# Impurity induced quantum chaos for an ultracold bosonic ensemble in a
double-well

Jie Chen<EMAIL_ADDRESS>Zentrum für Optische
Quantentechnologien, Fachbereich Physik, Universität Hamburg, Luruper Chaussee
149, 22761 Hamburg, Germany  Kevin Keiler  Zentrum für Optische
Quantentechnologien, Fachbereich Physik, Universität Hamburg, Luruper Chaussee
149, 22761 Hamburg, Germany  Gao Xianlong  Department of Physics, Zhejiang
Normal University, Jinhua 321004, China  Peter Schmelcher  Zentrum für
Optische Quantentechnologien, Fachbereich Physik, Universität Hamburg, Luruper
Chaussee 149, 22761 Hamburg, Germany  The Hamburg Centre for Ultrafast
Imaging, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany

###### Abstract

We demonstrate that an ultracold many-body bosonic ensemble confined in a one-
dimensional (1D) double-well (DW) potential can exhibit chaotic dynamics due
to the presence of a single impurity. The non-equilibrium dynamics is
triggered by a quench of the impurity-Bose interaction and is illustrated via
the evolution of the population imbalance for the bosons between the two
wells. While the increase of the post-quench interaction strength always
facilitates the irregular motion for the bosonic population imbalance, it
becomes regular again when the impurity is initially populated in the highly
excited states. Such an integrability to chaos (ITC) transition is fully
captured by the transient dynamics of the corresponding linear entanglement
entropy, whose infinite-time averaged value additionally characterizes the
edge of the chaos and implies the existence of an effective Bose-Bose
attraction induced by the impurity. In order to elucidate the physical origin
for the observed ITC transition, we perform a detailed spectral analysis for
the mixture with respect to both the energy spectrum as well as the
eigenstates. Specifically, two distinguished spectral behaviors upon a
variation of the interspecies interaction strength are observed. While the
avoided level-crossings take place in the low-energy spectrum, the energy
levels in the high-energy spectrum possess a band-like structure and are
equidistant within each band. This leads to a significant delocalization of
the low-lying eigenvectors which, in turn, accounts for the chaotic nature of
the bosonic dynamics. By contrast, those highly excited states bear a high
resemblance to the non-interacting integrable basis, which explains for the
recovery of the integrability for the bosonic species. Finally, we discuss the
induced Bose-Bose attraction as well as its impact on the bosonic dynamics.

##  I Introduction

Trapping of an ultracold many-body bosonic ensemble in a one-dimensional (1D)
double-well (DW) potential constitutes a prototype system for the
investigations of the correlated quantum dynamics DW_exp_1 ; DW_exp_2 ;
DW_exp_3 . Such a system represents a bosonic Josephson junction (BJJ), an
atomic analogy of the Josephson effect initially predicted for Cooper pair
tunneling through two weakly linked superconductors BJJ_1 ; BJJ_2 . Owing to
the unprecedented controllability of the trapping geometries as well as the
atomic interaction strengths cold_atom_rev , studies of the BJJ unveil various
intriguing phenomena which are not accessible for conventional superconducting
systems BJJ_Rabi_1 ; BJJ_Rabi_2 ; BJJ_Rabi_3 ; BJJ_Frag_1 ; BJJ_Frag_2 ;
BJJ_Squeeze_1 ; BJJ_Squeeze_2 . Examples are the Josephson oscillations
BJJ_Rabi_1 ; BJJ_Rabi_2 ; BJJ_Rabi_3 , fragmentations BJJ_Frag_1 ; BJJ_Frag_2
, macroscopic quantum self trapping DW_exp_3 ; BJJ_Rabi_1 ; BJJ_Rabi_2 ,
collapse and revival sequences BJJ_Rabi_3  as well as the atomic squeezing
state BJJ_Squeeze_1 ; BJJ_Squeeze_2 .

Under the explicit time-dependent driving forces, the BJJ can alternatively
turn into the quantum kicked top (QKT), a famous platform for the
investigations of quantum chaos as well as the classical-quantum
correspondence QKT_1 ; QKT_2 ; QKT_3 ; QKT_4 ; QKT_5 ; QKT_6 ; QKT_7 ; QKT_8 ;
QKT_9 ; QKT_10 ; QKT_11 ; QKT_12 ; QKT_13 . To date, related studies include
the spectral statistics QKT_2 , the entanglement entropy production QKT_3 ;
QKT_4 ; QKT_5 ; QKT_6 ; QKT_7 ; QKT_8 ; QKT_9 ; QKT_10 , the quantum
decoherence and quantum correlations QKT_11 ; QKT_12  as well as the border
between regular and chaotic dynamics QKT_13 . Moreover, by viewing the QKT as
a collective $N$-qubit system, the effects of the quantum chaos on the digital
quantum simulations have also been detailed discussed recently QKT_14 ; QKT_15
.

On the other hand, stimulated by the experimental progresses on few-body
ensembles few_exp1 ; few_exp2 ; few_exp3 ; few_exp4 ; few_exp5 ; few_exp6 ,
significant theoretical effort also focuses on the 1D few-body atomic systems
few_gs_1 ; few_gs_2 ; few_gs_3 ; few_gs_4 ; few_gs_5 ; few_gs_6 ; few_gs_7 ;
few_quench_1 ; few_quench_2 ; few_quench_3 ; few_bf_SC1 ; few_bf_SC2 ,
revealing for example the ground state few_gs_1 ; few_gs_2 ; few_gs_3 ;
few_gs_4 ; few_gs_5 ; few_gs_6 ; few_gs_7 ; few_bf_SC1 ; few_bf_SC2  as well
as the dynamical properties few_quench_1 ; few_quench_2 ; few_quench_3 , which
pave the way for the studies of the binary mixtures with large particle number
imbalance. Such hybridized systems are deeply related to the polaron physics
polaron_1 ; polaron_2 ; polaron_3  as well as the open quantum systems OQS_1
and are particularly interesting owing to the fact that one subsystem is in
the deep quantum regime while the other one can more or less be described by
the semi-classical physics. Note, however, that while most of the discussions
focus on impacts on the minority species from the majority bath, studies which
alternatively explore the feedback to the majority species due to the presence
of the minority one are still rare.

In the present paper, we investigate a binary ultracold atomic mixture made of
a single impurity and a non-interacting many-body bosonic ensemble that are
confined within a 1D DW potential. Unlike most of the previous studies where
the focuses are put on the weak-interacting regime, rendering the impurity
being restricted into the lowest two modes of the DW potential Impurity_BH_1 ;
Impurity_BH_2 ; Impurity_BH_3 ; Impurity_BH_4 ; Impurity_BH_5 ; Impurity_BH_6
, our discussions are not restricted to such a scenario. Specifically, we
study the onset of the chaos for the majority bosonic species due to the
presence of the impurity and put particular emphasis on the its dynamical
response upon a sudden quench of the impurity-Bose interaction strength. As an
exemplary observable, we monitor the quantum evolution of the population
imbalance for the bosons between the two wells starting from a balanced
particle population. While the increase of the post-quench interaction
strength always facilitates a chaotic motion for the bosonic population
imbalance, it becomes regular again when the impurity initially is prepared in
the highly excited states. In order to characterize such an integrability to
chaos (ITC) transition, we employ the linear entanglement entropy as a
signature of quantum chaos, which alternatively measures the decoherence for
the bosonic species. Depending on the degree of chaos, the transient dynamics
of the corresponding linear entanglement entropy can behave as either a rapid
growth or a slow variation with increasing time, whereas, its infinite-time
averaged value, in addition, captures the edge of quantum chaos, i.e., the
border between the integrable and the chaotic regions in the corresponding
classical phase space. Furthermore, by computing the infinite-time averaged
values of the linear entanglement entropy for various initial conditions, we
find a striking resemblance between its profile and a classical phase space
with attractive Bose-Bose interaction, which implies the existence of an
attractive interaction among the bosons induced by the impurity.

In order to elucidate the physical origin for the above observed ITC
transition, we perform a detailed spectral analysis with respect to both the
energy spectrum as well as the eigenstates of the mixture. Two distinguished
spectral behaviors upon a variation of the interspecies interaction strength
are observed. While the avoided level-crossings take place in the low-energy
spectrum, the energy levels in the high-energy spectrum possess a band-like
structure and are equidistant within each band. Consequently, this results in
a significant delocalization for those low-lying eigenstates which, in turn,
accounts for the chaotic nature of the bosonic non-equilibrium dynamics.
Remarkably, those highly excited states bear a striking resemblance to the
non-interacting integrable basis, which explains the recovery of the
integrability for the bosonic species. Finally, we also discuss the induced
Bose-Bose attraction and its impact on the bosonic dynamics.

