# Transition from Lorentz to Fano Spectral
Line Shapes in Non-Relativistic Quantum Electrodynamics

Davis M. Welakuh<EMAIL_ADDRESS>Harvard John A. Paulson School Of
Engineering And Applied Sciences, Harvard University, Cambridge, Massachusetts
02138, USA  Prineha Narang<EMAIL_ADDRESS>Harvard John A. Paulson
School Of Engineering And Applied Sciences, Harvard University, Cambridge,
Massachusetts 02138, USA

###### Abstract

Spectroscopic signatures associated with symmetric Lorentzian and asymmetric
Fano line shapes are ubiquitous. Distinct features of Fano resonances in
contrast with conventional symmetric resonances have found several
applications in photonics such as optical switching, sensing, lasing, and
nonlinear and slow-light devices. Therefore, it is important to have control
over the generation of these resonances. In this work, we show through _ab
initio_ simulations of coupled light-matter systems that Fano interference
phenomena can be realized in a multimode photonic environment by strong
coupling to the electromagnetic continuum. Specifically, we show that by
effectively enhancing the light-matter coupling strength to the photon
continuum in an experimentally feasible way, we can achieve a transition from
Lorentzian to Fano lines shapes for both electronic and polaritonic
excitations. An important outcome of switching between these spectral
signatures is the possibility to control the Purcell enhancement of
spontaneous emission alongside electromagnetically induced transparency which
is a special case of Fano resonances. Switching from Fano back to a Lorentzian
profile can be achieved by physically reducing the coupling strength to the
continuum of modes. Our results hold potential for realizing tunable Fano
resonances of molecules and materials interacting with the electromagnetic
continuum within multimode photonic environments.

##  I Introduction

In atomic, molecular and solid-state physics, the spectroscopic detection of
radiation characterizes the response of the system by probing physical
processes such as scattering, fluorescence, or absorption Flick  _et al._
(2018); Ruggenthaler  _et al._ (2018). The spectral line shapes that arise
provide fundamental information about these physical processes. A commonly
observed spectral line shape is the symmetric Lorentzian line shape that
represents the finite radiative lifetimes of excited states. In other cases,
asymmetric line shapes are usually observed such as Fano resonances in auto-
ionization in atoms Ott  _et al._ (2013). Such asymmetric line shapes occur
when a discrete quantum state interferes with a continuum band of states Fano
(1935, 1961). Fano resonances have been identified in different spectra
(scattering, emission or absorption) of several systems such as metamaterials
with weak asymmetric split ring resonators Luk’yanchuk  _et al._ (2010),
quantum dots and wires Franco  _et al._ (2003), plasmonic nanostructures
Luk’yanchuk  _et al._ (2010); Fan  _et al._ (2010), molecular systems Alhassid
_et al._ (2006), strong coupling between Mie and Bragg scattering in photonic
crystals Rybin  _et al._ (2009) and cavity electromagnonics Gollwitzer  _et
al._ (2021) and electrodynamics Geng  _et al._ (2021). Among the wide range of
applications of Fano interferences, some of the relevant applications are
chemical and biomedical sensing Hao  _et al._ (2008), lasing Yu  _et al._
(2017), switching Yu  _et al._ (2014), and directional scattering
Sheikholeslami  _et al._ (2011).

Recent work has considered the transition from Lorentzian to Fano resonances
Ott  _et al._ (2013); Tu  _et al._ (2015); Wang  _et al._ (2016) in the
context of weak light-matter coupling using a semi-classical description.
Light-matter interaction in a confined electromagnetic environment can be
controlled such that a regime is reached where the energy is coherently
exchanged between light and matter. In the case where strong light-matter
coupling is achieved, a photonic mode hybridizes with a matter resonance
giving rise to novel eigenstates commonly known as polaritons which are
separated in energy by the so-called Rabi splitting Skolnick  _et al._ (1998).
These new states of matter have attracted interest in enhancing or modifying
chemical landscapes Hutchison  _et al._ (2012); Feist  _et al._ (2018); Flick
and Narang (2020), electrical conductivities Laussy  _et al._ (2010); Orgiu
_et al._ (2015), biological processes Coles  _et al._ (2014), polariton lasing
Ramezani  _et al._ (2017) and enhance charge and energy transport Zhong  _et
al._ (2016); Feist and Garcia-Vidal (2015), among others. With such control of
physical and chemical processes, strong light-matter coupling is a timely
approach to investigate how Lorentzian line shapes can be tuned into Fano
resonances Limonov  _et al._ (2017).

