# Neutron Stars Mass-Radius relationship and
Electromagnetic follow-up of Kilonovae

D. Barba González  M. A. Pérez-García  Department of Fundamental Physics &
IUFFyM, University of Salamanca, E-37008 Spain

###### Abstract

When two Neutron Stars collide a multi-band electromagnetic emission, known as
Kilonova (KN), follows being powered by the radioactive decay of ejecta
products. In this contribution we discuss how the equation of state of dense
matter, impacts the mass and velocity in the KN ejecta and thus its light
curve. Using this information encoded in the stellar mass-radius relationship,
we ellaborate on how the future experimental observations in photon channels,
in addition to complementary multimessenger probes, could provide a new and
more detailed insight into the equation of state of nuclear matter.

##  1 Introduction

When two Neutron Stars (NSs) in a binary system, a BNS, coalesce and merge,
they are expected to produce a multimessenger signal: gravitational waves
(GW), photons from an electromagnetic (EM) counterpart and even neutrinos. One
example of such event was the observation of a rapidly fading EM transient in
the galaxy NGC 4993, which was spatially coincident[1] with GW170817. The
optical and infrared emission, known as Kilonova (KN) or macronova, was named
AT2017gfo. It was possible to identify line features in the spectra,
consistent with the presence of light r-process elements (atomic masses of
$A=90–140$) powering the emission. In addition, a gamma-ray burst GRB170817A,
was also detected. Since then it is now expected that many detections of this
kind or similar ones i.e. Black Hole and NS (BHNS) mergers will follow in the
future. An updated BNS rate[1] after the discovery of GW170817 leads to the
expectation of potentially detectable dozens of events per year by LIGO/Virgo
O4 and O5 runs.

In order to capture the challenging optical counterpart of a BNS merger, it is
crucial to develop an alert system allowing observers to point in the right
direction in the sky, obviously before it has faded away and ideally before
the time it peaks. The community has gained valuable experience thanks to the
historical event GW170817 localised optically in a sky area of around 30
square degrees within hours of its appearance. Typically, light curve peak
values are obtained at times $t_{\rm peak}$ ranging between tenths of a day
and a day. The alert system GraceDB[2], a communication node that connects
LIGO and Virgo analysis pipeline, sends an alert to astronomers when a
promising binary GW signal has been detected and located in the sky. Although
challenging, detecting these EM counterparts is key to shed light on some
relevant physical processes taking place, such as dynamical and post-merger
ejection, neutron capture and thermalization in the ambient matter. Existing
and future missions, like for example MAAT [3] in the GTC (Spain) will be able
to provide an agile response and deeper knowledge in this field. In this
contribution we discuss how KNe can serve yet as another additional
complementary constraint to the equation of state of neutron star matter,
particularly through the light curve and spectra as measured by existing and
future missions.

##  2 NS equation of state and key KN properties

The equation of state (EoS) of nuclear matter describes the behavior of the
matter inside NSs. Typically these objects have a mass ($M$) up to a maximum
value $M_{max}\sim 2M_{\odot}$ and a radius ($R$) in the $10-13$ km range.
Constraints on both quantities have been provided in a series of world-wide
experimental efforts [4]. These objects appear as solutions of the stellar
structure equations from General Relativity (GR) and their computed $M(R)$
values display some spread due to the poorly known high density matter EoS. In
particular, it is not yet known what are the actual degrees of freedom
suitable for its description, i.e. whether matter inside remains under the
form of nucleons or perhaps deconfined as quarks. NSs are born in the
aftermath of a Supernova explosion displaying a large velocity kick and
distribute mostly towards the center of the galaxies. Further into the NS
description, its layered structure can be mainly explained from two regions:
an external crust and a central core. It is believed that up to nearly $99\%$
of the NS mass resides in the latter under the form of nucleons, hyperons and
alike species although if high densities are in excess of $\sim 6\times
10^{14}$ $\rm g/cm^{3}$ a quark deconfined state may appear[5]. From the
population studies of Galactic double NSs (DNSs) it appears that $\simeq 98\%$
of all merging DNSs will have a mass ratio, $q$, close to unity[6] $q<1.1$.
Recent studies[7] find that the weighted mean masses of the primary and
companion stars are respectively $(1.439\pm 0.036)M\odot$ and $(1.239\pm
0.020)M\odot$. In addition, the surface magnetic field strength in the primary
stars of the DNSs is $B\sim 10^{10}$ G, and the spin period is $P\sim 50$ ms.
Due to the fact that the NS in the BNS events under scrutiny will share these
general characteristics it is important to see what information the EM
counterparts can yield. The gravitational and baryonic masses are two
quantities that determine the binding energy (BE) of the NS. Namely, the
gravitational mass is $m_{G}(R)=M=\int_{0}^{R}4\pi r^{2}\epsilon(r)dr$ where
$\epsilon(r)$ is the energy density at a radial distance $r$ from the center
and the baryonic mass $M_{B}=M^{*}=m_{0}\int_{0}^{R}4\pi
r^{2}\left[1-2Gm_{G}(r)/rc^{2}\right]^{-1/2}n(r)dr$ where $m_{0}$ is the mass
of a baryon and $n(r)$ is the baryon (nucleon) number density. The binding
energy of the NS is $BE=\left(M^{*}-M\right)c^{2}$ and typically $BE=|BE|$ is
taken as a positive quantity.

In the literature, some universal relations between the BE and $C$ have been
proposed, such as that[8] based on the radius of a $1.4M_{\odot}$ NS,
$R_{1.4}$, $M^{*}=M+R_{1.4}^{-1}\times M^{2}$, with $1.8\%$ relative error.
Note that the $1.4M_{\odot}$ selection is close to the most probable for
primary companions in DNSs. In order to illustrate our findings we consider
two EoS i.e. DD2 and SFHo [9], whose M(R) relationship is shown in Fig.(1)
left panel. For SFHo it is found a saturation density $n_{0}=0.1583$ $\rm
fm^{-3}$, nuclear incompressibility $K=245.4$ MeV, symmetry energy $J=31.57$
MeV and a logarithmic derivative of the symmetry energy $L=47.10$ MeV. For the
DD2 $n_{0}=0.1491\mathrm{fm}^{-3}$, $K=242.7$ MeV, $J=31.78$ MeV and $L=55.19$
MeV. Using existing data[10] from BNS simulations we obtain $R^{\rm
DD2}_{1.4}=13.5\pm 0.1$ km, $R^{\rm SFHo}_{1.4}=11.8\pm 0.1$ km, see Fig.(1)
right panel. Considering the sum of BE of both NSs described by a given EoS,
$BE_{1}+BE_{2}$, we arrive at values for DD2: $BE_{1}+BE_{2}=0.25\pm
0.04M_{\odot}$ and SFHo: $BE_{1}+BE_{2}=0.28\pm 0.04M_{\odot}$ which fulfill
the fitted form[11] $\frac{BE_{i}}{M_{i}}=-0.0130+0.618C_{i}+0.267{C^{2}_{i}}$
where $C_{i}=\frac{GM_{i}}{R_{i}c^{2}}$ is the compactness. They retain the
$M(R)$ relationship from $C$ and $M$ values.

