# Ground-state Properties and Bogoliubov Modes of a Harmonically Trapped One-
Dimensional Quantum Droplet

Xucong Du  Department of Physics and Key Laboratory of Optical Field
Manipulation of Zhejiang Province, Zhejiang Sci-Tech University, Hangzhou
310018, China  Yifan Fei  Department of Physics and Key Laboratory of Optical
Field Manipulation of Zhejiang Province, Zhejiang Sci-Tech University,
Hangzhou 310018, China  Xiao-Long Chen  Department of Physics and Key
Laboratory of Optical Field Manipulation of Zhejiang Province, Zhejiang Sci-
Tech University, Hangzhou 310018, China  Yunbo Zhang<EMAIL_ADDRESS>Department of Physics and Key Laboratory of Optical Field Manipulation of
Zhejiang Province, Zhejiang Sci-Tech University, Hangzhou 310018, China

###### Abstract

We study the stationary and excitation properties of a one-dimensional quantum
droplet in the two-component Bose mixture trapped in a harmonic potential. By
constructing the energy functional for the inhomogeneous mixture, we elaborate
the extended the Gross-Pitaevskii equation applicable to both symmetric and
asymmetric mixtures into a universal form, and the equations in two different
dimensionless schemes are in a duality relation, i.e. the unique parameters
left are inverse of each other. The Bogoliubov equations for the trapped
droplet are obtained by linearizing the small density fluctuation around the
ground state and the low-lying excitation modes are calculated numerically. It
is found that the confinement trap changes easily the flat-top structure for
large droplets and alters the mean square radius and the chemical potential
intensively. The breathing mode of the confined droplet connects the self-
bound and ideal gas limits, with the excitation in the weakly interacting Bose
condensate for large particle numbers lying in between. We explicitly show how
the continuum spectrum of the excitation is split into discrete modes, and
finally taken over by the harmonic trap. Two critical particle numbers are
identified by the minimum size of the trapped droplet and the maximum
breathing mode energy, both of which are found to decrease exponentially with
the trapping parameter.

##  I Introduction

The ultradilute quantum droplet is a novel quantum state whose self-bound
nature arises from the competition between two different interactions [1, 2].
Beyond-mean-field (BMF) interactions, also known as quantum fluctuations or
Lee-Huang-Yang (LHY) corrections [3], play a crucial role in such a system.
However, since the BMF term represents the next-order correction, it is
essentially a small quantity compared to its mean-field (MF) counterpart. The
interactions in a Bose gas are highly tunable, allowing the MF term to be
significantly reduced or even eliminated [4, 5, 6]. Therefore, multicomponent
or dipolar systems with bosons are promising platforms for the study of
quantum droplets [7]. Experimental realizations of quantum droplets have been
achieved in binary [8, 9, 10, 11] and dipolar [12, 13, 14] Bose systems.
Theoretically, the form of the BMF term in a two-component mixture depends on
the dimensionality of the system. Three-dimensional (3D) BMF term is
effectively repulsive, while in lower dimensions it can be either attractive
or repulsive. Especially in the one-dimensional (1D) case, the system can
exhibit soliton-like features as the interaction strength varies [15], and the
transition from soliton to droplet behavior has been studied [16, 9]. Recent
work focuses on the superfluidity and vortex states in 2D and 3D quantum
droplets [17, 18], collective excitations [19, 20], collision dynamics [21,
22, 23], dimensional crossover[24], and the universality and metastability in
the droplet system [25, 26]. In 1D self-bound droplet, the collective
excitation spectrum has been firstly calculated by means of linearizing the
eGPE around the ground state [15] and the rotational properties of reveal
simultaneous rigid-body and superfluid behavior when the droplet is put in a
ring-shaped confinement carrying an angular momentum [27]. Furthermore, the
quantum Monte Carlo method is applied to the 1D droplet system [28, 29] and
novel phases such as the pair superfluid droplets are found in the Bose-
Hubbard chain of droplets when loaded into a 1D optical lattice [30, 31]. Some
very elegant review articles summarize the latest developments at the
frontiers of this new state of matter [32, 33, 34].

The presence of an external potential is known to transform a simple BEC
system into a non-uniform one, thereby changing its properties [35, 36]. In
dipolar systems, the anisotropic nature of the interactions means that
different shapes of external potentials can lead to significantly different
results, such as the ground states and stability regions in the phase diagrams
[37, 38, 39, 40]. Several techniques, including hydrodynamic models [41, 42,
43, 44], variational approach [45, 46], and Bogoliubov theory [47, 15], are
adopted to examine how the introduction of interactions in many-body systems
affects the low-lying collective excitations of trapped BECs.

