# Intra- and interband excitations induced residue decay of the Bose polaron
in a one-dimensional double-well

Jie Chen<EMAIL_ADDRESS>Department of Physics, Center for
Optical Quantum Technologies, University of Hamburg, Luruper Chaussee 149,
22761 Hamburg, Germany  Simeon I. Mistakidis  ITAMP, Center for Astrophysics
$|$ Harvard $\&$ Smithsonian, Cambridge, MA 02138 USA  Peter Schmelcher
Department of Physics, Center for Optical Quantum Technologies, University of
Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany  The Hamburg Center for
Ultrafast Imaging, University of Hamburg, Luruper Chaussee 149, 22761 Hamburg,
Germany

###### Abstract

We investigate the polaronic properties of a single impurity immersed in a
weakly interacting bosonic environment confined within a one-dimensional
double-well potential using an exact diagonalization approach. We find that an
increase of the impurity-bath coupling results in a vanishing residue,
signifying the occurrence of the polaron orthogonality catastrophe. Asymptotic
configurations of the systems’ ground state wave function in the strongly
interacting regime are obtained by means of a Schmidt decomposition, which in
turn accounts for the observed orthogonality catastrophe of the polaron. We
exemplify that depending on the repulsion of the Bose gas, three distinct
residue behaviors appear with respect to the impurity-bath coupling. These
residue regimes are characterized by two critical values of the bosonic
repulsion and originate from the interplay between the intra- and the
interband excitations of the impurity. Moreover, they can be clearly
distinguished in the corresponding species reduced density matrices with the
latter revealing a phase separation on either the one- or the two-body level.
The impact of the interspecies mass-imbalance on the impurity’s excitation
processes is appreciated yielding an interaction shift of the residue regions.
Our results explicate the interplay of intra- and interband excitation
processes for the polaron generation in multiwell traps and for designing
specific polaron entangled states motivating their exposure in current
experiments.

##  I Introduction

Ultracold atomic gases provide pristine platforms to study quantum many-body
physics owing to their unprecedented controllability, e.g., in terms of the
involved trapping geometries and the atomic interactions cold_atom_rev ;
Feshbach_1 ; Feshbach_2 ; Feshbach_3 . Among the achieved milestones BH_exp_1
; BH_exp_2 ; BH_exp_3 , trapping of a many-body bosonic gas in a one-
dimensional (1D) double-well (DW) potential constitutes a prototype system for
unravelling the emergent complex quantum dynamics DW_exp_1 ; DW_exp_2 ;
DW_exp_3 . This system represents a bosonic Josephson junction (BJJ), namely
the atomic analogue of the Josephson effect initially predicted for tunneling
of Cooper pairs through two weakly linked superconductors BJJ_1 ; BJJ_2 .
Relevant investigations of the BJJ unveiled various intriguing phenomena
including, for instance, Josephson oscillations BJJ_Rabi_1 ; BJJ_Rabi_2 ;
BJJ_Rabi_3 , macroscopic quantum self-trapping events DW_exp_3 ; BJJ_Rabi_1 ;
BJJ_Rabi_2 , collapse and revival population sequences BJJ_Rabi_3 , and the
formation of an atomic squeezed state BJJ_Squeeze_1 ; BJJ_Squeeze_2 .
Moreover, the role of interparticle correlations has been examined BJJ_Frag_1
; BJJ_Frag_2 , revealing the existence of strongly correlated tunneling
processes in few-body systems BJJ_Few_1 ; BJJ_Few_2 ; BJJ_Few_3 ; BJJ_Few_4 ;
BJJ_Few_5 . The above mechanisms are not accessible in conventional
superconducting systems.

On the other hand, with the aid of sympathetic cooling as well as the
experimental progresses on realizing few-body ensembles mixture_exp_bf_1 ;
mixture_exp_bf_2 ; mixture_exp_bb_1 ; mixture_exp_bb_2 ; few_exp1 ; few_exp2 ;
few_exp3 ; few_exp4 ; few_exp5 ; few_exp6 , studies of ultracold atomic
mixtures featuring an appreciably large particle number imbalance have also
been put forward for both bosonic Bose_polaron_1 ; Bose_polaron_2  and
fermionic Fermi_polaron_1 ; Fermi_polaron_2 ; Fermi_polaron_3  settings. These
systems are inherently related to the so-called polaron concept Massignan_rev
; Schmidt_rev , which is originally introduced in the context of a mobile
impurity immersed in a quantum many-body environment with the former becoming
dressed by the excitations of the bath polaron_conmat_1 ; polaron_conmat_2 ;
polaron_conmat_3 ; polaron_conmat_4 . So far, notable investigations of the
emergent polaron properties in the ultracold atomic realm unveil, for example,
their effective mass polaron_eff_mass_1 ; polaron_eff_mass_2 ;
polaron_eff_mass_3 ; polaron_eff_mass_Mist_Vol ; Polaron_1D_exp , the
underlying excitation spectra polaron_ext_spec_1 ; polaron_ext_spec_2 ;
Mistakidis_rf ; Mistakidis_pump_probe ; Will , the existence of bath mediated
induced-interactions polaron_ind_int_1 ; polaron_ind_int_2 ; polaron_ind_int_3
; polaron_ind_int_4 ; polaron_ind_int_Mist_Vol ; Mukherjee_induced_int , as
well as the orthogonality catastrophe (OC) mechanism polaron_OC_1 ;
polaron_OC_2 ; polaron_OC_3 ; polaron_OC_4 ; Mistakidis_OC_2imp .

Interestingly, the above investigations focus on either homogeneous
polaron_eff_mass_1 ; polaron_eff_mass_2 ; polaron_eff_mass_3 ;
polaron_ext_spec_1 ; polaron_ext_spec_2 ; polaron_OC_1 ; polaron_OC_4 ; Will
or harmonically trapped mixtures polaron_eff_mass_Mist_Vol ; polaron_ind_int_1
; polaron_ind_int_2 ; polaron_ind_int_3 ; polaron_ind_int_4 ;
polaron_ind_int_Mist_Vol ; Mukherjee_induced_int ; polaron_OC_2 ; polaron_OC_3
; Mistakidis_OC_2imp ; Mistakidis_rf ; Mistakidis_pump_probe ; Polaron_1D_exp
. Studies considering a strong spatial inhomogeneity, e.g., by assuming that
both the impurity and the bath are trapped within a DW potential are still
rare polaron_few_1 ; polaron_few_2 . Indeed, current explorations have
predominantly dealt with the impurity transport in lattices Cai_transport1 ;
Johnson_transport2  and were mainly restricted to the lowest-band
approximation besides a few exceptions Theel_higherband . As such the
interplay of excitation processes within the same or between different
energetic bands as a result of the inhomogeneity is still far from being
completely understood even in the static properties of these systems. Since
the DW offers a toy-model of a lattice geometry where the band-structure is
important, it provides a testbed for examining the interplay of related
excitations and in particular their role in the polaron generation Yin_twoband
. Here, the polaronic behavior should strongly depend on the involved
interactions and an intriguing prospect would be to engineer specific
entangled polaron states in certain interaction regimes with an additional
knob provided by the interspecies mass-imbalance. Moreover, we should also
emphasize that these hybridized systems are of further interest due to the
fact that one subsystem (impurity) lies in the deep quantum regime while the
other one (medium) can be potentially described semi-classically Impurity_BH_1
; Impurity_BH_2 ; Impurity_BH_3 ; Impurity_BH_4 ; Impurity_BH_5 ;
Impurity_BH_6 . For instance, it has been demonstrated that following an
impurity-bath interaction quench leads to chaotic signatures in the dynamics
of the bath accompanied by significant coherence losses Impurity_BH_6 .

In the present work, we investigate the polaron properties of an atomic
mixture where a single impurity is embedded into a weakly interacting bosonic
gas confined within a 1D DW potential. Particular focus is placed on the
polaronic properties appearing in the ground state (GS) of the mixture upon
considering variations of the involved interaction strengths (both the intra-
and interspecies ones) as well as the interspecies mass-imbalance. To argue on
the emergent polaron generation and consequent behavior, we rely on the so-
called residue polaron_OC_1 ; polaron_OC_2 ; polaron_OC_3 ; polaron_OC_4 ,
being a measure of the overlap between the dressed polaronic state and the
initial non-interacting one. We analyze this composite system via the
numerically exact diagonalization (ED) method. The latter allows us to take
all correlations of the mixture into account as well as capture the impurity’s
higher-band excitation processes.

We find that an increase of the impurity-bath repulsion results in a vanishing
residue, signifying the occurrence of the polaron OC. Employing a Schmidt
decomposition of the many-body wave function, we obtain two asymptotic
configurations of the GS wave function for strong interspecies repulsions,
which in turn account for the observed OC of the polaron. We explicate that
depending on the repulsion strength of the Bose gas, three distinct residue
behaviors appear with respect to the impurity-bath coupling. More
specifically, with increasing interspecies repulsion the residue exhibits: i)
a sharp decrease, ii) an initial decrease followed by a pronounced revival at
intermediate repulsions and then vanishes for stronger impurity-bath coupling
or iii) a slow monotonous decay. These residue regimes are characterized by
two critical repulsion strengths of the Bose gas and originate from the
interplay between the intra- and the interband excitations of the impurity.
Moreover, they are clearly visible in the corresponding species reduced
density matrices, which alternatively reveal a spatial phase separation on
either the one- or the two-body level. The impact of the mass-imbalance on the
impurity’s excitation processes is also discussed yielding a shift of the
above-described residue regions with respect to the bosonic repulsion.

This work is organized as follows. In Sec. II, we introduce our setup and the
employed tight-binding description for the bath. Sec. III presents our main
observation: the existence of three interaction regions characterized by
distinct polaronic residue behaviors. We describe in detail each residue
regime in Sec. III.2, Sec. III.3 and Sec. III.4, respectively. Afterwards, a
spectral analysis is performed in Sec. III.5 unveiling the origin of the
underlying critical bosonic repulsions separating the aforementioned residue
regimes. An outlook is provided in Sec. IV containing some future perspectives
of our findings. In Appendix A we discuss the impact of the parity
conservation in the GS wave function on the shape of the bosonic density
configuration. Appendix B explicates the form of the asymptotic wave function
in terms of the Wannier basis representation, while Appendix C elaborates on
the ingredients of the species mean-field (SMF) approach.

