# Dynamical excitation processes and correlations of three-body
two-dimensional mixtures

G. Bougas<EMAIL_ADDRESS>Center for Optical Quantum
Technologies, Department of Physics, University of Hamburg, Luruper Chaussee
149, 22761 Hamburg Germany  S. I. Mistakidis  ITAMP, Center for Astrophysics
$|$ Harvard $\&$ Smithsonian, Cambridge, MA 02138 USA  Department of Physics,
Harvard University, Cambridge, Massachusetts 02138, USA  P. Giannakeas  Max-
Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, D-01187
Dresden, Germany  P. Schmelcher  Center for Optical Quantum Technologies,
Department of Physics, University of Hamburg, Luruper Chaussee 149, 22761
Hamburg Germany  The Hamburg Centre for Ultrafast Imaging, University of
Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany

###### Abstract

A scheme is proposed to dynamically excite distinct eigenstate superpositions
in three-body Bose-Fermi mixtures confined in a two-dimensional harmonic trap.
The system is initialized in a non-interacting state with a variable spatial
extent, and the scattering lengths are subsequently quenched spaning the
regime from weak to strong interactions. For spatial widths smaller than the
three-body harmonic oscillator length, a superposition of trimers and atom-
dimers is dynamically attained, otherwise trap states are predominantly
populated, as inferred from the frequency spectrum of the fidelity.
Accordingly, the Tan contacts evince the build-up of short range two- and
three-body correlations in the course of the evolution. A larger spatial
extent of the initial state leads to a reduction of few-body correlations,
endowed however with characteristic peaks at the positions of the avoided-
crossings in the energy spectra, thereby signalling the participation of atom-
dimers. Our results expose ways to dynamically excite selectively trimers,
atom-dimers and trapped few-body states characterized by substantial
correlations and are likely to be accessible within current experiments.

##  I Introduction

The appealing feature of ultracold physics is the controllability of
interactions that permits to study a plethora of phenomena, such as the
formation of droplets Petrov (2015); Luo  _et al._ (2021); Chomaz  _et al._
(2022) and polarons Massignan  _et al._ (2014); Ness  _et al._ (2020), and
understand in depth the build-up of few- and many-body correlations Mistakidis
_et al._ (2022). More specifically, the few-body correlations can be
quantified by Tan contacts. These stem from the short-range character of the
interatomic interactions Tan (2008a, b, c); Braaten  _et al._ (2011); Werner
and Castin (2012); Valiente  _et al._ (2012); Olshanii and Dunjko (2003); He
_et al._ (2016); Zhang  _et al._ (2017), and are experimentally probed through
radio-frequency (rf) spectroscopy Fletcher  _et al._ (2017); Sagi  _et al._
(2012); Wild  _et al._ (2012), time-of-flight expansion Stewart  _et al._
(2010) or Bragg spectroscopy Hoinka  _et al._ (2013); Kuhnle  _et al._ (2010).
Contacts interrelate macroscopic observables at equilibrium, such as energy
and pressure of a gas, in terms of few-body microscopic mechanisms Werner and
Castin (2012); Braaten  _et al._ (2008) addressing the properties of a gas
universally regardless of the atom number, the statistics, or the interaction
strength.

The recent realization of three-dimensional (3D) unitary Bose gases offers the
possibility to investigate the dynamical formation of few-body correlations in
strongly interacting ultracold matter Klauss  _et al._ (2017); Makotyn  _et
al._ (2014); Eigen  _et al._ (2017, 2018). Quenching the scattering length
from the non-interacting case to unitarity enables the experimental
observation of few-body states such as the Efimov ones, i.e. an infinite
geometric progression of three-body bound levels comprised of unbound two-body
subsystems Nielsen  _et al._ (2001); Efimov (1971). In addition, theoretical
efforts demonstrated that the quenched dynamics of such three-body systems
exhibits unique features in the population growth of Efimov trimers and atom-
dimers Colussi  _et al._ (2018); D’Incao  _et al._ (2018); Colussi  _et al._
(2019, 2020); Musolino  _et al._ (2022). For example, in Ref. Colussi  _et
al._ (2018) it was argued that the two-body Tan contact is enhanced during the
early stages of the dynamics, whereas the three-body contact increases
appreciably only when the interparticle spacing matches the size of an Efimov
state D’Incao  _et al._ (2018). However, the latter are typically short-lived
due to three-body recombination processes Greene  _et al._ (2017).

Promising candidates to mitigate such losses maintaining a high fraction of
trimer states, are two-dimensional (2D) gases D’Incao  _et al._ (2015);
Levinsen  _et al._ (2014). There the corresponding trimer wave functions have
a small amplitude at short distances suppressing three-body recombination
processes Kirk and Parish (2017); Levinsen  _et al._ (2014) as compared to 3D
systems. Additionally, theoretical studies in 2D three-body systems Bellotti
_et al._ (2013a); Pricoupenko and Pedri (2010); Bellotti  _et al._ (2011);
Bougas  _et al._ (2021); Sandoval  _et al._ (2016); Kartavtsev and Malykh
(2006); Bruch and Tjon (1979) have addressed their time-independent attributes
in terms of their eigenspectrum as well as their corresponding few-body
correlations via Tan contacts Bellotti  _et al._ (2014); Valiente  _et al._
(2011); Bellotti  _et al._ (2013b); Bougas  _et al._ (2021). In particular, it
was shown that mass-imbalanced mixtures support a multitude of trimer states
with amplified two- and three-body correlations compared to the mass-balanced
case Bougas  _et al._ (2021); Bellotti  _et al._ (2014).

In contrast to the 3D systems Kerin and Martin (2022a); Colussi  _et al._
(2019); D’Incao  _et al._ (2018); Colussi  _et al._ (2018); Kerin and Martin
(2022b), the dynamical response including the underlying excitation processes
and accompanying correlation mechanisms of 2D three-body systems is not well-
understood. Importantly, the study of these systems is thus far restricted to
their stationary correlation properties Bellotti  _et al._ (2014, 2013a,
2011); Valiente  _et al._ (2011) in the absence of external confinement. In
this work, a protocol is proposed for triggering specific excitation branches
in 2D harmonically trapped mixtures of two identical bosons or fermions
interacting with another atom. Apart from the particle statistics, our study
addresses the effect of unequal massed three-body collisions. In 3D gases, it
is known that highly mass-imbalanced systems exhibit rich resonant effects
Ulmanis  _et al._ (2016); Giannakeas and Greene (2021, 2018); Mikkelsen  _et
al._ (2015); Wacker  _et al._ (2016); Johansen  _et al._ (2017), or favor the
observation of multiple successive Efimov states Tung  _et al._ (2014); Pires
_et al._ (2014a); Kerin and Martin (2022), while offering unique platforms to
study reaction rates in atom-dimer and molecule-molecule collisions Gao and
Zhang (2018); Shalchi  _et al._ (2020, 2018); Rui  _et al._ (2017); Yang  _et
al._ (2019); Hoffmann  _et al._ (2018); Makrides  _et al._ (2020). Therefore,
the inclusion of unequal masses here provides a comprehensive description of
the dynamic properties of 2D three-body collisions ranging from light-light-
heavy (LLH) to heavy-heavy-light (HHL) systems.

Initially, the three-body mixture is considered in a non-interacting state
characterized by a parameter $w$, describing its spatial extent. Subsequently,
the interactions are turned on abruptly (interaction quench), resulting into
distinct dynamical response regimes characterized by specific excitation
mechanisms and correlations being imprinted in the fidelity spectrum. The
Hilbert space of the post-quench three-body system, at the final values of the
scattering lengths, is mainly partitioned into three generic types of
eigenstates: trimers, atom-dimers and trap states. For widths $w$ of the
initial state smaller than the harmonic oscillator length scale, we observe
that the dominant excitation branches identified in the fidelity spectrum
correspond to trimers and atom-dimers. In the case of HHL systems, however,
these states are prevalent over a relatively smaller range of scattering
lengths. For an increasing width $w$ of the initial state the trap states are
predominantly populated.

Furthermore, we show that the participation of distinct eigenstates impacts
strongly the dynamics of short-range correlations quantified by the Tan
contacts. In particular, both the two- and three-body correlations become
enhanced for initial state widths smaller than the spatial extent of the trap.
The correlations are suppressed as the width of the initial configuration is
increased since the population of trap states becomes more dominant. In
addition, distinct peaks in the few-body correlations are observed as the
scattering lengths vary. This structure arises from the narrow avoided-
crossings in the eigenspectrum where the atoms are in a superposition of trap
and atom-dimer states. The above mentioned features occur for both LLH and HHL
settings regardless the exchange symmetry of the particles. However, the
enhancements in the few-body contacts become narrower in the HHL case, as
compared to the LLH one, due to the existence of sharp avoided-crossings in
the respective energy spectrum Bougas  _et al._ (2021).

This work proceeds as follows: In Sec. II the adiabatic hyperspherical
formalism is briefly outlined, and in Sec. III the initial ansatz of the
three-body system and the time-evolved wave function are introduced.
Subsequently, the excitation spectra, associated modes and correlation
dynamics based on the fidelity spectrum and Tan contacts are unveiled for both
LLH systems in Sec. IV and HHL ones in Sec. V. In Sec. VI we briefly comment
on the possible experimental realization of our setup. Sec. VII lays out our
conclusions and provides an outlook. Moreover, Appendix A introduces the
adiabatic Hamiltonian and the 2D zero-range pseudopotential. Appendix B
provides the form of the hyperangular wave function for the non-interacting
initial state. In Appendix C, we elaborate on the excitation spectrum of the
BBX LLH system for widths of the initial state equal to the three-body
harmonic oscillator length.

##  II Adiabatic hyperspherical representation of the three-body mixture

In the following we consider three-body binary mass-imbalanced mixtures
trapped in a 2D harmonic oscillator of frequency $\omega$. They typically
consist of either two identical bosons (BBX) or two identical non-interacting
spin polarized fermions (FFX) interacting with a third distinguishable
particle. The underlying pairwise interactions are modeled with $s$-wave zero-
range pseudopotentials Olshanii and Pricoupenko (2001) characterized by 2D
scattering lengths $a_{FX}$ and $a_{BB}$, $a_{BX}$ for the FFX and BBX
systems, respectively. Here, $a_{\sigma\sigma^{\prime}}$ denotes the 2D
scattering length between a particle of species $\sigma$ and
$\sigma^{\prime}$, where $\sigma=$B, X or $\sigma=$F, X. Below, for
simplicity, we typically consider variations of $1/a_{FX}$ and $a_{BB}/a_{BX}$
where in the latter case $a_{BB}$ is kept fixed. The magnitude of the 2D
scattering lengths can in principle be adjusted via standard Fano-Feshbach
resonances Chin  _et al._ (2010), since they parametrically depend on their 3D
counterparts Petrov and Shlyapnikov (2001). Let us note that by definition the
2D scattering lengths can only be positive, a property stemming from the
existence of a two-body bound state always in 2D, and the non-interacting
limit occurs when they are either $0$ or $+\infty$ Liu  _et al._ (2010).
Moreover, depending on the mass ratio between the identical atom and the third
particle, i.e. $m_{B/F}/m_{X}$, we distinguish between LLH and HHL cases. In
particular, the employed mass ratios are $m_{B}/m_{X}=0.04,\,22.16$ for BBX
referring to mixtures of
${}^{7}\rm{Li}-^{7}\rm{Li}-^{173}\rm{Yb},\>^{133}\rm{Cs}-^{133}\rm{Cs}-^{6}\rm{Li}$
and $m_{F}/m_{X}=0.0451,\,24.71$ for FFX corresponding to
${}^{6}\rm{Li}-^{6}\rm{Li}-^{133}\rm{Cs},\>^{173}\rm{Yb}-^{173}\rm{Yb}-^{7}\rm{Li}$
systems.

