# Stochastic Dynamics and Bound States of Heavy Impurities in a Fermi Bath

Matteo Sighinolfi  INO-CNR BEC Center, 38123, Trento, Italy  Dipartimento di
Fisica, Università di Trento, 38123 Trento, Italy  Davide De Boni
Dipartimento di Fisica, Università di Trento, 38123 Trento, Italy  Alessandro
Roggero  Dipartimento di Fisica, Università di Trento, 38123 Trento, Italy
Trento Institute for Fundamental Physics and Applications, INFN, 38123,
Trento, Italy  Giovanni Garberoglio  European Centre for Theoretical Studies
in Nuclear Physics and Related Areas (FBK-ECT*), Trento, I-38123, Italy.
Trento Institute for Fundamental Physics and Applications, INFN, 38123,
Trento, Italy  Pietro Faccioli  Dipartimento di Fisica, Università di Trento,
38123 Trento, Italy  Trento Institute for Fundamental Physics and
Applications, INFN, 38123, Trento, Italy<EMAIL_ADDRESS>Alessio
Recati  INO-CNR BEC Center, 38123, Trento, Italy  Dipartimento di Fisica,
Università di Trento, 38123 Trento, Italy  Trento Institute for Fundamental
Physics and Applications, INFN, 38123, Trento, Italy<EMAIL_ADDRESS>
(August 27, 2024)

###### Abstract

We investigate the dynamics of heavy impurities embedded in an ultra-cold
Fermi gas by using a Generalized Langevin equation. The latter – derived by
means of influence functional theory – describes the stochastic classical
dynamics of the impurities and the quantum nature of the fermionic bath
manifests in the emergent interaction between the impurities and in the
viscosity tensor. By focusing on the two-impurity case, we predict the
existence of bound states, in different conditions of coupling and
temperature, and whose life-time can be analytically estimated. Our
predictions should be testable using cold-gases platforms within current
technology.

##  I Introduction

The concept of mediated interactions between particles due to the medium they
are immersed in is ubiquitous in physics. Notable examples include the phonon-
mediated interaction between electrons [1] –giving rise to
Bardeen–Cooper–Schrieffer superconductivity–, the interaction between cluster
or nuclear pasta structures mediated by the surrounding neutron fluid in the
inner crust of neutron stars [2, 3, 4], and the interaction between heavy
quarks mediated by a plasma of deconfined quarks and gluons, in super-hot
hadronic matter [5].

Highly imbalanced mixtures of ultra-cold gases provide clean and tunable
platforms where to study medium mediated interactions. In these systems, the
quasi-particles resulting from dressing impurities by the polarization of the
bath are usually referred to as polarons. The study of polaron physics in cold
gases was initiated by seminal experimental works on the normal-to-superfluid
phase transition in imbalanced Fermi-Fermi mixtures [6, 7] and the
identification of the normal phase as a weakly interacting gas of polarons, in
the spirit of Landau Fermi liquid theory (see, e.g., [8, 9] and reference
therein). Shortly after, also the case of impurities immersed in a Bose gas
was experimentally realized [10, 11].

Presently, the static and dynamical properties of a single polaron have been
relatively well understood, at least for the case of a degenerate polarized
Fermi bath, at zero temperature [12]. On the other hand, the experimental and
theoretical characterization of the effect of the mediated interaction between
impurities is, in general, much more challenging [13]. However, two very
recent experiments have measured the effect of the mediated interaction on a
Bose condensed gas in a Bose-Fermi mixture, in which the Fermi gas plays the
role of the bath [14, 15]

The present work aims at exploring the dynamics of heavy impurities in a Fermi
bath at finite temperature, within the framework of Generalized Langevin
equation (GLE) [16] that is derived from a chain of well-controlled
approximations, starting from the a microscopic Feynman–Vernon influence
functional [17]. In this way, we are able to provide semi-analytical
expressions for both the mediated inter-impurity interaction and the
configuration-dependent friction tensor.

Within our approach the effective stochastic dynamics of the haevy impurities
is treated at the classical level. On the other hand, quantum effects must be
fully taken into account when deriving the mediated interaction – which
corresponds to a finite temperature Ruderman–-Kittel-–Kasuya–-Yosida
(RKKY)[18, 19, 20] potential – and the friction tensor.

As a case study, we analyse the dynamics of two heavy impurities. By
numerically integrating their stochastic equations of motion starting from
configurations in which they are close to each other, we find evidence for the
formation of a transient bound state. Numerical estimates of the life-time of
this state at different temperatures agree well with the analytic calculations
of the dissociation rate performed within Kramers’ theory (see, e.g., [21] and
reference therein), thus demonstrating that the impurity pair dissociation is
a thermally activated rare event.

We also find that the position-dependent off-diagonal elements in the friction
tensor have important implications on the dynamics of the pair. In particular,
the relative motion of two close impurities is almost frictionless, yet the
presence of a longitudinal and a transverse friction leads to a rapid
dissipation of the relative orbital angular momentum (see Fig. 3 b)).

FIG. 1:  Two heavy impurities (red) in a bath of fermions (cyan): the bare
impurity-bath interaction $g$ is responsible for the induced forces between
impurities ${\bf F}_{1,2}$ and for the low friction region (yellow).

The paper is organized as follows: in Sec. II, we derive the Feynman–Vernon
influence functional for our system; in Sec. III, we describe the quantum
mediated interaction and friction and derive the GLE for the dynamics of the
impurities, focusing on the one- and two-impurity cases; in Sec. IV, we
discuss the numerical results obtained for the dynamics of two impurities
which are initially close to each other. A summary of our findings is the
content of Sec. V.

