# Mixture of two unequally charged superfluids in a magnetic field

S. Seyyare Aksu  Department of Physics, Mimar Sinan Fine Arts University,
Bomonti 34380, Istanbul, Turkey  A. Levent Subaşı  Department of Physics,
Istanbul Technical University, Maslak 34469, Istanbul, Turkey  Center for
Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico
87545, USA  Nader Ghazanfari<EMAIL_ADDRESS>Department of Physics,
Mimar Sinan Fine Arts University, Bomonti 34380, Istanbul, Turkey

###### Abstract

The artificial magnetic fields engineered for ultra cold gases depend on the
internal structure of the neutral atoms. Therefore the components of a mixture
composed of two atomic gases can exhibit a different response to an artificial
magnetic field. Such a mixture can be interpreted as a mixture of two atomic
gases, carrying different synthetic charges. We consider such mixtures of two
superfluids with unequal synthetic charges in a ring trap subject to a uniform
artificial magnetic field. The charge imbalance in such a mixture changes the
distribution of excited particles over angular momentum states compared to
that of an equally charged mixture. This microscopic difference exhibits
macroscopic consequences, such as the occurrence of an angular momentum
transfer between two unequally charged components. Due to the inter-fluid
atomic interactions in a ring, the angular momentum transfer can create a
counter flowing persistent current in the weakly charged superfluid. Even in
the limiting case of a charged and an uncharged superfluid mixture, a
persistent current can be induced in the uncharged superfluid, despite the
fact that it is not directly coupled to the magnetic field. The stability
analysis shows that the induction depends on the interplay between the inter-
fluid interaction and the applied magnetic field. We obtain instability
boundaries of the system and construct phase diagrams as a function of the
inter-fluid interaction and the magnetic field. We investigate these
properties employing the Bogoliubov approximation.

##  I Introduction

Over the last three decades experimental achievements and developments in the
field of ultracold atomic Bose and Fermi gases have provided a wide control
over different parameters of these systems. The interaction strength can be
finely tuned and different trap geometries can be generated in the experiments
for ultracold atomic and molecular systems. Another breakthrough achievement
in this field, which came almost a decade ago, has also allowed the coupling
of these charge neutral systems to artificial magnetic fields. This coupling
becomes possible via a light-induced synthetic magnetic field Lin  _et al._
(2009a, b).

Various schemes are used to generate synthetic magnetic fields Juzeliūnas  _et
al._ (2006); Zhu  _et al._ (2006); Spielman (2009); Günter  _et al._ (2009).
However, the general idea is based upon the coupling of a laser beam with the
internal states of the atom which gives rise to a geometric phase for the
center-of-mass motion. This phase corresponds to an effective vector
potential, which is varying in space, and hence can mimic the effect of a
magnetic field on a charged particle. The engineered artificial magnetic field
depends on the internal structure of the neutral atoms. Therefore, the
components of a mixture composed of two different atomic species or of similar
atoms with two different hyperfine states can exhibit a different response to
an applied artificial magnetic field. In other words, such a mixture can be
interpreted as a mixture of two atomic gases, each carrying different
synthetic charges subject to an external artificial magnetic field Ünal and
Oktel (2016). With the prospects of artificial gauge fields, the realization
of a mixture of two superfluids with different synthetic charges under an
artificial magnetic field is therefore an experimental possibility.

Superfluid mixtures have been studied and realized in different geometries
Ishino  _et al._ (2013); Shimodaira  _et al._ (2010); Ticknor (2013);
Ghazanfari  _et al._ (2014); Bandyopadhyay  _et al._ (2017); Ramanathan  _et
al._ (2011); Moulder  _et al._ (2012); Murray  _et al._ (2013); Beattie  _et
al._ (2013) including toroidal trapping potentials where the fluid can be
modeled as a one-dimensional system on a ring geometry. Different properties
of this system have been studied in great detail; persistent currents have
been created Rokhsar (1997); Bargi  _et al._ (2010); Baharian and Baym (2013);
Yakimenko  _et al._ (2013); Abad  _et al._ (2014); White  _et al._ (2016);
Gupta  _et al._ (2005); Ryu  _et al._ (2007); Halkyard  _et al._ (2010);
Ramanathan  _et al._ (2011); Moulder  _et al._ (2012); Murray  _et al._
(2013); Beattie  _et al._ (2013); Wright  _et al._ (2013a, b); Ryu  _et al._
(2013); Neely  _et al._ (2013); Jendrzejewski  _et al._ (2014); Corman  _et
al._ (2014); Ryu  _et al._ (2014); Eckel  _et al._ (2014) and even a long-
lived persistent flow of atoms has been observed Gupta  _et al._ (2005);
Moulder  _et al._ (2012); Murray  _et al._ (2013); Beattie  _et al._ (2013).
Moreover, rotational properties and different instabilities of these systems
have been scrutinized. However, here we consider a mixture of two unequally
charged superfluids for these systems. Motivated by Ünal and Oktel (2016) and
considering the recent experimental realizations of a persistent current in a
ring trap, we study the properties of the unequally charged superfluid
mixtures.

In a mixture of two unequally charged superfluids, the charge imbalance
qualitatively changes the distribution of the excited particles. As a result
of this change a highly charged superfluid can transfer the effect of Lorentz
force to the weakly charged superfluid via atomic interactions. Thus, the gas
which is weakly coupled to the magnetic field can be accelerated by the
strongly coupled one. Therefore, a transfer of angular momentum happens
between two superfluids. The process of the angular momentum transfer becomes
most striking when we consider the limiting case of a charged and an uncharged
superfluid mixture. Here, a vortex trapped in the ring, which is the state of
a persistent current around the ring, can be generated in the uncharged
superfluid despite the fact that it is not directly coupled to the magnetic
field. In the case of a ring trap the induction of a persistent current
corresponds to a non-dissipative drag as has been demonstrated for some
superfluids Andreev and Bashkin (1975); Tanatar and Das (1996); Fil and
Shevchenko (2005).

The above mentioned angular momentum transfer between two superfluids and
hence the possibility of inducing a persistent current to an uncharged
superfluid from a charged one can be studied under the Bogoliubov
approximation. In the Gross-Pitaevskii approach, where only density-density
interactions are considered, the phases of the superfluid wave functions do
not enter the energy expression. The absence of these phases prevents any sort
of interplay between the velocity fields of the superfluids. Therefore, at the
Gross-Pitaevskii level it is not possible to observe an angular momentum
transfer between two unequally charged superfluids. Consequently, the
induction of a vortex from a charged superfluid to an uncharged superfluid
under a magnetic field becomes impossible. However, the Bogoliubov approach
that takes excited particles into account allows for an angular momentum
transfer.