As shown in GR simulations[12] the ejected mass in a violent BNS merger is
mostly stripped from the inner and outer crust of the NSs and will be
determined by the energetics of the collision and the actual amount of matter
in the crust. In a NS the neutron rich crust displays a large set of
inhomogenous substructures ranging from lattice nuclei to pasta phases [13].
In the thin-crust approximation $M_{\rm{crust}}\approx\frac{4\pi
R^{3}P_{t}}{c^{2}C}[1-2C]$ where $P_{t}$ is the EoS dependent pressure
transition from the crust-core regions [14]. There is a large sensitivity to
the EoS in the crust mass value, being smaller for heavier stars. Recent
Bayesian analysis[15] provide values for the transition density
$n_{t}=0.072\pm 0.011$ $\rm fm^{-3}$ and the transition pressure
$P_{t}=0.339\pm 0.115$ $\rm MeV$ $\rm fm^{-3}$. The resulting crust masses
$M^{\rm DD2}_{\rm{crust}}=0.038M_{\odot}$ and $M^{\rm
SFHo}_{\rm{crust}}=0.024M_{\odot}$. In addition, the number of baryons must be
conserved $N^{*}_{1}+N^{*}_{2}=N^{*}_{remnant}+N^{*}_{ejec}$.

Figure 1: M(R) relationship for the DD2 and SFHo EoS (left) and BE as a
function of $M^{2}$ (right)

The complex dynamics of the BNS merger and its ejecta are studied using GR
numerical simulations with a number of approximations regarding neutrino
transport and microphysics. The output is obtained by determining the relevant
parameters modelling the KN emission, including the mass, $M_{\mathrm{ej}}$,
and velocity, $v_{\mathrm{ej}}$, of the ejecta, which are fitted[10] (with
some uncertainty, see parameterizations) to functional forms depending on both
objects in the BNS ($1,2$),

$\scriptstyle\frac{M_{\mathrm{ej}}^{\mathrm{fit}}}{10^{-3}M_{\odot}}=\left[a\left(\frac{M_{2}}{M_{1}}\right)^{1/3}\left(\frac{1-2C_{1}}{C_{1}}\right)+b\left(\frac{M_{2}}{M_{1}}\right)^{n}+c\left(1-\frac{M_{1}}{M_{1}^{*}}\right)\right]M_{1}^{*}+(1\leftrightarrow
2)+d,$ (1) $v_{\mathrm{ej}}=\sqrt{v_{\rho}^{2}+v_{z}^{2}},\quad
v_{\rho,z}=\left[a_{\rho,z}\left(\frac{M_{1}}{M_{2}}\right)\left(1+c_{\rho,z}C_{1}\right)\right]+(1\leftrightarrow
2)+b_{\rho,z}.$ (2)

We plot in Fig.(2) left panel, the ejected mass as a function of gravitational
masses $M_{1},M_{2}$ for $M_{i}\in[1.2,1.6]$ in our analysis with the DD2 EoS.
As a rule of thumb we see that smaller mass ratio $q\equiv M_{<}/M_{>}$
provide larger $M_{ej}$. The experimental measurement of the KN peak
bolometric luminosity and spectral time series usually depend on the
conditions of transparency after the ejection of matter parameterized through
the opacity, $\kappa$. For early emission, typically
$\kappa\sim{0.1\mathrm{\leavevmode\nobreak\
cm}^{2}\mathrm{\leavevmode\nobreak\ g}^{-1}}$ while for later r-powered
emission $\kappa$ could be a factor of 10 or more larger. The peak values and
are given [16] by $t_{{peak}}=4.9\mathrm{\leavevmode\nobreak\
d}\times\left(\frac{M_{ej}}{10^{-2}M_{\odot}}\right)^{\frac{1}{2}}\left(\frac{\kappa}{10\mathrm{\leavevmode\nobreak\
cm}^{2}\mathrm{\leavevmode\nobreak\
g}^{-1}}\right)^{\frac{1}{2}}\left(\frac{v_{\mathrm{ej}}}{0.1}\right)^{-\frac{1}{2}}$
. For a BNS the measurable magnitudes can be fitted to the Black Body power
spectrum [1] and from GW170817 some constraints appear for masses i.e. total
mass $M\approx 2.74M_{\odot}$, and individual masses of
$M_{1}\approx(1.36-1.6)M_{\odot}$ and $M_{2}\approx(1.17-1.36)M_{\odot}$. Also
from this analysis marginalized over the selection methods, it was obtained
that the radius of the progenitors $R_{1}\sim(10.8-11.9)\pm
2\mathrm{\leavevmode\nobreak\ km}$ and $R_{2}\sim(10.7-11.9)\pm
2\mathrm{\leavevmode\nobreak\ km}$ and the tidal deformability parameters
$\Lambda_{1}<\Lambda_{2},\Lambda_{1}<500$ and $\Lambda_{1}<1000$. Under this
additional input they find $\tilde{\Lambda}(310-345)_{-245}^{+691}$ and
$R_{1.4}=8.7-14.1\mathrm{\leavevmode\nobreak\ km},$ where the binary tidal
deformability $\tilde{\Lambda}$ is defined as
$\tilde{\Lambda}\equiv\frac{16}{13}\left[\frac{\Lambda_{1}M_{1}^{4}\left(M_{1}+12M_{2}\right)}{\left(M_{1}+M_{2}\right)^{5}}+1\leftrightarrow
2\right]$ which is most precisely derived.

Direct measurements from current and future missions will be able to provide
time spectral series. In Fig.(2), right panel, we plot the simulated fluxes of
two runs using POSSIS [17] as a function of the EM rest wavelength. Run 1 (Run
2) corresponds to a $q=1$ BNS merger of $1.4M_{\odot}$ at 40 Mpc using the
output of recent simulations[18] $M140140-LK$ of DD2 (SHFo) EoS using
polar/equatorial orientations. We notice that the early, agile KN follow up
would be key to further constrain the EoS. The soft SFHo EoS peaks at IR, and
is fainter than DD2 across the whole spectrum. Since the $q$ values remain the
same we rely to the EoS features (including secondary peak features) for
wavelengths less than $\sim 1\mu$m.

Figure 2: (left) Ejected mass from simulation data [10]. (right) Simulated
flux for a BNS merger using DD2 (run 1) and SFHo (run 2) EoS at 40 Mpc for
polar and equatorial view angles at 0.5 d (left) and 1 d (right) after the
merge.

To summarize, we have discussed the dependencies of the EM observables
obtained from KN light curves (peak values) with $q=1$ in order to help
indirectly constrain quantities such as the $M(R),C,BE,R_{1.4}$ and thus the
EoS of NS matter. In particular, the combination of optical/IR observations
from proposed missions such as MAAT in addition to GW signals from the
LIGO/Virgo collaboration, will provide complementary valuable input. Further
work is in progress.

## Acknowledgments

We acknowledge useful comments from C. Albertus, M. Bulla, L. Izzo and F.
Prada. Spectra in Fig. (2) are courtesy of M. Bulla. This research has been
supported by University of Salamanca, Junta de Castilla y León SA096P20,
Spanish Ministry of Science PID2019-107778GB-100 projects, MULTIDARK Spanish
network and PHAROS COST Action CA16214.