The stationary properties of these mixtures are straightforwardly addressed
within the adiabatic hyperspherical framework Nielsen  _et al._ (2001); Greene
_et al._ (2017); Naidon and Endo (2017); D’Incao (2018); Rittenhouse  _et al._
(2010), with the pairwise interactions modeled via contact pseudopotentials.
Owing to the decoupling of the center of mass, the hyperspherical coordinates
representation is employed and the relative position of the atoms is described
by a set of three hyperangles (which collectively are denoted by
$\bm{\Omega}$) and the hyperradius $R$ that controls the overall size of the
system. Hence, by employing the hyperspherical coordinates the relative three-
body Hamiltonian Bougas  _et al._ (2021) reads:

$H_{\rm{rel}}=-\frac{\hbar^{2}}{2\mu R^{3/2}}\frac{\partial^{2}}{\partial
R^{2}}R^{3/2}+\frac{1}{2}\mu\omega^{2}R^{2}+H_{{\rm ad}}(R;\bm{\Omega}).$ (1)

The first term refers to the kinetic energy, while the second one is the
external trapping potential. $H_{{\rm ad}}(R;\bm{\Omega})$ describes the
centrifugal motion of the three particles, and contains the pairwise $s$-wave
contact interactions, depending on the aforementioned 2D scattering lengths
[for more details see Appendix A]. Also, $\mu=m_{B/F}/\sqrt{2m_{B/F}/m_{X}+1}$
is the three-body reduced mass and $m_{B/F}$ stands for the mass of bosons or
fermions. Note that in the following we employ as a characteristic length
scale of the three-body system the quantity
$a_{\rm{ho}}=\sqrt{\hbar/\mu\omega}$, i.e. the three-body harmonic oscillator
length.

The eigenstates of the three-body system are determined as follows: First,
$H_{{\rm ad}}(R;\bm{\Omega})$ is diagonalized at fixed hyperradius $R$
Rittenhouse  _et al._ (2010) where the eigenvalues $s_{\nu}(R)$ are associated
with the adiabatic potential curves $\hbar^{2}(s_{\nu}^{2}(R)-1/4)/2\mu R^{2}$
and the corresponding eigenfunctions, i.e. $\Phi_{\nu}(R;\bm{\Omega})$, are
used as basis set for the three-body relative wave function. The latter in the
adiabatic hyperspherical representation is given by the expression
$\Psi(R,\bm{\Omega})=R^{-3/2}\sum_{\nu}F_{\nu}(R)\Phi_{\nu}(R;\bm{\Omega})$
111We note that in the following sections and appendices the wave functions
with the superscripts $\Psi(R,\bm{\Omega},t)$ or
$\Psi^{\rm{f}}(R,\bm{\Omega})$ indicate the time-evolved wave function at time
$t$ or the post-quench $\rm{f}$-th eigenstate respectively [see also Sec.
III].. $F_{\nu}(R)$ denotes the hyperradial component of $\Psi(R,\bm{\Omega})$
which satisfies the following system of coupled ordinary differential
equations:

$\displaystyle\Big{\\{}-\frac{\hbar^{2}}{2\mu}\frac{d^{2}}{dR^{2}}+U_{\nu}(R)\Big{\\}}F_{\nu}(R)$
$\displaystyle-\frac{\hbar^{2}}{2\mu}\sum_{\nu^{\prime}}\left[2P_{\nu\nu^{\prime}}(R)\frac{d}{dR}+Q_{\nu\nu^{\prime}}(R)\right]\,F_{\nu^{\prime}}(R)=EF_{\nu}(R).$
(2)

Here, $U_{\nu}(R)$ represents the $\nu$-th adiabatic potential curve including
the trap, whereas the $P_{\nu\nu^{\prime}}(R)$ and $Q_{\nu\nu^{\prime}}(R)$
terms denote the non-adiabatic coupling matrix elements. More specifically,
the adiabatic potential curves and the non-adiabatic coupling matrix elements
are given by the following expressions Bougas  _et al._ (2021); Rittenhouse
_et al._ (2010); Kartavtsev and Malykh (2006),

$\displaystyle U_{\nu}(R)$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{2\mu
R^{2}}\left(s^{2}_{\nu}(R)-\frac{1}{4}\right)+\frac{1}{2}\mu\omega^{2}R^{2}$
(3) $\displaystyle P_{\nu\nu^{\prime}}(R)$ $\displaystyle=$
$\displaystyle\braket{\Phi_{\nu}(R;\bm{\Omega})}{\frac{\partial\Phi_{\nu^{\prime}}(R;\bm{\Omega})}{\partial
R}}_{\bm{\Omega}}$ (4) $\displaystyle Q_{\nu\nu^{\prime}}(R)$ $\displaystyle=$
$\displaystyle\braket{\Phi_{\nu}(R;\bm{\Omega})}{\frac{\partial^{2}\Phi_{\nu^{\prime}}(R;\bm{\Omega})}{\partial
R^{2}}}_{\bm{\Omega}},$ (5)

where the symbol $\braket{\ldots}_{\bm{\Omega}}$ indicates that the
integration is over the hyperangles only. In the following, harmonic
oscillator units are adopted, unless stated otherwise, i.e.
$m_{B/F}=\hbar=\omega=1$, where $m_{B/F}$ is the mass of the identical bosons
or spin polarized fermions.

##  III Initialization and quench protocol

Initially the three atoms are prepared in a non-interacting state. This
situation in 2D translates to a scattering length either $0$ or $+\infty$,
which in the case of two harmonically trapped atoms is shown to reproduce the
corresponding non-interacting energy spectra Liu  _et al._ (2010); Bougas  _et
al._ (2019); Busch  _et al._ (1998). The state is characterized by
$1/a_{BX}=1/a_{BB}=0$ for BBX or $1/a_{FX}=0$ for FFX systems, while its
spatial extent is parametrized by $w$, see Fig. 1 (a). The initial three-body
wave function in the hyperspherical coordinate frame reads

$\displaystyle\Psi(R,\bm{\Omega},t=0)$ $\displaystyle=$
$\displaystyle\frac{R^{L}\sqrt{2}}{\sqrt{\Gamma(2+L)}w^{2+L}}e^{-\frac{R^{2}}{2w^{2}}}\Phi^{(0)}_{0}(\bm{\Omega}),$
(6)

where $\Gamma(\cdot)$ is the gamma function. Also,
$\Phi^{(0)}_{0}(\bm{\Omega})$ is the non-interacting ground state of $H_{{\rm
ad}}(R;\bm{\Omega})$ [Eq. (1)] (denoted by the $(0)$ superscript) taking into
account the total angular momentum $L$ and parity $\pi$ of the system
$L^{\pi}$. In particular, $L^{\pi}=0^{+}$ [$L^{\pi}=1^{-}$] for BBX [FFX]
systems. The independence of $\Phi^{(0)}_{0}(\bm{\Omega})$ on $R$ stems from
the independence of the hyperangular eigenvalues of the non-interacting
adiabatic Hamiltonian on this parameter, for more details see Appendix B. The
hyperradial part of $\Psi(R,\bm{\Omega},t=0)$ is the ground state of the
hyperradial equation [Eq. (2)] with zero non-adiabatic coupling matrix
elements, due to the independence of $\Phi_{0}^{(0)}(\bm{\Omega})$ on $R$, and
one potential curve, associated to this ground hyperangular state,
$U(R)=1/(2\mu R^{2})[(L+1)^{2}-1/4]+1/(2\mu w^{4})R^{2}$. Its energy reads
$(2+L)/(\mu w^{2})$, where $L=0\,(1)$ refers to the total angular momentum for
the BBX (FFX) system.

It should be noted that Eq. (6) is an eigenstate of the non-interacting
Hamiltonian Eq. 1 only in the case of $w=a_{\rm{ho}}$ coinciding with the non-
interacting ground trap state. The spatial extent $w$ can be adjusted
experimentally e.g. by means of a trap frequency quench [see Section VI for a
more detailed discussion], however in the following we treat it as a free
parameter. This permits us to investigate the role of the spatial extent of
the initial wave function on the post-quench dynamics. However, a detailed
argumentation on the interval of values of the width $w$ is provided in Sec.
VI. Nevertheless, for typical LLH settings that we shall consider below these
bounds yield, $w\geq 0.46$ while for HHL ones, $w\geq 1.16$.

###  III.1 Time-evolution of the wave function

To trigger the nonequilibrium dynamics of the three-body mixture we perform
quenches of the relevant 2D scattering lengths $a_{\sigma,\sigma^{\prime}}$.
Accordingly, their values are suddenly reduced at $t=0$ from their initial
non-interacting ones. Recall that this is experimentally feasible via
appropriate Feshbach resonances [for more details see also Section VI].
Specifically, a different quench scheme is applied for the BBX and FFX systems
since the former (latter) possesses two (one) scattering lengths, i.e.
$a_{BB}$ and $a_{BX}$ ($a_{FX}$). In the case of FFX mixtures, solely
$1/a_{FX}$ is quenched and the consequent dynamics is explored over a wide
range of post-quench $1/a_{FX}$ [Fig. 1 (a)]. On the other hand, for the BBX
system both the $1/a_{BB}$ and $1/a_{BX}$ are changed abruptly at $t=0$ from
their non-interacting values [Fig. 1 (a)] towards different post-quench
$1/a_{BX}$ and fixed $1/a_{BB}=1$. It is worth mentioning that by tuning the
magnetic field for the quench in the experiment, both $a_{BX}$ and $a_{BB}$
are affected, and hence broad (narrow) intraspecies (interspecies) resonances
are required such that the variation of $a_{BB}$ is very small compared to
that of $a_{BX}$ [see also Section VI]. We remark that $a_{BB}=1$ is chosen
such that the bosonic atoms have an intermediate repulsive interaction
strength 222The two-body interaction strength between the $\sigma=B,F$ and
$\sigma^{\prime}$ species Busch  _et al._ (1998); Bougas  _et al._ (2019);
Doganov  _et al._ (2013) is defined as
$g_{\sigma\sigma^{\prime}}=\left[\ln\left(2e^{-2\gamma}(1+m_{\sigma}/m_{\sigma^{\prime}})/a^{2}_{\sigma\sigma^{\prime}}\right)\right]^{-1}$,
where $\gamma=0.577$. This implies that when
$a_{\sigma\sigma^{\prime}}>(<)e^{-\gamma}\sqrt{2}\sqrt{1+\frac{m_{\sigma}}{m_{\sigma^{\prime}}}}$
attractive (repulsive) effective interaction regimes arise.. However, we have
checked that the dynamical processes and response of the LLH and HHL BBX
systems that are presented below [Section IV, Section V] do not change
substantially closer to the non-interacting limit, i.e. $a_{BB}>1$. The fact
that the qualitative features of the results remain the same towards the non-
interacting limit permits us to expose the role of the particle statistics
between BBX and FFX systems.