Following the discussion given above, in this paper we consider a mixture of
two superfluids with different synthetic charges in a ring trap subject to a
common uniform magnetic field. The paper is organized as follows: In Sec. II,
we review the instabilities and the angular momentum properties of a charged
superfluid in a magnetic field. Understanding the properties of a single
superfluid will be helpful when we study a mixture of two charged superfluids
in the next section. Then, in Sec. III, we introduce the mixture of two
unequally charged superfluids subject to a magnetic field and obtain the
excitation spectrum of the system. The energetic and dynamical instabilities
of different mixture types are studied in Sec. IV from the obtained excitation
spectrum. In Sec. V, we calculate the angular momentum properties of the
superfluid mixtures, and discuss the effect of charge imbalance on these
properties. Using the instability analysis illustrated in phase diagrams, the
conditions to induce a persistent current from a charged superfluid to an
uncharged superfluid is presented in Sec. VI. Finally, we summarize and
discuss our results in Sec. VII.

##  II A charged superfluid in a magnetic field

In this section we consider a single component superfluid consisting of $N$
particles each having mass $M$ and synthetic charge which is given as $Q$
times the unit charge $e$. The system is trapped in a quasi-one-dimensional
torus geometry with circumference $L=2\pi R$ (with $R$ being the radius of the
ring) and cross-section area $S=\pi r^{2}$, where the tube radius $r\ll R$.
The s-wave scattering length is assumed to be smaller than the trapping
dimensions, i.e. $a_{s}\ll r,R$, so that the low energy interactions can be
derived from the three-dimensional contact potential model as

$\mathpzc{g}^{1D}=\frac{4\pi\hbar^{2}a_{s}}{M}\frac{1}{S},$ (1)

and $\mathpzc{g}^{1D}N/L$ gives the mean interaction energy per particle. The
system is under a uniform magnetic field (${\bf B}=B_{0}\hat{e}_{z}$) along
the central axis of the ring, generated by a vector potential ${\cal
A}=\frac{B_{0}\rho}{2}|_{\rho=R}\hat{e}_{\theta}$. The many-body Hamiltonian
describing this system in the plane-wave basis along the circumference of the
ring can be written as

$\displaystyle\mathcal{H}=\sum_{k}\frac{\hbar^{2}}{2MR^{2}}\left(k-Q\Phi\right)^{2}a^{\dagger}_{k}a_{k}$
$\displaystyle+\;\frac{\mathpzc{g}^{1D}}{2L}\sum_{\begin{subarray}{c}k,k^{\prime},q\end{subarray}}a^{\dagger}_{k}a^{\dagger}_{k^{\prime}}$
$\displaystyle a_{k^{\prime}-q}a_{k+q}.$ (2)

Here, $\Phi=\pi R^{2}B_{0}/\Phi_{0}$ is the number of the magnetic flux quanta
passing through the area enclosed by the ring where $\Phi_{0}=h/e$ is the
magnetic flux quantum. Note that because of the periodic boundary condition
the momentum states carry a quantized angular momentum of $\hbar k$, where the
quantum number $k$ takes on integer values. The Hamiltonian can be non-
dimensionalized using the characteristic energy scale $\hbar^{2}/(2MR^{2})$
which corresponds to the energy difference between the ground and the first-
excited state of a single particle in the ring. Accordingly, the dimensionless
Hamiltonian becomes

$H=\sum_{k}\left(k-Q\Phi\right)^{2}a^{\dagger}_{k}a_{k}+\dfrac{g}{2}\sum_{k,k^{{}^{\prime}},q}a^{\dagger}_{k}a^{\dagger}_{k^{\prime}}a_{k^{\prime}-q}a_{k+q},$
(3)

where $g=2MR^{2}\mathpzc{g}^{1D}/\hbar L=4Ra_{s}/S$. As mentioned in Sec. I,
the Bogoliubov approximation provides the mechanism to investigate the
excitation spectrum of the system as well as its angular momentum properties.
According to the Bogoliubov approach a large fraction of the particles is
assumed to be condensed in the lowest-energy single-particle state and the
rest are distributed over higher-energy quantum states, so that

$\displaystyle N=\overline{N}_{0}+N_{ex}=\langle
a^{\dagger}_{k_{0}}a_{k_{0}}\rangle+\sum_{k\neq k_{0}}a^{\dagger}_{k}a_{k}.$
(4)

Here $k_{0}$ denotes the minimum energy single particle state where the
condensate resides and $N_{ex}\ll N_{0}$. This approximation leads us to a
quadratic Hamiltonian of the form

$\displaystyle H$ $\displaystyle=$
$\displaystyle\left(k_{0}-Q\Phi\right)^{2}N+\frac{gN^{2}}{2}+\sum_{q\neq
0}\left(\epsilon_{q}+U\right)a^{\dagger}_{k_{0}+q}a_{k_{0}+q}$ (5)
$\displaystyle+$ $\displaystyle\frac{U}{2}\sum_{q\neq
0}\left(a^{\dagger}_{k_{0}+q}a^{\dagger}_{k_{0}-q}+a_{k_{0}+q}a_{k_{0}-q}\right),$

for a fixed total number of particles $N$ with $U=gN$. Here
$\epsilon_{q}=q^{2}+2q\left(k_{0}-Q\Phi\right)$ and $q$ denotes the momentum
difference from the condensate momentum $k_{0}$. The constant term in the
Hamiltonian, i.e. $E_{0}=\left(k_{0}-Q\Phi\right)^{2}N+gN^{2}/2$ gives the
mean field or Gross-Pitaevskii ground state energy of the system. From this
energy expression one can obtain a critical magnetic field to generate a
persistent current around the ring. For example, as soon as $Q\Phi>1/2$, a
vortex state with angular momentum $k_{0}=1$ becomes energetically more
favorable against $k_{0}=0$. Note that this critical magnetic field is
determined at the Gross-Pitaevskii level, and the existence of the quantum
fluctuations brings a correction to this critical value.