The gaseous condensate is typically achieved in experiments by confining atoms
in harmonic potential. Even in experiments with self-bound droplets, as shown
in [9, 10, 8], researchers strive to reduce the strength of the trap as much
as possible to approximate the self-bound state in free space. Therefore, it
is crucial to study the influence of weak external potentials on 1D self-bound
droplets. Two recent papers have focused on the effect of the external
potential on the properties of Bose mixture droplets in the (quasi-) one-
dimensional case, one of which uses an extended Gross-Pitaevskii equation
(eGPE) that takes into account BMF corrections to study the bistable features
of symmetric trapped droplets‘ chemical potential varies with the number of
particles, and validates its results by dynamical evolution [48]. The other
adopts the non-perturbative many-body Hamiltonian to investigate the
correlated dynamics of collective excitations by quenching the trapping
frequency [49]. These are good attempt, however, a unified theory of
excitation spectrum is still lacking for symmetric and asymmetric 1D droplets,
and the stationary properties and how the collective excitation transitions
from a self-bound droplet to an ideal gas with the trapping strength and
particle number is still not clear. Starting with a 1D trapped droplet system,
in this paper we propose two different but complementary dimensionless
equations to bridge the gap between self-bound droplets and ideal gases. We
focus on how the particle number and trapping parameter affect the ground
state properties and the excitation modes of the system.

The remaining part of this paper is organized as follows. In next section II,
we introduce the appropriate time-dependent extended Gross-Pitaevskii equation
for this binary 1D quantum droplet in a harmonic external potential and
present two equations in mutually dual dimensionless forms. In section III,
the details of the numerical approach based on the eGPE for the ground states
and the Bogoliubov theory for low-lying excitation modes are introduced. We
show how the stationary properties in the ground state are affected by the
trapping potential in section IV and the excitation spectrum, especially the
breathing mode of trapped droplet, is analyzed within several energy scales
and the trend of breathing mode for weak and strong confinements and small and
large particle numbers in section V. Finally, we conclude our main results in
the last section VI.

##  II Model

The energy density of a 1D quantum droplet emerging in the homogeneous two-
component Bose mixture is written as [2, 15]

$\displaystyle{\cal E}_{\text{1D}}=$
$\displaystyle\frac{(g_{11}^{1/2}n_{1}-g_{22}^{1/2}n_{2})^{2}}{2}+\frac{g\delta
g(g_{22}^{1/2}n_{1}+g_{11}^{1/2}n_{2})^{2}}{(g_{11}+g_{22})^{2}}$
$\displaystyle-\frac{2\sqrt{m}}{3\pi\hbar}(g_{11}n_{1}+g_{22}n_{2})^{3/2},$
(1)

where $n_{1,2}$ denotes the density of the two components in different
hyperfine states of the same atom of mass $m$, $g_{11,22}$ and $g_{12,21}$
represent the repulsive intra- and attractive inter-component interaction
strengths with $g_{12}=g_{21}$, the parameters $g$ and $\delta g$ are related
to the coupling constants in the two components as $g=\sqrt{g_{11}g_{22}}$ and
$\delta g=g+g_{12}>0$, respectively.

The first two terms in Eq. (1) are the contribution of the MF theory, and the
last one is the LHY correction arising from the quantum fluctuation. Under the
condition of minimizing the dominant first MF term in Eq. (1), the density of
these two components will follow the ratio $n_{1}/n_{2}=\sqrt{g_{22}/g_{11}}$,
which is also true in the inhomogeneous mixture for smooth enough variation of
the total density. This allows us to reduce the mixture to an effective
single-component system characterized by the wavefunction $\psi(x)$, which is
related to that of each component as

$\displaystyle\psi_{1,2}(x)=\left(\frac{1}{2}\sqrt{\frac{g_{22,11}}{\mathcal{G}}}\right)^{1/2}\psi(x).$
(2)

where we further define $\mathcal{G}=(\sqrt{g_{11}}+\sqrt{g_{22}})^{2}/4$.
This definition assures that
$|\psi(x)|^{2}=|\psi_{1}(x)|^{2}+|\psi_{2}(x)|^{2}$, i.e. the density of the
effective single component is the sum of each component in the mixture
$n(x)=n_{1}(x)+n_{2}(x)$. The energy functional for the inhomogeneous mixture
is taken as

$\displaystyle{\cal E}[\psi(x),\psi^{*}(x)]=$ $\displaystyle\int
dx\left(\frac{\hbar^{2}}{2m}|\nabla\psi|^{2}+V(x)|\psi|^{2}\right.$
$\displaystyle\left.+\frac{g\delta
g}{4\mathcal{G}}|\psi|^{4}-\frac{2\sqrt{m}}{3\pi\hbar}g^{3/2}|\psi|^{3}\right).$
(3)

Minimizing the energy functional Eq. (3) with respect to independent
variations of $\psi(x)$ and its complex conjugate $\psi^{*}(x)$ subject to the
condition that the total number of particles $N=\int dx|\psi(x)|^{2}$ be
constant, we obtain the equation of motion of the system or the time-dependent
extended Gross-Pitaevskii equation (eGPE) for 1D droplet in a harmonic
trapping potential

$\displaystyle i\hbar\partial_{t}\psi=$
$\displaystyle\left[-\frac{\hbar^{2}}{2m}\nabla^{2}+\frac{1}{2}m\omega_{x}^{2}x^{2}\right.$
$\displaystyle\left.+\frac{g\delta
g}{2\mathcal{G}}|\psi|^{2}-\frac{\sqrt{m}}{\pi\hbar}g^{3/2}|\psi|\right]\psi,$
(4)

where $\omega_{x}$ is the harmonic trapping frequency. Note this equation
applies equally to the binary mixture for both symmetric ($g_{11}=g_{22}$) and
asymmetric ($g_{11}\neq g_{22}$) interaction strengths.