Figure 1: (a) Cartoon of the quench scenario. The system consists of either
two (red particles) identical bosons (BBX) or fermions (FFX) and a
distinguishable atom (blue particle). They are initialized ($t=0$) in a non-
interacting state with spatial extent $w$. The dynamics is induced by a sudden
change of the scattering lengths (interspecies denoted by springs and
intraspecies by wiggly lines) from their non-interacting to finite values. (b)
Schematic representation of a typical three-body energy spectrum. In region
III, below the BX or FX bare dimer threshold, (red dashed line), trimer states
can be formed, denoted by a circle. Region II signals the presence of atom-
dimers (dimers are marked by an ellipse), and in region I, trap states appear
along with atom-dimers. These two latter eigenstates feature avoided-
crossings, see for instance the dashed circle. The energy dependence of the
trimers, atom-dimers and trap states on the scattering length is schematically
presented by the blue, green and red dash-dotted lines respectively. Note that
the horizontal axis corresponds to a wide range of considered scattering
lengths, but does not however reach the zero limit.

To describe the quenched dynamics of the three-body system, the time-evolved
wave function is expressed as a projection of the initial state [Eq. (6)] onto
the interacting eigenstates of the post-quench 2D scattering lengths.
Specifically, it acquires the form

$\Psi(R,\bm{\Omega},t)=\sum_{\rm{f}}e^{-iE_{\rm{f}}t}c_{\rm{f},\rm{in}}\Psi^{\rm{f}}(R,\bm{\Omega}),$
(7)

where
$\Psi^{\rm{f}}(R,\bm{\Omega})=R^{-3/2}\sum_{\nu}F^{\rm{f}}_{\nu}(R)\Phi_{\nu}(R;\bm{\Omega})$
are the post-quench interacting eigenstates and $E_{\rm{f}}$ their
eigenenergies. Also, $c_{\rm{f},\rm{in}}=\int
dRd\bm{\Omega}\,R^{3}\Psi(R,\bm{\Omega},t=0)\left[\Psi^{f}(R,\bm{\Omega})\right]^{*}$
denote the overlap coefficients between the initial and the post-quench
eigenstates. The overlap coefficients are explicitly determined by the initial
state and hence its width $w$ for a fixed post-quench scattering length. This
leads to a $w$-dependent participation of specific post-quench eigenstates ,
i.e. depending on $w$ different eigenstates contribute in the dynamics, whose
distinct features dictate the dynamical response of the system, as it will be
demonstrated below [Section IV and Section V].

###  III.2 Classification of post-quench three-body eigenstates

A detailed knowledge of the three-body energy spectra Gharashi  _et al._
(2012); Bougas  _et al._ (2021); Liu  _et al._ (2010), will allow an in-depth
understanding of the emergent nonequilibrium dynamics of both the BBX and FFX
mixtures. The post-quench interacting eigenstates can be categorized into the
so-called trimers, atom-dimers and trap states Blume and Greene (2002);
Portegies and Kokkelmans (2011). Trimers are three-body bound states which
exist below the BX or FX dimer energies, see in particular the red-dashed line
and region III in Fig. 1 (b). In Ref. D’Incao  _et al._ (2015) it was shown
that in the absence of a trap the BX or FX dimer energy is given by $E_{\sigma
X}=-2e^{-2\gamma}(1+\mathcal{M})/a_{\sigma X}^{2}$. Here, $\sigma=B,F$,
$\gamma=0.577$ and $\mathcal{M}=m_{\sigma}/m_{X}$. For BBX systems there is
also the BB dimer energy determined by $E_{BB}=-4e^{-2\gamma}/a_{BB}^{2}$
which is constant since $a_{BB}=1$ remains fixed for all the post-quench
$a_{BX}$ scattering lengths333These relations are altered in the presence of a
trap only for scattering lengths comparable to or larger than the length scale
$\ell=\sqrt{\hbar/\mu_{2B}\omega}$ (with $\mu_{2B}$ being the two-body reduced
mass) Idziaszek and Calarco (2006). This effect depends also on the mass ratio
of the three-body system..

Region II of Fig. 1 (b) indicates the energies of the atom-dimer states which
are two-body bound states interacting with a third particle. The atom-dimer
states depend strongly on $a_{BB}/a_{BX}$ [$1/a_{FX}$] in the case of the BBX
[FFX] systems having a BX+B [FX+F] character. Moreover, the region I of Fig. 1
(b) depicts the energy regime of the trap states that are almost insensitive
to scattering length variations [see straight lines in Fig. 1(b)] referring to
three weakly interacting particles. Apparently, avoided-crossings occur
between BX+B or FX+F atom-dimers, also encountered in region I, and trap
states, designated by dashed circles in Fig. 1 (b). For BBX systems, apart
from the aforementioned states appearing in region I, BB+X atom-dimers arise
as well. Their eigenenergies experience only small variations with respect to
$a_{BB}/a_{BX}$, similarly to the trap states, since the post-quench $a_{BB}$
is kept fixed. A way to distinguish them from trap states is by inspecting
their stationary two-body BB short-range correlations, e.g. through the two-
body BB contact. In Ref. Bougas  _et al._ (2021) it was shown that the latter
is more pronounced in the case of BB+X atom-dimers than for trap states.

Notably, all three types of eigenstates display a different spatial extent in
terms of the hyperradius $R$. Therefore, the initial state described by Eq.
(6) will eventually screen out particular states or superpositions in the
time-evolution for different widths $w$ and this information is encoded in the
overlap coefficients $c_{\rm{f},\rm{in}}$ [see also Sections IV and V].

Figure 2: Average dynamical response, as captured by the time-averaged
fidelity $\langle\left|F\right|\rangle$, with respect to the scattering length
ratio (a) $a_{BB}/a_{BX}$ for the BBX and (b) $1/a_{FX}$ for the FFX LLH
systems. Cases of different widths $w$ (see legend) of the initial state are
presented. Apparently, in both settings the response is changed for widths
smaller or larger than $a_{\rm{ho}}=1.02$. In particular, it is enhanced for
wider initial states having $w>a_{\rm{ho}}$.

##  IV Quench dynamics of LLH settings

To obtain an overview of the system’s dynamical response for different widths
of the initial state and post-quench scattering lengths, we employ the time-
averaged fidelity Bougas  _et al._ (2019, 2020); Budewig  _et al._ (2019)

$\langle|F|\rangle=\lim_{T\to\infty}\frac{\int_{0}^{T}dt\,|F(t)|}{T}.$ (8)

The total time-evolution $T$ is considered to be long enough such that
$\langle|F|\rangle$ is converged.444Here we consider total evolution times
$T=800$, while the time-averaged fidelity for the LLH (HHL) settings saturates
already from $T=300$ ($T=500$). The fidelity, which essentially estimates the
deviation of the time-evolved state [Eq. (7)] from the initial one, reads

$\displaystyle F(t)$ $\displaystyle=$
$\displaystyle\langle\Psi(R,\bm{\Omega},t)|\Psi(R,\bm{\Omega},t=0)\rangle$ (9)
$\displaystyle=$
$\displaystyle\sum_{\rm{f}}\left|c_{\rm{f},\rm{in}}\right|^{2}e^{-iE_{\rm{f}}t}.$

Here, $c_{\rm{f},\rm{in}}$ are the overlap coefficients introduced in Eq. (7)
and $E_{\rm{f}}$ refer to the energies of the post-quench eigenstates. As a
function of the post-quench scattering length, the dynamical response of the
three-body system exhibits two distinct regimes mainly determined by the width
of the initial state with respect to the three-body harmonic oscillator
length, $a_{\rm{ho}}$. In this section the LLH setups that are considered have
a mass ratio $m_{B/F}/m_{X}=0.04$ yielding a three-body harmonic oscillator
length $a_{\rm{ho}}=1.02$.

Figure 3: Fidelity spectra of the quenched (a) BBX and (d) FFX LLH systems
with a narrow pre-quench state of $w/a_{\rm{ho}}=0.78$. The circles denote
frequencies associated to specific post-quench eigenstates. The interaction-
dependent excitation branches signal the dominant participation of trimer and
atom-dimer states in the dynamics and refer to their energy differences with
respect to trap states. Almost constant branches are related to trap
excitations. The energy spectra of the (b) BBX and (e) FFX LLH systems, where
a series of avoided-crossings among atom-dimers and trap states occurs, marked
by the dashed circles. The red dashed line indicates the bare BX or FX dimer
threshold. (c), (f) Profiles of the fidelity spectrum for the (c) BBX and (f)
FFX mixture at different scattering lengths (see legend).

Regarding the LLH BBX system, the time-averaged fidelity $\langle|F|\rangle$
in terms of $a_{BB}/a_{BX}$ is depicted in Fig. 2 (a) for various widths of
the initial state. Apparently, the qualitative behavior of $\langle|F|\rangle$
depends strongly on $w$. For instance, in the case of $w/a_{\rm{ho}}=0.78$ the
deviation from the initial state becomes larger for increasing
$a_{BB}/a_{BX}$. Such a decrease of $\langle|F|\rangle$ holds also when
$w/a_{\rm{ho}}=1$ as long as $a_{BB}/a_{BX}<4$ and beyond this interval it
shows a saturation trend, due to the amplified population of trap states, see
also Appendix C. The latter renders the response of the system more enhanced
compared to the $w/a_{\rm{ho}}=0.78$ case, since a larger number of post-
quench eigenstates contributes in the dynamics [see also Appendix C]. However,
considering an initial state with a width at $w/a_{\rm{ho}}=4.9$, the response
of the system is substantially enhanced as compared to the previous case and
in particular it is almost independent of $a_{BB}/a_{BX}$. This pattern, as
will be explained in Section IV.1, originates from the significant population
of trap states. In this sense, it becomes evident that there are two
characteristic response regimes of the system with respect to $a_{\rm{ho}}$.

A similar qualitative behavior of $\langle|F|\rangle$ occurs also for LLH FFX
settings [Fig. 2 (b)] at $w/a_{\rm{ho}}<1$ or $w/a_{\rm{ho}}>1$. Notably, for
$w/a_{\rm{ho}}=0.78$, $\langle|F|\rangle$ is almost constant in the region
$1/a_{FX}>3$. Such a response can be also observed for other widths
$w/a_{\rm{ho}}<1$, due to the participation of trap states for large
$1/a_{FX}$. For an initial state with $w/a_{\rm{ho}}=1$, we observe that the
response of the LLH FFX system is decreased for $1/a_{FX}>3$, meaning that the
deviation from the initial state reduces progressively. This mainly occurs due
to the smaller number of contributing states in the course of the evolution
(as thoroughly discussed in the Appendix A of Ref. Gorin  _et al._ (2006)),
since the participation of the first two atom-dimers reduces as $1/a_{FX}$ is
further tuned to larger values [see also Appendix C]. For $w/a_{\rm{ho}}=4.9$,
the time-averaged fidelity is practically constant due to the participation of
trap states during the dynamics, whose overlap coefficients do not depend
strongly on $1/a_{FX}$ [see also Section IV.2]. Note that the considered post-
quench $1/a_{FX}$ values do not include 0, and $\langle\left|F\right|\rangle$
therefore deviates from unity in the leftmost part in Fig. 2 (b) at
$w/a_{\rm{ho}}=1$. However, when $w\neq a_{\rm{ho}}$, even at $1/a_{FX}=0$ the
deviation would persist, since the initial state is not a non-interacting
eigenstate.

