# Quantum Droplets in Imbalanced Atomic Mixtures

T. A. Flynn<EMAIL_ADDRESS>L. Parisi  T. P. Billam  N. G. Parker  Joint
Quantum Centre (JQC) Durham–Newcastle, School of Mathematics, Statistics and
Physics, Newcastle University, Newcastle upon Tyne, NE1 7RU, United Kingdom

###### Abstract

Quantum droplets are a quantum analogue to classical fluid droplets in that
they are self-bound and display liquid-like properties — such as
incompressibility and surface tension — though their stability is the result
of quantum fluctuations. One of the major systems for observing quantum
droplets is two-component Bose gases. Two-component droplets are typically
considered to be balanced, having a fixed ratio between the densities of the
two component. This work goes beyond the fixed density ratio by investigating
spherical droplets in imbalanced mixtures. With increasing imbalance, the
droplet is able to lower its energy up to a limit, at which point the droplet
becomes saturated with the atoms of the majority component and any further
atoms added to this component cannot bind to the droplet. Analysing the
breathing mode dynamics of imbalanced droplets indicates that the droplet can
emit particles, as in balanced mixtures, but the imbalance leads to an
intricate superposition of multiple simultaneously decaying collective
oscillations.

##  I INTRODUCTION

Quantum gases are highly controllable systems due in part to their diluteness,
yielding a highly versatile platform for exploring quantum many-body physics
[1]. For a system of bosons, a mean-field model for quantum gases is Gross-
Pitaevskii (GP) theory [2, 3], which captures much of the physics of these
systems, whilst neglecting beyond-mean-field effects such as quantum
fluctuations [4, 5]. For a single-component homogeneous Bose gas stability is
governed by the atomic interactions, with the gas becoming unstable to
collapse for attractive interactions. This has been demonstrated
experimentally by using a Feshbach resonance to tune the interactions from
repulsive to attractive [6, 7]. For a two-component system, the stability
depends on both the inter- and intraspecies contact interactions, which can
likewise be tuned via Feshbach resonances [8, 9, 10]. For repulsive
intraspecies interactions — to ensure stability of each individual component —
and attractive interspecies interactions, mean-field theory once more predicts
an unstable gas. However, this attractive collapse can be stabilised at high
densities by quantum fluctuations [11] forming a self-bound quantum droplet;
this has inspired work in so called Lee-Huang-Yang (LHY) fluids [12, 13],
named for the correction describing quantum fluctuations to first-order. By
tuning mean-field interactions between the components to vanish, the
interactions of LHY fluids are described by quantum fluctuations alone [12,
13].

In free space quantum droplets exist in equilibrium with the vacuum and form a
seemingly counter-intuitive dilute liquid-like state [11, 14]. This adds to
the variety of properties that quantum gases can be used to investigate, as
the majority of experiments are inherently within the gas phase — a property
widely exploited in time-of-flight imaging [15] — whereas quantum droplets
open up the field to explore the properties of quantum liquids, such as
surface tension [11, 16, 17] and incompressibility [18, 19], within
controllable experimental conditions.

Quantum droplets have been experimentally observed in two systems: (i) dipolar
gases of Dy [20, 21, 22] and Er [23]; (ii) homonuclear 39K [24, 25, 26, 19]
and heteronuclear, 41K-87Rb and 23Na-87Rb [27, 28], two-component mixtures.
The underlying mechanism for droplet formation is the result of a vanishing
mean-field interspecies attraction tending to cause instability, that is
countered by the repulsive quantum fluctuations which become significant with
increasing density [11]. However, the mean-field interactions within a dipolar
Bose gas include both the two-body, short-ranged interactions of a non-dipolar
gas, and anisotropic, long-ranged interactions resulting from strong atomic
magnetic dipoles [29]. This anisotropy results in elongated droplet profiles
[30, 31, 32, 33, 34], which differentiates dipolar droplets from non-dipolar,
two-component droplets.

A defining property of three-dimensional (3D) two-component quantum droplets
is self-evaporation [11]. In certain regimes of the droplet’s phase diagram,
excitations will cause the droplet to shed atoms in order to relax to a system
of lower energy. This occurs when the energies of these excitations exceed
$-\mu$, the particle emission threshold, where $\mu$ is the chemical potential
of the droplet. Furthermore, the lowest energy monopole mode — the breathing
mode — decays across the largest proportion of the droplet’s phase diagram
relative to other collective modes [11, 35]. Self-evaporation is a non-
intuitive and remarkable property that is not exhibited by dipolar quantum
droplets [18], or for the breathing mode of a 1D two-component droplet [36],
and has not been experimentally observed, but it is crucial in understanding
the dynamics of these objects.

The formation of a two-component quantum droplet also has an interesting
property: density balancing. A key result from the pioneering work of Petrov
is the energetic favourability for the two component densities to maintain a
fixed ratio $n_{2}/n_{1}=\mathrm{const.}$ [11] where $n_{i}$ is the number
density of the $i$th component. By pairing this assumption with negligible
spin modes [11] — i.e., assuming only in-phase density oscillations — the
mixture can be modelled via a single macroscopic wavefunction. The majority of
the literature has focused on such balanced droplets, with theoretical studies
of imbalanced systems limited to dipolar mixtures [37, 38] and low-dimensions
[39, 40]. This work is a systematic study of the ground states and breathing
modes of 3D spherical imbalanced quantum droplets in homonuclear mixtures.
Adding atoms to one component of the mixture yields a lower energy
configuration than the associated balanced droplet. This forms a droplet with
a density imbalance in the core, and this imbalance can be increased to a
limit at which point any further atoms cannot bind to the droplet. By
investigating imbalanced droplets in the free space limit, this work explores
how the density profiles, chemical potentials and breathing mode dynamics of
the droplet are modified by the presence of a population imbalance.

This work begins with a discussion of the underlying theory in Section II.
This theory is then applied in Sections III and IV to first isolate imbalanced
droplet ground states and then to propagate these states in time, subject to
an initial perturbation, to analyse the droplet breathing modes. Finally the
main results and potential future research avenues are discussed in Section V.

##  II THE MODEL

A zero-temperature mixture of two weakly-interacting, dilute Bose gases can be
described by the energy functional [41, 11]

$\begin{split}E=\int&\bigg{[}\frac{\hbar^{2}}{2m_{1}}|\gradient{\Psi_{1}}|^{2}+\frac{\hbar^{2}}{2m_{2}}|\gradient{\Psi_{2}}|^{2}+\frac{g_{11}}{2}|\Psi_{1}|^{4}+\frac{g_{22}}{2}|\Psi_{2}|^{4}\\\
&+g_{12}|\Psi_{1}|^{2}|\Psi_{2}|^{2}+\frac{E_{\mathrm{LHY}}}{V}\bigg{]}\differential^{3}\mathbf{r},\end{split}$
(1)

in which $m_{i}$ are the atomic masses of the $i$-th component, $g_{ii}$ and
$g_{12}$ are the effective intra- and interspecies interaction strengths
respectively, and are related to the intra- and interspecies scattering
lengths by $g_{ii}=4\pi\hbar^{2}a_{ii}/m_{i}$ and
$g_{12}=2\pi\hbar^{2}a_{12}(1/m_{1}+1/m_{2})$. The first two terms of Equation
1 are the kinetic energy contributions; the next three terms describe the two-
body interactions within and between the components. The final term is the LHY
correction which, to first-order, describes the effects of quantum
fluctuations on the condensate [42]. For a Bose-Bose mixture, the LHY
correction takes the form [11]

$\frac{E_{\mathrm{LHY}}}{V}=\frac{8}{15\pi^{2}}\left(\frac{m_{1}}{\hbar^{2}}\right)^{3/2}\left(g_{11}n_{1}\right)^{5/2}f\left(z,u,x\right),$

where $f$ is a function, defined in [16, 43], with arguments $z=m_{1}/m_{2}$,
$u=g_{12}^{2}/(g_{11}g_{22})$ and $x=(g_{22}n_{2})/(g_{11}n_{1})$. The
function $f(z,u,x)$ means that the LHY term takes a relatively simple form in
homonuclear mixtures ($m_{2}=m_{1}$) [11], but takes a far more complex form
in heteronuclear mixtures ($m_{2}\neq m_{1}$) [11, 16]. The function argument,
$u$, reduces to one by assuming that the mixtures lies at the critical point
of attractive stability, i.e., $g_{12}^{2}=g_{11}g_{22}\implies u=1$, removing
the issue of complex contributions resulting from an unstable phonon mode [11,
44, 45]. It should be noted that this approximation is made only in this
derivation and does not imply any parameter choice in subsequent analyses.
Crucial to the formation of quantum droplets, quantum fluctuations stabilise
the attractive mixture against collapse, and furthermore are ubiquitous in
nature though often play a limited role in the physics of many quantum gas
experiments as discussed in Section I.