##  II Theoretical Setup

We consider a system composed by a bath of degenerate ultra-cold Fermi atoms
of mass $m$ and chemical potential $\mu$, interacting with $N$ impurities of
mass $m_{I}\gg m$. At the energy scales we consider, particles interact only
via s-wave scattering and therefore the interaction between the atoms of the
bath can be neglected. For the sake of clarity we also consider that there is
no direct interaction between the impurities. The interaction between the bath
and the impurities is characterized in the following by a contact potential
with strength $g$. We also assume that the system is at a temperature $T$ such
that the de Broglie thermal wavelength of the impurities
$\lambda\sim\hbar\sqrt{2\pi/m_{I}k_{B}T}$ is small compared to their typical
inter-particle distance. This will allow us to regard impurities as quasi-
classical particles. In order to obtain a stochastic equation of motion for
the impurities, it is convenient to describe them in first quantization and
coordinate representation. We rely on quantum field theory to describe the
dynamics of the degenarate fermionic bath. Our system can be modelled by the
following Hamiltonian:

$\hat{H}=\hat{H}_{F}+\hat{H}_{I}+\hat{V}\,,$ (1)

where

$\displaystyle\hat{H}_{I}=\sum_{i=1}^{N}\frac{\hat{\mathbf{p}}_{i}^{2}}{2m_{I}}$
(2) $\displaystyle\hat{H}_{F}=\int d{\bf x}\leavevmode\nobreak\
\hat{\Psi}^{\dagger}({\bf
x})\left(\frac{-\hbar^{2}}{2m}\hat{\nabla}^{2}-\mu\right)\hat{\Psi}({\bf x})$
(3) $\displaystyle\hat{V}=g\sum_{i=1}^{N}\int d{\bf x}\leavevmode\nobreak\
\hat{\Psi}^{\dagger}({\bf x})\leavevmode\nobreak\ \delta(\hat{\bf q}_{i}-{\bf
x})\leavevmode\nobreak\ \hat{\Psi}({\bf x}),$ (4)

where $\hat{\mathbf{p}}_{i}$ and $\hat{\mathbf{q}}_{i}$ denote the impurity
momentum and position operators, $\hat{\Psi}({\bf x})$ and
$\hat{\Psi}^{\dagger}({\bf x})$ are the annihilation and creation field
operators for the particles in the bath.

Let us consider a setup in which the impurities are initially decoupled from
the bath and localized at fixed positions ${\bf
Q}_{i}\equiv(\mathbf{q}_{1},\ldots,\mathbf{q}_{N})$. At time $t=0$, the
interaction with the bath is switched on and the system’s density matrix
begins to evolve according to the Hamiltonian (1). We are interested in the
diagonal elements of the reduced density matrix for the impurities, i.e. in
the probability of observing the impurities at ${\bf
Q}_{f}=(\mathbf{q}^{f}_{1},\ldots,\mathbf{q}^{f}_{N})$ at time $t_{f}$. Using
Feynman–Vernon path integral representation of the density matrix [17] we
obtain:

$P(\mathbf{Q}_{f},t|\mathbf{Q}_{i},0)=\int_{\mathbf{Q}_{i}}^{\mathbf{Q}_{f}}\mathcal{D}\mathbf{Q}\int\mathcal{D}\xi\int\mathcal{D}\xi^{*}e^{\frac{i}{\hbar}S[\mathbf{Q},\xi,\xi^{*}]}.$
(5)

In this equation, $\xi(t,{\bf x})$ and $\xi^{*}(t,{\bf x})$ are Grassmann
coherent field variables, while the functional at the exponent is

$\begin{split}S[\mathbf{Q},\xi,\xi^{*}]&=\int_{\mathcal{C}}dt^{\prime}\left\\{\frac{m_{I}}{2}\sum_{j=1}^{N}\dot{\mathbf{q}}^{2}_{j}(t^{\prime})+\right.\\\
&\left.\int d{\bf x}\xi^{*}(t^{\prime},{\bf
x})\left(i\hbar\partial_{t^{\prime}}-\frac{\hbar^{2}\nabla^{2}}{2m}-\mu-\rho(t^{\prime},{\bf
x})\right)\xi(t^{\prime},{\bf x})\right\\},\end{split}$ (6)

where $\rho(t,{\bf x})=g\sum_{i=1}^{N}\delta(\mathbf{q}_{i}(t)-{\bf x})$ is
the instantaneous impurity density and the time integral is defined over the
standard Keldysh contour $\mathcal{C}$ [22, 23].

The integral over the Grassmann fields $\xi,\xi^{*}$ can be carried out
analytically, leading to:

$P(\mathbf{Q}_{f},t|\mathbf{Q}_{i},0)=\int_{\mathbf{Q_{i}}}^{\mathbf{Q_{F}}}D\mathbf{Q}\leavevmode\nobreak\
e^{i\Phi_{\mathcal{C}}[\mathbf{Q}]}\leavevmode\nobreak\ \\\
e^{i\frac{m_{I}}{2\hbar}\sum_{j=1}^{N}\int_{\mathcal{C}}dt^{\prime}\dot{\mathbf{q}_{j}}^{2}},$
(7)

where $\Phi_{\mathcal{C}}[\mathbf{Q}]$ is the influence functional, which is
formally written as

$i\Phi_{\mathcal{C}}[\mathbf{Q}]=\text{Tr}\left[\log\left(i\hbar\partial_{t^{\prime}}-\frac{\hbar^{2}\nabla^{2}}{2m}-\mu-\rho(t^{\prime},{\bf
x})\right)\right].$ (8)

To obtain an explicit representation for $\Phi_{\mathcal{C}}[\mathbf{Q}]$, it
is convenient to deal separately with the upper and lower branches of the
Keldysh contour. In addition, we assume a low impurity density and perform a
functional expansion to second order in $\rho(t^{\prime},{\bf x})$. The $0$-th
order term is a constant that is reabsorbed in the definition of probability,
while the first order term is an energy shift that does not affect the
dynamics. The resulting expression for the transition probability density is

$\displaystyle P(\mathbf{Q}_{f},t|\mathbf{Q}_{i},0)=\int_{{\bf Q}_{i}}^{\bf
Q_{f}}\mathcal{D}{\bf Q^{\prime}}\int_{{\bf Q}_{i}}^{\bf Q_{f}}\mathcal{D}{\bf
Q^{\prime\prime}}$ $\displaystyle e^{i\Phi({\bf Q^{\prime}},{\bf
Q^{\prime\prime}})}\leavevmode\nobreak\
e^{i\frac{m_{I}}{2\hbar}\sum_{j=1}^{N}\int_{0}^{t}dt^{\prime}\left(\dot{\mathbf{q}}_{j}^{{}^{\prime}2}-\dot{\mathbf{q}_{j}}^{{}^{\prime\prime}2}\right)},$
(9)

where

$\displaystyle\Phi({\bf Q}^{\prime},{\bf
Q}^{\prime\prime})=\frac{i}{2}\sum_{a,b=1}^{2}\int_{0}^{t}\leavevmode\nobreak\
dt^{\prime}\int_{0}^{t}dt^{{}^{\prime\prime}}\int d{\bf x}\int d{\bf y}$
$\displaystyle\rho_{a}(t^{{}^{\prime}},{\bf
x})\Delta_{ab}(t^{{}^{\prime}}-t^{{}^{\prime\prime}},{\bf x}-{\bf
y})\leavevmode\nobreak\ \rho_{b}(t^{{}^{\prime\prime}},{\bf y}),$ (10)

where $a$, $b$ label the branches of the Keldysh contour ${\cal C}$, primed
variables lie on the forward branch of the contour and double-primed variables
lie on the backward branch. In particular in Eq. (10),

