# Andreev-Bashkin effect in superfluid cold gases mixtures

Jacopo Nespolo  e-mail<EMAIL_ADDRESS>Department of Physics, Arnold
Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität
München, Theresienstraße 37, D-80333 Munich, Germany, EU  Grigori E.
Astrakharchik  Departament de Física, Universitat Politècnica de Catalunya,
E-08034 Barcelona, Spain, EU  Alessio Recati  e-mail<EMAIL_ADDRESS>Department of Physics, Arnold Sommerfeld Center for Theoretical Physics,
Ludwig-Maximilians-Universität München, Theresienstraße 37, D-80333 Munich,
Germany, EU  INO-CNR BEC Center and Dipartimento di Fisica, Università di
Trento, Via Sommarive 14, I-38123 Povo, Italy, EU

###### Abstract

We study a mixture of two superfluids with density-density and current-current
(Andreev-Bashkin) interspecies interactions. The Andreev-Bashkin coupling
gives rise to a dissipationless drag (or entrainment) between the two
superfluids. Within the quantum hydrodynamics approximation, we study the
relations between speeds of sound, susceptibilities and static structure
factors, in a generic model in which the density and spin dynamics decouple.
Due to translational invariance, the density channel does not feel the drag.
The spin channel, instead, does not satisfy the usual Bijl-Feynman relation,
since the f-sum rule is not exhausted by the spin phonons. The very same
effect on one dimensional Bose mixtures and their Luttinger liquid description
is analysed within perturbation theory. Using diffusion quantum Monte Carlo
simulations of a system of dipolar gases in a double layer configuration, we
confirm the general results. Given the recent advances in measuring the
counterflow instability, we also study the effect of the entrainment on the
dynamical stability of a superfluid mixture with non-zero relative velocity.

###### pacs:

##  I Introduction

Mixtures of different kinds of miscible superfluids arise in various areas of
physics, starting from the first experiments on 3He-4He mixtures Edwards  _et
al._ (1965), through possible applications to astrophysical objects Alpar  _et
al._ (1984); Kobyakov and Pethick (2017), all the way to the more recent
developments in the fields of superconductivity Garaud  _et al._ (2014), cold
atoms Modugno  _et al._ (2002); Catani  _et al._ (2008); Roy  _et al._ (2017)
and exciton-polariton condensates Lagoudakis  _et al._ (2009).

The statistics of each component of the mixture can be arbitrary, and Bose-
Bose, Bose-Fermi, and Fermi-Fermi mixtures were all successfully realised
experimentally in cold gases. In these experiments, also the chemical nature
of the components can vary: the use of two different elements, of different
isotopes of the same element, and of different internal states of a common
isotope were demonstrated. The ability to reach simultaneous quantum
degeneracy in such a wide variety of atomic species in cold gases experiments
allows for the realisation of very diverse interactions between the two
component superfluids.

One of the most elusive effect of coupled superfluids is the existence of a
non-zero entrainment between them. The presence of mutual transport has been
pointed out for the first time in the 1970s by Andreev and Bashkin Andreev and
Bashkin (1976), correcting some previous work on three-fluid hydrodynamics
Khalatnikov (1957); Mineev (1975). The most prominent feature of such an
effect, nowadays known as Andreev-Bashkin effect (AB), is that the superfluid
current $\mathbf{j}_{i}$ of one component will in general depend also on the
superfluid velocity ${\bf v}_{j}$ of the other component, or, in other words,
that the superfluid density is a non-diagonal matrix, $\rho_{ij}$, namely,

${\bf j}_{i}=\rho_{ij}{\bf v}_{j},$ (1)

with the indices $i,j=\\{1,2\\}$ labelling the species and implicit summation
on repeated indices. We shall refer to the off-diagonal element $\rho_{12}$ of
the superfluid density matrix as the _superfluid drag_. Phenomenologically, a
nonzero $\rho_{12}$ carries important implication in the dynamics of vortices
of a superfluid mixture. Notably, it is predicted that the circulation leading
to the stable vortex configurations change abruptly as $\rho_{12}$ is varied,
giving rise to stable multiply circulating vortex configurations Meyerovich
(1984)

Despite its introduction was inspired by the problem of 3He and 4He superfluid
mixtures, the low miscibility of these two fluids makes this system hardly
achievable in experiments. The AB mechanism has instead recently found
applications in the domains of astrophysics and of cold atom systems. In the
astrophysical literature, it has been hypothesised that the AB effect could be
the source of several peculiar behaviours in neutron stars cores Alpar  _et
al._ (1984), which modern models predict to be composed of a mixture of
neutrons and protons, both in a superfluid phase (see Lattimer and Prakash
(2004); Babaev (2004) and reference therein). Cold atom experiments, on the
other hand, thanks to their flexibility and tunability, could open the way to
a direct measurement of superfluid drag, albeit this will still require some
careful analysis and mitigation of common drawbacks. For instance,
calculations within the Bogoliubov theory for Bose-Einstein condensates with
typical repulsive interaction—where quantum fluctuations are depressed—predict
the AB to be very small Fil and Shevchenko (2004, 2005). Quantum fluctuations
can be enhanced by increasing the interactions, but this would also intensify
three-body losses, which could be in turn suppressed by confining the system
in low dimensional geometries or introducing an optical lattice, as studied,
e.g., in Linder and Sudbø (2009); Hofer  _et al._ (2012). However, optical
lattices break translational invariance, thus strongly reducing the superfluid
density even a $T=0$. We recall that, in continuous space (and with time-
reversal symmetry), the superfluid density approaches the total density as $T$
is lowered to zero (see, e.g., Leggett (1998)).

Aside from making the AB mechanism efficient, a very important question is how
to measure experimentally its strength. The dynamical protocols typically
proposed require the ability to initialise a superfluid current in one
component and then observe the onset of dissipationless transport in the other
one, initially at rest. For best results, these kind of measurements would
likely require a ring geometry and can be of difficult interpretation, since a
number of decay processes are present Beattie  _et al._ (2013); Abad  _et al._
(2014).

In the present work, we address some of the above mentioned issues. In
particular, we derive some relations between the superfluid drag and other
measurable quantities, such as the susceptibilities of the system and the
speeds of sound. For systems with $\mathbb{Z}_{2}$ symmetry between the two
species, in which spin and density channels decouple, the density channel
follows the usual relations, whereas we show how the AB breaks the usual Bijl-
Feynman relation for the spin channel. Our findings open the way to measuring
the superfluid drag experimentally using standard static and dynamic
observables.

To provide support to our theoretical predictions, we study quantitatively a
specific model which can show large entrainment, i.e., a dipolar Bose gas
trapped in a bilayer configuration. Using the diffusion quantum Monte Carlo
method, we extract the dispersion relations, the susceptibilities, the
structure factors and the superfluid densities. We show that they satisfy, in
a proper regime, the expression derived in the general theory. In particular,
it is shown that the standard expression relating the square of the spin speed
of sound to the inverse of the susceptibility is inapplicable and it should be
corrected by a factor proportional to the superfluid drag. Another possible
quantity which could reveal the presence of a superfluid drag is the shift in
the position of the dynamical instability. We report a general stability
analysis of the mixture, and derive a simple analytical expression for the
onset of the dynamical instability to linear order in the drag.

As mentioned above, the AB mechanism could play a more prominent role in low
dimensionality. We discuss the modifications to Luttinger liquid theory
necessary to describe coupled one dimensional superfluids. We find that, in
analogy with the general description, the spin Luttinger parameters, as
derived by means of perturbative or ab-initio calculations, receive a
correction from the superfluid drag, which could become particularly relevant
in the strongly interacting (Tonks-Girardeau) regime.

The paper is organised as follows. In Sec. II we recall the main aspects of
the AB effect, which are then analysed within a minimal quantum hydrodynamic
toy model in Sec. III. The relations among experimentally relevant observables
are derived. The Luttinger liquid theory for one dimensional coupled
superfluid is corrected for the presence of AB in the same section. Numerical
evidence in support of our theoretical findings are reported in Sec. IV, where
we analyse the presence and magnitude of the superfluid drag in a bilayer
system of dipolar bosons. In Sec. V we study the dynamical instability of the
mixture with respect to the relative velocity between the two fluids.
Conclusions and future perspectives are drawn in Sec. VI. For the sake of
completeness, the derivations of some relations used in the main text are
postponed to the appendices without affecting the comprehension of the main
results.

##  II Andreev-Bashkin effect

Microscopically, the current drag originates from the interactions between two
superfluids, leading to the formation of quasi-particles with nonzero content
of either of the two species. It is then easy to understand that the transport
properties of the two components are not independent: the flow of one
component must be accompanied by mass transport of the other component Andreev
and Bashkin (1976).

Some important relations concerning the superfluid densities in Eq. (1) can be
easily obtained by considering the kinetic energy contribution in the
expansion of the ground state energy in terms of the superfluid velocities
Kaurov  _et al._ (2005). Due to Galilean invariance, if $\phi_{i}$ is the
phase of the superfluid order parameter for component $i$, of mass $m_{i}$,
its velocity is given by ${\bf v}_{i}=(\hbar/m_{i})\nabla\phi_{i}$ and the
energy due to the superfluid velocities can be written as

$\delta E=\int
d^{D}x\,\sum_{ij}\frac{\hbar^{2}}{2m_{i}m_{j}}\rho_{ij}\nabla\phi_{i}\cdot\nabla\phi_{j}.$
(2)

By performing a Galilean boost with velocity ${\bf V}$, the phases are shifted
to $\phi^{\prime}_{i}=\phi_{i}-(m_{i}/\hbar){\bf V}\cdot{\bf r}$, and the
energy change, to first order in ${\bf V}$, is $\delta E^{\prime}=\delta
E-\int{\bf P}\cdot{\bf V}\,d^{D}x$, with ${\bf
P}/\hbar=\sum_{i}(\rho_{i}+\rho_{12})/m_{i}\nabla\phi_{i}$ the momentum
density. Since on the other hand one must have ${\bf
P}=n_{1}\nabla\phi_{1}+n_{2}\nabla\phi_{2}$, with $n_{1,2}$ the number
densities, the superfluid densities must satisfy

$m_{i}n_{i}=\rho_{i}+\rho_{12}.$ (3)

Introducing the effective masses $m_{1,2}^{*}$ through $\rho_{ii}\equiv
n_{i}m_{i}^{2}/m_{i}^{*}$, we obtain

$\rho_{12}=n_{i}m_{i}\left(1-\frac{m_{i}}{m^{*}_{i}}\right),$ (4)

which provides the relation between the superfluid drag and the effective
masses and a constraint for the effective mass ratio.

##  III Quantum hydrodynamic model

In general, the presence of effective masses changes the relation between
static and dynamical properties of the system, and the possibility for some
excitation modes to exhaust the sum rules. Let us consider a superfluid
mixture with an energy density $e(n_{1},n_{2})$. Expanding the energy around
its ground state value to second order in the density fluctuations
$\Pi_{i}(x)$ and adding it to Eq. (2) we obtain a hydrodynamic Hamiltonian for
two miscible superfluids,

$H=\frac{1}{2}\sum_{ij}\int\left(\rho_{ij}\frac{\hbar\nabla\phi_{i}}{m_{i}}\cdot\frac{\hbar\nabla\phi_{j}}{m_{j}}+\alpha_{ij}\Pi_{i}\Pi_{j}\right)\,d^{D}x,$
(5)

where the matrix $\alpha_{ij}=\partial^{2}e/\partial n_{i}\partial n_{j}$
contains the information on inter- and intra-species interactions of the two
fluids. Hamiltonian (5) by requiring that the fields $\phi_{i}$ and $\Pi_{j}$
satisfy canonical commutation relations for bosons, i.e.,
$[\phi_{i}(x),\Pi_{j}(y)]=i\hbar\delta_{ij}\delta(x-y)$.