##  II Impurity setting and tight-binding description

We consider a highly particle imbalanced binary atomic mixture consisting of a
single impurity $N_{I}=1$ being immersed in a bosonic gas containing
$N_{B}=100$ bosons. As such for finite impurity-bath couplings the impurity is
dressed by the excitations of the bosonic gas forming a Bose polaron
Polaron_1D_exp , a behavior that has been exemplified in 1D by the emulation
of relevant spectroscopy schemes Mistakidis_rf ; Mistakidis_OC_2imp ;
Mistakidis_pump_probe  as well as by constructing related effective polaron
Hamiltonians polaron_eff_mass_Mist_Vol . The underlying many-body Hamiltonian
is given by $\hat{H}=\hat{H}_{I}+\hat{H}_{B}+\hat{H}_{IB}$, where

$\displaystyle\hat{H}_{I}$ $\displaystyle=\int
dx~{}\hat{\psi}^{\dagger}_{I}(x)\textit{h}_{I}(x)\hat{\psi}_{I}(x),$
$\displaystyle\hat{H}_{B}$ $\displaystyle=\int
dx~{}\hat{\psi}^{\dagger}_{B}(x)\textit{h}_{B}(x)\hat{\psi}_{B}(x)$
$\displaystyle+\frac{g_{BB}}{2}\int
dx~{}\hat{\psi}^{\dagger}_{B}(x)\hat{\psi}^{\dagger}_{B}(x)\hat{\psi}_{B}(x)\hat{\psi}_{B}(x),$
$\displaystyle\hat{H}_{IB}$ $\displaystyle={g_{IB}}\int
dx~{}\hat{\psi}^{\dagger}_{I}(x)\hat{\psi}^{\dagger}_{B}(x)\hat{\psi}_{B}(x)\hat{\psi}_{I}(x).$
(1)

In the above expressions,
$\textit{h}_{\sigma}(x)=-\frac{\hbar^{2}}{2m_{\sigma}}\frac{\partial^{2}}{\partial
x^{2}}+V_{DW}(x)$ is the single-particle Hamiltonian for the $\sigma=I(B)$
species being confined within a 1D symmetric DW potential
$V_{DW}(x)=a_{\sigma}(x^{2}-b_{\sigma}^{2})^{2}$. The parameters $a_{\sigma}$
and $b_{\sigma}$ control the central barrier height as well as the relative
distance between the two wells, respectively.
$\hat{\psi}_{\sigma}^{\dagger}(x)$ [$\hat{\psi}_{\sigma}(x)$] is the field
operator that creates (annihilates) a $\sigma$-species particle of mass
$m_{\sigma}$ at position $x$ Pitaevskii_book . Moreover, we assume that both
the intra- (Bose-Bose) and the interspecies (impurity-bath) interactions are
short-range i.e. of contact type characterized by the strengths $g_{BB}$ and
$g_{IB}$ respectively. This is an adequate approximation within the ultracold
temperature regime where $s$-wave interaction processes are the dominant ones
Feshbach_1 . In the following, we rescale the Hamiltonian of the mixture
$\hat{H}$ in harmonic units, with the energy and length being expressed in
terms of $\eta=\hbar\omega$ and $\xi=\sqrt{\hbar/m_{B}\omega}$, respectively.

Throughout this work, we assume that the atoms of both species are trapped in
the same DW geometry, i.e., $a_{I}=a_{B}=a_{DW}$ and $b_{I}=b_{B}=b_{DW}$. In
practice, we set $a_{DW}=2.0$ and $b_{DW}=1.5$ such that the low-lying single-
particle energy levels form a band-like doublet structure. The spatial
geometry of $V_{DW}(x)$ is depicted in Fig. 1 (see grey dashed line) together
with the lowest eight single-particle energy levels (see horizontal lines).
Moreover, we focus on repulsive interactions for both species and assume that
the bosonic bath is weakly interacting, i.e., $0<g_{BB}\ll 1$. We explore how
the GS polaronic nature is affected by considering variations of both
interaction strengths as well as the mass-imbalance $\beta=m_{I}/m_{B}$
between the two species. Let us note that such a 1D mixture is experimentally
accessible by imposing a strong transverse and a weak longitudinal confinement
to 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 that is made of
atoms residing in 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, constructing a
superlattice whose shells correspond to double-wells DW_exp_3 ; BJJ_2 .
Moreover, the contact interaction strengths $g_{IB}$ and $g_{BB}$ can be
controlled experimentally by tuning the $s$-wave scattering lengths via
Feshbach or confinement-induced resonances Feshbach_1 ; Feshbach_2 ;
Feshbach_3 .

Given that the bosons are weakly interacting and are confined within a tight
DW with a large energy gap $\Delta$ between the first and the second band
111For the employed DW parameters, $a_{DW}=2.0$ and $b_{DW}=1.5$, the energy
gap $\Delta$ between the first and the second band is approximately $10^{3}$
times larger than the width of the first band. [cf. Fig. 1], we further adopt
the two-mode approximation for the bosonic bath

$\hat{\psi}_{B}(x)=u_{L}(x)\hat{b}_{L}+u_{R}(x)\hat{b}_{R},$ (2)

with $u_{L,R}(x)$ being the Wannier states BJJ_2  localized in the left and
the right well, respectively [cf. Fig. 1]. They are chosen to be real valued
due to the preserved time-reversal symmetry of the mixture
quantum_mechanics_Landau . This ansatz leads to the low-energy effective
Hamiltonian for the bosonic species

$\hat{H}_{B}=-J_{B}(\hat{b}^{\dagger}_{L}\hat{b}_{R}+\hat{b}^{\dagger}_{R}\hat{b}_{L})+\frac{U_{B}}{2}\sum_{i=L,R}\hat{b}^{\dagger}_{i}\hat{b}^{\dagger}_{i}\hat{b}_{i}\hat{b}_{i},$
(3)

corresponding to the two-site Bose-Hubbard (BH) model 2_site_BH_1 ;
2_site_BH_2  with

$\displaystyle J_{B}$ $\displaystyle=\int
dx~{}u_{L}(x)\textit{h}_{B}(x)u_{R}(x),$ $\displaystyle U_{B}$
$\displaystyle=g_{BB}\int dx~{}|u_{i}(x)|^{4}~{}~{}~{}~{}~{}(i=L,R),$ (4)

representing the hopping amplitude and the on-site repulsion energy,
respectively.

Hereafter, we designate the GS of our mixture for $g_{IB}>0$ ($g_{IB}=0$) by
$|\Psi\rangle$ ($|\Psi_{0}\rangle$), and denote the eigenstates of the single-
species Hamiltonian $\hat{H}_{I}$ and $\hat{H}_{B}$ as
$\\{|\phi_{i}^{I}\rangle\\}$ and $\\{|\phi_{i}^{B}\rangle\\}$, respectively.
Moreover, we refer to the eigenstates of the bosonic single-particle
Hamiltonian $h_{B}$ as $\\{|\varphi^{B}_{i}\rangle\\}$. In this way, the above
introduced Wannier states $|u_{L,R}\rangle$ are expressed with respect to
$|\varphi^{B}_{0}\rangle$ and $|\varphi^{B}_{1}\rangle$ as
$|u_{L,R}\rangle=\frac{1}{\sqrt{2}}[|\varphi^{B}_{0}\rangle\pm|\varphi^{B}_{1}\rangle]$.
The preserved parity symmetry of the DW potential results in each eigenstate
of $\hat{H}$ or $\hat{H}_{\sigma}$ to possess a definitive parity.
Specifically, the GS $|\Psi\rangle$ ($|\Psi_{0}\rangle$) is even parity, while
$|\phi_{i}^{\sigma}\rangle$ for $i=0,2,4,\dots$ ($i=1,3,5,\dots$) possesses an
even (odd) parity.

Figure 1: (Color online) Schematic representation of the DW (grey dashed line)
characterized by $a_{DW}=2.0$, $b_{DW}=1.5$ and its energetically lowest
single-particle spectrum. The horizontal lines denote the lowest eight energy
levels, in which the solid (dashed) lines correspond to the even (odd) parity
states of the DW. The energy difference (gap) between the lowest two energetic
bands is denoted by $\Delta$. The red and blue solid lines represent the
Wannier states $u_{L}(x)$ and $u_{R}(x)$ in the lowest band of the left and
right well, respectively.

##  III Bose polaron residue response and its decay

In the following, we investigate the GS polaronic properties of the mixture.
In particular, we focus on how variations of both the intra- and the
interspecies interactions as well as the mass-imbalance impact the emergent
polaronic nature. To this end, we examine as a representative measure the so-
called polaronic residue, quantified by the overlap between the dressed
polaronic state and the initial non-interacting one polaron_OC_1 ;
polaron_OC_2 ; polaron_OC_3 ; polaron_OC_4 . As we shall argue below, the
increase of $g_{IB}$ always leads to a vanishing residue, signifying the
occurrence of the OC of the polaron. Interestingly, we shall also exemplify
that there exist two critical values of the bosonic repulsion, referred to as
$g_{BB}^{cI}$ and $g_{BB}^{cII}$ (depending also on the mass-imbalance
$\beta$) according to which three distinct residue response regions are
encountered for increasing $g_{IB}$. We perform a detailed investigation
demonstrating the properties of these regions and explicating the origin of
the aforementioned critical bosonic repulsion strengths.