The two component densities can each be related to a macroscopic wavefunction
or order parameter, $n_{i}=|\Psi_{i}|^{2}=\Psi_{i}^{\ast}\Psi_{i}$. A
variational approach,
$i\hbar\partialderivative*{\Psi_{i}}{t}=\functionalderivative*{E}{\Psi_{i}^{\ast}}$
[4], can then be used to derive the coupled extended GP equations,

$\displaystyle\begin{split}&i\hbar\partialderivative{\Psi_{1}}{t}=\bigg{[}-\frac{\hbar^{2}}{2m_{1}}\laplacian+g_{11}|\Psi_{1}|^{2}+g_{12}|\Psi_{2}|^{2}\\\
&+\frac{8}{15\pi^{2}}\left(\frac{m_{1}}{\hbar^{2}}\right)^{3/2}g_{11}^{5/2}\left\\{\frac{5}{2}f|\Psi_{1}|^{3}-\frac{g_{22}}{g_{11}}\partialderivative{f}{x}|\Psi_{1}||\Psi_{2}|^{2}\right\\}\bigg{]}\Psi_{1},\\\
&i\hbar\partialderivative{\Psi_{2}}{t}=\bigg{[}-\frac{\hbar^{2}}{2m_{2}}\laplacian+g_{22}|\Psi_{2}|^{2}+g_{12}|\Psi_{1}|^{2}\\\
&+\frac{8}{15\pi^{2}}\left(\frac{m_{1}}{\hbar^{2}}\right)^{3/2}g_{11}^{3/2}g_{22}\partialderivative{f}{x}|\Psi_{1}|^{3}\bigg{]}\Psi_{2}.\end{split}$
(2)

The computational complexity of these LHY terms can be minimised by assuming a
homonuclear mixture, giving the equal-mass coupled extended GP equations [11]

$\displaystyle i\hbar\partialderivative{\Psi_{1}}{t}=$
$\displaystyle\bigg{[}-\frac{\hbar^{2}}{2m}\laplacian+\frac{4\pi\hbar^{2}a_{11}}{m}|\Psi_{1}|^{2}+\frac{4\pi\hbar^{2}a_{12}}{m}|\Psi_{2}|^{2}$
$\displaystyle+\frac{128\sqrt{\pi}\hbar^{2}a_{11}}{3m}\left(a_{11}|\Psi_{1}|^{2}+a_{22}|\Psi_{2}|^{2}\right)^{3/2}\bigg{]}\Psi_{1},$
$\displaystyle i\hbar\partialderivative{\Psi_{2}}{t}=$
$\displaystyle\bigg{[}-\frac{\hbar^{2}}{2m}\laplacian+\frac{4\pi\hbar^{2}a_{22}}{m}|\Psi_{2}|^{2}+\frac{4\pi\hbar^{2}a_{12}}{m}|\Psi_{1}|^{2}$
$\displaystyle+\frac{128\sqrt{\pi}\hbar^{2}a_{22}}{3m}\left(a_{11}|\Psi_{1}|^{2}+a_{22}|\Psi_{2}|^{2}\right)^{3/2}\bigg{]}\Psi_{2}.$

Finally, the dimensional scalings $\mathbf{r}=\xi\mathbf{\tilde{r}}$,
$t=\tau\tilde{t}$ and $\Psi_{i}=\rho_{i}^{1/2}\tilde{\Psi}_{i}$ result in the
dimensionless, equal-mass coupled extended GP equations,

$\displaystyle\begin{split}i\partialderivative{\Psi_{1}}{t}=&\bigg{[}-\frac{1}{2}\laplacian+|\Psi_{1}|^{2}+\eta|\Psi_{2}|^{2}\\\
&+\alpha\left(|\Psi_{1}|^{2}+\beta|\Psi_{2}|^{2}\right)^{3/2}\bigg{]}\Psi_{1},\\\
i\partialderivative{\Psi_{2}}{t}=&\bigg{[}-\frac{1}{2}\laplacian+\beta|\Psi_{2}|^{2}+\eta\beta|\Psi_{1}|^{2}\\\
&+\alpha\beta^{2}\left(|\Psi_{1}|^{2}+\beta|\Psi_{2}|^{2}\right)^{3/2}\bigg{]}\Psi_{2}.\end{split}$
(3)

in which all tildes are hereafter neglected and the dimensionless parameters
are

$\eta=\frac{a_{12}}{\sqrt{a_{11}a_{22}}},$
$\alpha=\frac{32}{3}\left[\frac{2}{3\pi}\frac{|\delta
a|a_{11}^{5/2}n_{1}^{(0)}}{\sqrt{a_{11}}+\sqrt{a_{22}}}\right]^{1/2},\,\beta=\left(\frac{a_{22}}{a_{11}}\right)^{1/2},$

with dimensional parameters

$\displaystyle\xi=\sqrt{\frac{3}{8\pi}\frac{(\sqrt{a_{11}}+\sqrt{a_{22}})}{|\delta
a|\sqrt{a_{11}}n_{1}^{(0)}}},$
$\displaystyle\tau=\frac{3m}{8\pi\hbar}\frac{(\sqrt{a_{11}}+\sqrt{a_{22}})}{|\delta
a|\sqrt{a_{11}}n_{1}^{(0)}},$ $\displaystyle\rho_{1}=\frac{2}{3}\frac{|\delta
a|n_{1}^{(0)}}{\sqrt{a_{11}}(\sqrt{a_{11}}+\sqrt{a_{22}})},$
$\displaystyle\rho_{2}=\frac{2}{3}\frac{|\delta
a|n_{1}^{(0)}}{\sqrt{a_{22}}(\sqrt{a_{11}}+\sqrt{a_{22}})},$

where $\delta a=a_{12}+\sqrt{a_{11}a_{22}}$, and $n_{1}^{(0)}$ is the
equilibrium density of component-1 for the balanced mixture. The expression of
the equilibrium density is calculated in a homogeneous infinite system under
the criterion of a vanishing pressure — i.e., the droplet in equilibrium with
the vacuum — and takes the form [11]

$n_{1}^{(0)}=\frac{25\pi}{1024}\frac{(a_{12}+\sqrt{a_{11}a_{22}})^{2}}{a_{11}^{3/2}a_{22}(\sqrt{a_{11}}+\sqrt{a_{22}})^{5}}.$

The density scalings $\rho_{i}$ correspond to rescaled normalisation
constants, $\tilde{N_{i}}=N_{i}/(\rho_{i}\xi^{3})$, in which $N_{i}$ is the
normalisation constant of the $i$-th component wavefunction. In subsequent
sections the dimensionless forms of $N_{i}$, $\Psi_{i}$, $\mathbf{r}$ and $t$
are used, with tildes omitted for clarity.

By assuming a constant density ratio, $n_{2}/n_{1}=\sqrt{g_{11}/g_{22}}$, the
two component wavefunctions can be expressed in terms of a single
wavefunction, $\Psi_{i}=\sqrt{n_{i}}\phi$, neglecting any out-of-phase motion
between the components [11, 46, 47]. Equations (2) reduce to a single
equation,

$i\partialderivative{\phi}{t}=\left[-\frac{1}{2}\laplacian-3|\phi|^{2}+\frac{5}{2}|\phi|^{3}\right]\phi,$

with the system described by a single parameter, an effective atom number,
$\tilde{N}$, given by [11]

$\tilde{N}=\left(\frac{\sqrt{g_{22}}}{n_{1}^{(0)}(\sqrt{g_{11}}+\sqrt{g_{22}})}\right)\frac{N}{\xi^{3}},$
(4)

in which $N$ here is the total atom number $N=N_{1}+N_{2}$. Within this work,
balanced and imbalanced droplets are both modelled by Equations (3), though it
should be noted that, for a balanced droplet, the dimensionless parameters
$(N_{1},N_{2},\alpha,\beta,\eta)$ can be recast to $\tilde{N}$. In the
density-locked model, a given set of interaction strengths, $g_{ii}$ and
$g_{12}$, correspond to a fixed population number ratio,
$N_{2}/N_{1}=\sqrt{g_{11}/g_{22}}$, imposing that the atoms can only reside
bound to the droplet. By breaking the assumption of density-locking, it is
possible to imbalance population numbers such that
$N_{2}/N_{1}\neq\sqrt{g_{11}/g_{22}}$. The next section explores the effect of
this imbalance on the density structure and energy of 3D spherical droplet
ground states.

