# Stochastic-field approach to the quench dynamics of the one-dimensional Bose
polaron

J. Jager  R. Barnett  Department of Mathematics, Imperial College London,
London SW7 2AZ, United Kingdom

###### Abstract

We consider the dynamics of a quantum impurity after a sudden interaction
quench into a one-dimensional degenerate Bose gas. We use the Keldysh path
integral formalism to derive a truncated Wigner like approach that takes the
back action of the impurity onto the condensate into account already on the
mean-field level and further incorporates thermal and quantum effects up to
one-loop accuracy. This framework enables us not only to calculate the real
space trajectory of the impurity but also the absorption spectrum. We find
that quantum corrections and thermal effects play a crucial role for the
impurity momentum at weak to intermediate impurity-bath couplings.
Furthermore, we see the broadening of the absorption spectrum with increasing
temperature.

##  I Introduction

The interaction of a mobile impurity with a surrounding many-body quantum
system is one of the most prominent and oldest problems in condensed matter
physics. The polaron, initially considered by Landau and Pekar Pekar (1946);
Landau (1933) to describe an impurity electron interacting with the lattice
vibrations of a solid is a prototypical scenario to study quasi-particle
formation. In more recent years, neutral atoms immersed in a surrounding
ultra-cold gas have attracted widespread attention due to great experimental
controllability which enable the study of novel exotic regimes of the polaron.
For example, the impurity-bath coupling can be tuned via Feshbach resonances
Bässler and Köhler (2012). Here, the fundamental principles at work are the
same as in the original problem; the impurity forms a polaron through
interacting with the collective excitations of the surrounding superfluid.
While the Fermi polaron has been subject to extensive experimental study
Schirotzek  _et al._ (2009); Zhang  _et al._ (2012); Kohstall  _et al._
(2012); Koschorreck  _et al._ (2012); Scazza  _et al._ (2017); Cetina  _et
al._ (2015, 2016); Parish and Levinsen (2016); Massignan  _et al._ (2014), the
Bose polaron has only been realised in a few experiment Catani  _et al._
(2012); Jørgensen  _et al._ (2016); Hu  _et al._ (2016); Yan  _et al._ (2020);
Skou  _et al._ (2021). These experiments hint towards a delicate interplay
between equilibrium and out-of-equilibrium effects.

There has been extensive work addressing the Bose polaron in equilibrium Rath
and Schmidt (2013); Tempere  _et al._ (2009); Casteels  _et al._ (2011);
Grusdt  _et al._ (2017); Casteels and Wouters (2014); Christensen  _et al._
(2015); Ichmoukhamedov and Tempere (2019); Camacho-Guardian  _et al._ (2018).
In this work, we focus, in contrast, on the out-of-equilibrium Bose polaron.
More precisely, we consider quench dynamics, involving the abrupt immersion of
an impurity into a homogeneous Bose gas, at finite temperature. The quench can
be realised through a Feshbach resonant radiofrequency pulse Bässler and
Köhler (2012) that switches on the impurity-condensate interaction nearly
instantaneously. Such quench dynamics have been studied either at zero
temperatures, or focused on the (extended) Fröhlich models (or very similar
approximations) Drescher  _et al._ (2019); Shchadilova  _et al._ (2016);
Volosniev  _et al._ (2015); Nielsen  _et al._ (2019); Lampo  _et al._ (2018);
Dzsotjan  _et al._ (2020); Mistakidis  _et al._ (2019a, b); Boyanovsky  _et
al._ (2019); Drescher  _et al._ (2020); Ardila (2021); Guenther  _et al._
(2018); Lausch  _et al._ (2018); Charalambous  _et al._ (2019). In much of the
equilibrium and some of the out-of-equilibrium treatments, the Lee-Low-Pines
transformation Lee  _et al._ (1953), which eliminates the impurity from the
problem at the expense of adding an additional quartic vertex, has proven
extremely useful. However, when considering finite temperature, this requires
further approximations.

The vast majority of the existing literature has focused on three-dimensional
systems, where the (extended) Fröhlich model is widely applicable. In 1D, the
situation is different. In Grusdt  _et al._ (2017) it was shown that the
applicability of Fröhlich-type approximations are somewhat limited in 1D and
while the mass balanced case is integrable for the Fermi-gas Gamayun  _et al._
(2018) no such limit exist in the case of a Bose gas. Instead, it is natural
to incorporate the effects of the impurity on the condensate already at the
mean-field level. This can be done efficiently for a single impurity at zero
temperature using the LLP transformation, but this method does not extend to
several impurities and is also not straightforward to generalise to finite
temperatures. To circumvent those complications, several approaches do not
eliminate the impurity from the problem and can address finite temperature
Feynman (1955); Lampo  _et al._ (2018), and for example, treat the impurity in
a manner related to the coherent state representation of the impurity Blinova
_et al._ (2013). It has been shown in Jager  _et al._ (2020); Pastukhov
(2018); Mistakidis  _et al._ (2019b) that a product wave function within the
tree-level approximation, yields inconsistent results with those obtained in
the more accurate LLP-frame in 1D. Therefore, it is perhaps more appropriate
to treat the impurity in a position-momentum path-integral as it highlights
the particle nature of the impurity. Using the coherent state path integral
for the condensate has been shown to yield good results in 1D not only for the
polaron but also the bipolaron problem Will  _et al._ (2021); Panochko and
Pastukhov (2019); Mistakidis  _et al._ (2019b). This is conceptually close to
the approach originally developed by Feynman Feynman (1955) and applied to the
Bose polaron in Tempere  _et al._ (2009); Ichmoukhamedov and Tempere (2019)
with the main difference being that we do not expand the condensate around a
homogeneous density, and our focus lies on out-of-equilibrium phenomena.

In this work, we develop a conceptually simple and numerically tractable
approach to address quench dynamics at finite temperature in a general manner.
Ultimately, this is achieved by mapping the dynamics to a set of deterministic
differential equations with stochastic initial conditions. By averaging over
the different trajectories, expectation values can be calculated within one-
loop accuracy. We then use this methodology to study an impurity’s evolution
after a sudden interaction quench into the bath. We find that the impurity
delocalises quickly for weak impurity-bath couplings and that observables like
the impurity’s velocity crucially depend on the incorporation of quantum and
thermal effects. In the case of strong impurity-bath couplings, we observe
self-trapping and quantum corrections and thermal effects to be e considerably
less pronounced.

