# Polaritons for testing the universality of an impurity in a Bose-Einstein
condensate

A. Camacho-Guardian  Departamento de Física Química, Instituto de Física,
Universidad Nacional Autónoma de México, Apartado Postal 20-364, Ciudad de
México C.P. 01000, Mexico

###### Abstract

Universality is a fundamental concept in physics that allows for the
description of properties of systems that are independent of microscopic
details. In this work, we show that polaritons in a Bose-Einstein condensate
(BEC) are a suitable platform to probe in a non-demolition way the universal
high-energy spectrum of an impurity strongly coupled to a BEC. Based on a
field theory that includes the two-body correlations at the exact level we
demonstrate that under appropriate conditions, the damping rate of slow-light
propagation reveals the high-energy universal tail in the polaron spectrum.

##  I Introduction

The concept of universality is ubiquitous in physics, referring to properties
of systems that remain unchanged across interaction strengths, temperatures,
and microscopic details of certain classes of phenomena. Ultracold gases have
provided a platform in which to test universal features related to the
properties of strongly correlated Fermi Partridge  _et al._ (2005); Schirotzek
_et al._ (2008); Nascimbène  _et al._ (2010); Horikoshi  _et al._ (2010);
Stewart  _et al._ (2010); Kuhnle  _et al._ (2010); Bauer  _et al._ (2014);
Levinsen  _et al._ (2017a) and Bose gases Wild  _et al._ (2012); Makotyn  _et
al._ (2014); Lopes  _et al._ (2017); Eigen  _et al._ (2017).

The study of a quantum impurity coupled to a bosonic bath is a long-standing
problem that dates back to Landau’s and Pekar’s studies on the behavior of
electrons in a crystal Landau and Pekar (1948). The understanding of the so-
called Bose polaron has been renewed in light of new experiments with
ultracold gases Hu  _et al._ (2016); Jørgensen  _et al._ (2016); Ardila  _et
al._ (2019); Skou  _et al._ (2021a); Yan  _et al._ (2020), which has
encouraged the development of new theoretical and numerical approaches Li and
Sarma (2014); Christensen  _et al._ (2015); Shchadilova  _et al._ (2016);
Levinsen  _et al._ (2017b); Guenther  _et al._ (2018); Rath and Schmidt
(2013); Field  _et al._ (2020); Ardila and Giorgini (2015); Grusdt  _et al._
(2018); Drescher  _et al._ (2020); Massignan  _et al._ (2021); Guenther  _et
al._ (2021); Christianen  _et al._ (2022a, b); Yegovtsev  _et al._ (2022) far
beyond the regimes accessible in condensed matter systems. Despite this
interest, many aspects of the Bose polaron remain elusive today. In this
sense, the universal aspects of the Bose polaron are key features for
understanding properties that are invariant to the microscopic details of the
system.

One of these properties is the universal frequency tail of the spectrum of the
polaron, which is related to Tan’s contact Tan (2008a, b, c); Liu  _et al._
(2020a, b) and provides direct information on the short-range interaction and
initial dynamics of the polarons Braaten  _et al._ (2010); Fletcher  _et al._
(2017); Yoshida  _et al._ (2018); Skou  _et al._ (2021b, a, 2022).

Polaritons, hybrid quasi-particles of light and matter, have already
demonstrated their power in probing, measuring, and detecting strongly
correlated phases in solid-state systems Efimkin  _et al._ (2021); Ravets  _et
al._ (2018); Cotlet  _et al._ (2020); Imamoglu  _et al._ (2021); Cotleţ  _et
al._ (2019); Rana  _et al._ (2020); Pimenov  _et al._ (2017); Shahnazaryan
_et al._ (2020); Levinsen  _et al._ (2019); Navadeh-Toupchi  _et al._ (2019);
Camacho-Guardian  _et al._ (2021); Vashisht  _et al._ (2022). In the context
of atomic gases, dark-state polaritons have been studied in relation to
polaron-polaritons, leading to the prediction of intriguing phenomena such as
mass tuneable polaritons Grusdt and Fleischhauer (2016), lossless polariton
propagation above Landau’s critical velocity Nielsen  _et al._ (2020), and
strongly interacting polaritons Camacho-Guardian  _et al._ (2020). However,
compared to their counterparts in condensed matter systems, testing many-body
phases of matter with polaritons in ultracold gases remains a pending issue.
In particular, since polaron experiments typically rely on demolition
measurement protocols such as three-body losses or time-of-flight, polaritons
offer an alternative route to probe strongly correlated states of quantum
matter in a non-demolition way. Furthermore, single photon sources unfold an
alternative to probe polaron physics in the single impurity limit.

Figure 1: Diagram of the spectral response of an impurity coupled to a Bose-
Einstein condensate as a function of the inverse of the impurity-boson
scattering length and the frequency $\omega$. Universal high-energy tail
spectrum that scales with $\omega^{-5/2}$ (blue region) and weak-coupling
universal regime $a^{2}\omega^{-3/2}$ pink region. Many-body correlations form
within the green area. Dashed line are guides to the crossover to the
different scaling regimes which are smooth transitions given by: blue to pink:
$\omega=1/ma^{2}$, pink to green: $\omega=4\pi|a|n_{0}/m$ and green to blue:
$\omega=(4\pi n_{0}/m^{3})^{2/3}.$

In this manuscript, we propose a non-demolition probing scheme using dark-
state polaritons to measure the universal tail spectrum of the Bose polaron in
ultracold gases. A field-theoretical approach, which incorporates two-body
correlations of impurity-boson scattering, demonstrates that the damping rate
of the polariton contains information about the universal tail. This study
highlights the potential of polaritons as a tool to investigate universal
properties of strongly correlated systems in both atomic gases and condensed
matter.

##  II  Universal tail of the polaron

Before we enter into the study of slow-light and polaritons to test the
universal aspects of an impurity let us first briefly revise the problem of
the Bose polaron and its universal aspects.