In this _Article_ we investigate a transition between two seemingly different
spectroscopic signatures associated with symmetric Lorentzian and asymmetric
Fano line shapes in strongly coupled light-matter systems. Our first-
principles description of the coupled light-matter system is based on quantum
electrodynamical density-functional theory (QEDFT) Ruggenthaler  _et al._
(2018, 2014); Flick and Narang (2018); Flick  _et al._ (2015) in the linear-
response regime Flick  _et al._ (2019). Using this approach, we show
distinctively two cases where, on the one hand, electronic excitations are
tuned from Lorentzian into Fano line shapes and on the other hand, polaritonic
resonances that arise due to strong light-matter coupling are also tuned into
Fano line shapes. This seamless transition between spectroscopic signatures is
achieved by strongly coupling the bare and hybridized matter system composed
of discrete states to the continuum of states of the electromagnetic vacuum.
In order to confirm that the asymmetric line shapes that arise in the spectrum
are indeed Fano resonances, we employ the Fano formalism Limonov  _et al._
(2017); Ott  _et al._ (2013) that characterizes whether a profile is an
asymmetric Fano or symmetric Lorentz line shape. We obtain excellent agreement
between the fitted and _ab initio_ results which confirms that the asymmetric
line shapes that arise are indeed Fano resonances.

Next, we identify cases where the perturbative Fano formalism is inadequate to
describe the coupled system, thus strengthening the need for an _ab initio_
description of strongly coupled light-matter systems. Furthermore, a
particularly important physical outcome of strong coupling to the photon
continuum is the Purcell enhancement of spontaneous emission where the finite
radiative lifetimes of excited states get modified which is useful for
realizing optical quantum memory devices Lvovsky  _et al._ (2009). The
accompanying flexibility in tuning Fano resonances provides an approach to
create sensing devices with variable characteristics, valuable for
applications in photonics Limonov (2021); Limonov  _et al._ (2017). We also
show the special case of Fano resonances where electromagnetically induced
transparency is realized. The reverse process of switching from a Fano to a
Lorentzian profile can be achieved by reducing the coupling strength to the
continuum of modes. Our findings provide a valuable way to tune between the
two different line shapes and would be applicable to scenarios where strong
light-matter coupling can be enhanced within multimode photonic environments.

##  II Theoretical framework

For the quantum mechanical description of light-matter interaction, we
consider the Pauli-Fierz Hamiltonian in the long-wavelength limit
Cohen‐Tannoudji  _et al._ (1989); Craig and Thirunamachandran (1998). This
implies that for a coupled matter-photon system, the relevant photon modes
have wavelengths that are large compared to the size of a matter subsystem,
for instance, an atom or molecule. In this setting, the length form of the
Pauli-Fierz Hamiltonian in dipole approximation Rokaj  _et al._ (2018); Flick
_et al._ (2019); Rivera  _et al._ (2019); Ashida  _et al._ (2021) is given by

$\displaystyle\hat{H}$
$\displaystyle=\sum\limits_{i=1}^{N}\left(\frac{\hat{\textbf{p}}_{i}^{2}}{2m}+v_{\textrm{ext}}(\hat{\textbf{r}}_{i})\right)+\frac{e^{2}}{4\pi\epsilon_{0}}\sum\limits_{i>j}^{N}\frac{1}{\left|\hat{\textbf{r}}_{i}-\hat{\textbf{r}}_{j}\right|}$
$\displaystyle\quad+\sum_{\alpha=1}^{M}\frac{1}{2}\left[\hat{p}^{2}_{\alpha}+\omega^{2}_{\alpha}\left(\hat{q}_{\alpha}\\!-\\!\frac{\bm{\lambda}_{\alpha}}{\omega_{\alpha}}\cdot\hat{\textbf{R}}\right)^{2}\right].$
(1)

Here, the $N$ electrons are described by the electronic coordinates,
$\hat{\textbf{r}}_{i}$, and their corresponding momentum by the operator
$\hat{\textbf{p}}_{j}$. The external potential of the nuclei is represented by
$v_{\textrm{ext}}(\hat{\textbf{r}}_{i})$ and
$\hat{\textbf{R}}=\sum_{i=1}^{N}e\,\hat{\textbf{r}}_{i}$ represents the total
electronic dipole operator where $e$ is the electron charge. The quantized
electromagnetic field is described by harmonic oscillators which consists of
the displacement coordinate $\hat{q}_{\alpha}$ and canonical momentum operator
$\hat{p}_{\alpha}$ with associated mode frequency $\omega_{\alpha}$ for each
mode $\alpha$ of an arbitrarily large but finite number of photon modes $M$.
The physical coupling between light and matter is
$\bm{\lambda}_{\alpha}=\frac{1}{\sqrt{\epsilon_{0}}}\,S_{\alpha}(\textbf{r}_{0})\,\textbf{e}_{\alpha}$
where $\textbf{e}_{\alpha}$ is the transversal polarization vector of a
photonic mode $\alpha$ with wave vector $\textbf{k}_{\alpha}$ and
$\epsilon_{0}$ is the vacuum permittivity. The coupling strength depends on
the form of the mode functions $S_{\alpha}(\textbf{r}_{0})$ and the chosen
reference point for our matter subsystem Milonni (1993). Since we simulate the
continuum of states by explicitly treating many photon modes, it might become
necessary to use the bare mass $m$ instead of the already renormalized
physical mass $m_{e}$ (bare plus the electromagnetic masses) of the charged
particles Spohn (2004); Rokaj  _et al._ (2020). However, we make the common
assumption to treat explicitly the small part of the photon continuum that is
changed with respect to the free space case due to the multimode photonic
environment. This implies that the rest of the continuum of modes not affected
by the photonic cavity is subsumed in the usual renormalized physical mass
(i.e. $m_{e}=1$ in atomic units) of the charged particles Spohn (2004).