This work is organised as follows. In Section II.1 we derive the truncated
Wigner approximation from the Keldysh path integral representation of the
problem at hand. In this section, we also show how to obtain the absorption in
the language of semi-classical dynamics. We proceed by specifying the initial
Wigner function in Section II.2 and show how to regularise the divergences
that arise in the one-dimensional setting. We proceed by briefly outlining the
numerical considerations in Section II.3. To conclude, we discuss the results
for an impurity at rest and finite momentum in Section III and outline further
directions in Section IV.

##  II Methodology

###  II.1 The Keldysh formalism for the Bose polaron

In the section, we want to outline the derivation of the equations of motion
and discuss the truncated Wigner approximation for the polaron problem. We
start by considering the Hamiltonian,

$\displaystyle\hat{\mathcal{H}}=\frac{1}{2M}\hat{P}^{2}+\int_{x}\Bigg{\\{}\hat{{\phi}}^{\dagger}(x)\Big{[}\frac{-\partial_{x}^{2}}{2m}+\frac{g_{\mathrm{BB}}}{2}\hat{{\phi}}^{\dagger}(x)\hat{{\phi}}(x)$
(1)
$\displaystyle+g_{\mathrm{IB}}V\left(x-\hat{X}\right)\Big{]}\hat{{\phi}}(x)\Bigg{\\}},$

where $m$ ($M$) denotes the mass of the bosons (impurity atom),
$\hat{\phi}(x)$ is the Bose field operator, $g_{\mathrm{BB}}$
($g_{\mathrm{IB}}$) is the boson-boson (boson-impurity) interaction strength
and $\hat{X}$ ($\hat{P}$) denotes the position (momentum) operator of the
impurity. Moreover, we left the interaction potential between the impurities
and the condensate general instead of directly assuming s-wave scattering. We
will not employ a delta function here but a smoothed out version of it for
numerical reasons, that will be discussed later. In the following, we are
going to apply the Kedlysh formalism to (1). The expectation value of an
arbitrary observable (see Polkovnikov (2010) for a detailed discussion)
$\Omega(\hat{X},\hat{P},\hat{{\phi}}^{\dagger}(x),\hat{{\phi}}(x),t)$ is given
by

$\displaystyle\langle\Omega(\hat{X},\hat{P},\hat{{\phi}}^{\dagger}(x),\hat{{\phi}}(x)),t)\rangle=\int_{X_{0},P_{0},\phi_{0}(x),\phi_{0}^{*}(x)}W(X_{0},P_{0},\phi_{0}(x),\phi_{0}^{*}(x))\int\mathcal{D}[{X_{\mathrm{c}}}(\tau),{X_{\mathrm{q}}}(\tau),$
(2)
$\displaystyle{P_{\mathrm{c}}}(\tau),{P_{\mathrm{q}}}(\tau),{\phi_{\mathrm{q}}}(x,\tau),{\phi_{\mathrm{c}}}(x,\tau),{\phi^{*}_{\mathrm{q}}}(x,\tau),{\phi^{*}_{\mathrm{c}}}(x,\tau)]\exp\big{(}iS[{X_{\mathrm{c}}}(\tau),{X_{\mathrm{q}}}(\tau),{P_{\mathrm{q}}}(\tau),{P_{\mathrm{c}}}(\tau),$
$\displaystyle{\phi_{\mathrm{q}}}(x,\tau),{\phi_{\mathrm{c}}}(x,\tau),{\phi^{*}_{\mathrm{q}}}(x,\tau),{\phi^{*}_{\mathrm{c}}}(x,\tau)]\big{)}\,\Omega_{W}({{X_{\mathrm{c}}}(\tau)},{{P_{\mathrm{c}}}(\tau)},{{{\phi^{*}_{\mathrm{c}}}(x,\tau)}},{{{\phi_{\mathrm{c}}}(x,\tau)}},t),$

where $W$ is the Wigner function that depends on the initial density matrix
and will be specified below.
$\Omega_{W}({{X_{\mathrm{c}}}(\tau)},{{P_{\mathrm{c}}}(\tau)},{{{\phi^{*}_{\mathrm{c}}}(x,\tau)}},{{{\phi_{\mathrm{c}}}(x,\tau)}},t)$
is the Weyl ordered operator of the observable one wants to calculate (again
see Polkovnikov (2010) for more details). The subscript $c$ denotes the
classical field and the subscript $q$ the quantum field, which describes the
quantum fluctuation around the classical saddle point solution. Those two
fields arise when mixing forward and backward contour in the Keldysh
formalism. The $\mathcal{D}$ denotes the integration over all field
configurations in space and time. In contrast to that the first integral in
(2) can be understood as a normal integral in $X_{0}$ and $P_{0}$. The one-
loop approximation is now to drop all terms of order two and higher in
$\hbar$, which corresponds to an expansion up to second order in the quantum
fields. We then find that there are, in fact, only terms linear in the quantum
fields, and the action is given by

$\displaystyle S=\int_{\tau}\Bigg{\\{}\Big{[}-{X_{\mathrm{q}}}(\tau)\frac{\rm
d}{\rm d\tau}{P_{\mathrm{c}}}(\tau)+{P_{\mathrm{q}}}(\tau)\frac{\rm d}{\rm
d\tau}{X_{\mathrm{c}}}(\tau)$ (3)
$\displaystyle-\frac{{P_{\mathrm{q}}}(\tau){P_{\mathrm{c}}}(\tau)}{M}\Big{]}-\int_{x}\Big{[}({\phi^{*}_{\mathrm{c}}}(x,\tau)\big{[}-i\partial_{\tau}-\frac{\partial_{x}^{2}}{2m}$
$\displaystyle+g_{\mathrm{BB}}|{\phi_{\mathrm{c}}}(x,\tau)|^{2}+g_{\mathrm{IB}}V({X_{\mathrm{c}}}(\tau)-x)\big{]}{\phi_{\mathrm{q}}}(x,\tau)$
$\displaystyle+c.c.)+g_{\mathrm{IB}}\frac{\rm d}{\rm
dx}V(x-{X_{\mathrm{c}}}(\tau)){X_{\mathrm{q}}}(\tau)|{\phi_{\mathrm{c}}}(x,\tau)|^{2}\Big{]}\Bigg{\\}}.$