##  III GROUND STATES

This work considers spherically-symmetric droplets. Density is hence assumed
to be a function of radius only, reducing the computational problem to an
effective 1D system with the kinetic term becoming
$\laplacian\Psi_{i}\rightarrow[\partialderivative*[2]{(r\Psi_{i})}{r}]/r$.
Further assumptions within this work include: a homonuclear mixture and
balanced intraspecies scattering lengths ($a_{11}=a_{22}\implies\beta=1$), to
simplify the problem. Thus, the only differences between the components can
arise from an imposed atom number imbalance of $N_{1}=N_{2}+\delta N_{1}$.

###  III.1 Flat-top Droplet Limit

A simplified ansatz for the density of large droplets, as used in Ref. [11],
is a flat-top, step function in which kinetic energy contributions are
neglected. The two-components of the imbalanced droplet are modelled as
majority and minority components with $N+\delta N$ and $N$ atoms,
respectively. Substituting this ansatz into the two-component energy
functional gives the equilibrium energy of the flat-top droplet,
$E_{\mathrm{eqbm}}$, as a function of the imbalance, $\delta N$ (see the
Appendix Afor further details, including a schematic of the ansatz). The
results of this energy calculation are given in Figure 1 with the total
equilibrium energy given in the main plot, and the inset showing the
equilibrium energy per particle.

Firstly, the main plot of Figure 1 shows that there is a minimum at $\delta
N\neq 0$ in the total energy, indicating the droplet can lower its energy by
absorbing particles into one component. However, the equilibrium energy per
particle given in the inset of Figure 1 exhibits a minimum at $\delta N=0$
demonstrating that whilst the total energy of the droplet can be lowered by
single-component absorption, there is a corresponding increase in the energy
per particle resulting in a less stably bound droplet. Secondly, the Appendix
Ashows that the minimum in the equilibrium energy occurs at $\delta
N/N=2(\eta-2)+[(\eta-1)(4\eta-14)]^{1/2}$. That is, the energy of the droplet
decreases with imbalance up to this limit. The dependence of this quantity on
the scattering lengths is consistent with the prediction of Ref. [11] at
linear order 111Ref. [11] finds $\delta N/N\sim|\delta g|/g=-(1+\eta)/2$ for
$|\delta g|/g\ll 1$; expanding our result in the same limit yields $\delta
N/N\approx(1/3)|\delta g|/g$.. For the parameters of Figure 1 $\delta n\approx
0.458$, as highlighted by the orange vertical line.

At this limit the droplet becomes saturated with the majority component. Hence
if further majority component atoms are available, they cannot be absorbed
into the droplet and will form an unbound gas around the droplet (as described
in Ref. [11]). There are thus three droplet states: (1) balanced droplets; (2)
bound, imbalanced droplets; and (3) saturated, imbalanced droplets (which can
be immersed in a halo of unbound atoms).

Figure 1: Energy of a flat-top density ansatz as a function of imbalance,
$\delta N$, with parameters $N_{2}\approx 22524$, $\alpha\approx 0.0152$,
$\beta=1.0$ and $\eta=-1.2$. The equilibrium energy, $E_{\mathrm{eqbm}}$, is
shown in the main plot with the orange, vertical line indicating the location
of the minimum at $\delta N\approx 703$. The inset shows the equilibrium
energy per particle, $[E/(2N+\delta N)]_{\mathrm{eqbm}}$, with the only
minimum appearing at the origin.

The decrease in energy of an imbalanced droplet is consistent with the
predictions of Ref. [11] and stems from the basic argument that droplets
always seek to lower their energy by absorbing atoms. For example, in the
density-locked model, atoms can only be added to both components to ensure
that $N_{2}/N_{1}=\sqrt{g_{22}/g_{11}}$ is preserved, and by adding atoms to
both components, a lower energy state is recovered. One way to conceptualise
this is due to droplets having negative chemical potentials, meaning that it
is always energetically favourable to absorb more particles into both
components. However, the subtlety in the energy calculation given in the
Appendix Ais that the energy of a droplet can also be lowered by absorbing
particles into one component if there is an excess of this component
available, up to the saturation limit discussed above.

By using the simple flat-top density ansatz, key insights are gained into
effects of a population imbalance on the density structure and energy of a
two-component droplet. However, for smaller droplet sizes kinetic energy
cannot be neglected; thus, to extend this analysis to general, spherical
droplets, the coupled GP equations are solved numerically.

###  III.2 Numerical Solutions

To find ground state solutions, the coupled GP equations are propagated
numerically in imaginary time until the energy of the mixture is deemed
adequately converged. The numerical scheme is a $4^{\mathrm{th}}$-order Runge-
Kutta method, with centred finite-difference methods for the spatial
derivatives. Neumann boundary conditions
($\partialderivative*{\Psi_{i}}{r}=0$) are applied at the centre of the
computational box as the density in the droplet core is approximately
constant. The boundary at $r=L_{r}$, where $L_{r}$ is the radial computational
box size, has either Neumann or Dirichlet [$\Psi_{i}(r=L_{r})=0$] boundary
conditions, which are discussed further below with reference to Figure 2.

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Figure 2: Balanced and imbalanced ground state droplets, with fixed
parameters: $N_{2}\approx 22524$, $\alpha\approx 0.0152$, $\beta=1.0$ and
$\eta=-1.2$. (a), (b) and (c) Ground state density profiles of: a balanced
droplet ($\delta N_{1}=0$); a bound, imbalanced droplet ($\delta N_{1}\approx
450$); and a saturated, imbalanced droplet with an unbound halo of majority
component atoms ($\delta N_{1}\approx 5631$), respectively. The inset shows
the difference in component densities, $\Delta n(r)$. (d) The difference in
central densities as a function of imbalanced majority component atoms,
$\delta N_{1}$, with box size of $L_{r}=128$. At $\delta N_{1}=0$ the system
is balanced, with the leftmost vertical grey line corresponding to (a).
Increasing $\delta N_{1}$ leads to an approximately linear increase in $\Delta
n(r=0)$, there are however no atoms outside of the droplet as
$n_{1}(r=0.75L_{r})$, a measure of density outside of the droplet, is zero.
This regime corresponds to a bound, imbalanced droplet with the central
vertical grey line corresponding to (b). The linear increase in $\Delta
n(r=0)$ eventually plateaus to a regime in which the central densities in the
droplet cannot imbalanced further and the imbalanced droplet saturates with
the density outside of the droplet increasing, resulting in a non-zero density
of unbound atoms. The inset shows a measure of the difference in central
densities in the limit of a saturated imbalanced droplet, $\Delta
n^{\mathrm{sat}}(r=0)$, as a function of droplet size given by the effective
atom number $\tilde{N}$ in Equation 4, using the parameters for the $\delta
N_{1}=0$ case. The orange, dashed, horizontal line corresponds to $\delta
n\approx 0.458$, with the droplet size used in the main plot highlighted by
the grey, vertical line. (e) Neumann boundary conditions are applied at both
boundaries of the computational box, with $L_{r}=1024$. The main plot shows
the difference in central densities, recreating the results from (d), showing
that the behaviour is not a function of the boundary conditions. It likewise
demonstrates that the saturated droplet reaches a lower energy,
$E[\Psi_{1},\Psi_{2}]$, state than the balanced droplet. The inset shows an
example imbalance of $\delta N_{1}\approx 22524$, with varying box size
$L_{r}$. In the limit of large a computational box any background gas will be
effectively zero density, thus converging to a saturated droplet in free
space.