The improved critical magnetic field in a ring geometry is determined by the
energetic instability Rokhsar ; Anoshkin  _et al._ (2013). This instability is
associated with the zeros of the excitation spectrum. The above Hamiltonian
becomes diagonal following a usual Bogoliubov transformation,
$a_{k_{0}+q}=u_{q}\alpha_{q}-v_{q}\alpha_{-q}^{\dagger}$ with the
normalization condition $|u_{q}|^{2}-|v_{q}|^{2}=1$ to ensure that the bosonic
commutation relations of the new operators (quasi-particle operators) are
conserved. Without going in to the details of the diagonalization procedure,
we write the diagonalized Hamiltonian in terms of quasi-particle creation and
annihilation operators as

$H=E_{0}+\frac{1}{2}\sum_{q\neq
0}\left(\varepsilon_{q}-\epsilon_{q}-U\right)+\sum_{q\neq
0}\varepsilon_{q}\alpha^{\dagger}_{q}\alpha_{q}$ (6)

where the excitation energy is obtained as

$\varepsilon_{q}=2q\left(k_{0}-Q\Phi\right)+\sqrt{q^{4}+2Uq^{2}},$ (7)

which differs from the excitation energy of an uncharged superfluid only by
the additional term associated with the applied magnetic field. Note that only
positive excitations preserve the normalization of the Bogoliubov coefficients
$|u_{q}|^{2}-|v_{q}|^{2}=1$. The occurrence of negative solutions is a direct
signature of an energetic instability and indicates that there is a new state
with a lower energy. Supposedly, after the energetic instability the
superfluid occupies the new state. As seen from Eq. (7) for a charged
superfluid in a magnetic field the energetic instability happens when the
magnetic field exceeds a certain critical value. After that point, condensate
macroscopically occupies a new angular momentum state, which is the $k_{0}=1$
state. Therefore, the critical magnetic field to excite a persistent current
can be readily obtained from equating Eq. (7) to zero for $q=1$, i.e.

$Q\Phi_{c}=k_{0}+\frac{1}{2}\sqrt{1+2U}.$ (8)

Figure 1: (color online) (a) The dimensionless interaction energy $U$-magnetic
flux number $\Phi$ phase diagram of a charged superfluid coupled to an applied
magnetic field trapped in a ring-shaped geometry. The vertical lines are
associated with the instabilities calculated from the Gross-Pitaevskii energy
and the dashed lines show the energetic instabilities obtained from the
Bogoliubov excitation spectrum. (b) The angular momentum $L_{z}/(N_{0}\hbar)$
is plotted as a function of $\Phi$. At critical $\Phi$ and $U$ values a sudden
jump occurs in the angular momentum. (c) The distribution of the excited
particles per healing length $\xi=R/\sqrt{U}$ as a function of relative
momentum $q$ from the condensate.

Using the excitation spectrum we obtain a phase diagram, as shown in Fig.
1(a), which shows the energetic instability boundaries as a function of the
particle interaction strength and the magnetic field values. For very small
interactions, the critical magnetic field to excite a persistent current
around the ring is close to its value obtained from the Gross-Pitaevskii
energy (shown by vertical lines), but the difference becomes greater as the
interaction strength increases.

The transition of the condensate from one angular momentum state to another is
also signaled by the angular momentum expression which reveals that as soon as
the excitation energy becomes zero for a certain $q$ mode, a sudden jump
appears in the angular momentum expectation value Bloch (1973). The angular
momentum

$\displaystyle\frac{L_{z}}{\hbar}$ $\displaystyle=$
$\displaystyle\sum_{k}k\langle a^{\dagger}_{k}a_{k}\rangle$ (9)
$\displaystyle=$ $\displaystyle k_{0}N_{0}+\sum_{q\neq
0}(k_{0}+q)|v_{q}|^{2}(|u_{q}|^{2}-|v_{q}|^{2})$

is affected by the distribution of the non-condensed particles over single
particle excited states and the jump happens due to the transition of the
condensate to the next angular momentum state. Here

$v_{q}^{2}=\frac{1}{2}\left(\frac{q^{2}+U}{\sqrt{q^{2}(q^{2}+2U)}}-1\right).$
(10)

No change in the angular momentum is observed for a superfluid until the
condensed particles move to a new vortex state and a persistent current
appears inside the superfluid which stays constant until another jump, as seen
in Fig. 1 (b). Note that the Bogoliubov treatment allows for out of condensate
excitations, however the superfluid does not acquire a non-zero contribution
to the angular momentum due to the excited particles. Physically, this is
because the excitations result from particles scattering out of the condensate
with a momentum conserving interaction. Therefore, the distribution of the
excited particles over higher angular momentum states as a function of
relative momentum from the condensate, as shown in Fig. 1 (c), is symmetric
and the net contribution to angular momentum from Bogoliubov excitations is
always zero. We also note that because of the minimal coupling to the magnetic
field, the depletion of the condensate due to interactions is independent of
the magnetic field value, until an energetic instability.

From Eq. (9) we can explicitly confirm that this macroscopic occupation of a
new angular momentum state occurs at the energetic instability
($\varepsilon_{q}=0$). The energetic instability happens when a certain value
of magnetic field or interaction energy is exceeded for each angular momentum
mode. A larger magnetic field may excite vortices with higher angular momentum
quantum numbers Anoshkin  _et al._ (2013). In this picture, for a positively
charged superfluid subject to a magnetic field in the positive $z$-direction,
starting from a no-vortex state with $k_{0}=0$, no anti-vortex state appears
since for negative $q$ values the excitation energy is always positive.

##  III Mixture of two unequally charged superfluids

We now consider a mixture of two unequally charged superfluids, represented by
operators $a$, and $b$, with corresponding particle masses $M_{a}$ and
$M_{b}$, and synthetic particle charges $Q_{a}e$ and $Q_{b}e$, respectively.
The mixture is trapped in a ring whose geometry is defined in Sec. II. The
intra-component and inter-component interactions are modeled by the same
short-range contact interaction with coupling constants defined as in Eq. (1).
The system is under a uniform synthetic magnetic field (${\bf
B}=B_{0}\hat{e}_{z}$) along the central axis of the ring. In the plane wave
basis the many-body Hamiltonian describing this mixture can be written as