For the sake of clarity, we take the two superfluids to be equal:
$\rho_{ii}=\rho$, $\alpha_{ii}=\alpha$, $m_{i}=m$ and $n_{i}=n/2$, with
$n\equiv N/V$ the total number density of the system (see Appendix A for the
non-symmetric case). Due to the assumed $\mathbb{Z}_{2}$ symmetry, the
dynamics of this model decouples if we rewrite it in terms of the new fields

$\phi_{d(s)}=(\phi_{1}\pm\phi_{2})/\sqrt{2},\quad\Pi_{d(s)}=(\Pi_{1}\pm\Pi_{2})/\sqrt{2}.$
(6)

The fields $\Pi_{d}$ and $\phi_{d}$ represent the fluctuations in total
density and global phase, respectively. In a similar fashion, $\Pi_{s}$ and
$\phi_{s}$ encode the fluctuations of the difference in density of the two
species (magnetisation) and their relative phase (spin wave), respectively. We
use the labels $d(s)$ to indicate the density (spin) channel of the system’s
excitations. The new fields inherit the canonical commutation relations and
act as two independent hydrodynamic modes, obeying the Hamiltonian

$H=\frac{1}{2}\sum_{i=d,s}\int\left[\rho_{i}\left(\frac{\hbar\nabla\phi_{i}}{m}\right)^{2}+\alpha_{i}\Pi_{i}^{2}\right]\,d^{D}x,$
(7)

where $\rho_{d(s)}=\rho\pm\rho_{12}$ and $\alpha_{d(s)}=\alpha\pm\alpha_{12}$.
The Hamiltonian is now diagonal in the two channels, and the dispersion
relations for the two modes are linear in the momentum $k$, of the form

$\displaystyle(\hbar\omega)^{2}=\frac{\alpha_{i}\rho_{i}}{m^{2}}(\hbar
k)^{2},\qquad(i=d,s).$ (8)

The quantity $\alpha_{i}\rho_{i}/m^{2}$ can be identified with the speed of
sound of each mode. For the density mode, we have

$c_{d}^{2}=\frac{(\rho+\rho_{12})}{m^{2}}(\alpha+\alpha_{12})=\frac{n}{2m}(\alpha+\alpha_{12}),$
(9)

where in the last equality we used that, at $T=0$, the total superfluid
density is equal to the total mass density of the system. We note in
particular that $c_{d}$ is independent of the superfluid drag. On the other
hand, the spin speed of sound is

$c_{s}^{2}=\frac{(\rho-\rho_{12})}{m^{2}}(\alpha-\alpha_{12})=\frac{n}{2m}\left(\frac{2m}{m^{*}}-1\right)(\alpha-\alpha_{12}),$
(10)

which explicitly depends on $\rho_{12}$.

From the Hamiltonian (7), the static response to density and to spin probes
are simply given by $\alpha_{d(s)}$, which can be identified with the inverse
compressibility $\kappa_{d}^{-1}$ and inverse magnetic susceptibility
$\chi_{s}^{-1}$, respectively. We thus obtain the relations

$\displaystyle c_{d}^{2}$ $\displaystyle=\frac{n}{2m\kappa_{d}},$ (11)
$\displaystyle c_{s}^{2}$
$\displaystyle=\frac{\rho-\rho_{12}}{m^{2}\chi_{s}}=\frac{n}{2m\chi_{s}}\left(\frac{2m}{m^{*}}-1\right).$
(12)

These relations suggest that, by independently measuring $c_{s}$ and
$\chi_{s}$, it is possible to obtain the strength of the mass renormalisation,
i.e., the magnitude of the superfluid drag. Note that $c_{s}$ is expected to
vanish for $m^{*}=2m$, which imposes a bound $m^{*}\leq 2m$. From Eq. (4),
this bound translates into $\rho_{12}\leq mn/4$, thus anticipating result
(23), of which we will provide an additional derivation below.

The previous analysis has important consequences with respect to Bijl-Feynman
relations (f-sum rule) linking the dispersion relations to the static
structure factors (see, e.g., Giuliani and Vignale (2005)). From the above
discussion, it turns out that the f-sum rule for the density channel is
exhausted by the phonon mode, while for the spin mode this is not the case,
leading to the effective mass correction in the determination of the
dispersion relation. In particular, the zero temperature spin structure factor
at low momenta reads

$S_{s}(k)\stackrel{{\scriptstyle k\to
0}}{{=}}\frac{k}{2mc_{s}}\frac{(\rho-\rho_{12})}{m}=\frac{k}{2m}\sqrt{(\rho-\rho_{12})\chi_{s}},$
(13)

which does not satisfy the Bijl-Feynman relation. Notice that the linear term
in $k$ of the $S_{s}(k)$ can vanish, either because of a vanishing
susceptibility or because of a saturated drag, i.e., $\rho_{12}=\rho$. The
former (latter) case corresponds to a vanishing (diverging) spin speed of
sound. The fact that the drag and the interspecies interaction act
independently on the spin speed of sound [cf. Eq. (10)] is general, and
applies beyond the $\mathbb{Z}_{2}$ symmetry we assumed in this section. In
particular, the standard condition for the onset of phase separation (i.e.,
$\chi_{s}\rightarrow\infty$ for $\alpha=\alpha_{12}$) still holds (cf.
Appendix A).

On the other hand, due to translational invariance, the density structure
factor satisfy the Bijl-Feynman relation and it reads

$S_{d}(k)\stackrel{{\scriptstyle k\to
0}}{{=}}\frac{nk}{4mc_{d}}=\frac{k}{2m}\sqrt{2mn\kappa_{d}}.$ (14)

Let us conclude this section by briefly mentioning the effect of the AB
physics on the specific heat of the mixture. At low but finite temperature, we
may expect that thermal fluctuations do not change the low energy spectrum
significantly. Then the low temperature dispersion relations are still linear,
of the form, $\epsilon_{i}(k)=c_{i}k$, $(i=d,s)$, and we assume, within the
hydrodynamic picture, that the highest momentum that can be thermally excited
is $k_{T,i}=k_{B}T/c_{i}$. In the low temperature limit and $D$ spatial
dimensions, these assumptions lead to the specific heat

$C_{v}\propto T^{D}\left(\frac{1}{c_{d}^{D}}+\frac{1}{c_{s}^{D}}\right),$ (15)

which carries a dependence on $\rho_{12}$ through the sound velocity in the
spin channel. A large superfluid drag will therefore lead to a strong increase
of the specific heat.

###  III.1 One dimensional systems and Luttinger Liquid

Since the superfluid drag is due to quantum fluctuations, one can think about
increasing them by increasing the interactions, i.e., quantum depletion and
mutual dressing. This can be easily seen in the weakly interacting regime,
where analytical expressions for the superfluid drag have been nicely obtained
within a Bogoliubov approach by Fil and Shevchenko Fil and Shevchenko (2005,
2004). In a three-dimensional system, three-body losses strongly limit the
possible increase of the interaction strengths. On the other hand, in one
dimension, it is possible to reach strong quantum regimes, including the so-
called Tonks-Girardeau regime. The low-energy excitations of one-dimensional
gases are described in terms of Luttinger liquids Giamarchi (2004). For the
sake of simplicity, we consider two equal Luttinger liquids coupled together,
with speed of sound $c_{0}$ and Luttinger parameter $K_{0}\geq 1$. By
introducing both the density-density and the current-current couplings as a
perturbation, we can write

$\displaystyle H_{LL}$ $\displaystyle=$
$\displaystyle\sum_{i=1,2}\frac{c_{0}}{2}\int[K_{0}(\partial_{x}\phi_{i})^{2}+\frac{1}{K_{0}}\Pi_{i}^{2}]$
(16) $\displaystyle+$
$\displaystyle\int[\rho_{12}\frac{\partial_{x}\phi_{1}\partial_{x}\phi_{2}}{m^{2}}+g_{12}\Pi_{1}\Pi_{2}].$

As before, we can easily diagonalise the Hamiltonian (16) by introducing the
fields for the in-phase and out-of-phase fluctuations. We obtain a standard
expression for coupled Luttinger liquids

$H_{LL}=\sum_{i=d,s}\frac{c_{i}}{2}\int[K_{i}(\partial_{x}\phi_{i})^{2}+\frac{1}{K_{i}}\Pi_{i}^{2}],$
(17)

where the density parameters read

$\displaystyle 2mc_{d}^{2}$ $\displaystyle=n(c_{0}/K_{0}+g_{12}),$ (18)
$\displaystyle 2mK_{d}^{2}$ $\displaystyle=n/(c_{0}/K_{0}+g_{12}),$ (19)

and for the spin sector we get

$\displaystyle 2mc_{s}^{2}$
$\displaystyle=(n-4\rho_{12}/m)(c_{0}/K_{0}-g_{12}),$ (20) $\displaystyle
2mK_{s}^{2}$ $\displaystyle=(n-4\rho_{12}/m)/(c_{0}/K_{0}-g_{12}).$ (21)

In the above expressions, $n$ is the total density of the system, and we have
used the fact that, for a translationally invariant system, $c_{d}K_{d}=n/2m$
(see also Schulz  _et al._ (1998)), which implies that
$c_{0}K_{0}+\rho_{12}/m^{2}=n/m$. Therefore, $c_{d}$ and $K_{d}$ do not depend
on the off-diagonal superfluid density and for the compressibility we have
$\kappa=K_{d}/c_{d}=n/(2mc_{d}^{2})$. On the other hand, the spin channel
parameters acquire a dependence on $\rho_{12}$, as seen before in the general
case. In fact, the susceptibility reads
$\chi_{s}=K_{s}/c_{s}=(n-4\rho_{12})/(2mc_{s}^{2})$, to be compared with Eq.
(12). The correction due to AB in Eq. (20), in the strongly interacting limit
can therefore deeply modify the standard perturbative analysis Kleine  _et
al._ (2008a, b) and the RG flow for coupled Bose Luttinger liquids. Note, once
again, that the previous equations imply a bound on the value of the
superfluid drag, $\rho_{12}\leq nm/4$, which coincides with the one coming
from Eq. (12) in the previous section. Recent Monte-Carlo simulations on one
dimensional Bose gases confirm our results Parisi and Giorgini .

##  IV Magnitude of the drag and numerical evidence

The information on the superfluid drag can be extracted from quantum Monte-
Carlo (QMC) simulations based on the path integral formalism. In this
formalism, in fact, the superfluid density can be related to the statistics of
winding numbers of particles’ paths around the simulation domain Pollock and
Ceperley (1987). By extending this result to two species in the same
simulation box (see Appendix B for details), we obtain the relation

$\displaystyle\rho_{T}$ $\displaystyle=\frac{L^{2-D}}{\beta
D}\left[m_{1}^{2}\langle W_{1}^{2}\rangle+m_{2}^{2}\langle
W_{2}^{2}\rangle+2m_{1}m_{2}\langle W_{1}W_{2}\rangle\right]$
$\displaystyle=\rho_{1}+\rho_{2}+2\rho_{12},$ (22)

linking the total superfluid density $\rho_{T}$ to the winding numbers
$W_{1,2}$ of the two species. Here we are considering a simulation volume
$L^{D}$ at inverse temperature $\beta=1/T$. For zero temperature results, $T$
is taken smaller than all the other energy scales of the system and the
results are checked _a posteriori_ for convergence.

From Eq. (IV), the superfluid drag can be interpreted as the covariance
between the superfluid densities of the two components. Then, thanks to
Cauchy-Schwarz inequality, $\rho_{12}^{2}\leq\rho_{1}\rho_{2}$, and assuming
the symmetric case, in which $\rho_{1}=\rho_{2}$, we obtain an upper bound on
the magnitude of the drag,

$\rho_{12}\leq\frac{\rho_{T}}{4},\qquad(\rho_{1}=\rho_{2})$ (23)

which also bounds the effective mass to $m^{*}\leq 2m$. As already noted
above, the condition of saturation of this bound corresponds to a vanishing
speed of sound in the spin channel [cf. Eqs. (10)-(12)]. We shall also see
below that the saturation of this bound is a limiting case in the dynamic
stability of the mixture [see Eq. (42)].

###  IV.1 Quantum Monte Carlo results for bilayer dipolar Gases

Lattice simulations already showed evidence of superfluid drag effects Hofer
_et al._ (2012); Kaurov  _et al._ (2005). It was shown that the drag depends
on the lattice geometry, increases with the increase of interspecies
interactions and attains its maximum for non-equal masses of the two particle
species. The presence of the lattice explicitly breaks the translational
invariance, thus deeply modifying the mechanism leading to a dissipationless
drag. In particular, for incommensurate fillings, the drag between the two
fluids is essentially mediated by the presence of vacancies Kaurov  _et al._
(2005).

The magnitude of the superfluid drag, normalised by the total superfluid
density, spans the whole range allowed by bound (23). However, one must point
out that the presence of the lattice causes the depletion of the total
superfluid density; in particular, on a lattice, it is no longer true that the
total superfluid density coincides with the total particle density at zero
temperature Leggett (1998). In this context, it is noteworthy to mention the
analytical results of Ref. Linder and Sudbø (2009), which compute the
superfluid drag starting from the physical parameters of the lattice in a weak
coupling approximation. The authors report a superfluid drag $\rho_{12}/nm$,
normalised to the total mass density, of the order of $10^{-5}$–$10^{-4}$ for
weak to moderate intercomponent scattering amplitude. Quantitatively similar
QMC results are reported in Hofer  _et al._ (2012). It is important to keep in
mind that these low values are primarily due to the small total superfluid
density on the lattice.

Figure 1: Schematic representation of the bilayer dipolar bosonic model.
Particles obeying Bose-Einstein statistics are confined in two parallel
layers. The dipoles are pinned perpendicularly to the layers’ planes and
parallel to each other, yielding entirely repulsive on-plane interactions and
partially attractive out-of-plane interactions.