Figure 1: Schematic setup of a phenol molecule confined within a multimode
optical cavity of volume $V=L_{x}L_{y}L_{z}$. The photon field is polarized
along the $x$-axis with coupling strength $\bm{\lambda}_{\alpha}$ and the wave
vector is along the cavity axis of length $L_{z}$ in the $z$-direction. The
multimode cavity has a range of frequencies $\omega_{\alpha}$.

We schematically depict in Fig. (1) a physical setup to realize strong light-
matter coupling featuring a single phenol (C6H5OH) molecule confined within a
multimode optical cavity. Although we depict a standard Fabry-Pérot
configuration, several other multimode photonic setups for strong light-matter
coupling such as hybrid cavity-antenna configurations Zhang  _et al._ (2019),
nanoparticle arrays Wang  _et al._ (2018), or coupled photonic crystals Dousse
_et al._ (2019) are suitable for this study. Thus, the results presented here
are not restricted to any specific photonic environment and not limited to a
specific matter system (as demonstrated in App. A). To describe the coupling
of the phenol molecule to the electromagnetic continuum, we sample a wide
range of photon modes of a quasi one-dimensional mode space by employing the
coupling strength
$\bm{\lambda}_{\alpha}=\sqrt{\frac{2}{\epsilon_{0}L_{x}L_{y}L_{z}}}\text{sin}(k_{\alpha}\,z_{0})\textbf{e}_{x}$,
where $z_{0}=L_{z}/2$ is the position of the molecule inside the cavity,
$k_{\alpha}=\omega_{\alpha}/c$ and $\omega_{\alpha}=\alpha c\pi/L_{z}$ are the
wave vectors and frequencies of the modes, respectively. We assume a constant
mode function in the $x$\- and $y$-direction and choose the volume
$V=L_{x}L_{y}L_{z}$ (with $L_{z}=3250$ $\mu$m, $L_{x}=10.58$ Å and
$L_{y}=2.65$ Å) such that the sampled modes couple weakly to the molecular
system Flick  _et al._ (2019, 2017). The electromagnetic vacuum is sampled
with 100,000 photon modes with energies that cover densely a range from 0.19
meV for the smallest energy up to 38.14 eV for the energy cutoff with an equal
spacing of $\Delta\omega=0.38$ meV. Since we sample frequencies corresponding
to a one-dimensional mode space, the lifetimes will differ from the actual
three-dimensional lifetimes, but for the purpose of demonstrating the
possibilities of tuning between spectroscopic signatures, this is suﬃcient. In
this setting, the coupling strength $\bm{\lambda}_{\alpha}$ can be increased
by reducing the volume $V$ of the cavity which can be realized experimentally.
Since we sample a one-dimensional mode space with constant mode functions in
the $x$\- and $y$-direction, we choose to change the coupling strength by
varying the area $L_{x}L_{y}$.

As mentioned previously, when discrete quantum states interfere with a
continuum of states this give rise to Fano line shapes that manifest, for
example, in the absorption spectrum. This was realized in the original work of
Fano when atomic levels were coupled to the continuum Fano (1935, 1961). In
our setting, the discrete states can be electronic or polaritonic that couple
to the electromagnetic continuum, which makes it a viable setup to obtain Fano
resonances for a large variety of systems. One of the methods to identify and
characterize a Fano resonance in any spectra is to use the Fano formula Fano
(1935); Ott  _et al._ (2013), which for the photo-absorption cross-section at
energy $E=\hbar\omega$ is given as follows