To get a neat dimensionless form of the eGPE, we need to rescale the length,
time, and wave function, respectively, as

$\displaystyle x=x_{0}x^{\prime}\text{, }t=t_{0}t^{\prime}\text{,
}\psi=\psi_{0}\psi^{\prime},$ (5)

where the prime denotes the dimensionless quantities and the subscript 0
indicates the characteristic units. Note that all terms in the square bracket
of (4) are energies and the energy and time units can be derived from the
definition of the length unit $x_{0}$ using the following relationship:

$\displaystyle E_{0}=\frac{\hbar^{2}}{mx_{0}^{2}}=\frac{\hbar}{t_{0}}.$ (6)

For the purpose of making the final dimensionless equation contain as few
adjustable parameters as possible, we first require that the two nonlinear
terms have the same coefficients, i.e., they need to satisfy

$\displaystyle\frac{g\delta
g}{2\mathcal{G}}\psi_{0}^{2}=\frac{\sqrt{m}}{\pi\hbar}g^{3/2}\psi_{0},$ (7)

which results in a characteristic factor for the wave function as

$\displaystyle\psi_{0}=\frac{2\mathcal{G}\sqrt{mg}}{\pi\hbar\delta g}.$ (8)

Hence the crucial step is the choice of length unit $x_{0}$, for which there
are two different conventions to follow in the field of cold atoms and we will
refer to as $\lambda$-scheme and $\beta$-scheme, respectively. The
$\lambda$-scheme is commonly used in the self-bound quantum droplet systems as
suggested in [46, 15]: the length unit $x_{\lambda}$ is derived by equating
the kinetic energy and the interaction energy

$\displaystyle\frac{\hbar^{2}}{mx_{\lambda}^{2}}=\frac{\sqrt{m}}{\pi\hbar}g^{3/2}\psi_{0},$
(9)

which leads to

$\displaystyle x_{\lambda}=\frac{\pi\hbar^{2}}{mg}\sqrt{\frac{\delta
g}{2\mathcal{G}}}.$ (10)

Thus, a scaling factor for the number of particles is generated by the
normalization condition of the wave function

$\displaystyle\int|\psi^{\prime}|^{2}dx^{\prime}=\tilde{N},$ (11)

where $\tilde{N}=N/N_{\lambda}$ is the number of particles in unit of
$N_{\lambda}$ and

$\displaystyle
N_{\lambda}=\psi_{0}^{2}x_{\lambda}=\frac{1}{\pi}\left(\frac{2\mathcal{G}}{\delta
g}\right).$ (12)

The eGPE is then cast into the following dimensionless form

$\displaystyle
i\partial_{t}\psi=\left(-\frac{1}{2}\nabla^{2}+\frac{1}{2}\lambda^{2}x^{2}+|\psi|^{2}-|\psi|\right)\psi,$
(13)

where we have omitted all the prime symbols, and

$\displaystyle\lambda=\frac{\omega_{x}}{\omega_{\lambda}}$ (14)

is the trapping potential energy in units of the interaction energy in the
self-bound droplet $E_{\lambda}=\hbar\omega_{\lambda}$ [15] with

$\omega_{\lambda}=\frac{2mg^{2}\mathcal{G}}{\pi^{2}\hbar^{3}\delta g}.$ (15)

The dimensionless eGPE (13) obtained in the $\lambda$-scheme can describe the
influence of the external potential on the self-bound quantum droplet. When
the characteristic length of the harmonic potential is equal to that of a
self-bound droplet, i.e., $\lambda=1$, the external potential is considered
significant. At this point, the characteristic units defined in
$\lambda$-scheme may not be as appropriate, thus we give a second path.

Figure 1: Density profiles of the quantum droplet for the particle number
$\tilde{N}=1$ (red), $10$ (yellow), $20$ (blue), and trapping potential
$\lambda=0$ (a, top panel), $0.03$ (b, middle panel), $0.1$ (c, bottom panel),
in dimensionless units defined in Eqs. (10) and (8).

Once the trapping potential is strong enough, it is more reasonable to use the
characteristic length of the harmonic trapping potential as the unit of
length, which is actually by equating the kinetic energy and the harmonic
potential

$\displaystyle\frac{\hbar^{2}}{mx_{\beta}^{2}}=\frac{1}{2}m\omega_{x}^{2}x^{2},$
(16)

and

$\displaystyle x_{\beta}=\sqrt{\frac{\hbar}{m\omega_{x}}}.$ (17)

Similarly, the scaling factor for the number of particles in the
$\beta$-scheme is

$\displaystyle
N_{\beta}=\psi_{0}^{2}x_{\beta}=\frac{4\sqrt{m}g\mathcal{G}^{2}}{\pi^{2}\hbar^{3/2}\delta
g\sqrt{\omega_{x}}}.$ (18)

Then the dimensionless eGPE reads (where the primes are omitted)

$\displaystyle
i\partial_{t}\psi=\left[-\frac{1}{2}\nabla^{2}+\frac{1}{2}x^{2}+\beta(|\psi|^{2}-|\psi|)\right]\psi,$
(19)

with

$\displaystyle\beta=\frac{\omega_{\lambda}}{\omega_{\beta}}$ (20)

is the interaction energy in units of the trapping potential with
$\omega_{\beta}=\omega_{x}$. The dimensionless eGPE (19) obtained in the
$\beta$-scheme depends solely on one dimensionless parameter $\beta$, which
can equally be used to elucidate the role of the two types of interactions,
i.e. MF and LHY, on the properties of a trapped droplet.

