# Quantum droplets with magnetic vortices in spinor dipolar Bose-Einstein
condensates

Shaoxiong Li  Department of Engineering Science, University of Electro-
Communications, Tokyo 182-8585, Japan  Hiroki Saito  Department of Engineering
Science, University of Electro-Communications, Tokyo 182-8585, Japan

###### Abstract

Motivated by the recent experimental realization of a Bose-Einstein condensate
(BEC) of europium atoms, we investigate the self-bound droplet state of a
europium BEC with spin degrees of freedom. Under a sufficiently weak magnetic
field, the droplet has a torus shape with circulating spin vectors, which is
referred to as a magnetic vortex. The ground state transforms from the torus
to cigar shape through bistability with an increase in the magnetic field.
Dynamical change of the magnetic field causes the torus to rotate due to the
Einstein-de Haas effect. The magnetic vortices form a supersolid in a confined
system.

A magnetic flux-closure structure is a stable configuration of a ferromagnetic
material, in which magnetization vectors form closed loops to reduce the
magnetostatic energy. This structure can be observed in a ferromagnetic
material with multiple magnetic domains below the Curie temperature [1]. The
magnetic flux-closure structure has also been realized within nanoscale
particles [2, 3, 4, 5, 6, 7, 8, 9, 10], in which the magnetization vectors
circulate along a toroidal loop. This state is referred to as a magnetic
vortex. The nanoparticles with magnetic vortices can be used for, e.g., data
storage [11], cancer therapy [12], and neuromorphic computing [13]. Such an
isolated object with a magnetic vortex has so far been restricted to solid
materials. Is it possible to produce a liquid or gas analogue of this state of
matter, i.e., a self-bound droplet with a magnetic vortex? Although permanent-
magnetic liquid droplets have been produced recently [14], the magnetic-vortex
structure has not been observed.

Here we propose a self-bound superfluid droplet that contains a magnetic
vortex. Self-bound states of Bose-Einstein condensates (BECs) have attracted
much interest recently and are referred to as quantum droplets [15]. In a
quantum droplet, the attractive mean-field interaction balances with the
repulsive beyond-mean-field effect [16], which stabilizes the system against
collapse and expansion in free space. This novel state of matter was realized
in a BEC with magnetic dipole-dipole interaction (DDI) [17, 18, 19, 20, 21]
and a Bose-Bose mixture [22, 23, 24], and various theoretical studies have
been performed on these systems [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
36, 37, 38, 41, 42, 39, 40, 43]. However, in the experiments of quantum
droplets with DDI to date, the magnetization of the system has been frozen
into the direction of the strong external magnetic field. If the external
magnetic field is suppressed sufficiently, spin degrees of freedom in a
dipolar BEC are liberated, which allows for a spinor dipolar BEC [44, 45, 46,
47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65,
66, 67, 68, 69]. However, self-bound quantum droplets of spinor dipolar BECs
have not yet been studied.

In this paper, we will show that there exists a stable self-bound droplet of a
spinor dipolar BEC that contains a magnetic vortex with a torus-shaped density
distribution. This state is stable under a weak external magnetic field. As
the the magnetic field is increased, the ground state changes from the
magnetic vortex state to the well-known cigar-shaped droplet, and these two
states exhibit bistability. If the external magnetic field is suddenly
changed, the torus-shaped droplet begins to rotate to conserve the total
angular momentum, which resembles the Einstein-de Haas effect. The ground
state of a confined system exhibits periodic alignment of the torus-shaped
droplets, which can be regarded as a supersolid.

In the present study, we restrict ourselves to the BEC of ${}^{151}\text{Eu}$
atoms, which was recently realized experimentally [70]. A peculiar feature of
${}^{151}\text{Eu}$ is its wide range of the hyperfine spin ($F=1,\cdots,6$)
with small spin-dependent contact interactions, which could be smaller than
the DDI. The spin state in this system is therefore determined mainly by the
DDI under a weak external magnetic field, and the DDI-dominant spinor dipolar
phenomena can be investigated. To observe such phenomena, the magnetic
Feshbach resonance cannot be used to tune the contact interaction. Although
the $s$-wave scattering length of the $F=6$ hyperfine state measured in Ref.
[70] does not satisfy the condition for the droplet formation, the scattering
lengths for the other hyperfine spins $F$ and those for ${}^{153}\text{Eu}$
are unknown, for which the spinor dipolar droplet may be possible.
Furthermore, the contact interaction may be controllable using microwave-
induced Feshbach resonance [71], which enables the formation of the spinor
dipolar droplet.