$\displaystyle\sigma(E)=\sigma_{0}\frac{(q+\varepsilon)^{2}}{1+\varepsilon^{2}}\,,$
(2)

where $\sigma_{0}$ is the resonant cross-section, the dimensionless energy is
$\varepsilon=2(E-E_{0})/\hbar\Gamma$, with $E_{0}$ being the resonant Fano
energy and $\Gamma$ is the spectral line width. The parameter $q$ is the so-
called Fano parameter that describes the degree of asymmetry of a Fano line
shape. This parameter depends on the phase angle $\varphi$ as $q=\cot\varphi$
(and is periodic in $\pi$). The angle $\varphi$ describes the phase shift of
the continuum. In the limit $q\rightarrow\pm\infty$ when $\varphi\rightarrow
n\pi$ ($n$ is an integer) leads to Lorentzian line shapes and Fano resonances
are obtained between these extreme cases where $q$ is finite and non-zero
Limonov  _et al._ (2017); Limonov (2021). When $q=0$, the Fano profile is a
symmetric quasi-Lorentzian antiresonance which physically represents the case
when an external perturbation does not couple to the discrete state. The phase
shift $\varphi$ can be understood as an interference of two transition
amplitudes from initial to final state. In this context, the dimensionless
Fano parameter $q$ is the ratio of the amplitude representing a transition to
the discrete state and the amplitude describing the transition to the
continuum Miroshnichenko  _et al._ (2010). The Fano formula (2) is generally
applicable not only to absorption but also to different optical spectra (such
as transmission and scattering) in a variety of systems Limonov  _et al._
(2017). Here we study the spectra by considering the dipole strength function
which is related to the photo-absorption cross-section as
$S(E)=(m_{e}c/2\pi^{2}e^{2}\hbar^{2})\,\sigma(E)$ Yabana and Bertsch (1999).
In addition to Fano resonances that arise due to discrete to continuum
coupling, two other distinct interference effects occur: (i) a change of
spontaneous emission caused by the Purcell effect and (ii) a shift in energy
levels associated with the Lamb shift. These interference effects will be
discussed next.

##  III Results and Discussion

In this section, we investigate how to tune Lorentzian resonances into Fano
line shapes in a cavity QED setup of a molecule coupled to the electromagnetic
continuum. We study this physical effect in the absorption spectrum of the
coupled system where Lorentzian line shapes of electronic and polaritonic
excitations can be tuned. To study the coupled electron-photon system, we
solve the stationary eigenvalue problem of Eq. (1) by employing the
reformulation of linear-response theory for light-matter systems within the
framework of QEDFT Flick  _et al._ (2019). In particular, we use the electron-
photon Casida approach introduced in Ref. Flick  _et al._ (2019) while noting
that similar approaches as in Refs. Flick and Narang (2018); Welakuh  _et al._
(2022) can be used. For the spectrum considered here, the electronic and
polaritonic excitations (discrete states) are within the energy cutoff of the
photon continuum (large photonic vacuum density of states) which is necessary
to generate Fano resonances.

Figure 2: (a) _Ab initio_ calculation of the electronic excitation spectrum of
a phenol molecule weakly coupled to cavity modes of a multimode photonic
environment. Panels (b) and (c) show how the Lorentzian resonances are tuned
into Fano resonances by coupling strongly to the electromagnetic continuum.

In the first numerical example, we couple the phenol molecule weakly to the
photon continuum (as discussed in Sec. II). The results of this calculation
are shown in Fig. (2). We note that the peaks are not artificially broadened
in this calculation since we have sampled densely the photon continuum that
couples to the matter subsystem. This case specifies an _ab initio_ approach
on how to obtain lifetimes non-perturbatively from QEDFT linear-response
theory as discussed in Ref. Flick  _et al._ (2019). The color code in Fig. (2)
indicates the different contributions of each excitation in the response
function where from blue means more photonic to red more electronic. Since the
peak strengths are dominantly blue indicates that resonances are mainly
photonic in nature and the peak positions are mostly dominated by the matter
constituents. This result is in agreement with measurements of an absorption
(or emission) spectrum where a photon is absorbed (or emitted) and not the
matter constituents. In Fig. (2 a), we show the spectrum of the molecule where
we find Lorentzian line shapes for the different excitations. The light-matter
coupling strength in Fig. (2 a) corresponds to weak coupling to the photon
continuum (for $L_{x}L_{y}=28.0$ Å2). When the light-matter coupling strength
is effectively increased by reducing the volume of the cavity along the $x$\-
and $y$-direction, the molecule couples strongly to the photon continuum and
this changes the spectrum. For reducing $L_{x}L_{y}$ (and increasing
$\bm{\lambda}_{\alpha}$) as in Fig. (2 b,c), we find a transition of the
initially Lorentzian line shapes for the weak coupling to the continuum to
Fano line shapes for strong coupling to the photon continuum. The asymmetric
line shapes are due to the overlap of the discrete states of the electronic
system with the continuum states of the electromagnetic vacuum, where
destructive and constructive interference take place at close energy
positions.