It is now easy to see that all the quantum fields can be easily integrated out
and yield functional delta distributions enforcing the classical equations of
motion

$\displaystyle\frac{\rm d}{\rm d\tau}{X_{\mathrm{c}}}(\tau)$
$\displaystyle=\frac{{P_{\mathrm{c}}}(\tau)}{M}$ (4) $\displaystyle\frac{\rm
d}{\rm d\tau}{P_{\mathrm{c}}}(\tau)$
$\displaystyle=g_{\mathrm{IB}}\int_{x}\frac{\rm d}{\rm
dx}V(x-{X_{\mathrm{c}}}(\tau))|{\phi_{\mathrm{c}}}(x,\tau)|^{2}$ (5)
$\displaystyle i\partial_{\tau}{\phi_{\mathrm{c}}}(x,\tau)$
$\displaystyle=\Bigg{(}\frac{-\partial_{x}^{2}}{2m}+g_{\mathrm{BB}}|{\phi_{\mathrm{c}}}(x,\tau)|^{2}$
(6)
$\displaystyle\quad+g_{\mathrm{IB}}V\left({X_{\mathrm{c}}}(\tau)-x)\right)\Bigg{)}{\phi_{\mathrm{c}}}(x,\tau).$

The only challenge remaining at this point is to integrate over the initial
conditions weighted by the Wigner function. In practice this is done by
sampling initial conditions according to the Wigner function, solving the
classical equations of motion and then averaging the desired observable over
the calculated trajectories. In this framework it is now straightforward to
calculate the impurity dynamics. Before proceeding we would like to make some
remarks about the classical equations of motion and make contact with the
equilibrium case to show that even without taking the first-order correction
into account those equations give satisfactory results in the limiting case of
heavy impurities. For the equilibrium case we assume the impurity to travel at
constant velocity $\frac{\rm d}{\rm d\tau}{X_{\mathrm{c}}}(\tau)=v$, implying
$\frac{\rm d}{\rm d\tau}{P_{\mathrm{c}}}(\tau)=0$, which in turn tells us that
$|{\phi_{\mathrm{c}}}(x,\tau)|^{2}$ is symmetric around the impurity position.
Together with the equilibrium assumption this directly leads to the conclusion
that the bosonic field takes the following form
${\phi_{\mathrm{c}}}(x,\tau)=\phi_{c}(x-v\tau)=\phi_{c}(x-{X_{\mathrm{c}}}(\tau))$.
In the equilibrium setting we consequently find
$i\partial_{\tau}{\phi_{\mathrm{c}}}(x,\tau)=-iv\partial_{x}\phi_{c}(x-{X_{\mathrm{c}}}(\tau))$.
Putting it all together and defining $\tilde{x}=x-{X_{\mathrm{c}}}(\tau)$ we
find the equilibrium equation

$\left(\frac{-\partial_{\tilde{x}^{2}}}{2m}+g_{\mathrm{BB}}|\phi_{c}(\tilde{x})|^{2}+iv\partial_{\tilde{x}}+g_{\mathrm{IB}}V(\tilde{x})\right)\phi_{c}(\tilde{x})=0.$
(7)

We can now compare this equation with the one obtained by performing a Lee-
Low-Pines transformation and find that it has in fact the same form as the one
found in Jager  _et al._ (2020); Mistakidis  _et al._ (2019b); Volosniev and
Hammer (2017); Panochko and Pastukhov (2019), where it has been shown that
quantities obtained like the effective polaron mass or the polaron energy are
in excellent agreement with results obtained by quasi-exact quantum monte
carlo methods. The only difference is that instead of the reduced mass the
boson mass appears in front of $\partial_{x}^{2}$, which can be traced back to
the fact that in this derivation the effect of normal ordering is lost.
However, this effect is unimportant for heavy impurities. To summarize, we
showed that the equation obtained by employing a coherent state ansatz for the
bosonic field paired with a position momentum representation for the impurity
reduce to the correct mean-field equations in the equilibrium case if the
impurity mass is large.

Another quantity of great interest is the impurity Green’s function Dzsotjan
_et al._ (2020); Rath and Schmidt (2013); Shchadilova  _et al._ (2016) from
which the absorption spectrum can be calculated by taking the Fourier
transformation. The impurity Green’s function is defined by

$\displaystyle
G(t)=\mathrm{Tr}\left\\{\exp\left(i\hat{H_{0}}t\right)\hat{\rho}\exp\left(-i\hat{H}t\right)\right\\}$
(8)

where $\hat{H}_{0}$ stands for (1) with $V(x-\hat{X})=0$ and $\hat{\rho}$ is
the initial density matrix. We now observe that this has the same structure as
the trace that is considered to derive the Keldysh path integral, with the
only difference, that the forward and backward contour differ by an extra
interaction term. We can therefore perform the same steps as when deriving
(2). The only difference in $S$ is the resulting impurity boson interaction