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{k}\left[\left(k-Q_{a}\Phi\right)^{2}a^{\dagger}_{k}a_{k}+\mu\left(k-Q_{b}\Phi\right)^{2}b^{\dagger}_{k}b_{k}\right]$
(11) $\displaystyle+$
$\displaystyle\frac{1}{2}\sum_{\begin{subarray}{c}k,k^{\prime},q\end{subarray}}\left[g_{a}a^{\dagger}_{k}a^{\dagger}_{k^{\prime}}a_{k^{\prime}-q}a_{k+q}+g_{b}b^{\dagger}_{k}b^{\dagger}_{k^{\prime}}b_{k^{\prime}-q}b_{k+q}\right]$
$\displaystyle+$ $\displaystyle
g_{ab}\sum_{\begin{subarray}{c}k,k^{\prime},q\end{subarray}}a^{\dagger}_{k}b^{\dagger}_{k^{\prime}}b_{k^{\prime}-q}a_{k+q}$

where $\mu=M_{a}/M_{b}$ is the mass ratio between two superfluids and
$\Phi=\pi R^{2}B_{0}/\Phi_{0}$ is the number of magnetic flux quanta passing
through the area enclosed by the ring. All the relevant quantities in the
Hamiltonian are non-dimensionalized via scaling the Hamiltonian by the
characteristic energy $\hbar^{2}/(2M_{a}R^{2})$ so that
$g_{a}=4Ra_{s}^{(a)}/S$, $g_{b}=\mu 4Ra_{s}^{(b)}/S$, and
$g_{ab}=(1+\mu)4Ra^{(ab)}_{s}/S$. It is worth mentioning that by this scaling,
the mass ratio information enters into dimensionless coupling constants
$g_{b}$ and $g_{ab}$, and any changes in the mass ratio also implies a change
in the coupling constants. We note that the periodic boundary condition
imposes integral values for momentum $k$, i.e. $k=\pm 1,\pm 2,\ldots$.

In order to study the superfluid mixture, we employ the Bogoliubov
approximation as in the previous section. According to this approach we assume
that

$N_{a}=\overline{N}_{a}+\sum_{k\neq
k_{a}}a^{\dagger}_{k}a_{k}\quad\mathrm{and}\quad
N_{b}=\overline{N}_{b}+\sum_{k\neq k_{b}}b^{\dagger}_{k}b_{k},$ (12)

where $k_{a}$ and $k_{b}$ are the minimum energy states where condensates $a$
and $b$ reside, respectively. Here, $\overline{N}_{a}=\langle
a^{\dagger}_{k_{a}}a_{k_{a}}\rangle$ and $\overline{N}_{b}=\langle
b^{\dagger}_{k_{b}}b_{k_{b}}\rangle$ are macroscopically large so that the
total number of excited particles is very small compared to $\overline{N}_{a}$
and $\overline{N}_{b}$, respectively. The Bogoliubov approximation leads us to
a quadratic Hamiltonian of the form

$\displaystyle H$ $\displaystyle=$ $\displaystyle
E_{GP}+\frac{1}{2}\sum_{q>0}\left\\{\psi^{\dagger}_{q}\left({\begin{array}[]{cc}W&Z\\\
Z&W\\\ \end{array}}\right)\psi_{q}-\mathrm{Tr}[W]\right\\}$ (15)
$\displaystyle=$ $\displaystyle
E_{GP}+\frac{1}{2}\sum_{q>0}\left\\{\psi^{\dagger}_{q}{\cal
M}\psi_{q}-\mathrm{Tr}[W]\right\\}$ (16)

where

$\displaystyle E_{GP}$ $\displaystyle=$ $\displaystyle
N_{a}\left(k_{a}-Q_{a}\Phi\right)^{2}+\mu
N_{b}\left(k_{b}-Q_{b}\Phi\right)^{2}$ $\displaystyle+$
$\displaystyle\frac{1}{2}U_{a}N_{a}+\frac{1}{2}U_{b}N_{b}+U_{ab}\sqrt{N_{a}N_{b}}$

is the Gross-Pitaevskii ground state energy with $U_{i}=g_{i}N_{i}$ for
$i=a,b$, and $U_{ab}=g_{ab}\sqrt{N_{a}N_{b}}$. In Eq. (15) $\psi_{q}$ is given
by the vector operator

$\psi^{\dagger}_{q}=\left(a_{k_{a}+q}^{\dagger}~{}a_{k_{a}-q}^{\dagger}~{}b_{k_{b}+q}^{\dagger}~{}b_{k_{b}-q}^{\dagger}~{}a_{k_{a}+q}~{}a_{k_{a}-q}~{}b_{k_{b}+q}~{}b_{k_{b}-q}\right)$

and matrices $W$ and $Z$ can be written as

$\displaystyle
W=\left({\begin{array}[]{cccc}\epsilon^{a}_{q}+U_{a}&0&U_{ab}&0\\\
0&\epsilon^{a}_{-q}+U_{a}&0&U_{ab}\\\ U_{ab}&0&\epsilon^{b}_{q}+U_{b}&0\\\
0&U_{ab}&0&\epsilon^{b}_{-q}+U_{b}\\\ \end{array}}\right),$ (22)
$\displaystyle Z=\left({\begin{array}[]{*{20}c}0&U_{a}&0&U_{ab}\\\
U_{a}&0&U_{ab}&0\\\ 0&U_{ab}&0&U_{b}\\\ U_{ab}&0&U_{b}&0\\\
\end{array}}\right)$ (27)

where $\epsilon^{i}_{q}=q^{2}+2q\left(k_{i}-Q_{i}\Phi\right)$ for $i=a,b$.

We use the generalized Bogoliubov transformation in order to diagonalize the
Hamiltonian and introduce a linear canonical transformation
$\psi_{q}=T\phi_{q}$, where

$\phi^{\dagger}_{q}=\left(\alpha_{q}^{\dagger}~{}~{}\,\alpha_{-q}^{\dagger}~{}~{}\,\beta_{q}^{\dagger}~{}~{}\,\beta_{-q}^{\dagger}~{}~{}\,\alpha_{q}~{}~{}\,\alpha_{-q}~{}~{}\,\beta_{q}~{}~{}\,\beta_{-q}\right)$

is the vector of quasi-particle creation and annihilation operators. The
transformation can be explicitly written as