In the following, we focus on a system of dipolar Bose gases confined in a
bilayer geometry in continuous space, with the dipole orientation pinned
perpendicular to the planes, as sketched in Fig. 1. This system is similar to
the one studied in Ref. Terentjev and Shevchenko (1999), which pointed out the
presence of entrainment between the superfluid currents of two charged
superfluids in a bilayer configuration. The relative strength of interspecies
interactions as compared to intraspecies ones can be tuned by changing the
distance between the two layers. As it will be shown later in Fig. 3, for an
extended range of this control parameter, the superfluid drag can reach very
large values. Dipolar particles in a bilayer configuration are particularly
advantageous under a variety of aspects. Confining the molecules in a two-
dimensional geometry and imposing a repulsive dipolar interaction strongly
reduces the detrimental two-body chemical reactions Ni  _et al._ (2010). At
the same time it allows to exploit the anisotropy of the dipolar interaction,
which is partially attractive between particles on different layers.
Introducing the distance $h$ between the two layers, the interaction between
two particle of mass $m$ and dipole moment $d$ on different layers can be
written as

$V(r,h)=d^{2}\frac{r^{2}-h^{2}}{(r^{2}+h^{2})^{5/2}},$ (24)

where $r$ is the relative distance in the plane of motion. For dipolar gases,
it is very useful to introduce the characteristic length
$r_{0}=md^{2}/\hbar^{2}$. The various regimes of the system are characterised
by the interlayer parameter $h/r_{0}$ and the in-layer parameter
$n_{i}r_{0}^{2}$, with $n_{i}$ the single layer density. Static and dynamic
properties of this system were recently investigated in Macia  _et al._
(2014); Astrakharchik  _et al._ (2016). In particular, it has been found that
a transition from two coupled superfluid (atomic phase) to a pair superfluid
(molecular phase) takes place when the attractive interaction is strong
enough. We will show that, by approaching the transition point while remaining
in the atomic phase, the drag superfluidity becomes prominent.

To recover the description of Eq. (5), we point out that miscibility is here
to be intended with respect to the position of the particles projected in the
direction orthogonal to the layers’ planes. Dipolar Bose gases in a double
layer configuration do not show any phase separation Macia  _et al._ (2014).
This is intuitive, since any potential with $V(q)|_{q=0}\leq 0$ (in momentum
space) admits a bound state in $2D$. (The interlayer potential 24 has the
peculiarity to have $V(q)|_{q=0}=0$, which makes the bound and scattering
states of the system at weak coupling rather peculiar Klawunn  _et al._
(2010)). The dipolar bilayer Bose gas is moreover the first example of a two
component Bose gas which can form pairs without collapsing (i.e. forming
clusters) Macia  _et al._ (2014) as it occurs, e.g., in mixtures with contact
interaction only.

Besides serving as a testbed for the numerical study of the superfluid drag in
a homogeneous geometry and being a new system showing the AB physics, the
dipolar bilayer configuration can represent one of the best-case scenarios for
the experimental observation of the presence of superfluid drag. Recent
experiments using dipolar molecules consisting of two atoms of Erbium-168
demonstrated the availability of condensates with large magnetic moments, up
to $r_{0}\approx 1600\,{\rm a}_{0}$, with ${\rm a}_{0}$ the Bohr radius Frisch
_et al._ (2015). This value is still almost one order of magnitude smaller
with respect to the typical wavelength of the lasers used to confine the Er2
molecules in arrays of 2D layers. Experiments on stable polar Na-K molecules
Will  _et al._ (2016), which sport much larger $r_{0}$, may help in overcoming
this problem. The recent proposal of sub-wavelength confinement Nascimbene
_et al._ (2015) may further stretch the experimentally accessible range of
values of $h/r_{0}$, albeit it is not of easy implementation for dipolar
molecules. Therefore, experimental realisation of a bilayer system of strongly
interacting dipolar superfluids is reasonably within reach of current or near-
future technology.

We study the system by means of diffusion QMC, which allows us to extract both
the thermodynamics and the low energy spectrum of the system. Diffusion QMC is
based on solving the Schrödinger equation in imaginary time, thus projecting
out the ground state of the system (for a general introduction on the method
see, e.g., Boronat and Casulleras (1994)). The contributions of the excited
states are exponentially suppressed and the ground-state energy is recovered
in the limit of long propagation time. The simulations are performed for $60$
particles with the same parameters as in Refs. Macia  _et al._ (2014);
Astrakharchik  _et al._ (2016). For some quantities, this number of particles
is sufficiently large to be close to the thermodynamic limit; for some others,
residual finite-size corrections must be taken into account, as it will be
explained in more details later.

In this framework, a number of observables of interest can be obtained in a
straightforward way. The value of the gap $\Delta$ and of the spin
susceptibility $\chi_{s}$ are obtained from the dependence of the ground-state
energy on the polarisation $P=(N_{1}-N_{2})/N$. The latter is tuned by moving
particles from layer $1$ to layer $2$ while keeping the total number of
particles $N=N_{1}+N_{2}$ constant. In the limit of small polarisation $P$,
the energy can be expanded as

$E(P)=E(0)+N\Delta\cdot P+N\frac{n}{2\chi_{s}}\cdot P^{2}.$ (25)

In the gapless phase ($\Delta=0$) the dependence on the polarisation is
quadratic, while in the gapped phase it is linear. Similarly, the
compressibility $\kappa_{d}$ at $T=0$ can be obtained from the volume
dependence of the energy for an unpolarised gas,

$\kappa_{d}^{-1}=-\mathcal{V}\left(\frac{\partial^{2}E}{\partial\mathcal{V}^{2}}\right),$
(26)

where $\mathcal{V}$ is the $D$ dimensional volume of the system
($\mathcal{V}=L^{2}$ in the $2D$ geometry at hand).

The study of structure factors provides a way of accessing the dynamic
properties of the system. We use the technique of pure estimators Liu  _et
al._ (1974); Casulleras and Boronat (1995) to compute the intermediate
scattering function

$S_{\alpha\beta}({\bf k},\tau)=\frac{1}{N}\left\langle\rho_{\alpha}({\bf
k},\tau)\rho_{\beta}(-{\bf k},0)\right\rangle,$ (27)

with $\rho_{\alpha}({\bf k},\tau)=\sum_{j}^{N_{\alpha}}\exp\\{-i{\bf
k}\cdot{\bf r}_{j\alpha}(\tau)\\}$ and ${\bf r}_{j\alpha}$ the position of
particle $j$ in layer $\alpha$. The intermediate scattering function provides
information on the correlations in imaginary time $\tau$ and is the main
ingredient to compute the static structure factor $S_{\alpha\beta}({\bf
k})\equiv S_{\alpha\beta}({\bf k},0)$. We consider a balanced system with
$N_{A}=N_{B}$ and study the symmetric and antisymmetric structure factors,

$S_{d(s)}(k)=S_{11}(k)\pm S_{12}(k),$ (28)

corresponding to the density and spin channels of the discussion above,
respectively. The compressibility and the spin susceptibility can be compared
to the respective static structure factors in the low momentum limit, in order
to verify the sum rules. The structure factors further provide information on
the excitation spectra: their long imaginary time asymptotic behaviour can be
fitted to an exponential decay of the form

$S_{d(s)}({\bf k},\tau)\sim Ze^{-\omega_{d(s)}(k)\tau}.\qquad(\tau\to\infty)$
(29)

When phononic excitations are present, $\omega_{d(s)}(k)$ is linear for small
momenta, with the slope directly related to the speeds of sound of the density
and spin channels, respectively, through $\hbar\omega_{d(s)}(k)\simeq
c_{d(s)}k$.

Figure 2: Speeds of sound as a function of interlayer spacing $h$ for
$nr_{0}^{2}=1$, as extracted from different observables. (a) Speed of sound of
the density mode, $c_{d}$. (b) Speed of sound of the spin mode, $c_{s}$. The
Feynman method makes use of the static structure factor $S(k)=\hbar k/(2mc)$,
computed at the smallest $k$ compatible with periodic boundary conditions. The
speeds $c_{\alpha}[S_{\alpha}(k)]$, with $\alpha=\\{d,s\\}$ the channel index,
are computed with this method. The data show that the f-sum rule is exhausted
by the phonon mode in the density channel, whereas this does not hold in the
spin channel, the arrow indicating the divergence of $c_{s}[S_{s}(k)]$. The
speeds $c_{\alpha}(\omega_{\alpha})$, computed from the excitation spectrum,
assume a linear phononic dispersion relation $\omega(k)=ck$ with
$\omega_{\alpha}(k)$ obtained from Eq. (29). The speeds $c_{d}(\kappa_{d})$
and $c_{s}(\chi_{s})$ are computed from $mc_{d}^{2}=n\kappa_{d}^{-1}$ and
$mc_{s}^{2}=n\chi_{s}^{-1}$, with $\kappa_{d}$ and $\chi_{s}$ obtained from
Eqs. (26) and (25), respectively. The speed of sound in the atomic limit
coincides in the two channels. It is obtained from standard thermodynamic
relations using the equation of state $E(nr_{0}^{2})$ (taken from Ref.
Astrakharchik  _et al._ (2007)) of a single layer system with half the density
of the bilayer system.

It is instructive to show that, in a gapless system without the drag, exactly
the same information on the speeds of sound can be recovered from the static
structure factors $S_{d,s}(k)$, the low-momentum excitation spectra
$\omega_{d,s}(k)$, the compressibility $\kappa_{d}$ and the susceptibility
$\chi_{s}$. The speeds of sound obtained from the different methods are shown
in Fig. 2 for the density (a) and spin (b) modes. The density mode is gapless
for any value of the interlayer separation $h$. The speed of sound of this
channel, as obtained from structural, energetic and thermodynamic quantities,
always yields compatible values throughout the explored range of $h$. Finite-
size effects reduce the speed of sound, which for large $h$ (decoupled layers)
appears to lie below its asymptotic value. The latter is obtained from the
equation of state of a model with a single species at half the density (so
called “atomic” limit). In the computations, the dipolar interaction potential
was truncated at a distance equal to half the size of the simulation box. By
adding the missing “tail” correction to the compressibility, it is possible to
recover correct atomic limit asymptotics, as shown in the figure. The
situation is quite different for the spin channel, where the gap opens for
$h/r_{0}\lesssim 0.35$ and different methods cannot be consistent in that
parameter range. For large values of the interlayer separation,
$h/r_{0}\gtrsim 0.6$, we recover once again the atomic limit and different
quantities are consistent with one another. Manifestly, it is not the case for
parameter range $0.35\lesssim h/r_{0}\lesssim 0.6$, which still corresponds to
a gapless phase but is in the vicinity of the transition point. In this region
there is no consistency between the speeds of sound obtained with different
methods and, importantly, the f-sum rule is not satisfied. The reason for this
is appearance of the superfluid drag which we analyse in more details below.
It is interesting to note that the finite-size effects are more pronounced in
the density mode compared to the spin mode. One way to understand this is that
the spin mode probes the response to the polarisation, which does not change
the system volume, while the density (compression) mode is the response to a
change in volume. The tail correction, being sensitive to the change of the
volume, is able to account for the finite-size discrepancy.

Finally, in order to directly probe the superfluidity properties of the
system, we introduce the winding number related to species $\alpha$,

${\bf
W}_{\alpha}(\tau)=\sum_{i(\alpha)=1}^{N/2}\int_{0}^{\tau}d\tau^{\prime}\frac{d{\bf
r}_{i(\alpha)}(\tau^{\prime})}{d\tau^{\prime}},$ (30)

where $i(\alpha)$ indexes particles belonging to species $\alpha$ only. Taking
the limit of long propagation time, the statistics of the winding numbers are
related to the superfluid densities. In particular, we evaluate the symmetric
and antisymmetric combinations

$\rho_{1}+\rho_{2}\pm 2\rho_{12}=\lim_{\tau\to\infty}\frac{\langle[{\bf
W}_{1}(\tau)\pm{\bf W}_{2}(\tau)]^{2}\rangle}{2N\tau}.$ (31)

According to Eq. (IV), when the plus sign is considered, the quantity above is
an estimator of the total superfluid density of the system, $\rho_{T}$. In our
zero temperature simulation, this quantity is always compatible with the total
mass density, as it should be in continuous space, and in contrast with the
low total superfluidity observed in lattice simulations. When the minus sign
is considered, on the other hand, we directly probe the magnitude of the
superfluid drag. This observable asymptotically attains the value $\rho_{T}$
in the non interacting ($h\to\infty$) limit. In the symmetric mixture case,
borrowing the same notations of Sec. III, this quantity reduces to
$(\rho-\rho_{12})/nm$, and is reported in Fig. 3. For low interactions (large
$h$), it is compatible with unity and drops to zero as interactions are ramped
up. This corresponds to $\rho_{12}=\rho_{T}/4$, i.e., with bound (23). It must
be noted that, for $h<h_{c}\approx 0.35r_{0}$Macia  _et al._ (2014), the
system enters the molecular phase, so that not all the drop in
$\rho-\rho_{12}$ can be ascribed to an increase of the drag, but one must also
keep into account the emergence of the molecular condensate. Indeed, the
description we put forward holds only as long as the system is still in the
atomic phase. Interestingly, the saturation of the bound (23) (or,
alternatively $m^{*}\rightarrow 2m$) appears to coincide with the transition
to the molecular phase in which bound state physics start dominating.

Figure 3:  The quantity $(\rho-\rho_{12})$, extracted from diffusion QMC data
using different estimators. The direct winding number estimator (diamonds and
dashed line to guide the eye) is compared with indirect estimators. The latter
make use of the relations derived in the quantum hydrodynamic model of Sec.
III (cf. also Eq. (32)). The data sets are in satisfactory agreement with one
another, with the exception of the estimator $c_{s}^{2}[S_{s}(k)]\chi_{s}$,
which tends to diverge as the molecular phase is approached (shaded region).
The origin of the errorbars and of remaining finite-size effects are discussed
in the main text.