$\displaystyle\rho_{1}(t,{\bf x})=g\sum_{i}\delta({\bf
q}_{i}^{{}^{\prime}}(t)-{\bf x})$ $\displaystyle\rho_{2}(t,{\bf
x})=g\sum_{i}\delta({\bf q}_{i}^{{}^{\prime\prime}}(t)-{\bf x}),$ (11)

and $\Delta_{ab}$ are the entries of a $2\times 2$ matrix of Green’s
functions:

$\displaystyle\Delta_{11}(t,{\bf x})$ $\displaystyle=$
$\displaystyle\Delta_{F}(t,{\bf x})=iD_{F}(t,{\bf x})$
$\displaystyle\Delta_{12}(t,{\bf x})$ $\displaystyle=$
$\displaystyle-\Delta_{<}(t,{\bf x})=-iD_{<}(t,{\bf x})$
$\displaystyle\Delta_{21}(t,{\bf x})$ $\displaystyle=$
$\displaystyle-\Delta_{>}(t,{\bf x})=-iD_{>}(t,{\bf x})$
$\displaystyle\Delta_{22}(t,{\bf x})$ $\displaystyle=$
$\displaystyle\Delta_{\tilde{F}}(t,{\bf x})=iD_{\tilde{F}}(t,{\bf x}).$ (12)

Here, $D_{>}(t,{\bf x}),D_{<}(t,{\bf x})$ and $D_{F}(t,{\bf x})$ are the
standard fermionic polarization propagators of many-body theory [24, 25].

We emphasize that the expressions (9) and (10) follow directly from Eq. (7),
in the small impurity density limit.

In the next section, we shall introduce additional approximation which enable
us to efficiently compute this transition probability by integrating
stochastic differential equation of motions.

##  III Effective Stochastic Dynamics of Heavy Impurities

In this section, we introduce a chain of well-controlled approximations to
enable the sampling of the transition probability density (7).

###  III.1 Small Frequency Expansion

Since the mass of the impurities is much greater than that of the particles in
the bath the dynamics of the former is expected to be much slower. Then, it is
possible to perform a small frequency expansion of $\Delta_{ab}$ in Eq. (II):

$\displaystyle\Delta_{ab}(t,{\bf x})$ $\displaystyle=$
$\displaystyle\int\frac{d\omega}{2\pi}e^{-i\omega
t}\left(\sum_{n=0}^{\infty}\frac{\omega^{n}}{n!}\leavevmode\nobreak\
F_{ab}^{(n)}(x)\right)$ (13) $\displaystyle=$ $\displaystyle F^{(0)}_{ab}({\bf
x})+i\frac{d}{dt}\delta(t)F^{(1)}_{ab}({\bf x})+\ldots,$

where

$\displaystyle F^{(0)}_{ab}(\mathbf{x}-\mathbf{y})$ $\displaystyle\equiv$
$\displaystyle\Delta_{ab}(\omega=0,\mathbf{x}-\mathbf{y})$ (14) $\displaystyle
F^{(1)}_{ab}(\mathbf{x}-\mathbf{y})$ $\displaystyle\equiv$
$\displaystyle\lim_{\omega\to
0}\frac{d}{d\omega}\Delta_{ab}(\omega,\mathbf{x}-\mathbf{y}),$ (15)

and the dots denote higher order terms in the Taylor expansion.

Substituting Eqs. (13,11) into Eq. (10) we obtain

$\begin{split}\Phi({\bf Q}^{\prime},{\bf
Q}^{\prime\prime})&=\frac{ig^{2}}{2}\sum_{i,j=1}^{N}\int_{0}^{t}du\bigg{\\{}F^{(0)}_{F}(\mathbf{q}_{i}^{\prime}-\mathbf{q}_{j}^{\prime})+\\\
+&F^{(0)}_{\tilde{F}}(\mathbf{q}_{i}^{\prime\prime}-\mathbf{q}_{j}^{\prime\prime})-F^{(0)}_{<}(\mathbf{q}_{i}^{\prime}-\mathbf{q}_{j}^{\prime\prime})+\\\
-&F^{(0)}_{>}(\mathbf{q}_{i}^{\prime\prime}-\mathbf{q}_{j}^{\prime})-i\dot{\mathbf{q}}_{j1}\frac{\partial}{\partial\mathbf{q}_{j}^{\prime}}F_{>}^{(1)}(\mathbf{q}_{i}^{\prime\prime}-\mathbf{q}_{j}^{\prime})\\\
-&i\dot{\mathbf{q}}_{j2}\frac{\partial}{\partial\mathbf{q}_{j}^{\prime\prime}}F_{<}^{(1)}(\mathbf{q}_{i}^{\prime}-\mathbf{q}_{j}^{\prime\prime})\bigg{\\}}.\end{split}$
(16)

It is convenient to introduce the so-called complex potential
$\mathcal{V}(\mathbf{x}-\mathbf{y})$:

$i\mathcal{V}(\mathbf{x}-\mathbf{y})\equiv
F^{(0)}_{F}(\mathbf{x}-\mathbf{y})=V(\mathbf{x}-\mathbf{y})+iW(\mathbf{x}-\mathbf{y}).$
(17)

In appendix A, we show that the real and imaginary part of $\mathcal{V}$ can
be expressed in terms of the retarded polarization propagator in Fourier
space:

$\displaystyle
V(\mathbf{x}-\mathbf{y})=\text{Re}D^{R}(\omega=0,\mathbf{x}-\mathbf{y})$ (18)
$\displaystyle W(\mathbf{x}-\mathbf{y})=\frac{2}{\beta}\lim_{\omega\to
0}\frac{1}{\omega}\text{Im}D^{R}(\omega,\mathbf{x}-\mathbf{y}).$ (19)

We also show that for a bath of non-interacting fermions in 3 dimensions

$\displaystyle V(\mathbf{x}-\mathbf{y})=-\frac{mk_{F}}{4\pi^{4}\hbar^{2}}\int
dq\frac{\sin(qr)}{r}\int dk\,f_{FD}(k,T)k\times$
$\displaystyle\times\log\left|\frac{k+q/2}{k-q/2}\right|$ (20) $\displaystyle
W(\mathbf{x}-\mathbf{y})=-\frac{m^{2}}{2\pi^{3}\hbar^{3}\beta}\int
dq\,f_{FD}(q/2,T)q\frac{\sin(qr)}{qr},$ (21)

where $r=|\mathbf{x}-\mathbf{y}|$ and $f_{FD}$ is the Fermi-Dirac
distribution.

