0}}\>I(\mathbf{r}_{\mathrm{0}})e^{i\frac{\mathbf{r}_{\mathrm{0}}^{2}}{2}(\frac{k_{\mathrm{p}}}{d_{\mathrm{eff}}}-\frac{k_{\mathrm{c}}}{d_{\mathrm{eff}}^{\prime}})}e^{i\mathbf{r}_{\mathrm{0}}.(\frac{k_{\mathrm{c}}\mathbf{r}^{\prime}}{\overline{z}^{\prime}}-\frac{k_{\mathrm{p}}\mathbf{r}}{\overline{z}})}}$
$\displaystyle\times\displaystyle\int{\mathrm{d}\Delta\mathbf{r}_{0}\>C_{\mathrm{diff}}(\Delta\mathbf{r}_{0},\lambda_{\mathrm{p}},\lambda_{\mathrm{c}})\>e^{-i\frac{\Delta\mathbf{r}_{0}.\mathbf{r}_{\mathrm{0}}}{2}(\frac{k_{\mathrm{p}}}{d_{\mathrm{eff}}}+\frac{k_{\mathrm{c}}}{d_{\mathrm{eff}}^{\prime}})}}$
$\displaystyle\times
e^{i\frac{\Delta\mathbf{r}_{0}}{2}.(\frac{k_{\mathrm{p}}\mathbf{r}}{\overline{z}}+\frac{k_{\mathrm{c}}\mathbf{r}^{\prime}}{\overline{z}^{\prime}})}\>.$
(23)

###  A.3 3D monochromatic correlation close to the Fourier plane

In this part, we give the explicit calculation of the spatial autocorrelation
function of a monochromatic laser speckle field, i.e., at two different points
$\mathbf{r},\mathbf{r}^{\prime}$ but at the same wavelength
$\lambda_{\mathrm{p}}=\lambda_{\mathrm{c}}=\lambda$. Such derivation is very
well-known [37] but it will be used for the derivation of the bichromatic
correlation function in a simple form, which is an important result of the
paper. The two point autocorrelation function is defined as

$c_{\mathrm{3D}}(\Delta\mathbf{r}_{\perp},\Delta z)=\frac{\overline{\delta
I(\mathbf{r}_{\perp}+\Delta\mathbf{r}_{\perp},z+\Delta z)\delta
I(\mathbf{r}_{\perp},z)}}{\overline{\delta I^{2}}}$ (24)

with $\mathbf{r}_{\perp}=\\{x,y\\}$. Close to the Fourier plane and to the
optical axis, the numerator reads as $\overline{\delta
I(\Delta\mathbf{r}_{\perp},\Delta z)\delta I(\mathbf{0},0)}$ and is computed
using Wick’s theorem:

$\displaystyle\overline{\delta I(\Delta\mathbf{r}_{\perp},\Delta z)\delta
I(\mathbf{0},0)}=\lvert\Gamma_{\mathcal{A}}(\Delta\mathbf{r}_{\perp},\Delta
z)\rvert^{2}$
$\displaystyle=\left\lvert\overline{\mathcal{A}(\Delta\mathbf{r}_{\perp},\Delta
z)\mathcal{A}^{*}(\mathbf{0},0)}\right\rvert^{2}$
$\displaystyle=\left\lvert\frac{1}{\lambda^{2}f^{2}}\displaystyle\int{\mathrm{d}\mathbf{r}_{\mathrm{0}}\>I(\mathbf{r}_{\mathrm{0}})\>e^{-i\frac{\mathbf{r}_{\mathrm{0}}^{2}k\Delta
z}{2f^{2}}}\>e^{-i\frac{k\mathbf{r}_{\mathrm{0}}.\Delta\mathbf{r}_{\perp}}{f}}}\right.$
$\displaystyle\times\left.\displaystyle\int{\mathrm{d}\Delta\mathbf{r}_{0}\>C_{\mathrm{diff}}(\Delta\mathbf{r}_{0})\>e^{i\frac{\Delta\mathbf{r}_{0}.\mathbf{r}_{\mathrm{0}}k\Delta
z}{2f^{2}}}\>e^{i\frac{k\Delta\mathbf{r}_{0}.\Delta\mathbf{r}_{\perp}}{2f}}}\right\rvert^{2}\>.$
(25)