$\;L_{x}L_{y}$ (Å2) |  $\;\varphi$ (rad) |  $\;\hbar\Gamma$ (eV) | $\;q$
---|---|---|---
28.0 | $\pi$ | 0.003 | -8.17$\times 10^{15}$
3.11 | $0.972\pi$ | 0.015 | -11.43
1.12 | $0.906\pi$ | 0.041 | -3.27
Table 1: Fitted values for the electronic case of the phase shift $\varphi$,
line width $\hbar\Gamma$ and Fano parameter $q$ for the different cases of
changing the coupling strength to the continuum by varying the cavity mode
volume via $L_{x}L_{y}$. This corresponds to the Fano fit in Fig. 3.

To identify the physics of Fano resonances in the calculated spectra, we fit
the _ab initio_ results with the Fano formula of Eq. (2) by varying different
parameters (see Tab. 1 for fitted values). We do this specifically for the
peak around $5.6$ eV as shown in Fig. (3).

Figure 3: A Fano fit (orange line) to the peak around $5.6$ eV of Fig. (2)
characterizing a transition from a symmetric to an asymmetric line shape. (a)
Symmetric Lorentzian line shape occur in the weak coupling to the photon
continuum. In panel (b) and (c), the asymmetric line shapes emerge due to
strong coupling to the photon continuum as characterized by the Fano fit.

In Fig. (3 a), we find a symmetric Lorentzian line shape for the weak coupling
to the photon continuum as characterized with the Fano fit where the $\pi$
phase shift result to an infinite Fano parameter $q$. The symmetric Lorentzian
shape is due to the negligible transition amplitude for weak coupling to the
continuum. Fitting to the Lorentzian line shape is consistent with the Wigner-
Weisskopf theory Weisskopf and Wigner (1930); Flick  _et al._ (2019) as the
weakly interacting continuum modes merely describes the emission profile. For
the strong coupling case, the Fano fit agrees with the _ab initio_ results in
Fig. (3 b) and deviates slightly in Fig. (3 c). The asymmetric line shape
arises since the transition amplitude to the continuum increases for stronger
coupling such that both amplitudes (transition to discrete and continuum) are
different from zero, thus, giving a finite $q$. The competing interference
effect that characterizes the asymmetric Fano resonance manifests itself as an
increase in the cross section on one of the slopes of the profile
(constructive interference) and a decrease on the second slope (destructive
interference). This interference phenomena also explains the reduced
oscillator strength for increasing coupling to the continuum. As changes in
the electronic and photonic subsystem are self-consistent, the deviation of
the Fano fit in Fig. (3 c) will become more pronounced for stronger coupling
situations. This is expected since Fano used a perturbation approach to
explain the appearance of asymmetric resonances Fano (1961); Miroshnichenko
_et al._ (2010). Typically, since the photonic density of states now differ
from the vacuum density of states due to an increase in the coupling strength,
this usually result in the enhancement (or suppression) of the rate of
spontaneous emission Burstein and Weisbuch (1995). In our example, the line
width becomes broader (see Fig. (2)), implying a shortened lifetime which can
be obtained from Tab. (1) using $\tau=1/\Gamma$. The competing constructive
and destructive interference that leads to a decrease of the oscillator
strength gives rise to an enhancement of the radiative decay rate and a
broadened line width Rybin  _et al._ (2016). This has potential applications
for realizing a optical quantum memory device Lvovsky  _et al._ (2009).
Another interesting effect that arise is the Lamb shift due to the interaction
of discrete electronic states with the vacuum fluctuations of electromagnetic
field. It is not ideal to associate the shift of the calculated spectra from
the Fano fit in Fig. (2 b,c) since for one, the Fano formalism is
perturbative. However, the _ab initio_ results presented in this work are non-
perturbative and we note that the shift can be partly attributed to the Lamb
shift as discussed in Refs. Flick  _et al._ (2019); Welakuh (2021) and also to
the photon random-phase approximation applied to the transverse electron-
photon interaction terms.

Figure 4: (a) _Ab initio_ calculation of the absorption spectrum of phenol
capturing the lower and upper polaritonic peaks between 5 and 6 eV due to
resonant strong coupling where the peaks are Lorentzian. Panels (b) and (c)
shows how both polaritonic and electronic excitations can be tuned into Fano
resonances by strongly coupling to the photon continuum.