Evidently, regardless of the particle statistics we observe that the width of
the initial state plays a crucial role on the dynamical response of the three-
body system. Thus, in order to further address the physical origin of this
behavior in the following we will analyze the involved excitations, in terms
of the post-quench eigenstates, that contribute in the nonequilibrium
dynamics. Their identification is indeed, in general, tractable in few-body
setups Mistakidis  _et al._ (2014); Mistakidis and Schmelcher (2017). For this
purpose, we utilize the fidelity spectrum

$\left|F(\tilde{\omega})\right|=\left|\int\frac{dt}{\sqrt{2\pi}}\,e^{-i\tilde{\omega}t}\left|F(t)\right|\right|.$
(10)

It discloses information regarding the predominantly contributing final
eigenstates in the dynamics via the energy differences
$\tilde{\omega}_{\rm{f,f^{\prime}}}=E_{\rm{f}}-E_{\rm{f}^{\prime}}$ (recall
that we work with dimensionless units [Section II]), which are identified from
the energy spectra of BBX and FFX systems Bougas  _et al._ (2021). Below, we
elaborate on the excitation spectrum of both LLH BBX and FFX systems in the
two above-mentioned distinct response regimes.

###  IV.1 Excitations from narrow initial states with $w<a_{\rm{ho}}$

As a prototype LLH setup with an initial state width $w<a_{\rm{ho}}$ we use
the case of $w/a_{\rm{ho}}=0.78$. To understand the excitation processes of
the quenched system we inspect the respective fidelity spectrum together with
the energy eigenspectrum and the overlap coefficients. For the BBX system, the
fidelity spectrum $\left|F(\tilde{\omega})\right|$ and the three-body post-
quench eigenenergies are shown in Fig. 3 (a), and (b), respectively. Note that
the indexing of the eigenenergies e.g. in Fig. 3 (b) starts from the ground
state, which possesses an energy way below the displayed range, and increases
as we climb the energy ladder.

In Fig. 3, for $a_{BB}/a_{BX}<4$ the excited frequency branches appearing in
$\left|F(\tilde{\omega})\right|$ mainly refer to energy differences between
the second trimer state (first excited trimer) $\rm{f}=2$, and either the
first atom-dimer ($\rm{f}=3$) or the trap states ($\rm{f}=8$), see e.g
$\tilde{\omega}_{3,2}$ and $\tilde{\omega}_{8,2}$ respectively in Fig. 3 (a)
at $a_{BB}/a_{BX}=2.5$. In these frequency branches the most dominant
contribution in the coefficients $c_{\rm{f},\rm{in}}$ stems mainly from the
second trimer. This occurs since both the initial state and the second trimer
are well localized at small values of the hyperradius, i.e. for
$R<a_{\rm{ho}}$, yielding thus a large overlap. In particular, for the
frequency $\tilde{\omega}_{3,2}$ we observe that it remains constant as the
scattering length ratio $a_{BB}/a_{BX}$ varies. This arises from the fact that
the scattering length dependence of the second trimer and first atom-dimer
eigenenergies is similar as shown in Fig. 3(b), thus their energy difference
results into an almost constant frequency $\tilde{\omega}$.

As $a_{BB}/a_{BX}$ is tuned to larger values, the spatial extent of the post-
quench eigenstates changes drastically [Fig. 3 (b)], thus affecting their
overlap with the initial configuration. Indeed, the participation of the
second trimer state ($\rm{f}=2$) decreases for $a_{BB}/a_{BX}>4$. For these
scattering length ratios the trimer and the atom-dimer states become tightly
bound [see Fig. 3 (b)]. Accordinly, their wave functions are much narrower
than the initial one, which reduces the corresponding overlap coefficients. In
return, this results in a smaller amplitude of $\tilde{\omega}_{3,2}$, see
Fig. 3 (c) at $a_{BB}/a_{BX}=4.25$. This reduced contribution in the fidelity
spectrum is counterbalanced by the enhanced population of more trap states
giving rise to excitation branches whose values increase with larger
$a_{BB}/a_{BX}$, see e.g. the scaling of $\tilde{\omega}_{8,2}$ in Fig. 3 (a)]
555Note that even if the labels of the post-quench eigenstates are the same,
the frequency associated to them, $\tilde{\omega}_{8,2}$, acquires different
values depending on the scattering length [Fig. 3 (a)], since the energy
spectrum changes drastically with respect to $a_{BB}/a_{BX}$.. Their
increasing behavior reflects the growing energy difference between the second
trimer and trap states for $a_{BB}/a_{BX}>3$ [Fig. 3 (b)]. Also, the amplitude
$\tilde{\omega}_{8,2}$ increases with $a_{BB}/a_{BX}$ since the substantial
spatial extent of the trap wave functions yields larger overlap with the
initial state. Furthermore, a larger number of branches arises in the fidelity
spectrum as can be seen by comparing the profiles of
$\left|F(\tilde{\omega})\right|$ at $a_{BB}/a_{BX}=4.25$ and
$a_{BB}/a_{BX}=2.5$ illustrated in Fig. 3 (c). As a result, the response of
the time-averaged fidelity for $w/a_{\rm{ho}}=0.78$ is more enhanced (smaller
value of $\langle|F|\rangle$) for larger ratios of $a_{BB}/a_{BX}$ [Fig. 2
(a)]. Let us remark that time-dependent variation protocols of the scattering
lengths would be of great interest, since they could result in a significant
population of trimer states, even at the regimes where trap states acquire a
large contribution.

Figure 4: Fidelity spectra of a wide initial state, i.e. $w/a_{\rm{ho}}=4.9$,
for the (a) BBX and (b) FFX LLH systems following a quench of the scattering
length. The circles designate specific frequency peaks corresponding to
different post-quench eigenstates. The atoms reside in a superposition
consisting predominantly of trap states. The latter are imprinted as
excitation branches being insensitive to scattering length variations.

For the dynamical response of the LLH FFX system, we observe the appearance of
a larger number of excitations in the fidelity spectrum [Fig. 3 (d)] as
$1/a_{FX}$ increases. Notice that this behavior is already anticipated from
the enhanced response of $\langle|F|\rangle$ presented in Fig. 2 (b) for
$w/a_{\rm{ho}}=0.78$. However, the microscopic mechanisms behind this response
are different from the ones in the BBX system due to the distinct eigenenergy
spectra, compare in particular Fig. 3 (b) and (e). Evidently, in the case of
the LLH FFX system trimers do not form. Here, the major contribution for
$1/a_{FX}<2$ is shared among the first two atom-dimer states, $\rm{f}=1,2$,
possessing a small spatial extent and mostly localized at $R<a_{\rm{ho}}$.
This claim can be verified by the corresponding frequency peak
$\tilde{\omega}_{2,1}$ of $\left|F(\tilde{\omega})\right|$ shown in Fig. 3 (d)
and (f) as well as the contribution of the relevant overlap coefficients (with
total contribution $90\%-60\%$ for $1/a_{FX}\in[0.36,2]$). For large
scattering lengths ($1/a_{FX}>2$) the participation of atom-dimers diminishes
since their spatial extent further decreases. This results in their reduced
overlap with the initial state and consequently to a smaller amplitude of
$\tilde{\omega}_{2,1}$ as shown in Fig. 3 (f) for $1/a_{FX}=4$. In this case,
trap states acquire a non-negligible population leading to interaction-
dependent frequency branches which grow with respect to $1/a_{FX}$, see e.g.
$\tilde{\omega}_{8,1}$ in Fig. 3 (d).

###  IV.2 Response for wide initial configurations of $w>a_{\rm{ho}}$

Next, we examine the susceptibility of LLH three-body setups to quenches for
initial configurations characterized by $w>a_{\rm{ho}}$. As a representative
example of this kind we choose $w/a_{\rm{ho}}=4.9$ and first investigate BBX
mixtures. Recall that in this scenario the time-averaged response captured by
$\langle|F|\rangle$ [Fig. 2 (a)] is drastically enhanced as compared to
$w/a_{\rm{ho}}=0.78$ and experiences small variations with respect to
$a_{BB}/a_{BX}$.

To determine the microscopic origin of the involved excitations we resort
again to the fidelity spectrum $\left|F(\tilde{\omega})\right|$ provided in
Fig. 4 (a). The almost horizontal frequency branches stem from energy
differences between trap states, e.g. $\tilde{\omega}_{12,8}$. This is
verified by calculating the respective overlap coefficients and monitoring the
energy spectrum [Fig. 3 (b)]. Additionally, since $w/a_{\rm{ho}}=4.9\gg 1$ the
post-quench atom-dimers and trimers, being naturally narrow exhibit a reduced
overlap with the initial state. The dominant contribution in the course of the
evolution originates from the trap states whose overlap with
$\Psi(R,\bm{\Omega},t=0)$ is appreciable. Indeed, a multitude of trap states
is populated as can be inferred from the several frequency peaks of comparable
amplitude appearing in $\left|F(\tilde{\omega})\right|$ [Fig. 4 (a)]. This
fact, in turn, induces the enhanced response identified in $\langle|F|\rangle$
[Fig. 2 (a)] for $w/a_{\rm{ho}}=4.9$.

A similar overall phenomenology takes place also for LLH FFX systems, see Fig.
4 (b). Evidently, also here the respective excitation branches are almost
insensitive to $1/a_{FX}$ variations [Fig. 4 (b)]. Notably, the post-quench
eigenstates responsible for this behavior are again trap states, e.g.
$\tilde{\omega}_{6,3}$, although they are not the same as those identified in
the BBX scenario [Fig. 4 (a)]. The reason for this change can be traced back
to the different structure of the eigenspectrum between BBX and FFX LLH
systems, compare Figs. 3 (b) and (e).

Focusing on the underlying selection processes according to which specific
post-quench eigenstates are populated, it is instructive to carefully study
the respective overlap coefficients. Of immediate interest here are the ones
referring to pairs of post-quench eigenstates that experience avoided-
crossings [dashed circles in Fig. 3 (b), (e)], namely atom-dimers and trap
states, and in particular illuminate their dependence on the width $w$ of the
initial states. In the vicinity of the avoided-crossings, the spatial extent
of the involved eigenstates changes abruptly, since their character alters
between trap and atom-dimer states.

Figure 5: Overlap coefficients $|c_{\rm{f},\rm{in}}|^{2}$ between the initial
state of width $w$ and two post-quench eigenstates $\rm{f}$ as a function of
$w/a_{\rm{ho}}$ and the scattering length for the (a) BBX ($\rm{f}=5,6$) and
(b) FFX ($\rm{f}=3,4$) LLH system. In each case the presented pair of post-
quench eigenstates experiences an avoided-crossing in the respective energy
spectra [Fig. 3 (b), (e) with dashed circles]. A change of the character of
the state from a trap to an atom-dimer (atom-dimer to trap) state is signified
by a shift of its major contribution to smaller (larger) values of the width
of the pre-quench state $w/a_{\rm{ho}}$.