$\displaystyle
S_{\mathrm{int}}=\int_{\tau,x}\Bigg{[}\big{(}{\phi^{*}_{\mathrm{c}}}(x,\tau)\big{[}\frac{1}{2}g_{\mathrm{IB}}V^{(n)}(\tilde{x})\big{]}{\phi_{\mathrm{q}}}(x,\tau)$
(9)
$\displaystyle+c.c.\big{)}+{X_{\mathrm{q}}}(\tau)\frac{g_{\mathrm{IB}}}{2}V^{(1)}(\tilde{x})|{\phi_{\mathrm{c}}}(x,\tau)|^{2}$
$\displaystyle{X_{\mathrm{q}}}(\tau)+g_{\mathrm{IB}}V(\tilde{x})(|{\phi_{\mathrm{c}}}(x,\tau)|^{2}$
$\displaystyle+\frac{|{\phi_{\mathrm{q}}}(x,\tau)|^{2}}{4})+\frac{g_{\mathrm{IB}}}{8}|{\phi_{\mathrm{c}}}(x,\tau)|^{2}V^{(2)}(\tilde{x}){X_{\mathrm{q}}}(\tau)^{2}\Bigg{]},$

where we have introduced the notation $\frac{\rm d^{n}}{\rm
dx^{n}}V(x-{X_{\mathrm{c}}}(\tau))=V^{(n)}(\tilde{x})$ The resulting magnitude
of the interaction is changed by a factor of $1/2$ and a new purely classical
term arises . Lastly, we note that there is also a quadratic term in
${\phi_{\mathrm{q}}}(x,\tau)$ and ${X_{\mathrm{q}}}(\tau)$ now. If we want to
keep the accuracy up to one loop order this term has to be considered, which
somewhat complicates matters. An ad hoc approximation is to drop this term
altogether and therefore staying in a strictly semi-classical regime. To see
when this approximation is justified one can simply compare the arsing terms
and their order of magnitude. We note, that the
$|{\phi_{\mathrm{q}}}(x,\tau)|^{2}$ term directly competes with the
$|{\phi_{\mathrm{c}}}(x,\tau)|^{2}$ term. As long as one is within the
applicability region of a general c-field treatment,
$|{\phi_{\mathrm{q}}}(x,\tau)|^{2}$ will be small compared with
$|{\phi_{\mathrm{c}}}(x,\tau)|^{2}$, whenever the condensate deformation is
not large, which corresponds to small and intermediate $\eta$. For the
corrections in the impurity degrees of freedom one has to compare
$\frac{g_{\mathrm{IB}}}{2}V^{1}(\tilde{x})|{\phi_{\mathrm{c}}}(x,\tau)|^{2}$
with
$V^{(2)}(\tilde{x}){X_{\mathrm{q}}}(\tau)^{2}|{\phi_{\mathrm{c}}}(x,\tau)|^{2}$.
We note that through partial integration the derivative terms can be brought
onto bosonic field variables. One then realizes, that the second derivative of
the fields is going to be small compared to the first derivative for weak
coupling. Additionally, for large impurity masses $M$ the magnitude of $X_{q}$
will stay small. Those two considerations show that one would expect the
absorption spectrum yield reliable results for weak to intermediate impurity-
boson coupling and potentially a wider range of couplings if the impurity is
sufficiently heavy. We refer to section II.4 for more details on the validity
the approach presented here.

We also note that this approximation is only made when calculating the
absorption spectrum and the impurity Green’s function and all dynamical
results do not rely on this approximation. Henceforth, the additional
stochastic term will be dropped. This leaves us with the expression

$G(t)=\langle\exp\left(i\int_{\tau,x}g_{\mathrm{IB}}V({X_{\mathrm{c}}}(\tau)-x)|{\phi_{\mathrm{c}}}(x,\tau)|^{2}\right)\rangle_{W},$
(10)

where we denote the average with respect to the initial Wigner function as
$\langle...\rangle_{W}$. We are now in the position to calculate the real
space trajectories of the impurities for finite temperature as well as the
absorption spectrum.

###  II.2 The quench protocol and the initial Wigner function

In the following, we want to specify the quench protocol and the initial
Wigner function. We start by briefly outlining the initial state and then
specify the Wigner function for a 1D quasi condensate. Here, we will also
discuss all the regularisation necessary to arrive at a divergence-free quasi
condensate description. The quench protocol is the following: we start with a
free impurity and an interacting superfluid at temperature $T$. At $t=0$, the
interaction between the superfluid and the impurity is turned on instantly.
Experimentally this is realised through a Feshbach resonance Bässler and
Köhler (2012). We assume that the initial density matrix can be written as a
direct product of the state of the impurity (which is assumed to be pure) and
the thermal density matrix of the superfluid

$\displaystyle\hat{\rho}=\hat{\rho}_{\phi}\otimes|\psi\rangle\langle\psi|,$
(11)

As a consequence, the Wigner function also factorises, and we can sample the
initial conditions independently. For the condensate, we employ a quasi
condensate description; in 1D, this is best done by employing a density and
phase representation. We note that this has been used before in Martin and
Ruostekoski (2010), in the trapped gas case. Since we want to focus on a
homogeneous gas in continuum here, we need to regularise the non-condensed
part. In this representation, the condensate field operator can be written as

$\displaystyle\hat{\phi}(x,0)=\sqrt{n_{0}+\delta\hat{\rho}(x)}\exp(i\hat{\theta}(x)).$
(12)

The density operator and the phase operator can be expressed within the
Bogoliubov approximation Pitaevskii and Stringari (2016) as

$\displaystyle\hat{\theta}(x)$
$\displaystyle=\frac{-i}{2\sqrt{\rho_{0}}}\sum_{k}\Big{[}(u_{k}+v_{k})e^{ikx}\hat{a}_{k}-\mathrm{h.c.}\Big{]}$
(13) $\displaystyle\delta\hat{\rho}(x)$
$\displaystyle=\sqrt{n_{0}}\sum_{k}\Big{[}(u_{k}-v_{k})e^{ikx}\hat{a}_{k}+\mathrm{h.c.}\Big{]},$
(14)

where $u_{k}$ and $v_{k}$ are the usual Bogoliubov modes. For this treatment
to be valid one has to be in the vicinity of a weakly interacting Bose gas,
which can be characterised by the Tonks parameter
$\gamma=1/(2n_{0}^{2}\xi^{2})$ Girardeau (1960); Lieb and Liniger (1963) which
should be less than unity where $\xi=\frac{1}{\sqrt{2g_{\mathrm{BB}}n_{0}}}$
is the healing length. We refer to the section II.4 for a detailed discussion
of the validity of the presented approach. In the path integral this
corresponds to a shift of variables meaning that instead of integrating over
$\phi^{(*)}_{0}(x)$, we integrate over $a^{(*)}_{k}$, corresponding to the
operators $\hat{a}^{\dagger}_{k}$. In the standard way, we can now write down
the thermal Wigner function (within the coherent state representation) for the
$a_{k}$ Polkovnikov (2010)