By independently measuring $S_{s}(k)$, $\chi_{s}$ and $\omega_{s}$ (hence
$c_{s}$), we can show that the usual Bijl-Feynman approximation is not
applicable to systems in the presence of the superfluid drag. To this end, we
use Eqs. (12)-(13) to express $(\rho-\rho_{12})$ in terms of the observables
listed above, and then compare it with the direct winding number measurement.
Figure 3 reports the data corresponding to the three independent expressions

$\frac{\rho-\rho_{12}}{nm}=\frac{4mS^{2}_{s}(k)}{\chi_{s}k^{2}}=2\frac{c_{s}(\omega_{s})S_{s}(k)}{k}=c_{s}^{2}(\omega_{s})\chi_{s},$
(32)

where we are indicating by $c_{s}(\omega_{s})$ the speed of sound in the spin
channel as extracted from the fit to Eq. (29). The data show fair agreement
among the above expressions and between them and the direct measurement of
$\rho-\rho_{12}$. A notable exception are the data for
$c_{s}^{2}[S_{s}(k)]/\chi_{s}$, where $c_{s}[S_{s}(k)]$ is the speed of sound
in the spin channel, computed _as if_ Bijl-Feynman relation held. We observe
that this expression leads to the wrong behaviour in the region where
$\rho-\rho_{12}$ differs from one, i.e., where the drag effect is more
prominent.

In both Figs 2 and 3, the errobars in the quantities extracted from the static
structure factor and the winding number come from statistical averaging.
Spectral frequency, compressibility and susceptibility have additional
contributions to the error due to the use of fitting procedures, Eqs. (25) and
(29). The errors are effectively increased in some of the results, due to
cancellation of opposite trends as a function of $h$. The finite-size effects
are important for a quantitative agreement, as can be seen from Fig. 2a. It is
not obvious that different quantities have similar finite-size correction,
which might eventually be responsible for some remaining differences between
various estimations in Fig. 3.

##  V Dynamic stability

Given the recent advances in measuring the spin superfluidity and its critical
dynamics, we devote the final section to explore the consequences of the
presence of a superfluid drag term on such phenomena. Both long living spin
oscillations Bienaimé  _et al._ (2016) and critical spin superflow Kim  _et
al._ (2017) for Bose-Bose mixtures as well as for Fermi-Bose superfluid
mixtures Delehaye  _et al._ (2015) have been measured in the weakly
interacting regime. The agreement with the available estimates Castin  _et
al._ (2015); Abad  _et al._ (2015) is reasonable but rather far from being
quantitative.

In the following, we determine the critical relative velocity required to
trigger the dynamical instability of a binary mixture of superfluids at zero
temperature. To this end, we generalise Eqs. (2) and (5) and the results of
Refs. Kravchenko and Fil (2008, 2009); Abad  _et al._ (2015) by considering
the energy functional

$\displaystyle E[$ $\displaystyle n_{1},n_{2},\phi_{1},\phi_{2}]=$
$\displaystyle=\int\left\\{\frac{\hbar^{2}}{2}\sum_{\alpha}\left[\frac{m_{\alpha}n_{\alpha}-\rho_{12}}{m_{\alpha}^{2}}(\nabla\phi_{\alpha})^{2}\right]\right.$
$\displaystyle\left.\quad+\frac{\rho_{12}}{m_{1}m_{2}}\nabla\phi_{1}\cdot\nabla\phi_{2}+e(n_{1},n_{2})\,\right\\}d^{D}x$
(33)

with $e(n_{1},n_{2})$ the internal energy density. We subtracted $\rho_{12}$
from the diagonal kinetic terms, so that the condition (3) on the total
density is automatically satisfied. Considering $n_{\alpha}$ and
$\phi_{\alpha}$ as conjugate variables, the Hamilton equations
$m_{\alpha}\partial_{t}n_{\alpha}=\delta E/\delta\phi_{\alpha}$ and
$m_{\alpha}\partial_{t}\phi_{\alpha}=-\delta E/\delta n_{\alpha}$, yield
(implying $\alpha\neq\beta$)

$\displaystyle\partial_{t}n_{\alpha}=-\nabla(n_{\alpha}{\bf
v}_{\alpha})+m_{\alpha}^{-1}\nabla\left[\rho_{12}({\bf v}_{\alpha}-{\bf
v}_{\beta})\right],$ (34) $\displaystyle\partial_{t}{\bf
v}_{\alpha}=-\nabla\frac{v_{\alpha}^{2}}{2}+\nabla\left[\frac{\eta_{\alpha}}{m_{\alpha}}\left(\frac{v_{\alpha}^{2}}{2}-{\bf
v}_{1}\cdot{\bf v}_{2}\right)\right]-\frac{\nabla\mu_{\alpha}}{m_{\alpha}},$
(35)

where ${\bf v}_{\alpha}=(\hbar/m)\nabla\phi_{\alpha}$ are the superfluid
velocities, $\mu_{\alpha}=\partial\epsilon/\partial n_{\alpha}$ the chemical
potentials and we defined $\eta_{\alpha}\equiv\partial\rho_{12}/\partial
n_{\alpha}$, implicitly assuming that $\rho_{12}$ is a well-behaved function
of the densities. The previous system of hydrodynamic equations is satisfied
by a steady state solution of uniform velocity, matter and drag fields, such
that all gradient terms vanish. With a slight change of notation, we perturb
about this solution by expanding the density and velocity fields as

$\displaystyle n_{\alpha}\mapsto\bar{n}_{\alpha}+n_{\alpha}e^{i({\bf
qr}-\omega t)},$ (36) $\displaystyle{\bf v}_{\alpha}\mapsto\bar{{\bf
v}}_{\alpha}+{\bf v}_{\alpha}e^{i({\bf qr}-\omega t)},$ (37)

and only keep the first order in the fluctuations, whereas, since it is
already small with respect to $n_{\alpha}m_{\alpha}$, we only keep the zero-th
order in $\rho_{12}$. Substituting these expansions into Eqs. (34)-(35), we
obtain

$\displaystyle n_{\alpha}[\omega-\bar{{\bf v}}_{\alpha}\cdot{\bf
q}]=\bar{n}_{\alpha}{\bf v}_{\alpha}\cdot{\bf
q}-\frac{\rho_{12}}{m_{\alpha}}({\bf v}_{\alpha}-{\bf v}_{\beta})\cdot{\bf
q},$ (38) $\displaystyle{\bf v}_{\alpha}[\omega-\bar{{\bf
v}}_{\alpha}\cdot{\bf q}]=\frac{{\bf
q}}{m_{\alpha}}\sum_{\gamma=1,2}\mu_{\alpha\gamma}n_{\gamma},$ (39)

where we abbreviated
$\mu_{\alpha\beta}\equiv\partial\mu_{\alpha}/\partial\bar{n}_{\beta}$ (hence
$\mu_{12}=\mu_{21}$, as expected by the symmetry of interactions). In Eq.
(39), we neglected the term proportional to $\eta_{\alpha}$, as the leading
order is $O(\rho_{12}n_{\alpha},\rho_{12}v_{\alpha})$, and the zero-th order,
homogeneous by hypothesis, vanishes under spatial differentiation.

Eliminating $n_{\alpha}$ and $v_{\alpha}$ in the system of equations (38)-(39)
leads to

$\displaystyle\left\\{\Omega_{1}^{2}\Omega_{2}-\left[c_{1}^{2}\Omega_{2}-\rho_{12}\left(\frac{\mu_{11}\Omega_{2}}{m_{1}^{2}}-\frac{\mu_{21}\Omega_{1}}{m_{1}m_{2}}\right)\right]q^{2}\right\\}\left\\{\Omega_{2}^{2}\Omega_{1}-\left[c_{2}^{2}\Omega_{1}-\rho_{12}\left(\frac{\mu_{22}\Omega_{1}}{m_{2}^{2}}-\frac{\mu_{12}\Omega_{2}}{m_{1}m_{2}}\right)\right]q^{2}\right\\}=$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad=q^{4}\left[\bar{n}_{1}\frac{\mu_{12}\Omega_{2}}{m_{1}}-\rho_{12}\left(\frac{\mu_{12}\Omega_{2}}{m_{1}^{2}}-\frac{\mu_{22}\Omega_{1}}{m_{1}m_{2}}\right)\right]\left[\bar{n}_{2}\frac{\mu_{21}\Omega_{1}}{m_{2}}-\rho_{12}\left(\frac{\mu_{21}\Omega_{1}}{m_{2}^{2}}-\frac{\mu_{11}\Omega_{2}}{m_{1}m_{2}}\right)\right],$
(40)

where we set $\Omega_{\alpha}\equiv\omega-\bar{{\bf v}}_{\alpha}\cdot{\bf q}$
and $c_{\alpha}^{2}\equiv\bar{n}_{\alpha}\mu_{\alpha\alpha}/m_{\alpha}$, the
latter being the speed of sound of a single superfluid. We also define

$c_{12}^{4}\equiv\frac{\bar{n}_{1}\bar{n}_{2}}{m_{1}m_{2}}\mu_{12}^{2},$ (41)

which has got the dimension of a speed and measures the strength of inter-
species contact interactions. By assuming a linear dispersion $\omega=Cq$, the
one above is a sixth order equation for the speed of sound $C$ of the mixture,
which determines its stability. In particular, the presence of complex roots
flags a dynamical instability. By setting $\rho_{12}=0$, Eq. (40) reduces to
the problem of the stability of a mixture interacting only via contact
interactions, which was studied in depth in Abad  _et al._ (2015).

This equation simplifies considerably when rewritten in a symmetric frame of
reference (SFR) in which $v/2=\bar{{\bf v}}_{1}\cdot{\bf q}=-\bar{{\bf
v}}_{2}\cdot{\bf q}$, so that the projection of the relative velocity of the
two fluids along ${\bf q}$ is simply $v$. When $n_{1}m_{1}=n_{2}m_{2}$, this
choice corresponds to the frame of reference of the centre of mass of the
system. A brief analysis of the case $v=0$ is reported in the Appendix A.

###  V.1 Symmetric mixture

Some insight into the dynamic stability of the mixture can be gained by
considering a $\mathbb{Z}_{2}$ symmetric mixture, in which
$\bar{n}_{\alpha}=\bar{n}$, $m_{\alpha}=m$ and $\mu_{\alpha\alpha}$ are the
same for both species (hence also the speeds of sound coincide,
$c_{\alpha}=c$). The stability equation (40) then simplifies to a biquadratic
equation whose roots can be readily calculated. Restricting to positive
relative velocities, the condition that the two-fluid speed of sound be real
then yields the critical relative velocities for the stability of the mixture:

$\frac{v_{{\rm c}1}}{c}=2\sqrt{1-\frac{c_{12}^{2}}{c^{2}}},\quad\frac{v_{{\rm
c}2}}{c}=2\sqrt{\left(1+\frac{c_{12}^{2}}{c^{2}}\right)\left(1-4\frac{\rho_{12}}{\rho_{T}}\right)}.$
(42)

For $c_{12}=\rho_{12}=0$, the system is unstable for $v>2c$, confirming the
results of Refs. Kravchenko and Fil (2008, 2009); Castin  _et al._ (2015);
Abad  _et al._ (2015). When the relative velocity $v$ lies within $v_{{\rm
c}1}$ and $v_{{\rm c}2}$, the mixture becomes unstable, as schematically shown
in Fig. 4.

It is worth reminding that the stability analysis presented here is valid
within hydrodynamics, i.e., assuming a linear dispersion relation. The
inclusion of non-linear terms can give rise to finite momentum instabilities
above the upper critical velocity, as it was pointed out in the case of a
Bose-Bose mixture without superfluid drag Ishino  _et al._ (2011). Whether
such finite momentum instability occurs in the same way also in presence of AB
corrections is beyond the scope of the present paper and it will be discuss
elsewhere. Nevertheless for large enough drag,

$\frac{\rho_{12}}{\rho_{T}}>\frac{1}{4}\left[1-\frac{c^{2}-c_{12}^{2}}{c^{2}+c_{12}^{2}}\right].$
(43)

the lower critical velocity is $v_{{\rm c}2}$, which depends on the
entrainment: in this regime, it should be possible observe a shift in the
onset of the dynamical instability. In particular for $\rho_{12}=\rho_{T}/4$,
i.e., when the condition (23) is saturated and the spin speed of sound
vanishes, the mixture is unstable already at vanishing relative velocities.

For completeness, we compare the critical velocities for the dynamic stability
(42) with the speeds of sound of the density and spin modes, Eqs. (9)-(10),
respectively, which provide the critical velocity for the energetic (Landau)
instability. With the notation of this section, they read

$c_{d}^{2}=c^{2}+c_{12}^{2},\qquad
c_{s}^{2}=\left(1-4\frac{\rho_{12}}{\rho_{T}}\right)(c^{2}-c_{12}^{2}),$ (44)

and they are also reported in Fig. 4. Although it occurs at a lower critical
velocity, the energetic instability would not obfuscate the dynamical
instability. A counterflow experiment would only slightly trigger the Landau
instability, which in any case would develop much more slowly than the
dynamical one. Experimental methods based on the generation of soliton trains
Hoefer  _et al._ (2011); Hamner  _et al._ (2011); Kim  _et al._ (2017) were
shown to be able to cleanly identify dynamical instabilities of the kind
discussed in this section.