In the following, we use the rescaled imaginary potential $W_{R}$ as

$W_{R}(\mathbf{x}-\mathbf{y})=\frac{\beta}{2}W(\mathbf{x}-\mathbf{y}),$ (22)

for an easier understanding. Indeed, with this rescaling the term $1/\beta$ in
Eq. (21) disappears.

###  III.2 Classical Limit

We now take the classical limit for the dynamics of the impurities. In order
to implement this approximation, we first perform the change of variables

$\mathbf{r}_{i}=\frac{1}{2}(\mathbf{q}_{i}^{\prime}+\mathbf{q}_{i}^{\prime\prime})\qquad\mathbf{y}_{i}=\mathbf{q}_{i}^{\prime}-\mathbf{q}_{i}^{\prime\prime}.$
(23)

After an integration by parts, the free action of the impurity takes the form:

$e^{\frac{im_{I}}{\hbar}\sum_{i=1}^{N}\int_{0}^{t}\ddot{\bf r}_{i}\cdot{\bf
y}_{i}}.$ (24)

We expect the dominant contribution to the path integral to come from the
functional region where the time integral in the exponent is small or at most
of order unity. To estimate it, we note that $\int_{t_{i}}^{t_{f}}dt\ddot{\bf
r}_{i}\cdot{\bf y}_{i}\sim\sqrt{k_{B}T/m_{I}}|{\bf y}_{i}|$, where
$\sqrt{k_{B}T/m_{I}}$ is the average thermal velocity of the impurities. Then,
the stationary phase condition implies $|{\bf
y}_{i}|\lesssim\sqrt{1/m_{I}k_{B}T}.$ In the limit of heavy impurities,
fluctuations of $y(t)$ become small compared to all relevant length scales,
thus we can expand the influence functional to second order in ${\bf y}_{i}$,
leading to

$\begin{split}&P(\mathbf{R}_{f},t|\mathbf{R}_{i},0)=\int_{\mathbf{R}_{i}}^{\mathbf{R}_{f}}{\cal
D}\mathbf{R}\int_{0}^{0}\mathcal{D}\mathbf{Y}\\\
&\text{exp}\left\\{-\frac{i}{\hbar}\int_{0}^{t}dt^{\prime}\left[\mathbf{y}_{i}\left(m_{I}\ddot{\mathbf{r}}_{i}+\Gamma_{ij}(\mathbf{R})\dot{\mathbf{r}}_{j}-\mathbf{F}_{i}(\mathbf{R})\right)+\right.\right.\\\
&\left.\left.-\frac{1}{2}\mathbf{y}_{i}\frac{2}{\beta}\Gamma_{ij}(\mathbf{R})\mathbf{y}_{j}\right]\right\\},\end{split}$
(25)

where ${\bf R}=({\bf r}_{1},\ldots,{\bf r}_{N})^{T}$, ${\bf Y}=({\bf
y}_{1},\ldots,{\bf y}_{N})^{T}$ and the sum over repeated indices
$i,j=1,\ldots,N$ is understood. $F(\mathbf{R})$ and $\Gamma_{ij}(\mathbf{R})$
are defined as

$\displaystyle\mathbf{F}_{i}(\mathbf{R})=-g^{2}\sum_{j=i}^{N}\nabla
V(\mathbf{r}_{i}-\mathbf{r}_{j})$ (26)
$\displaystyle\Gamma_{ij}(\mathbf{R})=g^{2}\mathcal{H}_{W_{R}}(\mathbf{r}_{i}-\mathbf{r}_{j}),$
(27)

where $\mathcal{H}_{W_{R}}$ is the Hessian of $W_{R}$.

The Gaussian integral over ${\bf Y}$ can be evaluated analytically, leading to
our final path integral expression for the transition probability:

$\displaystyle
P(\mathbf{R}_{f},t|\mathbf{R}_{i},0)=\int_{\mathbf{R}_{i}}^{\mathbf{R}_{f}}{\cal
D}\mathbf{R}e^{-\int_{0}^{t}d\tau\left(m_{I}\ddot{\bf R}-m_{I}\Gamma({\bf
R})\dot{\bf R}-{\bf F}({\bf R})\right)^{2}}.$ (28)

Here, the probability for the impurities to go from ${\bf R}_{i}$ to ${\bf
R}_{f}$ in a time $t$ is written as a functional integral over all possible
trajectories connecting the initial and the final configuration. We note that
the functional at the exponent, which determines the relative statistical
weight of ${\bf R}(t)$ trajectories, does not explicitly depend on $\hbar$.
Indeed, it corresponds to an Onsager–Machlup action [26], which characterizes
path integral representation of propagator in classical Fokker–Planck
dynamics.

As a consequence, as explicitly shown in Ref. [27, 28], the same transition
probability density of Eq. (28) can be generated by the following GLE:

$m_{I}\ddot{\mathbf{r}}_{i}=-\Gamma_{ij}(\mathbf{R})\dot{\mathbf{r}}_{j}+\mathbf{F}_{i}(\mathbf{R})+\mathbf{\Psi}_{i}(\mathbf{R},t).$
(29)

The viscosity $\Gamma(\mathbf{R})_{ij}$ and the noise term
$\mathbf{\Psi}_{i}(\mathbf{R},t)$ satisfy the fluctuation-dissipation
relations

$\displaystyle\langle\mathbf{\Psi}_{i}(\mathbf{R},t)\rangle=0$ (30)
$\displaystyle\langle\mathbf{\Psi}_{i}(\mathbf{R},t)\otimes\mathbf{\Psi}_{j}(\mathbf{R},t^{\prime})\rangle=\frac{2}{\beta}\Gamma_{ij}(\mathbf{R})\delta(t-t^{\prime}).$
(31)

The noise $\mathbf{\Psi}_{i}{(\mathbf{R})}$ depends only the relative
distances between the $i$-th impurity and all the others impurities. To
conclude this section, we note that while the dynamics of the impurities has
been reduced to a classical diffusion process, the quantum nature of the bath
is still effectively encoded in the structure of the viscosity and force
terms, derived from Eqs. (20, 21) 111Within our approximations, the impurity
mass $m_{I}$ is not renormalized. However, this effect can be considered
performing the small frequency expansion at order $\omega^{2}$, where terms
proportional to $\ddot{{\bf r}}_{i}$ and to $\dot{{\bf r}}_{i}^{2}$ are
present. Moreover, in the limit of $m_{I}\gg m$ and in the range of
interaction considered is known to be small, see e.g. [46].  .