This paper is organized as follows. In Sec. II, we introduce our setup
including the Hamiltonian, the initial conditions as well as the quantities of
interests. In Sec. III, we present our main observation: the ITC transition
for the bosonic species. In Sec. IV, we perform a detailed spectral analysis
for the mixture with respect to both the energy spectrum as well as the
eigenstates, so as to elucidate the physical origin for the above observed ITC
transition. Finally, our conclusions and outlook are provided in Sec. V.

##  II Setup

###  II.1 Hamiltonian and angular-momentum representation

The Hamiltonian of our 1D ultracold impurity-Bose mixture is given by
$\hat{H}=\hat{H}_{I}+\hat{H}_{B}+\hat{H}_{IB}$, where

$\displaystyle\hat{H}_{\sigma}$ $\displaystyle=\int
dx_{\sigma}~{}\hat{\psi}^{\dagger}_{\sigma}(x_{\sigma})\textit{h}_{\sigma}(x_{\sigma})\hat{\psi}_{\sigma}(x_{\sigma}),$
$\displaystyle\hat{H}_{IB}$ $\displaystyle={g_{IB}}\int
dx~{}\hat{\psi}^{\dagger}_{I}(x)\hat{\psi}^{\dagger}_{B}(x)\hat{\psi}_{B}(x)\hat{\psi}_{I}(x),$
(1)

and
$\textit{h}_{\sigma}(x_{\sigma})=-\frac{\hbar^{2}}{2m_{\sigma}}\frac{\partial^{2}}{\partial
x_{\sigma}^{2}}+V_{DW}(x_{\sigma})$ is the single-particle Hamiltonian for the
$\sigma=I(B)$ species being confined within a 1D symmetric DW potential
$V_{DW}(x_{\sigma})=a_{\sigma}(x_{\sigma}^{2}-b_{\sigma}^{2})^{2}$. For
simplicity, we consider the atoms for both species are of the same mass
($m_{I}=m_{B}=m$) and are trapped by the same potential geometry, i.e.,
$a_{I}=a_{B}=a_{DW}$ and $b_{I}=b_{B}=b_{DW}$.
$\hat{\psi}_{\sigma}^{\dagger}(x_{\sigma})$
[$\hat{\psi}_{\sigma}(x_{\sigma})$] is the field operator that creates
(annihilates) a $\sigma$-species particle at position $x_{\sigma}$. Moreover,
we neglect the interactions among the bosons and assume the impurity-Bose
interaction is of zero range and can be modeled by a contact potential of
strength Feshbach_1 ; Feshbach_2 ; Feshbach_3 ; few_quench_3

$g_{IB}=\frac{2\hbar^{2}a_{3D}}{\mu
a_{\bot}^{2}}[1-C\frac{a_{3D}}{a_{\bot}}]^{-1}.$ (2)

Here $a_{3D}$ is the 3D impurity-Bose $s$-wave scattering length and $C\approx
1.4603$ is a constant. The parameter $a_{\bot}=\sqrt{\hbar/\mu\omega_{\bot}}$
describes the transverse confinement with $\mu=m/2$ being the reduced mass and
we assume the transverse trapping frequency $\omega_{\bot}$ to be equal for
both species. In the following discussions, we rescale the Hamiltonian of the
mixture $\hat{H}$ for the units of the energy, length and time as
$\eta=\hbar\omega_{\bot}$, $\xi=\sqrt{\hbar/m\omega_{\bot}}$ and
$\tau=1/\omega_{\bot}$, respectively. We focus on the repulsive interaction
regime, i.e., $g_{IB}\geqslant 0$ and set $a_{DW}=0.5$, $b_{DW}=1.5$, such
that the lowest two single-particle energy levels are well separated from the
others [see Fig. 1 (a), the spatial geometry of $V_{DW}(x)$ (black dashed
line) as well as the lowest six single-particle energy levels (grey solid
lines)]. Throughout this work, we explore a binary mixture made of a single
impurity and 100 bosons ($N_{I}=1$, $N_{B}=100$), and focus on the dynamical
response for the majority bosonic species upon a sudden quench of the
impurity-Bose interaction strength (see below). Let us note that such a 1D
mixture is experimentally accessible by imposing strong transverse and weak
longitudinal confinement for a binary e.g., Bose-Fermi mixture with two
different kinds of atoms mixture_exp_bf_1 ; mixture_exp_bf_2  or a Bose-Bose
mixture made of the same atoms with two different hyperfine states
mixture_exp_bb_1 ; mixture_exp_bb_2 . The DW potential can also be readily
constructed by imposing a 1D optical lattice on top of a harmonic trap
DW_exp_3 ; BJJ_2 . Moreover, the contact interaction strength $g_{IB}$ can be
controlled experimentally by tuning the $s$-wave scattering lengths via
Feshbach or confinement-induced resonances Feshbach_1 ; Feshbach_2 ;
Feshbach_3 .

Noticing further that the bosonic species is confined within a tight DW
potential with $\delta_{1}\gg\delta_{0}$ [c.f. Fig. 1 (a)], here $\delta_{i}$
denotes the energy difference between the $i$-th and the $(i+1)$-th single-
particle eigenstates. We adopt the two-mode approximation

$\hat{\psi}_{B}(x)=u_{L}(x)\hat{b}_{L}+u_{R}(x)\hat{b}_{R},$ (3)

with $u_{L,R}(x)$ being the Wannier-like states localized in the left and
right well, respectively. This leads to the low-energy effective Hamiltonian
for the bosonic species

$\hat{H}_{B}=-J_{0}(\hat{b}^{\dagger}_{L}\hat{b}_{R}+\hat{b}^{\dagger}_{R}\hat{b}_{L}),$
(4)

corresponding to the two-site Bose-Hubbard (BH) model with $J_{0}=0.071$ being
the hopping amplitude.

Before proceeding, it is instructive to express the above BH Hamiltonian in
the angular-momentum representation. To see this, we introduce three angular-
momentum operators BJJ_Rabi_3 ; BJJ_4

$\displaystyle\hat{J}_{x}$
$\displaystyle=\frac{1}{2}(\hat{b}_{L}^{\dagger}\hat{b}_{R}+\hat{b}_{R}^{\dagger}\hat{b}_{L}),~{}~{}~{}\hat{J}_{y}=-\frac{i}{2}(\hat{b}_{L}^{\dagger}\hat{b}_{R}-\hat{b}_{R}^{\dagger}\hat{b}_{L}),$
$\displaystyle\hat{J}_{z}$
$\displaystyle=\frac{1}{2}(\hat{b}_{L}^{\dagger}\hat{b}_{L}-\hat{b}_{R}^{\dagger}\hat{b}_{R}),$
(5)

obeying the SU(2) commutation relation
$[\hat{J}_{\alpha},\hat{J}_{\beta}]=i\epsilon_{\alpha\beta\gamma}\hat{J}_{\gamma}$.
The BH Hamiltonian in Eq. (4) thus can be rewritten as

$\hat{H}_{B}=-2J_{0}\hat{J}_{x},$ (6)

which describes the angular momentum precession of a single particle whose
spatial degrees of freedom (DOFs) are frozen. According to definitions for
$\hat{J}_{x}$ and $\hat{J}_{z}$ in Eq. (5), we note that the kinetic energy in
the BH model as well as the population imbalance for the bosons between the
two wells are in analogy to the magnetizations of this single particle along
the $x$ and the $z$ axes. Moreover, the particle number conservation in the
Hamiltonian (4) corresponds to the angular momentum conservation

$\hat{J}^{2}=\hat{J}_{x}^{2}+\hat{J}_{y}^{2}+\hat{J}_{z}^{2}=\frac{N_{B}}{2}(\frac{N_{B}}{2}+1)$
(7)

for the Hamiltonian (6).