Figure 4: Stability diagram for a superfluid mixture with contact interactions
such that $c_{12}=0.6$ and all speeds expressed in units of $c$. The spin
speed of sound $c_{s}$ is always below the density speed of sound $c_{d}$ (cf.
Eq. (44)). The former delimits the energetically unstable (EU) region (solid
filling). The dynamically unstable (DU) region (grid filling) lies between the
critical velocities $v_{{\rm c}1}$ and $v_{{\rm c}2}$, cf. Eq. (42). The
threshold relative velocities for both EU and DU vanish for
$\rho_{12}/\rho_{T}=1/4$.

##  VI Conclusions

In the present work, we studied the physics of a superfluid mixture in the
presence of current-current interactions, which lead to the so-called Andreev-
Bashkin effect (AB). Our minimal quantum hydrodynamic theory highlights the
consequences of the presence of the superfluid drag $\rho_{12}$ as an off-
diagonal coefficient of the superfluid density matrix. In particular, for
$\mathbb{Z}_{2}$ symmetric mixtures, in which the density and spin modes
decouple, we predict the density channel to remain unaffected; the spin
channel, on the other hand, exhibits corrections of linear order in
$\rho_{12}$ in both static quantities, such as the spin susceptibility
$\chi_{s}$, and in dynamic quantities, such as the speed of sound of the spin
mode, $c_{s}$. A prominent consequence of these corrections is that Bijl-
Feynman theory, relating $c_{s}$ to $\chi_{s}$, is no longer satisfied in the
presence of AB corrections, as shown by Eq. (13). In light of these results,
we identify the speeds of sound and the susceptibilities as being the most
promising observables in order to experimentally study the AB effect. The
quantum hydrodynamic theory further provides an upper bound to the value of
$\rho_{12}$, which must remain less than a fourth of the total superfluid
density $\rho_{T}=\rho_{1}+\rho_{2}+2\rho_{12}$ (alternatively, in the
language of the effective mass, $m^{*}\leq 2m$). The same bound appears also
in the path integral formulation of superfluid densities, extended to a
mixture of two superfluids, as well as a limiting case of the critical
velocity (42) for the dynamic stability of the mixture.

The minimal toy model employed in Sec. III also gives insight on low but
finite temperature properties such as the specific heat, which is expected to
depend on the drag through a factor $c_{s}^{-D}$ in $D$ dimensions. In
general, finite temperature has a detrimental effect on the possibility of
observing AB-related phenomena Terentjev and Shevchenko (1999); Fil and
Shevchenko (2004), due to the reduction of the superfluid density and to the
emergence of dissipative drag between the two components. In this respect, the
quantities we suggest to focus on to find experimental evidence of the
presence of entrainment are much more suitable than trying to directly detect
the current induced by one component on the other. It is worth mentioning that
a recent experiment Fava  _et al._ (2017) was able to record the dynamics of
both superfluid and normal fractions of a weakly interacting Bose-Bose
mixture, as well as the finite temperature polarisabilities. We believe that
experiments such as this one pave the way towards the detection of subtle
superfluid effects such as the AB drag.

As a check on the predictions of the quantum hydrodynamic model, we
numerically investigated a system of $\mathbb{Z}_{2}$ symmetric bilayer
dipolar bosons, with repulsive interactions within each layer and partially
attractive interactions between the two layers. The diffusion quantum Monte
Carlo method allows the simultaneous measurement of several observables,
separately on the density and spin channels. By inverting the theoretical
relations between the static and dynamic observables, and the superfluid drag,
we can compare indirect estimators of $\rho_{12}$ with the direct estimator
based on the winding numbers. Figure 3 shows a good agreement between direct
and indirect measurements of $(\rho-\rho_{12})/nm$. A notable exception is
when one extracts the speed of sound of the spin channel using the Bijl-
Feynman relation, which, as aforementioned, breaks down if $\rho_{12}\neq 0$.

We finally analysed the stability of the superfluid mixture. The condition for
the onset of the (static) phase separation instability only involves the
density channel, and hence is not modified by $\rho_{12}$. On the other hand,
when the two fluids are put in relative motion, for some values of the
relative velocity the mixture becomes dynamically unstable. The critical
velocities, reported in Eqs. (42), carry a dependence on the superfluid drag.
This result extends the previous ones Kravchenko and Fil (2008, 2009); Abad
_et al._ (2015); Castin  _et al._ (2015) which analysed a system with contact
interactions only.

In the light of recent experimental works Frisch  _et al._ (2015); Delehaye
_et al._ (2015); Roy  _et al._ (2017), it is reasonable to claim that, albeit
the AB is expected to be a subdominant effect, it may still be within reach of
current experimental technology. It further opens up the game to interesting
new phenomenological effects: for instance, if one is able to phase imprint a
vortex on one species of the mixture, one may expect at least part of the
vorticity to be transferred in a dissipationless fashion to the other species.
This _sympathetic stirring_ of a superfluid mixture could become a useful tool
in the study of vortex dynamics in superfluid mixtures. The stability
conditions we put forward in Sec. V and Appendix A may also have consequences
in astrophysics, and in particular on the rotation profiles of neutron star
cores Kobyakov and Pethick (2017).

###### Acknowledgements.

We acknowledge useful discussions with Giovanni Barontini, Stefano Giorgini,
Francesco Minardi, Lode Pollet and Sandro Stringari. G. E. A. acknowledges
partial financial support from the MICINN (Spain) Grant No.
FIS2014-56257-C2-1-P. The Barcelona Supercomputing Center (The Spanish
National Supercomputing Center – Centro Nacional de Supercomputación) is
acknowledged for the computational facilities provided. J. N. acknowledges
support from the European Research Council through FP7/ERC Starting Grant No.
306897. The authors gratefully acknowledge the Gauss Centre for Supercomputing
e.V. (gauss-centre.eu) for funding this project by providing computing time on
the GCS Supercomputer SuperMUC at Leibniz Supercomputing Centre (LRZ, lrz.de).

## References

  * Edwards  _et al._ (1965) D. O. Edwards, D. F. Brewer, P. Seligman, M. Skertic, and M. Yaqub, “Solubility of ${\mathrm{He}}^{3}$ in liquid ${\mathrm{He}}^{4}$ at $0^{\circ}$K,” Phys. Rev. Lett. 15, 773–775 (1965).
  * Alpar  _et al._ (1984) M. A. Alpar, S. A. Langer, and J. A. Sauls, “Rapid postglitch spin-up of the superfluid core in pulsars,” Astrophys. J. 282, 533–541 (1984).
  * Kobyakov and Pethick (2017) D. N. Kobyakov and C. J. Pethick, “Two-component superfluid hydrodynamics of neutron star cores,” Astrophys. J. 836, 203 (2017).
  * Garaud  _et al._ (2014) Julien Garaud, Karl A. H. Sellin, Juha Jäykkä, and Egor Babaev, “Skyrmions induced by dissipationless drag in U(1)$\times{}$U(1) superconductors,” Phys. Rev. B 89, 104508 (2014).
  * Modugno  _et al._ (2002) G. Modugno, M. Modugno, F. Riboli, G. Roati, and M. Inguscio, “Two atomic species superfluid,” Phys. Rev. Lett. 89, 190404 (2002).
  * Catani  _et al._ (2008) J. Catani, L. De Sarlo, G. Barontini, F. Minardi, and M. Inguscio, “Degenerate Bose-Bose mixture in a three-dimensional optical lattice,” Phys. Rev. A 77, 011603 (2008).
  * Roy  _et al._ (2017) Richard Roy, Alaina Green, Ryan Bowler, and Subhadeep Gupta, “Two-element mixture of Bose and Fermi superfluids,” Phys. Rev. Lett. 118, 055301 (2017).
  * Lagoudakis  _et al._ (2009) K. G. Lagoudakis, T. Ostatnický, A. V. Kavokin, Y. G. Rubo, R. André, and B. Deveaud-Plédran, “Observation of half-quantum vortices in an exciton-polariton condensate,” Science 326, 974–976 (2009).
  * Andreev and Bashkin (1976) A. F. Andreev and E. P. Bashkin, “Three-velocity hydrodynamics of superfluid solutions,” Sov. Phys.-JETP 42, 164–167 (1976).
  * Khalatnikov (1957) I. M. Khalatnikov, “Hydrodynamics of solutions of 2 superfluid liquids,” Sov. Phys.-JETP 5, 542–545 (1957).
  * Mineev (1975) V. P. Mineev, “Some problems in the hydrodynamics of solutions of two superfluid liquids,” Sov. Phys.-JETP 40, 338–341 (1975).
  * Meyerovich (1984) A. E. Meyerovich, “Dynamics of superfluid 3He in 3He-4He solutions,” Sov. Phys.-JETP 60, 741–747 (1984).
  * Lattimer and Prakash (2004) J. M. Lattimer and M. Prakash, “The physics of neutron stars,” Science 304, 536–542 (2004).
  * Babaev (2004) Egor Babaev, “Andreev-Bashkin effect and knot solitons in an interacting mixture of a charged and a neutral superfluid with possible relevance for neutron stars,” Phys. Rev. D 70, 043001 (2004).
  * Fil and Shevchenko (2004) D. V. Fil and S. I. Shevchenko, “Drag of superfluid current in bilayer Bose systems,” Low Temp. Phys. 30, 770–777 (2004).
  * Fil and Shevchenko (2005) D. V. Fil and S. I. Shevchenko, “Nondissipative drag of superflow in a two-component Bose gas,” Phys. Rev. A 72, 013616 (2005).
  * Linder and Sudbø (2009) Jacob Linder and Asle Sudbø, “Calculation of drag and superfluid velocity from the microscopic parameters and excitation energies of a two-component Bose-Einstein condensate in an optical lattice,” Phys. Rev. A 79, 063610 (2009).
  * Hofer  _et al._ (2012) Patrick P. Hofer, C. Bruder, and Vladimir M. Stojanović, “Superfluid drag of two-species Bose-Einstein condensates in optical lattices,” Phys. Rev. A 86, 033627 (2012).
  * Leggett (1998) A. J. Leggett, “On the superfluid fraction of an arbitrary many-body system at $T=0$,” J. Stat. Phys. 93, 927–941 (1998).
  * Beattie  _et al._ (2013) Scott Beattie, Stuart Moulder, Richard J. Fletcher, and Zoran Hadzibabic, “Persistent currents in spinor condensates,” Phys. Rev. Lett. 110, 025301 (2013).
  * Abad  _et al._ (2014) M. Abad, A. Sartori, S. Finazzi, and A. Recati, “Persistent currents in two-component condensates in a toroidal trap,” Phys. Rev. A 89, 053602 (2014).
  * Kaurov  _et al._ (2005) V. M. Kaurov, A. B. Kuklov, and A. E. Meyerovich, “Drag effect and topological complexes in strongly interacting two-component lattice superfluids,” Phys. Rev. Lett. 95, 090403 (2005).
  * Giuliani and Vignale (2005) Gabriele Giuliani and Giovanni Vignale,  _Quantum theory of the electron liquid_ (Cambridge University Press, 2005).
  * Giamarchi (2004) Thierry Giamarchi,  _Quantum Physics in One Dimension_ (Oxford University Press, 2004).
  * Schulz  _et al._ (1998) H. J. Schulz, G. Cuniberti, and P. Pieri, “Fermi liquids and Luttinger liquids,” (1998), arXiv:cond-mat/9807366 .
  * Kleine  _et al._ (2008a) A Kleine, C Kollath, I P McCulloch, T Giamarchi, and U Schollwöck, “Excitations in two-component Bose gases,” New J. Phys. 10, 045025 (2008a).
  * Kleine  _et al._ (2008b) A. Kleine, C. Kollath, I. P. McCulloch, T. Giamarchi, and U. Schollwöck, “Spin-charge separation in two-component Bose gases,” Phys. Rev. A 77, 013607 (2008b).
  * (28) L. Parisi and S. Giorgini, private communication .
  * Pollock and Ceperley (1987) E. L. Pollock and D. M. Ceperley, “Path-integral computation of superfluid densities,” Phys. Rev. B 36, 8343–8352 (1987).
  * Terentjev and Shevchenko (1999) S. V. Terentjev and S. I. Shevchenko, “On transfer of motion in a system of two-dimensional superfluid Bose-gases separated by a thin layer,” Low Temp. Phys. 25, 493–502 (1999).
  * Ni  _et al._ (2010) K.-K. Ni, S. Ospelkaus, D. Wang, G. Quemener, B. Neyenhuis, M. H. G. de Miranda, J. L. Bohn, J. Ye, and D. Jin, “Dipolar collisions of polar molecules in the quantum regime,” Nature 464, 1324–1328 (2010).
  * Macia  _et al._ (2014) A. Macia, G. E. Astrakharchik, F. Mazzanti, S. Giorgini, and J. Boronat, “Single-particle versus pair superfluidity in a bilayer system of dipolar bosons,” Phys. Rev. A 90, 043623 (2014).
  * Astrakharchik  _et al._ (2016) G. E. Astrakharchik, R. E. Zillich, F. Mazzanti, and J. Boronat, “Gapped spectrum in pair-superfluid bosons,” Phys. Rev. A 94, 063630 (2016).
  * Klawunn  _et al._ (2010) Michael Klawunn, Alexander Pikovski, and Luis Santos, “Two-dimensional scattering and bound states of polar molecules in bilayers,” Phys. Rev. A 82, 044701 (2010).
  * Frisch  _et al._ (2015) A. Frisch, M. Mark, K. Aikawa, S. Baier, R. Grimm, A. Petrov, S. Kotochigova, G. Quéméner, M. Lepers, O. Dulieu, and F. Ferlaino, “Ultracold dipolar molecules composed of strongly magnetic atoms,” Phys. Rev. Lett. 115, 203201 (2015).
  * Will  _et al._ (2016) Sebastian A. Will, Jee Woo Park, Zoe Z. Yan, Huanqian Loh, and Martin W. Zwierlein, “Coherent microwave control of ultracold ${}^{23}\mathrm{Na}^{40}\mathrm{K}$ molecules,” Phys. Rev. Lett. 116, 225306 (2016).
  * Nascimbene  _et al._ (2015) Sylvain Nascimbene, Nathan Goldman, Nigel R. Cooper, and Jean Dalibard, “Dynamic optical lattices of subwavelength spacing for ultracold atoms,” Phys. Rev. Lett. 115, 140401 (2015).
  * Boronat and Casulleras (1994) J. Boronat and J. Casulleras, “Monte Carlo analysis of an interatomic potential for He,” Phys. Rev. B 49, 8920–8930 (1994).
  * Liu  _et al._ (1974) K. S. Liu, M. H. Kalos, and G. V. Chester, “Quantum hard spheres in a channel,” Phys. Rev. A 10, 303–308 (1974).
  * Casulleras and Boronat (1995) J. Casulleras and J. Boronat, “Unbiased estimators in quantum Monte Carlo methods: Application to liquid ${}^{4}\mathrm{He}$,” Phys. Rev. B 52, 3654–3661 (1995).
  * Astrakharchik  _et al._ (2007) G. E. Astrakharchik, J. Boronat, J. Casulleras, I. L. Kurbakov, and Yu. E. Lozovik, “Weakly interacting two-dimensional system of dipoles: Limitations of the mean-field theory,” Phys. Rev. A 75, 063630 (2007).
  * Bienaimé  _et al._ (2016) Tom Bienaimé, Eleonora Fava, Giacomo Colzi, Carmelo Mordini, Simone Serafini, Chunlei Qu, Sandro Stringari, Giacomo Lamporesi, and Gabriele Ferrari, “Spin-dipole oscillation and polarizability of a binary Bose-Einstein condensate near the miscible-immiscible phase transition,” Phys. Rev. A 94, 063652 (2016).
  * Kim  _et al._ (2017) Joon Hyun Kim, Sang Won Seo, and Y. Shin, “Critical spin superflow in a spinor Bose-Einstein condensate,” Phys. Rev. Lett. 119, 185302 (2017).
  * Delehaye  _et al._ (2015) Marion Delehaye, Sébastien Laurent, Igor Ferrier-Barbut, Shuwei Jin, Frédéric Chevy, and Christophe Salomon, “Critical velocity and dissipation of an ultracold Bose-Fermi counterflow,” Phys. Rev. Lett. 115, 265303 (2015).
  * Castin  _et al._ (2015) Yvan Castin, Igor Ferrier-Barbut, and Christophe Salomon, “The Landau critical velocity for a particle in a Fermi superfluid,” Comptes Rendus Physique 16, 241–253 (2015).
  * Abad  _et al._ (2015) Marta Abad, Alessio Recati, Sandro Stringari, and Frédéric Chevy, “Counter-flow instability of a quantum mixture of two superfluids,” Eur. Phys. J. D 69, 126 (2015).
  * Kravchenko and Fil (2008) L. Yu. Kravchenko and D. V. Fil, “Critical velocities in two-component superfluid Bose gases,” J. Low Temp. Phys. 150, 612–617 (2008).
  * Kravchenko and Fil (2009) L. Y. Kravchenko and D. V. Fil, “Stationary waves in a supersonic flow of a two-component Bose gas,” J. Low Temp. Phys. 155, 219–234 (2009).
  * Ishino  _et al._ (2011) Shungo Ishino, Makoto Tsubota, and Hiromitsu Takeuchi, “Countersuperflow instability in miscible two-component Bose-Einstein condensates,” Phys. Rev. A 83, 063602 (2011).
  * Hoefer  _et al._ (2011) M. A. Hoefer, J. J. Chang, C. Hamner, and P. Engels, “Dark-dark solitons and modulational instability in miscible two-component Bose-Einstein condensates,” Phys. Rev. A 84, 041605 (2011).
  * Hamner  _et al._ (2011) C. Hamner, J. J. Chang, P. Engels, and M. A. Hoefer, “Generation of dark-bright soliton trains in superfluid-superfluid counterflow,” Phys. Rev. Lett. 106, 065302 (2011).
  * Fava  _et al._ (2017) E. Fava, T. Bienaimé, C. Mordini, G. Colzi, C. Qu, S. Stringari, G. Lamporesi, and G. Ferrari, “Spin superfluidity of a Bose gas mixture at finite temperature,” (2017), arXiv:1708.03923 .
  * Pitaevskii and Stringari (2016) Lev Pitaevskii and Sandro Stringari,  _Bose-Einstein condensation and superfluidity_ (Oxford University Press, 2016).