Consider a system formed by bosonic atoms

$\displaystyle\hat{H}_{b}=\sum_{\mathbf{k}}\epsilon_{\mathbf{k}}\hat{b}_{\mathbf{k}}^{\dagger}\hat{b}_{\mathbf{k}}+\frac{g_{bb}}{2V}\sum_{\mathbf{k},\mathbf{k}^{\prime},\mathbf{q}}\hat{b}^{\dagger}_{\mathbf{k}+\mathbf{q}}\hat{b}^{\dagger}_{\mathbf{k}^{\prime}-\mathbf{q}}\hat{b}_{\mathbf{k}^{\prime}}\hat{b}_{\mathbf{k}},$
(1)

where $\hat{b}_{\mathbf{k}}^{\dagger}$ creates a boson with momentum
$\mathbf{k}$ and energy $\epsilon_{\mathbf{k}}=k^{2}/2m$ where $m$ is the mass
of the bosons and $g_{bb}$ the boson-boson interaction strength. We assume
that the bosons form a weakly Bose-Einstein condensate (BEC) correctly
described by the Bogoliubov theory with a density $n_{0}.$ The energy and
momentum scale associated to the BEC are $E_{n}=k_{n}^{2}/2m$ and
$k_{n}=(6\pi^{2}n_{0})^{1/3},$ respectively.

A single impurity interacts with the BEC, described by the term

$\displaystyle\hat{H}_{I}=\frac{g}{V}\sum_{\mathbf{k},\mathbf{k}^{\prime},\mathbf{q}}\hat{b}^{\dagger}_{\mathbf{k}+\mathbf{q}}\hat{c}^{\dagger}_{\mathbf{k}^{\prime}-\mathbf{q}}\hat{c}_{\mathbf{k}^{\prime}}\hat{b}_{\mathbf{k}},$
(2)

where $\hat{c}_{\mathbf{k}}^{\dagger}$ creates an impurity atom with momentum
$\mathbf{k}$. The strength of the interaction $g=4\pi a/m$ in terms of the
boson-impurity scattering length that can be tuned on demand by means of a
Feshbach resonance. The volume of the system is denoted by $V$ whereas the
kinetic energy of the c-impurities is described by the standard term
$\hat{H}_{c}=\sum_{\mathbf{k}}\epsilon_{\mathbf{k}}^{(c)}\hat{c}^{\dagger}_{\mathbf{k}}\hat{c}_{\mathbf{k}}$
with $\epsilon_{\mathbf{k}}^{(c)}=k^{2}/2m+\epsilon_{c}$ being $\epsilon_{c}$
the energy off-set of the $c$-state with respect to the $b$-atoms.

The study of the many-body properties of the Bose polaron remains an
outstanding problem that is far beyond the scope of this manuscript. Instead,
in this article, we focus on the universal aspects of the Bose polaron, which
appear regardless of the theoretical approach employed. We write the spectral
function of the impurity as
$A(0,\omega)=A_{P}(0,\omega)+A_{\text{tail}}(\omega)$, where
$A_{\text{tail}}(\omega)$ represents the universal tail of the polaron
spectrum and $A_{P}(0,\omega)$ relates to the many-body correlations that may
form in the low-energy sector, which is not our current focus. The tail of the
spectrum is given by Braaten  _et al._ (2010)

$\displaystyle A_{\text{tail}}(0,\omega)=\frac{8\pi
n_{0}}{m^{3/2}\omega^{5/2}}\frac{m\omega a^{2}}{1+m\omega a^{2}}$ (3)

and holds for large $\omega.$ In Fig. (1), we illustrate the different regimes
of the polaron spectrum. It consists of a universal regime marked by the blue
area that scales as $\omega^{-5/2}$. For weak interactions, a weakly
interacting high-energy tail appears, marked by the pink area. The third
regime, illustrated by the green area, corresponds to energies at which the
polaron and many-body correlations form. Although there is still debate on the
properties of the green region - that is, how the spectral function looks when
many-body correlations arise- the tail of the polaron spectrum in Eq. (3) is
an exact result that stems from the analytical expression for the two-body
scattering

$\displaystyle
A_{\text{tail}}(0,\omega)=n_{0}\frac{A^{(0)}_{T}(\omega)}{\omega^{2}},$ (4)

this equation remarks that the tail of the polaron spectrum only relates to
the two-body scattering properties. Where
$A^{(0)}_{T}(\omega)=-2\text{Im}\mathcal{T}_{v}(0,\omega)$ is the spectral
function of the the two-body scattering matrix, in turn given by
$\mathcal{T}_{v}(0,\omega)=\frac{4\pi a}{m}/(1+ia\sqrt{m\omega}).$

##  III  Polaritons

We now turn our attention to the description of slow-light in the presence of
atomic interactions. We consider atoms with an internal three-level structure
as illustrated in Fig. (2) (inset) in a $\Lambda$ scheme. Atoms form a Bose-
Einstein condensate in the $|b\rangle$ atomic state, an excited state
$|e\rangle$ couples to a photon field $|a\rangle$ whereas a classical field
couples the excited state $|e\rangle$ and a third state $|c\rangle.$

The coupling of the atoms in the BEC to the photons and excited state
$|e\rangle$ is described by the standard light-matter Hamiltonian