For a BBX setup, a characteristic example regarding the dependence of the
overlap coefficients between the initial state and the $\rm{f}=5,6$
eigenstates as a function of $w$ and $a_{BB}/a_{BX}$ is displayed in Fig. 5
(a). A transition between the different types of eigenstates is apparent by
the complementary behavior of the respective overlap coefficients D’Incao  _et
al._ (2018). On the left of the first avoided-crossing shown in Fig. 3 (b) at
$a_{BB}/a_{BX}\simeq 3$ [dashed circle], the occupation of the trap state
$\rm{f}=5$ [see red color gradient in Fig. 5 (a)] prevails for a larger $w$
when compared to the atom-dimer $\rm{f}=6$ [see green color gradient in Fig. 5
(a)]. This behavior arises from the mere fact that the atom-dimer has a
smaller spatial extent compared to the trap state, thus the latter yields
larger overlap compared to the former. The opposite behavior takes place
within $a_{BB}/a_{BX}\in[3,3.5]$, since then the $\rm{f}=5,6$ states
interchange their character. After the second avoided-crossing at
$a_{BB}/a_{BX}\simeq 3.5$, these states are substantially occupied only for
$0.19<w/a_{\rm{ho}}<0.39$, since then both of them are atom-dimers [Fig. 3
(b)]. For larger $w/a_{\rm{ho}}\simeq 1$ and around $a_{BB}/a_{BX}\in[3.5,4]$
a significant contribution stems from a trap eigenstate ($\rm{f}=7$), not
shown in Fig. 5 (a). Similar transitions occur also for the FFX LLH system
[Fig. 5 (b)], where in this case the pair of eigenstates $\rm{f}=3,4$ exchange
character from a trap to an atom-dimer and vice versa through the avoided-
crossing at $1/a_{FX}\simeq 1.77$ [Fig. 3 (e) designated with a dashed
circle].

Figure 6:  Time-averaged (a) two-body
$\sqrt{{\langle\mathcal{D}_{2}^{BX}}\rangle}$, and (b) three-body contact
$\sqrt{\langle\mathcal{D}_{3}^{BBX}\rangle}$ of the BBX LLH setting and (c)
two-body contact $\sqrt{{\langle\mathcal{D}_{2}^{FX}}\rangle}$ of the FFX
setup. Correlations at all levels increase for larger $a_{BB}/a_{BX}$ or
$1/a_{FX}$ and their magnitude reduces for larger widths. The peak structure
at specific scattering lengths is an imprint of the participation of atom-
dimers. The widths of the initial state are provided in the legend.

Concluding, it is worth mentioning that upon considering a width of the
initial state being the same as the three-body harmonic oscillator length,
namely $w=a_{\rm{ho}}$, the original configuration corresponds to the non-
interacting ground trap state [Section III]. For this reason, the role of
trimers and atom-dimers is less important during the time-evolution and as
expected trap states have a somewhat larger population [for more details see
Appendix C]. This behavior holds for both BBX and FFX systems.

###  IV.3 Build-up of two- and three-body correlations

Having established an understanding regarding the contributing eigenstates for
different widths of the initial state, an intriguing question that arises is
how these states influence the associated short-range few-body correlations in
the course of the evolution. These correlations can be addressed by the
experimentally measurable Sagi  _et al._ (2012); Stewart  _et al._ (2010) two-
and three-body contacts Olshanii and Dunjko (2003); Tan (2008c); Werner and
Castin (2012); Colussi (2019); Bellotti  _et al._ (2013b); Bougas  _et al._
(2021). The latter are defined as coefficients in a high momentum expansion of
the $\sigma$-species one-body density in momentum space

$n_{\sigma}(\bm{p}_{\sigma},t)\simeq\frac{1}{N_{\sigma}p_{\sigma}^{4}}\sum_{\sigma^{\prime}}(1+\delta_{\sigma\sigma^{\prime}})\mathcal{D}_{2}^{\sigma\sigma^{\prime}}(t)+\frac{\ln^{3}p_{\sigma}}{p^{6}_{\sigma}}\mathcal{D}_{3}(t).$
(11)

This expansion pertains to the case where $p_{\sigma}$ is significantly larger
than the momentum scales provided by the inverse scattering lengths Bougas
_et al._ (2021). Here, $N_{\sigma}$ is the atom number belonging to the
$\sigma$-species, while $\mathcal{D}_{2}^{\sigma\sigma^{\prime}}(t)$ denotes
the time-dependent two-body contact between the species $\sigma$ and
$\sigma^{\prime}$. Note that only the three-body contact $\mathcal{D}_{3}(t)$
of BBX systems ($\mathcal{D}_{3}^{BBX}(t)$) is finite, since for FFX ones
three-body correlations are suppressed 666The three-body contact yields the
probability to detect three particles in close vicinity. As such, it is zero
by construction for FFX systems within the $s$-wave zero-range interaction
model, where the two identical and non-interacting fermions can not approach
one another due to the Pauli principle. due to the Pauli exclusion principle
Bellotti  _et al._ (2014). The main features of these few-body correlation
observables are captured by their time-averaged measure. Namely, the time-
averaged two-body contacts are described by the following expressions:

$\langle\mathcal{D}_{2}^{\sigma
X}\rangle=\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}dt\>\mathcal{D}_{2}^{\sigma
X}(t),\quad\sigma=B,F$ (12)

and the three-body ones read

$\langle\mathcal{D}_{3}^{BBX}\rangle=\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}dt\>\mathcal{D}_{3}^{BBX}(t).$
(13)

These quantities assess the overall degree of dynamical correlations for
various widths of the initial state and post-quench scattering lengths, see
Fig. 6. A detailed analysis of the stationary three-body FFX and BBX setups
reveals a hierarchy in terms of the degree of few-body correlations for the
different types of eigenstates. Namely, as shown in Refs. Bougas  _et al._
(2021); Colussi (2019); Blume  _et al._ (2018) trimer states possess more
enhanced two- and three-body correlations than those of the BX or FX atom-
dimer states and, similarly, the atom-dimer contacts are larger than those of
the trap states. This hierarchy will also be apparent here as the width of the
initial state changes and different eigenstates contribute in the dynamical
response. Indeed, as the width of the initial state [Eq. (6)] increases, the
magnitude of all the aforementioned correlations at any scattering length is
reduced [Fig. 6]. This occurs because for larger widths, a superposition of
trap states is predominantly populated [see also Fig. 4].

On the contrary, for $w/a_{\rm{ho}}=0.78$, the first two atom-dimers (second
trimer) provide the main contribution to the post-quench wave function Eq. (7)
of the FFX (BBX) system. This is confirmed through their dominant overlap
coefficients [see Section IV.1], enhancing few-body correlations compared to
cases where $w>a_{\rm{ho}}$ [Fig. 6]. Therefore, in the limit of small
$w<a_{\rm{ho}}$, correlations at the two- and three-body level are,
generically, enhanced due to the non-negligible involvement of trimer and
atom-dimer states. This amplification was also observed for a three-boson
setup in the quench dynamics at unitarity in 3D Colussi  _et al._ (2018),
especially when the width of the initial state matched the size of an Efimov
trimer.

Another remarkable feature of the correlations is their magnification at
particular scattering lengths for fixed $w$, see the individual peaks
displayed in Fig. 6. Their amplitudes become more prominent from the overall
two- and three-body contacts for increasing width $w$, where trap states
contribute substantially [see Section IV.2]. These peaks occur in the vicinity
of avoided-crossings present in Fig. 3 (b), (e) where the corresponding three-
body wave function is predominantly in a superposition of a trap and an atom-
dimer state. Therefore, in this range of scattering length ratios the overall
character of the wave function abruptly changes yielding in this manner an
enhanced $\langle\mathcal{D}_{2}^{BX}\rangle$,
$\langle\mathcal{D}_{3}^{BBX}\rangle$ and
$\langle\mathcal{D}_{2}^{FX}\rangle$. This particular property of the time-
averaged two- and three-body contacts can be utilized as an experimental probe
for the formation of atom-dimers in a 2D gas.

Furthermore, the appearance of enhanced peaks in the two-body contacts at the
avoided-crossings due to the atom-dimer component in the time-evolved wave
function is also a manifestation of the universal Tan relations. These
universal relations exemplify that the short-range two-body correlations are
proportional to the variation of the stationary energy spectra with respect to
the scattering length Werner and Castin (2012); Valiente  _et al._ (2011).
Therefore, close to the avoided-crossings the eigenenergies of the three-body
system [see Figs. 3 (b), (e)] strongly vary with the scattering length thus
yielding narrow peaked two-body correlations [Figs. 6 (a), (c)]. By this
token, we can address the main difference between the two-body contacts of BBX
and FFX systems in Fig. 6(a) and (c), respectively, where the former exhibits
broader peaks than the latter. This occurs because in the FFX eigenspectra
shown in Fig. 3(e) we observe much sharper avoiding-crossings than in the BBX
ones [see Fig. 3(b)]. Such a universal relation is absent in the case of the
three-body contact Bellotti  _et al._ (2013b) in 2D, and the peak structure is
attributed to the enhanced stationary three-body correlations Bougas  _et al._
(2021) of the atom-dimer component of the time-evolved wave function.

Figure 7: Time-averaged fidelity $\langle\left|F\right|\rangle$, of the three-
body (a) BBX and (b) FFX HHL mixture subjected to quenches of the interspecies
scattering length. Different widths of the initial state are considered (see
legend) whose values in terms of the oscillator length ($a_{\rm{ho}}=2.6$)
determine the degree of the system’s response. The substantial population of
atom-dimer and trimer (trap) states for $w<a_{\rm{ho}}$ ($w>a_{\rm{ho}}$)
leads to a strongly (weakly) interaction dependent response. In contrast to
the LLH case, trap states have also a small contribution for $w<a_{\rm{ho}}$
in addition to trimers and atom-dimers, and the larger number of participating
eigenstates compared to the $w>a_{\rm{ho}}$ scenario, enhances the response of
the system.

Moreover, it is also worth mentioning that a broadening of these correlation
peaks is evident for larger widths, see e.g. $w/a_{\rm{ho}}=4.9$ in Fig. 6. In
this case, as already discussed and observed in the fidelity spectrum [Fig. 4]
a large amount of trap states participates in the three-body time-evolved wave
function. This results into an agglomeration of avoided-crossings contributing
to the dynamics, which are slightly displaced horizontally from one another at
a fixed scattering length [see Fig. 3 (b), (e)]. The aforementioned
displacement then yields a range of scattering lengths over which the Tan
contacts display an enhanced behavior, manifested as a peak broadening.

##  V Dynamical response of HHL mixtures

In this section we address the role of the masses on the dynamical build up of
few-body correlations by considering HHL three-body mixtures. The intrinsic
dynamical behavior of this system is explored, for widths $w$ of the initial
state smaller or larger than the characteristic three-body harmonic oscillator
length $a_{\rm{ho}}=2.6$ [Eq. (6)]. As in the LLH case in Section IV, we
remark that initial states with a spatial extent smaller (larger) than
$a_{\rm{ho}}$ favors the participation of trimer and/or atom-dimer (trap)
states. Our analysis on the response of the 2D mixtures is based on the time-
averaged fidelity $\langle|F|\rangle$ given in Eq. (8).

Figure 8: Fidelity spectrum of the (a) BBX and (d) FFX HHL systems performing
an interaction quench of an initial state where $w/a_{\rm{ho}}=0.57$. The
arrows mark characteristic frequency branches $\tilde{\omega}_{f,f^{\prime}}$.
Excitation branches that alter with respect to the scattering length (see the
top left corners) correspond to energy differences between trimers, atom-
dimers and trap states. Otherwise, the almost fixed frequency branches refer
to trap states. Energy spectra of (b) BBX and (e) FFX HHL mixtures. Particular
eigenstates are denoted by circles and arrows. Specific profiles of the
fidelity spectrum of the (c) BBX and (f) FFX system at distinct scattering
lengths (see legends).