FIG. 2:  Some spatial dependences with $k_{F}a=0.19$ and $T/T_{F}=0.2$, $0.1$,
$0.05$ (dotted-dashed blue, dashed red, full green lines). Panel a): $V({\bf
s})$ at $k_{F}a=0.19$, the intersection between horizontal dashed lines and
potential curves determines the typical size of the bound state
$r_{\textrm{b}}$. Panel b): difference between the constant viscosity term and
the diagonal component of the viscosity matrix $\gamma-\Gamma_{12}^{ii}({\bf
s})$ in units of $\gamma_{T\to 0}$. Note that $|{\bf s}|k_{F}\gtrsim 6$ the
oscillations decay because the contribution of $\Gamma_{12}^{ii}$ vanishes. In
the inset is shown the behaviour for small $|{\bf s}|$. Panel c): sum between
the constant viscosity term and the diagonal component of the viscosity matrix
$\gamma-\Gamma_{12}^{ii}({\bf s})$ in units of $\gamma_{T\to 0}$. Panel d):
off-diagonal component of the viscosity matrix $\Gamma_{12}^{ij}$ multiplied
by $|{\bf s}|^{2}/s_{i}s_{j}$ in units of $\gamma_{T\to 0}$. Also for the off-
diagonal term the oscillatory behaviour decays for $|{\bf s}|k_{F}\gtrsim 6$.

###  III.3 Dynamics of a single impurity

It is instructive to first apply our formalism to the case of a single
impurity. For $N=1$, the GLE reduces to that of a standard Brownian particle,
with constant viscosity and white noise:

$m_{I}\ddot{\mathbf{r}}=-\gamma\dot{\mathbf{r}}+\mathbf{\Psi}(t),$ (32)

where we defined $\mathbf{r}=\mathbf{R}=\mathbf{r}_{1}$ and

$\gamma=\Gamma_{11}=-\frac{8m^{4}g^{2}}{3\hbar^{7}\pi^{3}}(k_{B}T)^{2}\text{Li}_{2}(-e^{\beta\mu(T)}),$
(33)

is the single impurity friction constant with Li2 is the dilogarithm [30]. At
finite temperature, Eqs. (32) and (33) yield the conventional Einstein’s
diffusion law, and the kinetic energy of the impurity thermalizes with the
bath. However, if temperature is much smaller then the bath Fermi temperature
$T_{F}=\varepsilon_{F}/k_{B}$, with the usual Fermi energy
$\varepsilon_{F}=\hbar^{2}k_{F}^{2}/2m$, Eq.(33) can be written as

$\gamma_{T\to 0}=\frac{4\hbar
k_{F}^{2}}{3\pi}\left(\frac{m}{m_{r}}k_{F}a\right)^{2}\left(1+\frac{T^{2}}{T_{F}^{2}}\frac{\pi^{2}}{3}\right).$
(34)

In this equation, $m_{r}=m_{I}m/(m_{I}+m)$ is the reduced mass and $g$ is
expressed in terms of the more physical $s$-wave scattering length $a$,
$g=2\pi\hbar^{2}a/m_{r}$. We note that the viscosity remains finite even at
zero temperature. This is possible because the impurity releases energy into
the bath, by inducing particle-hole excitations. The same result for the
viscosity Eq. (34) can also be obtained by considering the energy dissipation
of an infinite mass impurity moving in the bath (see Appendix B), as discussed
in Ref. [31] for the case of interacting Bose gases. Note that, in the latter
case (and for any superfluid system), the viscosity vanishes for $T\rightarrow
0$, due to the existence of the critical Landau velocity, which provides a
minimal velocity for the impurity to excite the system. Interestingly,
Schecter and Kamenev applied the same formalism we adopted in the present work
to compute the friction in a weakly interacting Bose gas, and found that it
scales as $\gamma_{BEC}\simeq T^{7}$ [32].

###  III.4 Dynamics of two impurities

Let us now consider the case of two impurities. The corresponding GLEs read:

$\displaystyle m_{I}\ddot{\mathbf{r}}_{1}$
$\displaystyle=-\left(\gamma\dot{\mathbf{r}}_{1}+\Gamma_{12}(\mathbf{r}_{1}-\mathbf{r}_{2})\dot{\mathbf{r}}_{2}\right)+$
$\displaystyle+{\bf
F}_{1}(\mathbf{r}_{1}-\mathbf{r}_{2})+\mathbf{\Psi}_{1}(\mathbf{r}_{1}-\mathbf{r}_{2},t)$
(35) $\displaystyle m_{I}\ddot{\mathbf{r}}_{2}$
$\displaystyle=-\left(\Gamma_{21}(\mathbf{r}_{1}-\mathbf{r}_{2})\dot{\mathbf{r}}_{1}+\gamma\dot{\mathbf{r}}_{2}\right)+$
$\displaystyle-{\bf
F}_{1}(\mathbf{r}_{1}-\mathbf{r}_{2})+\mathbf{\Psi}_{2}(\mathbf{r}_{1}-\mathbf{r}_{2},t),$
(36)

in strong analogy with the old result for heavy particles in incompressible
fluids [33]. It is convenient to rewrite the previous equations in terms of
the relative distance between the impurities,
$\mathbf{s}=\mathbf{r}_{1}-\mathbf{r}_{2}$, and the center of mass
$\mathbf{r}_{CM}=(\mathbf{r}_{1}+\mathbf{r}_{2})/2$:

$\displaystyle m_{I}\ddot{\mathbf{s}}$
$\displaystyle=-(\gamma-\Gamma_{12}({\bf s}))\leavevmode\nobreak\
\dot{\mathbf{s}}+2{\bf F}_{1}(\mathbf{s})+\mathbf{\eta}_{-}({\bf s},t)$ (37)
$\displaystyle m_{I}\ddot{\mathbf{r}}_{CM}$
$\displaystyle=-(\gamma+\Gamma_{12}({\bf s}))\leavevmode\nobreak\
\dot{\mathbf{r}}_{CM}+\frac{1}{2}\mathbf{\eta}_{+}({\bf s},t),$ (38)

where $\mathbf{\eta}_{+}({\bf s},t)$ and $\mathbf{\eta}_{-}({\bf s},t)$ are
two Gaussian noises,