For the case $g_{IB}=0$, the angular momentum dynamics can be simply
integrated out from the corresponding Heisenberg equations of motion, in which

$\displaystyle\hat{J}_{y}(t)$
$\displaystyle=\hat{J}_{z}(0)\text{cos}(2J_{0}t)-\hat{J}_{y}(0)\text{sin}(2J_{0}t),$
$\displaystyle\hat{J}_{z}(t)$
$\displaystyle=\hat{J}_{y}(0)\text{cos}(2J_{0}t)+\hat{J}_{z}(0)\text{sin}(2J_{0}t),$
(8)

being the harmonic oscillations with the frequency $\omega_{0}=2J_{0}$ and
$\hat{J}_{x}(t)=\hat{J}_{x}(0)$ is time-independent since
$[\hat{J}_{x},\hat{H}_{B}]=0$. Further introducing the normalized vector
$\hat{\vec{S}}(t)=\hat{S}_{x}(t)\vec{i}+\hat{S}_{y}(t)\vec{j}+\hat{S}_{z}(t)\vec{k}$
with $\hat{S}_{\gamma}(t)=\hat{J}_{\gamma}(t)/J$ for $\gamma=x,y,z$ and
$J=N_{B}/2$, together with the fact that

$\sum_{\gamma=x,y,z}\langle\hat{S}_{\gamma}\rangle^{2}(t)=\sum_{\gamma=x,y,z}\langle\hat{S}_{\gamma}\rangle^{2}(0),$
(9)

one can readily show that the motion of the vector $\hat{\vec{S}}(t)$ always
lies on the Bloch sphere with unit radius if, in addition, we choose the
initial state as the atomic coherent state (ACS) (see below).

###  II.2 Classical dynamics

The above angular momentum dynamics can alternatively be understood in a
classical manner. As we will show below, the periodic motions for
$\hat{J}_{y}(t)$ and $\hat{J}_{z}(t)$ [equivalently $\hat{S}_{y}(t)$ and
$\hat{S}_{z}(t)$] correspond to the periodic oscillation of a classical non-
rigid pendulum around its equilibrium position, while the conservation of
$\hat{J}_{x}(t)$ [$\hat{S}_{x}(t)$] relates to the energy conservation of this
pendulum. To this end, we first adopt the mean-field approximation as
$\hat{b}_{\beta}=b_{\beta}$ ($\beta=L,R$) with $b_{\beta}$ being a $c$-number
GPE_1 . The quantum operators $\hat{S}_{x}$, $\hat{S}_{y}$ and $\hat{S}_{z}$
then should be rewritten as

$\displaystyle S_{x}$
$\displaystyle=\frac{1}{2J}(b_{L}^{\ast}b_{R}+b_{R}^{\ast}b_{L}),~{}~{}~{}S_{y}=-\frac{i}{2J}(b_{L}^{\ast}b_{R}-b_{R}^{\ast}b_{L}),$
$\displaystyle S_{z}$
$\displaystyle=\frac{1}{2J}(b_{L}^{\ast}b_{L}-b_{R}^{\ast}b_{R}).$ (10)

Employing the phase-density representation for $b_{\beta}$ as
$b_{\beta}=\sqrt{N_{\beta}^{B}}e^{i\theta_{\beta}}$ and further introducing
the two conjugate variables

$Z=(N_{L}^{B}-N_{R}^{B})/N_{B},~{}~{}~{}~{}~{}~{}\varphi=\theta_{R}-\theta_{L},$
(11)

representing the relative population imbalance between the two wells and the
relative phase difference, respectively, we arrive at

$S_{x}=\sqrt{1-Z^{2}}\text{cos}\varphi,~{}~{}~{}S_{y}=\sqrt{1-Z^{2}}\text{sin}\varphi,~{}~{}~{}S_{z}=Z,$
(12)

whose dynamics are governed by the Hamiltonian

$H_{cl}=-J_{0}\sqrt{1-Z^{2}}\text{cos}\varphi,$ (13)

which, as aforementioned, describes a non-rigid pendulum with angular momentum
$Z$ whose length is proportional to $\sqrt{1-Z^{2}}$ BJJ_Rabi_1 ; BJJ_Rabi_2 ;
BJJ_Rabi_3 ; BJJ_Driven_1 . Comparing the Eq. (12) to the Eq. (13), we note
that $S_{y}$ and $S_{z}$, being the classical counterpart of the quantum
operators $\hat{S}_{y}$ and $\hat{S}_{z}$, now represent the horizontal
displacement and the angular momentum of this classical pendulum, while the
$S_{x}$ ($\hat{S}_{x}$) proportions to its total energy which is conserved
during the dynamics.

In this way, an one-to-one correspondence between the quantum and classical
dynamics is established in which the periodic motions for $\hat{S}_{y}(t)$ and
$\hat{S}_{z}(t)$ are mapped to the periodic oscillations for this classical
pendulum around its equilibrium position. Since our focus is put on the
dynamics of the population imbalance of the bosons, we compare the quantum
evolution $\hat{S}_{z}(t)$ for the case $g_{IB}=0$ to the classical dynamics
$Z(t)$ in Fig. 1 (b) and no discrepancies are observed among them. Hence, for
the case $g_{IB}=0$, we will always refer the classical $Z(t)$ dynamics as the
quantum $S_{z}(t)$ evolutions. However, it should also be emphasized that the
agreement between $\hat{S}_{z}(t)$ and $Z(t)$ takes place only for this non-
interacting case. For $g_{IB}>0$, on one side, the mixture has no classical
mapping, on the other side, the quantum correlations among the bosons come
into play, and, as a result, one can witness even a completely different
quantum dynamics as compared to the classical one, albeit the fact that the
bare Bose-Bose interaction always vanishes (see below).

The above classical interpretation provides us not only with a vivid picture
for visualizing the quantum dynamics in a classical manner, but also with the
profound physical insights with respect to its overall dynamical properties.
In particular, the periodic motions for $\hat{S}_{y}(t)$ and $\hat{S}_{z}(t)$
obtained from the quantum simulations are a direct consequence of the
integrability of the classical Hamiltonian $H_{cl}$. Owing to the energy
conservation for the case $g_{IB}=0$, $H_{cl}$ is completely integrable with
all the corresponding classical trajectories, characterized by
$[Z(t),\varphi(t)]$, being periodic in time PS . Such an integrability is also
transparently shown in the classical phase space [see Fig. 1 (c)]. Depending
on the initial condition, two distinguished types of motions are clearly
observed: a periodic trajectory orbiting around the fix point either located
at $(Z=0,\varphi=0)$ or $(Z=0,\varphi=\pi)$, referred as the zero- and the
$\pi$-phase mode for a 1D BJJ BJJ_2 .

Figure 1: (Color online) (a) Single-particle spectrum for the double-well
potential, in which the gray horizontal lines denote the lowest six energy
levels and the blue (red) arrows represent possible transitions that reverse
(preserve) the spatial parity of the impurity. (b) Real-time dynamics for the
bosonic population imbalance $S_{z}(t)$ for the initial state
$|\Psi(0)\rangle=|\phi_{0}\rangle\otimes|\pi/2,\pi/4\rangle$ and for the case
$g_{IB}=0$ (red solid line), together with the classical $Z(t)$ dynamics
starting from the phase point $(Z=0,\varphi=\pi/4)$ (blue dashed line). (c)
Classical phase space for $J_{0}=0.071$. The red dot denotes the phase space
point ($Z=0,\varphi=\pi/4$) corresponding to the ACS
$|\theta,\varphi\rangle=|\pi/2,\pi/4\rangle$.

###  II.3 Breaking of the integrability

In contrast to the above integrable limit, the presence of the impurity-Bose
interaction leads to the energy transport between the two species and, hence,
breaks the integrability for the bosonic species. In order to elaborate on
this process in more detail, we decompose the interspecies interaction into
various impurity-boson pair excitations

$\hat{H}_{IB}=\sum_{i,j=0}^{\infty}\sum_{\alpha,\beta=L,R}U_{ij\alpha\beta}\hat{a}_{i}^{\dagger}\hat{a}_{j}\hat{b}_{\alpha}^{\dagger}\hat{b}_{\beta},$
(14)

with $U_{ij\alpha\beta}=g_{IB}\int
dx~{}\phi_{i}(x)\phi_{j}(x)u_{\alpha}(x)u_{\beta}(x)$ and $\\{\phi_{i}(x)\\}$
being the single-particle basis for the DW potential. Moreover, $u_{L/R}(x)$,
being the above mentioned localized Wannier-like states, are constructed via a
linear superposition of the lowest two eigenstates $\phi_{0}(x)$ and
$\phi_{1}(x)$. Note that Eq. (14) is obtained by means of an expansion of the
field operator for the impurity
$\hat{\psi}_{I}(x)=\sum_{i=0}^{\infty}\phi_{i}(x)\hat{a}_{i}$, meanwhile, by
employing the two-mode approximation in Eq. (3) for the bosonic species.
Besides, all the eigenstate wavefunctions $\\{\phi_{i}(x)\\}$ are chosen to be
real due to the preserved time-reversal symmetry in the single-particle
Hamiltonian $\textit{h}_{\sigma}$.