We consider a BEC of ${}^{151}\text{Eu}$ atoms with hyperfine spin $F$ at zero
temperature using the beyond-mean-field approximation [25, 26]. The total
energy consists of five terms, $E=E_{\rm kin}+E_{s}+E_{\rm ddi}+E_{\rm
LHY}+E_{B}.$ The kinetic energy is given by $E_{\rm
kin}=\hbar^{2}/(2M)\sum_{m}\int
d\bm{r}\left|\nabla\psi_{m}(\bm{r})\right|^{2}$, where $\psi_{m}(\bm{r})$ is
the macroscopic wave function for the magnetic sublevels $m=-F,-F+1,\cdots,F$,
and $M$ is the mass of an atom. The wave function is normalized as
$\sum_{m}\int\left|\psi_{m}(\bm{r})\right|^{2}d\bm{r}=N$, where $N$ is the
total number of atoms. The spin-independent contact interaction has the form
$E_{s}=2\pi\hbar^{2}a_{s}M^{-1}\int\rho^{2}(\bm{r})d\bm{r}$, where $a_{s}$ is
the spin-independent $s$-wave scattering length and
$\rho(\bm{r})=\sum_{m}\left|\psi_{m}(\bm{r})\right|^{2}$ is the total density.
The DDI energy is given by $E_{\rm ddi}=\mu_{0}(g\mu_{B})^{2}/(8\pi)\int
d\bm{r}d\bm{r}^{\prime}\\{\bm{f}(\bm{r})\cdot\bm{f}(\bm{r}^{\prime})-3[\bm{f}(\bm{r})\cdot\bm{e}][\bm{f}(\bm{r}^{\prime})\cdot\bm{e}]\\}/|\bm{r}-\bm{r}^{\prime}|^{3}$,
where $\mu_{0}$ is the magnetic permeability of the vacuum, $g$ is the
hyperfine $g$ factor, $\mu_{B}$ is the Bohr magneton,
$\bm{f}(\bm{r})=\sum_{mm^{\prime}}\psi_{m}^{*}(\bm{r})(\bm{S})_{mm^{\prime}}\psi_{m^{\prime}}(\bm{r})$
with $\bm{S}$ being the spin matrix, and
$\bm{e}=(\bm{r}-\bm{r}^{\prime})/|\bm{r}-\bm{r}^{\prime}|$. The relative
strength of the DDI is characterized by $\varepsilon_{\rm dd}=a_{\rm
dd}/a_{s}$, where $a_{\rm dd}=\mu_{0}\mu^{2}M/(12\pi\hbar^{2})$ is the dipolar
length. The magnetic moment $\mu=g\mu_{B}F$ and the dipolar length $a_{\rm
dd}$ for each spin $F$ of ${}^{151}\text{Eu}$ are given in the Supplemental
Material. The spin distribution is mainly determined by the DDI, if $a_{\rm
dd}$ is much larger than the spin-dependent scattering lengths that consist of
the differences $\Delta a$ among the scattering lengths $a_{0,2,\cdots,2F}$ in
collisional spin channels. Since the values of $\Delta a$ are predicted to be
relatively small for the europium atoms [70, 72, 73], we ignore the spin-
dependent contact interaction.

As will be confirmed numerically, the spin state is almost fully polarized in
the droplet, and we can use the Lee-Huang-Yang (LHY) correction for the fully
polarized dipolar BEC. Under the local density approximation, the LHY
correction is written as [74, 75, 25, 26]

$E_{\rm
LHY}=\frac{2}{5}\frac{32}{3\sqrt{\pi}}\frac{4\pi\hbar^{2}}{M}a^{5/2}\chi(\varepsilon_{\rm
dd})\int\rho^{5/2}(\bm{r})d\bm{r},$ (1)

where $\chi(\varepsilon_{\rm dd})$ is the real part of
$\int_{0}^{\pi}d\theta\sin\theta[1+\varepsilon_{\rm
dd}(3\cos^{2}\theta-1)]^{5/2}/2$. In the presence of an external magnetic
field $\bm{B}(\bm{r})$, the linear Zeeman energy has the form
$E_{B}=-g\mu_{B}\int\bm{f}(\bm{r})\cdot\bm{B}(\bm{r})d\bm{r}$. The ratio of
the quadratic Zeeman energy to the linear Zeeman energy is estimated to be
$\mu B/\Delta_{\rm hf}\sim 10^{-4}$ at most for the present magnetic field
$\sim 0.1$ mG, and the quadratic Zeeman energy can be neglected even for the
relatively small hyperfine splitting $\Delta_{\rm hf}/\hbar\sim 100$ MHz of a
europium atom [76].