In the second study, in addition to the modes that sample the electromagnetic
vacuum, an additional mode is strongly coupled to an electronic excitation
(the peak around $5.6$ eV) that results to the Rabi splitting into the lower
and upper polariton peaks. Including an extra mode that couple strongly can be
realized, for example, in crossed cavity setups Heinz  _et al._ (2021);
Brekenfeld  _et al._ (2020). We show the results of this calculation in Fig.
(4). Typically, in this strong coupling regime, the continuum of weakly
interacting modes constitute the line width of the excitations and also
represent dissipation channels in the coupled system Welakuh  _et al._ (2021);
Flick  _et al._ (2019). Interestingly, we find in Fig. (4 a) a larger
broadening of the lower polaritonic peak when compared to that of the upper
polariton (see Fig. (5 a) for details) which implies the lower polariton state
has a longer radiative lifetime. When we effectively enhance the light-matter
coupling strength, this results to a transition of the initially Lorentzian
profiles for the polaritonic excitations into Fano resonances as shown in Fig.
(5 b,c).

$\;L_{x}L_{y}$ (Å2) |  $\;\varphi$ (rad) |  $\;\hbar\Gamma$ (eV) | $\;q$
---|---|---|---
28.0 | $\pi$ | 0.019 | -8.17$\times 10^{15}$
3.11 | $0.972\pi$ | 0.088 | -11.43
1.12 | $0.917\pi$ | 0.161 | -3.73
Table 2: Fitted values for the polaritonic case of the phase shift $\varphi$,
line width $\hbar\Gamma$ and Fano parameter $q$ for the different cases of
changing the coupling strength to the continuum by varying the cavity mode
volume via $L_{x}L_{y}$. This corresponds to the Fano fit in Fig. 5.

To prove that the asymmetric line shapes in Fig. (5) are indeed Fano
resonances, we again apply the Fano formula of Eq. (2) to the polaritonic
peaks between the energy range $5$ to $6$ eV (the fitted values are given in
Tab. (2)).

Figure 5: Calculated and Fano fit (orange line) to the polaritonic peaks in
Fig. (4) characterizing a transition from symmetric to asymmetric line shapes.
(a) Symmetric Lorentzian line shape occur in the weak coupling to the photon
continuum. Panels (b) and (c) show the asymmetric Fano line shapes that emerge
due to strong coupling to the photon continuum. The Fano fit become less
accurate for stronger coupling.

In Fig. (5 a), the Fano fit is in good agreement with the calculated spectrum
as it characterizes the Lorentzian line shapes corresponding to the lower and
upper polariton peaks. The symmetric line shapes are equally due to the
negligible transition amplitudes for coupling weakly to the continuum as in
electronic case. In contrast to Fig. (5 a), the Fano fit deviates from the
calculated spectra for increasing coupling to the continuum as shown in Fig.
(5 b,c). This deviation indicates the need for a self-consistent treatment of
light-matter interaction which cannot be captured by perturbative approaches.
We find that the line width of the polaritonic peaks are broader and leads to
a Purcell enhancement larger than the electronic case which could be useful in
an optical quantum memory devices Lvovsky  _et al._ (2009). Equally, even for
strong coupling we expect to find a Lamb shift of polaritonic excitations due
to interactions with vacuum fluctuations. It is likely the case that the Lamb
shift will increase when the vacuum density of states gets perturbed as in
this case as seen in Fig. (5 c). The reverse process of switching from a Fano
to a Lorentzian profile can be achieved by reducing the coupling strength
(which implies increasing the cavity volume) to the continuum of modes. Such a
change reverts the modified photonic density of states to that of the free-
space emission. A key interference effect that arises in setups that are
associated with Fano resonances is electromagnetically induced transparency
(EIT). This effect is a quantum interference phenomenon that allow light
pulses through an otherwise opaque medium Harris (1997). EIT is a special case
of Fano resonance that occur due to a destructive interference between the
discrete resonant state and a continuum state (i.e., when frequencies match)
such that $q=0$, which describes the resonant suppression of absorption
Limonov  _et al._ (2017). Our description of the coupled system captures the
EIT phenomenon as discussed in detail in App. A. The different examples
presented in this work make the cavity setup a versatile approach for
realizing distinct spectroscopic signatures of symmetric Lorentz and
asymmetric Fano resonances within the framework of non-relativistic quantum
electrodynamics.

