# Nonlinear optical processes in centrosymmetric systems by
strong-coupling-induced symmetry breaking

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

Nonlinear optical processes associated with even-order nonlinear
susceptibilities are critical for both classical and quantum technologies.
Inversion symmetry, however, prevents nonlinear optical responses mediated by
even-order susceptibilities in several material systems pertinent for
applications in nanophotonics. Here, we demonstrate induced nonlinear optical
processes, namely second- and fourth-harmonic generation that are naturally
forbidden in an inversion symmetric system, by strongly coupling to a photon
mode of a high-Q optical cavity. As an illustrative system with an inversion
symmetry, we consider a semiconductor quantum ring of GaAs that features a
single effective electron. For the coupled system, we control the inversion
symmetry breaking by changing the light-matter coupling strength which at the
same time allows to tune the nonlinear conversion efficiency. We find that the
harmonic generation yield can be significantly increased by increasing the
light-matter coupling strength in an experimentally feasible way. In the few-
photon limit where the incident pump field is a coherent state with just a few
photons, we find that the harmonic conversion efficiency is increased for
strong coupling as opposed to using intense pump fields. This new approach is
applicable to a wide variety of centrosymmetric systems as the symmetry
breaking rest on the properties of the photonic environment used to achieve
strong light-matter interaction. Our work constitutes a step forward in the
direction of realizing physically forbidden nonlinear optical processes in
centrosymmetric materials widely adopted for applications in integrated
photonics.

The interaction between light and matter is highly dependent on the symmetry
of the physical system, across atoms, molecules, and solids, which dictates
the linear and nonlinear optical response. Material systems that possess an
inversion symmetry center (that is, centrosymmetric systems) have a negligible
nonlinear response for processes associated with even-order susceptibilities
$\chi^{(n)}$ (where $n$ are even natural numbers) as these quantities vanish
Boyd (1992) by symmetry. Nonetheless, nonlinear optical processes associated
with even-order $\chi^{(n)}$ in centrosymmetric systems can still be observed
by breaking the symmetry. There exist several ways to achieve this symmetry
breaking, for example, by applying external strain Jacobsen  _et al._ (2006);
Cazzanelli  _et al._ (2012) or from twisted interfaces in van der Waals
heterostructures Yao  _et al._ (2021), to using DC electric fields Timurdogan
_et al._ (2017); Klein  _et al._ (2017), or using strong THz pulses Grishunin
_et al._ (2017); Vampa  _et al._ (2018), or even confining in optical Breunig
(2016); Gouveia  _et al._ (2013); Lettieri  _et al._ (2005); Zhang  _et al._
(2019) or a plasmonic cavities Li  _et al._ (2021). Importantly, these
different approaches of symmetry breaking require high or relatively high
Zhang  _et al._ (2019) pump powers for efficient harmonic generation.

Strong light-matter interaction within confined electromagnetic environments
such as optical cavities or plasmonic devices provide a parallel way to
control or modify material properties Flick  _et al._ (2018); Ruggenthaler
_et al._ (2018). In the strong coupling regime where energy is coherently
exchanged between light and matter, a mode hybridizes with a material
resonance leading to the formation of hybrid light-matter states known as
polaritons. This regime has been explored to control chemical reactions
Hutchison  _et al._ (2012), realize polariton lasing Kéna-Cohen and Forrest
(2010) and even the enhancement of harmonic generation from polaritonic states
in noncentrosymmetric systems Chervy  _et al._ (2016); Barachati  _et al._
(2018); Welakuh and Narang (2022a). Such control mediated by strong light-
matter coupling is suitable to investigate nonlinear optical processes
associated with even-order susceptibilities in centrosymmetric systems with
the possibility for control of the efficiency.

