# Moving binary Bose-Einstein condensates in a weak random potential

Abdelâali Boudjemâa  Department of Physics, Faculty of Exact Sciences and
Informatics, Hassiba Benbouali University of Chlef P.O. Box 78, 02000, Ouled
Fares, Chlef, Algeria<EMAIL_ADDRESS>
###### Abstract

We study the behavior of moving Bose-Bose mixtures in a weak disordered
potential in the realm of the Bogoliubov-Huang-Meng theory. Corrections due to
the quantum fluctuations, disorder effects and the relative motion of two
fluids to the glassy fraction, the condensed depletion, the anomalous density,
and the equation of state of each species are obtained analytically for small
velocity. We show that the intriguing interplay of the relative motion and the
disorder potential could not only change the stability condition, but destroy
also the localization process in the two condensates preventing the formation
of a Bose glass state. Unexpectedly, we find that the quantum fluctuations
reduce with the velocity of the two fluids. The obtained theoretical
predictions are checked by our numerical results.

Recently, quantum Bose mixtures have afforded a fascinating glimpse into the
cold atoms world since they display rich phase separation behavior. Mixtures
of two species, or two isotopes of a single species, or two-component Bose-
Einstein condensates (BECs) were all successfully realized experimentally Mya
; Hall ; Mad ; Cab ; Sem ; Pap ; Sug ; Mog ; McC ; Ler ; Pasq ; Wack ; Wang ;
Igor . One of the most elusive feature of such ensembles is their miscibility.
At zero temperature, the miscibility-immiscibility phase transition depends on
the ratio of the intra- and interspecies interactions via the miscibility
parameter $\Delta=g_{1}g_{2}/g_{12}^{2}$, where $(g_{1},g_{2})$ and $g_{12}$
are the intraspecies and interspecies scattering lengths, respectively. For
$\Delta>1$, the two condensates overlap in space while they phase separate
when $\Delta<1$. However, at finite temperature, deviation from this condition
has been very recently predicted in Refs.Boudj ; Boudj1 ; Ota .

It has been found that binary BECs Hall ; Boudj ; Sinatra ; Lee  have complex
motion that tended to preserve the total density but quickly damped to a
stationary state with non-negligible component overlap. These moving wave
packets of BECs have the possibility to collide and move through each other
Hall ; Koz ; Hag ; Mad ; Band . The influence of the relative motion of two
species on the stability condition of the system has been analyzed in Ref.Yuk
.

On the other hand, Bose-Bose mixtures in random potentials have attracted
immense attention in recent years Wehr ; Nied ; Xi ; Mard ; BB . Quite
recently, we have found that the competition between the interspecies
interactions and the disorder potential may affect the localization, the
stability condition, the thermodynamics, and the superfluid density in each
component of the mixture BB .

In this letter, we tackle the problem of a dirty moving symmetric Bose mixture
gas using the Bogoliubov-Huang-Meng theory HM . Such a system can shed a new
light on the quantum transport of collective excitations due to an
unprecedented control of interspecies interactions and the possibilities of
conception controlled random. Moreover, disordered Bose mixtures with moving
components may offer also a unique opportunity to study Anderson localization.
They also play an important role in the fundamental understanding of the
intra- and interspecies interactions effects on the diffusion of expanding
condensates.

The aim of this work is to study the impacts of a weak disorder potential with
delta-correlated disorder on the properties of moving Bose mixtures. The
velocity of each component is assumed to be considerably less than the speed
of sound in the mixture liquid. Based on the Bogoliubov theory, analytical
formulas for the relevant physical quantities such as the glassy density, the
quantum depletion, the anomalous density, and the equation of state (EoS) of
each component are derived. Condition under which the theory works is properly
established. Our results reveal that the relative motion of two BECs may
reduce the localization of atoms and the quantum fluctuations in each species.
This leads to prohibit the formation of the glass state and hence, allows the
emergence of weakly correlated and ultradilute BECs. It is shown that the
interplay between inter and intra-species interaction tends to increase the
glassy fraction and the condensed depletion, while it lowers the anomalous
density. We demonstrate in addition that the EoS rises with both the velocity
and the interactions. Our numerical calculations confirm this scenario for
parameters relevant to the recent experiments of Ref.Cab .