The overall response of a HHL BBX system characterized by $m_{B}/m_{X}=22.16$
is intensified in the case of $w/a_{\rm{ho}}=0.57$ [Fig. 7 (a)] as compared to
$w>a_{\rm{ho}}$ within $a_{BB}/a_{BX}\in[0.5,3]$. This is in contrast to the
susceptibility of LLH mixtures [Fig. 2 (a)]. Moreover, for $w/a_{\rm{ho}}=1$ a
strong dependence of $\langle\left|F\right|\rangle$ is observed with respect
to the scattering length ratio. This feature of $\langle\left|F\right|\rangle$
differs dramatically from the response for $w/a_{\rm{ho}}=1.92$, where it is
arguably almost insensitive within the interaction interval
$a_{BB}/a_{BX}\in[2,3]$. This behavior is related to the prominent
contribution of trap states. For $w/a_{\rm{ho}}=1$, the system becomes less
susceptible to the quench as compared to the case of $w/a_{\rm{ho}}=1.92$,
since fewer trap states contribute, especially for large $a_{BB}/a_{BX}>2$.
Notably, there is a series of peaks appearing in
$\langle\left|F\right|\rangle$ at specific scattering lengths, where avoided-
crossings among atom-dimer and trap states exist in the few-body eigenspectrum
[see also Fig. 8 (b), (e)]. Their importance, especially in the relevant few-
body correlations, will be discussed below.

Subsequently, the susceptibility of a HHL FFX system with $m_{F}/m_{X}=24.71$
is illustrated in Fig. 7 (b). Apparently, the FFX mixture becomes more
perturbed when considering $w/a_{\rm{ho}}=0.57$. For larger widths, e.g.
$w/a_{\rm{ho}}=1.92$, the system experiences a weak dependence on the
scattering length within the range $1/a_{FX}\in[1.5,2]$. This is linked to the
dominant presence of trap states during the time-evolution due to their large
spatial extent. Moreover, we note that similarly to the BBX HHL case [Fig. 7
(a)] the FFX mixture is less perturbed for $w/a_{\rm{ho}}=1.92$ than in the
$w/a_{\rm{ho}}=0.57$ scenario. However, in contrast to the HHL BBX system, for
$w/a_{\rm{ho}}=1$ the mixture develops a stronger response in comparison to
$w/a_{\rm{ho}}=1.92$, due to the more prominent population of trimers and
atom-dimers.

###  V.1 Excitation processes for $w<a_{\rm{ho}}$

Pre-quenched states with a spatial extent smaller than the three-body harmonic
oscillator length, apparently exhibit a larger overlap with the trimers and
atom-dimer states of the BBX and FFX HHL systems. The latter, contribute
significantly in the underlying dynamics compared to the case where
$w>a_{\rm{ho}}$. In the opposite regime ($w>a_{\rm{ho}}$) trap states become
substantially populated in the post-quench dynamics [see also Section IV.2], a
mechanism pertaining also to the HHL mixtures. The frequency spectra will be
analyzed for the $w<a_{\rm{ho}}$ scenario, since for $w>a_{\rm{ho}}$, the
underlying microscopic mechanisms resemble those presented in Section IV.2.
However, the differences present in $\langle\left|F\right|\rangle$ between LLH
[Fig. 2] and HHL setups [Fig. 7] for $w>a_{\rm{ho}}$ stem mostly from the
different number of participating trap states in the post-quench dynamics.
Moreover, in the HHL scenario, in addition to the participation of trap
states, there are a few contributing atom-dimer and trimer states especially
for small values of $1/a_{FX}$ and $a_{BB}/a_{BX}$. This results in further
perturbation of the system from the initial state compared to larger
scattering lengths. Recall also here the relevant discussion in Section IV
concerning LLH mixtures.

Figure 9: Time-averaged contacts revealing the enhancement of short range (a),
(c) two-body and (b) three-body correlations for larger inverse interspecies
scattering lengths of HHL (a), (b) BBX and (c) FFX settings. The existence of
peaks at individual scattering length ratios reveals the population of atom-
dimers due to the sharp avoided-crossing taking place at the eigenspectrum
[Fig. 8 (b), (e)]. The widths of the initial state are shown in the legend.

Inspecting the fidelity spectrum $\left|F(\tilde{\omega})\right|$ [Fig. 8 (a)]
together with the overlap coefficients and the energy spectrum [Fig. 8 (b)]
for the HHL BBX system, we can infer that for $a_{BB}/a_{BX}<1$ the second and
third trimer states are significantly populated. This gives rise to excitation
branches such as $\tilde{\omega}_{8,3}$, indicating the participation of the
third trimer and the $\rm{f}=8$ trap state [Fig. 8 (a)] for small
$a_{BB}/a_{BX}<1$. This frequency branch shows an appreciable growth with
larger $a_{BB}/a_{BX}$ due to the accompanied increasing energy difference
between trimer and trap states [Fig. 8 (a), (b)]. Note that the energies of
the $\rm{f}=2,\leavevmode\nobreak\ 3$ trimers are large in magnitude and
negative and therefore lie below the energy window presented in Fig. 8 (b).
Apart from trimer states, trap ones, e.g. $\rm{f}=8,\leavevmode\nobreak\ 10$,
are occupied as well but their respective energy differences depend weakly on
changes of $a_{BB}/a_{BX}$, see e.g. $\tilde{\omega}_{10,8}$ in Fig. 8 (a).

A further increase of the scattering length ratio $a_{BB}/a_{BX}>1$, leads to
a reduction of the amplitude and number of the higher-lying excitation
frequencies in comparison to $a_{BB}/a_{BX}<1$. This behavior can be readily
seen in the relevant profiles of the fidelity spectra depicted in Fig. 8 (c)
for $a_{BB}/a_{BX}=2.5$ and $a_{BB}/a_{BX}=0.8$. It stems from the suppressed
contribution of the two trimer states for $a_{BB}/a_{BX}>1$, resulting in a
less perturbed system as also reflected in $\langle\left|F\right|\rangle$
[Fig. 7 (a)] for $w=1.5$. Similarly to the case of $a_{BB}/a_{BX}<1$ trap
states are also populated here, imprinted in the spectrum as distinct almost
horizontal frequency branches e.g. $\tilde{\omega}_{52,50}$ in Fig. 8 (c)
777Apart from the horizontal excitation branches within
$a_{BB}/a_{BX}\in[2,3]$, there exist also faint ones having a ’V’ shape
dependence on the scattering length with tipping points located at
$a_{BB}/a_{BX}=2.25$ and $2.57$ [Fig. 8 (a)]. These are attributed to energy
differences between trap and atom-dimer states. At the tipping point of these
’V’ shaped branches the energies of participating states come close together
due to the avoided-crossings and are thus associated to small $\tilde{\omega}$
in $\left|F(\tilde{\omega})\right|$..

A qualitatively similar dynamical response to the BBX mixture is also observed
for the HHL FFX system, see $\left|F(\tilde{\omega})\right|$ illustrated in
Fig. 8 (d) for $w/a_{\rm{ho}}=0.57$. Here, the heavy fermions with respect to
the third particle favor trimer formation Pricoupenko and Pedri (2010), a
result that is in contrast to the corresponding LLH case. These trimer states
possess large negative energies Bougas  _et al._ (2021); Bellotti  _et al._
(2013a), lying beyond the values depicted in the energy spectrum provided in
Fig. 8 (e). Particularly, a superposition of the first two trimer states
($\rm{f}=1$ and $2$) is prevalent in the course of the evolution for
$1/a_{FX}<1$, leading to excitation branches such as $\tilde{\omega}_{2,1}$
[Fig. 8 (d)]. Moreover, similar to the BBX HHL system, trap states are also
present in the dynamical response of the corresponding FFX mixture, as
identified by the energy spectrum and the overlap coefficients. The frequency
branches associated to energy differences between these states are almost
independent of $1/a_{FX}$ [Fig. 8 (d)].

Tuning the inverse scattering length to larger values $1/a_{FX}>1$, a plethora
of trap states contributes in the time-evolved three-body wave function.
Accordingly, a multitude of excitation branches arise in
$\left|F(\tilde{\omega})\right|$ whose location is almost constant with
varying $1/a_{FX}$ [Fig. 8 (d)] and are clustering at low $\tilde{\omega}$ as
shown in Fig. 8 (f). The large number of contributing trap states for
$1/a_{FX}>1$ is linked to the enhanced response of the HHL FFX system, e.g.
captured by the time-averaged fidelity displayed in Fig. 7 (b) for
$w/a_{\rm{ho}}=0.57$.

###  V.2 Dynamical formation of few-body correlations

As already demonstrated in Section IV.3 for LLH systems, the build-up of few-
body correlations regardless of the particle statistics exhibit a peak
structure for scattering lengths in the vicinity of avoided-crossings
appearing in the post-quench eigenspectrum (see also Fig. 6). Similarly, in
this section we focus on HHL systems in order to showcase the role of
increased mass ratio on the time-averaged Tan contacts as illustrated in Fig.
9. In particular, the two-body BX species contact [Fig. 9 (a)] exhibits
sequences of narrow peaks at specific scattering length ratios in agreement
with Tan’s universal relation Tan (2008a); Werner and Castin (2012); Braaten
_et al._ (2011). Namely, at these post-quench scattering lengths, the
corresponding eigenspectrum possesses narrow avoided-crossings among trap
states and atom-dimers [Fig. 8 (b)], thus resulting into the strong
amplification of the two-body correlations. Moreover, the amplitude of the
peaks in the $\braket{\mathcal{D}_{2}^{BX}}$ decreases for large
$a_{BB}/a_{BX}$ independently of $w$. This suppression occurs for large
$a_{BB}/a_{BX}$ where the avoided-crossings become increasingly narrow [Fig. 8
(b)]. In this sense, they can not be well resolved leading to less pronounced
peaks compared to smaller $a_{BB}/a_{BX}$.

In the case of the three-body contact [Fig. 9 (b)] a multitude of peaks with
tiny amplitude appears as $w$ increases. This holds even for large
$a_{BB}/a_{BX}$ as $w$ increases, despite the narrow avoided-crossings present
in the HHL eigenspectra [Fig. 8 (b)]. Particularly, for increasing $w$ trap
states are predominantly populated, but in the vicinity of avoided-crossings
atom-dimers contribute as well. Therefore, the amplification of stationary
three-body correlations of the atom-dimer post-quench eigenstates compared to
trap states leads to the rise of peaks in
$\sqrt{\langle\mathcal{D}_{3}^{BBX}\rangle}$ at the locations of the avoided-
crossings. Moreover, equivalently to the two-body BX species contact [Fig. 9
(a)], the time-averaged three-body contact is reduced for larger $w$, due to
the significant participation of trap states, whose stationary three-body
correlations are greatly suppressed.

Furthermore, $\sqrt{\langle\mathcal{D}_{3}^{BBX}\rangle}$ at
$w/a_{\rm{ho}}=0.57$ has an overall maximum around $a_{BB}/a_{BX}\simeq 1.2$,
and then decreases for larger values of $a_{BB}/a_{BX}$. This behavior is
related to the significant population of the second trimer which specifically
possesses a population up to $16\%$ until $a_{BB}/a_{BX}\simeq 1.2$.
Subsequently, the corresponding overlap coefficient with the initial state
decreases for $a_{BB}/a_{BX}>1.2$, since in this range of scattering length
ratios the second trimer state is narrower than the initial one. HHL BBX
systems favor the existence of strongly bound trimer states, due to the
increased mass ratio Bellotti  _et al._ (2013a). The contribution of such a
trimer state (second) for $w<a_{\rm{ho}}$, results in an augmented three-body
contact, in contrast to the one presented in LLH setups [Fig. 6 (b),
$w/a_{\rm{ho}}=0.78$], where the small mass ratio inhibits the creation of
strongly bound trimers.