In this _Letter_ , we present a general approach to break the inversion
symmetry of material systems which allows for highly tunable even-order
harmonic generation naturally forbidden in such systems. Our approach relies
on a quantized treatment of the coupled light-matter system where the incident
(pump) field is treated as a quantized field in a coherent state. When the
light-matter system is strongly coupled, this leads to an induced symmetry-
breaking, thus, allowing for nonlinear optical processes associated with even-
order susceptibilities. An important feature of our mechanism is that we can
readily control the inversion symmetry-breaking. Such control is achieved by
increasing the coupling strength in an experimentally feasible way which
increases the degree of the induced asymmetry and simultaneously result in an
efficient ever-order harmonic yield from the strongly coupled light-matter
system. Several applications ranging from optical frequency conversion to
imaging and biosensing where control over efficiency is important Garmire
(2013) could leverage this mechanism. An advantage of our approach is that we
work in the few-photon limit realized in quantum optical experiments Chang
_et al._ (2014); Hacker  _et al._ (2019) which requires weak pump power as
opposed to methods that require high pump power Breunig (2016); Gouveia  _et
al._ (2013); Lettieri  _et al._ (2005). With this we show that the nonlinear
conversion efficiency increases using the few-photon limit with strong
coupling as opposed to using intense pump fields. An added merit of the
quantized treatment is that we can investigate the characteristics (photon
occupation and statistics) of the coherent incident field which indirectly
gives knowledge on the non-classical properties of the generated harmonics.
Our approach is therefore important as it is applicable to a wide variety of
centrosymmetric systems as the strong-coupling-induced symmetry breaking
depends on the properties of photonic environment used to achieve strong
light-matter interaction.

We start by describing the coupled light-matter system that considers
transverse fields with wavelengths that are substantially larger than the
matter system so that the long-wavelength limit Cohen‐Tannoudji  _et al._
(1989); Craig and Thirunamachandran (1998) is applicable. In this setting, the
non-relativistic dynamics of the coupled system is described by the Pauli-
Fierz Hamiltonian Rokaj  _et al._ (2018); Spohn (2004):

$\displaystyle\hat{H}$
$\displaystyle=\sum\limits_{i=1}^{N}\left(\frac{\hat{\textbf{p}}_{i}^{2}}{2m}+v_{\textrm{ext}}(\hat{\textbf{r}}_{i})-\frac{e}{m}\hat{\textbf{A}}\cdot\hat{\textbf{p}}_{i}+\frac{e^{2}}{2m}\hat{\textbf{A}}^{2}\right)$
$\displaystyle\quad+\sum\limits_{i>j}^{N}w(\hat{\textbf{r}}_{i},\hat{\textbf{r}}_{j})+\sum_{\alpha=1}^{M}\frac{1}{2}\left(\hat{p}_{\alpha}^{2}+\omega_{\alpha}^{2}\hat{q}^{2}\right)\,,$
(1)

where the $N$ electrons are described by the electronic coordinates,
$\hat{\textbf{r}}_{i}$, the momentum operator of the electrons by
$\hat{\textbf{p}}_{i}$, $w(\hat{\textbf{r}}_{i},\hat{\textbf{r}}_{j})$
describes the longitudinal interaction between the electrons and
$v_{\textrm{ext}}(\hat{\textbf{r}}_{i})$ represents the binding potential. The
last term of Eq. (1) is the energy of the quantized electromagnetic field
given in terms 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 vector potential is
$\hat{\textbf{A}}=\sum_{\alpha=1}^{M}\boldsymbol{\lambda}_{\alpha}\hat{q}_{\alpha}$
where
$\boldsymbol{\lambda}_{\alpha}=\sqrt{1/\epsilon_{0}V}\textbf{e}_{\alpha}$ is
the coupling strength and $V$ is the quantization volume of the field. We note
that in a standard investigation of harmonic generation based on a semi-
classical description, the incident field is not treated as a quantized field
but merely as an external time-dependent field that induces dynamics in the
usual matter-only constituents Timurdogan  _et al._ (2017); Klein  _et al._
(2017). Here, we treat the incident pump field as a quantized field and
investigate the changes the now self-consistently coupled light-matter system
has on the symmetry and nonlinear properties of the system.