## References

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# Quantum-Fluctuation-Driven Dynamics of Droplet Splashing, Recoiling and
Deposition in Ultracold Binary Bose Gases

Yinfeng Ma  Beijing National Laboratory for Condensed Matter Physics,
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
Xiaoling Cui<EMAIL_ADDRESS>Beijing National Laboratory for Condensed
Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing
100190, China  Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808,
China

###### Abstract

Droplet impact on a surface is practically relevant to a variety of fields in
nature and industry, while a complete control of its outcomes remains
challenging due to various unmanageable factors. In this work, we propose the
quantum simulation of droplet impact outcomes in the platform of ultracold
atoms. Specifically, we study the quantum-fluctuation-driven dynamics (QFDD)
of two-dimensional Bose-Bose mixtures from an initial Townes soliton towards
the formation of a quantum droplet. By tuning the fluctuation energy of the
initial Townes state through its size and number, the subsequent QFDD can
produce various outcomes including splashing, recoiling, and deposition,
similar to those in droplet impact dynamics. We have utilized the Weber number
to identify the thresholds of splashing and recoiling, and further established
a universal scaling law between the maximum spreading factor and the Weber
number in the recoiling regime. In addition, we show that the residual QFDD in
the deposition regime can be used to probe the collective breathing modes of a
quantum droplet. Our results reveal a mechanism for the droplet impact
outcomes, which can be directly tested in cold-atom experiments and can pave
the way for exploring intriguing droplet dynamics in a clean and fully
controlled quantum setting.

##  I  Introduction

Given the wide practical relevance to both nature and industry [1, 2, 3, 4, 5,
6, 7], droplet impact dynamics on a surface has attracted much attention ever
since the first study by Washington [8, 9]. Various impact outcomes-including
splashing, receding/recoiling, rebound, and deposition-have been observed
successfully in experiments [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21,
22, 23, 24, 25]. In general, these dynamics were characterized by two physical
observables, namely, the maximum spreading factor ($\beta$) [12, 16, 23] and
the splashing threshold(${\cal K}$) [11, 13, 14, 15, 17, 18, 19, 20, 21, 22,
25], which were shown not only to depend on the properties of the droplet
itself (size, density, surface tension, viscosity, impact velocity), but also
to be strongly influenced by the surface condition(roughness, wettability) and
surrounding gas (pressure, composition). Because of the complexities
associated with various unmanageable factors, it is extremely challenging to
deterministically parametrize $\beta,\ {\cal K}$ and fully control the impact
outcomes. In this situation, a common practice is to assume an ideal droplet
impact (on a smooth solid surface at atmospheric condition) and then to
quantify the actual dynamics by the Weber and Reynolds numbers, which,
respectively, describe the relative strength of droplet inertia with respect
to capillary and viscous forces [26, 27]. Various scaling laws between
$\beta,\ {\cal K}$ and these numbers have been proposed in the literature [26,
27], based on different models or empirical fitting from experimental data.

In the past few decades, ultracold atoms have emerged as an ideal platform for
quantum simulation, given their extremely clean environment and the high
controllability on the species, number, dimension, interaction strength etc
[28, 29]. In particular, a recent important achievement in this field was the
realization of a quantum droplet in both dipolar gas [30, 31, 32, 33, 34, 35,
36] and alkali bosonic mixtures [37, 38, 39, 40, 41, 42], with extremely
dilute densities ($\sim 10^{14}-10^{15}{\rm cm}^{-3}$) that can be eight
orders of magnitude lower than water. In forming these gaseous droplets,
quantum fluctuations play an essential role in providing the repulsive force
for their stabilization, for which they are called quantum droplets [43]. To
date, the idea of a quantum droplet has been successfully extended to various
atomic systems, including low dimensional ones [44, 45, 46, 47, 48, 49], Bose-
Fermi mixtures [50, 51, 52, 53, 54, 55] and multi-component dipolar or alkali
atomic mixtures [56, 57, 58]. The non-equilibrium properties of quantum
droplets have also been investigated in terms of their dynamical formations
[59, 60, 61] and collisions [62, 63]. These developments offer an
unprecedented opportunity for simulating droplet impact dynamics in ultracold
atoms, particularly, at the microscopic quantum level and in a highly
controllable manner.

In this work, we demonstrate the capability of using ultracold Bose gases to
simulate the droplet impact outcomes in a fully controlled quantum setting.
Contrary to the conventional droplet impact setup, here there is no impact
surface for the droplet, and the driving force of its dynamics is purely from
its intrinsic energy due to quantum fluctuations. Specifically, we study the
dynamical property of a two-dimensional (2D) Bose-Bose mixture with repulsive
intraspecies and attractive inter-species couplings, whose ground state is a
quantum droplet. To highlight the quantum effect in the dynamics, we have
chosen the initial state as the Townes soliton generalized from the single-
species case [64], which features a zero mean-field energy with continuous
scale invariance as recently confirmed in experiments [65, 66, 67, 68]. In
this way, the dynamics here is purely driven by quantum fluctuations and thus
can be called quantum-fluctuation-driven dynamics (QFDD). It is found that by
tuning the fluctuation energy of the initial Townes state through its
size($\sigma_{0}$) and number($N$), the subsequent QFDD can produce various
outcomes, including splashing, recoiling and deposition, similar to those in
droplet impact dynamics. We have mapped out the dynamical phase diagram in the
($\sigma_{0},N$) plane and employed the Weber number to characterize different
phases. The splashing and recoiling thresholds are identified, and a universal
scaling law is established between the maximum spreading factor and the Weber
number for the recoiling dynamics, which is applicable for a considerably
large parameter regime. Finally, we show that the long-time QFDD in the
deposition regime can be used to extract the collective breathing modes of
quantum droplet. These results can be directly tested in the current cold
atoms experiments. In the Appendixes, we provide more details on the
derivation of the generalized Townes soliton with unequal masses, as well as
on the numerical simulations.

##  II  Model

We start from the energy functional of two-species bosons in 2D ($\hbar=1$ for
brevity):

$\displaystyle\cal{E}(\bm{\rho})$ $\displaystyle=$
$\displaystyle-\sum_{i=1,2}\phi_{i}^{*}(\bm{\rho})\frac{\nabla_{\bm{\rho}}^{2}}{2m}\phi_{i}(\bm{\rho})+\sum_{ij}\frac{g_{ij}}{2}n_{i}(\bm{\rho})n_{j}(\bm{\rho})$
(1) $\displaystyle+{\cal E}_{\rm LHY}(n_{i}(\bm{\rho}));$

here $\bm{\bm{\rho}}=(x,y)$ is the 2D coordinate; $\phi_{i}$ is the
wavefunction of the $i$-th species and $n_{i}=|\phi_{i}|^{2}$ is its density;
$g_{ij}$ is the bare coupling between $i$\- and $j$-species, which can be
expressed as $g_{ij}=4\pi a_{ij}/(ml_{z})$ in quasi-2D geometry, with $a_{ij}$
the s-wave scattering length and $l_{z}$ the characteristic length along the
confined ($z$) direction; and ${\cal E}_{\rm LHY}$ is the Lee-Huang-Yang(LHY)
correction from quantum fluctuations, and for quasi-2D bosons near the mean-
field instability point ($\delta a\equiv a_{12}+\sqrt{a_{11}a_{22}}\sim 0$) it
reads [44, 47]

${\cal E}_{\rm LHY}=\frac{\eta^{2}\ln(\eta\sqrt{e})}{8\pi ml_{z}^{4}},\ \ \ \
{\rm with}\ \ \eta=4\pi l_{z}(a_{11}n_{1}+a_{22}n_{2}).$ (2)

It has been shown that this LHY term can balance with the mean-field force and
result in a self-bound droplet as the ground state [44]. Given the
$n^{2}\ln(n)$ dependence of LHY energy, the 2D droplet can exist at both mean-
field collapse and stable regimes [44] and with any infinitesimal atom number
[49]. At a sufficiently large number, the 2D droplet develops a flat-top
structure in its density profile, similar to the 3D case [43].