Consider two-component BECs with equal masses, $m_{1}=m_{2}=m$, in a weak
disorder potential $U({\bf r})$, labeling the components by the index $j$. A
quantitative condition for weak disorder will be determined later on.
Bogoliubov quasiparticles and disorder-induced fluctuations decouple in the
lowest order HM , implying that at zero temperature and in the dilute regime,
the dynamics of such a system can be governed by the coupled Gross-Pitaevskii
equations (GPE) BB

$\displaystyle i\hbar\frac{\partial\phi_{j}({\bf r},t)}{\partial t}$
$\displaystyle=\bigg{(}\frac{-\hbar^{2}}{2m}\nabla^{2}+U({\bf
r})+g_{j}|\phi_{j}({\bf r},t)|^{2}$ (1) $\displaystyle+g_{12}|\phi_{3-j}({\bf
r},t)|^{2}\bigg{)}\phi_{j}({\bf r},t),$

where the intra- and the interspecies coupling constants are given,
respectively in terms of the intra- and interspecies $s$-wave scattering
length by $g_{j}=4\pi\hbar^{2}a_{j}/m_{j}$ and $g_{12}=2\pi\hbar^{2}a_{12}/m$.
The disorder potential should fulfil the following statistical properties
$\langle U(\mathbf{r})\rangle=0$, and $\langle
U(\mathbf{r})U(\mathbf{r}^{\prime})\rangle=R(\mathbf{r},\mathbf{r}^{\prime})$,
where $\langle\bullet\rangle$ denotes the disorder ensemble average and
$R(\mathbf{r},\mathbf{r}^{\prime})$ is the disorder correlation function.

Now we assume a mixture of moving components, where each component moves with
the same velocity $v_{1}=v_{2}=v$. The motion of the mixture is done by means
of the Galilean transformation:

$\phi({\bf r},t)=\phi({\bf r},t)\exp{\left(\frac{i}{\hbar}m_{j}{\bf v}.{\bf
r}\right)}.$ (2)

Introducing Eq.(2) into Eq.(1), the above coupled GPE can be rewritten as
follows:

$\displaystyle i\hbar\frac{\partial\phi_{j}({\bf r},t)}{\partial t}$
$\displaystyle=\bigg{(}\frac{p^{2}}{2m_{j}}+{\bf p}.{\bf
v}+\frac{1}{2}m_{j}v^{2}+U({\bf r})$ (3) $\displaystyle+g_{j}|\phi_{j}({\bf
r},t)|^{2}+g_{12}|\phi_{3-j}({\bf r},t)|^{2}\bigg{)}\phi_{j}({\bf r},t),$

where ${\bf p}=-i\hbar{\bf\nabla}$ is the kinetic-energy operator.

The calculation of the Bogoliubov spectrum and quantum fluctuations for moving
Bose mixtures amounts to solving the Bogoluibov-de-Gennes equations. To do so,
we linearize Eqs.(3) around static solutions $\phi_{0j}$ using the
transformation:

$\phi_{j}({\bf r},t)=\left[\phi_{0j}({\bf r})+\delta\phi_{j}({\bf
r},t)\right]\exp{\left(-i\mu_{j}t/\hbar\right)},$ (4)

where $\mu_{j}$ are chemical potentials related with bosonic components, and
$\delta\phi_{j}({\bf r},t)\ll\phi_{0j}$ are small quantum fluctuations. Let us
consider the case of a uniform mixture and focus on the situation where the
external potential is very weak, hence $U({\bf r})$ describes the local
spatial fluctuations around the homogeneous background Gaul . Substituting
Eq.(4) into Eq.(3) and keep in mind that the wavefunctions are real-valued
($\phi_{0j}=\phi_{0j}^{*}=\sqrt{n_{j}}$, and
$\phi_{0\,3-j}=\phi_{0\,3-j}^{*}=\sqrt{n_{3-j}}$), we obtain up to zeroth-
order

$\displaystyle\mu_{j}\sqrt{n_{j}}$
$\displaystyle=\bigg{[}\frac{p^{2}}{2m_{j}}+{\bf p}.{\bf
v}+\frac{1}{2}m_{j}v^{2}+U+g_{j}n_{j}$ (5)
$\displaystyle+g_{12}n_{3-j}\bigg{]}\sqrt{n_{j}}.$