$\displaystyle\\!\\!\\!a_{k_{a}+q}=u_{11}(q)\alpha_{q}\\!-\\!v_{11}(q)\alpha^{\dagger}_{-q}\\!+\\!u_{12}(q)\beta_{q}\\!-\\!v_{12}(q)\beta^{\dagger}_{-q}$
$\displaystyle\\!\\!\\!b^{\dagger}_{k_{b}+q}=u_{21}(q)\alpha^{\dagger}_{q}\\!-\\!v_{21}(q)\alpha_{-q}\\!+\\!u_{22}(q)\beta^{\dagger}_{q}\\!-\\!v_{22}(q)\beta_{-q}$
(28)

which brings the Hamiltonian to the following form

$\displaystyle H$ $\displaystyle=$ $\displaystyle
E_{GP}+\frac{1}{2}\sum_{q>0}\left\\{\phi^{\dagger}_{q}T^{\dagger}{\cal
M}T\phi_{q}-\mathrm{Tr}[W]\right\\}$ (29) $\displaystyle=$ $\displaystyle
E_{GP}+\frac{1}{2}\sum_{q>0}\left\\{\phi^{\dagger}_{q}\eta T^{-1}\eta{\cal
M}T\phi_{q}-\mathrm{Tr}[W]\right\\},$

since for bosons $T^{\dagger}=\eta T^{-1}\eta$. Here

$\eta=\left({\begin{array}[]{*{20}c}I&0\\\ 0&-I\\\ \end{array}}\right)$ (30)

is a diagonal matrix defined by the bosonic commutation relations obeyed by
operators, i.e., $\eta_{ij}=\left[\psi_{q,i},\psi^{\dagger}_{q,j}\right]$ and
$I$ is the $4\times 4$ identity matrix. Note that the new quasi-particle
creation and annihilation operators should obey the same commutation
relations, which gives

$\displaystyle|u_{11}|^{2}+|u_{12}|^{2}-|v_{11}|^{2}-|v_{12}|^{2}=1$
$\displaystyle|u_{21}|^{2}+|u_{22}|^{2}-|v_{21}|^{2}-|v_{22}|^{2}=1.$ (31)

The transformation leads us to the eigenvalue equation of form $|\eta{\cal
M}-\lambda I|=0$, or in a more explicit form

$\displaystyle\left|\begin{array}[]{cccc}W-\lambda I&Z\\\ Z&W+\lambda
I\end{array}\right|=0.$ (34)

Consequently, the diagonalized Hamiltonian becomes

$H=E_{B}+\sum_{q\neq
0}\left(\varepsilon_{q}^{\alpha}\alpha_{q}^{\dagger}\alpha_{q}+\varepsilon_{q}^{\beta}\beta_{q}^{\dagger}\beta_{q}\right).$
(35)

where

$E_{B}=E_{GP}+\sum_{q\neq
0}\left(\varepsilon_{q}^{\alpha}+\varepsilon_{q}^{\beta}-\epsilon_{q}^{a}-\epsilon_{q}^{b}-U_{a}-U_{b}\right)$
(36)

denotes the Bogoliubov ground state energy.

For the general case, there are some methods to diagonalize the system
analytically Sun and Pindzola (2010); Anoshkin  _et al._ (2013) and the
eigenvalue problem in Eq. (34) can conveniently be solved numerically. The
excitation spectra of some mixtures yield simple analytical forms as well
which can be used to develop useful insight. For example, the excitation
spectrum of an equal mass ($\mu=1$) mixture not subject to a magnetic field is
given by

$\varepsilon_{q}^{\alpha,\beta}\\!\\!=\\!\\!\sqrt{\\!q^{4}\\!+\\!(U_{a}+U_{b})q^{2}\pm
q^{2}\sqrt{4U_{ab}^{2}\\!+\\!(U_{a}-U_{b})^{2}}}.$ (37)

This is a well known spectrum for mixtures with an always positive mode and a
potentially complex mode for repulsive intra-component interactions.

In the presence of a magnetic field, we consider the limit of equal intra-
component interactions but different inter-component interactions, i.e.
$U_{a}=U_{b}=U$, and $U_{ab}\neq U$ with $\mu=1$. We call this case the
interaction-balanced case for later reference. The Bogoliubov modes in this
case can be written as

$\varepsilon_{q}^{\alpha,\beta}=A_{+}+\sqrt{B\pm 2\sqrt{C}},$ (38)

where

$\displaystyle\\!\\!\\!\\!\\!\\!A_{\pm}=q\left[(k_{a}-Q_{a}\Phi)\pm(k_{b}-Q_{b}\Phi)\right],$
(39) $\displaystyle\\!\\!\\!\\!\\!\\!B=q^{4}+2Uq^{2}+A_{-}^{2},$
$\displaystyle\\!\\!\\!\\!\\!\\!C=U_{ab}^{2}q^{4}+q^{2}(2U+q^{2})A_{-}^{2}.$

The excitations have an always positive mode and a potentially negative or
complex mode for repulsive intra-component interactions for which we discuss
the relevant parameter regions in the next section. Note that the spectrum
reduces to the spectrum of the single component superfluid in a magnetic field
given by Eq. (7) by taking $k_{a}=k_{b}=k_{0}$, $Q_{a}=Q_{b}=Q$, and
$U_{ab}=0$.

Figure 2: (color online) The Gross-Pitaevskii energies for a charged-uncharged
superfluid mixture as a function of magnetic field for different condensates
$\left(k_{a},k_{b}\right)$ with equal number of particles $N_{a}=N_{b}=N$ and
equal intra-component interactions $U_{a}=U_{b}=U$. It is seen that the energy
of a system with a vortex in one superfluid and a(n) (anti-)vortex in the
other is higher than the energy of a system with only a(n) (anti-)vortex in
the charged superfluid.

In Sec. I we qualitatively discussed that the transfer of angular momentum
from a strongly charged superfluid to a weakly charged one is impossible under
the Gross-Pitaevskii approximation. This also rules out the possibility of
inducing a persistent current from a charged superfluid to the uncharged one.
Here, we quantitatively show that Gross-Pitaevskii ground state energies of
different superfluids also reveal the same result. From now on, we focus on
the interaction-balanced mixtures with charged and uncharged superfluid
($Q_{b}=0$) components and $N_{a}=N_{b}=N$. Figure 2 shows the Gross-
Pitaevskii ground state energies of different condensates in $(k_{a},k_{b})$
angular momentum states as a function of the magnetic field for given
particle-particle interactions. It is seen that the energy of a system with a
vortex in one superfluid and a(n) (anti-)vortex in the other is higher than
the energy of a system with only a(n) (anti-)vortex in the charged superfluid,
i.e. ($E_{GP}\left(k_{a}=\pm 1,k_{b}=0\right)<E_{GP}\left(k_{a}=\pm
1,k_{b}=\mp 1\right)=E_{GP}\left(k_{a}=\pm 1,k_{b}=\pm 1\right)$). This means
that observing a persistent current induction from a charged superfluid to an
uncharged one is not possible in the context of the Gross-Pitaevskii
approximation. Throughout the next section we study the energetic and
dynamical instabilities of such a system, and in particular, we determine the
critical parameters to obtain a persistent current in the uncharged superfluid
using the Bogoliubov modes.