Given Eq.(1), the dynamics of $\\{\phi_{i}\\}$ is governed by the time-
dependent Gross-Pitaevskii(GP) equations:

$i\partial_{t}\phi_{i}=\left(-\frac{\nabla_{\bm{\rho}}^{2}}{2m}+\sum_{j}g_{ij}n_{j}+\frac{\partial{\cal
E}_{LHY}}{\partial n_{i}}\right)\phi_{i}.$ (3)

In this work, we will focus on the solution with zero angular momentum, since
the ground state and the Townes soliton both stay within this sector, and
different angular momentum sectors are decoupled from each other. In this
sector, we can replace the coordinate $\bm{\bm{\rho}}$ simply by its magnitude
$\rho=|\bm{\rho}|$. More details of solving Eq.(3) in the discretized
coordinate and time space have been given in the Appendixes.

Throughout the paper, we specifically consider the two hyperfine states of 39K
atoms, $|1\rangle\equiv|F=1,m_{F}=-1\rangle,\
|2\rangle\equiv|F=1,m_{F}=0\rangle$, as has been well studied in the droplet
experiments [37, 38, 39]. In this case, $a_{11}=33.5a_{B},\ a_{12}=-53a_{B}$
($a_{B}$ is the Bohr radius), and $a_{22}$ is highly tunable by magnetic
field. The confinement length is chosen as $l_{z}=0.08\mu m$.

##  III  Generalized Townes soliton

For spinless bosons in 2D, it is known that the kinetic term and mean-field
attraction can support a special stationary solution called the Townes soliton
[64]. This state features zero energy and continuous scale invariance, and it
can only exist when the boson number and coupling strength satisfy
$N|g|=5.85/m$. Such a special solution has been successfully observed in both
non-linear optics [65] and ultracold atoms [66, 67, 68]. In these experiments,
the LHY correction takes little effect as it is much smaller than the mean-
field part. In the following, we will show that the Townes soliton can be
generalized to two-species bosons if equally neglecting the LHY correction.

By omitting the LHY term in (3), we can see that the two GP equations for
$\\{\phi_{1},\phi_{2}\\}$ can support a single-mode solution
$\phi_{i}=\sqrt{N_{i}}\phi\exp(-i\mu_{i}t)$ as long as

$\frac{N_{1}}{N_{2}}=\frac{g_{22}-g_{12}}{g_{11}-g_{12}},$ (4)

where $\mu_{1}=\mu_{2}\equiv\mu$, and the single mode $\phi$ satisfies:

$\left(-\frac{\nabla_{\bm{\rho}}^{2}}{2m}+Ng_{\rm
eff}|\phi|^{2}\right)\phi=\mu\phi.$ (5)

Here $N=N_{1}+N_{2}$ is the total number, and the effective interaction
$g_{\rm eff}$ is given by

$g_{\rm eff}=\frac{g_{11}g_{22}-g_{12}^{2}}{g_{11}+g_{22}-2g_{12}}.$ (6)

Apparently we have $g_{\rm eff}<0$ in the mean-field collapse regime
($g_{12}<-\sqrt{g_{11}g_{22}}$). It is then straightforward to check that
under the condition

$Nm|g_{\rm eff}|=5.85,$ (7)

there exists a sequence of zero-energy eigenstates with continuous scale
invariance, i.e., the eigenstate nature and zero-energy property will not
change under an arbitrary scaling transformation
$\phi(\bm{\rho})\rightarrow\lambda\phi(\lambda\bm{\rho})$ (accordingly
$E\rightarrow\lambda^{2}E$). These stationary solutions are the generalized
Townes soliton for two-species bosons. Note that a similar generalization also
works for the case of unequal masses, with slight modifications in Eqs.(4,6)
as shown in the Appendixes.

Figure 1: (Color online) Test of generalized Townes soliton for two-species
bosons (neglecting the LHY correction). (a) A stationary Townes profile when
the total number $N$ and number ratio $N_{1}/N_{2}$ satisfy (7) and (4)
simultaneously. (b) Density profile at different times when the total number
is changed to $0.8N$, while the number ratio is the same as in (a). (c)
Density profile at different times when the total number is $1.2N$, with the
same number ratio as in (a). (d) Density profile at different times with the
same total number $N$ as in (a) while keeping $N_{1}/N_{2}=1$. Here we take
two-species 39K atoms with $a_{11}=33.5a_{B},\ a_{12}=-53a_{B}$ and
$a_{22}=83.1a_{B}$, which gives $N=6500$ for a stationary Townes state
according to Eqs.(6,7); the initial size of the Townes profile is taken as
$7\mu m$.

In figure 1(a), we confirm the stationary Townes soliton for two-species
bosons once the total number $N$ satisfies (7) and the number ratio
$N_{1}/N_{2}$ satisfies (4). In comparison, if we change $N$ to be smaller or
larger, the original profile will shrink (figure 1(b)) or expand (figure 1(c))
with time. The profile is also unstable if $N_{1}/N_{2}$ deviates from (4),
see figure 1(d). In a word, both conditions (7) and (4) are required in
supporting a stationary two-species Townes solution.

##  IV  Quantum-fluctuation driven dynamics

A crucial difference between the single- and two-species bosons is that
quantum fluctuations play an important role in the latter, which can lead to
droplet formation as a ground state. It then follows that starting from the
generalized Townes soliton, which is the mean-field stationary solution for
two-species bosons, the quantum fluctuation can destabilize it strongly and
drive its time evolution towards the droplet formation. Such dynamics can be
called the quantum-fluctuation driven dynamics (QFDD), also in light of the
fact that the total energy of Townes soliton is purely given by the LHY part,
$E(t=0)=E_{\rm LHY}$.

Figure 2: (Color online) Quantum-fluctuation driven dynamics(QFDD) of
deposition (b1), recoiling (b2) and splashing (b3) starting from initial
Townes solitons with different sizes. The parameters of $\\{a_{ij}\\}$ and
initial $N$ are the same as in figure 1. (a) Total energy per particle
($E/N=E_{\rm LHY}/N$) for the initial Townes state as functions of its size
($\sigma$). The color on the curve denotes the corresponding Weber number. The
dashed line shows the energy from Gaussian ansatz, and the lower dotted line
shows the energy of ground state droplet. The energy unit is $1/(ml_{0}^{2})$
with $l_{0}=1\mu m$. (b1-b3) Three typical QFDD starting from different
initial Townes states, which respectively simulate the droplet impact outcomes
of deposition (b1), recoiling (b2) and splashing(b3). The initial sizes of
Townes solitons are respectively $\sigma_{0}(\mu m)=4.75(b1),\ 2.5(b2),\
2(b3)$, as located by circles in (a). (c) Time evolution of the cloud size,
$\sigma(t)$, for the dynamics in (b1-b3).