The first-order terms yield

$\displaystyle i\hbar\frac{\partial\delta\phi_{j}({\bf r},t)}{\partial t}$
$\displaystyle=\bigg{(}\frac{p^{2}}{2m_{j}}+{\bf p}.{\bf
v}+\frac{1}{2}m_{j}v^{2}+U+2g_{j}n_{j}$ (6)
$\displaystyle+g_{12}n_{3-j}-\mu\bigg{)}\delta\phi_{j}({\bf
r},t)+g_{j}n_{j}\delta\phi_{j}^{*}({\bf r},t)$
$\displaystyle+g_{12}\sqrt{n_{j}n_{3-j}}\left[\delta\phi_{3-j}^{*}({\bf
r},t)+\delta\phi_{3-j}({\bf r},t)\right].$

To calculate the energy of elementary excitations, we introduce
transformation:

$\displaystyle\delta\phi_{j}({\bf r},t)$ $\displaystyle=u_{jp}\exp({i{\bf
p.r}/\hbar-i\varepsilon_{p}t/\hbar})$ (7) $\displaystyle+v_{jp}\exp({i{\bf
p.r}/\hbar+i\varepsilon_{p}t/\hbar})-\beta_{jp},$

where $u_{jp}$ and $v_{jp}$ are the Bogoliubov amplitudes,
$\beta_{jp}=\sqrt{n_{j}/V}\left(|u_{jp}+v_{jp}|^{2}/\varepsilon_{p}\right)U_{p}$
Yuk , where $U_{p}$ is the Fourier transform of the external potential $U({\bf
r})$ and $V$ denotes the volume. For two decoupled BECs, Eq.(7) reduces to the
standard Huang-Meng transformation HM . Thus, the chemical potential takes the
form:

$\mu_{j}=\frac{1}{2}m_{j}v^{2}+U+g_{j}n_{j}+g_{12}n_{3-j}.$ (8)

Inserting Eqs.(8) and (7) into Eq.(6), we obtain the coupled Bogoluibov-de-
Gennes equations for a mixture with moving species:

$\displaystyle\varepsilon_{p}u_{jp}$ $\displaystyle=\left(E_{p}+{\bf p}.{\bf
v}+g_{j}|\phi_{j}|^{2}\right)u_{jp}+g_{j}n_{j}v_{jp}$ (9)
$\displaystyle+g_{12}\sqrt{n_{j}n_{3-j}}(u_{jp}+v_{jp}),$

and

$\displaystyle-\varepsilon_{p}v_{jp}$ $\displaystyle=\left(E_{p}+{\bf p}.{\bf
v}+g_{j}|\phi_{j}|^{2}\right)v_{jp}+g_{j}n_{j}u_{jp}$ (10)
$\displaystyle+g_{12}\sqrt{n_{j}n_{3-j}}(u_{jp}+v_{jp}),$

where $E_{p}=p^{2}/2m$ is the free particle energy.

From now on we assume a symmetric mixture with $n_{1}=n_{2}=n$ and
$g_{1}=g_{2}=g$. A straightforward calculation leads to the following
expressions for the quasiparticle amplitudes $u_{p\pm},v_{p\pm}$:

$u_{p\pm},v_{p\pm}=\frac{1}{2}\left(\sqrt{\frac{\varepsilon_{p\pm}}{E_{p}+pv\cos\theta}}\pm\sqrt{\frac{E_{p}+pv\cos\theta}{\varepsilon_{p\pm}}}\right),$
(11)

and the gapless excitations spectrum of the system reads

$\varepsilon_{p\pm}=\sqrt{\left(E_{p}+pv\cos\theta\right)^{2}+2mc_{s\pm}^{2}\left(E_{p}+pv\cos\theta\right)},$
(12)

where $c_{s+}=c_{s}\sqrt{\delta g_{+}/g}$, and $c_{s-}=c_{s}\sqrt{\delta
g_{-}/g}$ are the sound velocities in the density and spin channels Ota19 ,
respectively, with $c_{s}=\sqrt{gn/m}$ being the usual sound velocity of a
single BEC, $\delta g_{\pm}=g(1\pm g_{12})$, and $\theta$ is the angle between
the two vectors ${\bf p}$ and ${\bf v}$. For $v=0$, one reproduces the
standard density and spin excitations for immovable Bose mixtures:
$\varepsilon_{p\pm}=p\sqrt{(p/2m)^{2}+c_{s\pm}}$ Boudj2 . The stability of the
spectrum (12) requires the conditions: $\delta g_{\pm}>0$ and
$\theta=2\ell\pi$, with $\ell$ being an integer or equivalently, $\delta
g_{\pm}<0$ and $\theta=(2\ell+1)\pi$. Note that the stability condition could
also be modified as a result of the relative motion of two liquids Yuk . For
quantum droplets with $\delta g_{+}<0$, the $\varepsilon_{p+}$ density energy
is imaginary. In this regime, the mean-field theory would result in energy
$\propto n^{2}$ leading to a collapse of a homogeneous state. The beyond mean-
field Lee-Huang-Yang (LHY) corrections which provide a repulsive term $\propto
n^{5/2}$ eliminates the mechanical instability (see e.g. Petrov ; Boudj2  and
references therein).