Next, we group different pair excitations with respect to their bosonic
indices as

$\displaystyle\hat{H}_{IB}$
$\displaystyle=\left[\sum_{i,j=0}^{\infty}U_{ijLR}\hat{a}_{i}^{\dagger}\hat{a}_{j}\hat{b}_{L}^{\dagger}\hat{b}_{R}+\sum_{i,j=0}^{\infty}U_{ijRL}\hat{a}_{i}^{\dagger}\hat{a}_{j}\hat{b}_{R}^{\dagger}\hat{b}_{L}\right]$
$\displaystyle+\left[\sum_{i,j=0}^{\infty}U_{ijLL}\hat{a}_{i}^{\dagger}\hat{a}_{j}\hat{b}_{L}^{\dagger}\hat{b}_{L}+\sum_{i,j=0}^{\infty}U_{ijRR}\hat{a}_{i}^{\dagger}\hat{a}_{j}\hat{b}_{R}^{\dagger}\hat{b}_{R}\right].$

By noticing the fact that

$U_{ijLR}=U_{ijRL},~{}~{}U_{ijLL}=\eta U_{ijRR}$ (16)

with $\eta=1$ ($\eta=-1$) for $n_{e,o}=|i-j|$ being an even (odd) number,
together with the definitions given in Eq. (5), we finally arrive at

$\displaystyle\hat{H}_{IB}$
$\displaystyle=\left(\sum_{i,j=0}^{\infty}U_{ij}^{(1)}\hat{a}_{i}^{\dagger}\hat{a}_{j}\right)2\hat{J}_{x}+\left(\sum_{|i-j|=n_{e}}^{\infty}U_{ij}^{(2)}\hat{a}_{i}^{\dagger}\hat{a}_{j}\right)\hat{N}_{B}$
$\displaystyle+\left(\sum_{|i-j|=n_{o}}^{\infty}U_{ij}^{(3)}\hat{a}_{i}^{\dagger}\hat{a}_{j}\right)2\hat{J}_{z}$
$\displaystyle=\hat{H}_{IB}^{(1)}+\hat{H}_{IB}^{(2)}+\hat{H}_{IB}^{(3)}.$ (17)

Here $U_{ij}^{(1)}=U_{ijLR}=U_{ijRL}$, $U_{ij}^{(2)}=U_{ijLL}=U_{ijRR}$ and
$U_{ij}^{(3)}=U_{ijLL}=-U_{ijRR}$. Let us emphasize that the Eq. (16) relies
on the fact that the DW potential is spatially symmetric, as a result, all its
single-particle eigenstates $\\{\phi_{i}\\}$ respect the spatial parity
symmetry.

Equation (17) transparently elaborates how the interspecies interaction
$\hat{H}_{IB}$ breaks the integrability for the Hamiltonian $\hat{H}_{B}$.
Since both $\hat{H}_{IB}^{(1)}$ and $\hat{H}_{IB}^{(2)}$ commute with
$\hat{H}_{B}$ [c.f. Eq. (6)], it is the non-commutativity between
$\hat{H}_{IB}^{(3)}$ and $\hat{H}_{B}$ that results in the energy non-
conservation for the bosonic species, and breaks its integrability for
$g_{IB}=0$. Further inspecting the $\hat{H}_{IB}^{(3)}$ term in more detail,
we notice that it corresponds to all the different single-particle excitations
that reverse the impurity’s spatial parity [see Fig. 1 (a) for a schematic
illustration]. With this, we conclude that those parity non-conservation
transitions of the impurity leads to the integrability breaking for the
majority bosonic species.

###  II.4 Initial condition

We prepare our impurity-Bose mixture initially as
$|\Psi(0)\rangle=|\phi_{n}\rangle\otimes|\theta,\varphi\rangle$, being a
product state between the two species. Here $|\phi_{n}\rangle$ is the $n$-th
single-particle eigenstate for the impurity and $|\theta,\varphi\rangle$
denotes the ACS given by ACS_1 ; ACS_2

$\displaystyle|\theta,\varphi\rangle$
$\displaystyle=\frac{1}{\sqrt{N_{B}!}}\left[\text{cos}(\frac{\theta}{2})\hat{b}^{\dagger}_{L}+\text{sin}(\frac{\theta}{2})e^{i\varphi}\hat{b}^{\dagger}_{R}\right]^{N_{B}}~{}|vac\rangle$
$\displaystyle=\sum_{N^{B}_{L}=0}^{N_{B}}\left(\begin{array}[]{c}N_{B}\\\
N^{B}_{L}\end{array}\right)^{1/2}\text{cos}^{N^{B}_{L}}(\theta/2)~{}\text{sin}^{N^{B}_{R}}(\theta/2)~{}e^{iN^{B}_{R}\varphi}~{}|N^{B}_{L},N^{B}_{R}\rangle,$
(20)

which is the linear superposition of all the number states
$\\{|N^{B}_{L},N^{B}_{R}\rangle\\}$ and fulfills the completeness relation

$(N_{B}+1)\int\frac{d\Omega}{4\pi}|\theta,\varphi\rangle\langle\theta,\varphi|=1$
(21)

with $d\Omega=\text{sin}\theta d\theta d\varphi$ being the volume element.
Physically, the ACS $|\theta,\varphi\rangle$ corresponds to the classical
state $(Z,\varphi)$ in such a way that
$\text{cos}\theta=(N^{B}_{L}-N^{B}_{R})/N_{B}=Z$ controls the initial
population difference for the bosons and $\varphi$, possessing the same
meaning with its classical counterpart, determines the phase difference
between the two wells BJJ_4 . For a given ACS $|\theta,\varphi\rangle$, the
mean values for the angular-momentum operators introduced in Eq. (5) are
BJJ_Rabi_3

$\langle\hat{S}_{x}\rangle=\text{sin}\theta\text{cos}\varphi,~{}~{}~{}\langle\hat{S}_{y}\rangle=\text{sin}\theta\text{sin}\varphi,~{}~{}~{}\langle\hat{S}_{z}\rangle=\text{cos}\theta,$
(22)

which satisfies the normalization condition
$\langle\hat{S}_{x}\rangle^{2}+\langle\hat{S}_{y}\rangle^{2}+\langle\hat{S}_{z}\rangle^{2}=1$.
Together with the Eqs. (8) and (9), we conclude that, for the case $g_{IB}=0$,
the motion of the $\hat{\vec{S}}(t)$ vector starting from an arbitrary ACS
always lies on a Bloch sphere with unit radius. Even for the case $g_{IB}>0$,
where the vector $\hat{\vec{S}}(t)$ can jump out of the Bloch sphere
significantly, the use of the ACS still allows us to visualize the quantum
trajectory in a classical manner (see below), which simplifies the analysis of
the complex quantum dynamics to a large extent, meanwhile, provides insights
for the classical-quantum correspondence. Finally, let us note that the ACS
has been implemented in recent ultracold experiments in a controllable manner.
Tuning a two-photon transition between two hyperfine states of
${}^{87}\textrm{Rb}$ atoms, allows for preparing an ACS with arbitrary
$|\theta,\varphi\rangle$ ACS_3 ; ACS_4 .

In this paper, we aim at exploring the dynamical response of the majority
bosonic species to the presence of the impurity. To this end, we quench at
$t=0$ the impurity-Bose interaction strength from initial $g_{IB}=0$ to some
finite value $g_{IB}>0$, and monitor the quantum evolution of the bosonic
population imbalance starting from a balanced population. While the initial
state for the mixture is
$|\Psi(0)\rangle=|\phi_{n}\rangle\otimes|\theta,\varphi\rangle$, without other
specifications, we always choose the bosonic part being
$|\theta,\varphi\rangle=|\pi/2,\pi/4\rangle$. The corresponding $S_{z}(t)$
dynamics for this initial ACS and for the case $g_{IB}=0$ has been detailed
discussed above and is presented in Fig. 1 (b) (red solid line). Furthermore,
we also consider the scenarios for various initial impurity states
$|\phi_{n}\rangle$, so as to explore its impact on the bosonic dynamics.