In figure 2(a), we show that the total energy ($=E_{\rm LHY}$) of two-species
Townes soliton can be conveniently tuned by its size $\sigma$, taking a
typical combination of $\\{N,g_{ij}\\}$ that satisfies (4,7). In particular,
$E_{\rm LHY}$ varies non-monotonically with $\sigma$ and shows a minimum at
certain finite $\sigma$. To understand this behavior, we employ a Gaussian
ansatz to approximate the single mode
$\phi(\rho)=\exp[-\rho^{2}/(2\sigma^{2})]/(\sqrt{\pi}\sigma)$, which leads to
the total energy $E=\int 2\pi\rho d\rho{\cal E}(\rho)$ as

$\begin{split}E=\frac{N}{2m\sigma^{2}}+\frac{N^{2}g_{\rm
eff}}{4\pi\sigma^{2}}+\frac{N^{2}\bar{a}^{2}}{ml_{z}^{2}\sigma^{2}}\ln\frac{4N\bar{a}l_{z}}{\sigma^{2}},\end{split}$
(8)

with $\bar{a}\equiv\sqrt{a_{11}a_{22}}$. We can see that the first two terms
can support zero-energy states with arbitrary $\sigma$ under the condition
$Nm|g_{\rm eff}|=2\pi$, which are just the simplified Gaussian version of the
Townes soliton. However, when including the third LHY term, the total
energy($=E_{\rm LHY}$) will deviate from zero, and the Townes profile is no
longer stationary. Given the expression $E_{\rm
LHY}\sim-\ln(\sigma^{2})/\sigma^{2}$, we can easily arrive at a non-monotonic
$E_{\rm LHY}\sim\sigma$ dependence with energy minimum $E_{\rm
min}=-N\bar{a}/(4l_{z}^{3}e)$ at $\sigma_{\rm min}=\sqrt{4N\bar{a}l_{z}e}$. As
shown in figure 2(a), $E_{\rm LHY}$ from a Gaussian ansatz provides a
qualitatively good prediction to the $E\sim\sigma$ lineshape of real Townes
solutions.

Given the easily tunable fluctuation energy of the initial Townes state, the
subsequent QFDD can exhibit rich dynamical outcomes. In figure 2(b1,b2,b3), we
show the time evolution of density profiles for three typical QFDDs:

(I) Deposition. When the initial size $\sigma_{0}$ is close to $\sigma_{\rm
min}$ and $E_{\rm LHY}$ is small, the QFDD shows a typical deposition behavior
(figure 2(b1)). Specifically, as time passes the system repels a small
proportion of atoms outside, and the rest automatically follows the profile of
a ground-state droplet with additional periodic oscillations. As discussed
later, such residual oscillation can be used to probe the collective breathing
modes of a quantum droplet.

(II) Recoiling. As $\sigma_{0}$ deviates more from $\sigma_{\rm min}$ and
$E_{\rm LHY}$ gets larger, the system enters the recoiling regime. As shown in
figure 2(b2), at early times a considerable portion of atoms are repelled
outside, while at some point they stop to spread and flow back to merge with
the central part. Such back-flow (or recoiling) can be attributed to the
competition between the surface tension and the kinetic energy of the cloud
during the dynamics.

(III) Splashing. When $\sigma_{0}$ deviates significantly from $\sigma_{\rm
min}$ and $E_{\rm LHY}$ is large enough, the system shows a rapid splashing
dynamics, see figure 2(b3). In this case, the large $E_{\rm LHY}$ causes a
drastic change of the initial Townes profile in a short time, i.e., the cloud
quickly splits into two pieces. In this process, $E_{\rm LHY}$ converts to the
large kinetic energy of the outgoing part, such that it completely separates
from the central part and flows away forever.

The above dynamics can be well distinguished by monitoring the mean size of
the dynamical system, $\sigma(t)=\sqrt{\langle\bm{\rho}^{2}\rangle_{t}}$. As
shown in figure 2(c), at longer times $\sigma(t)$ is almost static in the
deposition regime, while it shows a non-monotonic behavior in the recoiling
regime and a continuous increase in the splashing regime.

To this end, we have shown that the QFDD starting from the Townes states can
produce rich dynamical phases, including the deposition, recoiling, and
splashing, which perfectly mimic the droplet impact outcomes as studied in the
literature [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25].
However, different from these existing studies of droplet impact on a surface,
in our case the driving force of the dynamics is purely from the intrinsic
energy contributed by quantum fluctuations. This introduces a mechanism for
these fluid dynamics. Meanwhile, in QFDD there are no complexities caused by
the impact surface or environment, and the dynamical outcome can be fully
controlled by adjusting the size $(\sigma_{0})$ and the number $(N)$ of the
initial state. Therefore, the ultracold atoms provide an extremely clean and
convenient platform to simulate droplet dynamics, where the quantum effect can
be well manipulated and the fluid mechanics can be understood in a more
deterministic way.

##  V Weber number and dynamical phase diagram

We now quantify various dynamical phases in QFDD by the Weber number. Note
that the Reynolds number is irrelevant here because of the zero viscosity of
Bose condensates. In conventional droplet impact dynamics [26, 27], the Weber
number is defined as $W=\rho_{d}Dv^{2}/\gamma$, where $\rho_{d},\ D,\ v,\
\gamma$ respectively denote the droplet density, diameter, impact velocity,
and surface tension. It measures the relative strength of droplet inertia with
respect to capillary force. Here, we generalize the definition of $W$ to
describe the quantum dynamics in general:

$W=\frac{\Delta E}{D\gamma},$ (9)

where $\Delta E$ is the energy difference between the initial state (here the
Townes soliton) and the true ground state given the same initial parameters
($N,\ g_{ij}$); $D$ and $\gamma$ are the same as before, i.e., the droplet
diameter and surface tension. Specifically, we have $\gamma=\int
d\bm{\bm{\rho}}\left({\cal
E}(\bm{\bm{\rho}})-\mu_{1}|\phi_{1}|^{2}-\mu_{2}|\phi_{2}|^{2}\right)$, with
$\mu_{i}$ the chemical potential for the i-th species.

As shown in the color plot in figure 2(a), $W$ defined in (9) can well
characterize different dynamical phases in QFDD. Namely, a low $W$ corresponds
to the deposition dynamics, where the small $\Delta E$ can be well absorbed by
the surface change of the droplet; as $W$ increases, the system enters the
recoiling regime and finally end up at splashing, where the large $\Delta E$
overwhelms the capacity of the droplet surface and causes it to change
drastically. Hereafter, we refer to the critical $W$ at the recoiling-
splashing boundary as the splashing threshold (${\cal K}_{s}$), and that at
the deposition-recoiling boundary as the recoiling threshold (${\cal K}_{r}$).

Figure 3: (Color online) Characterization of different dynamical phases and
their boundaries in-between. (a) Contour plot of Weber number in the parameter
plane of initial Townes size ($\sigma_{0}$) and number($N$). The splashing,
recoiling and deposition phases are respectively marked by ’S’, ’R1,2’ and ’D’
in the diagram. The white squares with white dashed lines mark their
boundaries. Gray dashed line shows the location of energy minimum for initial
Townes state. (b) Splashing threshold (${\cal K}_{s}$) and two recoiling
thresholds (${\cal K}_{r1},{\cal K}_{r2}$) as functions of $N$ along the phase
boundaries in (a). Note that when changing $N$, the coupling constant $a_{22}$
of 39K atoms also changes according to the constraint (6,7) for initial Townes
states. The other couplings $\\{a_{11},a_{12}\\}$ are the same as in figure 1.