$\displaystyle\hat{H}_{l-m}=\sqrt{n_{0}}g_{p}\sum_{\mathbf{k}}\left(\hat{e}^{\dagger}_{\mathbf{k}}\hat{a}_{\mathbf{k}}+\hat{a}^{\dagger}_{\mathbf{k}}\hat{e}_{\mathbf{k}}\right),$
(5)

where the single photon coupling is denoted by $g_{p}.$ The ideal dispersion
of the photons and the excited state are given by
$\hat{H}_{a}=\sum_{\mathbf{k}}\omega_{\mathbf{k}}\hat{a}^{\dagger}_{\mathbf{k}}\hat{a}_{\mathbf{k}}$
and
$\hat{H}_{e}=\sum_{\mathbf{k}}\epsilon_{\mathbf{k}}^{(e)}\hat{e}^{\dagger}_{\mathbf{k}}\hat{e}_{\mathbf{k}}$
respectively. As usual, the photons have a linear dispersion $\omega_{k}=ck$
with $c$ the speed of light, while the dispersion of the excited state
$\epsilon_{\mathbf{k}}^{(e)}=k^{2}/2m+\epsilon_{e}-i\gamma_{e},$ where
$\epsilon_{e}$ is the energy of the excited state relative to the $b$-state
and $\gamma_{e}$ its damping rate. The $\Lambda$-scheme is completed by a
classical control field that couples the excited and metastable state and is
given by the Hamiltonian

$\displaystyle\hat{H}_{cl}=\Omega\sum_{\mathbf{k}}(\hat{c}^{\dagger}_{\mathbf{k}-\mathbf{k}_{\text{cl}}}\hat{e}_{\mathbf{k}}e^{i\omega_{\text{cl}t}}+\hat{e}^{\dagger}_{\mathbf{k}}\hat{c}_{\mathbf{k}-\mathbf{k}_{\text{cl}}}e^{-i\omega_{\text{cl}}t}),$
(6)

with $\omega_{\text{cl}}$ and $\mathbf{k}_{\text{cl}}$the energy-momentum of a
classical field and $\Omega$ the Rabi frequency.

To treat the light-matter coupling together with the atomic interactions we
introduce the imaginary Green’s function

$G_{\alpha\beta}(\mathbf{k},\tau)=-\langle
T_{\tau}\\{\psi_{\alpha\mathbf{k}}(\tau)\psi^{\dagger}_{\beta\mathbf{k}}(0)\\}\rangle,$

where the indices $\alpha,\beta=\\{c,e,a\\}$ for the metastable, excited, and
photon field respectively, here $\tau$ is the imaginary time and $T_{\tau}$
the time-ordering operator. The Dyson’s equation in energy-momentum space
governing the photon field is

$\displaystyle
G^{-1}_{aa}(\mathbf{k},\omega)=G^{(0)}_{aa}(\mathbf{k},\omega)^{-1}-\Sigma_{aa}(\mathbf{k},\omega)$
(7)

with the bare photonic Green’s function
$G^{(0)}_{aa}(\mathbf{k},\omega)^{-1}=\omega-c|\mathbf{k}|$. First, the
diagonal self-energy of the photons is given by

$\displaystyle\Sigma_{aa}(\mathbf{k},\omega)=n_{0}g_{p}^{2}G_{ee}(\mathbf{k},\omega),$
(8)

The Green’s function of the excited state $|e\rangle$ is

$\displaystyle
G^{-1}_{ee}(\mathbf{k},\omega)=G^{(0)}_{ee}(\mathbf{k},\omega)^{-1}-\Omega^{2}G_{cc}(\mathbf{k}-\mathbf{k}_{cl},\omega),$
(9)

where
$G^{(0)}_{ee}(\mathbf{k},\omega)^{-1}=\omega-\epsilon_{\mathbf{k}}^{(e)},$
includes the induced damping and energy shifty due to the coupling to the $b$
state as described by the Weisskopf-Wigner theory Weisskopf and Wigner (1997).
We consider a weak probe photon with momentum $\mathbf{p}_{r}$ we introduce
the one- photon detuning
$\Delta=c|\mathbf{p}_{r}|-\text{Re}\epsilon_{\mathbf{p}_{r}}^{(e)},$ and
define the two-photon detuning as
$\delta=\epsilon_{c}+\omega_{\text{cl}}-cp_{r}.$

In the absence of atomic interactions, we can use the bare propagator of the
$c$-state in Eq. (9):
$G^{-1}_{cc}(\mathbf{k},\omega)=\omega-\epsilon_{\mathbf{k}}^{(c)}.$ In the
rotating frame
$\hat{c}^{\dagger}_{\mathbf{k}}\rightarrow\hat{c}^{\dagger}_{\mathbf{k}}e^{i\omega_{\text{cl}}t}$
, the energy of the $c$ state is given by
$\epsilon_{\mathbf{k}}^{(c)}=k^{2}/2m+\delta+cp_{r}.$ In this case, when the
two-photon detuning vanishes, light propagates in the form of a dark-state
polariton Fleischhauer and Lukin (2000, 2002); Hau  _et al._ (1999), which has
a dramatically reduced group velocity of
$v_{g}/c=1/(1+n_{0}g_{p}^{2}/\Omega^{2})$ and propagates without losses in an
opaque medium, a phenomenon also coined electromagnetically induced
transparency (EIT) Boller and Imamoğlu (1991); Fleischhauer  _et al._ (2005).

We now include the boson-impurity interactions. Although the predicted state
of the polaron may depend on the theoretical framework used, in this study, we
focus only on the high-energy spectrum of the polaron. Therefore, we account
for a formalism that includes this high-energy tail exactly using the so-
called ladder approximation Rath and Schmidt (2013). Within this
approximation, the self-energy of the $c$-Green’s function is given by

$\displaystyle\Sigma_{cc}(\mathbf{k},\omega)=n_{0}\mathcal{T}(\mathbf{k},\omega).$
(10)

Here, the scattering matrix $\mathcal{T}$ describes the interaction between a
single boson in the $c$ state with a boson of the BEC, that is,

$\displaystyle\mathcal{T}(\mathbf{k},\omega)=\frac{1}{\frac{1}{g}-\Pi(\mathbf{k},\omega)},$
(11)

with

$\displaystyle\Pi(\mathbf{k},\omega)=-\sum_{\mathbf{p},i\omega_{\nu}}G_{\text{BEC}}(\mathbf{p},i\omega_{\nu})G^{(0)}_{cc}(\mathbf{k}-\mathbf{p},\omega-i\omega_{\nu}).$
(12)