Similarly to the single disordered BEC, the spectrum (12) is independent of
the disorder potential. This independence justifies the neglect of the
noncondensed and anomalous fluctuations in the Bogoliubov quasiparticle
spectrum. Such an assumption is necessary to guarantee that the excitations
spectrum has no energy gap HM ; Mich . Obviously, if one takes into
consideration higher-order quantum fluctuations in the spectrum Gaul ; Boudj ;
Boudj1 ; Yuk , it would depend on the random potential.

Figure 1: (a) Bogoliubov excitations energy $\varepsilon_{p+}$ from Eq.(12),
for $\theta=\pi$, $a_{12}/a=0.5$ and $v/c_{s}=0.5$. Solid lines correspond to
the real part of the spectrum. Dashed lines correspond to the imaginary part
of the spectrum. (b) The same as Fig.1.(a) but for $v/c_{s}=1.5$. (c)
Bogoliubov excitations energy $\varepsilon_{p+}$ for $\theta=2\pi$,
$a_{12}/a=0.5$ and several values of $v/c_{s}$. Black line: $v/c_{s}=0$. Red
dotted line: $v/c_{s}=0.5$. Blue dashed line: $v/c_{s}=1.5$. Figs.1.(d)-(f)
are the same as Figs.1.(a)-(c) but for $\varepsilon_{p-}$.

In Fig.1 we plot the spectrum (12) for different values of the relative
velocity and the angle $\theta$. The dispersion $\varepsilon_{p\pm}$ has an
imaginary part for $\theta=\pi$ regardless of the value of the velocity and
the strength of the interspecies interaction. The spatially modulated
perturbations grow with ratio $v/c_{s}$ as is seen in Figs.1.(a), (b), (d) and
(e). The fastest growth occurs for the wave number
$p_{max}=\partial\varepsilon_{p\pm}/\partial p=0$ that gives a maximum of
$\varepsilon_{p\pm}$. One should stress that the interspecies interaction
$a_{12}/a$ may also affect the modulation of $\varepsilon_{p\pm}$. Figures
1.(c) and (f) show that for $\theta=2\pi$, the Bogoliubov spectrum
$\varepsilon_{p\pm}=\sqrt{\left(E_{p}+pv\right)^{2}+2mc_{s\pm}^{2}\left(E_{p}+pv\right)}$
is stable in the whole range of the momentum $p$. We observe also that the
relative velocity $v/c_{s}$ rises the spectrum $\varepsilon_{p\pm}$ leading to
diminish the density of noncondensed atoms and the anomalous density (see
below).

We assume henceforth that $\delta g_{\pm}>0$ and $\theta=2\ell\pi$. In this
case, the minimum value possible for $v$ which must be kept positive, can be
obtained in lowest value of $p$ i.e in the phonon regime:
$v_{c}=c_{s}\sqrt{\delta g_{\pm}/g}$, below which there is no solution for
$v$.

At zero temperature, the condensate depletion and the anomalous density for
each component are defined, respectively as Boudj5 :
$\tilde{n}_{\pm}=V^{-1}\sum_{\bf
p}[v_{p\pm}^{2}+\langle|\beta_{p}|^{2}\rangle]$, and
$\tilde{m}_{\pm}=V^{-1}\sum_{\bf
p}[-u_{p\pm}v_{p\pm}+\langle|\beta_{p}|^{2}\rangle]$. Working in the
thermodynamic limit, the sum over $p$ can be replaced by the integral
$\sum_{\bf p}\rightarrow V\int_{0}^{\infty}d{\bf p}/(2\pi\hbar)^{3}$. Thus,
the noncondensed and anomalous densities take the form :