##  Appendix A Quantum hydrodynamics for a non-symmetric mixture

We consider Eq. (5) in the more general case in which the two species are not
symmetric ($\rho_{11}\neq\rho_{22}$, $\alpha_{11}\neq\alpha_{22}$ and
$m_{1}\neq m_{2}$). Hamilton equations lead to the equation of motion for
phase fluctuations,

$\ddot{\phi_{i}}(x)=-\left(\frac{\hbar^{2}}{m_{i}m_{l}}\alpha_{il}\rho_{lj}\right)k^{2}\phi_{j}(x),$
(45)

and the quantity in parentheses can be identified with the matrix $C_{ij}$ of
the squared speeds of sound.

Written with respect to an eigenbasis of $C_{ij}$, the system exhibits two
non-interacting modes which obey a linear dispersion relation
$\omega_{\pm}=c^{2}_{\pm}k^{2}$. The speeds of sound $c_{\pm}^{2}$,
eigenvalues of $C_{ij}$, generalise Eqs. (9)-(10). Obviously, contrary to Sec.
III, the two independent modes are not “pure” density and spin channels, due
to the lack of $\mathbb{Z}_{2}$ symmetry.

In terms of the coefficients of matrix $C_{ij}$, the eigenspeeds of sound read

$\displaystyle c^{2}_{\pm}=$ $\displaystyle\frac{C_{11}+C_{22}}{2}$
$\displaystyle\pm\frac{1}{2}\sqrt{(C_{11}+C_{22})^{2}-4(C_{11}C_{22}-C_{12}C_{21})},$
(46)

The system is stable as long as $c_{\pm}^{2}$ are positive, which leads to the
inequality $C_{11}C_{22}-C_{12}C_{21}>0$, or, in terms of the model’s
coefficients,

$(\alpha_{11}\alpha_{22}-\alpha_{12}^{2})(\rho_{11}\rho_{22}-\rho_{12}^{2})>0$
(47)

The second factor in this equation is always positive, thanks to the bound on
the magnitude of the superfluid drag (cf. discussion leading to Eq. (23)). We
thus obtain a condition on the strength of the density-density interactions,
namely $(\alpha_{11}\alpha_{22}-\alpha_{12}^{2})>0$, thus recovering the well
known criterion for the onset of the phase separation instability Pitaevskii
and Stringari (2016).

##  Appendix B Relation between winding numbers and superfluid densities

Following Pollock and Ceperley (1987), we imagine a slab of fluid sandwiched
between two parallel boundaries, moving with respect to the fluid with
velocity ${\bf v}$. We can write the density matrix of a two-species system in
a frame comoving with the boundaries as

$\displaystyle\rho_{v}(R,R^{\prime},\beta)=\langle R^{\prime}|e^{-\beta
H_{v}}|R\rangle,$ (48) $\displaystyle
H_{v}=\sum_{\alpha=\\{1,2\\}}\sum_{j=1}^{N_{\alpha}}\frac{({\bf
p}_{j}-m_{\alpha}{\bf v})^{2}}{2m_{\alpha}}+V,$ (49)

where $V$ comprises both inter- and intra-species interactions, both assumed
to depend on positions only. (Greek indices identify the species, while Latin
indices count particles). The change in total momentum in the frame of
reference of the boundaries is due to the response of the normal component
$\rho_{N}$ of the fluid, hence, calling $\rho_{0}=\rho_{v=0}$, the density
matrix in the frame of rest of the fluid,

$\frac{\rho_{N}}{\rho_{0}}(N_{1}m_{1}+N_{2}m_{2}){\bf
v}=\frac{\rho_{N}}{\rho_{0}}M{\bf v}=\langle{\bf
P}_{v}\rangle=\frac{\operatorname{tr}[{\bf
P}_{v}\rho_{v}]}{\operatorname{tr}[\rho_{v}]},$ (50)

and is equal to one minus the total superfluid fraction, which can then be
written in terms of the variation of the free energy as

$\frac{\rho_{T}}{\rho_{0}}=\frac{\partial\mathcal{F}_{v}}{\partial(\frac{1}{2}Mv^{2})}.$
(51)

We can write the density matrix in Fourier space and relate the density matrix
$\rho_{0}$ in the frame of rest to $\rho_{v}$,

$\rho_{v}=\exp\left\\{i{\bf
v}\cdot\sum_{\alpha,j}(r_{j\alpha}-r^{\prime}_{j\alpha})m_{\alpha}\right\\}\tilde{\rho}_{0}.$
(52)

Under periodic permutations of the particles,

$\rho_{v}=e^{i{\bf v}\cdot m_{1}{\bf W_{1}}L}e^{i{\bf v}\cdot m_{2}{\bf
W_{2}}L}\tilde{\rho}_{0},$ (53)

whence

$\displaystyle e^{-\beta\Delta\mathcal{F}_{v}}$
$\displaystyle=\frac{1}{\mathcal{Z}}\int\tilde{\rho}_{0}\,e^{i(m_{1}{\bf
W_{1}}+m_{2}{\bf W_{2}})\cdot{\bf v}L}$ $\displaystyle=\langle e^{i(m_{1}{\bf
W_{1}}+m_{2}{\bf W_{2}})\cdot{\bf v}L}\rangle.$ (54)

Expanding to leading order the previous expression, one finally gets

$\displaystyle\Delta\mathcal{F}_{v}=\frac{1}{2}\frac{v^{2}L^{2}}{d\beta}\langle(m_{1}W_{1}+m_{2}W_{2})^{2}\rangle$
(55)

and, using Eq. (51) and $\rho_{0}=ML^{-d}$,

$\displaystyle\rho_{s}^{tot}$
$\displaystyle=\frac{L^{2-d}}{d\beta}\langle(m_{1}W_{1}+m_{2}W_{2})^{2}\rangle,$
(56)

yielding Eq. (IV).


11institutetext: Nanjing University of Science and Technology 22institutetext:
University of Science and Technology of China
22email<EMAIL_ADDRESS>
# U-shaped Vision Mamba for Single Image Dehazing

Zhuoran Zheng 11 Chen Wu 22

###### Abstract

Currently, Transformer is the most popular architecture for image dehazing,
but due to its large computational complexity, its ability to handle long-
range dependency is limited on resource-constrained devices. To tackle this
challenge, we introduce the U-shaped Vision Mamba (UVM-Net), an efficient
single-image dehazing network. Inspired by the State Space Sequence Models
(SSMs), a new deep sequence model known for its power to handle long
sequences, we design a Bi-SSM block that integrates the local feature
extraction ability of the convolutional layer with the ability of the SSM to
capture long-range dependencies. Extensive experimental results demonstrate
the effectiveness of our method. Our method provides a more highly efficient
idea of long-range dependency modeling for image dehazing as well as other
image restoration tasks. The URL of the code is https://github.com/zzr-idam.

###### Keywords:

Transformer Image dehazing UVM-Net SSMs.

##  1 Introduction

Haze is a common atmospheric phenomenon that interferes with people’s daily
life as well as their judgment of goals, and heavy haze can even impact
traffic safety. For computer vision, haze reduces the quality of images in
most cases. It affects the model reliability in advanced vision tasks, further
misleading machine systems, such as autonomous driving. Therefore, to perform
efficient image dehazing through algorithms is a necessary work.