$\eta_{\pm}({\bf
s},t)=\bm{\Psi}_{1}(\mathbf{s},t)\pm\bm{\Psi}_{2}(\mathbf{s},t).$ (39)

Using Eqs. (20, 22), the explicit expression for the force and the viscosity
matrix can be respectively written as

$\displaystyle{\bf F}_{1}^{i}(\mathbf{s})$ $\displaystyle=$
$\displaystyle\frac{mg^{2}}{16\pi^{4}\hbar^{2}}\frac{s_{i}}{s^{2}}$
$\displaystyle\int_{0}^{\Lambda}\\!\\!\\!\\!dq\;qh(q,s)\int_{0}^{\infty}\\!\\!\\!\\!dk\leavevmode\nobreak\
kf_{FD}(k/2)\log\left|\frac{k+q}{k-q}\right|$

and

$\displaystyle\Gamma_{12}^{ij}(\mathbf{s})$ $\displaystyle=$
$\displaystyle-\frac{m^{2}g^{2}}{4\pi^{3}\hbar^{3}s^{2}}\int_{0}^{\infty}\\!\\!\\!\\!dq\leavevmode\nobreak\
q\left\\{h(q,s)\left(\delta_{ij}-\frac{s_{i}s_{j}}{s^{2}}\right)\right.$
$\displaystyle\left.-(2h(q,s)+qs\sin(qs))\frac{s_{i}s_{j}}{s^{2}}\right\\}f_{FD}(q/2),$

where $h(q,s)=\cos(qs)-\sin(qs)/(qs)$. Note that in Eq.(III.4) we have
introduced a UV momentum cutoff $\Lambda$. To be consistent with the physical
interaction characterized by the $s$-wave scattering length $a$, the coupling
constant must satisfy [34]:

$4k_{F}a=\left(\frac{\hbar^{2}}{2m_{r}}\frac{\pi}{gk_{F}}+\frac{\Lambda}{\pi
k_{F}}\right)^{-1}.$ (42)

The presence of a second impurity significantly modifies the stochastic
equations of motion. In particular, the relative motion experiences the effect
of an external force, which provides the finite temperature generalization of
the RKKY interaction [18, 19, 20]. In this case, the friction matrix depends
on the relative distance between the pair, distinguishing between friction in
the directions collinear and transverse to the relative distance ${\bf s}$. In
particular, the relative motion becomes underdamped in the limit in which the
distance between the impurities is small. On the other hand, the center of
mass diffuses according to a simple Brownian motion, with a friction matrix
that depends on the relative coordinate only.

The conservative potential and the friction matrix elements are plotted in
Fig. 2, which also shows that the temperature dependence is very weak and that
non-collinear friction is small.

FIG. 3:  Two representative trajectories obtained by integrating the GLEs
starting from a configuration with relative impurity distance
$r_{\textrm{b}}$, $k_{F}a=0.19$ and $T/T_{F}=0.05$. The components $s_{x}$,
$s_{y}$ of the distance ${\bf s}$ are shown and the color map labels time. The
motion of the impurities is also not confined on a plane due to thermal
fluctuations. In trajectory reported in panel a), the impurities remain in a
bound state throughout the entire simulation time. In panel b) the bound state
starts to dissociates for $t\simeq 1000\,t_{F}$. In both cases, the relative
motion in the bound state becomes quasi one-dimensional, because of angular
momentum dissipation induced by the large transverse viscosity.

FIG. 4:  Panel a): time evolution of $s=|\mathbf{s}|$ at $k_{F}a=0.19$ for
different temperatures: full green line is $T/T_{F}=0.05$, dashed red line is
$T/T_{F}=0.1$ and blue dotted-dashed line is $T/T_{F}=0.2$. For $T/T_{F}=0.2$
and $T/T_{F}=0.1$ also the dissociation of the bound state at time $\tau$ is
visible and it is indicated with an arrow. Panel b): Numerically observed
life-times $\tau$ (in units of Fermi time $t_{F}$) at different $T/T_{F}$ and
$k_{F}a$.

##  IV Numerical results: bound states dynamics and life-time

As a case study for the dynamics of two heavy impurities in a free Fermi gas,
we focus on the presence of localised (bound) solutions due to the mediated
interaction and estimate its life-time under the effect of the stochastic
noise. Firstly, we consider the typical value of the bound state size,
$r_{\textrm{b}}$: this can be estimated by matching the average kinetic energy
provided by the coupling with the bath with the strength of the mediated
interaction: $(k_{F}a)^{2}V(r_{\textrm{b}})\propto 3k_{B}T/2$. As expected, an
increase in the scattering length (temperature) leads to a smaller (larger)
$r_{\textrm{b}}$, as shown in Fig. 2 a).

We solve Eq. (29) using a stochastic Verlet algorithm [35] and simulate the
time evolution of 2 impurities at different temperature, with scattering
length $k_{F}a$. The two impurities start at rest with an initial random
position ${\bf s}_{0}$ subject to the constraint $|{\bf s}_{0}|=r_{b}$. We
average over 1000 independent simulations, with a mass ratio set to
$m_{I}/m=30$, which is comparable to that of typical experimental setups (for
example, in 133Cs-6Li mixtures [36] one has $m_{I}/m\simeq 22$).

In all simulations, we find that in the long-time regime, impurities drift
apart and eventually diffuse according to the single-impurity Brownian
dynamics described by Eq.(32). In Fig. 3, two representative trajectories of
the short-time regime are shown: in panel a) the impurities remain within a
distance comparable to $r_{\textrm{b}}$ throughout the entire simulation time,
signaling the existence of a bound state. In panel b) the impurities
eventually dissociate and begin an independent Brownian diffusion.

An interesting feature that can be inferred from these trajectories is that
the relative motion of the two impurities in the bound state tends to become
quasi one-dimensional. This is due to the presence of the transverse component
of the friction that leads to the dissipation of the internal orbital angular
momentum.

The life-time of the bound state $\tau$ is defined as the average dissociation
time. In the low temperature regime (i.e. when dissociation is a thermally
activated process), $\tau$ can be calculated using Kramers’ theory [21]:

$\tau=2\pi\sqrt{\frac{K}{K_{a}}}\frac{2m_{I}e^{\beta
U}}{\sqrt{\gamma^{2}+4Km_{I}}-\gamma},$ (43)

Here, the viscosity $\gamma$ is estimated from Eq. (33) by taking the limit of
vanishing distance, $K$ and $K_{a}$ are the curvature of the potential at the
top and bottom of the potential energy barrier, and $U$ is the height of the
barrier.