###  III.1 Overview of the polaronic residue

The polaronic residue is given by $Z=|\langle\Psi_{0}|\Psi\rangle|^{2}$, which
measures the overlap between the dressed polaronic state and the initial non-
interacting one. Here,
$|\Psi_{0}\rangle=|\phi^{I}_{0}\rangle|\phi^{B}_{0}\rangle$ is the GS of the
mixture for fixed $g_{BB}$ and $\beta$ and vanishing impurity-medium coupling
($g_{IB}=0$), while $|\Psi\rangle$ denotes the GS for $g_{IB}>0$. In Fig. 2,
we present the residue $Z$ as a function of both the Bose-Bose and the
impurity-bath interaction strengths for various mass ratios, namely
$\beta=0.5$ [Fig. 2 (a,b)], $\beta=1.0$ [Fig. 2 (c,d)] and $\beta=2.0$ [Fig. 2
(e,f)]. As it can be seen, for fixed $\beta$ and $g_{BB}$, an increase of
$g_{IB}$ leads eventually to the suppression of the polaronic residue, which
in turn implies the decay of the polaronic nature. In particular, the residue
becomes negligible for $g_{IB}\rightarrow 1$, signifying that the OC of the
polaron takes place in this strongly interacting regime 222Note that the
mechanism of the polaron OC, discussed herein, is different from the original
Anderson OC polaron_conmat_4 . The latter is strictly defined in the
thermodynamic limit while our finite atom number mixture is also spatially
inhomogeneous due to the DW. Instead, we rely on the fact that for a GS with
vanishing polaronic nature its residue becomes $Z=0$, a behavior that it is
here partially caused by the trap-induced spatial inhomogeneity. Albeit this
differentiation, as we shall see below, the polaronic residue in our system
exhibits a similar behavior to the one obtained in the Anderson’s original
argument. Namely, for an increasing bath particle number
$N_{B}\rightarrow\infty$ the residue $Z\rightarrow 0$.. Moreover, for all
different mass ratios, there exist two critical bosonic repulsion strengths
$g_{BB}^{cI}$ and $g_{BB}^{cII}$ [see the white dashed horizontal lines in
Fig. 2]. Depending on $g_{BB}$ being below or above each of these critical
values, three distinct residue regions are found for increasing $g_{IB}$. For
instance, in the equal-mass case we have $g_{BB}^{cI}=0.0035$ and
$g_{BB}^{cII}=0.035$ [Fig. 2 (c)]. For $g_{BB}=0$ and increasing $g_{IB}$, the
residue, starting from $Z=1$, sharply drops to $Z\approx 0$ for $g_{IB}=0.003$
and remains almost constant for further increasing $g_{IB}$ [cf. Fig. 2 (d)
red line]. This behavior essentially reveals that, for $g_{BB}<g_{BB}^{cI}$,
the OC of the polaron takes place even for an extremely weak impurity-bath
repulsion. In contrast, once we ramp up the bosonic repulsion e.g., to
$g_{BB}=0.008$, the residue initially decreases to $Z=0.05$ and then exhibits
a pronounced peak for larger $g_{IB}$ [cf. Fig. 2 (d) blue line], which
signifies the revival of the polaron for intermediate impurity-bath
interactions and within the region $g_{BB}\in[g_{BB}^{cI},g_{BB}^{cII}]$.
Turning to $g_{BB}>g_{BB}^{cII}$, a relatively slower monotonous decrease of
$Z$ with respect to $g_{IB}$ is observed, suggesting that the mixture enters a
regime with a slow polaron decay [see black line in Fig. 2 (d) for the case
$g_{BB}=0.05$].

Interestingly, while $g_{BB}^{cI}$ appears to be inert with respect to
variations of the mass ratio, $g_{BB}^{cII}$ depends strongly on $\beta$,
e.g., $g_{BB}^{cII}=0.008$ for $\beta=0.5$ while $g_{BB}^{cII}=0.035$ for
$\beta=1.0$ and $g_{BB}^{cII}=0.045$ for $\beta=2.0$. Based on the above
observations, we will refer to the aforementioned three distinct regions as
type-I ($g_{BB}\in[0,g_{BB}^{cI}]$), type-II
($g_{BB}\in[g_{BB}^{cI},g_{BB}^{cII}]$) and type-III polaron decay
($g_{BB}>g_{BB}^{cII}$), respectively. Moreover, in order to analyze the
properties and origin of these regions we will solely focus on the equal-mass
case [Sec. III.2 to Sec. III.4], while the influence of the mass-imbalance
will be discussed later on in Sec. III.5.

Figure 2: (Color online) Polaronic residue $Z$ with respect to variations of
the involved interaction strengths and particular mass ratios. The latter
refer to (upper panels) (a) $\beta=0.5$, (c) $\beta=1.0$ and (e) $\beta=2.0$.
Three different residue regimes in terms of $g_{BB}$ are identified and they
are separated by the white dashed horizontal lines, which denote the critical
bosonic repulsion strengths corresponding to (a) $g_{BB}^{cI}=0.003$,
$g_{BB}^{cII}=0.008$, (c) $g_{BB}^{cI}=0.0035$, $g_{BB}^{cII}=0.035$ and (e)
$g_{BB}^{cI}=0.004$, $g_{BB}^{cII}=0.045$. Lower panels: same with the upper
panels but for specific bosonic repulsions (see legends). The residue decay
designating the OC of the Bose polaron for strong repulsive $g_{IB}$ is
evident.

###  III.2 Type-I polaron decay

To understand the polaron decay in the interaction region
$g_{BB}<g_{BB}^{cI}$, we first investigate the $\sigma$-species single-
particle density
$\rho^{\sigma}_{1}(x)=\langle\Psi|\hat{\psi}^{\dagger}_{\sigma}(x)\hat{\psi}_{\sigma}(x)|\Psi\rangle/N_{\sigma}$
for $\sigma=I(B)$. It provides the probability of finding a $\sigma$-species
particle at position $x$ dmat_1 ; dmat_2  and is experimentally accessible via
an average over a sample of single-shot measurements cold_atom_rev ;
Disipative_motion_imp . Figures 3 (a), (b) depict the spatial distributions of
$\rho^{\sigma}_{1}(x)$ for fixed $g_{BB}=0$ and varying $g_{IB}$. Due to the
DW confinement, the $\rho^{\sigma}_{1}(x)$ configurations in the case of
$g_{IB}=0$ exhibit a two-hump structure, which reflects the spatial
inhomogeneity of the mixture [cf. Fig. 3 (b) red dash-dotted line].
Interestingly, for increasing $g_{IB}$ the profiles of $\rho^{\sigma}_{1}(x)$
for both species remain un-altered [cf. Fig. 3 (a)]. We note that the behavior
of $\rho^{B}_{1}(x)$ can be understood via the adopted two-mode approximation
[cf. Eq. (2)]. Due to the negligible spatial overlap between the two Wannier
states, i.e. $u_{L}(x)u_{R}(x)\approx 0$, it holds that
$\rho^{B}_{1}(x)\approx\sum_{i=L,R}u_{i}(x)u_{i}(x)\langle\Psi|\hat{b}^{\dagger}_{i}\hat{b}_{i}|\Psi\rangle/N_{B}$.
Together with the property
$\hat{b}^{\dagger}_{L}\hat{b}_{L}=\hat{b}^{\dagger}_{R}\hat{b}_{R}=N_{B}/2$,
originating from the parity symmetry (see Appendix A and the discussions
below), $\rho^{B}_{1}(x)$ is thus invariant against variations of both
$g_{IB}$ and $g_{BB}$ 333This result relies on the fact that the two Wannier
states have negligible spatial overlap $u_{L}(x)u_{R}(x)\approx 0$, which is
only valid for a DW with an adequately large barrier. For a shallow DW, the
bosonic density indeed depends on both $g_{IB}$ and $g_{BB}$ (results are not
shown).. In contrast, the interaction dependence of the impurity’s density is
not explained straightforward. As we shall discuss below, this behavior can
also be understood if the impurity is predominately restricted to the lowest
band of the DW.

Despite the fact that the single-particle densities of the two species remain
miscible, with increasing $g_{IB}$, we note that a phase separation occurs on
the two-body level. This becomes evident by inspecting the two-body
interspecies density distribution, i.e.
$\rho^{IB}_{2}(x_{I},x_{B})=\langle\Psi|\hat{\psi}^{\dagger}_{I}(x_{I})\hat{\psi}_{I}(x_{I})\hat{\psi}^{\dagger}_{B}(x_{B})\hat{\psi}_{B}(x_{B})|\Psi\rangle/(N_{I}N_{B})$,
which refers to the probability of detecting the impurity at position $x_{I}$
while one boson is at position $x_{B}$ dmat_1 ; dmat_2 . In this way, it
naturally incorporates impurity-bath spatial correlations. For $g_{IB}=0$, the
two species are fully decoupled, namely both the impurity and the bosons can
freely tunnel between the two wells. As a result, $\rho^{IB}_{2}(x_{I},x_{B})$
exhibits four dominant peaks within the spatial regions ($x_{I},x_{B})=(\pm
1.5,\pm 1.5$), i.e., around the minima of the DW [cf. Figs. 1 and 3 (c)]. With
increasing $g_{IB}$, the emergent repulsion renders such a two-body density
distribution energetically unfavorable. Consequently, the impurity and the
bosons tend to become anti-bunched, leading to an increase (decrease) of
$\rho^{IB}_{2}(x_{I},x_{B})$ values in the vicinity of the off-diagonal
(diagonal) [cf. Fig. 3 (d)]. Moreover, in the case of $g_{IB}=1.0$, the two
species are fully anti-bunched, leaving a probability distribution solely
along the off-diagonal [see Fig. 3 (e)]. As a matter of fact, the mixture in
turn resides in a superposition of two equally-weighted configurations and
thus forms a Schrödinger-cat state Impurity_BH_2 , manifesting the dominant
role of impurity-bath entanglement (see also below). Specifically, each of the
two configurations represents a scenario where the impurity lies in one well
while all the bosons are located in the other well. In this way, a two-body
interspecies phase separation takes place in space (see also the discussion
below), a phenomenon that has already been observed in fermionic mixtures
Erdmann_2b_separation . Comparing the residue behavior [Fig. 2 (d), red solid
line] to the respective two-body densities [Fig. 3 (c)-(e)], it becomes clear
that the OC of the polaron is directly related to the phase separation between
the two species. This has been indeed confirmed independently by recent
studies on the polaron dynamics, where this phase separation leads to a
dissipative motion of the impurity accompanied by an energy transfer to the
bath species, and hence diminishes its polaronic nature polaron_OC_3 ;
Disipative_motion_imp .

Having investigated the impact of $g_{IB}$ on the $\sigma$-species density
distributions, we now analyze the structure of the many-body wave function in
order to gain deeper insights into the above-described polaron decay. To this
end, we first employ a Schmidt decomposition and express the GS wave function
of the mixture in the form Schmidt

$|\Psi\rangle=\sum_{i=1}^{\infty}\sqrt{\lambda_{i}}~{}|\psi_{i}^{I}\rangle|\psi_{i}^{B}\rangle,$
(5)

where $\lambda_{i}$, being real positive values, are the Schmidt numbers with
$\lambda_{1}>\lambda_{2}>\cdots$. They obey the constraint
$\sum_{i}\lambda_{i}=1$ which originates from the normalization of the wave
function. $|\psi_{i}^{\sigma}\rangle$ denotes the $i$th Schmidt orbital for
the $\sigma$-species and can always be expanded as a linear superposition of
the $\hat{H}_{\sigma}$ eigenstates

$|\psi_{i}^{\sigma}\rangle=\sum_{k=0}^{D_{\sigma}-1}C^{\sigma}_{i,k}|\phi_{k}^{\sigma}\rangle.$
(6)

Here, $D_{\sigma}$ is the dimension of the Hilbert space corresponding to the
$\sigma$-species Hamiltonian $\hat{H}_{\sigma}$. Regarding the bosonic
species, $D_{B}$ is finite due to the employed two-mode approximation and we
have $D_{B}=N_{B}+1=101$. For the impurity, we truncate $D_{I}$ to some finite
large values depending on the interaction strengths in order to guarantee the
convergence of our results. It is important to note that the Schmidt numbers
$\\{\lambda_{i}\\}$ directly reveal the presence of interspecies entanglement
Schmidt . In particular, if $\lambda_{1}=1$ while the remaining
$\lambda_{i\neq 1}=0$, the mixture is non-entangled with the corresponding
wave function being of a simple product form, i.e.
$|\Psi\rangle=|\psi^{I}\rangle|\psi^{B}\rangle$ (for simplicity, we have
neglected the subscript of each Schmidt orbital for this product form). As we
shall discuss later, the mixture in this special limit is fully captured by
the SMF description, where the mutual impact of the species is merely an
effective potential polaron_ind_int_3 .