$\displaystyle\tilde{n}_{\pm}$ $\displaystyle=\tilde{n}_{\pm}^{0}+n_{R\pm}$
(13)
$\displaystyle=\int\frac{p^{2}dp}{4\pi^{2}\hbar^{3}}\left[\frac{E_{p}+pv+mc_{s\pm}^{2}}{\varepsilon_{p\pm}}-1\right]+n_{R\pm},$

and

$\tilde{m}_{\pm}=\tilde{m}_{\pm}^{0}+n_{R\pm}=-\int\frac{p^{2}dp}{4\pi^{2}\hbar^{3}}\frac{mc_{s\pm}^{2}}{\varepsilon_{p\pm}}+n_{R\pm},$
(14)

where the glassy fraction $n_{R\pm}$ reads

$n_{R\pm}=\frac{1}{V}\sum_{p}\langle|\beta_{p\pm}|^{2}\rangle=n\int_{0}^{\infty}\frac{p^{2}dp}{4\pi^{2}\hbar^{3}}\frac{R_{p}(E_{p}+pv)^{2}}{\varepsilon_{p\pm}^{4}},$
(15)

where $R_{p}=\langle|U_{p}|^{2}\rangle$. The first term in Eqs.(13) and (14)
originates from the quantum fluctuations. Whereas, $n_{R}$ defined in Eq.(15)
accounts for the disorder fluctuation known also as glassy fraction is analog
to the Edwards-Anderson order parameter of a spin glass Edw ; Yuk1 . It arises
from the accumulation of density near the potential minima and density
depletion around the maxima. The emergence of glassy fraction in Bose systems
does not mean that the whole system is transformed into into the Bose glass
phase Yuk1 . Note that the validity of the present approach requires the
inequality: $\tilde{n}\ll n$ HM ; Axel ; Boudj3 .

In quantum Bose gases, the EoS is important since it characterizes the
thermodynamic properties of the system. Corrections to the EoS owing to the
Lee-Huang-Yang (LHY) quantum fluctuations for Bose mixtures can be given via
the relation Boudj ; Boudj2 :

$\mu_{\text{LHY}\pm}=\frac{g}{V}\sum_{p}[v_{p\pm}^{2}-u_{p\pm}v_{p\pm}]=g(\tilde{n}_{\pm}+\tilde{m}_{\pm}),$
(16)

where $\tilde{n}_{\pm}$ and $\tilde{m}_{\pm}$ are defined in Eqs.(13) and
(14).

To be concrete, we consider in what follows the white noise random potential,
which assumes a delta distribution

$R({\bf r}-{\bf r^{\prime}})=R_{0}\delta({\bf r}-{\bf r^{\prime}}),$ (17)

where $R_{0}$ is the disorder strength with dimension (energy)2 $\times$
(length)3. The model (17) is valid when the correlation length of the
correlation function $R({\bf r}-{\bf r^{\prime}})$ is sufficiently shorter
than the healing length (see e.g. Yuk1 ; Axel ; Boudj3 ; Boudj4 ).

Integrals (13)-(15) are infinite for the Bogoliubov energy (12). Assuming that
the velocity of the mixture is small ($v\ll v_{c}=c_{s}\sqrt{\delta
g_{\pm}/g}$) we expand the above integrals in powers of $v$ up to first-order.
The zero-order terms give exactly the usual expressions for the condensate
depletion, the anomalous density, and the EoS of immovable Bose mixtures.

After a straightforward calculation, we get for the glassy fraction:

$n_{R\pm}\simeq n_{\text{HM}\pm}\bigg{(}1-\frac{2v}{\pi c_{\pm}}\bigg{)},$
(18)

where $n_{\text{HM}\pm}=nmR_{0}/(4\pi\hbar^{3}{c_{s\pm}})$. For $v=0$, one can
reproduce the result of immovable dirty binary BECs obtained in our recent
work BB . Equation (18) shows that $n_{R\pm}$ is decreasing with increasing
$v/c_{\pm}$ indicating that the velocity of two BECs lead to tune the disorder
fluctuations ensuring the existence of the condensate.