##  III Bosonic ITC transition

###  III.1 Onset of quantum chaos

Let us first focus on the case where the impurity is initially prepared in its
ground state. The many-body initial state for the mixture is then given by
$|\Psi(0)\rangle=|\phi_{0}\rangle\otimes|\pi/2,\pi/4\rangle$. Fig. 2 depicts
the real-time population imbalance for the bosonic species $S_{z}(t)$ for
various fixed postquench impurity-Bose interaction strengths $g_{IB}=0.01$
[Fig. 2 (a)], $g_{IB}=0.1$ [Fig. 2 (b)] and $g_{IB}=1.0$ [Fig. 2 (c)],
together with the classical $Z(t)$ dynamics (all blue dashed lines) which, as
aforementioned, equivalents to the $S_{z}(t)$ for $g_{IB}=0$. For a weak
impurity-Bose interaction ($g_{IB}=0.01$), the $S_{z}(t)$ dynamics is only
slightly perturbed by the presence of the impurity, as a result, it leads to
the small deviations of the population imbalance between the quantum and the
classical simulations [c.f. Fig. 2 (a), red solid line and blue dashed line].
For a larger time scale ($t>5000$), a “collapse-and-revival” behavior for
$S_{z}(t)$ is observed (result is not shown here), manifesting its near
integrability in this weak interacting regime BJJ_Rabi_3 ; BJJ_Driven_1 .
Further increasing the interaction strength, the quantum $S_{z}(t)$ evolution
becomes much more complicated and large discrepancies between $S_{z}(t)$ and
${Z}(t)$ are observed with respect to both the oscillation amplitude and the
frequencies. For the case $g_{IB}=1.0$, the quantum $S_{z}(t)$ dynamics
finally becomes completely irregular [c.f. Fig. 2 (c), red solid line],
signifying the onset of quantum chaos for the bosonic species.

In order to diagnose such an ITC transition, meanwhile, to quantify the degree
of the above observed quantum chaos, we employ the linear entanglement entropy
(EE)

$S_{L}=1-\text{tr}\hat{\rho}_{1B}^{2}$ (23)

for the bosonic species, which represents the bipartite entropy between the
single boson and the $N_{B}-1$ bosons after tracing out the impurity QKT_7 ;
QKT_8 . Here $\hat{\rho}_{1B}$ stands for the reduced one-body density matrix
for the bosonic species dma1_1 ; dma1_2 ; BJJ_4 . Before proceeding, let us
point out the reason for not using the spectral statistics as an indicator for
the quantum chaos. Similar to the situation for a single particle in a 1D
harmonic trap, the single DOF of the Hamiltonian $\hat{H}_{B}$ for a fixed
particle number violates the Berry-Tabor conjecture, which states that the
energy level spacing distribution follows the universal Poisson form for an
integrable system level_1 ; level_2 ; level_3 . As a result, the variation of
the level distribution for our mixture upon the increase of $g_{IB}$ can
behave largely different as compared to other systems level_3 , and hence, it
is insufficient to capture the quantum chaos. Upon a spectral decomposition of
the reduced density matrix $\hat{\rho}_{1B}$, $S_{L}$ in Eq. (23) can be
expressed, with respect to the natural populations $\\{n_{1},n_{2}\\}$, as
$S_{L}=1-\sum_{i=1}^{2}n^{2}_{i}$. In this way, the linear EE alternatively
measures the degree of the decoherence for the bosonic species. Note that the
two-mode expansion employed in the Eq. (3) renders the single-particle
Hamiltonian $\textit{h}(x)$ being restricted to a two-dimensional Hilbert
space and thus gives rise to only two natural populations obtained from the
spectral decomposition BJJ_4 . For the case where all the bosons reside in the
same single-particle state, the bosonic species is of complete coherence, as a
result, we have $S_{L}=0$. By contrast, for the case of maximal decoherence we
have $n_{1}=n_{2}=1/2$, which gives rise to the upper bound for the linear EE
as $S_{L}=1/2$.

The linear EE has been extensively used in the QKT systems as a signature of
the quantum chaos QKT_7 ; QKT_8 . Depending on whether the corresponding
classical trajectory is regular or chaotic, the linear EE behaves as either as
rapidly growing or a slowly varying in a short time (referred to as the
Ehrenfest time). On the other hand, the infinite-time averaged values of the
linear EE for various initial ACSs additionally characterize the edge of the
quantum chaos, denoted as the border between the integrable and the chaotic
region in the corresponding classical phase space QKT_7 ; QKT_8 . Fig. 3 (a)
reports the transient dynamics of the linear EE for the cases examined in Fig.
2 (a-c). At short times ($t<200$), the $S_{L}(t)$ evolution for a stronger
interaction exhibits a more rapid growth as compared to the cases with a
smaller $g_{IB}$. This is particularly obvious for the case $g_{IB}=1.0$,
where we observe the linear EE surges to the value $S_{L}=0.38$ at $t=10$,
while it only reaches to $S_{L}=0.02$ ($S_{L}=0.0007$) for the case
$g_{IB}=0.1$ ($g_{IB}=0.01$). With this knowledge, we conclude that the
different transient dynamical behaviors of the linear EE fully capture the ITC
transition that is observed in the dynamics of the bosonic population
imbalance. Besides, we shall also note that the linear EE for $t=0$ trivially
vanishes since all the bosons are initially condensed into the same single-
particle state [c.f. Eq. (20)].

Having investigated the transient dynamics of the linear EE for a specific
ACS, let us now explore its asymptotic behaviors with respect to different
ACSs, which, as aforementioned, characterize the edge of the quantum chaos. To
this end, we compute the infinite-time averaged value of the linear EE (ITEE)
for the initial state
$|\Psi(0)\rangle=|\phi_{0}\rangle\otimes|\theta,\varphi\rangle$,

$\overline{S}_{L}(\theta,\varphi)=lim_{T\rightarrow\infty}~{}\frac{1}{T}\int_{0}^{T}dt~{}S_{L}(t).$
(24)

Note that, the impurity initially always occupies the ground state
$|\phi_{0}\rangle$ and in our practical numerical simulations the time average
is performed up to $t=10^{4}$, being much larger than any other time scales
involved in the dynamics. Before proceeding, let us point out the geometrical
interpretation of the ITEE value. To show it, we first of all rewrite the
linear EE in Eq. (23) for time $t$ as QKT_7 ; QKT_8

$S_{L}(t)=\frac{1}{2}\left[1-\sum_{\gamma=x,y,z}\langle\hat{S}_{\gamma}\rangle^{2}(t)\right],$
(25)

where we have used the relation

$\hat{\rho}_{1B}=\frac{1}{2}\left[1+\sum_{\gamma=x,y,z}\langle\hat{S}_{\gamma}\rangle\hat{\sigma}_{\gamma}\right],$
(26)

with $\\{\hat{\sigma}_{\gamma}\\}$ being the Pauli matrices. Since $S_{L}(t)$
is proportional to the instant distance of the vector $\hat{\vec{S}}(t)$ to
the Bloch sphere, $\overline{S}_{L}$ thus measures its averaged distance for
the entire dynamics. From the results in the QKT systems QKT_7 ; QKT_8 , we
note that there exists a clear correspondence between the ITEE values and the
classical phase space structure. Regions of low ITEE correspond to regular
trajectories, while regions of high EE correspond to the chaotic trajectories.
Moreover, a sudden change of the ITEE value takes place as one crosses the
border between the integrable and the chaotic region, which, as
aforementioned, characterizes the edge of the quantum chaos. Fig. 3 (b)
depicts the computed ITEE values for various ACSs for the case $g_{IB}=1.0$.
Note that, we have rescaled the $\theta$ axis to $\text{cos}\theta$ since
$\text{cos}\theta=Z$ [see discussions in Sec. II.4]. Varying the initial ACS,
the ITEE value varies accordingly. In particular, regions close to
($\text{cos}\theta=0,\varphi=\pi$) and ($\text{cos}\theta=\pm
0.8,\varphi=0,2\pi$) possess significant low ITEE values as compared to the
other places. Such a $\overline{S}_{L}(\theta,\varphi)$ profile significantly
deviates from the structure of the non-interacting classical phase space.
Instead, it bears a striking resemblance to the phase space with an attractive
Bose-Bose interaction with the positions for those fixed points precisely
match with those low ITEE regions [c.f. Fig. 3 (c), red stars]. Hence, we note
that it indicates an effective Bose-Bose attraction existing among the bosons.
In Sec. IV.3, we will discuss this induced interaction in detail as well as
its impact on the bosonic dynamics.