The Gross-Pitaevskii (GP) equation is given by the functional derivative of
the total energy as $i\hbar\partial\psi_{m}/\partial t=\delta
E/\delta\psi_{m}^{*}$. To obtain the ground state or metastable state, the GP
equation is propagated in imaginary time, in which $i$ on the left-hand side
of the GP equation is replaced with $-1$. The GP equation is numerically
solved using the pseudospectral method with typical spatial and time steps
$dx\sim 0.01$ $\mu{\rm m}$ and $dt\sim 0.1$ $\mu{\rm s}$.

Figure 1:  Self-bound ground state with a magnetic vortex for $B=0$. The
system has translational and rotational symmetry, and the origin and $z$ axis
are taken as the center and symmetry axis of the torus, respectively. (a)
Density distribution on the $z=0$ plane for $F=1$, $N=15000$, and
$\varepsilon_{\rm dd}=1.2$. The arrows represent the magnetization $\bm{f}$.
The inset shows the isodensity surface. (b) Phase distributions of components
$m=1,0,-1$ on the $z=0$ plane. (c) Distributions of $\rho$ and $|\bm{f}|$
along the $x$ axis. (d) Density distributions on the $z=0$ plane (upper
panels) and $y=0$ plane (lower panels) for $(F,N,\varepsilon_{\rm
dd})=(1,15000,1.2)$, $(1,80000,1.2)$, and $(6,15000,1.3)$. The unit of density
in (a), (c), and (d) is $N\mu{\rm m}^{-3}$.

First, we consider the case in which the external magnetic field $B$ is zero.
Figure 1(a) shows a typical ground state, which has a torus shape, in contrast
to the usual cigar-shaped droplet in a strong magnetic field [17, 18, 19, 20,
21]. The magnetization vectors $\bm{f}$ circulate along the torus, as shown by
the arrows in Fig. 1(a). Although such a magnetic vortex state has already
been proposed for a trapped spinor dipolar BEC [49, 51], this is the first
example of a self-bound droplet of a fluid containing a magnetic vortex.

In the LHY energy in Eq. (1), we assumed that the spin state is fully
polarized, and here we examine the validity of this assumption. Figure 1(c)
shows the distributions of the atomic density $\rho$ and magnetization density
$|\bm{f}|$, which indicates that the spin is almost fully polarized, i.e.,
$|\bm{f}|/\rho\simeq F=1$, except near the center. This result justifies the
use of Eq. (1), since the LHY correction is important only in the high-density
region to counteract the collapse. The central hole of the torus is mainly
occupied by the $m=0$ component, since the $m\neq 0$ components have the
topological defects at the center, as shown in Fig. 1(b).

Figure 1(d) shows the parameter dependence of the density profile. The size of
the droplet increases with the number of atoms $N$, while the aspect ratio
between the major and minor radii appears almost unchanged. For larger spin
$F=6$, on the other hand, the hole of the torus is enlarged and the aspect
ratio is significantly changed. This is due to the kinetic energy that arises
from the spin winding, which is proportional to $F$. Such behavior is analyzed
in the Supplemental Material using the variational method.

We employ the variational wave function as,

$\bm{\Psi}_{v}(\bm{r})=\sqrt{\rho_{v}(r,z)}e^{-i{S}_{z}\phi}\bm{\zeta}^{(y)},$
(2)

where $(r,\phi,z)$ is the cylindrical coordinate, and $\bm{\zeta}^{(y)}$
represents the spin state fully polarized in the $y$ direction with
$\sum_{m}|\zeta_{m}^{(y)}|^{2}=1$. The $z$ axis is taken as the symmetry axis
of the torus. The matrix $e^{-i{S}_{z}\phi}$ rotates the spin vector to make a
magnetic vortex. We propose a torus-shaped variational density as

$\rho_{v}(r,z)=\frac{N}{\pi^{3/2}\sigma_{r}^{2\lambda+2}\sigma_{z}\Gamma(\lambda+1)}r^{\lambda}e^{-\frac{r^{2}}{\sigma_{r}^{2}}-\frac{z^{2}}{\sigma_{z}^{2}}},$
(3)

where $\sigma_{r}>0$, $\sigma_{z}>0$, and $\lambda>0$ are variational
parameters and $\Gamma$ is the gamma function. The variational energy for Eqs.
(2) and (3) is derived in the Supplemental Material, which is numerically
minimized with respect to the variational parameters using the Newton-Raphson
method.