In figure 3(a), we map out the dynamical phase diagram in the ($\sigma_{0},N$)
parameter plane. Distinct dynamical outcomes of QFDD, including deposition
(’D’), recoiling (’R’) and splashing (’S’), are identified by monitoring the
mean size $\sigma(t)$ of the cloud during expansion (see figure 2(c)). In
addition, we show the contour plot of $W$ in the ($\sigma_{0},N$) plane, and
one can see clearly that the ’D’, ’R’ and ’S’ phases respectively correspond
to the small, intermediate and large $W$ regions. Due to the non-monotonic
dependence of $W$ on $\sigma_{0}$ (as shown in figure 2(a)), there are two
recoiling regions in the diagram, as marked by ’R1’ and ’R2’. In figure 3(b),
we extract the two recoiling thresholds(${\cal K}_{r1},{\cal K}_{r2}$) and the
splashing threshold(${\cal K}_{s}$) along the phase boundaries as varying $N$.
One can see that ${\cal K}_{r1}\approx{\cal K}_{r2}$ are given by a constant
$\sim 0.12$, while ${\cal K}_{s}$ is a much larger value and continuously
increases with $N$.

##  VI  Maximum spreading factor

Another important physical quantity to characterize the droplet impact
dynamics is the maximum spreading factor $\beta=\sigma_{\rm max}/\sigma_{0}$,
as defined by the ratio between the maximum spreading radius ($\sigma_{\rm
max}$) and the initial one ($\sigma_{0}$). Clearly one has $\beta\sim 1$ for
the deposition dynamics and $\beta\rightarrow\infty$ for splashing. An
interesting behavior of $\beta$ shows up in the recoiling regime, where
$\beta$ is finite and varies sensitively with $W$.

Figure 4: (Color online) Universal scaling between the maximum spreading
factor ($\beta$) and the Weber number ($W$) in the recoiling regime. Here we
take four given atom numbers of the initial Townes state with tunable sizes,
as following the trajectory along the horizontal lines in Fig.3(a). The
discrete points are from numerical calculations and the lines are fitting
functions according to Eq.(10).

In figure 4, we extract $\beta$ as a function of $W$ along the horizontal
lines in figure 3(a), i.e., by varying $\sigma_{0}$ at several fixed $N$ in
the recoiling regime. Apart from the region near the phase boundaries
($\beta\rightarrow 1$ or large enough), the data of $\beta$ for any given $N$
well follow the scaling relation:

$\beta-1\propto W^{\alpha},\ \ \ {\rm with}\ \ \alpha=1.59.$ (10)

As shown in figure 4, such scaling works well for a wide parameter regime with
$N/10^{4}\in(0.5,2)$, $\beta\in(1.5,10)$ and $W\in(0.2,1)$. Therefore, we
expect the scaling law in (10) to reflect a very robust intrinsic property of
the ultracold fluid during the recoiling QFDD.

##  VII  Breathing modes

Finally, we demonstrate that the deposition regime of QFDD can be used to
probe the breathing modes of quantum droplet. The breathing mode can be
theoretically obtained as follows. Assuming a small fluctuation mode
$\delta\phi_{i}$ for the i-th species boson, and only keeping the lowest-order
fluctuations in the GP equations(3), we obtain the following equations for
$\\{\delta\phi_{i}\\}$:

$\begin{split}i\partial_{t}\delta\phi_{1}&=\left(-\frac{\nabla^{2}_{\bm{\bm{\rho}}}}{2m}+g_{11}n_{1}+g_{12}n_{2}+\frac{\partial{\cal
E}_{LHY}}{\partial
n_{1}}\right)\delta\phi_{1}+g_{11}n_{1}(\delta\phi_{1}+\delta\phi_{1}^{*})+g_{12}\phi_{1}\phi_{2}(\delta\phi_{2}+\delta\phi_{2}^{*})\\\
&+\frac{\partial^{2}{\cal E}_{LHY}}{\partial
n_{1}^{2}}n_{1}(\delta\phi_{1}+\delta\phi_{1}^{*})+\frac{\partial^{2}{\cal
E}_{LHY}}{\partial n_{1}\partial
n_{2}}\phi_{1}\phi_{2}(\delta\phi_{2}+\delta\phi_{2}^{*}),\\\ \end{split}$
(11)
$\begin{split}i\partial_{t}\delta\phi_{2}&=\left(-\frac{\nabla^{2}_{\bm{\bm{\rho}}}}{2m}+g_{12}n_{1}+g_{22}n_{2}+\frac{\partial{\cal
E}_{LHY}}{\partial
n_{2}}\right)\delta\phi_{2}+g_{22}n_{2}(\delta\phi_{2}+\delta\phi_{2}^{*})+g_{12}\phi_{1}\phi_{2}(\delta\phi_{1}+\delta\phi_{1}^{*})\\\
&+\frac{\partial^{2}{\cal E}_{LHY}}{\partial n_{1}\partial
n_{2}}\phi_{1}\phi_{2}(\delta\phi_{1}+\delta\phi_{1}^{*})+\frac{\partial^{2}{\cal
E}_{LHY}}{\partial n_{2}^{2}}n_{2}(\delta\phi_{2}+\delta\phi_{2}^{*}).\\\
\end{split}$ (12)

According to the standard Bogoliubov analysis, we search for solutions of the
form:

$\delta\phi_{i}=\exp(-i\mu_{i}t)\sum_{j}\left(u_{ij}(\bm{\rho})\exp(-i\omega_{j}t)+v_{ij}^{*}(\bm{\rho})\exp(i\omega_{j}t)\right).$
(13)

Here $\omega_{j}$ is the $j$-th collective (eigen-)mode of the system. These
modes can be extracted from the following coupled equations for
$u_{ij}(\bm{\rho})$ and $v_{ij}(\bm{\rho})$:

$\left(\begin{matrix}L_{1}+M_{1}&M_{12}&M_{1}&M_{12}\\\
M_{12}&L_{2}+M_{2}&M_{12}&M_{2}\\\ -M_{1}&-M_{12}&-(L_{1}+M_{1})&-M_{12}\\\
-M_{12}&-M_{2}&-M_{12}&-(L_{2}+M_{2})\end{matrix}\right)\left(\begin{matrix}u_{1j}\\\
u_{2j}\\\ v_{1j}\\\
v_{2j}\end{matrix}\right)=\omega_{j}\left(\begin{matrix}u_{1j}\\\ u_{2j}\\\
v_{1j}\\\ v_{2j}\end{matrix}\right),$ (14)

where

$\begin{split}L_{i}&=-\frac{\nabla^{2}_{\bm{\bm{\rho}}}}{2m}+\sum_{j}g_{ij}|\phi_{j}|^{2}+\frac{\partial{\cal
E}_{LHY}}{\partial n_{i}}-\mu_{i}\\\
M_{i}&=g_{ii}|\phi_{i}|^{2}+\frac{\partial^{2}{\cal E}_{LHY}}{\partial
n_{i}^{2}}n_{i}\\\ M_{12}&=g_{12}\phi_{1}\phi_{2}+\frac{\partial^{2}{\cal
E}_{LHY}}{\partial n_{1}\partial n_{2}}\phi_{1}\phi_{2}.\\\ \end{split}$ (15)

In Fig.5, we show the lowest two breathing modes ($\omega_{1},\omega_{2}$)
below the atom emission threshold ($-\mu_{1},-\mu_{2}$), see solid lines. In
our numerics, we have obtained $\omega_{j}$ by discretizing the coordinate
space and exactly diagonalizing the resulted large matrix in the left-hand-
side of (14). Note that the breathing modes, by definition, stay in the same
zero angular momentum sector as the Townes state and the ground state droplet.
More details on the numerics are given in the Appendixes.