It is interesting that the dimensionless eGPE from these two schemes are in a
duality relation, i.e. the parameters are inverse of each other

$\beta=1/\lambda,$ (21)

from which the length unit and the particle number unit are related as

$\displaystyle x_{\lambda}=\frac{x_{\beta}}{\sqrt{\beta}}\text{,
}N_{\lambda}=\frac{N_{\beta}}{\sqrt{\beta}}.$ (22)

Therefore, in order to better present the results, all physical quantities in
the figures presented in this paper are dimensionless and characterized via
the $\lambda$-scheme. It is worth noting that, by setting $g_{11}=g_{22}$, we
naturally arrive at $\mathcal{G}=g=g_{11,22}$ and the system enters the
familiar symmetric case (be careful with the definition of $n$) which are
discussed in detail in Refs. [46, 2, 33].

##  III Bogoliubov theory

In this section, we use the Bogoliubov method to study the low-lying
collective excitations of the system. To quantitatively determine the density
profile of the ground state and the collective excitations of the quantum
droplet, it is necessary to solve the stationary eGPE for the ground state
wave function

$\displaystyle\hat{\mathcal{H}}\psi_{g}=\mu\psi_{g},$ (23)

with $\mu$ the chemical potential in unit of $E_{0}$ and the operator
$\hat{\mathcal{H}}$ denoted in the $\lambda$ or $\beta$-scheme as

$\displaystyle\hat{\mathcal{H}}_{\lambda}=$
$\displaystyle-\frac{1}{2}\nabla^{2}+\frac{1}{2}\lambda^{2}x^{2}+\psi_{g}^{2}-\psi_{g},$
(24) $\displaystyle\hat{\mathcal{H}}_{\beta}=$
$\displaystyle-\frac{1}{2}\nabla^{2}+\frac{1}{2}x^{2}+\beta(\psi_{g}^{2}-\psi_{g}),$
(25)

and then solve the Bogoliubov equations for small-amplitude excitations around
the condensate

$\displaystyle\begin{bmatrix}\hat{\mathcal{H}}-\mu+\hat{\mathcal{M}}&\hat{\mathcal{M}}\\\
-\hat{\mathcal{M}}&-\hat{\mathcal{H}}+\mu-\hat{\mathcal{M}}\end{bmatrix}\begin{bmatrix}u_{j}\\\
v_{j}\end{bmatrix}=\omega_{j}\begin{bmatrix}u_{j}\\\ v_{j}\end{bmatrix},$ (26)

where we have defined the operators,

$\displaystyle\hat{\mathcal{M}}_{\lambda}$ $\displaystyle=$
$\displaystyle\psi_{g}^{2}-\frac{1}{2}\psi_{g},$ (27)
$\displaystyle\hat{\mathcal{M}}_{\beta}$ $\displaystyle=$
$\displaystyle\beta(\psi_{g}^{2}-\frac{1}{2}\psi_{g}),$ (28)

and used the fact that the stationary ground state $\psi_{g}$ is real. The
Bogoliubov equations are obtained by assuming a small density fluctuation
around the ground state, i.e. using the following ansatz

$\displaystyle\psi(x,t)=\bigg{\\{}\psi_{g}(x)+\sum_{j}\left[u_{j}(x)e^{-i\omega_{j}t}+v_{j}^{*}(x)e^{i\omega_{j}t}\right]\bigg{\\}}e^{-i\mu
t},$ (29)

to expand the eGPE to the first order of $u_{j}$ and $v_{j}$. We emphasize
that the dimensionless Bogoliubov equations (26) can be obtained either from
the dimensionaless eGPEs (13) or (19) by inserting back the wave function with
fluctuation (29), or from the original eGPE (4) assuming a similar fluctuated
wave function and performing the dimensionless after. The excitation energy
(frequency) $\omega_{j}$ in units of $E_{0}$ and the amplitudes $u_{j},v_{j}$
in units of $\psi_{0}$ are labeled by an integer $j$ with $j=0,1,2,\cdots$.
This linearization process is a standard method for studying low-lying
excitations.

In the numerical workload for obtaining the collective excitation spectrum, we
first evolve the stationary eGPE in imaginary time to obtain the ground state
wave function. This allows us to construct the Bogoliubov matrix, which is
then diagonalized using Arnoldi’s method [50] to obtain the eigenvalues
representing the collective excitation spectrum. The implicitly restarted
Arnoldi method, which is implemented in the ARPACK software package, enables
finding the $M$ largest or smallest eigenvalues of the operator matrix in
(26), where $M$ is selected by the user. In the program we take $M=40$, which
contains 20 pairs of opposite eigenvalues, and in the end only 20 of these
positive values are retained. The sine-spectral method [51, 52] is used to
deal with the kinetic term, which is known for its higher accuracy compared to
the conventional finite difference method. Specifically this method introduces
the sine interpolation of a function $u(x)$ to approximate its second order
derivative $\partial_{xx}u(x)$ at grid points which can be formulated as a
matrix-vector multiplication.