$\displaystyle
W(a^{*}_{k},a_{k})=\frac{2}{\pi}\tanh\bigg{(}\frac{\omega_{k}}{2T}\bigg{)}\exp\Big{\\{}-2|a_{k}|^{2}\
\tanh\bigg{(}\frac{\omega_{k}}{2T}\bigg{)}\Big{\\}},$ (15)

with the Bogoliubov dispersion $\omega_{k}$. From this, it can be seen
immediately, that the average condensate particle number is given by
$N_{0}/L=\rho_{0}$. In order to account for the quantum and thermal depletion,
we fix the total particle number $N$ and then choose $N_{0}$ according to
$N_{0}=N-N_{d}$ (this is done for every realisation), where after proper
regularisation (see Salasnich and Toigo (2016); Stoof  _et al._ (2009))

$\displaystyle
N_{d}=\sum_{k}\Big{[}\frac{e_{k}-\omega_{k}}{2\omega_{k}}+\frac{\mu}{2e_{k}+2\mu}+\frac{e_{k}}{\omega_{k}}\big{(}a^{*}_{k}a_{k}-\frac{1}{2}\big{)}\Big{]},$
(16)

with the single-particle dispersion $e_{k}=k^{2}/(2m)$. Here, a first-order
T-matrix approximation was employed, and in line with the Bogoliubov theory up
to one loop, $\mu$ is the chemical potential within the Bogoliubov
approximation and hence is not temperature-dependent. The $1/2$ is needed to
cancel the extra factor that stems from the symmetric ordering. After
averaging, this reproduces the expected result for a thermal quasi condensate
in 1D. It should be noted that $\mu$ has to be chosen consistently with the
total particle number, which can be done by fixing one reference point, where
the total particle number is known. Henceforth we assume a mean-field density
at $T=0$. This then fixes $\mu$ in the Bogoliubov approximation through
$\mu=g_{\mathrm{BB}}n_{0}(T=0)$ We can now use (16) to determine the total
particle number which remains fixed throughout the calculation. We can now
sample individual realisations of the condensate, whose description is free of
IR and UV divergences. Upon closer inspection, one might realise that even
though the mean of the phase and density corrections are zero, the variance
scales up to linearly with the system size $L$. A direct result of this is
that the computational time needed to achieve convergence also scales with the
system size. This computational challenge can be tackled by increasing the
system size gradually until the results are independent of the system size and
then validating certain data points with larger system size. Furthermore, this
restricts the discretisation of space, as outlined in Castin (2004), which
will be discussed in the next section. Because the effect of the impurity is
local, we find relatively low dependence on the system size already for small
systems. Lastly, we assume that the impurities are not entangled (namely can
be represented by a product wave function) and are localised in space around
$x_{0}$ or equivalently in momentum space localised around $q$ at $t=0$. It is
therefore natural to choose a wave packet as their initial wave function

$\displaystyle\langle
x|\psi\rangle=\frac{1}{\pi^{1/4}\sqrt{a_{0}}}\exp\big{(}-(x-x_{0})^{2}/(4a_{0}^{2})\big{)}.$
(17)

Here $a_{0}$ is an external parameter that determines how localised the
initial state is. The Wigner function in this setting is well known to be
Polkovnikov (2010)

$\displaystyle
W(X_{0},P_{0})=2\exp\left(-2a_{0}^{2}(P_{0}-q)^{2}-(X_{0}-x_{0})^{2}/(2a_{0}^{2})\right).$
(18)

###  II.3 Numerical considerations

In this section, we will show that three quantities can describe the whole
parameter space, and we will briefly outline the discretisation of space and
all the subtleties involved. We now define the following scaled parameters

$\displaystyle\tilde{\tau}=\tau/\tau_{s},\quad\tilde{x}=x/x_{s},\quad\tilde{\phi}(\tilde{x})=\sqrt{x_{s}}\phi(x),$
(19) $\displaystyle\quad\tilde{P}=Px_{s},\quad\tilde{X}=X/x_{s},$

where we choose $\tau_{s}=\tilde{\xi}/c$, $x_{s}=\sqrt{2}\xi=\tilde{\xi}$, and
where the speed of sound is given by
$c=\sqrt{\frac{g_{\mathrm{BB}}n_{0}}{m}}$. Dropping the twiddle we then find
the Hamiltonian

$\displaystyle\hat{\mathcal{H}}=\alpha\hat{P}^{2}+\int_{x}\Big{\\{}\hat{{\phi}}^{\dagger}(x)\big{[}\frac{-\partial_{x}^{2}}{2}+\frac{\sqrt{\gamma}}{2}\hat{{\phi}}^{\dagger}(x)\hat{{\phi}}(x)$
(20)
$\displaystyle+\eta\sqrt{\gamma}V(x-\hat{X)}\big{]}\hat{{\phi}}(x)\Big{\\}},$

where $\eta$ remains unchanged and $\alpha=\frac{m}{M}$. From here the
following equations of motion can be obtained

$\displaystyle\frac{\rm d}{\rm
d\tau}{X_{\mathrm{c}}}(\tau)=\alpha{P_{\mathrm{c}}}(\tau)$ (21)
$\displaystyle\frac{\rm d}{\rm
d\tau}{P_{\mathrm{c}}}(\tau)=\eta\sqrt{\gamma}\int_{x}\frac{\rm d}{\rm
dx}V(x-{X_{\mathrm{c}}}(\tau))|{\phi_{\mathrm{c}}}(x,\tau)|^{2}$ (22)
$\displaystyle
i\partial_{\tau}{\phi_{\mathrm{c}}}(x,\tau)=\Big{(}\frac{-\partial_{x}^{2}}{2}+\sqrt{\gamma}|{\phi_{\mathrm{c}}}(x,\tau)|^{2}$
(23)
$\displaystyle\quad+\eta\sqrt{\gamma}V({X_{\mathrm{c}}}(\tau)-x)\Big{)}{\phi_{\mathrm{c}}}(x,\tau).$