Here $G_{\text{BEC}}^{-1}(\mathbf{k},\omega)=\omega-k^{2}/2m$ corresponds to
the Green’s function of the BEC which for simplicity we assume to be ideal,
and $i\omega_{\nu}$ is a bosonic Matsubara frequency. On the other hand,
$G_{cc}(\mathbf{k},\omega)$ corresponds to the $c$ propagator, which is now,
dressed by the light-matter coupling

$\displaystyle[G^{(0)}_{cc}(\mathbf{k}-\mathbf{k}_{\text{cl}},\omega)]^{-1}=\omega-\epsilon^{(c)}_{\mathbf{k}-\mathbf{k}_{\text{cl}}}-\frac{\Omega^{2}}{\omega-\epsilon^{(e)}_{\mathbf{k}}-\frac{n_{0}g_{p}^{2}}{\omega-
ck}}.$ (13)

The scattering matrix depends therefore on the coupling between the different
states to light, and as we shall reveal, the properties of the
$\mathcal{T}$-matrix reflect on the propagation of the dark-state polaritons.

##  IV Probing polaron universality with polaritons

Before delving into technical and numerical details, let us first discuss some
limits that will guide the physical interpretation of our results.

Under perfect EIT conditions, light can propagate without losses in an
otherwise opaque BEC. In this study, we use the losses induced by atomic
interactions on the dark-state polariton to probe the universal tail of the
polaron spectrum. We begin by defining the damping rate of the polariton,
which will serve as a starting point for our analysis

$\displaystyle\Gamma_{a}=-Z_{a}\text{Im}\Sigma_{aa}(\mathbf{p}_{r},cp_{r}),$
(14)

where $Z_{a}$ is the residue of the polariton
$Z_{a}=1/(1+\frac{n_{0}g_{p}^{2}}{\Omega^{2}}).$

The main result of our work is summarised in the following equation

$\displaystyle\Gamma_{a}\approx
A_{\text{tail}}\left(0,\omega_{l}\right)\omega_{l}^{2},$ (15)

where $\omega_{l}=-\Omega^{2}/\Delta$ with $\Delta<0$. This expression relates
the losses of the polariton to the universal tail of the polaron spectrum. It
provides a non-demolition probing scheme, allowing for the measurement of the
high-energy spectrum of a strongly interacting impurity coupled to a BEC via
polaritons. That is, by tuning the light-matter parameters $\Omega$ and
$\Delta$, one can measure the tail of the polaron spectrum as in Eq. (15).
Equation (15) links the tail of the two-body scattering to the damping rate of
the polariton as it can be re-written as $\Gamma_{a}\approx
n_{0}A^{(0)}_{T}\left(0,\omega_{l}\right).$

Figure 2: The damping rate of the dark-state polaritons is plotted as a
function of $\Omega/\gamma_{e}$ for several values of the inverse of the
scattering length $1/k_{n}a$. The red lines represent unitarity
($1/k_{n}a=0$), with the solid curve giving the exact numerical result and the
dashed line corresponding to Eq. (15). The black lines correspond to
$1/k_{n}a=-1$, with the solid line illustrating the full numerical solution
and the dashed curve showing the asymptotic behavior. The blue lines represent
weak interactions ($1/k_{n}a=-2$). Inset: $\Lambda$ scheme of the atoms as
detailed in the main text.

To understand the physical origin and limitations of Eq. (15), let us look at
the photon Green’s function in Eq. (7) evaluated for an incoming photon with
energy-momentum $(\mathbf{p}_{r},cp_{r}),$

$\displaystyle
G^{-1}_{aa}(\mathbf{p}_{r},cp_{r})=-\frac{n_{0}g_{p}^{2}}{\Delta+i\gamma_{e}+\frac{\Omega^{2}}{n_{0}\mathcal{T}(\mathbf{0},cp_{r})}}.$
(16)

If the relevant polaron energies in the term $n_{0}\mathcal{T}(0,cp_{r})$
remain smaller than the width of the EIT:
$\frac{\Omega^{2}}{\sqrt{\gamma_{e}^{2}+\Delta^{2}}n_{0}\mathcal{T}(\mathbf{0},cp_{r})}\gg
1,$ then we can estimate the imaginary part of the dressed photon propagator
as
$\text{Im}G^{-1}_{aa}(\mathbf{p}_{r},cp_{r})\approx\frac{n_{0}g_{p}^{2}}{\Omega^{2}}n_{0}\text{Im}\mathcal{T}(0,cp_{r})\approx\frac{1}{Z_{a}}n_{0}A^{(0)}_{T}\left(0,\omega_{l}\right).$
This condition shows that only under a certain regime of the light-matter
parameters one can link the propagation of the dark-state polariton with the
tail of the impurity-boson scattering, as we will explicitly show.

To test our theoretical analysis, we now obtain the photon propagator
numerically. For this purpose we take $\gamma_{e}=10$MHz, a one-photon
detuning of $\Delta=-10\gamma_{e}$, $k_{\text{cl}}=0.1k_{n},$ a BEC with
$E_{n}=500\text{kHz},$ and $\sqrt{n_{0}}g_{p}/\gamma_{e}=30,500$ Lampis  _et
al._ (2016). We probe the zero-momentum polaron, that is
$\mathbf{k}_{\text{cl}}=\mathbf{p}_{r}$ and take a zero two-photon detuning.
To interpret our results, we note that a photon propagating in a BEC with
$\Omega=0$ spends an amount $t^{(0)}_{a}=L/c$ of time in the condensate of
length $L$. Thus, it is convenient to introduce the dimensionless quantity
termed optical depth $\text{OD}_{0}=\Gamma^{(0)}_{a}t^{(0)}$ where
$\Gamma_{a}^{(0)}=n_{0}g_{p}^{2}/\gamma_{e},$ is the damping rate of the
photon in the two-level medium ($\Omega=0).$ Comparing the same quantity for
slow-light $\text{OD}=\Gamma_{a}t_{a}$ and $t_{a}=L/v_{g}$ we obtain that the
ratio $\text{OD}/\text{OD}_{0}=\Gamma_{a}/\sigma_{e}$ can be written in terms
of $\sigma_{e}=\Omega^{2}/\gamma_{e},$ which is the normalization we will use.
Under strict EIT conditions $\Gamma_{a}=0$.