Figure 2: Mean square radius $\sqrt{\Braket{x^{2}}}$ of the droplet as
functions of the number of particles $\tilde{N}$ for various trapping
parameters $\lambda$ from the self bound case $\lambda=0$ to a strong
confinement $\lambda=1$ as indicated in the legend. The minimum values marked
with red crosses are critical particle numbers $\tilde{N}_{c}$ where the
droplet has a minimum size.

##  IV Stationary properties

We first consider the effect of the applied confinement potential on the
static properties of the quantum droplet from the numerical solutions of the
stationary eGPE. It is already known that the density of a self-bound droplet
takes a gaussian-like profile for a small number of particles, while it
exhibits a flat-top structure when the number of particles is large. Here we
do the numerical simulations in the $\lambda$-scheme for the reduced particle
numbers in the range $\tilde{N}=0.01$ to $50$ and consider three
representative values for the parameter $\lambda$: $0$, $0.03$, and $0.1$.

The density profiles of the system are shown in Fig. 1. The case $\lambda=0$
describes the self-bound droplets in free space whose density profiles undergo
a smooth transition from the flat-top structure at large particle numbers
$\tilde{N}=20$ to the Gaussian shape for a few atoms $\tilde{N}=1$ (Fig. 1a).
The presence of an external potential introduces a slight inhomogeneity into
the system and especially the central region of the droplet feels this more
sensitively: more particles are pushed towards the center and the maximum
density starts to increase, as shown in Fig. 1b for $\lambda=0.03$. The flat-
top structures are easily destroyed and the droplet shows the bell-like
profile as the strength of the external potential increases further (Fig. 1c
for $\lambda=0.1$), a phenomenon that also occurs in the 3D case [19]. We
observe that in the 1D case the flat-top structure is more fragile and hard to
observe than its 3D counterpart, as a very weak external potential
$\lambda\sim 10^{-3}$ is enough to spoil the flat-top while it is still robust
for a confinement as strong as $\lambda\sim 0.03$ in 3D [19]. Note that the
spatial extension in Fig. 1 gradually shrink inward and the maximum value of
density is squeezed to a relatively high value as the confinement becomes
tighter.

Figure 3: Chemical potential $\mu$ of the droplet as a function of $\tilde{N}$
in different traps with different parameters $\lambda$ as indicated in the
legend. The dashed line represents the equilibrium value corresponding to the
spatially uniform state when $\tilde{N}\rightarrow+\infty$, and the dot-dashed
line is the negative-to-positive transition line $\mu=0$ which intersect with
the chemical potential for increasing parameters $\lambda$ at fewer and fewer
number of particles marked by the red corsses. The chemical potential is
always positive for trapping parameter larger than $\lambda_{c}\sim 0.47$
(dot-dashed curve). Figure 4: The ratios $\omega_{B}/|\mu|$ (left axis) and
$\omega_{B}/\lambda$ (right axis) of the droplet as a function of $\lambda$
for $\tilde{N}=0.1$ (a), $1$ (b), $10$ (c), and $20$ (d). The black dashed
line represents the results for the self-bound system [15], the dot-dashed
lines $\omega_{B}=2\lambda$ and the dashed lines $\omega_{B}=\sqrt{3}\lambda$
are the prediction of an ideal gas and the interacting BEC system,
respectively. The grey dotted curve in (d) represents the excitation energy
for a very large droplet $\tilde{N}=50$ with a bottom very close to the value
of 1D dilute bose gas.

It is helpful to have a closer look at how the confinement would change the
minimum size of the droplet. In Fig. 2 we present the numerical result for the
mean-square radius $\sqrt{\Braket{x^{2}}}$ of the droplet with

$\left<x^{2}\right>=\frac{\int^{+\infty}_{-\infty}x^{2}|\psi_{g}|^{2}dx}{\int^{+\infty}_{-\infty}|\psi_{g}|^{2}dx},$
(30)