The life-time $\tau$ of the bound state can also be directly inferred from the
numerical simulations. Typical evolution of the inter-impurity distance is
shown in Fig 4 a) for $k_{F}a=0.19$ and for $T/T_{F}=0.2$ (dotted-dashed blue
line), $T/T_{F}=0.1$ (dashed red line) and for $T/T_{F}=0.05$ (green line).
For the latter the dissociation occurs at $t>300$ tF. In Fig. 4 b) the life-
time $\tau$ for different scattering lengths is reported and they perfectly
agree with the ones predicted by Kramers’ Eq. (43). This implies that, at
these temperatures, the dissociation of the bound states is a thermally
activated event. The range of temperatures we consider is experimentally
accessible. In addition, typical Fermi time $t_{F}=\hbar/\varepsilon_{F}$ in
recent experiments, see e.g. [37], is of order $10^{-2}$ ms, thus dissociation
times between 100 and 1000 $t_{F}$ should be experimentally detectable. We
stress that an agreement between Kramers’ theory predictions and experimental
dissociation times would represent a validation of the classical approach
developed in this work.

##  V Conclusions

In a fermionic bath, the stochastic dynamics of impurities is strongly
influenced by the effective interaction and friction induced by the coupling
to the medium. Under a well-controlled chain of approximations, the bath
degrees of freedom can be traced out and the impurities’ dynamics can be
described through an effective stochastic dynamics. In this scheme, the
impurities obey classical GLEs, and the quantum nature of the system is
encoded only in the induced force and viscosity terms.

In this work, we focused on the dynamics of a system consisting of two
impurities. We found that, in the short time regime, the interplay between
induced interaction and thermal fluctuations leads to the formation of a bound
state characterized by a radius $r_{\textrm{b}}$ and a life-time $\tau$.

Two experimental realizations of mixtures of Bose-Einstein condensates and
Fermi gas have been achieved so far [14, 15]. However, the density of heavy
impurities in these systems is relatively high, so that a description in terms
of heavy particles independently diffusing in the medium may not be accurate.
An important question to address is whether it is feasible to experimentally
probe systems with lower impurity densities, using the existing technology.

Although more demanding, cold gases could also be the proper platform to
obtain a direct experimental evidence of a non-collinear friction for the
impurities.

We note that the same approach adopted in the present work was applied by some
of us to investigate the dynamics of heavy quarks diffusing in a ultra-
relativistic quark-gluon plasma Ref. [27]. That analysis was based on an
effective finite temperature Abelian gauge theory, to describe the dynamics in
the deconfined plasma. In that approach, heavy quarks and anti-quarks played
the role of two distinct types of impurities, while light quarks and anti-
quarks formed the thermal bath. All quarks in the systems were coupled via a
Debye-screened Coulomb-type interaction. As a consequence of these features,
the sign of $\Gamma_{12}$ was found to be different from that of the present
Fermi system. Namely, the center of mass motion experiences a very reduced
effective friction, while the relative internal motion of the quark-antiquark
pair is overdamped. As an outlook, it could be interesting to device a cold
atom system that can mimic such a model. This may be done by properly
selecting two different hyperfine levels or two different atomic species that
couple with opposite sign to the particles in the Fermi bath. The extension of
the present simulation strategy to a superfluid fermionic bath and to many-
body systems of impurities would also be extremely valuable to understand the
properties of the outer layers of neutron stars, such as entrainment effects
caused by the presence of the medium (see e.g. [38]) and modifications to
transport properties of the crust like the thermal conductivity [39, 40] and
the neutrino opacity [41, 42, 43].

## Acknowledgements

We thank J. P. Blaizot for useful discussions. Financial support from the
Italian MIUR under the PRIN2017 project CEnTraL (Protocol Number 20172H2SC4),
from the Provincia Autonoma di Trento and from Q@TN, the joint lab between
University of Trento, FBK- Fondazione Bruno Kessler, INFN- National Institute
for Nuclear Physics and CNR- National Research Council is acknowledged.

##  Appendix A Complex potential

In Sec. II we introduced the matrix of polarization propagators $\Delta_{ab}$.
Here we demonstrate that with a bath in thermal equilibrium all of these
functions are related and we only need one, $\Delta^{R}$, to derive the
complex potential needed for the dynamics.

In position space, $\Delta^{R}$ is defined as

$\Delta^{R}(\mathbf{x}-\mathbf{y})=\Delta^{F}(\mathbf{x}-\mathbf{y})-\Delta^{<}(\mathbf{x}-\mathbf{y}).$
(44)

The procedure to derive Eqs. (18, 19) of the main text is modelled on Ref.
[27]. Complex potential in small frequency approximation is

$\displaystyle i\mathcal{V}(\mathbf{q})=\lim_{\omega\to
0}(\Delta^{R}(\omega,\mathbf{q})+\Delta^{<}(\omega,\mathbf{q}))=$
$\displaystyle=\lim_{\omega\to
0}(\text{Re}\Delta^{R}(\omega,\mathbf{q})+i\text{Im}\Delta^{R}(\omega,\mathbf{q})+\Delta^{<}(\omega,\mathbf{q})).$
(45)

The real part of $\Delta^{R}$ is related to the spectral density $\sigma$.
Indeed, we have that
$2\text{Re}\Delta^{R}(\omega,\mathbf{q})=\sigma(\omega,\mathbf{q})$. In small
frequency approximation therefore

$\displaystyle\text{Re}\Delta^{R}(\omega,\mathbf{q})=A^{R}(\mathbf{q})+\omega
B^{R}(\mathbf{q})+o(\omega^{2}).$ (46)

Spectral density is odd in $\omega$, i.e.
$\sigma(-\omega,\mathbf{q})=-\sigma(\omega,\mathbf{q})$. Therefore $A(q)=0$
and

$\text{Re}\Delta^{R}(\omega,\mathbf{q})=\omega
B^{R}(\mathbf{q})+o(\omega^{2})=\frac{1}{2}\sigma(\omega,\mathbf{q}).$ (47)

Exploiting the fluctuation-dissipation relation (FDR), that is valid for a
bath at equilibrium, we have that

$\Delta^{<}=\frac{2}{e^{\beta\omega}-1}\text{Re}\Delta^{R}(\omega,\mathbf{q}).$
(48)

In this last equality an algebraic relation between $\Delta^{<}$ and
$\Delta^{R}$ is established. Thanks to this, we will be able to write ${\cal
V}$ only in terms of $\Delta^{R}$.