Single image dehazing aims to estimate the sharp image given a hazy input,
which is a highly ill-posed problem. Conventional approaches are physically
inspired and apply various sharp image priors [20, 12, 52, 2] to regularize
the solution space. However, these methods depend on manual priors, and real-
world haze is more complex than the preset priors. Recently, deep learning
approaches have shown promising performance [4, 38, 50, 26, 39, 30, 5, 11, 9,
10, 36, 45, 42, 40], especially image dehazing methods based on Transformer’s
architecture [41].

Although these approaches achieve state-of-the-art results on a single image
dehazing task, their models usually have a large size of parameters and are
computationally expensive. Note that self-attention mechanisms are
quadratically proportional to the size of the input image, thus making them
resource-intensive, especially for hazy images with high resolution that are
usually available. Therefore, how to have efficient modeling of long-range
dependencies of images remains an unresolved issue. Recently, State Space
Sequence Models (SSMs) [16, 19], especially Structured State Space Sequence
Models (S4) [18], have emerged as efficient building blocks (or layers) for
constructing deep networks, and have gained promising performances in the
analysis of continuous long sequence data [15, 18]. Recently, mamba [17]
demonstrated excellent modeling capabilities for long-range dependencies, and
it performed well for image [34] and video [23] classification, and medical
image segmentation [31, 43].

Inspired by U-Mamba [31] and Mamba-UNet [43], we present a U-shaped Vision
Mamba (UVM-Net), which forms a deep network based on the U-Net structure with
both local capture capability and efficient long-range modeling. Unlike
U-Mamba and Mamba-UNet, we propose the Bi-SSM module, where we scroll the
feature maps over the channel domain to fully utilize the long-range modeling
capability of SSM. U-Mamba and Mamba-UNet do not built a long-range dependency
on another dimension of the feature map (the non-channel domain). Extensive
experimental results demonstrate that our method performs well on the task of
image dehazing. UVM-NET paves the way for future advances in network design
that can show additional capabilities in other image restoration tasks.

---
Figure 1: Overview of the UVM-Net architecture. UVM-Net employs the encoder-
decoder framework with UVM-Net blocks in the encoder and convolution blocks in
the decoder, together with skip connections. In UVM-Net block, our feature
maps are first applied to a convolution operation, then the unfolded pixels
are modeled over SSM, and the size of the final feature is reshaped to the
size of the input information.

##  2 Method

UVM-Net follows the encoder-decoder network architecture that captures both
local features and long-range information efficiently. Fig. 1 shows an
overview of the UVM-Net block and the whole network architecture. Next, we
first introduce the Mamba block followed by illustrating the details of UVM-
Net.

Table 1:  Quantitative comparison of various dehazing methods trained on the RESIDE datasets.  Methods | ITS | OTS | RESIDE-6K | RS-Haze | Overhead
---|---|---|---|---|---
SOTS-indoor | SOTS-outdoor | SOTS-mix | RS-Haze-mix
PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | #Param | MACs
(TPAMI’10) DCP [20] | 16.62 | 0.818 | 19.13 | 0.815 | 17.88 | 0.816 | 17.86 | 0.734 | - | -
(TIP’16) DehazeNet [4] | 19.82 | 0.821 | 24.75 | 0.927 | 21.02 | 0.870 | 23.16 | 0.816 | 0.009M | 0.581G
(ECCV’16) MSCNN [38] | 19.84 | 0.833 | 22.06 | 0.908 | 20.31 | 0.863 | 22.80 | 0.823 | 0.008M | 0.525G
(ICCV’17) AOD-Net [26] | 20.51 | 0.816 | 24.14 | 0.920 | 20.27 | 0.855 | 24.90 | 0.830 | 0.002M | 0.115G
(CVPR’18) GFN [39] | 22.30 | 0.880 | 21.55 | 0.844 | 23.52 | 0.905 | 29.24 | 0.910 | 0.499M | 14.94G
(WACV’19) GCANet [5] | 30.23 | 0.980 | - | - | 25.09 | 0.923 | 34.41 | 0.949 | 0.702M | 18.41G
(ICCV’19) GridDehazeNet [30] | 32.16 | 0.984 | 30.86 | 0.982 | 25.86 | 0.944 | 36.40 | 0.960 | 0.956M | 21.49G
(CVPR’20) MSBDN [10] | 33.67 | 0.985 | 33.48 | 0.982 | 28.56 | 0.966 | 38.57 | 0.965 | 31.35M | 41.54G
(ECCV’20) PFDN [11] | 32.68 | 0.976 | - | - | 28.15 | 0.962 | 36.04 | 0.955 | 11.27M | 50.46G
(AAAI’20) FFA-Net [36] | 36.39 | 0.989 | 33.57 | 0.984 | 29.96 | 0.973 | 39.39 | 0.969 | 4.456M | 287.8G
(CVPR’21) AECR-Net [45] | 37.17 | 0.990 | - | - | 28.52 | 0.964 | 35.69 | 0.959 | 2.611M | 52.20G
DehazeFormer-T [41] | 35.15 | 0.989 | 33.71 | 0.982 | 30.36 | 0.973 | 39.11 | 0.968 | 0.686M | 6.658G
DehazeFormer-S [41] | 36.82 | 0.992 | 34.36 | 0.983 | 30.62 | 0.976 | 39.57 | 0.970 | 1.283M | 13.13G
DehazeFormer-B [41] | 37.84 | 0.994 | 34.95 | 0.984 | 31.45 | 0.980 | 39.87 | 0.971 | 2.514M | 25.79G
DehazeFormer-M [41] | 38.46 | 0.994 | 34.29 | 0.983 | 30.89 | 0.977 | 39.71 | 0.971 | 4.634M | 48.64G
DehazeFormer-L [41] | 40.05 | 0.996 | - | - | - | - | - | - | 25.44M | 279.7G
Ours | 40.17 | 0.996 | 34.92 | 0.984 | 31.92 | 0.982 | 39.88 | 0.972 | 19.25M | 173.55G

###  2.1 UMV-Net

Mamba has demonstrated impressive results on a variety of discrete data, but
it remains underexplored in the field of image dehazing. Here, we take
advantage of Mamba’s linear scaling to enhance the long-range dependency
modeling of traditional U-Net.

As shown in Fig. 1, each building block contains two successive convolution
blocks [21] followed by a UVM-net block. The convolution block contains the
standard convolutional layer ($\times 3$) with Leaky ReLU [32]. Image features
with a shape of $(B,C,H,W)$ are then flattened and transposed to $(B,L,C)$
where $L=H\times W\times D$. After passing the Layer Normalization [1], the
features enter the Mamba block [17] that contains two parallel branches. In
the first branch, the features are expanded to $(B,L,C)$ by a SSM layer, next,
the roll transforms into an $(B,C,L)$ before entering an SSM. In the second
branch, the features are also expanded to $(B,L,C)$ by a SSM layer, next, the
feature map goes through softmax to form an attention map. Then, the features
from the two branches are merged with the Hadamard product. Finally, the
features are projected back to the original shape $(B,L,C)$ and then reshaped
and transposed to $(B,C,H,W)$. The entire model down-sampling and up-sampling
of the feature map is consistent with standard U-Net, notably the bilinear
interpolation used for up-sampling.

##  3 Experiments

The setup of the experiment follows exactly Dehazeformer’s [41] configuration
in the paper, including the number of iterations, the learning rate, and the
batch size. This is because it is currently the SOTA in the field of single-
image dehazing and our structure is similar to it. For this paper, we used
PyTorch 1.8 and the GPU was a 3090RTX shader with 48G RAM. We use the number
of parameters (#Param) and multiply-accumulate operations (MACs) to measure
the overhead. MACs are measured on $256\times 256$ images.

###  3.1 Quantitative Comparison

We quantitatively compare the performance of UVM-Net and baselines, and the
results are shown in TABLE 1. Here we bold the results where UVM-Net exceeds
them. Overall, our proposed UVM-Net outperformed these baselines.

###  3.2 Qualitative Comparison

We compared with dehazeformer on the publicly available dataset and the real-
world image with haze, where dehazeformer is over-enhanced on the publicly
available dataset.

Figure 2:  Qualitative comparison of image dehazing methods on SOTS mix set,
where the first rows are outdoor images, and the second row is indoor images.
The third and fourth rows are real-world images. The first column is the hazy
images and the last is the corresponding ground truth.

###  3.3 Ablation Study

We perform ablation studies on the RESIDE-Full’s indoor scene. We conducted
two ablation experiments where 1) we removed the SSM to use 1D convolution
instead, and 2) we removed the SSM to use scaled dot product instead. The
experimental results show the effectiveness of our method.

Table 2:  Ablation study on SSM module.  Setting | PSNR | SSIM | #Param | MACs
---|---|---|---|---
Ours | 39.88 | 0.972 | 19.25M | 173.55G
1D conv | 35.11 | 0.950 | 10.66M | 88.93G
SDP | 38.25 | 0.968 | 23.88M | 229.93G
Table 3: Compare quantization with the state-of-the-art image enhancement
methods in LOL and MIT-Adobe FiveK (MIT5K).

| BIMEF | CRM | LIME | RetinexNet | MBLLEN | DSLR | DRBN | ZeroDCE++ | KinD++ | TFR | Ours
---|---|---|---|---|---|---|---|---|---|---|---
LOL | PSNR $\uparrow$ | 13.88 | 17.20 | 16.76 | 16.77 | 17.56 | 18.24 | 20.13 | 19.43 | 21.30 | 22.92 | 24.32
SSIM $\uparrow$ | 0.577 | 0.644 | 0.564 | 0.567 | 0.736 | 0.787 | 0.802 | 0.768 | 0.822 | 0.837 | 0.859
NIQE $\downarrow$ | 7.69 | 8.02 | 9.13 | 9.73 | 3.46 | 4.11 | 4.63 | 7.79 | 5.11 | 4.03 | 3.92
MIT5K | PSNR $\uparrow$ | 18.67 | 13.99 | 11.21 | 20.81 | 16.42 | 17.02 | 20.95 | 16.46 | 22.01 | 25.03 | 25.78
SSIM $\uparrow$ | 0.693 | 0.674 | 0.667 | 0.687 | 0.851 | 0.750 | 0.794 | 0.766 | 0.832 | 8.521 | 0.925
NIQE $\downarrow$ | 3.87 | 4.22 | 4.50 | 4.48 | 4.19 | 3.90 | 5.44 | 3.92 | 4.15 | 3.49 | 3.40

Table 4: Comparison of quantitative results on five benchmark datasets. Best
and second best values are indicated with bold text and underlined text
respectively.

Datasets | Test100 | Rain100H | Rain100L | Test1200 | Test2800 | Average
---|---|---|---|---|---|---
Methods | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM
DerainNet [13] | 22.77 | 0.810 | 14.92 | 0.592 | 27.03 | 0.884 | 24.31 | 0.861 | 23.38 | 0.835 | 22.48 | 0.796
SEMI [44] | 22.35 | 0.788 | 16.56 | 0.486 | 25.03 | 0.842 | 24.43 | 0.782 | 26.05 | 0.822 | 22.88 | 0.744
DIDMDN [51] | 22.56 | 0.818 | 17.35 | 0.524 | 25.23 | 0.741 | 28.13 | 0.867 | 29.95 | 0.901 | 24.58 | 0.770
UMRL [48] | 24.41 | 0.829 | 26.01 | 0.832 | 29.18 | 0.923 | 29.97 | 0.905 | 30.55 | 0.910 | 28.02 | 0.880
RESCAN [28] | 25.00 | 0.835 | 26.36 | 0.786 | 29.80 | 0.881 | 31.29 | 0.904 | 30.51 | 0.882 | 28.59 | 0.857
PReNet [37] | 24.81 | 0.851 | 26.77 | 0.858 | 32.44 | 0.950 | 31.75 | 0.916 | 31.36 | 0.911 | 29.42 | 0.897
MSPFN [24] | 27.50 | 0.876 | 28.66 | 0.860 | 32.40 | 0.933 | 32.82 | 0.930 | 32.39 | 0.916 | 30.75 | 0.903
MPRNet [49] | 30.27 | 0.897 | 30.41 | 0.890 | 36.40 | 0.965 | 33.64 | 0.938 | 32.91 | 0.916 | 32.73 | 0.921
KiT [25] | 30.26 | 0.904 | 30.47 | 0.897 | 36.65 | 0.969 | 33.85 | 0.941 | 32.81 | 0.918 | 32.81 | 0.929
DGUNet [33] | 30.32 | 0.899 | 30.66 | 0.891 | 37.42 | 0.969 | 33.68 | 0.938 | 33.23 | 0.920 | 33.06 | 0.923
IDT [46] | 29.69 | 0.905 | 29.95 | 0.898 | 37.01 | 0.971 | 33.38 | 0.937 | 31.38 | 0.908 | 32.28 | 0.924
HINet [6] | 30.26 | 0.905 | 30.63 | 0.893 | 37.20 | 0.969 | 33.87 | 0.940 | 33.01 | 0.918 | 33.00 | 0.925
DCT [29] | 30.91 | 0.912 | 30.74 | 0.892 | 38.19 | 0.974 | 33.89 | 0.941 | 33.57 | 0.926 | 33.46 | 0.929
SFNet [8] | 31.47 | 0.919 | 31.90 | 0.908 | 38.21 | 0.974 | 33.69 | 0.937 | 32.55 | 0.911 | 33.56 | 0.929
Ours | 32.11 | 0.915 | 30.66 | 0.909 | 39.66 | 0.985 | 34.92 | 0.942 | 33.89 | 0.952 | 33.98 | 0.940

##  4 Other task

Low light enhancement. We evaluate the performance of our method on two paired
public datasets, including the LOL dataset [7] and the MIT-Adobe FiveK dataset
[3] (see Table 3). We randomly select 450 pairs divided in the dataset for
training and the other 50 pairs for testing. Each pair contains a low-light
image and its corresponding well-exposed reference image. MIT-Adobe FiveK
contains 5000 images recorded with a DSLR camera, and the tonal attributes of
all images are manually adjusted by five photographic experts (A/B/C/D/E).
Reference with [22], we use the retouched results of expert C as the ground
truth. We manually divide the MIT-Adobe FiveK dataset into two parts at a
ratio of 8:2, using the first 4000 images to train the model and randomly
selecting 500 images from the remaining images for testing. We uniformly
convert the raw images in the above dataset to PNG format for training and
testing. The Implementation of the environment uses the PyTorch [35] open-
source framework, trained and tested on Intel(R) Xeon(R) Gold 5218 CPU @
2.30GHz, 128G RAM, and TITAN RTX3090 GPU with 24G RAM. We set the batch size
to 12 and trained our model using the AdamW optimizer with $\beta_{1}$ = 0.9,
$\beta_{2}$ = 0.999, and $\epsilon=10^{-8}$. The learning rate is initialized
to 0.0001, and 200 epochs are enforced, decaying by a factor of 0.5 in every
50 epochs.