as a function of the reduced particle number $\tilde{N}$. It can be observed
that, for both the self-bound droplet and the droplet in the trapping
potential, the size of the droplets exhibits a minimum value, which, marked by
red crosses in Fig. 2, in the self-bound droplet is near $\tilde{N}=1$ and
moves gradually toward smaller particle numbers with increasing confinement
$\lambda$. While in the self-bound case the size of a smaller droplet diverges
at $\tilde{N}\rightarrow 0$ and in the opposite limit the size of large
droplets grows linearly with $\tilde{N}$, we find that the confinement changes
this picture, i.e. the size of a smaller droplet no longer diverges but tends
to be a constant in units of $x_{0}$ for increasingly strong traps, and at
large particle numbers the droplet size still grows linearly with $\tilde{N}$,
but the growth rate becomes very slow. The minimal values define a critical
particle number $\tilde{N}_{c}$ where the droplet has the minimum size. For
the self-bound case it separates two different density profiles, i.e. the
Gaussian-like profile for $\tilde{N}\ll\tilde{N}_{c}$ and the flap-top
structure for $\tilde{N}\gg\tilde{N}_{c}$. But it is no longer true for the
droplet in the trap as the flat-top is very fragile and easily lost for 1D
trap. This critical number of particle $\tilde{N}_{c}$ depends almost linearly
on the trapping parameter $\lambda$. This phenomenon is likely attributed to
the dominance of the kinetic term when the number of particle is small (see
[46]). It can be explained as follows. In the self-bound system the quantum
pressure generated by the kinetic term serves to balance the interaction
energy. The presence of the external trapping potential then compensates for
the potential energy, leading to a reduction in the required interaction
energy, which also corresponds to the need for fewer particles. As a result,
the minimum value of $\sqrt{\Braket{x^{2}}}$ gradually shifts towards lower
particle numbers. This behavior is also reflected in the monopole oscillation
of the collective excitation, which will be discussed in detail in the next
section.

We note that the confinement will also alter the chemical potential $\mu$
greatly. For 1D self-bound droplet, the chemical potential always takes a
negative value that approaches $-2/9$ for large number of particles, as shown
in Fig. 3, implying that the state is self-bound in the equilibrium in
consistent with previous findings [2, 46]. In this case, the chemical
potential $\mu$ can be regarded as the threshold of particle emission, making
$|\mu|$ a suitable unit for excitation energy. The trapping potential
contributes to the chemical potential of the 1D trapped droplet that will not
only increase with the number of particles $\tilde{N}$ but also with the
confinement parameter $\lambda$. For a weaker potential ($\lambda<0.2$), the
chemical potential maintains the same downward trend as the self-bound
droplets for small particle numbers, but begins to increase after a certain
number of particles until its value becomes positive. For a stronger potential
($\lambda>0.5$), the chemical potential is already positive for small particle
numbers and increases rapidly with $\tilde{N}$. We observe a critical trapping
parameter $\lambda_{c}\sim 0.47$, above which the chemical potential is never
negative. This transition of the chemical potential from negative to positive
causes the ratio of excitation energy to chemical potential, e.g.
$\omega_{B}/|\mu|$, to diverge at the transition points $\mu=0$. Marked by the
red crosses in Fig. 3, these transition points occur for smaller particle
numbers for stronger trapping potentials. To avoid this divergence, in both
the strong confinement limit and the large particle number limit, we plot the
ratio of the breathing mode frequency $\omega_{B}$ and the trapping frequency
$\lambda$ in the following.

##  V Breathing mode of trapped droplet

Figure 5: The first three non-trivial modes $\omega_{j=2,3,4}$ of the
Bogoliubov excitation spectrum and the chemical potential (with minus sign) of
the self-bound droplet and trapped droplet with different parameters
$\lambda=0.001$ (a), $0.01$ (b), and $0.05$ (c) as a function of $\tilde{N}$.
The dashed and solid lines are the result for the chemical potential and the
excitation modes, respectively. The breathing mode $\omega_{B}$ is the third
mode, i.e. $\omega_{j=2}$.

In the context of quantum droplets in cold atoms, the monopole oscillation,
also known as the breathing mode, refers to a collective oscillation of the
droplet’s density profile or size, which can be calculated by means of several
different methods. For example, the low-energy excitation spectrum of the
system can be obtained by solving the hydrodynamic equations, where the lowest
monopole mode is the breathing mode [53, 54]. In the variational approach,
using the time-dependent ansatz, one can define an effective potential by the
Euler-Lagrange equation for the width parameter and assume that it has the
form of a harmonic potential to obtain the frequency of the breathing mode
[46, 55, 56, 57, 20]. For the time-independent situation the elegant sum-rule
approach is adopted in order to evaluate the collective frequencies in the
intermediate regimes where the hydrodynamic equations are not analytically
soluble [19, 58, 59, 60]. In addition, the breathing mode can also be excited
by introducing a small interaction perturbation [20] or by driving a single
droplet out of equilibrium with an initial excitation in the norm $\tilde{N}$
and study the ensuing dynamics by simulating the eGPE [46]. Here we
numerically solve the Bogoliubov equation (26) to extract the breathing mode
of the system and check how it is affected by the confinement parameter
$\lambda$. It is noteworthy that the breathing mode for the 1D droplet
corresponds to the third eigenvalue of the Bogoliubov equations (26), i.e.
$j=2$.