Figure 3: (Color online) Spatial profiles of the impurity’s reduced densities
for the type-I residue decay. (a) The single-particle density
$\rho^{I}_{1}(x)$ of the impurity as a function of $g_{IB}$ with $g_{BB}=0$
showing an equal population of the two wells. The bosonic gas density
$\rho^{B}_{1}(x)$ exhibits exactly the same shape (not shown). (b) Profiles of
$\rho^{I}_{1}(x)$ ($\rho^{B}_{1}(x)$) for $g_{BB}=0$ and for either $g_{IB}=0$
(red dash-dotted line) or $g_{IB}=1.0$ (blue dashed line). (c)-(e) The
impurity-bath two-body density distributions $\rho^{IB}_{2}(x_{I},x_{B})$ in
the case of $g_{BB}=0$ and (c) $g_{IB}=0.0$, (d) $g_{IB}=0.003$ and (e)
$g_{IB}=1.0$. A two-body phase-separation process occurs for increasing
$g_{IB}$.

Figure 4 (a) showcases the first two Schmidt numbers with respect to $g_{IB}$
for the case of $g_{BB}=0$. Note, however, that all other Schmidt numbers are
explicitly excluded due to their negligible values ($\lambda_{i}<10^{-5}$ for
$i>2$). For $g_{IB}=0$, the GS is simply
$|\Psi_{0}\rangle=|\phi_{0}^{I}\rangle|\phi_{0}^{B}\rangle$, and hence the two
Schmidt numbers are $\lambda_{1}=1$ and $\lambda_{2}=0$. Upon switching on the
impurity-bath repulsion, we observe that $\lambda_{1}$ ($\lambda_{2}$) quickly
decreases (increases) to the value $\lambda_{1}=0.52$ ($\lambda_{2}=0.48$) for
$g_{IB}=0.003$. Afterwards, it exhibits an extremely slow decrease (increase)
towards $\lambda_{1}=\lambda_{2}=0.5$ until $g_{IB}=0.8$ and retains this
value for $g_{IB}>0.8$. In this regard, we can deduce the asymptotic form of
the GS wave function in the region $g_{BB}<g_{BB}^{cI}$ and for a strong
impurity-bath repulsion (namely $g_{IB}>0.8$)

$|\Psi_{T-I}\rangle=\frac{1}{\sqrt{2}}\left[|\psi_{1}^{I}\rangle|\psi_{1}^{B}\rangle+|\psi_{2}^{I}\rangle|\psi_{2}^{B}\rangle\right],$
(7)

with the prefactor $1/\sqrt{2}$ stemming from the fact that
$\lambda_{1}=\lambda_{2}=1/2$ for large $g_{IB}$ [see also Fig. 4 (a)].
Further inspecting the first two Schmidt orbitals of the impurity, we find
that

$|\psi_{1}^{I}\rangle\approx|\phi_{0}^{I}\rangle,~{}~{}~{}|\psi_{2}^{I}\rangle\approx|\phi_{1}^{I}\rangle.$
(8)

Together with Eq. (7), this immediately yields the single-particle (SP)
occupations for the impurity as $\hat{n}_{0}^{sp}=\hat{n}_{1}^{sp}\approx 1/2$
and $\hat{n}_{i>1}^{sp}\approx 0$, reflecting the fact that the impurity is
predominately occupying the lowest band of the DW [cf. Fig. 4 (b) and the
discussions below]. Here,
$\hat{n}_{i}^{sp}=(\hat{a}_{i}^{I})^{\dagger}\hat{a}_{i}^{I}$ denotes the
occupation number of the $i$th single-particle state $|\phi^{I}_{i}\rangle$ of
the impurity.

On the other hand, the corresponding Schmidt orbitals for the bosonic species
can be written as

$|\psi_{1}^{B}\rangle=\sum_{k=0,2,4,\dots}C^{B}_{1,k}|\phi_{k}^{B}\rangle,~{}~{}~{}~{}|\psi_{2}^{B}\rangle=\sum_{k=1,3,5,\dots}C^{B}_{2,k}|\phi_{k}^{B}\rangle,$
(9)

with $C^{B}_{1,k}$ and $C^{B}_{2,k}$ being the corresponding expansion
coefficients and $|\phi_{k}^{B}\rangle$ denoting the $k$th eigenstate of
$\hat{H}_{B}$ for the case $g_{BB}=0$ [cf. Eq. (6)] . As it can be seen in
Fig. 5 (a), for $g_{IB}=1.0$, $C^{B}_{1,k}$ ($C^{B}_{2,k}$) obtained from the
ED method vanishes exactly for the odd (even) parity eigenstates
$|\phi_{k}^{B}\rangle$. This is the reason for the index notation
$k=0,2,4,\dots$ ($k=1,3,5,\dots$) introduced in Eq. (9). Physically, this
observation is a direct consequence of the preserved parity symmetry of the
wave function of the mixture. Indeed, since the GS wave function
$|\Psi\rangle$ is of even parity, each product state
$|\psi_{i}^{I}\rangle|\psi_{i}^{B}\rangle$ in Eq. (7) then needs to has an
even parity as well. As a result, the bosonic Schmidt orbital
$|\psi_{i}^{B}\rangle$ shares the same parity symmetry with the impurity.
According to Eq. (8), the Schmidt orbital $|\psi_{1}^{B}\rangle$
($|\psi_{2}^{B}\rangle$) for the bosonic species hence has an even (odd)
parity. Moreover, the even parity of the GS wave function $|\Psi\rangle$
ensures that the distribution of the bosons between the two wells obeys
$\hat{b}^{\dagger}_{L}\hat{b}_{L}=\hat{b}^{\dagger}_{R}\hat{b}_{R}=N_{B}/2$
(see Appendix A). This leads to the invariant nature of $\rho^{B}_{1}(x)$
against variations of the interaction strengths. Importantly, we note that
such a property of $\rho^{B}_{1}(x)$ holds for all different types of residue
decay. Apart from that, we also notice that both $|C^{B}_{1,k}|^{2}$ and
$|C^{B}_{2,k}|^{2}$ follow a binomial distribution with their maximal values
located at $k=50$ and $k=49$, respectively [cf. Fig. 5 (a) red solid line and
blue dashed line]. We note that, this result directly stems from the above-
described interspecies two-body phase separation [see also Fig. 3 (e)].

In Appendix B, we show that $|\Psi_{T-I}\rangle$ [see also the results from
Eq. (7) to Eq. (9)] is exactly equivalent to

$|\Psi_{T-I}\rangle=\frac{1}{\sqrt{2N_{B}!}}\left[\hat{a}_{L}^{\dagger}(\hat{b}_{R}^{\dagger})^{N_{B}}+\hat{a}_{R}^{\dagger}(\hat{b}_{L}^{\dagger})^{N_{B}}\right]|0\rangle,$
(10)

with $\hat{a}_{L,R}=\frac{1}{\sqrt{2}}(\hat{a}_{0}^{I}\pm\hat{a}_{1}^{I})$ and
$\hat{a}_{i}^{I}$ ($i=0,1$) referring to the annihilation operator acting on
the $i$th single-particle state $|\phi_{i}^{I}\rangle$ of the impurity.
Remarkably, Eq. (10) essentially represents a Schrödinger-cat state being a
superposition of two macroscopic many-body states. Each of them corresponds to
a configuration where the impurity resides in one well while all the bosons
are located in another well. Thus it reflects the above-described two-body
phase separation between the two species. We remark that such a Schrödinger-
cat state has been reported in Ref. Impurity_BH_2  by restricting the impurity
in the lowest band of the DW (see below). It has also been argued that this
state is extremely sensitive to the environment meaning that an arbitrary
small perturbation can lead to a collapse onto one of the two macroscopic
configurations. This should be indeed the case upon the experimental
measurement of this configuration which is a manifestation of the
entanglement. The obtained asymptotic GS wave function $|\Psi_{T-I}\rangle$
allows for further explicating the OC of the polaron for strong impurity-bath
repulsion: since for the impurity
$\langle\psi_{2}^{I}|\phi_{0}^{I}\rangle=\langle\phi_{1}^{I}|\phi_{0}^{I}\rangle=0$
[cf. Eq. (8)], $|\psi_{2}^{I}\rangle|\psi_{2}^{B}\rangle$ is trivially
orthogonal to the ground state
$|\Psi_{0}\rangle=|\phi_{0}^{I}\rangle|\phi_{0}^{B}\rangle$ for $g_{IB}=0$.
Moreover, the binomial distribution of the coefficients $|C^{B}_{1,k}|^{2}$
together with the normalization condition $\sum_{k}|C^{B}_{1,k}|^{2}=1$ render
444The maximal value of the binomial distribution of $|C^{B}_{1,k}|^{2}$ is
located at $k=N_{B}/2$. For $N_{B}\rightarrow\infty$, the $C^{B}_{1,k}$ value
for both $k\ll N_{B}/2$ and $k\gg N_{B}/2$ approaches $C^{B}_{1,k}\to 0$. This
can also been seen in Fig. 5 (a). $C^{B}_{1,0}\rightarrow 0$ for
$N_{B}\rightarrow\infty$. As a result, we have
$\langle\psi_{1}^{B}|\phi_{0}^{B}\rangle=0$ [cf. Eq. (6)] and
$|\psi_{1}^{I}\rangle|\psi_{1}^{B}\rangle$ is orthogonal to $|\Psi_{0}\rangle$
as well. In this way, we explicate the above OC of the polaron.