By enforcing the Dirichlet boundary condition at $r=L_{r}$, Figure 2(a), (b)
and (c) show the three different ground state solutions (discussed above) by
varying $\delta N_{1}$, i.e., by varying the imbalance of the mixture. Figure
2(a) presents an example of a balanced droplet ($N_{1}=N_{2}\implies\delta
N_{1}=0$) with an inset of the density difference $\Delta
n(r)=n_{1}(r)-n_{2}(r)$, showing that the two component densities are
identical. By increasing the population imbalance to $\delta N_{1}\approx
450$, as shown in Figure 2(b), the two component densities start to split
within the droplet core, with both an increase and decrease in the majority
and minority components’ central densities, respectively. Hence, the two
components are no longer identical but notably the density differences in the
inset shows that this imbalanced droplet has no unbound atoms around the
droplet. This implies that this state is a bound, imbalanced droplet. Driving
the imbalance higher to $\delta N_{1}\approx 5631$, Figure 2(c) demonstrates a
more pronounced density splitting within the droplet core, but crucially also
exhibits a non-zero gas density outside of the droplet [highlighted in the
inset of (c)]. This ground state hence corresponds to a saturated, imbalanced
droplet at the centre of the box with an unbound gas cloud outside of the
droplet core.

For a set droplet size — i.e., fixed $\alpha$, $\eta$ and $N_{2}$ — the only
free parameter is $\delta N_{1}$ and thus Figure 2(d) shows two measures of
the droplet ground states for varying $\delta N_{1}$. The two measures are the
central density difference, $\Delta n(r=0)=n_{1}(r=0)-n_{2}(r=0)$, and the
majority component density, $n_{1}$, at a fixed radius outside of the droplet.
The chosen fixed radius is $r=0.75L_{r}$, which is chosen to be relatively far
from the droplet surface but not too close to $r=L_{r}$, to avoid the density
drop resulting from the Dirichlet boundary condition. These measures are used
to illustrate the transition between bound, imbalanced droplets, and
saturated, imbalanced droplets. Figure 2(d) shows that for increasing $\delta
N_{1}$ from zero, there is an immediate density splitting within the droplet
corresponding to the approximately linear increase in $\Delta n(r=0)$. It
should be noted that in this regime, $n_{1}(r=0.75L_{r})$ stays fixed at zero,
meaning that this relatively small $\delta N_{1}$ regime corresponds to bound,
imbalanced droplets. However, increasing $\delta N_{1}$ further eventually
leads to the formation of a shoulder in $\Delta n(r=0)$ which is the
saturation of the imbalanced droplet, i.e., the droplet is reaching a limit of
how many majority component atoms can be absorbed. Beyond this shoulder
$\Delta n(r=0)$ then approaches a constant value [labelled here as $\Delta
n^{\mathrm{sat}}(r=0)$]. Once in the saturated droplet regime, there is a
corresponding increase in $n_{1}(r=0.75L_{r})$, indicating that the excess
atoms reside outside of the droplet. The phase diagram in Figure 2(d)
corresponds to the three ground state densities in (a), (b) and (c) given by
the vertical grey lines. Furthermore, the inset of Figure 2(d) illustrates how
the central density splitting of the saturated droplet varies with droplet
size, given by $\tilde{N}$, the effective atom number of the balanced droplet
in Equation 4 (from Ref. [11]). To vary droplet size, the fixed parameters are
$\alpha$ and $\eta$, thus $N_{2}$ (and hence $N_{1}$) is varied for the
different droplet sizes, with $\tilde{N}$ calculated for the balanced droplet
$(\delta N_{1}=0)$ parameters. The central density difference of the saturated
droplet increases with droplet size and asymptotically approaches a constant
value of $\delta n\approx 0.458$ [given by the orange, dashed, horizontal line
in the inset of (d)] in the limit of an large, flat-top droplet as discussed
above.

Figure 2(c) and (d) show that in the limit of the saturated droplet, there
exists the unbound gas. This work seeks to explore imbalanced droplets, in the
absence of this unbound gas, to probe the physics of imbalancing the droplet
in free space. Thus, rather than imposing a Dirichlet boundary condition at
$r=L_{r}$, instead the same Neumann boundary condition used at $r=0$, is used
at $r=L_{r}$, also. This gives a numerical approximation to a free space
system. Figure 2(e) uses equivalent parameters as in (d), with a slightly
increased $\delta N_{1}$ range, and exhibits the same central density
difference behaviour as with the Dirichlet boundary conditions at $r=L_{r}$.
From this point on, Neumann boundary conditions are applied at both
boundaries.

Figure 2(e) also shows the energy of the mixture which uses the dimensionless
form of Equation 1, by defining $\tilde{E}=E/\epsilon$ where
$\epsilon=[4/(3\pi^{2}\alpha)][\hbar^{2}/(m\xi^{2})]$, in which tildes are
subsequently neglected. The energy decreases with imbalance — as predicted by
the analytic calculated presented in the Appendix A— until again reaching the
saturation limit. Note again that a saturated, imbalanced droplet cannot
absorb further majority component atoms and thus the energy of the droplet
does not vary with $\delta N_{1}$. However, this is an effect of the large
computational boxes used, in that the density of the unbound atoms is
negligibly small such that once in this saturation limit the system is
essentially identical with changing $\delta N_{1}$. In smaller computational
boxes — or equivalently, exploring much larger imbalances — the density of the
background gas becomes significant [e.g. Figure 2(c)], which is explored in
the inset of (e). By fixing $\delta N_{1}\approx 22524$, i.e., an imbalanced
droplet in the saturation limit, the size of the computational box is varied.
In small boxes, the density of the unbound gas is non-zero and thus has a
significant, positive energy contribution. The existence of the background gas
likewise modifies the internal structure of the droplet as can be seen by the
increased value of $\Delta n(r=0)$. In the limit of large $L_{r}$ (or
equivalently small $1/L_{r}$) the energy contribution of the background
approaches zero as the density of the gas approaches zero. Thus in the limit
of a large box, all of the ground states converge to the saturated droplet in
free space, as can be seen from the convergence of both the energy and $\Delta
n(r=0)$ in the inset of Figure 2(e).

Further evidence of the energy saturation is given by the two component
chemical potentials in Figure 3(a). For increasing $\delta N_{1}$ the majority
component chemical potential (the dark orange curve) converges towards zero,
implying there is no energetic favourability for the droplet to absorb more
majority component atoms. Additionally, if more atoms of the minority
component were made available to the droplet, then these atoms will likewise
be absorbed into the droplet, reducing the energy of the droplet even further.

In summary, droplets will always seek to absorb atoms to reduce the system
energy. If there is an imbalance of atoms from
$N_{2}/N_{1}=\sqrt{g_{22}/g_{11}}$ then the droplet will absorb more of the
majority component but this reaches a limit in which the droplet is saturated
with the majority component atoms. It should be noted that in the case of a
non-zero density unbound gas, this limiting behaviour will be modified by the
positive energy contribution of the majority component density tails, similar
to submerged droplets observed in 1D imbalanced droplets [39, 40]. These
results have been presented both from an analytic energy calculation, and from
numerical simulations, the latter of which is to be used next to explore the
dynamical stability of these imbalanced droplets across their parameter space.

##  IV BREATHING MODE

Self-evaporation is a key property of 3D two-component quantum droplets [11].
Recalling that the density-locked droplet can be described by a single
parameter, $\tilde{N}$, there are three main regimes of interest for the
droplet collective modes: 1) all modes exceed the particle emission threshold
and hence are evaporated ($\tilde{N}\lesssim 94.2$); 2) the monopole mode
evaporates but other non-zero angular momentum modes are stable
($94.2\lesssim\tilde{N}\lesssim 933.7$); 3) the monopole is stabilised
($\tilde{N}\gtrsim 933.7$) [11, 35]. This first phase is the principal
argument behind the self-evaporative property of a two-component droplet,
since a finite-temperature droplet is inherently excited by contributions
from, for example, the non-condensate components. Thus, these excitations are
evaporated and the droplet relaxes to a lower energy configuration, akin to
self-cooling [11].

In this work, the system is restricted to spherically symmetric droplets,
meaning that the only observable mode is the breathing mode. The restriction
of spherical symmetry is applied to reduce computational cost. This is due to
particle shedding of self-evaporative droplets resulting in reflections from
the computational box boundaries which become a substantial issue for long-
time dynamics. To avoid this issue, very large box sizes — $L_{r}=8192$,
approximately $500$ times the droplet sizes considered here — are used to
observe the dynamics of the droplets without interference from reflected
particles. If simulating a general 3D droplet, these box sizes would quickly
become infeasible. Note that the use of large computational boxes is crucial
to the focus of this paper, namely imbalanced droplets in free space, i.e.,
without the external effects of unbound atoms. By excluding any non-zero
angular momentum modes, the spherical symmetry yields two regimes: a decaying
and a stable breathing mode, thus this work analyses the stability of
breathing modes in the presence of an imbalance.