In an equal fashion to the time-averaged two-body BX contact,
$\sqrt{\langle\mathcal{D}_{2}^{FX}\rangle}$ [Fig. 9 (c)] showcases small
amplitude peaks, arising mostly for $w/a_{\rm{ho}}=0.57$. Their magnitude
again drops for increasing scattering length ratio $1/a_{FX}$ since sharper
avoided-crossings are encountered in the eigenspectrum of the HHL FFX system
than the ones appearing in the LLH case [compare Fig. 3 (e) and Fig. 8 (e)].

##  VI Experimental parameters for the realization of the dynamics of the
three-body mixture

In an experimental environment 2D gases are realized in quasi-2D trapping
potentials where the confinement in the transversal direction of the 2D plane
is tighter than the radial one. This transversal trapping component is
characterized by a frequency $\omega_{\perp}$ chosen such that the atomic
motion is energetically restricted to the radial confinement potential with
frequency $\omega_{r}$ Petrov and Shlyapnikov (2001); He and Zhou (2019). A
comparison of the low-lying energy states of two interacting particles in 3D
and in a pure 2D geometry Idziaszek and Calarco (2006), revealed that the
aspect ratio in a quasi-2D setup required to attain the 2D character of the
relative motion of the two particles Petrov and Shlyapnikov (2001) should
satisfy $\omega_{r}/\omega_{\perp}<1/10$. This is corroborated by typical
quasi-2D experiments Kwon  _et al._ (2021); Holten  _et al._ (2018); Murthy
_et al._ (2019). For our setup, this energy requirement translates to $1/(\mu
w^{2})\leq 0.1\omega_{\perp}$, and furthermore assuming $\omega_{\perp}=50$
Makhalov  _et al._ (2014), it reduces to $w\geq 1/\sqrt{5\mu}$. As such, for
the typical LLH settings that we have considered this condition yields $w\geq
0.4559$, while for HHL ones it yields $w\geq 1.16$.

The dynamical protocol outlined in Section III relies on the realization of a
non-interacting three-body system with a tunable spatial extent $w$, and the
subsequent quench of the relevant 2D scattering lengths. The latter are
related to their 3D counterparts Petrov and Shlyapnikov (2001), which can be
tuned by means of Feshbach resonances Chin  _et al._ (2010). For the BBX
systems, in particular, the coexistence of broad and narrow intra- and
interspecies resonances in a magnetic field window ensures a regime where the
post-quench scattering length $a_{BB}$ remains almost constant while $a_{BX}$
varies in magnitude and sign. For instance, for the HHL BBX system of
${}^{133}\rm{Cs}-^{133}\rm{Cs}-^{6}\rm{Li}$, such a magnetic field window
exists for $[840,845]\rm{G}$, i.e. around the interspecies resonance Berninger
_et al._ (2013); Pires  _et al._ (2014b); Repp  _et al._ (2013). Also, in the
vicinity of $\simeq 880\,\rm{G}$ both 3D scattering lengths vanish, thus
materializing a non-interacting state.

The parameters of interest for the trapping potential are
$\omega_{r}=2\pi\times 65\,\rm{kHz}$ and $\omega_{\perp}=50\,\omega_{r}$
Makhalov  _et al._ (2014). Also, regarding the 3D counterparts of the 2D post-
quench scattering lengths used herein, we discern the following values
displayed in Table 1. Note that in the considered intervals of the 3D
scattering length (in atomic units), there is a sign change due to a
resonance.

Our analysis in the previous sections illustrated the role of the width $w$ of
the initial state in the dynamical response of the three-body system. This $w$
parameter can be experimentally adjusted by the following procedure. The two
identical particles (B or F) together with the third distinguishable atom (X)
are confined in a trap with a planar frequency $\omega_{\rm{in}}$, which are
initialized in their non-interacting ground state. A simple relation can be
established between the initial state’s width and the planar frequency, i.e.
$\mu\omega_{\rm{in}}=w^{-2}$, where $\mu$ is the three-body reduced mass [see
also Sec. III]. Prior to the quench on the scattering lengths, a quench on the
trap frequency from $\omega_{\rm{in}}$ to $\omega_{\rm{f}}$ is performed. This
allows for the preparation of initial states that possess widths different
from the length scale of the trap with final frequency $\omega_{\rm{f}}$ where
the interaction quench dynamics will take place. By setting the final radial
trapping frequency at $\omega_{\rm{f}}=2\pi\times 65\,\rm{kHz}$, the initial
frequency is determined from the relation
$\omega_{\rm{in}}=\omega_{\rm{f}}\,a_{\rm{ho}}^{2}/w^{2}$. Thus, for the LLH
settings in Sec. IV, the widths $w/a_{\rm{ho}}=0.78,\leavevmode\nobreak\ 4.9$
correspond to $\omega_{\rm{in}}=2\pi\times(105.5,2.7)\,\rm{kHz}$. For the HHL
setup (Sec. V), the initial widths $w/a_{\rm{ho}}=0.57,\leavevmode\nobreak\
1.92$, are obtained for $\omega_{\rm{in}}=2\pi\times(194.5,17.5)\,\rm{kHz}$.

| $1/a_{FX}$ | $a^{3D}_{FX}\>(a_{0})$
---|---|---
LLH | $[0.36,2.77]\,([4,5])$ | $[-246,-3000]\,([3000,1343])$
HHL | $[0.36,6]\,([0.82,2.5])$ | $[-715,-2976]\,([2991,380])$
| $1/a_{BX}$ | $a^{3D}_{BX}\>(a_{0})$
LLH | $[2,2.81]\,([3.94,4.65])$ | $[-994,3000]\,([2995,1497])$
HHL | $[0.85,3]$ | $[3000,380]$
| $a_{BB}$ | $a^{3D}_{BB}\>(a_{0})$
LLH | $1$ | $-421$
HHL | $1$ | $1578$
Table 1: Mapping of the 2D BX, FX and BB post-quench scattering lengths to
their 3D counterparts (in atomic units with $a_{0}$ denoting the Bohr radius)
for both LLH and HHL setups. The radial and transversal trapping frequencies
utilized herein are $\omega_{r}=2\pi\times 65\,\rm{kHz}$, and
$\omega_{\perp}=50\omega_{r}$.

##  VII Summary and Outlook

The quench dynamics of mass-imbalanced three-body mixtures with either bosonic
or fermionic constituents interacting with a third atom is investigated.
Depending on the mass ratio, we distinguish between the LLH and HHL cases.
Initially the mixture is confined in a 2D harmonic trap, and assumed to be
non-interacting. The spatial extent of the initial state and the post-quench
scattering length are exploited as parameters in order to map out the build-up
of two- and three-body correlations via distinct microscopic excitation
mechanisms.

In particular, the interactions are abruptly switched on triggering a distinct
dynamical response depending on the width of the initial state. A complete
knowledge of the energy spectra in conjunction with the fidelity spectrum,
allows us to identify the prevalent microscopic mechanisms in terms of
specific post-quench eigenstates. It is found that if the initial state width
is smaller than the three-body harmonic oscillator length $a_{\rm{ho}}$,
trimers and atom-dimers contribute predominantly in the dynamics. In contrast,
for larger widths trap states are those which are significantly populated
regardless the mass imbalance of the system. However, in HHL ensembles for
narrow widths, the participation of trimers and atom-dimers prevails in a
relatively smaller range of scattering lengths as compared to LLH mixtures.

Interestingly, the participating eigenstates have a distinct imprint on the
dynamics of the underlying few-body short-range correlations, as captured by
the Tan contacts. It is explicated that for an increasing width of the initial
state, the magnitude of both the overall time-averaged two- and the three-body
correlations decreases for a fixed 2D scattering length. For small widths,
these correlations are found to be enhanced as a result of the involvement of
trimer states and atom-dimers. The respective amplification of the Tan
contacts, due to the participation of such states, was also independently
reported following the quench dynamics of three-body systems at unitarity in
3D Colussi  _et al._ (2018). Strikingly, for widths larger than the three-body
harmonic oscillator length, few-body correlations display sharp peaks at
certain scattering lengths. This behavior is directly linked to the presence
of avoided-crossings among trap and atom-dimer states taking place in the few-
body eigenspectrum and signify the non-negligible cooperation of atom-dimers
in the time-evolution.

Overall, our work proposes a scheme to dynamically excite distinct
superpositions of eigenstates in three-body mixtures. Specifically, it was
demonstrated that depending on the interplay between the three-body harmonic
oscillator length and the width of the initial state, all three types of
eigenstates, that is trimers, atom-dimers and trap states, are possible to be
dominantly populated during the nonequilibrium dynamics. Moreover, temperature
effects are expected to mitigate few-body correlations as shown in Yan and
Blume (2013); Bougas  _et al._ (2021). In this sense, the investigation of
possible smearing effects of the identified peak structures building upon the
time-averaged contacts for large $w>a_{\rm{ho}}$ is a compelling perspective
for further research.

In addition, an interesting question that arises for future studies is how to
efficiently populate individual target states, and in particular trimers.
Their properties such as lifetimes are usually studied indirectly via three-
body recombination loss mechanisms Pires  _et al._ (2014a); Ulmanis  _et al._
(2016). However, many questions remain open especially regarding their
dynamical formation in a gas Klauss  _et al._ (2017). A promising route
towards achieving this goal would be to utilize time-dependent protocols, in
order to activate individual target states instead of superpositions of them
generated by quenches. There is currently active research for the dynamical
creation of the macroscopic population of trimer states in cold gases Klauss
_et al._ (2017); Musolino  _et al._ (2022); Colussi  _et al._ (2018). A first
step has already been accomplished in Ref. Klauss  _et al._ (2017), where an
abrupt tuning of interactions to unitarity and a subsequent sweep to weak
repulsion was shown to be able to produce a $8\%$ population of trimers.

###### Acknowledgements.

G. B. acknowledges financial support by the State Graduate Funding Program
Scholarships (Hmb-NFG). S.I.M. acknowledges support from the NSF through a
grant for ITAMP at Harvard University. This work is supported (P.S.) by the
Cluster of Excellence ‘The Hamburg Center for Ultrafast Imaging’ of the
Deutsche Forschungsgemeinschaft (DFG)-EXC 1074- project ID 194651731. The
authors thank G. M. Koutentakis for insightful discussions, M.T. Eiles for his
comments on the manuscript, and Lydia Schollmeier for the collaboration and
discussions in the early stages of this project.

##  Appendix A Adiabatic Hamiltonian and $s$-wave pseudopotential in two-
dimensions

The adiabatic Hamiltonian $H_{{\rm ad}}(R;\bm{\Omega})$ as introduced in Eq.
(1) is expressed in the following way Rittenhouse  _et al._ (2010)

$H_{{\rm ad}}(R;\bm{\Omega})=\frac{\hbar^{2}\Lambda^{2}(\bm{\Omega})}{2\mu
R^{2}}+\frac{3\hbar^{2}}{8\mu R^{2}}+\sum_{k}V_{k}(R;\bm{\Omega}^{(k)}),$ (14)

where $\Lambda^{2}(\bm{\Omega})$ is the hyperangular operator referring to the
centrifugal motion of the three particles Avery (1989); Das (2016). Also, the
three-body reduced mass is $\mu=m_{B/F}/\sqrt{2m_{B/F}/m_{X}+1}$ with
$m_{B/F}$ denoting the mass of the bosons or the fermions depending on the
type of the mixture.