Figure 2: (Color online) Time evolution of the bosonic population imbalance
$S_{z}(t)$ (red solid lines) for the initial state
$|\Psi(0)\rangle=|\phi_{0}\rangle\otimes|\pi/2,\pi/4\rangle$ and for various
fixed impurity-Bose interaction strengths, in which (a) $g_{IB}=0.01$, (b)
$g_{IB}=0.1$ and (c) $g_{IB}=1.0$. For comparisons, the classical $Z(t)$
dynamcis is depicted as well (all blue dashed lines). Figure 3: (Color online)
(a) The linear EE evolutions for the initial state
$|\Psi(0)\rangle=|\phi_{0}\rangle\otimes|\pi/2,\pi/4\rangle$ and for the post-
quench interaction strengths $g_{IB}=0.01$ (red solid line), $g_{IB}=0.1$
(green solid line) and $g_{IB}=1.0$ (blue solid line). (b) Infinite-time
averaged values for the linear EE for $g_{IB}=1.0$ and for various ACSs. (c) A
typical classical phase space for the BJJ with an attractive on-site
interaction, where the red stars denote the corresponding classical fixed
points.

###  III.2 Recovery of the integrability

In this section, we investigate the scenario where the impurity is initially
pumped into a highly excited state. The out-of-equilibrium dynamics again is
triggered by a sudden quench of the impurity-Bose interaction strength. Here,
our main aim is to show that the integrability of the bosonic species is
recovered by means of preparing the impurity in a highly excited state. The
initial condition of the impurity, therefore, provides an additional DOF for
controlling the ITC transition of the majority bosonic species. Here, we note
that the employed notion of “integrability” specifically refers to how close
the bosonic dynamics in the interacting cases ($g_{IB}>0$) is to the one in
the non-interacting integrable case ($g_{IB}=0$), which is different from the
commonly used context in which it is uniquely associated to the system’s
Hamiltonian.

For an illustrative purpose, we consider the impurity is initially at
$|\phi_{150}\rangle$, being the 150-th excited state, and focus on the case
for the post-quench interaction strength $g_{IB}=1.0$. The many-body state for
$t=0$ is again given by
$|\Psi(0)\rangle=|\phi_{150}\rangle\otimes|\pi/2,\pi/4\rangle$. The
corresponding quantum evolution of the bosonic population imbalance $S_{z}(t)$
is depicted in Fig. 4 (a) (red solid line). As compared to the classical
$Z(t)$ dynamics [Fig. 4 (a), blue dashed line], we find a good agreement
between them with negligible discrepancies. Interestingly, these discrepancies
are even much smaller than the ones between $S_{z}(t)$ and $Z(t)$ for the case
$g_{IB}=0.01$ [c.f. Fig. 2 (a)]. Besides, we also note that the negligible
increment of the corresponding linear EE in the course of the dynamics
alternatively signifies the recovery of the integrability for the bosonic
species [c.f. Fig. 4 (b)].

Figure 4: (Color online) Time evolution of the bosonic population imbalance
$S_{z}(t)$ for $g_{IB}=1.0$ and for the initial state
$|\Psi(0)\rangle=|\phi_{150}\rangle\otimes|\pi/2,\pi/4\rangle$ (red solid
line), together with the classical $Z(t)$ dynamics (blue dashed line) which
corresponds to the $S_{z}(t)$ dynamics for $g_{IB}=0$. (b) The evolution of
the linear EE for the corresponding case.

##  IV Spectral analysis and induced interaction

In order to shed light on the physics for the above-analyzed bosonic dynamics,
hereafter, we perform a detailed spectral analysis for the mixture with
respect to both the energy spectrum and the eigenstates via a numerically
exact diagonalization (ED). In particular, we would like to unveil the
physical origin for the observed ITC transition for the bosonic species
manifested by the corresponding dynamics of the population imbalance.
Moreover, we will discuss the presented Bose-Bose attraction induced by the
impurity as well as its impact on the bosonic dynamics.

###  IV.1 Spectral structure

Let us begin with the case for $g_{IB}=0$. In the absence of the interspecies
interaction, the two species are completely decoupled. As a result, the
eigenenergy of the mixture is trivially given by
$E=\epsilon_{k}+\epsilon^{B}_{l}$ with $\epsilon_{k}$ and $\epsilon^{B}_{l}$
being the $k$-th and $l$-th eigenvalue for the subsystem Hamiltonians
$\hat{H}_{I}$ and $\hat{H}_{B}$, respectively. Owing to the neglected Bose-
Bose interaction, the many-body spectrum for $\hat{H}_{B}$ is always
equidistant with the energy difference $2J_{0}$ between the two successive
levels, which accounts for the harmonic oscillation of the $S_{z}(t)$ dynamics
for the case $g_{IB}=0$ [c.f. Fig. 1 (b)]. As for the impurity, due to the
rapid growth of the energy difference between two successive eigenstates, the
single-particle spectrum is inhomogeneous in which the high-energy part is
much more sparse as compared to the low-energy one [c.f. Fig. 1 (a)]. An
important consequence for such a spectral structure on the mixture’s many-body
spectrum is the following. For $\delta_{i}>\Delta_{B}$, with
$\delta_{i}=\epsilon_{i+1}-\epsilon_{i}$ being the energy difference between
the $i$-th and the $(i+1)$-th single-particle eigenstates for the DW potential
(see also the discussions in Sec. II.1), and $\Delta_{B}$ representing the
width of the spectrum for the Hamiltonian $\hat{H}_{B}$, a band-like structure
is naturally formed in the high-energy part of the many-body spectrum with the
band gap being $\delta_{i}-\Delta_{B}$, meanwhile, the energy levels within
each band are equidistant.

This simple picture, however, ceases to be valid upon the variation of the
impurity-Bose interaction. Indeed, the inclusion of the interspecies
interaction introduces additional coupling between the two subsystems and, as
a result, our spectral analysis needs to be performed with respect to the
complete mixture. Figure 5 showcases the many-body spectrum as a function of
the interspecies interaction strength $g_{IB}$. Owing to the preserved spatial
parity symmetry in the Hamiltonian $\hat{H}$, we present here only half of the
spectrum which corresponds to the even parity eigenstates. With the increase
of $g_{IB}$, the low-energy spectrum shows many avoided-crossings among the
energy levels, which is in sharp contrast to the high-energy spectrum where
only a linear growth of their values is observed [c.f. Figs. 5 (a) and (c)].
Moreover, for the high-energy spectrum, features like the band-like structure
as well as the equidistant energy levels within each band that are present in
the non-interacting limit are retained in the interacting cases as well.

The above two distinguished spectral behaviors can roughly be understood via
the structure of the impurity’s single-particle spectrum [c.f. Fig. 1 (a)].
Owing to the large energy separations among those highly excited states, the
transitions for the impurity among those states are significantly prohibited.
From a many-body perspective, the resulting high-energy effective Hamiltonian
of the mixture reads
$\hat{H}^{\prime}=\hat{H}_{I}+\hat{H}_{B}+\hat{H}_{IB}^{\prime}$, with

$\displaystyle\hat{H}_{I}$ $\displaystyle=\sum_{i\gg
1}\epsilon_{i}\hat{a}_{i}^{\dagger}\hat{a}_{i},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hat{H}_{B}=-2J_{0}\hat{J}_{x},$
$\displaystyle\hat{H}_{IB}^{\prime}$ $\displaystyle\approx\sum_{i\gg
1}2U_{i}^{(1)}\hat{J}_{x}+U_{i}^{(2)}\hat{N}_{B}\approx\sum_{i\gg
1}U_{i}^{(2)}\hat{N}_{B}.$ (27)

Here $U_{i}^{(1)}=U_{iiLR}=U_{iiRL}$, $U_{i}^{(2)}=U_{iiLL}=U_{iiRR}$ and we
notice that $U_{i}^{(1)}=g_{IB}\int
dx~{}\phi_{i}(x)\phi_{i}(x)u_{L}(x)u_{R}(x)\approx 0$, due to the negligible
spatial overlap between the two localized states $u_{L}(x)$ and $u_{R}(x)$.
Before proceeding, we note the validity condition for the above high-energy
effective Hamiltonian as: $\delta_{i}\gg\epsilon_{IB}$ and
$\delta_{i}\gg\Delta_{B}$ with $\epsilon_{IB}$ being the interspecies
interaction energy per particle. Eq. (27) explains the observed high-energy
spectral behaviors as follows: since the interspecies interaction
$\hat{H}_{IB}^{\prime}$ now becomes the “zero-point” energy of the mixture,
the increment of the $g_{IB}$ thus only raises the energy level for those
highly excited states. As a result, the band-like structure as well as
equidistant nature that are formed in the non-interacting case are naturally
preserved.