The breathing mode frequencies of the balanced, density-locked droplet are a
function of the single parameter, $\tilde{N}$ [11]. However, for an imbalanced
system, a further degree of freedom is introduced, the size of the imbalance.
As shown in the previous section, the central density differences, chemical
potentials and energies, as functions of imbalance, reach the saturation limit
and any further imbalance has no significant effect on the internal structure
of the droplet. In order to observe these collective modes, the ground state
solutions found via imaginary time propagation, shown in Section III, are then
evolved in real time via Equations (3). To trigger a breathing mode in the
droplet, a harmonic phase is imprinted on the ground state wavefunction, i.e.,
$e^{i\epsilon r^{2}}$ where $\epsilon$ is small (here $\epsilon=10^{-5}$) [4,
49]. This phase is always imprinted onto the minority component, though
breathing modes are an in phase oscillation so the subsequent dynamics are not
dependent on this choice of component.

Figure 3: Chemical potentials and breathing modes as a function of majority
component imbalance, $\delta N_{1}$, in the self-evaporative regime
(equivalent to a balanced droplet, $N_{1}=N_{2}$, with $\tilde{N}\approx
649$). (a) The chemical potentials (light orange — minority component, dark
orange — majority component) and early-time breathing mode frequencies (light
and dark purple dashed lines) as a function of imbalance. These results
correspond to the range $0\leq\delta N_{1}\lesssim 4257$, $N_{2}\approx
17027$, $\alpha\approx 0.00657$ and $\eta\approx-1.11$. At $\delta N_{1}=0$,
the chemical potentials are equal but diverge as the imbalance is increased
eventually reaching a saturation point where the chemical potential of the
majority component is approximately zero. The minority component chemical
potential likewise saturates, to an increased value. (b) A highlighted
simulation of $\delta N_{1}\approx 1447$ [corresponding to the bold vertical
line in (a)] at early times, with an inset of the associated power spectrum.
This indicates that at early times, there is a single rapidly decaying mode.
(c) The same $\delta N_{1}\approx 1447$ simulation at late times in which
there is there is a superposition of two modes corresponding to the two
chemical potentials, given by the vertical dashed lines in the inset power
spectrum of (c).

###  IV.1 Self-Evaporative Regime

The first modes to consider are within the self-evaporative regime
($\tilde{N}\lesssim 933.7$). Breathing modes of spherical, balanced droplets
have been quite extensively investigated [11, 50, 35]. Dynamically, within
this regime, the droplet begins to oscillate at a frequency exceeding the
particle emission threshold, $-\mu$, and rapidly decays with the decay rate
asymptotically tending to zero and the oscillation frequency tending to the
particle emission threshold [50]. The initial rapid decay is due to a high
dissipation of energy through particle emission, though in the long-time limit
this corresponds to limited particle emissions at the energy of the chemical
potential. This asymptotically decaying behaviour is likewise recovered in the
imbalanced case though with three modes instead of one due to the imbalance
yielding two chemical potentials. Figure 3(a) describes the three super-
imposed modes: the early-time, rapidly decaying mode — see purple dashed lines
in Figure 3(a) denoting the extracted equal frequencies of the two components
— that varies marginally with imbalance, which can be considered as the
intrinsic droplet breathing mode; at late times there are two further modes
replacing the early-time mode. The late-time modes arise from the splitting of
the chemical potentials which diverge with increasing imbalance before again
reaching the saturated droplet [see orange lines in Figure 3(a)].

To visualise the single early mode and two late modes, Figure 3(b) and (c)
represent a measure of the droplet central density,
$\bar{n}_{i}(t)=n_{i}(r=0,t)-\langle n_{i}(r=0)\rangle_{t}$, with insets
showing the associated power spectra $|\mathcal{F}^{\prime}[\bar{n}_{i}]|^{2}$
in which $\mathcal{F}^{\prime}[\cdot]$ denotes the power spectrum rescaled by
the mean, and all negative frequencies set to zero purely for data
visualisation. The early-time mode is a high-amplitude, rapidly decaying mode
that is given in Figure 3(b), with an inset of the associated power spectrum
highlighting the frequency (vertical dashed line) corresponding to the purple
dashed lines in Figure 3(a). As this mode decays, it is then replaced by the
two late-time modes corresponding to the split chemical potentials. The late-
time dynamics is thus a superposition of a higher and lower frequency mode, as
can be seen in Figure 3(c), once more with the associated power spectra
showing peaks at the two chemical potentials (given by the vertical dashed
lines). It should be noted that evidence of these late-time modes can even be
seen at early times, such as the shorter peak in the inset of Figure 3(b),
which roughly corresponds to the chemical potential of the majority component.

In summary, at early times the droplet oscillates with an unstable, high-
amplitude mode that decays rapidly due to energy dissipation from particle
emission. In the long-time limit however, the particle emission is
considerably reduced, with particles only emitted at energies of the two
chemical potentials. Hence, these late-time modes decay at much slower rates
than the initial mode. This is analogous to the density-locked mixture [50],
in which the mode frequency asymptotically converges to the particle emission
threshold, with a vanishing decay rate. Physically this vanishing decay rate
is the result of late-time emitted particles of energy $\approx-\mu$, which
thus have a negligible effect on the kinetic energy of the droplet.
Equivalently, for an imbalanced droplet, there are now two chemical potentials
which each have associated particle emission and hence associated residual
long-lived breathing modes [50].

###  IV.2 Non-Self-Evaporative Regime

The second set of breathing modes are within the non-self-evaporative regime
($\tilde{N}\gtrsim 933.7$). For a balanced droplet in this regime the
breathing mode frequency is lower than the particle emission threshold
resulting in a stable oscillation [11, 50]. In the self-evaporative regime,
the dynamics of the imbalanced droplet is highly reminiscent of the balanced
case but with further modes corresponding to the split chemical potentials,
whereas the non-self-evaporative region of the droplet phase has some greater
subtleties.

Figure 4: Chemical potentials and breathing modes as a function of majority
component imbalance, $\delta N_{1}$, in the non-self-evaporative regime
(equivalent to a balanced droplet, $N_{1}=N_{2}$, with $\tilde{N}\approx
1502$), using the same parameters as in Figure 2 with $0\leq\delta
N_{1}\lesssim 5631$. (a) The chemical potentials (light orange — minority
component, dark orange — majority component) and early-time breathing mode
frequencies (light and dark purple dashed lines) as a function of the
imbalance. The top inset shows the majority component chemical potential
crossing the breathing mode frequencies. This is shown in the lower insets in
which the mode is either stable (left) or is decayed (right). (b) A
highlighted simulation with $\delta N_{1}\approx 1014$ [corresponding to the
bold vertical line in (a)] at early times, with the associated power spectrum
(inset). The early time dynamics are dominated by a single damped breathing
mode. (c) The same $\delta N_{1}\approx 1014$ simulation as in (b), at late
times showing a superposition of three modes, with the lowest and highest
energy modes corresponding to the majority and minority component chemical
potentials, respectively. The third mode [the central peak in the inset of
(c)] is the early-time mode still decaying.

Figure 4(a) shows that, as in the self-evaporative case, the chemical
potentials diverge with increasing imbalance until reaching the saturated,
imbalanced droplet. The dynamics are again dominated by a high-amplitude mode
in both components — given by the central purple dashed lines of Figure 4(a) —
that is relatively constant with the changing imbalance. This mode can again
be considered the intrinsic droplet breathing mode, as described in [51]. In
the balanced case, the stability of the breathing mode for droplets of this
size is due to the particle emission threshold exceeding the mode frequency.
With increasing imbalance however, as the chemical potentials split, the
majority component chemical potential [i.e., the lower chemical potential
branch in Figure 4(a)] eventually crosses over the frequency of the stable
high-amplitude mode, at which point this mode will begin to decay. This mode
crossing implies that the imbalanced non-self-evaporative regime can instead
be split into two regions: 1) a stable breathing mode; 2) a decaying breathing
mode. This behaviour is highlighted in the upper inset of Figure 4(a),
focusing on smaller imbalances. Decay of the high-amplitude, intrinsic mode
occurs when the frequency exceeds the negative of the majority component
chemical potential, validating the idea that stability of the mode is entirely
dependent on the mode frequency exceeding the droplet’s particle emission
threshold. The lower left inset of Figure 4(a) shows that the mode is stable,
for an imbalance of $\delta N_{1}\approx 45$, as the frequency lies beneath
both chemical potentials. However, the lower right inset shows that this mode
becomes unstable, for an imbalance of $\delta N_{1}\approx 180$, and decays as
the frequency exceeds one of the chemical potential branches. The critical
imbalance between stability and instability of this mode is $\delta
N_{1}\approx 124$, for this specific droplet. Thus for a sufficiently small
imbalance, it is possible to have an imbalanced droplet with a stable
breathing mode.