As an illustrative system with an inversion symmetry, we consider a well-
studied semiconductor quantum ring of GaAs that features a single effective
electron (see SI. Section I for further details). Due to the inversion
symmetry of the system, nonlinear processes associated with even-order
susceptibilities are forbidden Boyd (1992). However, we will demonstrate that
the inversion symmetry of the system can be broken and controlled by coupling
to a quantized photon mode which gives rise to efficient nonlinear optical
processes like even-order harmonic generation. To achieve this, the matter
system is placed inside an electromagnetic environment, for example, a high-Q
optical cavity. The pump mode is treated as a quantized field that couples to
the matter system as in equation (1). We perform a numerically exact ab-initio
simulation of the coupled light-matter system by solving the time-dependent
Schrödinger equation for the Hamiltonian of Eq. (1) (see SI. Section IV for
details). As an initial state, we choose a factorizable product state of the
ground-state of the matter system and the photon mode (pump field) initially
prepared in a coherent state with $n=|\xi|^{2}=25$ photons where $\xi$ is the
amplitude of the coherent state. The importance of a few-photon quantized pump
field is that it presents an alternative way to achieve qualitatively the same
result with small pump intensity (see SI. Section III for details). We note
that the choice of describing the pumping process with a coherent state yields
quantitatively the same results as the case where the coupled system is driven
by an external potential or charge current with the same amount of photons as
demonstrated in Ref. Welakuh  _et al._ (2021).

Figure 1: Induced SHG and FHG processes in an inversion symmetric system for a
quantized treatment of the pump field in a coherent state. The inset shows
that the harmonic intensity of the FHG is larger than the SHG process which is
attributed to the strong light-matter coupling. $\omega_{f}$ are the
fundamental (pump) frequencies for SHG and FHG.

For the coupled light-matter system, we now investigate individually the
nonlinear optical processes of second-harmonic generation (SHG) and fourth-
harmonic generation (FHG) that are normally forbidden in such a system. To
realize these nonlinear processes in a quantized light-matter system, the
fundamental pump frequency is chosen to be one-half (SHG) and one-fourth (FHG)
the transition energy (24.65 meV) between the electronic states
$|\varphi_{1}^{0}\rangle\leftrightarrow|\varphi_{7}^{1}\rangle$ (see SI.
Section I). The results of this calculation is shown in Fig. (1). As the
quantized pump mode breaks the centrosymmetry of the matter system due to
strong light-matter coupling (see SI. Section I for details), this allow for
forbidden nonlinear processes such as SHG and FHG to now exist in the coupled
system. Thus, this highlights a strong-coupling-induced symmetry breaking in
centrosymmetric systems. In the standard semi-classical description of
harmonic generation, for a fixed pump field the intensity of the generated
harmonics (second-, third-, fourth-harmonics, etc) decrease as a result of the
associated nonlinear susceptibilities Boyd (1992). Interestingly, we find in
the fully quantized treatment that for a fixed amplitude of the coherent state
$\xi=5$, the intensity of the FHG is larger than the SHG process. This
increase is attributed to the light-matter coupling strength

$\displaystyle
g_{\alpha,ij}^{(kl)}=\frac{e}{m}\sqrt{\frac{\hbar}{2\epsilon_{0}\omega_{\alpha}V}}\langle\varphi_{i}^{k}|\textbf{e}_{\alpha}\cdot\hat{\textbf{p}}|\varphi_{j}^{l}\rangle\,,$
(2)

where $|\varphi_{j}^{l}\rangle$ are the eigenstates of the matter system and
$\omega_{\alpha}$ in this case represents the fundamental frequency of the SHG
and FHG processes. Typically, in a strong light-matter coupling regime there
is an emergence of polariton states due to a resonant coupling of a photon
mode with an electronic resonance. In our investigation the contribution from
polariton states do not play a significant role since the transition
amplitudes involving these states are negligible as the fundamental frequency
is off-resonance for the respective harmonic generation processes. Clearly,
for a fixed volume $V$ the coupling $g_{\alpha,ij}^{(kl)}$ (and vector
potential) is greater for FHG than it is for the SHG process. The difference
in the light-matter coupling for these different processes accounts for the
FHG having an intensity larger than the SHG as shown in Fig. (1). It is
important to note that we cannot just increase the coupling strength arbitrary
by reducing the frequency to zero since there is a natural renormalization in
the lower frequency that arises as an intrinsic frequency cutoff Rokaj  _et
al._ (2022); Spohn (2004). This intrinsic cutoff which has a dependence on the
number of particles ensures that no infrared divergence appears. However,
unphysical divergences may arise if this natural renormalization of the
frequencies is not taken properly into account.