Figure (2) shows the damping rate of the dark-state polariton as a function of
$\Omega$ for different values of the boson-impurity scattering length. The
solid lines are the result of the full numerical solutions, while the dashed
line corresponds to the asymptotic tail in Eq.(15). Overall, we see that the
polariton damping agrees well with the high-energy tail of Eq.(15).

For strong interactions, in particular, at unitarity $1/k_{n}a$ (red lines),
we observe that the polariton damping joins the universal tail at
$\Omega/\gamma_{e}\approx 1$ and seems to agree very well with the tail for
large $\Omega$. On the other hand, as $\Omega$ is decreased, we obtain
significant deviations from Eq. (15) until the comparison no longer makes
sense. If at unitarity the typical polaron energies are of the order of
$E_{n}$, for $\Omega/\gamma_{e}=1.0$, we obtain that for the parameters taken,
$\Omega^{2}/(\sqrt{\Delta^{2}+\gamma_{e}^{2}}E_{n})=2$. Thus, in accordance
with our analysis, we expect to start agreeing with the universal tail close
to this value and higher values of $\Omega$.

At intermediate coupling strengths ($1/k_{n}a=-1$), we also observe agreement
between the high-energy tail and the numerical results. However, the agreement
shifts to larger values of $\Omega$. Physically, this can be understood from
Fig. (1): as the interaction strength decreases, the universal tail shifts to
even higher energies. Thus, in our case, we are required to tune $\Omega$ to
higher values.

Finally, for weak interactions ($1/k_{n}a=-2$), we find that as the universal
tail is pushed further in energies, the agreement between the numerics and Eq.
(15) occurs only at large values of $\Omega$, consistent with our previous
discussion.

Figure 3:  Universail tail from the damping rate of polaritons. In red we show
the unitary regime $1/k_{n}a=0$, in black we have $1/k_{n}a=-0.2$ while the
blue line gives $1/k_{n}a=-1$. Solid curves give the numerical results, the
orange line gives the universal limit. The remaining parameters as in Fig.
(2).

Our analysis indicates that the damping rate of the dark-state polariton
scales with the inverse of the Rabi frequency, $\Gamma_{a}\approx
n_{0}/\Omega$. However, it is difficult to directly confirm this hypothesis
from Fig. (2). In order to demonstrate this relationship, we present
$\Gamma_{a}\Omega$ on a logarithmic scale in Fig. (3). We observe that for
strong interactions, the value of $\Gamma_{a}\Omega$ approaches the unitary
value of $\Gamma_{a}\Omega\approx\frac{n_{0}}{m^{3/2}}\sqrt{\Delta}$ (dashed
orange line). At unitarity (solid red line), we observe an evident change of
slope close to $\Omega/\gamma_{e}=1$, when
$\Omega^{2}/(\sqrt{\Delta^{2}+\gamma_{e}^{2}}E_{n})=1$. As the interaction is
decreased, we find that the high-energy tail is reached at larger values of
$\Omega$, as discussed previously. It is important to note that the values of
$\Delta$ and $\gamma_{e}$ are crucial and according to our analysis, if
$\Delta$ is tuned to be too small, the excited state may imprint additional
losses on the polariton, leading to deviations from the universal tail.

We have demonstrated that the damping rate of the polariton encodes the
universality of the polaron and can thus be used as a non-demolition probing
scheme. Unlike polaron experiments with highly-imbalanced population mixtures,
where the signal of the impurity typically scales linearly with the density of
the impurities, polaritons can be realized with single-photon sources, and
thus genuinely test single-impurity physics.

##  V  Conclusions

Ultracold gases stand as a powerful platform to study how universality
reflects in physical observables. The high-energy physics of an impurity
coupled to a degenerate quantum gas is perhaps the most simple yet
experimentally and theoretically challenging example with many facets to be
unveiled. In this work we demonstrated that the propagation of light in form
of slow-light can be used as a non-demolition scheme to probe the universality
of the Bose polaron in the high-energy spectrum.

Our results illustrate that under correct conditions the damping rate of the
polariton is determined by the high-energy polaron spectrum. In particular, we
show that for atoms in a $\Lambda$ scheme level configuration, the damping of
the polariton scales as $\Gamma_{a}\Omega\approx n_{0}\sqrt{\Delta}/m^{3/2}.$

Our study reflects a rich interplay between well-established polariton physics
and high-energy polaron physics. The vast playground of these two phenomena
may serve as a benchmark for future experiments and theories to explore
polariton and polaron physics beyond the linear regime Camacho-Guardian  _et
al._ (2018); Camacho-Guardian and Bruun (2018); Will  _et al._ (2021);
Camacho-Guardian  _et al._ (2022).

Acknowledgments.- A. C. G. thanks valuable discussions with G. M. Bruun which
motivated this work and comments to the manuscript. A.C.G. acknowledges
financial support from UNAM DGAPA PAPIIT Grants No. IA101923 and UNAM DGAPA
PAPIME Grants No. PE101223.