Figure 4(b) and (c) focus on the early and late times of an unstable breathing
mode in this regime. At early times, the initial high-amplitude mode dominates
the system and oscillates at approximately the frequency of the balanced case.
This is highlighted in the power spectrum shown in the inset of Figure 4(b)
with the frequency given by the vertical dashed line. As found in the self-
evaporative regime, the energy of the droplet is dissipated through atom
shedding which then leads to the evaporation of the initial mode. The
oscillations at late times — shown in Figure 4(c) — are instead dominated by
two other modes, with frequencies corresponding to the chemical potentials of
each component given by the two outer vertical dashed lines. There are still
residual oscillations from the decaying initial mode which hence explains the
interference seen in the oscillations of Figure 4(c) and the central peak
shown in the power spectrum.

Thus, in both the self-evaporative and non-self-evaporative regimes, providing
there is decay of the initial high-amplitude mode, there are two main regions
of the dynamics: early times — where the dynamics is dominated principally by
a high-amplitude, intrinsic mode that is related to the balanced droplet; late
times — where there is a superposition of two modes corresponding to each
chemical potential. All of these modes are decaying but over different
timescales due to the rate of atom shedding. The initial mode decays
relatively rapidly due to a high dissipation of energy from the particle
emission. However, at late times the particle emission is considerably
reduced, with slower emission of particles. This slower emission of particles,
at energies of the two chemical potentials, is negligible relative to the
droplet kinetic energy and hence the dynamics decay asymptotically [50].

Finally, it should be noted that the breathing modes of imbalanced droplets
have a larger range of instability than balanced droplets, which have an
unstable breathing mode in the self-evaporative regime only. This is
consistent with the discussions in both Section III and the Appendix A,of how
the energy, and energy per particle, varies with imbalance. The key point from
these discussions is that increasing imbalance corresponds to an increasing
energy per particle and thus a less stably bound droplet. Hence, it would be
expected that a less stably bound object would be more susceptible to particle
shedding when perturbed, as is shown in the main results of this section.

##  V DISCUSSION AND CONCLUSIONS

By solving the spherically symmetric, coupled generalised GP equations, this
work has systematically investigated the spherical ground states and breathing
modes of imbalanced droplets across the size of the droplet and imbalance, in
free space. Droplets can lower their energy by absorption of atoms, either by
absorbing a symmetric number of atoms into each component, under the condition
$N_{2}/N_{1}=\sqrt{g_{22}/g_{11}}$, or, as has been investigated here, if
there exists an asymmetric quantity of atoms available, the droplet will
absorb the available atoms forming a majority and minority component. This
modifies the core structure of the droplet by splitting the two central
component densities, yielding a bound, imbalanced droplet. If further majority
component atoms become available then the imbalanced droplet saturates and
thus any excess majority components will not bind to the droplet and reside
outside of the droplet as an unbound gas. To investigate the density profiles,
chemical potentials and breathing dynamics of these imbalanced droplets,
without the external effects of the unbound gas, large computational boxes
were used such that the background gas density is effectively zero.

One of the main experimental probes to justify the observation of a quantum
droplet is measuring a constant width of the atomic cloud, after switching off
all traps, i.e., the object is self-bound in free space [25, 27, 28]. In this
scenario any unbound gas would be lost, though the imbalanced droplet could
still remain stable in free space as demonstrated in this work. The
observation of the three breathing mode frequencies could be experimentally
achieved in homonuclear mixtures, for which the number of atoms in each
component can be tuned by radio-frequency pulses on the cloud [26], and there
have been recent experimental observations of imbalanced heteronuclear
mixtures [52]. However these system will likewise be affected by three- and
higher-body losses.

An immediate next question of this work is to explore how the application of
trapping potentials effects imbalanced droplets. Isotropic harmonic traps
could be applied to the spherically symmetric system considered here, to
investigate the modification of both the ground states and breathing mode
dynamics. The breathing mode dynamics of trapped, spherically symmetric
droplets has been investigated in Ref. [51] and so there are natural
extensions of these results for imbalanced droplets. Further ideas include
investigating the spherically symmetric ground states and dynamics of a
heteronuclear mixture, as the differing kinetic energy contributions of the
two components may lead to novel dynamics. This is computationally challenging
however, due to the form of the two-component LHY correction of a
heteronuclear mixture [11, 16]. There are approximations for this term, though
they are least accurate at the droplet centres [43]. This presents an issue
for accurate analysis of imbalanced ground state solutions and dynamics,
especially for the superimposed, time-dependent collective excitations shown
here.

With the continuing development of quantum droplet experiments — for example
through the realisation of new mixtures [53]— it will become increasingly
important to understand how these mixtures can minimise their energy, from
self-evaporating excitations to modifying their internal structure.

The data presented in this paper are available [54].

###### Acknowledgements.

The authors acknowledge support from the UK Engineering and Physical Sciences
Research Council (Grant No. EP/T015241/1). T. A. F. also acknowledges support
from the UK Engineering and Physical Sciences Research Council (Grant No.
EP/T517914/1). The authors also thank Jakub Kopyciński, Richard Tattersall and
Nick Keepfer for useful discussions and patience.

*

##  Appendix A Density difference of imbalanced, infinite droplets

The density profile of a droplet in the limit of large $\tilde{N}$ approaches
a flat-top, step function, i.e., a constant density in the droplet core and a
steep drop to zero density at the surface [11]. This approximate form is a
useful model for infinite sized droplets as it allows for kinetic energy
contributions to be neglected. Figure 5 shows a schematic of the step function
model of a large imbalanced droplet, used in this derivation. The minority
component is modelled as having volume $V=\frac{4}{3}\pi R^{3}$, whilst the
majority component has volume $V+\delta V=\frac{4}{3}\pi(R+\delta R)^{3}$. The
step functions hence correspond to the central densities $n$ and $n+\delta n$
for the minority and majority components, respectively, with the two
components normalised to $N$ and $N+\delta N$ in which $\delta N\geq 0$,
imposing the population imbalance.

Figure 5: Flat-top density profiles as an ansatz for each component of an
imbalanced droplet. The orange component is the higher-density, majority
component and the purple component is the lower-density, minority component.

The dimensionless form of the energy functional given in Equation 1, with
$\beta\equiv\sqrt{a_{22}/a_{11}}=1$, can be written in terms of these constant
imbalanced densities giving,

$\begin{split}E=\int&\Bigg{[}\frac{(n+\delta
n)^{2}}{2}+\frac{n^{2}}{2}+\eta(n+\delta n)n\\\
&+\frac{2}{5}\alpha\left(2n+\delta
n\right)^{5/2}\Bigg{]}\differential^{3}\mathbf{r}.\end{split}$ (5)

The schematic in Figure 5 states nothing about the sign of $\delta R$, i.e.,
the volume of the majority component, $V+\delta V$, can be larger or smaller
than the minority component volume, $V$. However, it can be shown that the
only physical system is that where $\delta V=0$, which is demonstrated in the
following calculation.