Figure 2: Intensity dependence of even-order harmonic generation where the
harmonic intensity increases for increasing light-matter coupling. Here
$g_{\alpha}$ increases by decreasing $V$ while $\omega_{\alpha}$ have values
for the pump frequencies for SHG and FHG. The intensity of FHG is larger when
compared to SHG due to stronger coupling $g_{\alpha}/\hbar\omega_{\alpha}$.

Another means by which the light-matter coupling can be increased for the
single emitter case here is by reducing the volume $V$ as normally done in
other theoretical works Flick  _et al._ (2019); Welakuh  _et al._ (2022);
Welakuh and Narang (2022b) and realized in experiments Barachati  _et al._
(2018). The controlled strong-coupling-induced symmetry breaking in the
coupled system is linked to changing the coupling strength by reducing the
cavity volume (see App. I). As the interaction between the photon field and
matter breaks the inversion symmetry for a chosen amplitude of the incident
field, a new possibility arises to control the degree of the induced asymmetry
by strongly coupling the system. This is accompanied by a significant increase
of the intensity of the induced harmonic generation process. To illustrate
this, we show the peak intensity dependence of the harmonic generation
processes on the dimensionless ratio
$g_{\alpha,ij}^{(kl)}/\hbar\omega_{\alpha}$ which provides an indication for
relative light-matter coupling strengths. In Fig. (2), for the same decreasing
values of $V$ in Eq. (2) (and increasing
$g_{\alpha,ij}^{(kl)}/\hbar\omega_{\alpha}$) for both processes, we find that
the peak intensities increase significantly for both SHG and FHG processes.
This result shows clearly how the SHG and FHG processes can be made more
efficient by increasing the light-matter coupling strength. An added merit of
the quantized treatment is that we can investigate the characteristics (photon
occupation and statistics) of the coherent pump field which indirectly
provides information on the non-classical properties of the generated
harmonics as discussed in App. II. It is important to note that in the
decoupling limit of the system, i.e., in the limit where $g_{\alpha}$ tends to
zero (free space coupling) for very large volumes, the inversion symmetry can
no longer be broken and the induced nonlinear optical processes do not occur.

Figure 3: The pump field dependence on the intensity of the induced even-order
harmonic generation processes in an inversion symmetric system for a quantized
treatment of the coupled system. The electric field has a quadratic (quartic)
dependence on the peak intensities of SHG (FHG).

In the next step, we verify that the even-order harmonic processes in Fig. (1)
are truly the SHG and FHG processes. To do this, we calculate the dependence
of their peak intensities on the incident field by varying the pump amplitude
$\xi$ from $1.4$ to $6.2$ with an equidistant spacing $\Delta\xi=0.4$ for a
fixed cavity volume. In Fig. (3), we show the peak intensity dependence of the
SHG (FHG) on the electric field of the pump mode which has a quadratic
(quartic) dependence, confirming that these processes are indeed the SHG and
FHG. Of particular interest is the dipole radiation pattern of a system whose
inversion symmetry has been broken (reduced), in this case, due to coupling to
a quantized pump field. For this, we calculate the polarization dependence
spectra of the even-order harmonic generation in Fig. (1). The polarization of
the incident field which was originally chosen such that the angle $0^{\circ}$
corresponds to the polarization direction along the $x$-axis is now varied. In
Fig. (4) we show a polar plot of the calculated peak intensity dependence on
the pump polarization where we find that the induced SHG and FHG is maximum
when the radiated field is parallel to the pump field. Also, as the pump and
harmonic processes are collinearly polarized, we obtain a radiation pattern
where the intensities varies as $I_{\text{SHG,FHG}}\propto\cos^{2}(\theta)$
where $\theta$ is the incident polarization angle. This is expected for a
linearly polarized pump field along a single direction (axis).