Deraining. we conduct extensive experiments on the Rain13K training dataset
which contains 13700 clean/rain image pairs. For the testing, five synthetic
benchmarks (Test100 [27], Rain100H [47], Rain100L [47], Test2800 [14], and
Test1200 [14]) are considered for evaluation. For each image used, we use BLIP
to generate corresponding text descriptions. We calculate the PSNR and SSIM
along the Y channel in the YCbCr color space as quantitative comparisons. We
compare our method with 14 image deraining approaches, including DerainNet
[13], SEMI [44], DIDMDN [51], UMRL [48], RESCAN [28], PReNet [37], MSPFN [24],
MPRNet [49], KiT [25], DGUNet [33], IDT [46], HINet [6], DCT [29] and SFNet
[8]. In Tab. 4, we report the quantitative comparison results among different
methods. We conduct experiments in PyTorch on NVIDIA GeForce RTX 3090 GPUs.
The network is trained for a total of 300k iterations with a batch size of 8.
The initial learning rate is $2\times 10^{-4}$, and the Adam optimizer is used
to decay with the cosine annealing schedule. During training, we utilize
cropped patches of size 128 $\times$ 128 as input, and to augment the training
data, random horizontal and vertical flips are applied to the input images.

##  5 Discussion

Our model is a potentially large visual model (3.9G weight file). Limited by
the consumer GPU’s RAM, the fixed weight of 64 can be increased to 256 if the
number of GPUs is sufficient.

##  6 Conclusion

In this paper, we present a new architecture, UVM-Net, for single image
dehazing, which integrates the advantages of local mode recognition of CNN and
global context understanding of SSM. The results show that UVM-Net has the
promise to be the backbone of promising image restoration networks for the
next generation.

## References

  * [1] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016)
  * [2] Berman, D., Avidan, S., et al.: Non-local image dehazing. In: CVPR. pp. 1674–1682 (2016)
  * [3] Bychkovsky, V., Paris, S., Chan, E., Durand, F.: Learning photographic global tonal adjustment with a database of input/output image pairs. In: CVPR 2011. pp. 97–104. IEEE (2011)
  * [4] Cai, B., Xu, X., Jia, K., Qing, C., Tao, D.: Dehazenet: An end-to-end system for single image haze removal. IEEE TIP 25(11), 5187–5198 (2016)
  * [5] Chen, D., He, M., Fan, Q., Liao, J., Zhang, L., Hou, D., Yuan, L., Hua, G.: Gated context aggregation network for image dehazing and deraining. In: WACV. pp. 1375–1383 (2019)
  * [6] Chen, L., Lu, X., Zhang, J., Chu, X., Chen, C.: Hinet: Half instance normalization network for image restoration. In: CVPR. pp. 182–192 (2021)
  * [7] Chen Wei, Wenjing Wang, W.Y.J.L.: Deep retinex decomposition for low-light enhancement. In: British Machine Vision Conference (2018)
  * [8] Cui, Y., Tao, Y., Bing, Z., Ren, W., Gao, X., Cao, X., Huang, K., Knoll, A.: Selective frequency network for image restoration. In: ICLR (2022)
  * [9] Deng, Q., Huang, Z., Tsai, C.C., Lin, C.W.: Hardgan: A haze-aware representation distillation gan for single image dehazing. In: ECCV. pp. 722–738. Springer (2020)
  * [10] Dong, H., Pan, J., Xiang, L., Hu, Z., Zhang, X., Wang, F., Yang, M.H.: Multi-scale boosted dehazing network with dense feature fusion. In: CVPR. pp. 2157–2167 (2020)
  * [11] Dong, J., Pan, J.: Physics-based feature dehazing networks. In: ECCV. pp. 188–204. Springer (2020)
  * [12] Fattal, R.: Dehazing using color-lines. ACM TOG 34(1), 1–14 (2014)
  * [13] Fu, X., Huang, J., Ding, X., Liao, Y., Paisley, J.: Clearing the skies: A deep network architecture for single-image rain removal. TIP 26(6), 2944–2956 (2017)
  * [14] Fu, X., Huang, J., Zeng, D., Huang, Y., Ding, X., Paisley, J.: Removing rain from single images via a deep detail network. In: CVPR. pp. 3855–3863 (2017)
  * [15] Goel, K., Gu, A., Donahue, C., Ré, C.: It’s raw! audio generation with state-space models. In: International Conference on Machine Learning. pp. 7616–7633 (2022)
  * [16] Gu, A.: Modeling Sequences with Structured State Spaces. Phd thesis, Stanford University (2023), proQuest Document ID: 2880853867
  * [17] Gu, A., Dao, T.: Mamba: Linear-time sequence modeling with selective state spaces. arXiv preprint arXiv:2312.00752 (2023)
  * [18] Gu, A., Goel, K., Re, C.: Efficiently modeling long sequences with structured state spaces. In: International Conference on Learning Representations (2021)
  * [19] Gu, A., Johnson, I., Goel, K., Saab, K., Dao, T., Rudra, A., Ré, C.: Combining recurrent, convolutional, and continuous-time models with linear state space layers. Advances in Neural Information Processing Systems 34, 572–585 (2021)
  * [20] He, K., Sun, J., Tang, X.: Single image haze removal using dark channel prior. IEEE TPAMI 33(12), 2341–2353 (2010)
  * [21] He, K., Zhang, X., Ren, S., Sun, J.: Deep residual learning for image recognition. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. pp. 770–778 (2016)
  * [22] Hu, Y., He, H., Xu, C., Wang, B., Lin, S.: Exposure: A white-box photo post-processing framework. ACM Trans. Graph. 37(2), 1–17 (2018)
  * [23] Islam, M.M., Bertasius, G.: Long movie clip classification with state-space video models. In: European Conference on Computer Vision. pp. 87–104 (2022)
  * [24] Jiang, K., Wang, Z., Yi, P., Chen, C., Huang, B., Luo, Y., Ma, J., Jiang, J.: Multi-scale progressive fusion network for single image deraining. In: CVPR. pp. 8346–8355 (2020)
  * [25] Lee, H., Choi, H., Sohn, K., Min, D.: Knn local attention for image restoration. In: CVPR. pp. 2139–2149 (2022)
  * [26] Li, B., Peng, X., Wang, Z., Xu, J., Feng, D.: Aod-net: All-in-one dehazing network. In: ICCV. pp. 4770–4778 (2017)
  * [27] Li, S., Araujo, I.B., Ren, W., Wang, Z., Tokuda, E.K., Junior, R.H., Cesar-Junior, R., Zhang, J., Guo, X., Cao, X.: Single image deraining: A comprehensive benchmark analysis. In: CVPR. pp. 3838–3847 (2019)
  * [28] Li, X., Wu, J., Lin, Z., Liu, H., Zha, H.: Recurrent squeeze-and-excitation context aggregation net for single image deraining. In: ECCV. pp. 254–269 (2018)
  * [29] Li, Y., Lu, J., Chen, H., Wu, X., Chen, X.: Dilated convolutional transformer for high-quality image deraining. In: CVPR. pp. 4198–4206 (2023)
  * [30] Liu, X., Ma, Y., Shi, Z., Chen, J.: Griddehazenet: Attention-based multi-scale network for image dehazing. In: ICCV. pp. 7314–7323 (2019)
  * [31] Ma, J., Li, F., Wang, B.: U-mamba: Enhancing long-range dependency for biomedical image segmentation. arXiv preprint arXiv:2401.04722 (2024)
  * [32] Maas, A.L., Hannun, A.Y., Ng, A.Y., et al.: Rectifier nonlinearities improve neural network acoustic models. In: International Conference on Machine Learning. vol. 28 (2013)
  * [33] Mou, C., Wang, Q., Zhang, J.: Deep generalized unfolding networks for image restoration. In: CVPR. pp. 17399–17410 (2022)
  * [34] Nguyen, E., Goel, K., Gu, A., Downs, G., Shah, P., Dao, T., Baccus, S., Ré, C.: S4nd: Modeling images and videos as multidimensional signals with state spaces. In: Advances in Neural Information Processing Systems. vol. 35, pp. 2846–2861 (2022)
  * [35] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. arXiv preprint arXiv:1912.01703 (2019)
  * [36] Qin, X., Wang, Z., Bai, Y., Xie, X., Jia, H.: Ffa-net: Feature fusion attention network for single image dehazing. In: AAAI. pp. 11908–11915 (2020)
  * [37] Ren, D., Zuo, W., Hu, Q., Zhu, P., Meng, D.: Progressive image deraining networks: A better and simpler baseline. In: CVPR. pp. 3937–3946 (2019)
  * [38] Ren, W., Liu, S., Zhang, H., Pan, J., Cao, X., Yang, M.H.: Single image dehazing via multi-scale convolutional neural networks. In: ECCV. pp. 154–169 (2016)
  * [39] Ren, W., Ma, L., Zhang, J., Pan, J., Cao, X., Liu, W., Yang, M.H.: Gated fusion network for single image dehazing. In: CVPR. pp. 3253–3261 (2018)
  * [40] Shao, Y., Li, L., Ren, W., Gao, C., Sang, N.: Domain adaptation for image dehazing. In: CPVR. pp. 2808–2817 (2020)
  * [41] Song, Y., He, Z., Qian, H., Du, X.: Vision transformers for single image dehazing. IEEE Transactions on Image Processing 32, 1927–1941 (2023)
  * [42] Wang, C., Shen, H.Z., Fan, F., Shao, M.W., Yang, C.S., Luo, J.C., Deng, L.J.: Eaa-net: A novel edge assisted attention network for single image dehazing. KBS 228, 107279 (2021)
  * [43] Wang, Z., Zheng, J.Q., Zhang, Y., Cui, G., Li, L.: Mamba-unet: Unet-like pure visual mamba for medical image segmentation. arXiv preprint arXiv:2402.05079 (2024)
  * [44] Wei, W., Meng, D., Zhao, Q., Xu, Z., Wu, Y.: Semi-supervised transfer learning for image rain removal. In: CVPR. pp. 3877–3886 (2019)
  * [45] Wu, H., Qu, Y., Lin, S., Zhou, J., Qiao, R., Zhang, Z., Xie, Y., Ma, L.: Contrastive learning for compact single image dehazing. In: CVPR. pp. 10551–10560 (2021)
  * [46] Xiao, J., Fu, X., Liu, A., Wu, F., Zha, Z.J.: Image de-raining transformer. TPAMI (2022)
  * [47] Yang, W., Tan, R.T., Feng, J., Liu, J., Guo, Z., Yan, S.: Deep joint rain detection and removal from a single image. In: CVPR. pp. 1357–1366 (2017)
  * [48] Yasarla, R., Patel, V.M.: Uncertainty guided multi-scale residual learning-using a cycle spinning cnn for single image de-raining. In: CVPR. pp. 8405–8414 (2019)
  * [49] Zamir, S.W., Arora, A., Khan, S., Hayat, M., Khan, F.S., Yang, M.H., Shao, L.: Multi-stage progressive image restoration. In: CVPR. pp. 14821–14831 (2021)
  * [50] Zhang, H., Patel, V.M.: Densely connected pyramid dehazing network. In: CVPR. pp. 3194–3203 (2018)
  * [51] Zhang, H., Patel, V.M.: Density-aware single image de-raining using a multi-stream dense network. In: CVPR. pp. 695–704 (2018)
  * [52] Zhu, Q., Mai, J., Shao, L.: A fast single image haze removal algorithm using color attenuation prior. IEEE TIP 24(11), 3522–3533 (2015)