Figure 5: (Color online) Extracting collective breathing modes from QFDD in
the deposition regime. Here we take $a_{22}=83.8a_{B}$ and the other
parameters are the same as in Figure 1; the initial atom number is
$N(10^{4})=7.3$. (a) and (b) show the boson densities at different locations
$\bm{\rho}=0$ (center), $16.4\mu m$ in the time and frequency domains (related
by Fourier transformation), staring from an initial Townes state with size
$\sigma_{0}=18\mu m$. The two red arrows in (a) mark the time interval for
Fourier transformation. (c) Collective excitation spectrum as a function of
droplet number $N_{d}$. The lines are theoretical results based on Bogoliubov
analysis, and the discrete data with error bar are extracted from the real
density oscillations in the deposition dynamics. In the latter case the
residual number $N_{d}$ can be effectively tuned by setting different sizes of
initial Townes state, and here we take two sizes $\sigma_{0}(\mu m)=16,\ 18$
which lead to $N_{d}(10^{4})=6.5,\ 6.2$ respectively. The energy unit is
$1/(ml_{0}^{2})$ with $l_{0}=1\mu m$.

In figure 5, we show how to extract collective breathing modes from the QFDD
in the deposition regime for a given set of coupling strengths $\\{a_{ij}\\}$
and initial atom number $N$. Figure 5(a) shows the typical density
oscillations with time at different locations. One can see that such
oscillations are synchronic for both species and at different locations, and
therefore their Fourier transformations give the same peak frequencies; see
figure 5(b). In figure 5(c), we plot out two sets of extracted peak
frequencies ($\omega_{1},\omega_{2}$) with different residual droplet numbers
$N_{d}$. Here $N_{d}$ can be tuned effectively by setting different sizes of
the initial Townes state. For all the extracted data we have collected 30
samples by varying the real-space locations or the time intervals in Fourier
transformation, from which we obtain both the mean value and the variance. One
can see that the extracted results match very well with theoretical
predictions of collective breathing modes from Eq.14 (solid lines).

We note that a previous study has extracted one branch of a collective mode
from the formation dynamics of a quantum droplet [59], while the dynamics
there is not QFDD. Here, we have shown in figure 5 that two branches of
collective modes can be simultaneously extracted from the residual
oscillations in QFDD. Moreover, we have checked that by taking different
$\\{a_{ij}\\}$ and initial $N$, one can also get access to the other regime
with only a single breathing mode. In comparison to the single-mode case, here
the main complexity brought by the presence of two modes is the appearance of
additional frequency peaks at $\omega_{2}\pm\omega_{1}$ due to their
interference. To correctly identify the breathing modes $\\{\omega_{j}\\}$
from the multiple frequency peaks, it is important to note that these modes
are associated with the most pronounced peaks among all others. This can be
understood by analyzing the spin densities
$n_{i}=|\phi_{i}+\delta\phi_{i}|^{2}$, with $\delta\phi_{i}$ given in Eq.(13).
Obviously, one can see that the Fourier transform of $n_{i}$ results in
multiple peaks at $\\{\omega_{j}\\}$, $\\{2\omega_{j}\\}$ and
$\\{\omega_{i}\pm\omega_{j}\\}$, while the ones at $\\{\omega_{j}\\}$ have the
largest magnitudes, which depend linearly on the strength of the fluctuation
mode (the others all have a quadratic dependence). Based on this principle,
one can easily identify the correct breathing modes ($\omega_{1},\omega_{2}$)
as marked in figure 5(b).

##  VIII  Experimental relevance

Our results can be readily tested in the current cold-atom experiments. The
initial Townes profile with a given amplitude and size can be imprinted by
properly designing the optical potential applied to the atoms, as successfully
implemented in previous experiments on two-component Bose gases [68, 69]. The
subsequent dynamics of the system can then be measured through the in situ
image, and various dynamical outcomes can be distinguished typically within
tends of milliseconds, as shown in Fig.2. Within this time scale severe atom
loss in the 39K droplet can be effectively avoided [37]. On the other hand,
for long time dynamics, the atom loss will play an essential role and may even
impede the breathing mode detection, which requires a typical time scale of
hundreds of milliseconds, as shown in Fig.5(a1,a2). For that, one can resort
to the heteronuclear mixtures of 41K and 87Rb, whose lifetime has been shown
to be much longer ($\sim 1s$) due to the low droplet density therein to
suppress the atom loss [40, 41]. Since the Townes soliton can be well extended
to boson mixtures with unequal masses (see the Appendixes), and the
controllability of LHY energy by the size and number generally applies to
these systems, we expect that various dynamical outcomes and the breathing
mode extraction can be tested equally in heteronuclear Bose gases.

##  IX  Summary and discussion

In summary, we have demonstrated the quantum simulation of droplet impact
outcomes in ultracold boson mixtures. Various dynamical phases including
splashing, recoiling, and deposition, have been revealed. A remarkable
difference here is that these dynamics are purely driven by quantum
fluctuations, instead of the mechanical impact force in previous studies.
Given the easy manipulation of initial states with tunable fluctuation
energies, the current cold-atom platform provides complete control of these
dynamics in the microscopic quantum level. To characterize different dynamics,
we have introduced the Weber number and examined two important physical
quantities, namely, the splashing/recoiling threshold and the maximum
spreading factor. We have also proposed to extract the collective breathing
modes from the residual dynamics in the deposition regime. These results are
directly relevant to ongoing cold-atom experiments.

Furthermore, we remark that our work is in distinct contrast to the previous
studies of QFDD in ultracold atoms, where a visible dynamics can only be
achieved for small condensates [70, 71, 72, 73]. This is because usually the
energy difference per particle between the mean-field and the true quantum
ground states decays rapidly as the number $N$ increases [74, 75, 76, 77] ,
and thus for large $N$ the quantum fluctuation energy is negligibly small as
compared to the mean-field one. However, this is not the case for a quantum
droplet. For a static droplet, quantum fluctuations have been shown to provide
an indispensable repulsive force for its stabilization, without which the
whole system will collapse [43]. In parallel, our current work reveals the
equally significant role of quantum fluctuations played in the non-equilibrium
dynamics of a quantum dropet, regardless of whether the droplet is small or
large.

Finally, we anticipate that the dynamical outcomes of splashing, recoiling,
and deposition revealed in this work may be equivalent with other initial
conditions when the dynamics is not purely driven by quantum fluctuations. In
fact, the generalized Weber number as defined in Eq.(9) can be applied to any
initial state, and the key issue is whether the additional energy of such a
state (as compared to the true ground state) can be absorbed by the surface
tension of a droplet during dynamics. In this sense, Eq.(9) may serve as a
unified quantity to understand the different dynamics of quantum droplets with
various initial conditions.