By comparing the different phase terms, we can neglect the exponentials of the
second integral. After normalization by
$\lvert\overline{I(\mathbf{0},0)}\rvert^{2}$, and identifying a Fourier
transform, we obtain the usual result [37]:

$c_{\mathrm{3D}}(\Delta\mathbf{r}_{\perp},\Delta
z)=\frac{{\left\lvert\mathrm{FT}\left[I(\mathbf{r}_{\mathrm{0}})\>e^{-i\frac{\mathbf{r}_{\mathrm{0}}^{2}k\Delta
z}{2f^{2}}}\right]_{\frac{k\Delta\mathbf{r}_{\perp}}{f}}\right\rvert}^{2}}{{\left\lvert\displaystyle\int{\mathrm{d}\mathbf{r}_{\mathrm{0}}\>I(\mathbf{r}_{\mathrm{0}})}\right\rvert}^{2}}\>.$
(26)

###  A.4 Bichromatic correlation function of the speckle

We consider now bichromatic correlation function
$c_{2\lambda}(\mathbf{r}_{\perp},z,\lambda_{\mathrm{p}},\lambda_{\mathrm{c}})$
between two speckle fields created at different wavelength speckle pattern at
a single point located by $\mathbf{r}=\\{{\mathbf{r}_{\perp},z\\}}$ compared
to the center of the Fourier plane. As discussed in the text, the correlation
between the two speckles disorder potentials is directly linked to this
function, see Eq. (11). It is defined as

$c_{2\lambda}(\mathbf{r}_{\perp},z,\lambda_{\mathrm{p}},\lambda_{\mathrm{c}})=\frac{\overline{\delta
I(\mathbf{r}_{\perp},z,\lambda_{\mathrm{p}})\delta
I(\mathbf{r}_{\perp},z,\lambda_{\mathrm{c}})}}{\overline{I(\mathbf{r}_{\perp},z,\lambda_{\mathrm{p}})}\>\overline{I(\mathbf{r}_{\perp},z,\lambda_{\mathrm{c}})}}\>.$
(27)

The correlation function in the numerator is computed using Wick’s theorem
$\overline{\delta I(\mathbf{r}_{\perp},z,\lambda_{\mathrm{p}})\delta
I(\mathbf{r}_{\perp},z,\lambda_{\mathrm{c}})}=\left\lvert\Gamma_{\mathcal{A}}(\mathbf{r}_{\perp},z,\lambda_{\mathrm{p}},\lambda_{\mathrm{c}})\right\rvert^{2}$,
with:

$\displaystyle\left\lvert\Gamma_{\mathcal{A}}(\mathbf{r}_{\perp},z,\lambda_{\mathrm{p}},\lambda_{\mathrm{c}})\right\rvert^{2}=\left\lvert\overline{\mathcal{A}(\mathbf{r}_{\perp},z,\lambda_{\mathrm{p}})\mathcal{A}^{*}(\mathbf{r}_{\perp},z,\lambda_{\mathrm{c}})}\right\rvert^{2}$
$\displaystyle=\left\lvert\frac{1}{\lambda_{\mathrm{p}}\lambda_{\mathrm{c}}z^{2}}\displaystyle\int{\mathrm{d}\mathbf{r}_{\mathrm{0}}\>I(\mathbf{r}_{\mathrm{0}})\>e^{-i\frac{\mathbf{r}_{\mathrm{0}}^{2}z}{2f^{2}}(k_{\mathrm{p}}-k_{\mathrm{c}})}\>}\right.$
$\displaystyle\times
e^{i\frac{\mathbf{r}_{\perp}.\mathbf{r}_{\mathrm{0}}}{f}(k_{\mathrm{c}}-k_{\mathrm{p}})}\displaystyle\int{\mathrm{d}\Delta\mathbf{r}_{0}\>C_{\mathrm{diff}}(\Delta\mathbf{r}_{0},\lambda_{\mathrm{p}},\lambda_{\mathrm{c}})}$
$\displaystyle\times\left.e^{i\frac{\Delta\mathbf{r}_{0}.\mathbf{r}_{\mathrm{0}}z}{f^{2}}\frac{k_{\mathrm{p}}+k_{\mathrm{c}}}{2}}\>e^{i\frac{\Delta\mathbf{r}_{0}.\mathbf{r}_{\perp}}{f}\frac{k_{\mathrm{p}}+k_{\mathrm{c}}}{2}}\right\rvert^{2}\>.$
(28)