Beginning with the $\delta V>0$ case, Equation 5 can be written in terms of
the component volumes and population numbers as,

$\begin{split}E&=\frac{(N+\delta N)^{2}}{2(V+\delta
V)}+\frac{N^{2}}{2V}+\eta\frac{N(N+\delta N)}{V+\delta V}\\\
&+\frac{2}{5}\alpha\left\\{V\left[\frac{N+\delta N}{V+\delta
V}+\frac{N}{V}\right]^{5/2}+\delta V\left[\frac{N+\delta N}{V+\delta
V}\right]^{5/2}\right\\}.\end{split}$

To analyse how the energy of this model varies with $\delta V$, the energy
above can be minimised with respect to the difference in component volumes,
giving,

$\begin{split}\partialderivative{E}{(\delta V)}&=-\frac{(N+\delta
N)^{2}}{2(V+\delta V)^{2}}-\eta\frac{N(N+\delta N)}{(V+\delta V)^{2}}\\\
&+\frac{2}{5}\alpha\Bigg{\\{}-\frac{5}{2}\frac{V(N+\delta N)}{(V+\delta
V)^{2}}\left[\frac{N+\delta N}{V+\delta V}+\frac{N}{V}\right]^{3/2}\\\
&+\left[\frac{N+\delta N}{V+\delta V}\right]^{5/2}-\frac{5}{2}\delta
V\frac{(N+\delta N)^{5/2}}{(V+\delta V)^{7/2}}\Bigg{\\}}.\end{split}$

Taking the limit of $\delta V\rightarrow 0^{+}$ reduces to,

$\begin{split}\partialderivative{E}{(\delta V)}\evaluated{}_{\delta
V\rightarrow 0^{+}}&=-\frac{(N+\delta N)^{2}}{2V^{2}}-\eta\frac{N(N+\delta
N)}{V^{2}}\\\ &+\alpha\Bigg{\\{}-\frac{(N+\delta N)}{V}\left[\frac{2N+\delta
N}{V}\right]^{3/2}\\\ &+\frac{2}{5}\left[\frac{N+\delta
N}{V}\right]^{5/2}\Bigg{\\}},\end{split}$

which can be written in the form,

$\partialderivative{E}{(\delta V)}\evaluated{}_{\delta V\rightarrow
0^{-}}=-\frac{N+\delta N}{2V^{2}}\left[(1+2\eta)N+\delta
N\right]+\mathcal{O}(\alpha).$

Recall from Section II that $\delta a=a_{12}+\sqrt{a_{11}a_{22}}$, and that
defining $\delta a<0$ implies dominantly attractive interactions, which can be
written as $\eta<-1$ in the dimensionless parameters used here. Thus, with
$\delta N$ small, the mean-field terms becomes dominantly positive whilst the
LHY contributions are small due to $\alpha\ll 1$. Hence, the limit of $\delta
V\rightarrow 0^{+}$ yields, $\partialderivative*{E}{(\delta V)}\geq 0$.

The next case to check is $\delta V<0$, for which Equation 5 takes the form,

$\begin{split}E&=\frac{(N+\delta N)^{2}}{2(V+\delta
V)}+\frac{N^{2}}{2V}+\eta\frac{N(N+\delta N)}{V}\\\
&+\frac{2}{5}\alpha\left\\{(V+\delta V)\left[\frac{N+\delta N}{V+\delta
V}+\frac{N}{V}\right]^{5/2}-\delta
V\left[\frac{N}{V}\right]^{5/2}\right\\}.\end{split}$

This expression is again minimised with respect to the difference in component
volumes,

$\begin{split}\partialderivative{E}{(\delta V)}&=-\frac{(N+\delta
N)^{2}}{2(V+\delta V)^{2}}+\frac{2}{5}\alpha\Bigg{\\{}\left[\frac{N+\delta
N}{V+\delta V}+\frac{N}{V}\right]^{5/2}\\\ &-\frac{5}{2}\frac{(N+\delta
N)}{V+\delta V}\left[\frac{N+\delta N}{V+\delta
V}+\frac{N}{V}\right]^{3/2}-\left(\frac{N}{V}\right)^{5/2}\Bigg{\\}}.\end{split}$

Taking the limit of $\delta V\rightarrow 0^{-}$ yields,

$\begin{split}\partialderivative{E}{(\delta V)}\evaluated{}_{\delta
V\rightarrow 0^{-}}&=-\frac{(N+\delta
N)^{2}}{2V^{2}}+\alpha\Bigg{\\{}\frac{2}{5}\left[\frac{2N+\delta
N}{V}\right]^{5/2}\\\ &-\frac{N+\delta N}{V}\left[\frac{2N+\delta
N}{V}\right]^{3/2}-\frac{2}{5}\left(\frac{N}{V}\right)^{5/2}\Bigg{\\}},\end{split}$

which can be written in the form,

$\partialderivative{E}{(\delta V)}\evaluated{}_{\delta V\rightarrow
0^{-}}=-\frac{(N+\delta N)^{2}}{2V}+\mathcal{O}(\alpha).$

Hence, the limit of $\delta V\rightarrow 0^{-}$ results in,
$\partialderivative*{E}{(\delta V)}\leq 0$. Thus, the conditions

$\partialderivative{E}{(\delta V)}\evaluated{}_{\delta V\rightarrow 0^{+}}\geq
0,\,\mathrm{and},\,\partialderivative{E}{(\delta V)}\evaluated{}_{\delta
V\rightarrow 0^{-}}\leq 0,$

imply that $\delta V=0$ is either a smooth minimum or a cusp. This is
physically realistic in the limit of large droplets, as this implies that the
radii of the two components are equal, i.e., no single-component atoms reside
outside of the droplet core. However, in the finite droplet limit this is not
the case as kinetic energy contributions cannot be neglected, as is
highlighted in the ground state profiles in Figure 2 (b) and (c). The
imbalanced droplet hence appears to slowly approach the flat-top density
profile, relative to the rate at which the large, balanced droplet approaches
the flat-top density limit.

Taking the case of $\delta V=0$, the energy can be written as,

$\begin{split}\frac{E}{2N}&=\frac{(N+\delta
N)^{2}}{4NV}+\frac{N}{4V}+\eta\frac{N+\delta N}{2V}\\\
&+\frac{2}{5}\alpha\left[\frac{V}{2N}\left(\frac{2N+\delta
N}{V}\right)^{5/2}\right].\end{split}$

A factor of $\left(1+\delta N/2N\right)$ can be factored out of this
expression, allowing for the above equation to be written in powers of
$(N/V)$,

$\frac{E}{2N}=\left(1+\frac{\delta
N}{2N}\right)\left[\mathcal{A}\left(\frac{N}{V}\right)+\mathcal{B}\left(\frac{N}{V}\right)^{3/2}\right],$

where

$\mathcal{A}=\left(1+\frac{\delta
N}{2N}\right)+(\eta-1)+\frac{1-\eta}{2}\left(\frac{1}{1+\frac{\delta
N}{2N}}\right),$

and

$\mathcal{B}=\frac{4\sqrt{2}\alpha}{5}\left(1+\frac{\delta
N}{2N}\right)^{3/2}.$

Section II discusses the key property that droplets exist in equilibrium with
the vacuum, i.e., they have zero pressure, $\partialderivative*{E}{V}=0$. With
$N$ constant, this expression for the zero pressure droplet can be rewritten
as $\partialderivative*{(E/2N)}{(N/V)}=0$, yielding,

$\left(\frac{N}{V}\right)^{1/2}=-\frac{2\mathcal{A}}{3\mathcal{B}}\implies
n=\left(\frac{N}{V}\right)=\frac{4\mathcal{A}^{2}}{9\mathcal{B}^{2}},$

giving an expression for the equilibrium density. This results in the
equilibrium energy, given by,

$E_{\mathrm{eqbm}}=\left(2N+\delta
N\right)\left[\frac{4\mathcal{A}^{3}}{27\mathcal{B}^{2}}\right],$ (6)

or equivalently as the equilibrium energy per particle,

$\left(\frac{E}{2N+\delta
N}\right)_{\mathrm{eqbm}}=\frac{4\mathcal{A}^{3}}{27\mathcal{B}^{2}}.$ (7)

For fixed $\alpha$, $\eta$ and $N$, Equations 6 and 7 are plotted as functions
of $\delta N$ in Figure 1. The main plot in Figure 1 shows Equation 6 varying
with $\delta N$. By differentiating Equation 6, the minimum is located at

$\frac{\delta N}{N}=2(\eta-2)+[(\eta-1)(4\eta-14)]^{1/2}.$

For the parameters of Figure 1 the minimum is at $\delta N\approx 703$, which
using the expression,

$\delta n=\frac{\delta
N}{N}\left(\frac{4\mathcal{A}^{2}}{9\mathcal{B}^{2}}\right),$

corresponds to $\delta n\approx 0.458$, i.e., the orange, dashed, horizontal
line in the inset of Figure 2(d).