##  Acknowledgment

The work is supported by the National Key Research and Development Program of
China (2018YFA0307600), the National Natural Science Foundation of China
(12074419, 12134015, 12205365), and the Strategic Priority Research Program of
Chinese Academy of Sciences (No. XDB33000000).

##  Appendix A Townes soliton for two-species bosons with unequal masses

Neglecting the LHY correction, we write down the Gross-Pitaevskii(GP) equation
for two-species bosons with unequal mass ($m_{1},m_{2}$):

$\displaystyle
i\partial_{t}\phi_{1}=\left(-\frac{\nabla_{\bm{\rho}}^{2}}{2m_{1}}+g_{11}|\phi_{1}|^{2}+g_{12}|\phi_{2}|^{2}\right)\phi_{1};$
(16) $\displaystyle
i\partial_{t}\phi_{2}=\left(-\frac{\nabla_{\bm{\rho}}^{2}}{2m_{2}}+g_{12}|\phi_{1}|^{2}+g_{22}|\phi_{2}|^{2}\right)\phi_{2}.$
(17)

In the case $m_{1}\neq m_{2}$, we define the mass-imbalance parameter as
$w=m_{2}/m_{1}$. By comparing the two GP equations, we can see that they can
support a single-mode solution $\phi_{i}=\sqrt{N_{i}}\phi\exp(-i\mu_{i}t)$ as
long as

$\frac{N_{1}}{N_{2}}=\frac{wg_{22}-g_{12}}{g_{11}-wg_{12}},$ (18)

where $\mu_{1}=w\mu_{2}$, and the single mode $\phi$ satisfies:

$\left(-\frac{\nabla_{\bm{\rho}}^{2}}{2m_{1}}+Ng_{\rm
eff}|\phi|^{2}\right)\phi=\mu_{1}\phi.$ (19)

Here $N=N_{1}+N_{2}$ is the total number, and the effective interaction
$g_{\rm eff}$ is given by

$g_{\rm eff}=\frac{w(g_{11}g_{22}-g_{12}^{2})}{g_{11}+wg_{22}-(w+1)g_{12}}.$
(20)

Again we have $g_{\rm eff}<0$ in the mean-field collapse regime
($g_{12}<-\sqrt{g_{11}g_{22}}$), and the Townes solution occurs when

$Nm_{1}|g_{\rm eff}|=5.85.$ (21)

For the equal mass case ($w=1$), these equations automatically reduce to
Eqs.(4-7) in the main text.

##  Appendix B Numerical Method

In our numerical simulations, we have considered the solution with zero
angular momentum. This choice is because the angular momentum is preserved by
the Hamiltonian, and both the ground state and the Townes soliton stay exactly
in the zero angular momentum sector. Moreover, the breathing mode, by
definition, also stays in the zero angular momentum sector. Given this
constraint, in the following we simply replace the coordinate $\bm{\bm{\rho}}$
by its magnitude $\bm{\rho}=|\bm{\bm{\rho}}|$.

In solving the GP equation of the form
$i\partial_{t}\phi(\rho,t)=H\phi(\rho,t)$, we have discretized the coordinate
and time as $\rho_{j}=j\delta,\ t_{n}=n\tau$, where $j,n$ are all integers and
we have taken small intervals $\delta=0.005\mu m,\ \tau=2.74\times 10^{-3}ms$.
Then starting from a given state $\phi_{n}$ at time $t_{n}$, we obtained the
new wavefunction $\phi_{n+1}$ at the next time $t_{n+1}$ as follows.

First, we split $H$ into the local and non-local parts: $H=H_{1}+H_{2}$, where
the derivatives of $\rho$ are all contained in $H_{2}$. In this case, we
simply have $H_{2}$ as the kinetic energy
$H_{2}=-\frac{1}{2m}\left[\frac{\partial^{2}}{\partial\rho^{2}}+\frac{1}{\rho}\frac{\partial}{\partial\rho}\right]$,
and $H_{1}$ is the rest part that solely depends on local densities.

As the first step of the update, the local $H_{1}$ produces an intermediate
state $\phi^{\prime}$ from $\phi_{n}$:

$\phi^{\prime}=\exp(-iH_{1}\tau)\phi_{n}.$ (22)

Then, we perform the time evolution generated by $H_{2}$ with the Crank-
Nichilson scheme:

$\frac{\phi_{n+1}-\phi^{\prime}}{-i\tau}=\frac{H_{2}}{2}\left(\phi_{n+1}+\phi^{\prime}\right),$
(23)

which gives

$\phi_{n+1}=\frac{1-i\tau H_{2}/2}{1+i\tau H_{2}/2}\phi^{\prime}.$ (24)

Specifically, in the discretized coordinate space we have

$H_{2}\phi_{j}=-\frac{1}{2m}\left(\frac{\phi_{j+1}+\phi_{j-1}-2\phi_{j}}{\delta^{2}}+\frac{1}{j\delta}\frac{\phi_{j+1}-\phi_{j-1}}{2\delta}\right).$
(25)

Here we have simplified $\phi(\rho_{j})$ as $\phi_{j}$. To this end, (24)
gives the updated wavefunction $\phi_{n+1}$ at time $t_{n+1}$.

In above we have shown the details on the numerical simulation of GP
equations. By transforming $\tau\to-i\tau$, i.e., the imaginary time
evolution, one can also obtain the ground state of the system (droplet
solution). To make sure that the wavefunction is reduced to zero well before
touching the boundary of the system, we have chosen a sufficiently large size
in the simulation with the maximum radius ranging from 50$\mu m$ to 200$\mu
m$. In this way one can avoid the boundary effect to dynamical and static
properties of the system.

In solving the breathing modes from Eq.(14) in the main text, we have again
discretized the coordinate space and transformed it to a large matrix
equation. The breathing modes are then obtained by exactly diagonalizing the
matrix.

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Wavelet Adaptive Proper Orthogonal Decomposition for Large Scale Flow Data

Wavelet Adaptive Proper Orthogonal Decomposition

Philipp Krah Thomas Engels Kai  Schneider Julius Reiss

Philipp Krah,
Technische Universität Berlin,
Institute of Mathematics,
Straße des 17. Juni 136, 10623 Berlin, Germany

Thomas Engels,
Department of Animal Physiology, Institute of Biological Sciences, University of Rostock, Albert-Einstein-Str. 3, Rostock 18059, Germany

Kai Schneider,
Aix-Marseille Université, CNRS, Centrale Marseille, Institut de Mathématiques de Marseille (I2M), 39 rue Joliot-Curie, 13453 Marseille cedex 13, France

Julius Reiss,
Technische Universität Berlin,
Institute of Fluid Mechanics and Engineering Acoustics,
Müller-Breslau-Str. 15,
10623 Berlin, Germany

The authors gratefully acknowledge the support of the Deutsche Forsch-ungs- gemeinschaft (DFG) as part of GRK2433 DAEDALUS.
The authors were granted access to the HPC resources of IDRIS under the Allocation No. 2018-91664 attributed by GENCI (Grand Équipement National de Calcul Intensif).
Centre de Calcul Intensif d’Aix-Marseille Université is acknowledged for granting access to its high performance computing resources.

§ CONFLICT OF INTEREST

The authors declare that they have no conflict of interest.

§ CODE AND DATA AVAILABILITY

Software can be found under <cit.>. All scripts to reproduce the results are available at <cit.>.