Using (i) the same approximation as in Eq. (A.3) for the phase exponentials of
the second integral, (ii) the normalization by the average intensity profile
around the center $\lvert\overline{I(\mathbf{0},0)}\rvert^{2}$, and (iii)
$\delta\lambda=\lvert\lambda_{\mathrm{c}}-\lambda_{\mathrm{p}}\rvert\ll\lambda_{\mathrm{p,c}}\sim\lambda$,
we obtain:

$\displaystyle
c_{2\lambda}(\mathbf{r}_{\perp},z,\lambda_{\mathrm{p}},\lambda_{\mathrm{c}})$
$\displaystyle=\frac{\left\lvert\displaystyle\int{\mathrm{d}\Delta\mathbf{r}_{0}\>C_{\mathrm{diff}}(\Delta\mathbf{r}_{0},\lambda_{\mathrm{p}},\lambda_{\mathrm{c}})}\right\rvert^{2}}{\displaystyle\int{\mathrm{d}\Delta\mathbf{r}_{0}\>C_{\mathrm{diff}}(\Delta\mathbf{r}_{0},\lambda_{\mathrm{p}})}\displaystyle\int{\mathrm{d}\Delta\mathbf{r}_{0}\>C_{\mathrm{diff}}(\Delta\mathbf{r}_{0},\lambda_{\mathrm{c}})}}$
$\displaystyle\times\frac{\left\lvert\displaystyle\int{\mathrm{d}\mathbf{r}_{\mathrm{0}}\>I(\mathbf{r}_{\mathrm{0}})\>e^{-ik\frac{\mathbf{r}_{\mathrm{0}}^{2}z}{2f^{2}}\frac{\delta\lambda}{\lambda}}\>e^{ik\frac{\mathbf{r}_{\perp}.\mathbf{r}_{\mathrm{0}}}{f}\frac{\delta\lambda}{\lambda}}}\right\rvert^{2}}{\left\lvert\displaystyle\int{\mathrm{d}\mathbf{r}_{\mathrm{0}}\>I(\mathbf{r}_{\mathrm{0}})}\right\rvert^{2}}$
(29)

and consists of two terms. The first one only features the diffuser while the
second describes the free space propagation after the diffuser. The first term
can be computed using:

$\int{\mathrm{d}\Delta\mathbf{r}_{0}\>C_{\mathrm{diff}}(\Delta\mathbf{r}_{0},\lambda)}=2\pi
r_{\mathrm{diff}}^{2}$ (30)

The normalized bichromatic correlation function finally reads as:

$\displaystyle
c_{2\lambda}(\mathbf{r}_{\perp},z,\lambda_{\mathrm{p}},\lambda_{\mathrm{c}})$
$\displaystyle=C_{\mathrm{diff}}^{2}(\mathbf{0},\lambda_{\mathrm{p}},\lambda_{\mathrm{c}})\frac{\left\lvert\mathrm{FT}\left[I(\mathbf{r}_{\mathrm{0}})\>e^{-ik\frac{\mathbf{r}_{\mathrm{0}}^{2}z}{2f^{2}}}\right]_{\frac{k\mathbf{r}_{\perp}}{f}\frac{\delta\lambda}{\lambda}}\right\rvert^{2}}{\left\lvert\displaystyle\int{\mathrm{d}\mathbf{r}_{\mathrm{0}}\>I(\mathbf{r}_{\mathrm{0}})}\right\rvert^{2}}$
$\displaystyle=e^{-\sigma_{\Delta\phi}^{2}}\>c_{\mathrm{3D}}\left(\mathbf{r}_{\perp}\frac{\delta\lambda}{\lambda},z\frac{\delta\lambda}{\lambda}\right)$
(31)

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