## References

  * Partridge  _et al._ (2005) G. B. Partridge, K. E. Strecker, R. I. Kamar, M. W. Jack, and R. G. Hulet, “Molecular probe of pairing in the bec-bcs crossover,” Phys. Rev. Lett. 95, 020404 (2005).
  * Schirotzek  _et al._ (2008) André Schirotzek, Yong-il Shin, Christian H. Schunck, and Wolfgang Ketterle, “Determination of the superfluid gap in atomic fermi gases by quasiparticle spectroscopy,” Phys. Rev. Lett. 101, 140403 (2008).
  * Nascimbène  _et al._ (2010) S. Nascimbène, N. Navon, K. J. Jiang, F. Chevy, and C. Salomon, “Exploring the thermodynamics of a universal fermi gas,” Nature 463, 1057–1060 (2010).
  * Horikoshi  _et al._ (2010) Munekazu Horikoshi, Shuta Nakajima, Masahito Ueda, and Takashi Mukaiyama, “Measurement of universal thermodynamic functions for a unitary fermi gas,” Science 327, 442–445 (2010), https://www.science.org/doi/pdf/10.1126/science.1183012 .
  * Stewart  _et al._ (2010) J. T. Stewart, J. P. Gaebler, T. E. Drake, and D. S. Jin, “Verification of universal relations in a strongly interacting fermi gas,” Phys. Rev. Lett. 104, 235301 (2010).
  * Kuhnle  _et al._ (2010) E. D. Kuhnle, H. Hu, X.-J. Liu, P. Dyke, M. Mark, P. D. Drummond, P. Hannaford, and C. J. Vale, “Universal behavior of pair correlations in a strongly interacting fermi gas,” Phys. Rev. Lett. 105, 070402 (2010).
  * Bauer  _et al._ (2014) Marianne Bauer, Meera M. Parish, and Tilman Enss, “Universal equation of state and pseudogap in the two-dimensional fermi gas,” Phys. Rev. Lett. 112, 135302 (2014).
  * Levinsen  _et al._ (2017a) Jesper Levinsen, Pietro Massignan, Shimpei Endo, and Meera M Parish, “Universality of the unitary fermi gas: a few-body perspective,” Journal of Physics B: Atomic, Molecular and Optical Physics 50, 072001 (2017a).
  * Wild  _et al._ (2012) R. J. Wild, P. Makotyn, J. M. Pino, E. A. Cornell, and D. S. Jin, “Measurements of tan’s contact in an atomic bose-einstein condensate,” Phys. Rev. Lett. 108, 145305 (2012).
  * Makotyn  _et al._ (2014) P. Makotyn, C. E. Klauss, D. L. Goldberger, E. A. Cornell, and D. S. Jin, “Universal dynamics of a degenerate unitary bose gas,” Nature Physics 10, 116–119 (2014).
  * Lopes  _et al._ (2017) Raphael Lopes, Christoph Eigen, Adam Barker, Konrad G. H. Viebahn, Martin Robert-de Saint-Vincent, Nir Navon, Zoran Hadzibabic, and Robert P. Smith, “Quasiparticle energy in a strongly interacting homogeneous bose-einstein condensate,” Phys. Rev. Lett. 118, 210401 (2017).
  * Eigen  _et al._ (2017) Christoph Eigen, Jake A. P. Glidden, Raphael Lopes, Nir Navon, Zoran Hadzibabic, and Robert P. Smith, “Universal scaling laws in the dynamics of a homogeneous unitary bose gas,” Phys. Rev. Lett. 119, 250404 (2017).
  * Landau and Pekar (1948) LD Landau and SI Pekar, “Effective mass of a polaron,” Zh. Eksp. Teor. Fiz 18, 419–423 (1948).
  * Hu  _et al._ (2016) Ming-Guang Hu, Michael J Van de Graaff, Dhruv Kedar, John P Corson, Eric A Cornell, and Deborah S Jin, “Bose polarons in the strongly interacting regime,” Physical review letters 117, 055301 (2016).
  * Jørgensen  _et al._ (2016) Nils B Jørgensen, Lars Wacker, Kristoffer T Skalmstang, Meera M Parish, Jesper Levinsen, Rasmus S Christensen, Georg M Bruun, and Jan J Arlt, “Observation of attractive and repulsive polarons in a bose-einstein condensate,” Physical review letters 117, 055302 (2016).
  * Ardila  _et al._ (2019) LA Peña Ardila, NB Jørgensen, T Pohl, S Giorgini, GM Bruun, and JJ Arlt, “Analyzing a bose polaron across resonant interactions,” Physical Review A 99, 063607 (2019).
  * Skou  _et al._ (2021a) Magnus G Skou, Thomas G Skov, Nils B Jørgensen, Kristian K Nielsen, Arturo Camacho-Guardian, Thomas Pohl, Georg M Bruun, and Jan J Arlt, “Non-equilibrium quantum dynamics and formation of the bose polaron,” Nature Physics 17, 731–735 (2021a).
  * Yan  _et al._ (2020) Zoe Z Yan, Yiqi Ni, Carsten Robens, and Martin W Zwierlein, “Bose polarons near quantum criticality,” Science 368, 190–194 (2020).
  * Li and Sarma (2014) Weiran Li and S Das Sarma, “Variational study of polarons in bose-einstein condensates,” Physical Review A 90, 013618 (2014).
  * Christensen  _et al._ (2015) Rasmus Søgaard Christensen, Jesper Levinsen, and Georg M. Bruun, “Quasiparticle properties of a mobile impurity in a bose-einstein condensate,” Phys. Rev. Lett. 115, 160401 (2015).
  * Shchadilova  _et al._ (2016) Yulia E. Shchadilova, Richard Schmidt, Fabian Grusdt, and Eugene Demler, “Quantum dynamics of ultracold bose polarons,” Phys. Rev. Lett. 117, 113002 (2016).
  * Levinsen  _et al._ (2017b) Jesper Levinsen, Meera M. Parish, Rasmus S. Christensen, Jan J. Arlt, and Georg M. Bruun, “Finite-temperature behavior of the bose polaron,” Phys. Rev. A 96, 063622 (2017b).