Figure 2:  (a) Density distribution $\rho(r,z=0)$ obtained by the imaginary-
time evolution of the GP equation (solid lines) and the variational method
(dashed lines) for $(F,N,\varepsilon_{\rm dd})=(1,15000,1.2)$,
$(1,80000,1.2)$, and $(6,15000,1.3)$. (b) Lines: the critical number of atoms
above which the droplet is stable, obtained by the variational method. The six
lines represent $F=1$, $\cdots$, 6 from left to right. The circles (crosses)
indicate that the droplet is stable (unstable) for $F=1$ (blue or dark gray)
and $F=6$ (red or light gray), obtained by the GP equation. The inset shows a
magnification of the main panel.

Figure 2(a) compares the density distributions $\rho(r,z=0)$ obtained by the
GP equation and by the variational method. The two distributions agree well
with each other. For $N=80000$, the distribution of the GP result becomes
broader than that of the variational method, since the flat-top tendency of a
large droplet is not taken into account in Eq. (3). The density at $r=0$ must
vanish for the fully-polarized assumption in Eq. (2), whereas the center is
slightly occupied for the GP results. Figure 2(b) shows the critical number of
atoms above which the droplet is stable, where the lines are obtained by the
variational method and the plots by the GP equation. The variational method
can predict the critical number of atoms very well, which facilitates the
study of this system, because the numerical cost for the GP simulation is much
higher than that for the variational method, especially near the critical line
of the stability where convergence of the imaginary-time evolution is slow.

Figure 3:  Effects of magnetic field $B_{z}$ obtained by the GP equation for
$F=1$ and $\varepsilon_{\rm dd}=1.2$. (a) Cross-sectional $\rho$ and $\bm{f}$
distributions and isodensity surface of the ground state for $N=50000$ and
$B_{z}=0.2$ mG. (b) Stability diagram. Parameter sets marked by circles and
squares respectively indicate that the torus-shaped and cigar-shaped droplets
are stable or metastable. (c) $B_{z}$ dependence of the energy $E$ and
averaged magnetization $F_{z}$ of the torus-shaped and cigar-shaped droplets
with $N=50000$. The inset shows $\rho$ and $\bm{f}$ distributions of the
torus-shaped droplet for $B_{z}=0.16$ mG, where the cross section is taken for
the symmetry plane ($y=0$ plane).

We next examine the effect of the external magnetic field applied in the $z$
direction. Figure 3(a) shows a typical ground state for a large magnetic field
$B_{z}$, where the droplet has a cigar-shape, as observed experimentally. The
spin is almost polarized in the $z$ direction, whereas it is slightly tilted
around both edges of the cigar-shape and exhibits the flower-like structure
[49]. Figure 3(b) shows the stability diagram with respect to $N$ and $B_{z}$.
There is a critical magnetic field, above which the torus-shaped droplet
becomes unstable, whereas the cigar-shaped droplet becomes unstable below some
critical magnetic field. In Fig. 3(b), there is a bistability region in which
both torus-shaped and cigar-shaped droplets are stationary (both circles and
squares are marked). Figure 3(c) reveals the bistability with plots of the
energy $E$ and the averaged magnetization in the $z$ direction $F_{z}=\int
f_{z}d\bm{r}/N$ for both droplets. The bistability ranges from $B_{z}\simeq
0.03$ to $\simeq 0.17$ mG, and the energies of the two droplets cross at
$B_{z}\simeq 0.14$ mG. The direction of the torus-shaped droplet is fixed by
the magnetic field in such a way that the toroidal plane is parallel to the
$z$ direction, as shown in the inset of Fig. 3(c). In this inset, the right-
hand side of the torus becomes slightly thicker than the left-hand side, which
results in the increase of $F_{z}$.

Figure 4:  Einstein-de Haas effect of the torus-shaped droplet for $F=1$,
$\varepsilon_{\rm dd}=1.2$, and $N=15000$. The initial state is the ground
state for zero magnetic field (the same state as Fig. 1(a), except that the
symmetry axis of the torus is taken to be the $y$ direction). (a) Time
evolution of the density distribution on the $z=0$ plane (main panels) and
isodensity surface observed from the $-y$ direction (insets), where the
magnetic field $B_{z}=0.1$ mG is switched on at $t=0$. The unit of density is
$N\mu{\rm m}^{-3}$. See the Supplemental Material for the movie of the
dynamics. (b) Time evolution of the orbital angular momentum $L_{z}$ (solid
lines), spin angular momentum $F_{z}$ (dotted lines), and total angular
momentum $F_{z}+L_{z}$ (dashed lines) for $B_{z}=0.05$ (blue or dark gray) and
0.1 mG (red or light gray). The first 1 ms is magnified.