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11institutetext: Berlin University of Applied Sciences and Technology
22institutetext: Einstein Center Digital Future, Berlin, Germany
22email<EMAIL_ADDRESS>
# Automated Extraction of Fine-Grained Standardized Product Information from
Unstructured Multilingual Web Data

Alexander Flick 11 0000-0001-5273-0679 Sebastian Jäger 11 0000-0001-9420-8571
Ivana Trajanovska 11 0000-0003-0374-1210
Felix Biessmann 1122 0000-0002-3422-1026

###### Abstract

Extracting structured information from unstructured data is one of the key
challenges in modern information retrieval applications, including e-commerce.
Here, we demonstrate how recent advances in machine learning, combined with a
recently published multilingual data set with standardized fine-grained
product category information, enable robust product attribute extraction in
challenging transfer learning settings. Our models can reliably predict
product attributes across online shops, languages, or both. Furthermore, we
show that our models can be used to match product taxonomies between online
retailers.

###### Keywords:

product information extraction e-commerce

##  1 Introduction

Recent research achievements in the field of machine learning (ML) [1, 13]
have the potential to improve automated information extraction in applications
such as e-commerce. However, the translation of these ML innovations into
real-world application scenarios is impeded by the lack of publicly available
data sets. Here we demonstrate that recent advances in ML can be translated
into automated information extraction applications when leveraging carefully
curated data. To better assess the contribution of this study, we first
highlight some relevant data sets and methods that aim at the automated
extraction of structured data in the field of e-commerce.

### Public E-commerce Data Sets

We summarize publicly e-commerce data sets used for the automated extraction
of product information in Table 1. To leverage the potential of ML, large and
diverse data sets that follow a fine-grained product taxonomy are favorable. A
common and detailed taxonomy is the Global Product Classification (GPC)
standard, which ”classifies products by grouping them into categories based on
their essential properties as well as their relationships to other
products”[4]. For example, multiple _Brick_ s (shirts and shorts) can belong
to the same _Family_ (clothing) but are different _Class_ es (upper and lower
body wear)111See the GPC Browser for more examples: https://gpc-
browser.gs1.org/.

Table 1: Comparison of e-commerce data sets used for product attribute extraction and classification. Column _GPC_ means whether or not the data set follows the GPC taxonomy.  | regular | multi- | GPC | size
---|---|---|---|---
updated | lingual | shop | family
Farfetch product meta data [9] | ✗ | ✗ | ✗ | ✗ | ✗ | 400K
Product details on Flipkart [3] | ✗ | ✗ | ✗ | ✓ | ✗ | 20K
Amazon browse node classification [2] | ✗ | ✗ | ✗ | ✓ | ✗ | 3M
Amazon product-question answering [16] | ✗ | ✗ | ✗ | ✓ | ✗ | 17.3GB
Rakuten data challenge [10] | ✗ | ✗ | ✗ | ✓ | ✗ | 1M
MAVE [18] | ✗ | ✗ | ✗ | ✓ | ✗ | 2.2M
Innerwear from victoria’s secret & co [15] | ✗ | ✗ | ✓ | ✗ | ✗ | 600K
WDC-MWPD [19] | ✗ | ✗ | ✓ | ✗ | ✓ | 16K
WDC-25 gold standard [14] | ✗ | ✗ | ✓ | ✓ | ✓ | 24K
GreenDB [7] | ✓ | ✓ | ✓ | ✓ | ✓ |  $>$576K

### Multilingual Fine-Grained Product Classification

There are few recent studies investigating automated extraction of
standardized product information in text corpora. Brinkmann et al. [1] study
how hierarchical product classification benefits from domain-specific language
modeling. They report an improvement of 0.012 weighted F1 score by using
schema.org product222Website: https://schema.org/Product annotations for pre-
training. Peeters et al. [12] study cross-language learning for entity
matching and demonstrate that multilingual transformers outperform single-
language models (German BERT) by 0.143 F1 when trained on a single language
(German) and tested on multiple (German and English). Furthermore, using
additional training data for the second language (English) improves the
performance by another 0.038 weighted F1.

These studies highlight the potential of modern ML methods for automated
product attribute extraction. In this work, we show that transfer learning
helps to extract structured information (product category) from unstructured
data (product name and description) and to find reliable taxonomy mappings.

##  2 Experiments

We evaluate three transfer learning scenarios for product classification:

  1. 1.

Language Transfer: training on data of one language, test on other language
data

  2. 2.

Shop Transfer: training on data of one shop, test on other shop data

  3. 3.

Language and Shop Transfer: training on data of one shop and one language,
test on data of different shops and languages

Furthermore, we study whether ML methods can be used to find reliable taxonomy
mappings. For this, we apply a model trained for a _target taxonomy_ to data
that uses a _source taxonomy_. For each source category, the majority of
predicted target categories define the mapping from source to target taxonomy.

### Data Sets

In our experiments, we use two data sets, the GreenDB [6] and the Farfetch
data set [9]. The GreenDB333We use GreenDB version 0.2.2 available at
https://zenodo.org/record/7225336 is a multilingual data set covering 5
European shops with about 576k unique products of the 37 most important
product categories following the GPC taxonomy. It covers categories from the
GPC segments Clothing, Footwear, Personal Accessories, Home Appliances, Audio
Visual/Photography, and Computing. A recent publication [8] presents the
GreenDB’s high quality and usefulness for information extraction tasks. The
Farfetch data set has about 400k unique products from a single shop. It does
not follow a public taxonomy and covers only fashion products.

### ML Model

The experiment implementation is based on autogluon’s [17] TextPredictor and
uses _mDeBERTav3_ [5] as the backbone model. For training, we use the GreenDB
and apply Cleanlab [11] to find and remove miss-classified products (211 were
found). Our models use the product’s name and description to predict their
product category. $model_{baseline}$ is trained on the entire GreenDB (all
shops), $model_{ZaDE}$ on the German, $model_{ZaFR}$ on the French, and
$model_{ZaALL}$ on the German, French, and English Zalando products contained
in the GreenDB.

### Online Demo

To demonstrate the transfer capabilities, we published an online demo
available: https://product-classification.demo.calgo-lab.de. As shown in
Figure 1, it automatically downloads the HTML of a given URL, extracts the
products’ name and description, and uses $model_{baseline}$ to predict its GPC
category.

Figure 1: Online demo overview. Automated extraction of schema.org information
(product name and description) from HTML, used for product classification.

##  3 Results

The baseline performance ($model_{baseline}$) shows a strong 0.99 weighted F1
score on a GreenDB test set.

### Transfer Tasks

$model_{ZaDE}$ demonstrates language transfer when it is applied to other
languages of the same shop. It achieves weighted F1 scores of 0.898 for
English and 0.873 for French. Applying $model_{ZaFR}$ and $model_{ZaDE}$ on
other shops demonstrates shop transfer with weighted F1 scores from 0.648 to
0.836. If the model is fine-tuned on multi-lingual data ($model_{ZaALL}$),
almost all shops benefit, see Table 2 for details. The language and shop
transfer is even more challenging and performs worse for all shops.
Transferring across data sets, i.e., applying $model_{baseline}$ to Farfetch
data, achieves a 0.924 weighted F1 score.

Table 2: Weighted F1 scores for shop transfer experiments. Scores from 0.648 to 0.836 demonstrate robust shop transfer. Shop transfer profits from additional data in other languages.  | Model | FR | DE
---|---|---|---
Asos | H&M | Otto | Amazon
Shop Transfer | $model_{ZaFR}$ | 0.836 | 0.678 | - | -
$model_{ZaDE}$ | - | - | 0.777 | 0.648
$model_{ZaALL}$ | 0.842 | 0.717 | 0.762 | 0.739
Shop & Language Transfer | $model_{ZaFR}$ | - | - | 0.614 | 0.449
$model_{ZaDE}$ | 0.795 | 0.666 | - | -

### Taxonomy Matching

Using $model_{baseline}$ to map products’ categories from Farfetch to GreenDB
(GPC taxonomy) results in 41 out of 46 ($>$89%) correctly mapped categories.

##  4 Conclusion

We demonstrate that combining rich multilingual data sets and modern ML
methods enables fine-grained standardized product information extraction from
unstructured data. We investigate several transfer learning settings when
training and testing on data from different shops and languages, even in zero-
shot scenarios when no data from another shop and language was available in
the training data.

Acknowledgements This research was supported by the Federal Ministry for the
Environment, Nature Conservation and Nuclear Safety based on a decision of the
German Bundestag.

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