##  IV Instabilities of superfluid mixtures

There are two kinds of instabilities exhibited by superfluids. The first one
is the energetic instability and is observed in all superfluids. As we
discussed in Sec. II, this instability is related to the critical magnetic
field value to excite a vortex and is associated with the onset of negative
eigenvalues. The second instability, which is called dynamical instability, is
characterized by the occurrence of complex eigenvalues in the excitation
spectrum Pethick and Smith (2008). This instability is observed in non-uniform
superfluids, such as ultracold atomic condensates, and indicates that such a
condensed state is subject to decay. For a single component condensate the
dynamical instability is observed with attractive interactions Dalfovo  _et
al._ (1999) which can be identified from the excitation spectrum in Eq. (7)
for $U<-1/2$. For two components the dynamical instability signals the spatial
separation of the mixture’s components Pethick and Smith (2008). In our
instability analysis we concentrate on the two cases for which analytical
expressions for the excitation spectrum are given earlier in the previous
section.

First, for the dynamical instability of a mixture of two atomic superfluids
with no magnetic field Smyrnakis  _et al._ (2009); Anoshkin  _et al._ (2013)
one obtains from Eq. (37) the condition for which the Bogoliubov excitations
become complex valued as

$\left(U_{a}+\frac{1}{2}q^{2}\right)\left(U_{b}+\frac{1}{2}q^{2}\right)<U_{ab}^{2}.$
(40)

For the radius $R$ being much greater than the coherence lengths
$\xi_{i}=R/\sqrt{U_{i}}$ the above relation reduces to $g_{a}g_{b}<g_{ab}^{2}$
for the homogeneous system. For a finite system, the minimum value of inter-
component interaction $U_{ab}$ leading to such an instability is greater than
that of the homogeneous case. Beyond this critical inter-component
interaction, the superfluids constituting the mixture become spatially
separated.

Second, we consider the mixture with repulsive intra-component interactions
($U>0$) subject to a magnetic field. Considering Eq. (39) it can be easily
seen that $C\geq 0$ for all values of interactions, momentum modes, and
charges. Moreover, $B\geq 0$ for $U>0$ and we do not consider superfluids with
attractive intra-component interactions, i.e $U<0$. Then, one obtains the
condition for the onset of the energetic instability as

$A_{+}+\sqrt{B-2\sqrt{C}}<0,$ (41)

and the condition for the dynamical instability as

$B<2\sqrt{C}.$ (42)

As seen from the above relations the applied magnetic field splits the onset
of the two instabilities. Accordingly, for a given magnetic field value we
define two different inter-component interaction values, one associated with
energetic instability and the other one associated with the dynamical
instability and called $U_{ab}^{E}$ and $U_{ab}^{D}$, respectively. These
critical inter-component interactions are

$\displaystyle
U_{ab}^{E}=\pm\frac{1}{2}\sqrt{\left[A_{+}^{2}+A_{-}^{2}-(1+2U)\right]^{2}+4A_{+}^{2}A_{-}^{2}},$
$\displaystyle U_{ab}^{D}=\pm\frac{1}{2}\left(1+2U-A_{-}\right)$ (43)

Alternatively, for fixed interactions, these give implicit equations for two
critical values of the magnetic field giving rise to energetic and dynamical
instabilities, respectively. These critical values play a very important role
in our analysis throughout this paper. At $U_{ab}^{E}$ a persistent current
appears around the ring, and at $U_{ab}^{D}$ the superfluids become spatially
separated. We analyze these two states, i.e., the vortex and the phase
separated states, in the following sections.

Figure 3: (color online) Phase diagram showing the energetic and dynamical
instabilities associated with equally and unequally charged mixtures in the
inter-component interaction $U_{ab}$ \- magnetic flux number $\Phi$ plane. The
critical values $U_{ab}^{E}$ (solid lines) and $U_{ab}^{D}$ (dashed lines) are
obtained from the onset of zero and complex eigenvalues in the excitation
spectrum, respectively.

The energetic instabilities as given in Eq. (IV) and shown by solid lines in
Fig. 3 are determined by an interplay between the magnetic field and the
inter-component interaction. For the equally charged and the charged-uncharged
mixtures initially settled at $(k_{a},k_{b})=(0,0)$ states, it is seen that
the phase boundary with a persistent current phase is crossed at smaller
magnetic fields for larger inter-component interactions. In other words, with
smaller inter-component interaction the energetic instability occurs at a
larger magnetic field. Moreover, the charge imbalance enlarges the stability
regions of the mixtures in the $\Phi-U_{ab}$. Note that in the presence of a
magnetic field the energetic instabilities always happen before the dynamical
instabilities shown by the dashed lines in Fig. 3 indicating a persistent
current state before superfluids become spatially separated.

For an equally charged mixture the dynamical instability also given in Eq.
(IV) is independent of the value of magnetic field. This behavior is expected,
since in a mixture of two equally charged superfluids, both components feel
the same Lorentz force and the transition to phase separation is driven only
by increasing the inter-component interactions ($U_{ab}>U$). On the other
hand, as a charge imbalance is introduced the dynamical instability becomes
magnetic field dependent as shown by the dotted line in Fig. 3 for
$(Q_{a}=1,Q_{b}=0)$. The charge imbalance breaks the symmetry between the
components under a magnetic field and the phase separation boundary becomes
magnetic field dependent.

From Fig. 3 it is also seen that the behavior of the instabilities is
symmetric when the magnetic field direction is reversed. A similar symmetry is
observed in one-dimensional ring under the exchange of the nature of inter-
particle interactions switching from repulsive ($U_{ab}<0$) to attractive
($U_{ab}>0$).