##  IV Summary and outlook

In conclusion, we present an approach that allows switching between two
spectroscopic signatures associated with symmetric Lorentzian and asymmetric
Fano line shapes. To achieve such control, we consider a setting for realizing
strong light-matter interaction where the photonic environment in which the
matter subsystem is immersed is of a multimode nature. In this cavity setup,
we show that by strongly coupling the bare matter and hybridized system to the
continuum of transverse electromagnetic modes, the initial spectroscopic
signatures of symmetric Lorentzian line shapes are tuned into asymmetric Fano
line shapes. We further confirm that these asymmetric line shapes are indeed
Fano resonances by fitting the Fano formalism to the calculated spectra where
we find good agreement for the case of discrete electronic states interacting
strongly with the continuum. In the case where we coupled polaritonic states
to the continuum of modes, we find important differences between the fit and
_ab initio_ results, highlighting the need for a first-principles description
of the interference phenomenon of Fano resonances where the perturbative Fano
formalism breaks down. From the Fano fit, we are able to characterize the
degree of asymmetry of the Fano resonances as well as deduce directly the
finite radiative lifetimes of excited states including the Purcell enhancement
of spontaneous emission due to a change in the photonic density of states. We
also showed that our cavity setup for realizing interference effects also
captures the special case of Fano resonance of EIT. This work presents a step
forward in the physics of Fano resonances as the asymmetric signatures can be
tuned and controlled by effectively enhancing the light-matter coupling to the
electromagnetic continuum. Finally, the results presented here show the new
perspectives that _ab initio_ QED approaches now provide to analyze and
predict quantum nanophotonics in regimes of strong coupling.

##  V Acknowledgments

We acknowledge helpful discussions with Marissa Weichman and Stephane Kena-
Cohen. This work is supported by the Office of Naval Research (ONR) MURI
Program under grant number ONR N00014-21-1-2537 as well as the Quantum Science
Center (QSC), a National Quantum Information Science Research Center of the
U.S. Department of Energy (DOE). P.N. acknowledges support as a Moore Inventor
Fellow through Grant No. GBMF8048 and gratefully acknowledges support from the
Gordon and Betty Moore Foundation as well as support from a NSF CAREER Award
under Grant No. NSF-ECCS-1944085.

##  Appendix A Electromagnetically induced transparency (EIT)

Electromagnetically induced transparency (EIT) is a quantum interference
effect that allows the propagation of light pulses through an otherwise opaque
medium Harris  _et al._ (1990); Harris (1997). EIT is a special case of Fano
resonance that occur due to a destructive interference between the discrete
resonant state and a continuum state (i.e., when frequencies match) such that
the Fano parameter vanishes at $q=0$, which describes the resonant suppression
of absorption Limonov  _et al._ (2017). One major application of EIT is in
light delay and storage, where light propagating through a medium is slowed
down Haugland  _et al._ (2020) and even stopped xi Liu  _et al._ (2005);
Welakuh and Dikandé (2017). We now consider how to realize EIT in the cavity
setup depicted in Fig. (1). In the following, we replace the phenol molecule
with a single benzene molecule. By so doing, we highlight that the
interference effects presented in this work is independent of the matter
system. To realize EIT in the current setup, we enhance the coupling of one of
the sampled continuum of modes to the $\pi-\pi^{*}$ excitation of benzene
around $6.855$ eV. The rest of the continuum modes couple weakly as discussed
in Sec. II where the cavity area is $L_{x}L_{y}=28.0$ Å2. The results of this
calculation are shown in Fig. (6) where we capture the $\pi-\pi^{*}$
excitation of benzene around $6.855$ eV as shown in Fig. (6 a). When the
frequency of the enhanced cavity mode matches that of the $\pi-\pi^{*}$
excitation, this induces a window of transparency at which the system can no
longer absorb light as shown in Fig. (6 b-d).

Figure 6: (a) The absorption spectra of a single benzene molecule capturing
the $\pi-\pi^{*}$ excitation. Panels (b)-(d) show the onset of a window of
transparency for an increasing coupling strength to a continuum mode. The
quasi-Lorentzian antiresonance Fano fit characterizes the transmission spectra
at the transparency window.

The Fano fit is in good agreement to the computed spectra and the fitted
values are given in Tab. (3).

$g_{\alpha}$ (eV) | $g_{\alpha}/\hbar\omega_{\alpha}$ |  $\;\varphi$ (rad) |  $\;\hbar\Gamma$ (eV) | $\;q$
---|---|---|---|---
0.0011 | 0.00016 | $\pi$ | 0.019 | -8.16$\times 10^{15}$
0.0087 | 0.0013 | $\pi/2$ | 0.011 | 0
0.010 | 0.0015 | $\pi/2$ | 0.014 | 0
0.0122 | 0.0018 | $\pi/2$ | 0.016 | 0
Table 3: Fitted values for EIT for the phase shift $\varphi$, line width
$\hbar\Gamma$ and Fano parameter $q$ for the different coupling strength of an
enhanced resonant bath mode. These values corresponds to the Fano fit in Fig.
(6).