The excitation energy of the breathing mode $\omega_{B}$ of the droplet is
shown in Fig. 4 as a function of the trapping parameter $\lambda$ for several
typical atomic numbers. We use two different energy units for nearly self-
bound droplet on one side and very tightly confined droplet on the other side,
i.e. the ratio $\omega_{B}/|\mu|$ and $\omega_{B}/\lambda$ are plotted,
respectively, as explained in the last section. To illustrate more clearly the
excitation energies in the two limits the logarithm coordinates are used for
$\lambda$. Numerically for weak confinement $\lambda\sim 10^{-4}-10^{0}$ we
use the $\lambda$-scheme to diagonalize the Bogoliubov equations with
operators $\hat{\mathcal{H}}_{\lambda}$ and $\hat{\mathcal{M}}_{\lambda}$ and
to extract the third eigenvalues, i.e. the excitation energy $\omega_{B}$ of
the breathing mode, which are shown in black solid lines in Fig. 4. For small
or large droplets the breathing modes fall back to their self-bound values
[15] (the horizontal dotted lines) when the confinement is as weak as
$\lambda\sim 10^{-2}$ as expected. For strong confinement $\lambda\sim
10^{0}-10^{4}$, on the other hand, the numerical $\beta$-scheme is preferred
and it is the operators $\hat{\mathcal{H}}_{\beta}$ and
$\hat{\mathcal{M}}_{\beta}$ that play the role in the diagonalization of
Bogoliubov matrix and the numerical results are shown in red solid lines. We
find that in very strong confinement the droplets all behave like ideal gases
with the excitation energy showing a smooth convergence to the value
$\omega_{B}/\lambda=2$ [36]. However, the excitation energies for larger
droplets undergo a downward process with a minimum value around $\lambda\sim
1$ and eventually approach the ideal gas value from below. The downward moving
of this minimum in the excitation energy is attributed to the competition
among the trapping potential ($\sim N$), the mean-field ($\sim N^{2}$) and LHY
($\sim N^{5/2}$) interaction in the droplet and more particles would push the
bottom further down to the value of weakly interacting condensate, i.e.
$\omega_{B}/\lambda=\sqrt{3}$ [42], denoted in dashed lines in Fig. 4, and the
case for a relatively large droplet of $\tilde{N}=50$ is additionally shown in
Fig. 4d.

Figure 6: The dependence of the breathing mode $\omega_{B}$ of the trapped
droplet on $\tilde{N}$ for several larger parameters $\lambda$. Just as in the
self bound case, the breathing mode of trapped droplet exhibits a maximum
excitation energy for each parameter $\lambda$ and determines a critical
number of particles $\tilde{N}_{c}$ of $\omega_{B}$.

The ”divergence” in the excitation energy seems unreasonable, which is due to
either the vanishing of the chemical potential at the transition points for
$\omega_{B}/|\mu|$ or the disappearance of the trap for $\omega_{B}/\lambda$.
To cure this, the dependence of the non-trivial excitation modes including the
breathing mode, as well as the chemical potential, on the droplet’s size is
presented in unit of energy scale $E_{\lambda}$ in Fig. 5. It is already known
[46] that for the self-bound droplet this dependence is non-monotonous with
the largest excitation energy (stiffness) reached around
$\tilde{N}_{c}=1.2776$, which is very close to $\tilde{N}_{c}$ of the mean
square radius when the droplet has the minimal size $\tilde{N}_{c}=0.8330$. It
is also argued that the “autocooling” mechanism, which lose atoms until all
excitations are gone and droplets are generated in the true ground state for
3D droplet, is no longer applicable in 1D geometry, as the energy of breathing
mode is always less than the absolute value of the chemical potential, i.e.
$\omega_{B}<|\mu|$. It is worth to note that there is no longer the particle
emission threshold to continuum in the presence of the external trap as all
excitations become bound modes with discrete frequency [19]. We find that for
the 1D droplet the excitation energy is indeed lower than $-\mu$ in very weak
traps as in the self-bound case [46] \- we see no difference in the excitation
spectrum for $\lambda\sim 10^{-3}$ and the self bound droplet. Note that
$-\mu$ actually will approach its equilibrium value $2/9$ corresponding to the
spatially uniform state when $\tilde{N}\rightarrow+\infty$. No intersection
point is found in the lowest breathing mode and the chemical potential, while
the higher modes $\omega_{j=3,4,\cdots}$ would intersect with $-\mu$ at their
branching points, below which the corresponding excitations lie in the
continuum spectrum, as shown in Fig. 5a. The presence of the external trap,
e.g. a very weak one with $\lambda=0.01$, immediately splits the continuum
into discrete modes, which are likely to be peeled off layer by layer from the
threshold $-\mu$, and allows the excitation modes of smaller droplets to
increase first to exeed $-\mu$, so that we observe the intersection of the
lowest breathing mode with $-\mu$, and branching points for higher modes are
now intersections with $-\mu$, moving toward larger numbers of particles in
Fig. 5b. In the mean time, $-\mu$ for large $\tilde{N}$ is bent downward, e.g.
in a trap $\lambda=0.05$ in Fig. 5c, and two intersections appear in the
competition of breathing mode and the chemical potential, but no intersections
for higher modes, at the same time the discrete modes are gradually taken over
by the harmonic trapping potential, i.e. for small or large $\tilde{N}$ the
excitation spectrum tends to be equidistance with a characteristic gap
proportional to $\lambda$.

Figure 7: The critical numbers of particles $\tilde{N}_{c}$ determined by the
minimum values of the mean square radius $\sqrt{\Braket{x^{2}}}$ (light blue
dots) and by the maximum values of the breathing mode excitation energy
$\omega_{B}$ (black dots) as functions of trapping parameter $\lambda$. A
simple fit is shown by the dashed lines.