Before closing this section, let us elaborate on the polaron decay for
$0<g_{IB}<0.8$ in more detail. As we have already mentioned above, for all
values of $g_{IB}$ only the first two Schmidt orbitals are populated in the GS
[see Fig. 4 (a) as well as its inset providing a magnification within the
region $0<g_{IB}<0.01$]. The GS wave function correspondingly takes the form

$|\Psi\rangle=\sqrt{\lambda_{1}}|\psi_{1}^{I}\rangle|\psi_{1}^{B}\rangle+\sqrt{\lambda_{2}}|\psi_{2}^{I}\rangle|\psi_{2}^{B}\rangle.$
(11)

For the impurity, we have also verified that the Schmidt orbitals follow the
relations in Eq. (8) exactly for varying $g_{IB}$. This observation, on the
one hand, reveals that the impurity is predominately restricted within the
lowest band of the DW which, in turn, explicates the invariant profile of
$\rho^{I}_{1}(x)$ for different $g_{IB}$ [cf. Fig. 3 (a), (b) and the
accompanied discussion above]. On the other hand, it also results in an even
(odd) parity bosonic Schmidt orbital $|\psi_{1}^{B}\rangle$
($|\psi_{2}^{B}\rangle$) [see the expansion coefficients $|C^{B}_{1,k}|^{2}$
and $|C^{B}_{2,k}|^{2}$ in Fig. 5 (b), (c)]. Moreover, we note that the GS of
the mixture deforms from $|\Psi_{0}\rangle$ with respect to $g_{IB}$ as
follows: For $0<g_{IB}<0.003$, a larger $g_{IB}$ leads to the intraband
excitations of the impurity within the lowest-band of the DW identified by an
SP transition of the impurity between the states $|\phi_{0}^{I}\rangle$ and
$|\phi_{1}^{I}\rangle$ [see Fig. 4 (b)]. This process is accompanied by a
rapid growth of the bath species populations onto the low-lying excited states
$\\{|\phi_{k}^{B}\rangle\\}$ of $\hat{H}_{B}$ for $k\leqslant 12$ [cf. Fig. 5
(c)]. Consequently, it leads to the sharp decrease of the polaronic residue as
depicted in Fig. 2 (d). Further increasing the impurity-bath repulsion to
$g_{IB}=0.8$, the impurity ceases to be excited to higher bands. Instead, the
bosonic species gradually populate its higher-lying excited states
$\\{|\phi_{k\gg 1}^{B}\rangle\\}$ [cf. Fig. 5 (b)]. Notice, however, that the
bosons are always restricted to the lowest-band of the DW due to the employed
two-mode approximation [cf. Eq. (2)]. The excited states are
$\\{|\phi_{k}^{B}\rangle\\}$ strictly referred to as the eigenstates of the BH
Hamiltonian $\hat{H}_{B}$. Owing to the fact that the impurity always remains
in the lowest band of the DW (when $g_{BB}\approx 0$) during the whole process
of the polaron decay, hereafter, we refer to this regime as the intraband
excitation induced polaron decay.

Figure 4: (Color online) Schmidt numbers and SP occupations of the impurity
quantifying interspecies entanglement and the excitations of the impurity. (a)
The first two Schmidt numbers for $g_{BB}=0.0$ and varying $g_{IB}$. The red
solid (blue dashed) line denotes the value of $\lambda_{1}$ ($\lambda_{2}$).
The inset provides a magnification within the region $0<g_{IB}<0.01$. (b) The
SP occupation $n_{i}^{sp}$ of the impurity when considering $g_{BB}=0$ as a
function of $g_{IB}$. The red solid (blue dashed) line refers to $n_{0}^{sp}$
($n_{1}^{sp}$). The inset depicts the SP occupations for
$\sum_{i>1}n_{i}^{sp}$. Figure 5: (Color online) Expansion coefficients for
the first two bosonic Schmidt orbitals for $g_{BB}=0.0$ and (a) $g_{IB}=1.0$,
(b) $g_{IB}=0.05$ and (c) $g_{IB}=0.003$. In all cases, the red solid and the
blue dashed line denote $|C^{B}_{1,k}|^{2}$ and $|C^{B}_{2,k}|^{2}$,
respectively. The black solid line in (a) depicts the expansion coefficients
$|\tilde{C}^{B}_{1,k}|^{2}$ and $|\tilde{C}^{B}_{2,k}|^{2}$ obtained from Eq.
(30) given in Appendix B.

###  III.3 Type-II polaron decay

Next, we turn into the description of the type-II polaron decay, which occurs
in the interaction regime $g_{BB}\in[g_{BB}^{cI},g_{BB}^{cII}]$. For
$\beta=1$, we have $g_{BB}^{cI}=0.0035$ and $g_{BB}^{cII}=0.035$. As a
characteristic example, we present the density distributions of the impurity
for fixed $g_{BB}=0.008$ and different values of $g_{IB}$ [cf. Figs. 6 (a) and
6 (b)]. Before proceeding, let us remark that the corresponding bosonic
density $\rho^{B}_{1}(x)$ has the same shape as the one depicted in Fig. 3
(b), i.e. it distributes among the individual wells. On the contrary, the
spatial profile of the impurity’s density in this regime of $g_{BB}$ depends
strongly on $g_{IB}$ [cf. Figs. 3 (a) and 6 (a)]. Instead of maintaining the
initial two-hump structure, for increasing $g_{IB}$, the bosons considerably
expel the impurity into the center of the DW rendering a highly localized
$\rho^{I}_{1}(x)$ distribution around the region $x\in[-1,1]$ for
$g_{IB}>0.1$. This correspondingly leads to a negligible spatial overlap
between $\rho^{I}_{1}(x)$ and $\rho^{B}_{1}(x)$ for strong impurity-bath
repulsions [see red dash-dotted and black solid lines in Fig. 6 (b) referring
to the cases $g_{IB}=0$ and $g_{IB}=1.0$. Moreover, note that the profile of
$\rho^{I}_{1}(x)$ for $g_{IB}=0$ is same with the one of $\rho^{B}_{1}(x)$].
The above process illustrates another type of phase separation, which is
clearly visible on the single-particle densities. It is for this reason that
below we shall refer to it as the one-body phase separation
polaron_eff_mass_Mist_Vol . Complementary, this phase separation is also
captured by the corresponding two-body interspecies density since the latter
contains all the information of the single-particle density. For increasing
$g_{IB}$, we observe that $\rho^{IB}_{2}(x_{I},x_{B})$ quickly evolves from a
checkerboard pattern with both the diagonal and the off-diagonal elements
occupied for weak $g_{IB}$ [cf. Fig. 6 (c), (d)] towards a configuration with
dominant populations around the spatial regions $x_{I}\in[-1,1]$ and
$x_{B}\approx\pm 1.5$ for $g_{IB}=1.0$ [cf. Fig. 6 (e)]. This reflects a
configuration where the impurity lies at the center of the DW, while the
bosons are located in the outer two wells. In this regard, it manifests the
above-mentioned one-body phase separation.

Figure 6: (Color online) The impurity’s reduced densities regarding the type-
II residue decay. (a) The single-particle density of the impurity as a
function of $g_{IB}$ for $g_{BB}=0.008$. (b) Profiles of $\rho^{I}_{1}(x)$ for
$g_{BB}=0.008$ and $g_{IB}=0$ (red dash-dotted line), $g_{IB}=0.2$ (blue
dashed line) and $g_{IB}=1.0$ (black solid line). The impurity localizes
around the central region of the DW for stronger $g_{IB}$ experiencing a one-
body phase separation with the medium. (c-e) The corresponding two-body
impurity-medium density spatial distributions for the parameters showcased in
(b).

Interestingly, the distinct density behavior of $\rho^{I}_{1}(x)$ in this
regime is inherently related to a richer excitation behavior of the impurity
as compared to the former region of the type-I polaron decay. In contrast to
the case discussed in Sec. III.2, where the impurity features only the
intraband excitations here a larger $g_{IB}$ triggers also interband
processes. To demonstrate the latter, we resort again to the above introduced
SP occupation $\hat{n}_{i}^{sp}$, which is alternatively related to the
underlying band occupation for the impurity as
$\hat{n}_{1}^{b}=\hat{n}_{0}^{sp}+\hat{n}_{1}^{sp}$,
$\hat{n}_{2}^{b}=\hat{n}_{2}^{sp}+\hat{n}_{3}^{sp}$, and so on. For
$g_{IB}<0.1$, the impurity predominately populates the lowest band of the DW,
leaving a band occupation $\hat{n}_{1}^{b}>0.98$ [see Figs. 7 (a) and 7 (b)
red solid lines]. This is also manifested by restricting the impurity into the
lowest band, i.e., by considering the ansatz
$\hat{\psi}_{I}(x)=\phi_{0}^{I}(x)\hat{a}_{0}^{I}+\phi_{1}^{I}(x)\hat{a}_{1}^{I}$
and calculate the corresponding polaronic residue $Z$ for different $g_{IB}$.
As compared to the numerically exact results, we find a good agreement between
each other [cf. Fig. 8 (a), compare blue solid and green dash-dotted lines].
Importantly, we remark that this observation further implies that the
corresponding GS wave function $|\Psi\rangle$ takes the form introduced in Eq.
(11). In fact, since each Schmidt orbital of the impurity becomes a linear
superposition of the SP states $|\phi_{0}^{I}\rangle$ and
$|\phi_{1}^{I}\rangle$, to preserve its orthogonality, i.e.,
$\langle\psi_{i}^{I}|\psi_{j}^{I}\rangle=0$ for $i\neq j$, there are at most
two linearly independent Schmidt orbitals. On the contrary, increasing the
impurity-bath repulsion towards $g_{IB}=0.2$ leads to a sharp decrease
(increase) of the SP occupation $\hat{n}_{i}^{sp}$ for $i=1$ ($i=2,4$) as well
as a slow decrease of $\hat{n}_{0}^{sp}$ [cf. Figs. 7 (b), (c) and (e), red
solid lines]. This behavior essentially reveals that the impurity experiences
transitions among the SP states
$|\phi_{1}^{I}\rangle\rightarrow|\phi_{2}^{I}\rangle$ and
$|\phi_{1}^{I}\rangle\rightarrow|\phi_{4}^{I}\rangle$. Hence, interband
excitation processes of the impurity become relevant. Interestingly, combining
our observations for the impurity’s SP occupations [cf. Fig. 7] and its
density [cf. Fig. 6 (a)] we conclude that the triggered interband excitations
are accompanied by the occurrence of one-body phase separation between the two
species.