##  V The transfer of angular moementum between two superluids

In this section we discuss the effect of the charge imbalance on the angular
momentum properties of the mixture. The angular momenta of the components
about the axis of the ring are given by

$\displaystyle\\!\\!\\!\frac{L_{z}^{a}}{\hbar}\\!\\!=\\!\\!\sum_{k}k\langle
a_{k}^{\dagger}a_{k}\rangle=k_{a}\overline{N}_{a}+\sum_{q\neq
0}(k_{a}+q)\langle a_{k_{a}+q}^{\dagger}a_{k_{a}+q}\rangle$
$\displaystyle\\!\\!\\!\frac{L_{z}^{b}}{\hbar}\\!\\!=\\!\\!\sum_{k}k\langle
b_{k}^{\dagger}b_{k}\rangle=k_{b}\overline{N}_{b}+\sum_{q\neq
0}(k_{b}+q)\langle b_{k_{b}+q}^{\dagger}b_{k_{b}+q}\rangle.$ (44)

and using the Bogoliubov transformation in Eq. (III) can be calculated as

$\frac{L_{z}^{i}}{\hbar}=k_{i}\overline{N}_{i}+\sum_{q\neq
0}(k_{i}+q)\left[|v_{i1}(q)|^{2}+|v_{i2}(q)|^{2}\right],$ (45)

for $i=a,b$. The first term above gives the angular momentum of the condensed
part and the sum containing the Bogoliubov coefficients $v_{i1}(q)$ and
$v_{i2}(q)$ stands for the angular momentum carried by the excited particles.

In order to bring forth the effects of the charge imbalance, we focus on the
interaction-balanced case ($U_{ab}\neq U_{a}=U_{b}=U$) and compare the angular
momentum properties of the equally charged and the charged-uncharged mixtures.
For both types of mixture, we plot the angular momentum distribution of the
excited particles relative to the condensates $(k_{a},k_{b})=(0,0)$ and the
total angular momentum of each component in the mixture as a function of the
magnetic field in Fig. 4.

For an equally charged mixture, the angular momentum distribution as shown in
Fig. 4(a) is symmetric about $q=0$ for both components, which is similar to
the response of a single component superfluid discussed in Sec. II [see
Fig.1(b)]. The collisions giving rise to the depletion of the condensates
result from two condensed particles of either condensate scattering into
opposite momentum states and these excitations have the same energy when the
momenta of the colliding particles are exchanged. Therefore, the net
contribution of these excited particles to the total angular momentum is zero
and the total angular momentum of each superfluid remains constant and equal
to the angular momentum of the condensed part as shown in Fig. 4(b).

We note in passing that the symmetric picture discussed above remains
qualitatively the same when there is an imbalance between the intra-component
interactions or the masses (the mass ratio $\mu$ between two superfluids is
also inside the scaled parameters $U_{a}$ and $U_{b}$, as mentioned in Sec.
III), i.e. the distributions remain symmetric about $q=0$ but the depletion in
each component may be different.

Figure 4:  (color online) The angular momentum distribution of excited
particles per healing length $\xi$ for charge-balanced and charge-imbalanced
mixtures for equal intra-component interactions $U=10$ and inter-component
interaction $U_{ab}=8$ are plotted in (a) and (c), respectively. The charge
imbalance breaks the symmetry about $q=0$ for $(k_{a},k_{b})=(0,0)$ mixture.
The total angular momentum of the components of the two mixtures for $U=10$
and $U_{ab}=8$ are plotted as a function of the magnetic flux in (b) and (d),
respectively. The sudden jump at the critical values indicates the creation of
persistent currents. For the charge-imbalanced mixture, the difference in the
excitation spectrum leads to finite angular momenta for both components before
the critical field and counter-flowing persistent currents after the critical
field as shown in the inset in (d).

For two unequally charged superfluids, we note that the inter-component
interaction causes a qualitative change in the distributions of non-condensed
particles of both components compared to those of two equally charged
superfluids. The charge imbalance breaks the symmetry of excitations. The
resulting angular momentum distributions are shown in Fig. 4(c). In this case,
the nature of excitations is again such that when a particle from the highly
charged superfluid is excited, another one from the weakly charged superfluid
gets excited to an angular momentum state with the same magnitude, but
opposite direction. However, excited particles from the highly charged
superfluid rotate more in the same direction with the field and excited
particles from the weakly charged superfluid more in the opposite direction
which explains the asymmetry of the momentum distributions about $q=0$ for
each superfluid.

As a result of the asymmetry in the momentum distributions, the net relative
angular momentum of the excited particles in each superfluid becomes non-zero
and proportional to the applied magnetic field. This is shown in the inset in
Fig. 4(d) as a function of increasing magnetic field. Unlike the charge
balanced case, the total angular momentum of each superfluid changes when a
magnetic field is applied. We can therefore interpret the result as a transfer
of angular momentum from the highly charged superfluid to the weakly charged
one albeit in the opposite direction (an anti-drag effect). The total angular
momentum of the system is conserved as the excess angular momenta of both
superfluids cancel each other.

We check the validity of the Bogoliubov approximation by calculating the
number of excited particles $N_{\mathrm{ex}}$ to ensure that this number stays
small with respect to the total number of particles. It is convenient to
express this number per coherence length $\xi=R/\sqrt{U}$, and it should be
smaller than one Pethick and Smith (2008). The density of the excited
particles can be written as

$n_{\mathrm{ex}}^{i}=\\!\frac{1}{2\pi R}\sum_{q\neq
0}\left[|v_{i1}(q)|^{2}+|v_{i2}(q)|^{2}\right]\\!=\frac{1}{2\pi\sqrt{U}}\frac{N^{i}_{\mathrm{ex}}}{\xi}.$
(46)

We note that the excitation number does not depend on the magnetic field for
equally charged mixtures because of the symmetry between the dispersion
relations of the components. For unequally charged mixtures there is a slight
dependency as the symmetry between the components is broken. For example, for
$U=10$ we find $N_{\mathrm{ex}}\simeq 2$ for both components until the
energetic instability which corresponds to $\xi n_{\mathrm{ex}}\simeq 0.1$.
Assuming a scattering length of $a_{s}=$10\text{\,}\mathrm{nm}$$, $U=10$
requires $N=10^{3}$ and results in a fraction of
$N_{\mathrm{ex}}/N\simeq$2\text{\times}{10}^{-3}\text{\,}$$. The distribution
of excited particles per healing length $\xi$ is shown in Figs. 4(a), and
4(c). For the same scattering length and $U=100$, the corresponding numbers
are $N=10^{4}$, $N_{\mathrm{ex}}\simeq 12$, $\xi n_{\mathrm{ex}}\simeq 0.2$
and $N_{\mathrm{ex}}/N\simeq$1.2\text{\times}{10}^{-3}\text{\,}$$,
respectively.

It is worth mentioning that the behavior of the instability conditions and the
angular momentum transfer between two superfluids in a one-dimensional ring
remains unchanged when the sign of the inter-component interaction is
reversed. The attractive interaction acts similarly to the repulsive one. We
believe that considering the finite width of the superfluids will change this
picture, which will be the subject of our future studies.