Interestingly, for even stronger confinement, e.g. $\lambda=0.1$, the lowest
excitation mode will exceed $-\mu$ completely and we further illustrate the
size-dependent frequency of the breathing mode for large trapping parameters
in Fig. 6, where an additional shift $2\lambda$ has been made for each mode
$\omega_{B}$ in units of $E_{\lambda}$. The reason why we shift this
$2\lambda$ is due to the fact that for small enough number of particles the
excitation mode is more or less near the ideal gas limit $\omega_{B}=2\lambda$
as shown in Fig. 4a. For each trapping parameter, there exists a peak value of
number of particles, which defines another critical particle number
$\tilde{N}_{c}$ where the excitation energy of the droplet $\omega_{B}$ has
the maximum energy, marked by the red crosses in Fig. 6. Recall that the
critical number of particles $\tilde{N}_{c}$ of the mean square radius in Fig.
2, we find both critical numbers of particles $\tilde{N}_{c}$ of
$\sqrt{\Braket{x^{2}}}$ and $\tilde{N}_{c}$ of $\omega_{B}$ decrease with the
trapping parameter $\lambda$ exponentially, as shown in Fig. 7, where the
self-bound values of $\tilde{N}_{c}=1.2776$ and $0.8330$ are denoted by two
red crosses. The discrepancy of the critical numbers in the self-bound droplet
was roughly carried over into the trapped droplets, which are compared in
detail in Fig. 7. We find that clearly the existence of the minimum size of
the droplet is accompanied by the appearance of the maximum excitation energy
of breathing mode. It’s important to note that the study of excitation modes
in quantum droplets is an active and rapidly evolving field of research, and
the understanding of their properties, including the breathing mode, is still
developing. For example, similar minimum size for the 3D self-bound droplet
occurs at $\tilde{N}\sim 30$, while the variational Gaussian ansatz for the
breathing mode gives an estimate of $\tilde{N}\sim 80$ and the numerical
result from eGPE is instead $\tilde{N}\sim 10^{3}$ [19]. The specific details
of the droplet’s composition, interparticle interactions, and external
conditions can significantly influence the relationship between droplet size
and the excitation energy of the breathing mode in cold atom systems.

Finally let us check the reliability of the two numerical schemes adopted in
this paper. Note that in the chosen characteristic units the parameters in
these two schemes are related by equation (21), i.e. the only parameters
$\lambda$ and $\beta$ left in the dimensionless equations are inverse to each
other. The immediate problem is that, when $\lambda$ and $\beta$ are assigned
relatively large values during the numerical procedure, the corresponding
trapping or interaction energy terms will be very large in units of the chosen
characteristic wave function $\psi_{0}$ while the step sizes of spatial and
temporal discretization in numerical computation are defined in terms of units
$x_{0}$ and $t_{0}$, making it difficult to obtain an accurate ground state
and thus reliable small-amplitude excitations. Thus we choose $\lambda$-scheme
to carry out the calculation for very weak traps as the case of self-bound
droplet $\lambda=0$ may serve as the benchmark, while it is more suitable to
adopt the $\beta$-scheme for very strong confinement and the trapped ideal gas
limit $\omega_{B}=2\lambda$ is a good reference. In this context, we have
checked the breathing modes calculated by these two schemes over the
intermediate regions of the parameter $\\{\lambda,\beta\\}\sim\\{0.1,10\\}$
from a numerical perspective. The results of the two eGPEs are nearly
identical when either $\lambda$ or $\beta$ is not greater than 10.

##  VI Conclusion

In conclusion, our study provides a comprehensive analysis of the stationary
and excitation properties of a one-dimensional quantum droplet by emphasizing
the role of a harmonic trapping potential. To address different energy scales,
we have introduced two different dimensionless time-dependent Gross-Pitaevskii
equations (GPE) with a parametric duality that allows us to study the system
under weak and strong trapping potentials, respectively. By means of these
equations, we have explored the ground-state properties such as the density,
mean-square radius and chemical potential, etc. We find the fragile feature of
the flap-top shape in the density distribution with respect to the external
potential, and a minimum size in terms of the mean-square radius with respect
to the normalization.

Furthermore, after applying the Bogoliubov theory or adding a small
fluctuation in the GPE, we have carefully discussed the low-lying elementary
collective modes, especially the breathing mode in the excitation energy
spectrum. By varying the external potential strength, we have shown an
intriguing non-monotonic behavior in the breathing-mode frequency which can
effectively recovers the results of a 1D uniform quantum droplet, an ideal gas
and a conventional trapped BEC in the specific parameter limits. In addition,
the breathing-mode frequency is depicted as a function of the particle number
and exhibits a maximum value at a critical position, which is closely
associated with the minimum in the mean-square radius. The critical particle
number in both breathing-mode frequency and the radius is further shown as an
exponentially decaying function of the external potential strength. The
predicted stationary and excitation properties are directly accessible with
current techniques in ultracold quantum gases experiments [8, 9, 10, 11].

###### Acknowledgements.

We acknowledge support from the NSF of China (Grant Nos. 12074340 and
12204413) and the Science Foundation of Zhejiang Sci-Tech University (Grant
Nos. 20062098-Y and 21062339-Y).

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