Further increasing the impurity-bath interaction to $g_{IB}>0.2$, a negligible
(large) SP occupation of the odd (even) parity states of the impurity is
observed. Particularly, a slow decrease (increase) of the population
$\hat{n}_{0}^{sp}$ and $\hat{n}_{2}^{sp}$ ($\hat{n}_{4}^{sp}$) takes place.
The interband excitations of the impurity are consequently dominated by the
transitions e.g. among the SP states
$|\phi_{0}^{I}\rangle\leftrightarrow|\phi_{2}^{I}\rangle$,
$|\phi_{0}^{I}\rangle\leftrightarrow|\phi_{4}^{I}\rangle$ and
$|\phi_{2}^{I}\rangle\leftrightarrow|\phi_{4}^{I}\rangle$, which preserve the
parity symmetry of the impurity’s wave function (see also the discussion
below). Importantly, the Schmidt numbers for $g_{IB}>0.2$ become as
$\lambda_{1}\approx 1$ and $\lambda_{i>1}\approx 0$. Hence, the GS wave
function $|\Psi\rangle$ possesses a simple product form

$|\Psi\rangle=|\psi^{I}\rangle|\psi^{B}\rangle.$ (12)

Therefore, our mixture can be fully captured within the SMF description where
the mutual impact of the species is merely an effective potential
polaron_ind_int_3 . The SMF description reduces the interspecies correlated
setting into a single-species problem whose effective Hamiltonian reads

$\hat{H}_{\text{eff}}^{\sigma}=\hat{H}_{\sigma}+\langle\psi^{\bar{\sigma}}|\hat{H}_{IB}|\psi^{\bar{\sigma}}\rangle,$
(13)

with $\bar{\sigma}=B(I)$ for $\sigma=I(B)$. Moreover, the GS of
$\hat{H}_{\text{eff}}^{\sigma}$ is equivalent to the Schmidt orbital
$|\psi^{\sigma}\rangle$ given by Eq. (12). In this way, the $\sigma$-species
particle experiences, instead of the initial DW, an effective potential due to
the presence of the other species. For the impurity, this effective potential
takes the form

$V_{\text{eff}}^{I}(x)=V_{DW}(x)+g_{IB}~{}N_{B}\rho_{1}^{B}(x),$ (14)

which is simply the DW superimposed to a potential proportional to the bosonic
density. With increasing $g_{IB}$, we observe that $V_{\text{eff}}^{I}(x)$
deforms from the initial DW towards a triple-well structure for $g_{IB}=1.0$
with two dominant barriers located at the positions $x\approx\pm 1.5$ [cf.
Fig. 8 (b)]. The central barrier of the DW is gradually smeared out in the
resulting effective potential $V_{\text{eff}}^{I}(x)$ due to the growth of the
potential term $g_{IB}~{}N_{B}\rho_{1}^{B}(x)$ for larger $g_{IB}$ 555Note
that the corresponding zero-point energies are excluded from the depicted
effective potentials in Fig. 8 (b) and thus $V_{\text{eff}}^{I}(x)$ have the
same minimum independently of $g_{IB}$.. As a result, the impurity becomes
significantly localized at the center of the DW [cf. Fig. 6 (b), black solid
line]. Before proceeding, we note that within the parameter space that we have
examined, the effective potential for the bosonic species only slightly
deviates from the initial DW due to the large particle number imbalance.
Hence, the corresponding bosonic Schmidt orbital $|\psi^{B}\rangle$ resembles
the GS of the Hamiltonian $\hat{H}_{B}$, i.e.,

$|\psi^{B}\rangle\approx|\phi_{0}^{B}\rangle.$ (15)

Note, however, that the case $g_{IB}\rightarrow\infty$ will result in a
bosonic effective potential significantly different from the DW. This could
subsequently render the employed two-mode approximation insufficient for
describing the bosonic gas [cf. Eq. (2)]. A discussion of this limit, is
therefore, beyond the scope of our study.

The SMF description not only introduces a significant simplification in the
study of mixtures but also allows for profound insights concerning the polaron
decay as well as its OC. Indeed, the results provided in Eq. (12) and Eq. (15)
suggest that the residue $Z$ for $g_{IB}>0.2$ simply becomes the overlap
between the Schmidt orbital $|\psi^{I}\rangle$, i.e., the GS of the effective
potential $V_{\text{eff}}^{I}(x)$, and the initial SP state
$|\phi_{0}^{I}\rangle$ [cf. Fig. 8 (a) blue solid and red dashed lines]. Upon
the increase of $g_{IB}$, the deformation of the density profile
$\rho^{I}_{SMF}(x)=|\psi^{I}(x)|^{2}$ from
$\rho^{I}_{0}(x)=|\phi_{0}^{I}(x)|^{2}$ [cf. Fig. 8 (c)] suggests the
subsequent decrease of the overlap between $|\psi^{I}\rangle$ and
$|\phi_{0}^{I}\rangle$, and hence, leads to the corresponding residue decay
[see Eq. (6) and Fig. 8 (a)]. Based on the behavior of the SP occupations
presented in Fig. 7, we know that the impurity is dominated by interband
transitions during this process. In this regard, we refer to this decay
mechanism as the interband excitation induced polaron decay.

Turning to the case of $g_{IB}=1.0$, which corresponds to the OC of the
polaron we note that the negligible spatial overlap between
$\rho^{I}_{SMF}(x)$ and $\rho^{I}_{0}(x)$ [cf. Fig. 8 (c), blue solid line and
black dashed line] indicates that $\langle\phi_{0}^{I}|\psi^{I}\rangle\approx
0$. Together with the fact that $\rho^{I}_{0}(x)=\rho^{B}_{1}(x)$, the above
result implies that $\rho^{I}_{SMF}(x)\rho^{B}_{0}(x)\approx 0$, which
reflects the above-mentioned one-body interspecies phase separation [cf. Figs.
6 (a), (b) and (e)]. In this way, we obtain the physical connection between
the OC of the polaron and the one-body phase separation. With this knowledge,
let us point out the asymptotic form of the GS wave function corresponding to
this strongly interacting regime

$|\Psi_{T-II}\rangle=|\psi^{I}\rangle|\psi^{B}\rangle~{}~{}~{}\text{and}~{}~{}~{}\rho^{I}_{1}(x)\rho^{B}_{1}(x)=0,$
(16)

which accounts for the above OC of the polaron. Interestingly, for a fixed
large $g_{IB}$ (e.g., $g_{IB}=1.0$) and for increasing $g_{BB}$, a transition
between the GS wave function $|\Psi\rangle=|\Psi_{T-I}\rangle$ for
$g_{BB}<g_{BB}^{cI}$ and $|\Psi\rangle=|\Psi_{T-II}\rangle$ for
$g_{BB}>g_{BB}^{cI}$ is realized. This crossover is also accompanied by the
deformation of the impurity’s density profile from the $\rho^{I}_{1}(x)$ in
Fig. 3 (b) (blue dashed line) to the one depicted in Fig. 6 (b) (black solid
line). In Sec. III.5, we will elaborate on the physical origin of this
crossover via a spectral analysis of the wave functions $|\Psi_{T-I}\rangle$
and $|\Psi_{T-II}\rangle$. Moreover, it is worth to be mentioned that in the
limiting case $N_{B}\rightarrow\infty$ the effective potential in Eq. (14) can
be approximated by $V_{\text{eff}}^{I}(x)\approx
g_{IB}~{}N_{B}\rho_{1}^{B}(x)$. The impurity thus feels two infinitely high
potential barriers located at positions $x\approx\pm 1.5$, which corresponds
to the two-hump structure of the bosonic density [cf. Fig. 3 (b)]. As a
result, it leads to a vanishing overlap between $\rho^{I}_{SMF}(x)$ and
$\rho^{B}_{0}(x)$ leaving the residue $Z\rightarrow 0$. Another important
consequence of the SMF description is that, due to the preservation of the
parity symmetry in the effective potential picture, the Schmidt orbital
$|\psi^{I}\rangle$ has always even parity. As a result, it leads to the
negligible SP occupations $\hat{n}_{i}^{sp}$ of all the odd parity states
$|\phi_{i}^{I}\rangle$ for $g_{IB}>0.2$ [see the red solid lines in Fig. 7
(b), (d), (f)].

Figure 7: (Color online) SP occupations of the impurity for the type-II and
type-III residue decay regions. The SP occupation $n_{i}^{sp}$ of the impurity
with respect to $g_{IB}$ for $g_{BB}=0.008$ (red solid line) and $g_{BB}=0.05$
(blue dashed line). The left (right) panels correspond to all even (odd)
parity single-particle states. Figure 8: (Color online) (a) The polaronic
residue $Z$ as a function of $g_{IB}$ as predicted within ED for fixed
$g_{BB}=0.008$ (blue solid line). The green dash-dotted line represents the
resulting residue $Z$ by restricting the impurity within the lowest band of
the DW. The red dashed line denotes the overlap between $|\psi^{I}\rangle$ and
$|\phi_{0}^{I}\rangle$ with respect to $g_{IB}$ for $g_{BB}=0.008$ thus using
the SMF method (no entanglement). It can be concluded that the residue
predictions of the lowest-band (SMF) are in good agreement with ED for weak
(strong) impurity-medium couplings but both approaches fail to predict the
correct residue behavior in the intermediate interaction regime. (b) The
effective potential of the impurity for $g_{BB}=0.008$ and $g_{IB}=0.2$ (red
dash-dotted line) and $g_{IB}=1.0$ (blue solid line). The black dashed line
represents the DW. (c) The corresponding impurity density distributions
obtained from the Schmidt orbital $\rho^{I}_{SMF}(x)=|\psi^{I}(x)|^{2}$ (red
dash-dotted and blue solid line) for the cases examined in (b). The black
dashed line shows the impurity density $\rho^{I}_{0}(x)=|\phi_{0}^{I}(x)|^{2}$
for $g_{IB}=0$ (see main text). (d) The pair-correlation function for
$g_{BB}=0.008$ and $g_{IB}=0.1$ obtained from ED simulations, explicating the
emergent two-body impurity-medium correlations.

###  III.4 Type-III polaron decay

Finally we explore the polaron decay in the interaction region
$g_{BB}>g_{BB}^{cII}$. For a fixed $g_{IB}>0.2$, the negligible discrepancies
between the residue values for the case $g_{BB}=0.008$ and $g_{BB}=0.05$
suggest that the GS wave function takes the same product form as introduced in
Eq. (12) [cf. Fig. 2 (d) blue solid and black solid line]. This is indeed
confirmed by inspecting the $g_{IB}$-dependence of the Schmidt numbers finding
that $\\{\lambda_{i\geqslant 2}\\}$ have negligible values for $g_{BB}=0.05$
and $g_{IB}>0.2$ (results not shown). It turns out that the above relation
holds for all cases with $g_{IB}<0.2$ as well. As a matter of fact, the
mixture in the region $g_{BB}>g_{BB}^{cII}$ is fully described by the SMF
ansatz. The corresponding polaron decay with increasing $g_{IB}$ can thus be
readily interpreted via the above introduced effective potential. As expected,
it gives rise to the same asymptotic form of the GS wave function
$|\Psi_{T-II}\rangle$ as introduced in Eq. (16) for large $g_{IB}$ as well as
the negligible SP occupations $\hat{n}_{i}^{sp}$ for all odd parity states
$|\phi_{i}^{I}\rangle$ [see the blue dashed lines in Fig. 7 (b), (d) and (f)].
Due to the small variations of the effective potential for bosons with respect
to the initial DW, we note that for the bosonic species the relation in Eq.
(15) holds as well, indicating the negligible probability for the bosons to
populate the excited states $\\{|\phi_{i>0}^{B}\rangle\\}$.