In order to tame the extensive variance and ensure numerical stability, we
have to choose the discretisation $l=L/N_{\mathrm{grid}}$ as outlined in
Castin (2004), namely $l$ has to be large enough to satisfy $\rho_{0}l\gg 1$,
while at the same time ensuring that the energy cut-off introduced by $l$ does
not alter the physics. This translates to $l<\xi,\lambda$, where $\xi$ is the
healing length that sets a natural length scale for our problem and $\lambda$
is the thermal de Broglie wavelength. Lastly, we note that we make the
following choice for the interaction potential

$V(x)=\frac{1}{\sqrt{2\pi l^{2}}}e^{-x^{2}/(2l)},$ (24)

which converges to the delta distribution as $l\rightarrow 0$, but has the
advantage of being smoother than $\delta_{x}/l$, where $\delta_{x}$ is the
Kronecker delta.

###  II.4 Validity of formalism

In this subsection we will address the regime of validity of the formalism
used in this work. We start by giving some general arguments for the validity
of the approach followed by a term-by-term discussion of the higher-order
corrections and their order of magnitude. While for the absorption spectrum we
have to restrict our considerations to weak or moderate boson-impurity
couplings, we would like to stress that for all dynamical properties
calculated (excluding the absorption spectrum) the result is strictly non
perturbative in $g_{\mathrm{BB}}$ and $g_{\mathrm{IB}}$. Hence we can safely
say that the method presented is valid for $\gamma\leq 1$, which is equivalent
to the range for the Gross-Pitaevskii equation, which corresponds to the tree
level approximation of (1). The same reasoning applies to $g_{\mathrm{IB}}$,
leading to the conclusion that our results, at least for short times are valid
across the whole range from weak to strong impurity-boson couplings. For
longer times one has to consider the deformation of the condensate. Another
interesting point to consider is that as soon as $\alpha$ is small (meaning
the impurity is heavy) the accuracy of the presented approach will improve
further, since the impurity will behave more classically. In fact $\alpha$ can
be seen as a strict control parameter for the corrections arising due to
higher order terms in ${X_{\mathrm{q}}}(\tau)$. Combining those considerations
with the known fact that in general the truncated Wigner approximation is
exact for short time scales Polkovnikov (2010), we can conclude that for short
to intermediate time scales our results are trustworthy for all values of
boson-boson and boson-impurity coupling. For weak coupling it is ab initio
reasonable to assume that the presented results hold for longer time scales,
since higher order quantum corrections should accumulate slowly if at all.
However, no such universal statement can be made for strong couplings. There
are two contributions of order $\hbar^{2}$ or higher, which were neglected in
our approach and would have to be added to (9) if one wanted to solve the
problem exactly, the first one coming from the boson-boson interaction and the
second one being due to the impurity-boson interaction. The first one takes
the form

$\sqrt{\gamma}\left(|\phi_{q}(x,\tau)|^{2}\phi_{q}(x,\tau)\phi^{*}_{c}(x,\tau)+{\rm
c.c.}\right).$ (25)

We note that this term is closely related to the standard Bogoliubov
approximation, with the main difference being that the classical field here is
taken to be the deformed field. We note that in no point of our simulations
the expectation value of $\langle|\phi^{*}_{c}(x,\tau)|^{2}\rangle$ falls
below the value of $3\tilde{\xi}$, meaning that the it is safe, in the spirit
of the well-established Bogoliubov approximation, to neglect higher-order
terms in the quantum fields, which only scale linearly in $\phi_{c}$ and are
of $\mathcal{O}(\phi_{c}\phi^{3}_{q})$ which is certainly small compared to
$\mathcal{O}(|\phi_{c}(x,\tau)|^{3}\phi_{q})$, as long as the density of the
condensate stays larger than the healing length. We note that in the case of
very low density gases and strong boson-impurity coupling the above argument
comes under question and it is a priori not clear whether the outlined method
is reliable in this regime. This regime was not investigated in the present
work. This argument is further underlined by the remarkable accuracy of pure
c-field methods (which do not take first-order quantum corrections into
account) for the equilibrium polaron, see Jager  _et al._ (2020); Drescher
_et al._ (2020); Volosniev and Hammer (2017); Will  _et al._ (2021), where the
c-field approximation was shown to also hold for low-density gases and strong
coupling, and the fact that the boson-boson interaction to all orders in
$g_{\mathrm{BB}}$ and to first order in $\hbar$ are already taken into account
in our calculations. A somewhat more complicated expression is obtained when
considering the impurity-boson interaction term. Here, one finds that already
in (9) all orders of ${X_{\mathrm{q}}}(\tau)$ are present and the higher order
terms take the following form

$\displaystyle\eta\sqrt{\gamma}\Bigg{(}-\frac{|{\phi_{\mathrm{q}}}(x,\tau)|^{2}}{4}\sum^{\infty}_{n=0}V^{(2n+1)}(\tilde{x})\frac{X^{(2n+1)}_{q}}{4^{n}(2n+1)!}$
(26)
$\displaystyle-|{\phi_{\mathrm{c}}}(x,\tau)|^{2}\sum^{\infty}_{n=1}V^{(2n+1)}(\tilde{x})\frac{X^{2n+1}_{q}}{4^{n}(2n+1)!}$
$\displaystyle+({\phi^{*}_{\mathrm{c}}}(x,\tau){\phi_{\mathrm{q}}}(x,\tau)+c.c.)\sum^{\infty}_{n=1}V^{(2n)}(\tilde{x})\frac{X^{2n}_{q}}{4^{n-1}(2n)!}\Bigg{)}.$

Again for approximating the impact of those terms it is convenient to bring
the derivative terms to the bosonic field operators. It then becomes clear
that all corrections can be understood as a gradient expansion in the bosonic
field operators, whose impact is certainly small for short time scales and
also for longer time scales as long as the coupling stays moderate. Besides
the gradient terms we also note that each correction is accompanied by an
increasing power of ${X_{\mathrm{q}}}(\tau)$ terms, which are also small at
short times and whose impact can be controlled by $\alpha$.