  * Guenther  _et al._ (2018) Nils-Eric Guenther, Pietro Massignan, Maciej Lewenstein, and Georg M Bruun, “Bose polarons at finite temperature and strong coupling,” Physical review letters 120, 050405 (2018).
  * Rath and Schmidt (2013) Steffen Patrick Rath and Richard Schmidt, “Field-theoretical study of the bose polaron,” Physical Review A 88, 053632 (2013).
  * Field  _et al._ (2020) Bernard Field, Jesper Levinsen, and Meera M Parish, “Fate of the bose polaron at finite temperature,” Physical Review A 101, 013623 (2020).
  * Ardila and Giorgini (2015) LA Peña Ardila and S Giorgini, “Impurity in a bose-einstein condensate: Study of the attractive and repulsive branch using quantum monte carlo methods,” Physical Review A 92, 033612 (2015).
  * Grusdt  _et al._ (2018) Fabian Grusdt, Kushal Seetharam, Yulia Shchadilova, and Eugene Demler, “Strong-coupling bose polarons out of equilibrium: Dynamical renormalization-group approach,” Physical Review A 97, 033612 (2018).
  * Drescher  _et al._ (2020) Moritz Drescher, Manfred Salmhofer, and Tilman Enss, “Theory of a resonantly interacting impurity in a bose-einstein condensate,” Phys. Rev. Research 2, 032011 (2020).
  * Massignan  _et al._ (2021) Pietro Massignan, Nikolay Yegovtsev, and Victor Gurarie, “Universal aspects of a strongly interacting impurity in a dilute bose condensate,” Phys. Rev. Lett. 126, 123403 (2021).
  * Guenther  _et al._ (2021) Nils-Eric Guenther, Richard Schmidt, Georg M. Bruun, Victor Gurarie, and Pietro Massignan, “Mobile impurity in a bose-einstein condensate and the orthogonality catastrophe,” Phys. Rev. A 103, 013317 (2021).
  * Christianen  _et al._ (2022a) Arthur Christianen, J. Ignacio Cirac, and Richard Schmidt, “Chemistry of a light impurity in a bose-einstein condensate,” Phys. Rev. Lett. 128, 183401 (2022a).
  * Christianen  _et al._ (2022b) Arthur Christianen, J. Ignacio Cirac, and Richard Schmidt, “Bose polaron and the efimov effect: A gaussian-state approach,” Phys. Rev. A 105, 053302 (2022b).
  * Yegovtsev  _et al._ (2022) Nikolay Yegovtsev, Pietro Massignan, and Victor Gurarie, “Strongly interacting impurities in a dilute bose condensate,” Phys. Rev. A 106, 033305 (2022).
  * Tan (2008a) Shina Tan, “Energetics of a strongly correlated fermi gas,” Annals of Physics 323, 2952–2970 (2008a).
  * Tan (2008b) Shina Tan, “Large momentum part of a strongly correlated fermi gas,” Annals of Physics 323, 2971–2986 (2008b).
  * Tan (2008c) Shina Tan, “Generalized virial theorem and pressure relation for a strongly correlated fermi gas,” Annals of Physics 323, 2987–2990 (2008c).
  * Liu  _et al._ (2020a) Weizhe Edward Liu, Zhe-Yu Shi, Jesper Levinsen, and Meera M. Parish, “Radio-frequency response and contact of impurities in a quantum gas,” Phys. Rev. Lett. 125, 065301 (2020a).
  * Liu  _et al._ (2020b) Weizhe Edward Liu, Zhe-Yu Shi, Meera M. Parish, and Jesper Levinsen, “Theory of radio-frequency spectroscopy of impurities in quantum gases,” Phys. Rev. A 102, 023304 (2020b).
  * Braaten  _et al._ (2010) Eric Braaten, Daekyoung Kang, and Lucas Platter, “Short-time operator product expansion for rf spectroscopy of a strongly interacting fermi gas,” Phys. Rev. Lett. 104, 223004 (2010).
  * Fletcher  _et al._ (2017) Richard J. Fletcher, Raphael Lopes, Jay Man, Nir Navon, Robert P. Smith, Martin W. Zwierlein, and Zoran Hadzibabic, “Two- and three-body contacts in the unitary bose gas,” Science 355, 377–380 (2017), https://www.science.org/doi/pdf/10.1126/science.aai8195 .
  * Yoshida  _et al._ (2018) Shuhei M. Yoshida, Shimpei Endo, Jesper Levinsen, and Meera M. Parish, “Universality of an impurity in a bose-einstein condensate,” Phys. Rev. X 8, 011024 (2018).
  * Skou  _et al._ (2021b) Magnus G. Skou, Thomas G. Skov, Nils B. Jørgensen, and Jan J. Arlt, “Initial dynamics of quantum impurities in a bose–einstein condensate,” Atoms 9 (2021b), 10.3390/atoms9020022.
  * Skou  _et al._ (2022) Magnus G. Skou, Kristian K. Nielsen, Thomas G. Skov, Andreas M. Morgen, Nils B. Jørgensen, Arturo Camacho-Guardian, Thomas Pohl, Georg M. Bruun, and Jan J. Arlt, “Life and death of the bose polaron,” Phys. Rev. Res. 4, 043093 (2022).
  * Efimkin  _et al._ (2021) Dmitry K Efimkin, Emma K Laird, Jesper Levinsen, Meera M Parish, and Allan H MacDonald, “Electron-exciton interactions in the exciton-polaron problem,” Physical Review B 103, 075417 (2021).
  * Ravets  _et al._ (2018) Sylvain Ravets, Patrick Knüppel, Stefan Faelt, Ovidiu Cotlet, Martin Kroner, Werner Wegscheider, and Atac Imamoglu, “Polaron polaritons in the integer and fractional quantum hall regimes,” Physical review letters 120, 057401 (2018).
  * Cotlet  _et al._ (2020) Ovidiu Cotlet, Dominik S Wild, Mikhail D Lukin, and Atac Imamoglu, “Rotons in optical excitation spectra of monolayer semiconductors,” Physical Review B 101, 205409 (2020).