Working out the quantum contribution following the same procedure, the
condensate depletion of each species takes the form:

$\tilde{n}_{\pm}^{0}\simeq\frac{m^{3}c_{s\pm}^{3}}{3\pi^{2}\hbar^{3}}\bigg{(}1-\frac{3v}{2c_{\pm}}\bigg{)}.$
(19)

Again the increase in velocity of two fluids in turn reduces the density of
noncondensed atoms $\tilde{n}_{\pm}^{0}$ and hence, augments the density of
condensed atoms. One may argue that noncondensed particles screen the motion
of atoms through the system.

Figure 2: (Top) Glassy fraction from Eq.(15). (Middle) Noncondensed fraction
of a cleaned Bose mixture from Eq.(13). (Bottom) Anomalous fraction of a
cleaned Bose mixture from Eq.(14) as a function of the mixture velocity
$v/c_{s}$ and the interspecies interaction $a_{12}/a$. Parameters are :
$R^{\prime}=0.05$, and $na^{3}\sim 10^{-5}$ Cab .

Corrections to the anomalous density due to the relative motion contribution
is the same as for the noncondensed density. The only difference comes from
the quantum term which diverges at high momentum. This ultraviolet divergence
is well-known and arises from the contact potential. One strategy to overvome
this problem is the dimensional regularization which is valid for weak
interacions Yuk ; Boudj . After regularization we obtain for the anomalous
density
$\tilde{m}_{\pm}^{0}\simeq(mc_{s\pm})^{3}/\pi^{2}\hbar^{3}\big{(}1-3v/2c_{\pm}\big{)}$,
which is larger than $\tilde{n}_{\pm}^{0}$.

Collecting all terms together, we obtain useful expressions for the total
depletion $\tilde{n}=\sum_{\pm}\tilde{n}_{\pm}$ and the total anomalous
density $\tilde{m}=\sum_{\pm}\tilde{m}_{\pm}$ as a function of the small
parameter of the theory:

$\displaystyle\frac{\tilde{n}}{n}$
$\displaystyle\simeq\frac{8}{3}\sqrt{\frac{na^{3}}{\pi}}\sum_{\pm}\left(\frac{\delta
g_{\pm}}{g}\right)^{3/2}\bigg{\\{}\left[1+\frac{3\pi}{4}\frac{R^{\prime}}{(\delta
g_{\pm}/g)^{2}}\right]$ (20)
$\displaystyle-\frac{3}{2}\left[1+\frac{R^{\prime}}{(\delta
g_{\pm}/g)^{2}}\right]\frac{v}{c_{s}\sqrt{\delta g_{\pm}/g}}\bigg{\\}},$

and

$\displaystyle\frac{\tilde{m}}{n}$ $\displaystyle\simeq
8\sqrt{\frac{na^{3}}{\pi}}\sum_{\pm}\left(\frac{\delta
g_{\pm}}{g}\right)^{3/2}\bigg{\\{}\left[1+\frac{\pi}{4}\frac{R^{\prime}}{(\delta
g_{\pm}/g)^{2}}\right]$ (21)
$\displaystyle-\frac{1}{2}\left[1+\frac{R^{\prime}}{(\delta
g_{\pm}/g)^{2}}\right]\frac{v}{c_{s}\sqrt{\delta g_{\pm}/g}}\bigg{\\}},$

where $R^{\prime}=R_{0}n/m^{2}c_{s}^{4}$ is a dimensionless disorder strength
for each component. Obviously, for $R^{\prime}=v=0$, $\tilde{n}$ and
$\tilde{m}$ reduce to those obtained for symmetric cleaned immovable Bose
mixtures Boudj ; Boudj2 .

Substituting Eqs.(20) and (21) into Eq.(16), the corrected EoS for Bose
mixtures with moving components reads

$\displaystyle\frac{\mu_{\text{LHY}}}{mc_{s}^{2}}$
$\displaystyle=\frac{32}{3}\sqrt{\frac{na^{3}}{\pi}}\sum_{\pm}\left(\frac{\delta
g_{\pm}}{g}\right)^{3/2}\bigg{\\{}\left[1+\frac{3\pi}{8}\frac{R^{\prime}}{(\delta
g_{\pm}/g)^{2}}\right]$ $\displaystyle-\left[1+\frac{R^{\prime}}{(\delta
g_{\pm}/g)^{2}}\right]\frac{v}{c_{s}\sqrt{\delta g_{\pm}/g}}\bigg{\\}}.$ (22)

For $v=R^{\prime}=0$, we reproduce the standard LHY EoS for cleaned Bose
mixtures Boudj2 . Evidently, for $v=0$, Eq.(Moving binary Bose-Einstein
condensates in a weak random potential) reduces to that of a dirty immovable
mixture BB . Again corrections due to the relative motion effects may lead to
decrease the chemical potential of the system. In such a regime the system
behaves as a particle-like object with a negative mass
$M\sim-m\sqrt{na^{3}/\pi}\sum_{\pm}\left[\left(\delta
g_{\pm}/g\right)+R^{\prime}/(\delta g_{\pm}/g)\right]$ results in any friction
force, if exist, would accelerate the mixture.