To summarize, while the validity of this approach for very large couplings and
long time scales can not be judged a priori, we note that for short time
scales the results here hold regardless of impurity-boson coupling strength
and also note that $\alpha$ serves as a control parameter for the
approximation in the impurity degrees of freedom.

Figure 1: The superfluid density, the position variance and the variance of
the velocity for different repulsive interactions and temperatures. In all
plots the parameters are $\alpha=0.2$, $\gamma=0.04$, $n_{0}=5\tilde{\xi}$ and
$a_{0}=3/\tilde{\xi}$. As expected, we can clearly see that the impurity
deforms the condensate over time. It also becomes apparent that the impurity
delocalises over time and that this effect is slowed down with increasing
$\eta$, which can be understood by a self-trapping argument given in the main
text.

##  III Results

###  III.1 Post-quench density profile

Figure 2: The superfluid density and position variance and velocity variance
for different attractive interactions and temperatures. In all plots the
parameters are as follows $\alpha=0.2$, $\gamma=0.04$, $n_{0}=5\tilde{\xi}$
and $a_{0}=3/\tilde{\xi}$. We can see that the impurity attracts surrounding
particles, which in turn depletes the condensate. As can be seen in (b), this
appears as a repulsively interacting polaron.

In this subsection, we focus on the density of the condensate for repulsive
and attractive impurity-bath couplings at different times after the quench and
supplement those findings with the variance of the position
$\sigma^{2}_{X}=\langle(\hat{X}-\langle\hat{X}\rangle)^{2}\rangle$ and the
variance of the velocity
$\sigma^{2}_{V}=\langle(\hat{P}/M-\langle\hat{P}/M\rangle)^{2}\rangle$. We
find a dynamically distinct behaviour for weak and strong couplings on the
repulsive and the attractive side.

In Fig. 1, the condensate density at different times and the evolution of the
variance of the position and the variance of the velocity for repulsive
interactions are shown. Before discussing the results in more detail we notice
that even for $\eta=50$ the minimum of the condensate density is still larger
than $3\tilde{\xi}$, indicating that the approximations made later for the
absorption spectrum are justified. We note that for weak coupling, the
impurity delocalises faster than a free impurity would. For stronger
couplings, the impurity stays localised much longer in time, indicating self-
trapping. The velocity variance saturates after a finite time, and the time
scale is inversely proportional to the impurity boson coupling. We note that
this can be explained by realising that two competing effects are determining
the dynamics. Namely, the impurity tends to distribute the repulsion equally
throughout the condensate, causing the impurity to delocalise and the opposing
effect of self-trapping, where the impurity deforms the condensate and then
self-traps in the deformation. It is intuitively clear that self-trapping will
not occur for weak couplings, which explains the different behaviours seen in
Fig. 1. It can also be seen that the variance of the impurity velocity can
exceed the speed of sound, which is associated with the emission of non zero
energy excitation, indicating an energy transfer from the impurity to the
bath, which has also been observed in Mistakidis  _et al._ (2019c). The
temperature influences the value with which the position variance and velocity
variance saturate; it does not significantly influence the timescale. For
strong coupling, the temperature dependence becomes relatively weak, which can
be explained by noting that the impurity-boson scattering length determines
the relevant energy scale, which is much larger than the thermal length in
this case.

In Fig 2 the same situation for attractive couplings is shown. Here the
difference between strongly attractive and moderate attractive couplings
becomes obvious. We observe that in the case of moderately attractive
couplings, the impurity not only diffuses but also forms a purely attractive
polaron. In contrast, for strongly attractive couplings, an attractive polaron
with repulsive interactions is observed, and the time scales of the polaron
formation are prolonged. This difference is a dynamical effect and can be
understood by noting that when the interaction is turned on, particles from
the condensate start to accumulate around the impurity, which in turn depletes
the condensate around the impurity. The superfluid is interacting itself, and
therefore the depletion is filled by the particles around it, with the time
scale being set by the boson-boson interaction. Meanwhile, the impurity-boson
interaction strength determines the number of particles that can accumulate
around the impurity before the boson-boson action prevents further
accumulation. At the same time the impurity delocalises, which prevents the
formation of a well-defined peak around the impurity and the interaction can
thus seem repulsive for a long time scale. Ultimately this is of course only a
metastable state. This process continues for a longer time when the impurity-
boson interaction is large, resulting in a polaron that looks repulsive which
can be observed in Fig 2(b).

###  III.2 Impurity velocity

Another quantity that is of great interest is the impurity velocity. The out-
of-equilibrium case gives insight into the polaron formation and the time
scales at work. It is also of great importance for the equilibrium case since
it can be used to calculate the effective mass of the polaron Grusdt and
Demler (2015).

Here, the impurity is not at rest when the quench occurs but instead carries
some finite momentum. The sudden quench of the impurity-boson interaction
leads to a momentum transfer from the impurity to the surrounding particles,
and we expect a slow down of the impurity. The time evolution for the velocity
of the impurity is depicted in Fig. 3. Here it becomes apparent that quantum
corrections have a noticeable impact on the evolution of the velocity at weak
to intermediate coupling. This can easily be understood by noting that the
impurity is treated as a point particle on the mean-field level, and as
observed in Fig 1, which is not a valid approximation for weak couplings. This
also explains why the MF velocity is lower than the corrected solution. We
also note that the steady-state velocity increases with the temperature, which
can be understood by noting that the surrounding gas has a higher average
squared velocity for increasing temperature, and therefore the momentum
transfer will be smaller. In contrast, for strong couplings, the impurity
stays localised and approximating it as a point particle is, therefore, less
of a simplification. The same behaviour can be observed for the temperature
dependence, which is more critical for weak and intermediate couplings, which
is again due to the scattering length being larger than the thermal length. We
also note that the impurity transfers some of its momentum to the Bose-gas and
then relatively quickly reaches an equilibrium velocity for not too strong
interactions. For stronger interactions, we observe different behaviour. After
an initial abrupt slow down of the impurity, the impurity velocity changes
sign, which directly results from the back action from the condensate. We note
that a similar abrupt slowdown has been observed in the three dimensional
setting in Grusdt  _et al._ (2018). The velocity then performs a damped
oscillation around its final velocity.