The limit $\omega\to 0$ of Eq. (48) is

$\lim_{\omega\to
0}\Delta^{<}(\omega,\mathbf{q})=\frac{2}{\beta}B^{R}(\mathbf{q}).$ (49)

We now perform the limit in Eq. (45) and we obtain

$\displaystyle\mathcal{V}(\mathbf{q})=\text{Im}\Delta^{R}(\omega=0,\mathbf{q})-i\frac{2}{\beta}B^{R}(\mathbf{q}).$
(50)

In terms of $D^{R}=-i\Delta^{R}$ the real and imaginary part of the complex
potential now are

$\displaystyle V(\mathbf{q})=\text{Re}D^{R}(\omega=0,\mathbf{q}),$ (51)
$\displaystyle W(\mathbf{q})=\frac{2}{\beta}\lim_{\omega\to
0}\frac{\text{Im}D^{R}(\omega,\mathbf{q})}{\omega}.$ (52)

The full expressions of $\text{Re}D^{R}$ and $\text{Im}D^{R}$ in momentum
space are

$\displaystyle\text{Re}D(\omega,\mathbf{q})=-\frac{m}{2\pi^{2}\hbar^{2}}\int
dkf_{FD}(k,T)\frac{k}{2q}\times$
$\displaystyle\times\left(\log\left|\frac{k/k_{F}-\nu_{-}}{k/k_{F}+\nu_{-}}\right|-\log\left|\frac{k/k_{F}-\nu_{+}}{k/k_{F}+\nu_{+}}\right|\right),$
(53)
$\displaystyle\text{Im}D(\omega,\mathbf{q})=-\frac{mk_{F}}{2\pi\hbar^{2}}\left[\frac{\omega}{v_{F}q}+\frac{1}{\beta
v_{F}q}\log\left(\frac{1+e^{\beta(\nu_{-}^{2}\varepsilon_{F}-\mu)}}{1+e^{\beta(\nu_{+}^{2}\varepsilon_{F}-\mu)}}\right)\right],$
(54)

where $\beta=1/k_{B}T$, $\nu_{\pm}=\omega/qv_{F}\pm q/2k_{F}$ and
$v_{f}=k_{F}/m$.

##  Appendix B Zero temperature friction

As shown in Ref. [31], friction can be understood also in terms of energy
dissipated by an impurity moving at velocity $V$, $\dot{E}=-F_{V}V$, with
$F_{V}$ the velocity dependent drag force. Following the convention of [44],
Sec. 7 we have that the energy dissipated per unit time and unit particle when
a contact interaction of strength $g$ is considered is

$\displaystyle\dot{E}$ $\displaystyle=-\int_{-\infty}^{\infty}\frac{d{\bf
k}}{(2\pi)^{3}}\\!\\!\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}2\pi
S(\omega,{\bf k})\frac{n}{2N}\omega 2\pi g^{2}\delta(\omega-k_{z}V)=$
$\displaystyle=-\frac{ng^{2}}{2N}\frac{1}{(2\pi)^{2}}\int_{-\infty}^{\infty}d{\bf
k}\leavevmode\nobreak\ S(k_{z}V,{\bf}k)k_{z}V=-F_{V}V.$ (55)

Now we focus on the drag force $F_{V}$

$\displaystyle F_{V}$
$\displaystyle=\frac{ng^{2}}{8N\pi^{2}}\int_{-\infty}^{\infty}\\!d{\bf
k}S(k_{z}V,{\bf k})k_{z}=$
$\displaystyle=\frac{ng^{2}}{4N\pi}\iint_{-\infty}^{\infty}\\!\\!dk_{\perp}dk_{z}k_{\perp}k_{z}S(k_{z}V,\sqrt{k_{z}^{2}+k_{\perp}^{2}}).$
(56)

In order to perform the integration in (56), we use the expression of the
dynamical structure factor $S(k_{z}V,\sqrt{k_{z}^{2}+k_{\perp}^{2}})$ given in
Ref. [45], Sec. 2. This expression is

$S(\omega,{\bf k})=\frac{\nu(0)}{2}\frac{\omega}{kv_{F}}\quad\text{if}\quad
0\leq\omega\leq kv_{F}-\frac{k^{2}}{2m},$ (57)

that in the small velocity limit gives the conditions $0\leq k_{z}\leq
2mv_{F}$ and $0\leq k_{\perp}\leq\sqrt{(2mv_{F})^{2}-k_{z}^{2}}$. Performing
the integration we obtain

$\frac{\nu(0)V}{2v_{F}}\int_{0}^{2mv_{F}}\\!\\!dk_{z}\leavevmode\nobreak\
k_{z}^{2}\int_{0}^{\sqrt{(2mv_{F})^{2}-k_{z}^{2}}}\\!\\!dk_{\perp}\frac{k_{\perp}}{\sqrt{k_{z}^{2}+k_{\perp}^{2}}}=\\\
=\frac{8}{3}\nu(0)mVk_{F}^{3}.$ (58)

Finally, for $F_{V}$ we obtain

$\displaystyle F_{V}$
$\displaystyle=\frac{ng^{2}}{4N\pi}\frac{8}{3}\nu(0)mVk_{F}^{3}=\frac{3mN}{k_{F}^{2}}\frac{k_{F}^{3}}{6\pi^{2}}\frac{g^{2}}{N\pi}\frac{2}{3}mk_{F}^{3}V=$
$\displaystyle=\frac{m^{2}k_{F}^{4}}{3\pi^{3}}g^{2}V=\frac{4k_{F}^{2}}{3\pi}\left(k_{F}a\frac{m}{m_{r}}\right)^{2}V=\gamma_{T=0}V.$
(59)

In this derivation, we used $\nu(0)=3mN/k_{F}^{2}$, see [45], and
$n=k_{F}^{3}/6\pi^{2}$. Now, comparing (59) with (34) we see that we recovered
the same result for the friction coefficient at $T=0$ (the missing $\hbar$
factor is due to the fact that in this Appendix we set $\hbar=1$).

This connection between the statistical structure factor $S(\omega,{\bf k})$
gives also a useful insight on why $\gamma$ vanishes for a Bose gas or
generally for a phononic spectrum at $T=0$. The dynamical structure factor in
presence of single low energy phonon mode reads $S(\omega,{\bf q})=S_{\bf
k}\delta(\omega-c|\bf k|)$, where $c$ is the speed of sound. Therefore the
drag force vanishes for any impurity speed $V<c$ (obviously in agreement with
the Landau criterion for superfludity).

On the other hand, having the fermions a continuum of particle-hole
excitations at low energy, a moving object will release energy to the bath at
whatever speed $V$ it moves.

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