Albeit the fact that, for a fixed $g_{IB}$, there are no significant
differences among the reduced densities $\rho^{\sigma}_{1}(x)$ and
$\rho^{IB}_{2}(x_{I},x_{B})$ between the case of $g_{BB}>g_{BB}^{cII}$ and the
one for $g_{BB}\in[g_{BB}^{cI},g_{BB}^{cII}]$, prominent deviations are
imprinted in the corresponding pair-correlation function. The latter is
defined as

$g_{2}(x_{I},x_{B})=\frac{\rho^{IB}_{2}(x_{I},x_{B})}{\rho^{I}_{1}(x_{I})\rho^{B}_{1}(x_{B})}.$
(17)

Notice that, through the division by the single-particle densities the $g_{2}$
function, as compared to the two-body density $\rho^{IB}_{2}(x_{I},x_{B})$,
naturally excludes the impact of the spatial inhomogeneity. It can be also
readily deduced that within the SMF description $g_{2}$ is simply unity due to
the product form of the many-body wave function. In contrast, for $g_{IB}=0.1$
and $g_{BB}=0.008$ the pair-correlation function deviates significantly from
unity with $g_{2}<1$ ($g_{2}>1$) along the diagonal (off-diagonal) region.
This pattern signifies the presence of strong anti-correlations between the
impurity and the majority bosons [cf. Fig. 8 (d)].

###  III.5 Origin of the critical bosonic repulsion

So far, we have discussed the properties of the three different types of
residue decay in terms of $g_{BB}$ upon increasing the repulsive impurity-bath
interaction strength. It has been found that the corresponding residue regimes
are characterized by two critical bosonic repulsion strengths $g_{BB}^{cI}$
and $g_{BB}^{cII}$. In addition, two asymptotic GS wave function forms namely
$|\Psi_{T-I}\rangle$ and $|\Psi_{T-II}\rangle$ were obtained for strong
impurity-bath repulsion, which in turn account for the OC of the polaron.
Interestingly, for a fixed large $g_{IB}$ (e.g., $g_{IB}=1.0$) and increasing
$g_{BB}$, we observe an abrupt transition of the GS wave function from
$|\Psi\rangle=|\Psi_{T-I}\rangle$ for $g_{BB}<g_{BB}^{cI}$ towards
$|\Psi\rangle=|\Psi_{T-II}\rangle$ for $g_{BB}>g_{BB}^{cI}$. Hereafter, we
will first perform a spectral analysis with respect to the above two
asymptotic wave functions. This allows us to unveil the origin of the critical
bosonic repulsion $g_{BB}^{cI}$. Afterwards, we comment on the existence of
$g_{BB}^{cII}$ and demonstrate how the impurity-medium mass ratio affects
their values.

To begin with, we note that for both $|\Psi_{T-I}\rangle$ and
$|\Psi_{T-II}\rangle$ the corresponding interspecies interaction energies
given by $\langle\Psi_{T-I}|\hat{H}_{IB}|\Psi_{T-I}\rangle$ and
$\langle\Psi_{T-II}|\hat{H}_{IB}|\Psi_{T-II}\rangle$ can be safely neglected.
In fact, due to the phase separation occurring on either the one-body or the
two-body level in the limit $g_{IB}\rightarrow 1$, atoms from different
species have negligible probability to reside in the vicinity of one another.
Together with the employed contact form of the interspecies interaction, it
thereby results in a negligible interspecies interaction energy. In this
regard, we solely need to compute the energies associated with the single-
species Hamiltonians $\hat{H}_{I}$ and $\hat{H}_{B}$ [Eq. (1)]. For
$|\Psi_{T-I}\rangle$, we have

$E_{T-I}=\langle\Psi_{T-I}|\hat{H}_{I}|\Psi_{T-I}\rangle+\langle\Psi_{T-I}|\hat{H}_{B}|\Psi_{T-I}\rangle=E_{SW}^{I}+E_{SW}^{B},$
(18)

with $E_{SW}^{I}$ ($E_{SW}^{B}$) being the energy corresponding to the
configuration where the impurity (all the bosons with a fixed $g_{BB}$) are
restricted in either the left (right) or the right (left) well. On the other
hand, for $|\Psi_{T-II}\rangle$ the resulting energy reads

$E_{T-II}=\langle\Psi_{T-II}|\hat{H}_{I}|\Psi_{T-II}\rangle+\langle\Psi_{T-II}|\hat{H}_{B}|\Psi_{T-II}\rangle=E_{SMF}^{I}+E_{0}^{B},$
(19)

in which $E_{SMF}^{I}=\langle\psi^{I}|\hat{H}_{I}|\psi^{I}\rangle$ represents
the impurity energy obtained from the SMF Schmidt orbital $|\psi^{I}\rangle$
corresponding to a strong impurity-bath repulsion and fixed
$g_{BB}>g_{BB}^{cI}$. Here, we set $g_{IB}=1.0$ and $g_{BB}=0.008$. We note
that as long as $g_{BB}>g_{BB}^{cI}$, different values of $g_{BB}$ do not
impact the shape of the impurity’s effective potential, thus resulting in the
same Schmidt orbital $|\psi^{I}\rangle$ (see also the discussion in Sec.
III.3). Also, $E_{0}^{B}$ is the GS energy of the Hamiltonian $\hat{H}_{B}$
for a fixed $g_{BB}$.

Eqs. (18) and (19) essentially reveal that both $E_{T-I}$ and $E_{T-II}$
depend on the bosonic repulsion strength $g_{BB}$. Therefore, the competition
between the two energies upon the increase of $g_{BB}$ directly determines the
form of the GS wave function $|\Psi\rangle$ in the asymptotic limit of strong
$g_{IB}$. In the case of $g_{BB}\approx 0$, the negligible width of the lowest
band of the DW leads to $E_{SW}^{I}\approx E_{0}^{I}$ and $E_{SW}^{B}\approx
E_{0}^{B}$ with $E_{0}^{I}$ being the GS energy of $\hat{H}_{I}$. Since
$E_{SMF}^{I}$ is obviously larger than $E_{0}^{I}$, it thus results in
$E_{T-I}<E_{T-II}$. Thereby, the wave function $|\Psi_{T-I}\rangle$ is
energetically more favorable. In contrast, with increasing $g_{BB}$, the rapid
growth of the on-site bosonic repulsion renders the energy $E_{SW}^{B}$
dominant as compared to all other energies. As a result, the GS of the mixture
turns out to be $|\Psi\rangle=|\Psi_{T-II}\rangle$. Based on this knowledge,
we can deduce that the critical bosonic repulsion strength $g_{BB}^{cI}$
corresponds to the case where $E_{T-I}=E_{T-II}$ holds [cf. Fig. 9]. Turning
to $g_{BB}^{cII}$, it corresponds to the situation that the GS of the mixture
always acquires the product form
$|\Psi\rangle=|\psi^{I}\rangle|\psi^{B}\rangle$ for different $g_{IB}$, in
which $|\psi^{B}\rangle\approx|\phi_{0}^{B}\rangle$ [cf. Eqs. (12) and (15)].
Thus, the bosonic species is hardly excited irrespectively of the variations
of the interspecies coupling. Accordingly, it becomes clear that only a large
enough $g_{BB}$ can result in corresponding large energy differences between
the GS and the excited states of $\hat{H}_{B}$. This fact effectively
prohibits the excitations among the bosons.

Finally, let us demonstrate how the variations of mass-imbalance impact the
values of the above-discussed bosonic repulsions. For a light impurity, the
associated SP wave functions $\\{\phi_{i}^{I}(x)\\}$ are much more spatially
extended as compared to the ones for a heavy impurity (results are not shown).
This facilitates the interband excitations of the impurity quantified by the
transition amplitude $T^{I}=\sum_{i>1}U_{i}$ with

$U_{i}=\sum_{j=0,1}\sum_{k,l=0,1}g_{IB}\int
dx~{}|\phi_{i}^{I}(x)\phi_{j}^{I}(x)\varphi_{k}^{B}(x)\varphi_{l}^{B}(x)|^{2}.$
(20)

It represents the magnitude of the transition amplitude of the impurity, due
to the impurity-bath coupling, between the energetically lowest and higher
bands of the DW after averaging out different configurations of the bosons.
Recall that, from the discussions in Sec. III.3 and Sec. III.4, it is known
that $g_{BB}^{cII}$ essentially corresponds to the regime where the impurity
is dominated by interband excitation processes upon increasing $g_{IB}$.
Hence, it gives rise to a smaller $g_{BB}^{cII}$ in the case of $\beta=0.5$
when compared to $\beta=2.0$. Regarding $g_{BB}^{cI}$, it essentially
characterizes the transition of the GS wave function from $|\Psi_{T-I}\rangle$
to $|\Psi_{T-II}\rangle$ for larger $g_{IB}$. As we have already discussed
above, the interspecies interaction energy in this limit always vanishes.
Accordingly, changes of $T^{I}$ lead to a negligible impact on this GS
crossover. It is for this reason that $g_{BB}^{cI}$ is inert to variations of
the mass-imbalance $\beta$.

Figure 9: (Color online) Energy competition between two asymptotic GS wave
functions (see main text). The energies $E_{T-I}$ (red solid line) and
$E_{T-II}$ (blue solid line) obtained within ED as a function of $g_{BB}$.
Here, $E_{T-I}=\langle\Psi_{T-I}|\hat{H}_{I}+\hat{H}_{B}|\Psi_{T-I}\rangle$
and $E_{T-II}=\langle\Psi_{T-II}|\hat{H}_{I}+\hat{H}_{B}|\Psi_{T-II}\rangle$.