Figure 3: The impurity velocity over time for different temperatures. The
parameters are $\alpha=0.2$, $\gamma=0.04$, $n_{0}=5\tilde{\xi}$ and
$a_{0}=3/\tilde{\xi}$. We note that the quantum corrections have a big
influence for weak impurity-bath interactions.

###  III.3 The absorption spectrum

Figure 4: The absorption spectrum
$A(\omega)=2\mathrm{Re}\int^{\infty}_{0}G(t)e^{i\omega t}\,\mathrm{d}t$ for
different temperatures $T$ calculated using the truncated Wigner approximation
for $\alpha=0.2$, $\gamma=0.04$, $n_{0}=5\tilde{\xi}$ and
$a_{0}=3/\tilde{\xi}$

Next we turn our attention to the (injection) absorption spectrum
$A(\omega)=2\mathrm{Re}\int^{\infty}_{0}G(t)e^{i\omega t}\,\mathrm{d}t$ Cetina
_et al._ (2016); Massignan  _et al._ (2014), which can be measured using
Ramsey spectroscopy Jørgensen  _et al._ (2016); Cetina  _et al._ (2015, 2016).
It can be calculated by taking the Fourier transform of the impurity green’s
function, which characterises the dephasing of the system and is closely
related to the Lochschmidt echo Vaníček and Cohen (2016). The absorption
spectrum gives essential information about the polaron formation and can be
used to estimate the polaron energy and lifetime Kain and Ling (2014); Mahan
(2000). At this point, we also want to stress that pure mean-field (MF)
calculations in position momentum basis are not sufficient to calculate the
impurity Green’s function. This can be seen by noting that there is no
averaging for a classical calculation, which means there is no dephasing
between different trajectories, and therefore one always obtains $|G|=1$ in
purely classical calculations.

Our results are depicted in Fig. 4, it can be observed that the quasi-particle
peak widens with increasing temperature, which is also reported in 3D Dzsotjan
_et al._ (2020). However, in contrast to the 3D case, Shchadilova  _et al._
(2016), where the extended Fröhlich model was considered, we do not find
several peaks on the repulsive side. We also note that the overall amplitude
decreases with $\eta$, and the quasi-particle peak gets washed out with
increasing $\eta$, which is a direct consequence of the orthogonality
catastrophe Fogarty  _et al._ (2020); Guenther  _et al._ (2021); Campbell  _et
al._ (2014), hence the emergence of a clear quasi-particle peak comes under
question. We note that the absorption spectrum shows a functional dependence
associated with quasi-particle behaviour, and we do not find an infrared
dominated regime as observed in other one dimensional systems Kain and Ling
(2014). Those findings are supplemented by the overlap $G(t)$. Here we can see
another vastly different feature of the one-dimensional case, compared to the
three-dimensional case Dzsotjan  _et al._ (2020); Shchadilova  _et al._
(2016), namely that $|G|$, approaches zero even for moderate couplings,
signalling the onset of the orthogonality catastrophe. As expected, the
dephasing becomes more rapid with increased temperature and increased
$g_{\mathrm{IB}}$.

##  IV Summary & outlook

In summary, by leveraging the Keldysh formalism, we derived a truncated Wigner
approach to study dynamical properties of the bose polaron in 1D. This allowed
us to reduce the problem to simulating semi-classical equations of motion with
stochastic initial conditions. We showed how to adequately account for
temperature effects of the surrounding bath by sampling the phase and density
of the condensate. We also discussed how to regularise the arising divergences
that typically occur in such one dimensional systems. The method presented
here takes the back action of the impurity onto the condensate into account
and is therefore applicable from weak to strong impurity bath couplings.

We then used this framework to calculate the dynamics of an impurity after
sudden immersion in a surrounding bath and the absorption spectrum. By
considering the condensate density and the position/velocity variance, we
showed that there is a distinct dynamical behaviour associated with the strong
and weak coupling regime, namely our results indicate self-trapping of the
impurity for strongly repulsive interactions, and we also find a repulsive
polaron on the attractive side. We also investigated the temperature
dependence of the polaron formation and found a substantial influence of
quantum corrections on dynamical properties like the velocity of the impurity,
showing the necessity to go beyond pure mean-field considerations. Lastly, we
considered the absorption spectrum and the impurity Green’s function. Here, we
observed a clear quasi-particle peak for weak to intermediate couplings. In
contrast to that, we see that the quasi-particle peak is washed out for strong
couplings and that temperature effects widen the quasi-particle peak. In
contrast to the higher dimensional case, the impurity Green’s function
approaches zero even for weak couplings.

At this point, we want to stress that our approach is neither limited to 1D
nor a single impurity. It could therefore serve as an exciting starting point
to explore higher dimensional systems, as well as the interplay of several
impurities. While the generalization to several impurities is quite
straightforward we want to stress that, as pointed out for example in Ardila
_et al._ (2020); Khan  _et al._ (2021); Lingua  _et al._ (2018), the
generalisation to higher dimensions is highly non-trivial in general. First we
note that in higher dimensions it is not possible to use bare contact
interactions for the boson-boson interaction and the boson-impurity
interaction simultaneously when employing the approach. One has to resort to
using more realistic interaction potentials for at least one of them as has,
for example, been done in the three dimensional context in Drescher  _et al._
(2020); Guenther  _et al._ (2021). Another major challenge is a purely
numerical one, as it becomes more costly to sample the Bose fields in higher
dimensions. Nevertheless we expect that the presented method may be paired
with some small approximations to address higher dimensional systems.

## Acknowledgments

We would like to thank Michael Fleischhauer and Martin Will for fruitful
discussions. J.J. is grateful for support from EPSRC under Grant EP/R513052/1.
R.B. is grateful for support from a Cecilia Tanner Research Impulse Grant.

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