Figure 4: Polar plot of the polarization dependence of the induced even-order
harmonic generation intensities. The radiation pattern for different
polarization angles $\theta$ shows the intensities varies as
$I_{\text{SHG,FHG}}\propto\cos^{2}(\theta)$. The polar angle $\theta$ is
defined with respect to the $x$-axis of the quantum ring.

We now consider an important aspect of the harmonic generation which is the
nonlinear conversion efficiency. It is clear that the SHG and FHG processes
are efficient from a centrosymmetric system with broken symmetry as opposed to
the case where the symmetry remains intact since $\chi^{(n)}=0$ (for even
$n$). Beyond this observation, we show that our approach (which is the few-
photon limit and strong coupling) is favorable over the standard approach
which requires strong fields. Towards this, we consider the case of SHG where
the nonlinear conversion efficiency is defined as
$\eta_{\text{SHG}}=I_{2\omega}/I_{\omega}^{2}$ where $I_{\omega}$ is the peak
intensity of the fundamental frequency. To compare between the two methods, we
choose the pump amplitude $\xi=5$ (which gives an electric field strength
$E_{0}=18.54$ kV/m) and vary the coupling strength
$g_{\alpha}/\hbar\omega_{\alpha}$ for the strong coupling case and for the
strong field case, we fix the coupling
$g_{\alpha}/\hbar\omega_{\alpha}=0.0017$ and vary the electric field strength
by increasing $\xi$. The results of this comparison are shown in Fig. (5)
where we find the efficiency increases for the strong coupling case but
decreases for increasing electric field strengths of the pump. This result
highlights that we do not necessarily need an intense pump field for the
harmonic generation process but rather need strong light-matter coupling as
also demonstrated for the case of the down-conversion process Welakuh  _et
al._ (2021). It equally validates the long-standing goal in optical science to
implement nonlinear optical effects (here the harmonic generation) at
progressively lower pump powers or pulse energies Chang  _et al._ (2014).

Figure 5: (a) Induced SHG efficiency dependence on the coupling strength which
shows an increase for increasing $g_{\alpha}$. (b) A decrease in the
conversion efficiency of SHG for increasing pump field strengths which shows
the efficiency will eventually saturate. A comparison between (a) and (b)
indicates strong coupling is favorable over using strong fields.

To conclude, in this work we have demonstrated a general approach to break the
inversion symmetry of material systems, thus enabling induced and highly
tunable nonlinear optical processes such as SHG and FHG, typically physically
forbidden in centrosymmetric systems. Our approach relies on a quantum
mechanical treatment of the coupled light-matter system where the pump field
is treated as a quantized field in a coherent state. We demonstrate that the
inversion symmetry can be broken by strongly coupling the light-matter, thus
allowing for even-order harmonic generation processes to occur. In particular,
we show how the strong-coupling-induced symmetry breaking can be controlled by
increasing the light-matter coupling strength which simultaneously results to
efficient SHG and FHG. Such control is of great interest for several
applications ranging from optical frequency conversion to imaging and bio-
sensing where high efficiency is required Garmire (2013). Our choice of a
matter system with a single effective electron interacting with the photon
field was to illustrate in more detail the strong-coupling-induced symmetry
breaking that results to even-order processes like SHG and FHG in
centrosymmetric systems. In the case where we have an ensemble of emitters or
a general many-electron system, the light-matter coupling strength increases
by a square-root of the number of identical emitters Garraway (2011). In such
multi-electron systems, the efficiency and induced harmonic yield of the SHG
and FHG will increase significantly due to collective strong coupling. Such
many-electron systems interacting with the quantized electromagnetic field can
be treated with already existing first-principles methods for strongly coupled
light-matter systems Flick and Narang (2018); Svendsen  _et al._ (2021);
Haugland  _et al._ (2020); Jestädt  _et al._ (2020). Our approach is
applicable to a wide variety of material systems since the strong-coupling-
induced symmetry breaking relies primarily on the ability to tune properties
of the electromagnetic environment to attain strong light-matter interaction.
Further, this approach is not limited to optical cavities but applicable in
other photonic environments used to achieve strong light-matter coupling. We
note that strong-coupling induced symmetry breaking in centrosymmetric systems
allows for other nonlinear optical effects associated with even-order
susceptibilities other than the harmonic generation process. Finally, this
approach can be applied to reduce other symmetries of material systems to
control physical and chemical processes.