The last term of Eq. (14) stands for the three (two) pairwise $s$-wave contact
interactions among the particles in a BBX (FFX) system. The $V_{k}$ potential
refers to the interaction between the $i$ and $j$ particles (also known as
odd-man-out notation where the $i$, $j$ or $k$ indices refer to interaction
pairs of the remaining two indices Rittenhouse  _et al._ (2010)). In
particular, the $V_{k}$ interaction is modeled by a 2D pseudopotential which
reads Olshanii and Pricoupenko (2001); Kanjilal and Blume (2006)

$\displaystyle V_{k}(R;\bm{\Omega}^{(k)})$
$\displaystyle=-\frac{\hbar^{2}\delta(\alpha^{(k)})}{\mu\sin(2\alpha^{(k)})R^{2}\ln(A\lambda
a^{(k)})}$
$\displaystyle\times\left[1-\ln\left(A\lambda\sqrt{\mu/\mu_{k}}R\sin(\alpha^{(k)})\right)\alpha^{(k)}\frac{\partial}{\partial\alpha^{(k)}}\right],$

where $\alpha^{(k)}\in[0,\pi/2]$ is the hyperangle describing the relative
position of two particles compared to the third one. For instance, if
$\alpha^{(k)}=0$, then the particles $i$ and $j$ are on top of each other,
whereas for $\alpha^{(k)}=\pi/2$, all three particles are collinear. Moreover,
$\mu_{k}=\frac{m_{i}m_{j}}{m_{i}+m_{j}}$ is the reduced two-body mass and
$A=0.5\,e^{\gamma}$ with $\gamma\approx 0.577$ being the Euler-Mascheroni
constant. Importantly, $a^{(k)}\equiv a_{ij}$ is the 2D scattering length
between the $(i,j)$ pair of particles. The factor $\lambda$ is an ultraviolet-
cutoff for the zero-range pseudopotential, setting an upper bound in momentum
space. However, it does not affect any observable as argued in Refs. Olshanii
and Pricoupenko (2001); Pricoupenko and Olshanii (2007).

##  Appendix B Hyperangular wave function of the non-interacting initial state

The hyperangular wave function of the non-interacting initial state (denoted
by the $(0)$ superscript) can be expressed Nielsen  _et al._ (2001); Volosniev
_et al._ (2014) as follows

$\displaystyle\Phi^{(0)}_{n}(\bm{\Omega})$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{3}\sum_{\begin{subarray}{c}m_{1},m_{2}\\\
\left|m_{1}+m_{2}\right|=L\end{subarray}}C^{(k)}\mathcal{N}^{(m_{1},m_{2})}_{n}\sin^{\left|m_{1}\right|}\alpha^{(k)}\cos^{\left|m_{2}\right|}\alpha^{(k)}Y_{m_{1}}(\theta_{1}^{(k)})Y_{m_{2}}(\theta_{2}^{(k)})\frac{\Gamma(1+n+\left|m_{1}\right|)}{\Gamma(1+\left|m_{1}\right|)n!}$
$\displaystyle\times_{2}F_{1}\left(1+\left|m_{1}\right|+\left|m_{2}\right|+n,-n;\left|m_{1}\right|+1;\sin^{2}\alpha^{(k)}\right),$

where
$\mathcal{N}^{(m_{1},m_{2})}_{n}=\sqrt{\frac{(2n+1+\left|m_{1}\right|+\left|m_{2}\right|)\Gamma(n+1)\Gamma(n+1+\left|m_{1}\right|+\left|m_{2}\right|)}{2\Gamma(n+1+\left|m_{1}\right|)\Gamma(n+1+\left|m_{2}\right|)}}$
are normalization coefficients. The above eigenfunction is the $n$-th
eigenstate ($n$ is a non-negative integer) of the hyperangular operator
$\Lambda^{2}(\bm{\Omega})$ Avery (1989); Das (2016) with eigenvalues
$\lambda_{n}(\lambda_{n}+2)$ where

$\lambda_{n}=2n+\left|m_{1}\right|+\left|m_{2}\right|,$ (17)

and $L=\left|m_{1}+m_{2}\right|$ being the total angular momentum of the
three-body system. It is expressed in terms of the angular quantum numbers
$m_{1}$, $m_{2}$ related to the polar angles $\theta_{1}^{(k)}$ and
$\theta_{2}^{(k)}$. The polar angles $\theta_{1}^{(k)}$ and $\theta_{2}^{(k)}$
refer to the orientation of the Jacobi vectors $\bm{\rho}_{1}^{(k)}$,
$\bm{\rho}_{2}^{(k)}$ in the 2D plane, respectively, where
$\bm{\rho}_{1}^{(k)}$ is the relative distance of the $(i,j)$-pair and
$\bm{\rho}_{2}^{(k)}$ is the relative vector of the $k$ spectator particle
relative to the $(i,j)$-pair’s center of mass. The summation running over
these angular quantum numbers is restricted by the condition
$L=\left|m_{1}+m_{2}\right|$. Note that in the case of three identical
particles, $n=1$ gives an unphysical solution and therefore it is not allowed
D’Incao and Esry (2014). Additionally, ${}_{2}F_{1}(a,b;c;\cdot)$ is the Gauss
hypergeometric function Abramowitz and Stegun (1965) and
$Y_{m}(x)=e^{imx}/\sqrt{2\pi}$ are plane waves. The angle $\alpha^{(k)}$
determines the ratio of the measure of the two Jacobi vectors via the relation
$\tan\alpha^{(k)}=\rho_{1}^{(k)}/\rho_{2}^{(k)}$ [see also Appendix A].

The particle statistics of the above wave function is properly taken into
account by the first summation and the $C^{(k)}$ coefficients. These read
explicitly $(C_{1},-C_{1},0)$ and $(C_{1},C_{1},C_{2})$ for FFX and BBX
systems respectively, with the $C_{1}$ and $C_{2}$ terms being normalization
coefficients. The hyperangular wave functions $\Phi_{\nu}(R;\bm{\Omega})$,
(which are eigenstates of $H_{\rm{ad}}(R;\bm{\Omega})$) correspond to the
interacting post-quench eigenstates and have angular quantum numbers
$(m_{1},m_{2})=(0,\pm L)$ due to the $s$-wave zero-range pseudopotential. As
such, the relevant subset in the summation [Eq.
(LABEL:Eq:Hyperangular_Gaussian)] will also be $(0,\pm L)$. Indeed, the
remaining terms in the summation have a zero contribution in the overlap
coefficients, $c_{\rm{f},\rm{in}}$, since the plane-waves $Y_{m}(\cdot)$ are
orthonormal. Here, we focus on $n=0$, that is the ground state. Note that the
hyperangular wave function does not depend on the hyperradius $R$ since in the
non-interacting case $H_{\rm{ad}}(R;\bm{\Omega})$ does not depend on $R$, as
all interaction terms $V_{k}(R;\bm{\Omega}^{(k)})$ drop [see also Appendix A].

##  Appendix C Quench dynamics of the LLH BBX mixture for initial states with
$w=a_{\rm{ho}}$

Figure 10: (a) Fidelity spectrum ($\left|F(\tilde{\omega})\right|$) for the
LLH BBX system subjected to a quench of $a_{BB}/a_{BX}$ from an initial non-
interacting state with $w/a_{\rm{ho}}=1$. (b) Profiles of
$\left|F(\tilde{\omega})\right|$ at different scattering length ratios
$a_{BB}/a_{BX}$ (see legend). The excitation processes involve majorly trimer
and atom-dimer states which are imprinted in the spectrum as branches that are
sensitive to the scattering length. Notice that the participation of the
second trimer is reduced compared to the $w/a_{\rm{ho}}=0.78$ case, resulting
in different branches than in $\left|F(\tilde{\omega})\right|$ depicted in
Fig. 3 (c).

For completeness, we shall also analyze the excitation spectrum of three-body
mixtures starting from a pre-quench state of width $w=a_{\rm{ho}}$. As
characteristic system for this investigation we consider a LLH BBX system
whose fidelity spectrum [Eq. (10)] is illustrated in Fig. 10 for varying post-
quench $a_{BB}/a_{BX}$.

Recall that for $w/a_{\rm{ho}}=0.78$, the second trimer state $\rm{f}=2$
contributes the most in the quench dynamics of the LLH BBX setting, see also
the discussion in Sec. IV.1. The predominant population of the second trimer
yields, in particular, excitation branches that are strongly influenced by
$a_{BB}/a_{BX}$ [Fig. 3 (a)]. This is a consequence of the fact that the
branches associated to these transitions refer to energy differences between
the $\rm{f}=2$ trimer and the trap states and are increasing as
$a_{BB}/a_{BX}$ is tuned to larger values.

These excitation branches are still present even for an initial state width
$w/a_{\rm{ho}}=1$ as shown in Fig. 10 (a). Here, the almost constant frequency
branch located around $\tilde{\omega}\simeq 2$, stemming from the transition
among the second trimer ($\rm{f}=2$) and the first atom-dimer ($\rm{f}=3$)
states, is more enhanced than in the case where $w/a_{\rm{ho}}=0.78$ (compare
$\tilde{\omega}_{3,2}$ in Fig. 10 (b) and Fig. 3 (c)). This difference is
attributed to the fact that the occupation of the first atom-dimer state is
larger when $w=a_{\rm{ho}}$, while the one from the second trimer is reduced,
a result that is supported by the corresponding overlap coefficients
$c_{\rm{f},\rm{in}}$. To be more precise, the population of the $\rm{f}=2$
trimer as long as $w/a_{\rm{ho}}=0.78$ ($w/a_{\rm{ho}}=1$) ranges from $73\%$
($57\%$) to $35\%$ ($21\%$) within the interval $a_{BB}/a_{BX}\in[2,4.6]$.
Apart from the enhanced population of the first atom-dimer, the contribution
of trap states, similar to the ones populated also for $w/a_{\rm{ho}}=0.78$,
increases as well with respect to $w/a_{\rm{ho}}=0.78$. This is imprinted in
the spectrum by the larger number of faint excitation branches, compare in
particular Fig. 10 (a) where $w/a_{\rm{ho}}=1$ with Fig. 3 (a) for which
$w/a_{\rm{ho}}=0.78$.

Similar observations to the above can be made for the other type of mixtures
utilized in the main text. Regarding the LLH FFX system, the contribution of
the first two atom-dimer states at $w/a_{\rm{ho}}=1$ remains the same in
comparison to $w/a_{\rm{ho}}=0.78$ for $1/a_{FX}<1$. Otherwise, it reduces
further from the value obtained for $w/a_{\rm{ho}}=0.78$ ($18\%$ versus $25\%$
at $1/a_{FX}=4.5$). This reduction is compensated by an increasing population
of a few trap states. Due to the reduced number of participating post-quench
eigenstates compared to smaller $1/a_{FX}$, the time-averaged fidelity
possesses a smaller magnitude for $1/a_{FX}>3$ [see Fig. 2 (b) for
$w/a_{\rm{ho}}=1$]. In a similar way, the population of trimers and first
atom-dimers also drops when considering $w/a_{\rm{ho}}=1$ for the HHL mixtures
(both BBX and FFX systems) as compared to the scenario where
$w/a_{\rm{ho}}=0.57$.

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