Upon increasing the coupling strength leads to an increase in the transparency
window. With these results, EIT phenomenon is now for the first time realized
from first-principles using QEDFT linear-response framework and our results
show the new perspectives that can be achieved with ____ab initio____ QEDFT
methods Flick  _et al._ (2019); Flick and Narang (2018); Welakuh  _et al._
(2022). We note that EIT has also been studied using a time-dependent density-
functional theory (TDDFT) approach that includes a local potential which
accounts for the coupling to the electromagnetic continuum Schäfer and
Johansson (2021).

It is important to note that the splitting of the absorption peak that occur
at the transparency window (see Fig. (6 b-d)) differs from the strong coupling
regime that leads to the formation of polariton peaks (see Fig. (7)). The
difference between these two effects is that, EIT occurs when the decay rate
from the resonant state is greater than the coupling strength $g_{\alpha}$
(see Sec. B for details) and $g_{\alpha}$ is greater than the decay rate of
the continuum while for strong coupling, $g_{\alpha}$ is greater than both
decay rates Limonov  _et al._ (2017). To demonstrate this, we compare the
values of the first column to the fourth column of Tab. (3) where we find that
the EIT condition is satisfied since $g_{\alpha}$ will be greater than the
decay rate of the continuum. The strong coupling case is immediately satisfied
since $g_{\alpha}=0.174$ eV is greater than the respective decay rates of the
EIT case. Therefore, increasing the coupling to the resonant mode allows a
transition between EIT to the strong coupling regime.

Figure 7: (a) _Ab initio_ calculation of the absorption spectrum of benzene
capturing the lower and upper polaritonic peaks within the range $6.5$ to
$7.5$ eV due to strong coupling to the $\pi-\pi^{*}$ excitation. The line
shapes are symmetric Lorentzians. Panels (b) and (c) shows how both
polaritonic and electronic excitations can be tuned into Fano resonances by
strongly coupling to the photon continuum.

##  Appendix B Numerical details

We now give the numerical details of the calculations presented in this work.
To calculate the electronic structure of the phenol molecule, we use a
cylindrical real space grid of $8$ Å length with the radius of $6$ Å in the
$x$-$y$ plane, and a spacing of $\Delta x=\Delta y=\Delta z=0.22$ Å. We
explicitly describe the 36 valence electrons which amounts to 18 doubly
occupied orbitals, while the core atoms are considered implicitly by the
local-density approximation (LDA) Troullier-Martins pseudopotentials Troullier
and Martins (1991). In the excited state manifold, we include $500$ unoccupied
states in the pseudo-eigenvalue calculation. This number amounts to
$N_{v}*N_{c}=18*500=9000$ pairs of occupied-unoccupied states where $N_{v}$
and $N_{c}$ denote the number of occupied and unoccupied Kohn-Sham orbitals,
respectively. Using the electron-photon Casida equation, the resulting
dimension of the coupled but truncated matrix is
$\left((N_{v}*N_{c}+M)\times(N_{v}*N_{c}+M)\right)$ where we sampled
$M=100,000$ photon modes in the first example and $M=100,001$ in the second
case. To describe the electron-electron and electron-photon interaction in the
linear-response QEDFT framework Flick  _et al._ (2019), we apply the adiabatic
LDA (ALDA) to the Hartree exchange-correlation kernel (i.e.
$f^{n}_{\text{Hxc}}\rightarrow f^{n}_{\text{Hxc,ALDA}}$) and the photon
random-phase approximation to the photon exchange-correlation kernels (i.e.
$f^{n}_{\text{pxc}},f^{q_{\alpha}}_{\text{pxc}}\rightarrow
f^{n}_{\text{p}},f^{q_{\alpha}}_{\text{p}}$), respectively. We define the
electron-photon coupling from the bilinear interaction term of Eq. (1) as is
Ref. Welakuh (2021):

$\displaystyle
g_{\alpha}^{(ij)}=\sqrt{\frac{\hbar\omega_{\alpha}}{\epsilon_{0}V}}\langle\varphi_{i}|\textbf{e}_{\alpha}\cdot\hat{\textbf{R}}|\varphi_{j}\rangle\,,$
(3)

where $|\varphi_{j}\rangle$ are the eigenstates of the bare matter system and
$\langle\varphi_{i}|\textbf{e}_{\alpha}\cdot\hat{\textbf{R}}|\varphi_{j}\rangle$
is the transition dipole moment. We compute the $x$-contribution of the dipole
strength function Flick  _et al._ (2019) with respect to the polarization of
the cavity modes.

We numerically describe the Benzene molecule coupled to photons in App. A
using the same details as the phenol molecule presented here. The only
difference is that benzene has 30 valence electrons which amounts to 15 doubly
occupied orbitals. In this case, the number of occupied-unoccupied pairs is
$N_{v}*N_{c}=15*500=7500$.

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