  * Imamoglu  _et al._ (2021) Atac Imamoglu, Ovidiu Cotlet, and Richard Schmidt, “Exciton–polarons in two-dimensional semiconductors and the tavis–cummings model,” Comptes Rendus. Physique 22, 1–8 (2021).
  * Cotleţ  _et al._ (2019) Ovidiu Cotleţ, Falko Pientka, Richard Schmidt, Gergely Zarand, Eugene Demler, and Atac Imamoglu, “Transport of neutral optical excitations using electric fields,” Physical Review X 9, 041019 (2019).
  * Rana  _et al._ (2020) Farhan Rana, Okan Koksal, and Christina Manolatou, “Many-body theory of the optical conductivity of excitons and trions in two-dimensional materials,” Physical Review B 102, 085304 (2020).
  * Pimenov  _et al._ (2017) Dimitri Pimenov, Jan von Delft, Leonid Glazman, and Moshe Goldstein, “Fermi-edge exciton-polaritons in doped semiconductor microcavities with finite hole mass,” Physical Review B 96, 155310 (2017).
  * Shahnazaryan  _et al._ (2020) V Shahnazaryan, VK Kozin, IA Shelykh, IV Iorsh, and O Kyriienko, “Tunable optical nonlinearity for transition metal dichalcogenide polaritons dressed by a fermi sea,” Physical Review B 102, 115310 (2020).
  * Levinsen  _et al._ (2019) Jesper Levinsen, Francesca Maria Marchetti, Jonathan Keeling, and Meera M. Parish, “Spectroscopic signatures of quantum many-body correlations in polariton microcavities,” Phys. Rev. Lett. 123, 266401 (2019).
  * Navadeh-Toupchi  _et al._ (2019) M. Navadeh-Toupchi, N. Takemura, M. D. Anderson, D. Y. Oberli, and M. T. Portella-Oberli, “Polaritonic cross feshbach resonance,” Phys. Rev. Lett. 122, 047402 (2019).
  * Camacho-Guardian  _et al._ (2021) A. Camacho-Guardian, M. A. Bastarrachea-Magnani, and G. M. Bruun, “Mediated interactions and photon bound states in an exciton-polariton mixture,” Phys. Rev. Lett. 126, 017401 (2021).
  * Vashisht  _et al._ (2022) Amit Vashisht, Maxime Richard, and Anna Minguzzi, “Bose polaron in a quantum fluid of light,” SciPost Physics 12, 008 (2022), arXiv:2107.04424 [cond-mat.quant-gas] .
  * Grusdt and Fleischhauer (2016) Fabian Grusdt and Michael Fleischhauer, “Tunable polarons of slow-light polaritons in a two-dimensional bose-einstein condensate,” Phys. Rev. Lett. 116, 053602 (2016).
  * Nielsen  _et al._ (2020) K. Knakkergaard Nielsen, A. Camacho-Guardian, G. M. Bruun, and T. Pohl, “Superfluid flow of polaron polaritons above landau’s critical velocity,” Phys. Rev. Lett. 125, 035301 (2020).
  * Camacho-Guardian  _et al._ (2020) A. Camacho-Guardian, K. Knakkergaard Nielsen, T. Pohl, and G. M. Bruun, “Polariton dynamics in strongly interacting quantum many-body systems,” Phys. Rev. Research 2, 023102 (2020).
  * Weisskopf and Wigner (1997) Victor Weisskopf and EP Wigner, “Berechnung der natürlichen linienbreite auf grund der diracschen lichttheorie,” Part I: Particles and Fields. Part II: Foundations of Quantum Mechanics , 30–49 (1997).
  * Fleischhauer and Lukin (2000) M. Fleischhauer and M. D. Lukin, “Dark-state polaritons in electromagnetically induced transparency,” Phys. Rev. Lett. 84, 5094–5097 (2000).
  * Fleischhauer and Lukin (2002) M. Fleischhauer and M. D. Lukin, “Quantum memory for photons: Dark-state polaritons,” Phys. Rev. A 65, 022314 (2002).
  * Hau  _et al._ (1999) Lene Vestergaard Hau, Stephen E Harris, Zachary Dutton, and Cyrus H Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
  * Boller and Imamoğlu (1991) K-J Boller and A Imamoğlu, “Observation of electromagnetically induced transparency,” Physical Review Letters 66, 2593 (1991).
  * Fleischhauer  _et al._ (2005) Michael Fleischhauer, Atac Imamoglu, and Jonathan P Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Reviews of modern physics 77, 633 (2005).
  * Lampis  _et al._ (2016) Andreas Lampis, Robert Culver, Balázs Megyeri, and Jon Goldwin, “Coherent control of group index and magneto-optical anisotropy in a multilevel atomic vapor,” Optics Express 24, 15494–15505 (2016).
  * Camacho-Guardian  _et al._ (2018) A. Camacho-Guardian, L. A. Peña Ardila, T. Pohl, and G. M. Bruun, “Bipolarons in a bose-einstein condensate,” Phys. Rev. Lett. 121, 013401 (2018).
  * Camacho-Guardian and Bruun (2018) A. Camacho-Guardian and Georg M. Bruun, “Landau effective interaction between quasiparticles in a bose-einstein condensate,” Phys. Rev. X 8, 031042 (2018).
  * Will  _et al._ (2021) M. Will, G. E. Astrakharchik, and M. Fleischhauer, “Polaron interactions and bipolarons in one-dimensional bose gases in the strong coupling regime,” Phys. Rev. Lett. 127, 103401 (2021).
  * Camacho-Guardian  _et al._ (2022) A. Camacho-Guardian, M. Bastarrachea-Magnani, T. Pohl, and G. M. Bruun, “Strong photon interactions from weakly interacting particles,” Phys. Rev. B 106, L081302 (2022).