##  VI Induction of a persistent current from a charged superluid to an
uncharged superfluid

From angular momentum calculations of both equally and unequally charged
superfluid mixtures presented in Figs. 4(b) and 4(d), we observe that after a
critical value of the magnetic flux $\Phi$, a sudden and simultaneous change
occurs in the angular momentum of both components. This sudden macroscopic
change in the angular momentum, which happens at the energetic instability of
the mixture, shows the appearance of persistent currents. Consequently, the
initial mean field collapses and the calculations should be done with a new
mean field, i.e. this time the superfluids residing in new $k_{a}$ and $k_{b}$
values, until the next instability.

The process of exciting persistent currents in both components in a charged-
uncharged superfluid mixture is interesting. Here, the angular momentum
transfer creates a persistent current in the uncharged superfluid which is not
directly coupled to the applied magnetic field. In other words, this is the
induction of a vortex from a charged superfluid coupled to a magnetic field to
an uncharged superfluid. The uncharged superfluid indirectly becomes coupled
to the magnetic field via the interactions between two superfluids. As a
result, two counter-flowing currents can appear in the ring. In order to
discuss the conditions for the induction of a persistent current in an
uncharged superfluid we need to explore the stability diagrams of the mixtures
in more detail.

Figure 5: (color online) The energetic (solid lines) and dynamical (dashed
lines) instability boundaries for different condensates $(k_{a},k_{b})$ of
equally (left column) and unequally (right column) charged superfluid mixtures
in terms of the inter-component interactions $U_{ab}$ and the magnetic flux
number $\Phi$. For unequally charged mixtures, a stable region for two
counter-flowing superfluids exists beyond the energetic instability of the
initially at rest superfluids. The increase in the magnitude of the intra-
component interaction from $U=10$ (upper panels) to $U=100$ (lower panels)
shifts the energetic instability of the mixture to higher magnetic field
values while the qualitative behavior remains the same.

The phase diagrams in Fig. 5 show the energetic (solid lines) and dynamical
(dashed lines) stability boundaries determined by the interplay between the
inter-component interaction strength and the magnetic field for charge
balanced and charge imbalanced mixtures, respectively. The dynamical
instability as we discussed in Sec. IV for a mixture of two equally charged
superfluids is independent of the applied magnetic field and appears when a
certain value of the inter-component interaction is exceeded. For a mixture of
two unequally charged superfluids this instability becomes magnetic field
dependent. However, it always appears after the energetic instability.

For both equally (Fig. 5, left column) and unequally (Fig. 5, right column)
charged mixtures the instability boundaries show that the mixture of two
superfluids with a persistent current created in each is locally stable. For a
mixture of two equally charged superfluids this stable region corresponds to
the state of two persistent currents flowing in the same direction, i.e. two
superfluids residing in $(k_{a},k_{b})=(1,1)$ angular momentum states.
However, for a mixture of a charged and an uncharged superfluid, a state of
two counter-flowing superfluids, i.e. superfluids occupying the states
$(k_{a},k_{b})=(1,-1)$ is stable. This phase which emerges as a result of
inducing a persistent current to the uncharged superfluid from charged
superfluid coupled to the magnetic field happens before the dynamical
instability of the initial non-rotating mixture and has energetic and
dynamical stability. Then, the induction of two persistent currents happens
before superfluids are spatially separated.

The parameters used to obtain these results are compatible with the parameters
used in recent experiments. The radius of the ring $R$ ranges from
$12\text{\,}\mathrm{\SIUnitSymbolMicro m}25\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ and the cross-sectional radius $r$ of the toroidal trapping potential is
around $5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ Henderson  _et al._ (2009);
Ramanathan  _et al._ (2011); Beattie  _et al._ (2013); Jendrzejewski  _et al._
(2014); Ryu  _et al._ (2014). An intra-component interaction strength of
$U=10$ with $N=10^{3}$ particles corresponds to an s-wave scattering length of
$10\text{\,}\mathrm{nm}$, which is reasonable for the experimental setups.
Note that the qualitative behavior of the results presented is not very
sensitive to the strength of the intra-component interaction (compare the
upper and lower panels in Fig. 5). The increase in the magnitude of the intra-
component interaction only shifts the energetic instability of the mixture to
higher magnetic field values. Therefore, a scaled interaction strength of
$U=100$ with $N=10^{4}$ particles corresponds to an s-wave scattering length
of again $10\text{\,}\mathrm{nm}$, however with an instability predicted at a
different magnetic field.

##  VII Conclusion

We considered a mixture of two unequally charged superfluids in a ring trap
subject to a uniform magnetic field. The charge imbalance in this mixture
causes qualitative changes in the excitation properties of the system compared
to those of a two equally charged superfluid mixture. In a mixture of two
equally charged interacting superfluids, the distribution of the excited
particles over angular momentum states for each component is similar to that
of a single component superfluid. Due to the symmetry between the components,
the excited particles are distributed symmetrically with respect to the
condensate momenta and carry zero net angular momentum. The excitation of
persistent currents is determined by the internal dynamics of each component
in this mixture.

For a mixture of two unequally charged superfluids the distribution of the
excited particles over angular momentum states changes completely which causes
a change in the angular momentum properties of the system. In this case, we
calculate a finite angular momentum associated with the contribution of the
excited particles to the total angular momentum of each superfluid. These
finite angular momenta are equal in magnitude but opposite in direction so
that the total angular momentum of the system is conserved. Moreover, a finite
angular momentum appears in both superfluids even when one of the superfluids
is uncharged, and hence not directly coupled to the magnetic field. In other
words, we show that a transfer of angular momentum occurs from a charged
superfluid to an uncharged one. This transfer of angular momentum causes
counter-flowing persistent currents around the ring.

The conditions for the persistent current induction are studied through the
instabilities of the mixtures. According to the instability analysis the
induction of a persistent current from a charged superfluid to an uncharged
one is allowed energetically and dynamically. The persistent currents flow in
the opposite directions regardless of the attractive or repulsive nature of
the inter-component interactions, since for a one-dimensional ring geometry
the attractive inter-component interaction acts similarly to the repulsive
one. We believe that considering a two-dimensional harmonic trap will change
this picture, which will be the subject of our future studies.

###### Acknowledgements.

The authors thank M. Ö. Oktel for initial motivation and useful discussions.
This work is supported by TÜBİTAK under Project No. 117F469. A.L.S. is
supported by the TÜBİTAK BİDEB 2219 scholarship program and by the Laboratory
Directed Research and Development program of Los Alamos National Laboratory
under Project No. 20180045DR.

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