In contrast, the densely distributed low-energy (single-particle) spectrum of
the impurity facilitates the transitions among different (low-lying) many-body
eigenstates caused by the interspecies interaction $\hat{H}_{IB}$ [c.f. Eq.
(17)]. With increasing $g_{IB}$, this results in the above observed avoided
level-crossings among the low-energy many-body spectrum QKT_1 .

###  IV.2 Eigenstate delocalization

The avoided level-crossings in the low-energy spectrum impact the
characteristics of the corresponding eigenstates as well. Specifically, it
results in a significant delocalization for those low-lying eigenvectors with
respect to an integrable basis (see below), which, in turn, accounts for the
chaotic nature of the bosonic non-equilibrium dynamics. To demonstrate this,
we introduce the Shannon entropy

$S^{S}_{j}=-\sum_{k}c_{j}^{k}\ln{c_{j}^{k}}$ (28)

for a many-body eigenstate $|\Phi_{j}\rangle$ of the mixture as a measure of
the delocalization IPR_1 ; IPR_2 . Here
$c_{j}^{k}=|\langle\psi_{k}|\Phi_{j}\rangle|^{2}$ with
$\\{|\psi_{k}\rangle\\}$ being the eigenbasis for the Hamiltonian
$\hat{H}_{B}$ that are used as the “integrable basis”. The Shanon entropy
thereby measures the number of this integrable basis vectors that contribute
to each eigenstate. As a result, the lower the Shanon entropy value is the
closer this eigenstate $|\Phi_{j}\rangle$ is to a non-interacting eigenvector.
From the random matrix theory (RMT), for a chaotic system described by the
gaussian orthogonal ensemble (GOE), the amplitudes $c_{j}^{k}$ are independent
random variables and all eigenstates are completely delocalized QKT_1 .
However, due to the spectral fluctuations the weights $\\{c_{j}^{k}\\}$
fluctuate around $1/D$, yielding the averaged value
$S_{\text{GOE}}=\ln{(0.48D)}$ IPR_1 ; IPR_2 . Here, we refer to $D=N_{B}+1$ as
the Hilbert space dimension for the bosonic species, which is different from
the single-species cases IPR_1 ; IPR_2 .

Figure 6 (a) presents the Shannon entropy of the many-body eigenstates as a
function of their quantum numbers $j$ (sorted in the ascending order with
respect to the energy) for the case $g_{IB}=1.0$. The distinguished
localization nature between the low-lying and the highly excited eigenvectors
are clearly exhibited. While those low-energy eigenvectors are delocalized
with the corresponding Shannon entropy values close to the result from the GOE
$S_{\text{GOE}}=3.8812$, for increasing $j$, a decrease of the $S^{S}_{j}$
value is clearly observed, indicating those high-energy eigenvectors are
significantly localized. Thus, we may further conjecture that
$S^{S}_{j}\rightarrow 0$ for $j\rightarrow\infty$. Physically, the avoided
level-crossings in the low-energy spectrum results in a strong mixing of
different eigenstates with respect to their physical properties. In this way,
an eigenstate from the non-interacting basis can be largely delocalized after
experiencing a serious of avoided level-crossings QKT_1 . On the other hand,
the localized nature for those high-lying excited states can also be readily
seen from the effective Hamiltonian in Eq. (27). Since here
$\hat{H}_{IB}^{\prime}$ corresponds to the “zero-point ” energy of the
mixture, it is not surprising that the interacting basis (eigenstates of the
mixture for $g_{IB}>0$) is similar to the non-interacting integrable basis.

Before proceeding, let us highlight that the degree of the localization for an
eigenstate $|\Phi_{j}\rangle$ also reflects the degree of the encoded
entanglement between the impurity and the majority bosons. To see this, we
employ the von Neumann entropy for an eigenstate $|\Phi_{j}\rangle$ few_gs_7 ;
Schmidt ,

$S^{V}_{j}=-\text{tr}(\hat{\rho}_{j}\ln{\hat{\rho}_{j}})$ (29)

with $\hat{\rho}_{j}=|\Phi_{j}\rangle\langle\Phi_{j}|$ being the corresponding
density matrix. For the case where the two species are non-entangled, the
eigenstate $|\Phi_{j}\rangle$ is simply of a product form with respect to the
wavefunctions of the two species. Correspondingly, it gives rise to the von
Neumann entropy $S^{V}_{j}=0$. By contrast, any existing entanglement between
the two species will lead to an increase of the von Neumann entropy,
therefore, one can anticipate large $S^{V}_{j}$ values for those highly
entangled eigenstates. The corresponding von Neumann entropies for various
eigenstates for the case $g_{IB}=1.0$ are shown in Fig. 6 (b). As compared to
the Fig. 6 (a), a striking resemblance between the $S^{V}_{j}$ and $S^{S}_{j}$
distributions are transparently observed, manifesting the existence of the
correspondence between a delocalized (localized) eigenstate to a large (small)
von Neumann entropy value. Based on this knowledge, we refer to the above
eigenstate delocalization as the entanglement induced delocalization.

Finally, let us discuss the impact of the eigenstate delocalization to the
bosonic non-equilibrium dynamics. For the case
$|\Psi(0)\rangle=|\phi_{0}\rangle\otimes|\pi/2,\pi/4\rangle$, the initial
state is mainly a linear superposition of those low-lying eigenvectors for
both $g_{IB}=0$ and $g_{IB}=1.0$ [c.f. Fig. 6 (c), the left part]. Owing to
the delocalization nature for the eigenstates of the mixture for large
interspecies interactions, the expansion coefficients $\\{A_{j}^{1}\\}$ for
$g_{IB}=1.0$ are broadly distributed as compared to the ones
($\\{A_{j}^{0}\\}$) for $g_{IB}=0.0$, reflecting the fact that much more
eigenstates are involved in the bosonic dynamics. Since the energy levels for
the interacting case are no longer equidistant, it thus gives rise to the
completely irregular behaviors for the above $S_{z}(t)$ dynamics [c.f. Fig. 2
(c)]. In contrast, those highly excited states in the interacting basis
preserve the main features of the non-interacting basis, leaving a similar
distribution of the corresponding expansion coefficients [c.f. Fig. 6 (c), the
right part]. Together with the equidistant nature for those high-lying energy
levels, it thereby accounts for the integrable $S_{z}(t)$ motion for the
initial state $|\Psi(0)\rangle=|\phi_{150}\rangle\otimes|\pi/2,\pi/4\rangle$
and for the case $g_{IB}=1.0$.

Figure 5: (Color online) Energy spectrum of the mixture as a function of
impurity-Bose interaction strength $g_{IB}$. (a) High-energy part of the
spectrum, (b) A zoom-in view of the high-energy spectrum, (c) Low-energy part
of the spectrum, (d) A zoom-in view of the low-energy spectrum.  Figure 6:
(Color online) (a) Shannon entropy $S^{S}_{j}$ for the many-body eigenstates
as a function of quantum number $j$ for the case $g_{IB}=1.0$. The red dashed
line denotes the Shannon entropy from the GOE $S_{\text{GOE}}=3.8812$. (b) Von
Neumann entropy $S^{V}_{j}$ for the eigenstates for the case $g_{IB}=1.0$. (c)
Expansion coefficients $A_{j}=|\langle\Psi(0)|\Phi_{j}\rangle|^{2}$ with
respect to eigenstates for initial states
$|\Psi(0)\rangle=|\phi_{0}\rangle\otimes|\pi/2,\pi/4\rangle$ (left part) and
$|\Psi(0)\rangle=|\phi_{150}\rangle\otimes|\pi/2,\pi/4\rangle$ (right part)
and for the cases $g_{IB}=0.0$ (red solid line and are denoted as $A_{j}^{0}$)
and $g_{IB}=1.0$ (blue dashed line and are denoted as $A_{j}^{1}$).