##  IV Conclusions and Outlook

We have investigated the polaronic properties in the ground state of a binary
atomic mixture consisting of a single impurity and bosonic gas confined in a
1D DW. Due to the trap-induced spatial imhomogeneity, three distinct polaronic
residue regions in terms of the bosonic repulsion have been identified with
respect to the impurity-bath interaction strength. They are characterized by
two critical values of the bosonic repulsion denoted as $g_{BB}^{cI}$ and
$g_{BB}^{cII}$. Moreover, the increase of $g_{IB}$ always results in a
suppressed residue, signifying the occurrence of the OC of the polaron.
Depending on the value of $g_{BB}$ being below or above $g_{BB}^{cI}$ or
$g_{BB}^{cII}$, the residue exhibits a sharp decrease, an initial decrease
followed by a pronounced revival or a slow monotonous decay, respectively. The
presence of these interaction regions stems from the interplay between the
intra- and interband excitations of the impurity and are clearly imprinted in
the structure of the corresponding reduced density matrices. The latter
essentially reveal a phase separation on either the one- or the two-body level
depending on the bosonic repulsion. Additionally, it is found that the
interspecies mass ratio affects the values of the $g_{BB}^{cII}$ critical
bosonic repulsion while $g_{BB}^{cI}$ appears to be almost un-affected. It is
argued that this behavior can be understood in terms of the respective
interband transition amplitudes.

To provide deeper insights into the polaron decay and the existence of the
interaction regions where the residue exhibits a distinct behavior, we perform
a detailed analysis of the structure of the many-body wave function at
specific limits. By utilizing a Schmidt decomposition of the many-body wave
function, it allows us to construct two asymptotic configurations of the GS
wave function for strong interspecies repulsion. Importantly, they also
account for the observed OC of the polaron. Finally, by means of a spectral
analysis, we elucidate the physical origin of the observed two critical
bosonic repulsions and demonstrate how the mass-imbalance impacts their
values.

Possible future research directions include the investigation of the polaronic
properties arising in the ground state of the binary atomic mixture in the
presence of long-range e.g. dipolar interactions. Another interesting
perspective is to study the polaron dynamics following an interaction quench.
Here, the impact of higher-band excitations of the impurity or a beyond the
two-site Bose-Hubbard description for the bosonic species would be worthwhile
to pursue.

###### Acknowledgements.

This work has been funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) - SFB 925 - project 170620586. S.I.M. gratefully
acknowledges financial support from the NSF through a grant for ITAMP at
Harvard University. The authors acknowledge fruitful discussions with G.M.
Koutentakis.

##  Appendix A Impact of an even parity GS wave function $|\Psi\rangle$ to the
bosonic single-particle density

Let us demonstrate that an even parity GS wave function $|\Psi\rangle$
essentially implies that the bosonic spatial distribution between the two
wells obeys the condition
$\hat{b}^{\dagger}_{L}\hat{b}_{L}=\hat{b}^{\dagger}_{R}\hat{b}_{R}=N_{B}/2$.
Therefore, it leads to an invariant $\rho^{B}_{1}(x)$ profile against
variations of the involved interaction strengths. To showcase this behavior,
we first rewrite the bosonic annihilation operator as
$\hat{b}_{L,R}=\frac{1}{\sqrt{2}}(\hat{a}_{0}^{B}\pm\hat{a}_{1}^{B})$, with
$\hat{a}_{i}^{B}$ being the annihilation operator acting on the $i$th SP state
$|\varphi_{i}^{B}\rangle$ of the DW. As a result, the occupation number of the
bosons on the left (right) well becomes

$N_{L,R}^{B}=\langle\Psi|\hat{b}_{L,R}^{\dagger}\hat{b}_{L,R}|\Psi\rangle=\frac{N_{B}}{2}\pm\frac{1}{2}\langle\Psi|(\hat{a}_{0}^{B})^{\dagger}\hat{a}_{1}^{B}+(\hat{a}_{1}^{B})^{\dagger}\hat{a}_{0}^{B}|\Psi\rangle,$
(21)

where we have used the fact that
$(\hat{a}_{0}^{B})^{\dagger}\hat{a}_{0}^{B}+(\hat{a}_{1}^{B})^{\dagger}\hat{a}_{1}^{B}=N_{B}$,
due to the particle number conservation and the employed two-mode
approximation for the bosons. Further employing a Schmidt decomposition on the
GS wave function $|\Psi\rangle$, Eq. (21) reduces to

$N_{L,R}^{B}=\frac{N_{B}}{2}\pm\frac{1}{2}\sum_{i=1}^{\infty}\lambda_{i}\left[\langle\psi_{i}^{B}|(\hat{a}_{0}^{B})^{\dagger}\hat{a}_{1}^{B}+(\hat{a}_{1}^{B})^{\dagger}\hat{a}_{0}^{B}|\psi_{i}^{B}\rangle\right],$
(22)

with the last term in Eq. (22) corresponding to a parity-flipping operation
acting on the bosonic species. Since the GS wave function $|\Psi\rangle$ is of
even parity, each product state $|\psi_{i}^{I}\rangle|\psi_{i}^{B}\rangle$ in
Eq. (7) then has an even parity as well, which renders the bosonic Schmidt
orbital $|\psi_{i}^{B}\rangle$ possessing a definitive parity. As a result,
the last term in Eq. (22) vanishes and thus we have $N_{L,R}^{B}=N_{B}/2$.

##  Appendix B Expressing the asymptotic wave function $|\Psi_{T-I}\rangle$ in
the Wannier-basis representation

In this part, we demonstrate that the wave function $|\Psi_{T-I}\rangle$ of
Eq. (7) is equivalent to

$|\Psi_{T-I}\rangle=\frac{1}{\sqrt{2N_{B}!}}\left[\hat{a}_{L}^{\dagger}(\hat{b}_{R}^{\dagger})^{N_{B}}+\hat{a}_{R}^{\dagger}(\hat{b}_{L}^{\dagger})^{N_{B}}\right]|0\rangle,$
(23)

where

$\hat{a}_{L,R}=\frac{1}{\sqrt{2}}(\hat{a}_{0}^{I}\pm\hat{a}_{1}^{I}),~{}~{}~{}~{}\hat{b}_{L,R}=\frac{1}{\sqrt{2}}(\hat{a}_{0}^{B}\pm\hat{a}_{1}^{B}),$
(24)

with $\hat{a}_{i}^{I}$ ($\hat{a}_{i}^{B}$) being the annihilation operator
acting on the $i$th SP state $|\phi_{i}^{I}\rangle$
($|\varphi_{i}^{B}\rangle$) of the impurity and the bosonic species,
respectively. By substituting Eq. (24) into Eq. (23), it results in
$|\Psi_{T-I}\rangle=\frac{1}{\sqrt{2}}\left[|\phi_{0}^{I}\rangle|\tilde{\psi}_{1}^{B}\rangle+|\phi_{1}^{I}\rangle|\tilde{\psi}_{2}^{B}\rangle\right]$,
where

$\displaystyle|\tilde{\psi}_{1}^{B}\rangle$
$\displaystyle=\sum_{k=0,2,4,\dots}^{N_{B}}\tilde{C}^{B}_{1,k}|\phi_{k}^{B}\rangle=\sum_{k=0,2,4,\dots}^{N_{B}}(\frac{1}{\sqrt{2}})^{N_{B}-1}\left(\begin{array}[]{c}N_{B}\\\
k\end{array}\right)^{1/2}|\phi_{k}^{B}\rangle,$ (27)
$\displaystyle|\tilde{\psi}_{2}^{B}\rangle$
$\displaystyle=\sum_{k=1,3,5,\dots}^{N_{B}}\tilde{C}^{B}_{2,k}|\phi_{k}^{B}\rangle=\sum_{k=1,3,5,\dots}^{N_{B}}-(\frac{1}{\sqrt{2}})^{N_{B}-1}\left(\begin{array}[]{c}N_{B}\\\
k\end{array}\right)^{1/2}|\phi_{k}^{B}\rangle.$ (30)

Here, $|\phi_{k}^{B}\rangle$ denotes the $k$th eigenstate of $\hat{H}_{B}$ for
$g_{BB}=0$. The expansion coefficients $|\tilde{C}^{B}_{1,k}|^{2}$ and
$|\tilde{C}^{B}_{2,k}|^{2}$ in Eq. (30) follow a binomial distribution and
match exactly to the $|C^{B}_{1,k}|^{2}$ and $|C^{B}_{2,k}|^{2}$ of Eq. (9)
[cf. Fig. 5 (a) black solid line, red solid line and blue dashed line]. In
this way, we show that $|\Psi_{T-I}\rangle$ in Eq. (7) is equivalent to the
form of Eq. (23).

##  Appendix C The species mean-field description

The species mean-field description assumes the wave function of the mixture to
have a simple product form, i.e.
$|\Psi\rangle=|\psi^{I}\rangle|\psi^{B}\rangle$. This leads to the
corresponding Lagrangian of the mixture

$\mathcal{L}=\langle\Psi|\hat{H}|\Psi\rangle+\sum_{\sigma=I,B}\mu_{\sigma}\left[1-\langle\psi^{\sigma}|\psi^{\sigma}\rangle\right],$
(31)

where $\mu_{\sigma}$ is the associated Lagrange multiplier under the
constraint of norm conservation of $|\psi^{\sigma}\rangle$. By utilizing a
variational principle with respect to each orbital $|\psi^{\sigma}\rangle$, we
immediately obtain a Schrödinger-type equation for the $\sigma$-species
$\hat{H}^{\sigma}_{\text{eff}}|\psi^{\sigma}\rangle=\mu_{\sigma}|\psi^{\sigma}\rangle$,
in which the effective Hamiltonian $\hat{H}^{\sigma}_{\text{eff}}$ reads

$\hat{H}^{\sigma}_{\text{eff}}=\hat{H}_{\sigma}+\langle\psi^{\bar{\sigma}}|\hat{H}_{IB}|\psi^{\bar{\sigma}}\rangle,$
(32)

with $\bar{\sigma}=B(I)$ for $\sigma=I(B)$. In this way, we note that the GS
of $\hat{H}^{\sigma}_{\text{eff}}$ is equivalent to the Schmidt orbital
$|\psi^{\sigma}\rangle$. Moreover, the last term in Eq. (32) represents a
potential

$\hat{V}^{\sigma}=\langle\psi^{\bar{\sigma}}|\hat{H}_{IB}|\psi^{\bar{\sigma}}\rangle=g_{IB}N_{\bar{\sigma}}\int
dx~{}\rho_{1}^{\bar{\sigma}}(x)~{}\hat{\psi}^{\dagger}_{\sigma}(x)\hat{\psi}_{\sigma}(x),$
(33)

with
$\rho_{1}^{\bar{\sigma}}(x)=\langle\psi^{\bar{\sigma}}|\hat{\psi}^{\dagger}_{\bar{\sigma}}(x)\hat{\psi}_{\bar{\sigma}}(x)|\psi^{\bar{\sigma}}\rangle/N_{\bar{\sigma}}$
being the one-body density of the $\bar{\sigma}$-species. From Eqs. (32) and
(33), we conclude that within the species mean-field description the mutual
impact of the species is merely an additional potential experienced by the
other species.

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