The increase of the magnetization $F_{z}$ with the magnetic field $B_{z}$
implies the emergence of the Einstein-de Haas effect [77, 78]; if the applied
magnetic field $B_{z}$ is increased dynamically, the spin angular momentum
$F_{z}$ will increase, which must be accompanied by a decrease in the orbital
angular momentum
$L_{z}=-i\int\sum_{m}d\bm{r}\psi_{m}^{*}(x\partial_{y}-y\partial_{x})\psi_{m}^{*}$
to conserve the total angular momentum. Figure 4 demonstrates the dynamics of
the Einstein-de Haas effect, where the initial droplet state is prepared for
zero magnetic field with the symmetry axis in the $y$ direction, and the
magnetic field $B_{z}$ is turned on at $t=0$. As expected, the droplet begins
to rotate around the $z$ axis, where the total angular momentum $F_{z}+L_{z}$
is maintained to be zero. We note that such mechanical rotation of the torus
is a clearer manifestation of the Einstein-de Haas effect than that in the
trapped system [45].

Figure 5:  Ground state for $F=1$, $N=4\times 10^{5}$, $\varepsilon_{\rm
dd}=1.4$, and $B_{z}=0$ in a harmonic potential
$M(\omega_{x}^{2}x^{2}+\omega_{y}^{2}y^{2}+\omega_{z}^{2}z^{2})/2$ with
$(\omega_{x},\omega_{y},\omega_{z})=2\pi\times(100,1500,6000)$ Hz. (a)
Isodensity surface and (b) distributions of the density $\rho$ and
magnetization $\bm{f}$ on the $z=0$ plane. The unit of density is $N\mu{\rm
m}^{-3}$.

Next we consider the ground state of the system confined in a “surfboard-
shaped” trap [79], where the trap is weak, moderate, and tight in the $x$,
$y$, and $z$ directions, respectively. Figure 5 shows the density and spin
distributions of the ground state, in which multiple droplets with magnetic
vortices are aligned along the $x$ axis with alternate circulations of the
magnetic vortices. This state can be called a one-dimensional supersolid,
because the ground state has the (quasi)periodicity that breaks the
(quasi)translation symmetry (in the $x$ direction), while each droplet is
connected with adjacent droplets, which enables superflow between them. We
note that the dipolar supersolid of cigar-shaped droplets [80, 81, 82, 83, 84,
85, 86, 87, 88, 89, 90, 91, 92, 93, 94] also requires confinement in one or
two directions, i.e., restricted geometry is required for the dipolar BEC to
split into multiple droplets. If the trap in the $y$ or $z$ direction in Fig.
5 is removed, the ground state becomes a large single droplet with a magnetic
vortex.

The torus-shaped droplet may be generated experimentally by the following
procedure. First, the condensate atoms are prepared in the $|F,m=0\rangle$
hyperfine state in an optical trap. Since this nonmagnetized state is
dynamically and energetically unstable, spontaneous magnetization occurs [55].
If the BEC is confined to the expected droplet size, the spontaneous
magnetization will form the magnetic vortex with the lowest energy [49]. After
some relaxation time, the optical trap is switched off, which results in the
self-bound torus-shaped droplet. When the hyperfine spins with higher energies
($F\neq 6$) are used, the above experimental procedure must be accomplished
within the lifetime due to hyperfine exchanging collisions, which has not been
measured for a europium BEC.

To summarize, we have investigated a self-bound droplet in a spinor dipolar
BEC. For a large number of atoms, large $\varepsilon_{\rm dd}$, and a small
magnetic field, there exists a stable self-bound torus-shaped droplet that
contains a magnetic vortex (Fig. 1). For some range of the magnetic field, the
system exhibits bistability between the torus and cigar-shaped droplets (Fig.
3). The torus-shaped droplet exhibits the Einstein-de Haas effect by the
change of the applied magnetic field (Fig. 4). Multiple droplets form a
supersolid in a surfboard-shaped trap (Fig. 5).

This work was supported by JSPS KAKENHI Grant No. JP23K03276.

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