Figure 3:  The LHY corrections to the EoS
$\mu_{\text{LHY}}=\sum_{\pm}\mu_{\text{LHY}\pm}$ as a function of $v/c_{s}$
for two values of $a_{12}/a$. Parameters are: $R^{\prime}=0.05$, and
$na^{3}\sim 10^{-5}$ Cab .

The disorder potential can be considered as weak if the condensate depletion
due to the disorder is much smaller than the total density $n_{R}\ll n$. This
leads to the criterion for the critical disorder strength:

$\displaystyle R_{c}\simeq(na^{3}/\pi)^{-1/2}\sum_{\pm}\frac{\delta
g_{\pm}/g}{\sqrt{\delta g_{\pm}/g}-2v/(\pi c_{s})}.$ (23)

For $v=0$, Eq.(23) reduces to $R_{c}\simeq\sum_{\pm}\sqrt{(\delta
g_{\pm}/g)/(na^{3}/\pi)}$. The Bogoliubov theory is no longer applicable for
$R^{\prime}>R^{\prime}_{c}$. Therefore, the investigation of disordered moving
binary BECs in such a regime requires to go beyond the perturbative theory or
using numerical quantum Monte Carlo simulation.

In order to gain additional insights into the behavior of moving binary BECs
in random potentials and to verify our analytical results, we perform
numerical simulations of the integrals (13)-(15). Then, we consider two 39K
BECs with the interaction parameter of each component is $na^{3}\sim 10^{-5}$
Cab , which is sufficient to ensure that the system meets the requirement of
weakly interacting gas. The interspecies interactions can be adjusted by means
of Feshbach resonance. The disorder strength has been chosen
$R^{\prime}=0.05$. However, for such a value the condensate fluctuations due
to the disorder is $\lesssim 1\%$ of the total density (see Fig.2 (top)).
Evidently, the disorder fraction $n_{R}/n$ becomes important for large
$R^{\prime}$.

We observe also that the glassy fraction in both components decreases with
$v/c_{s}$ and increases with $a_{12}/a$. This indicates that the competition
of the relative motion of the mixture fluid, the interspecies interactions and
the disorder potential may strongly modify the localization of atoms (see
Fig.2 (top)). This is in contrast to the immovable disordered BEC where the
disorder tends to fragment the condensate and hence, reduce both the condensed
and the superfluid fractions. The same holds true for the behavior of the
quantum contribution to the noncondensed fraction $\tilde{n}^{0}/n$ (see Fig.2
(middle)). The situation is quite different for the anomalous fraction
$\tilde{m}^{0}/n$, where its modulus decreases with both $v/c_{s}$ and
$a_{12}/a$ (see Fig.2 (bottom)). We are thus treating ultradilute and weakly
correlated Bose mixture liquids.

Figure.3 depicts that the LHY corrected EoS is decreasing with $v/c_{s}$
whatever the value of the ratio $a_{12}/a$ in agreement with our analytical
results. This is attributed to the interplay of the disorder potential and the
relative motion of the mixture.

In conclusion, we systematically investigated the implications of a weak
random potential on moving binary BECs with a combined analytical and
numerical methods. Our calculations are based on the Bogoliubov-Huang-Meng
approach which is valid for weak interactions, weak disorder and small
velocity. We showed that the relative motion of two components plays a key
role in reducing the glassy fraction, the quantum depletion, the anomalous
correlations, and the EoS. One can expect that when the condensate reaches the
Landau critical velocity where dissipation is more efficient, the condensate
depletion would increase. Furthermore, we pointed out that corrections due to
the interspecies interaction, disorder and relative motion may lead to the
occurrence of stable ultradilute Bose mixtures. We anticipate that our
findings will motivate new avenues of research in quantum transport for
disordered Bose mixtures.

We would like to thank V.I. Yukalov for careful reading of the manuscript.

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