###  IV.3 Induced Bose-Bose attraction

The presence of the impurity not only brings the bosonic species into the
chaotic regime, yielding an irregular behavior for the corresponding
$S_{z}(t)$ motion, but also fundamentally changes its dynamical properties. As
we will show below, the impurity effectively induces an attractive Bose-Bose
interaction, which, in turn, leads to a completely different quantum
trajectory as compared to the integrable case. To show it, we employ the time-
averaged Husimi distribution (TAHD) TAHD_1 ; BJJ_4 ; QKT_7

$\overline{Q}_{H}(\theta,\varphi)=lim_{T\rightarrow\infty}~{}\frac{1}{T}\int_{0}^{T}Q_{H}(\theta,\varphi,t)dt,$
(30)

with

$Q_{H}(\theta,\varphi,t)=\frac{N_{B}+1}{4\pi}\langle\theta,\varphi|\hat{\rho}_{B}(t)|\theta,\varphi\rangle,$
(31)

and $\hat{\rho}_{B}(t)$ being the reduced density matrix for the bosonic
species after tracing out the impurity. According to the Eq. (21),
$Q_{H}(\theta,\varphi,t)$ satisfies the normalization condition $\int
Q_{H}(\theta,\varphi,t)d\Omega=1$. Physically, the TAHD represents the
probability for the bosons to locate at a specific ACS
$|\theta,\varphi\rangle$ averaged over the entire dynamics, which, with
respect to its physical meaning, resembles to the probability density function
(PDF) for a classical trajectory. In this sense, we note that the TAHD
represents a quantum trajectory in an averaged manner.

The computed TAHD for the initial state
$|\Psi(0)\rangle=|\phi_{0}\rangle\otimes|\pi/2,\pi/4\rangle$ and for the case
$g_{IB}=0$ is depicted in Fig. 7 (a), together with the classical trajectory
governed by the Hamiltonian $H_{cl}$ and starting from the phase point
$(Z=0,\varphi=\pi/4)$ (black solid line). Compared to the classical
trajectory, we note that the TAHD profile fully captures its main
characteristic with those high $\overline{Q}_{H}(\theta,\varphi)$ regions
precisely matching the positions for this classical trajectory, which
additionally manifests the agreement between the quantum $S_{z}(t)$ and
classical $Z(t)$ dynamics for the case $g_{IB}=0$ [c.f. Fig. 1 (b)]. The TAHD
for $g_{IB}=1.0$, however, deviates from the non-interacting case
significantly and bears a striking resemblance to the classical trajectory
corresponding to the BH Hamiltonian in Eq. (4) with an on-site attraction
[c.f. Figs. 7 (b) and 3 (c)]. In this sense, we conjecture an effective Bose-
Bose attraction is induced by the impurity in the dynamics which, in turn,
alters the corresponding quantum trajectory.

This expectation is indeed confirmed by analyzing the pair-correlation
function few_gs_6 ; few_gs_7 ; GPE_1

$g_{2}(\alpha,\beta)=\frac{\rho_{2}^{B}(\alpha,\beta)}{\rho_{1}^{B}(\alpha)\rho_{1}^{B}(\beta)},$
(32)

for the bosons, with $\rho_{2}^{B}(\alpha,\beta)$ and $\rho_{1}^{B}(\alpha)$
being the reduced two- and one-body density for the bosonic species and
$\alpha,\beta=L,R$. Physically, $\rho_{2}^{B}(L,R)$ denotes a measure for the
joint probability of finding one boson at the left well while the second is at
the right well. Through the division by the one-body densities, the $g_{2}$
function excludes the impact of the inhomogeneous density distribution and
thereby directly reveals the spatial two-particle correlations induced by the
interaction few_gs_6 ; few_gs_7 . Based on this knowledge, let us first
elaborate the $g_{2}$ function for the non-interacting case, which corresponds
to the TAHD depicted in Fig. 7 (a). Since there is no interaction among the
particles, all the bosons thus can independently hop between the two wells,
hence, it always results in $g_{2}^{o}=g_{2}^{d}=1$, with
$g_{2}^{o}=g_{2}(\alpha,\alpha)$ [$g_{2}^{d}=g_{2}(\alpha,\beta\neq\alpha)$]
being the two-particle correlations within the same well (between the two
wells). By contrast, the presence of the impurity-Bose interaction largely
changes the above $g_{2}$ profile. As shown in Fig.7 (c), the $g_{2}$ function
quickly deviates from the initial values $g_{2}^{o}=g_{2}^{d}=1$ to
$g_{2}^{o}>1$ and $g_{2}^{d}<1$ for $t<5$ and persistently oscillate around
their asymptotic values $g_{2}^{o}=1.28$ and $g_{2}^{d}=0.72$, respectively.
Physically, such an evolution of the $g_{2}$ function indicates that the
bosons are in favor of bunching together with a collective tunneling between
the wells in the dynamics, which evidently manifests the existence of the
Bose-Bose attraction induced by the impurity-Bose repulsion.

Figure 7: (Color online) Time-averaged Husimi distribution for the initial
state $|\Psi(0)\rangle=|\phi_{0}\rangle\otimes|\pi/2,\pi/4\rangle$ and for (a)
$g_{IB}=0.0$ and (b) $g_{IB}=1.0$. Moreover, the black solid line in (a)
denotes the classical trajectory starting from the phase point
$(Z=0,\varphi=\pi/4)$. (c) The evolution of the pair-correlation function
$g_{2}^{o}(t)$ (blue solid line) and $g_{2}^{d}(t)$ (red solid line) for the
case examined in (b).

##  V Conclusions and Outlook

We have demonstrated that a non-interacting ultracold many-body bosonic
ensemble confined in a 1D DW potential can exhibit a chaotic nature due to the
presence of a single impurity. We trigger the non-equilibrium dynamics by
means of a quench of the impurity-Bose interaction and monitor the evolution
of the population imbalance for the bosons between the two wells. While the
increase of the post-quench interaction strength always facilitates the
chaotic motion for the bosonic population imbalance, it becomes regular again
for the cases where the impurity is initially prepared in a highly excited
state. Employing the linear entanglement entropy, it not only enables us to
characterize such an ITC transition but also implies the existence of an
effective Bose-Bose attraction in the dynamics induced by the impurity. In
order to elucidate the physical origin for the above observed ITC transition,
we perform a detailed spectral analysis for the mixture with respect to both
the energy spectrum as well as the eigenstates. In particular, two
distinguished spectral behaviors upon a variation of the interspecies
interaction strength are observed: while the avoided level-crossings take
place in the low-energy spectrum, the energy levels in the high-energy
spectrum possess the main features of the integrable limit. Consequently, it
results in a significant delocalization for the low-lying eigenvectors which,
in turn, accounts for the chaotic nature of the bosonic dynamics. In contrast,
those highly excited states bear a high resemblance to the non-interacting
integrable basis, rendering the recovery of the integrability for the bosonic
species. Finally, we discuss the induced Bose-Bose attraction as well as its
impact on the bosonic dynamics.

Possible future investigations include the impact on the bosonic dynamics with
the inclusion of several additional impurities and/or the bare Bose-Bose
repulsion. Since for the latter there exists a competition between the bare
Bose-Bose repulsion and the induced attractive interaction, this may
significantly affect the bosonic ITC transition. Another interesting
perspective is the study of the chaotic dynamics for an atomic mixture
consisting of atomic species with different masses. The impact of the higher
bands of the DW potential, beyond the two-site BH description for the bosonic
species, is also an interesting perspective.

###### Acknowledgements.

The authors acknowledge fruitful discussions with A. Mukhopadhyay and X.-B.
Wei. This work has been funded by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) - SFB 925 - project 170620586. K. K. gratefully
acknowledges a scholarship of the Studienstiftung des deutschen Volkes. G. X.
acknowledges support from the NSFC under Grants No. 11835011 and No. 11774316.

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