_Acknowledgments:_ We acknowledge helpful discussions with Nicolas Tancogne-
Dejean, Jonathan Curtis, Stephane Kena-Cohen, and Aaron Lindenberg. This work
is primarily supported through the Department of Energy BES QIS program on
‘Van der Waals Reprogrammable Quantum Simulator’ under award number DE-
SC0022277 for the work on nonlinearity in low dimensional systems, as well as
partially supported by the Quantum Science Center (QSC), a National Quantum
Information Science Research Center of the U.S. Department of Energy (DOE) on
cavity-control of quantum matter. 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.

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Supplemental Information:
Nonlinear optical processes in centrosymmetric systems by strong-coupling-
induced symmetry breaking

Davis M. Welakuh and Prineha Narang

Harvard John A. Paulson School Of Engineering And Applied Sciences, Harvard
University, Cambridge, Massachusetts 02138, USA

##  I 2D semiconductor quantum ring

The matter system is a 2D semiconductor quantum ring of finite width that
features a single effective electron confined in two-dimensions in real-space
($\hat{\textbf{r}}=\hat{x}\textbf{e}_{x}+\hat{y}\textbf{e}_{y}$). The system
is described by the Hamiltonian

$\hat{H}_{\text{M}}=-\frac{\hbar^{2}}{2m}\left(\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial
y^{2}}\right)+\underbrace{\frac{1}{2}m\omega_{0}^{2}\hat{\textbf{r}}^{2}+V_{0}e^{-\hat{\textbf{r}}^{2}/d^{2}}}_{v_{\text{ext}}(\textbf{r})},$
(S1)

where the potential part with $\hat{\textbf{r}}^{2}=\hat{x}^{2}+\hat{y}^{2}$
introduces a parabolic confinement including a Gaussian peak located at the
center as shown in Fig. (S1 a). The parameters of the potential are chosen
such that it reflects the energy and length scales of a semiconductor quantum
ring of GaAs as used in experiments Fuhrer  _et al._ (2001); Ihn  _et al._
(2005) where $\hbar\omega_{0}=10$ meV, $d=10$ nm, $m=0.067m_{e}$, and
$V_{0}=200$ meV. In this regard, we work in scaled effective atomic units:
$\text{Ha}^{*}=(m/\epsilon^{2})\text{Ha}\approx 11.30$ meV,
$a_{B}^{*}=(m/\epsilon)a_{0}\approx 10.03$ nm, and
$u_{t}^{*}=\hbar/\text{Ha}^{*}\approx 58.23$ fs where we made use of the
dielectric constant $\epsilon=12.7\epsilon_{0}$ of GaAs Räsänen  _et al._
(2007). The eigenstates $|\varphi_{j}^{l}\rangle$ are labeled by the angular
momentum $l=0,\pm 1,\pm 2,\pm 3,...\,$ and the index $j=|l|+1$ enumerates over
the energy levels (see Fig. (S2)). Due to symmetry of the system, dipole-
allowed transitions can only occur between states with consecutive angular
momenta Räsänen  _et al._ (2007); Flick  _et al._ (2015); Welakuh  _et al._
(2021).

Figure S1: (a) Real-space 2D potential of the quantum ring with a Gaussian
peak at the center. The potential is symmetric with respect to its center of
inversion. (b) Comparison of the bare quantum ring potential to the case when
coupled to a quantized field along the $x$-direction for decreasing cavity
volumes (increasing coupling strength).

To understand how the photon mode breaks the symmetry, we consider the
electron-photon system in the unitary equivalent length gauge Rokaj  _et al._
(2018). For the quantum ring coupled to one mode, the Hamiltonian has the
form:

$\displaystyle\hat{H}_{\text{L}}$
$\displaystyle=-\frac{\hbar^{2}}{2m}\left(\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial
y^{2}}\right)+\frac{1}{2}\left(\hat{p}^{2}+\omega^{2}\hat{q}^{2}\right)$
$\displaystyle\quad+\underbrace{\frac{1}{2}m\omega_{0}^{2}\hat{\textbf{r}}^{2}+V_{0}e^{-\hat{\textbf{r}}^{2}/d^{2}}-e\,\omega\,\hat{q}\,\boldsymbol{\lambda}\cdot\hat{\textbf{r}}+\frac{1}{2}\left(e\,\boldsymbol{\lambda}\cdot\hat{\textbf{r}}\right)^{2}}_{v_{\text{ext}}^{(ph)}(\textbf{r})}\,.$
(S2)

The dipole self-energy (last term) adds to the harmonic binding potential
(i.e. $\frac{1}{2}m\omega_{0}^{2}\hat{\textbf{r}}^{2}$) and the bilinear
interaction term (i.e.
$e\,\omega\,\hat{q}\,\boldsymbol{\lambda}\cdot\hat{\textbf{r}}$) breaks the
symmetry of the quantum ring for a specific polarization of the cavity mode.
This is clear since $v_{\text{ext}}^{(ph)}(-\textbf{r})\neq
v_{\text{ext}}^{(ph)}(\textbf{r})$. Besides breaking the symmetry, there is
the possibility to control the degree of asymmetry by reducing the cavity
volume. In Fig. (S1 b), we show how to control the asymmetry by changing the
cavity volume $V$.

Figure S2: Energy spectrum of the 2D quantum ring showing the two lowest
radial bands with degenerate and nondegenerate excited states and a
nondegenerate ground state for different angular momentum $l$. The blue and
red arrows indicate the SHG and FHG which yields a harmonic field (purple)
which is a multiple of their respective pump energies.

##  II Light-matter coupling strength and the quantized pump field

Provided we can obtain all electronic eigenenergies $E_{i}$ and eigenstates
$|\psi_{i}\rangle$ of a matter system, for example through numerical exact
diagonalization, we can write velocity gauge Hamiltonian Eq. 1 as:

$\displaystyle\hat{H}$
$\displaystyle=\sum\limits_{i=1}\hbar\omega_{i}|\psi_{i}\rangle\langle\psi_{i}|+\sum_{\alpha=1}^{M}\frac{1}{2}\left(\hat{p}_{\alpha}^{2}+\omega_{\alpha}^{2}\hat{q}_{\alpha}^{2}\right)$
$\displaystyle\quad-\sum_{i,j=1}\sum_{\alpha=1}^{M}g_{\alpha}^{(ij)}\left(\hat{a}_{\alpha}+a_{\alpha}^{\dagger}\right)|\psi_{i}\rangle\langle\psi_{j}|+\frac{e^{2}}{2m}\hat{\textbf{A}}^{2}\,.$
(S3)

The the electron-photon coupling term in the above equation is defined to be

$\displaystyle
g_{\alpha}^{(ij)}=\frac{e}{m}\sqrt{\frac{\hbar}{2\epsilon_{0}\omega_{\alpha}V}}\langle\psi_{i}|\textbf{e}_{\alpha}\cdot\hat{\textbf{p}}|\psi_{j}\rangle\,,$
(S4)

Here, we consider a linearly polarized field with polarization
$\textbf{e}_{\alpha}=\cos\theta\textbf{e}_{x}+\sin\theta\textbf{e}_{y}$. From
the operator of the quantized vector potential

$\displaystyle\hat{\textbf{A}}$
$\displaystyle=\sum_{\alpha=1}^{M}\sqrt{\frac{\hbar}{2\epsilon_{0}\omega_{\alpha}V}}\left(\hat{a}_{\alpha}+\hat{a}_{\alpha}^{\dagger}\right)\,\textbf{e}_{\alpha}\,,$
(S5)

we obtain the vector potential at the initial time by taking the expectation
with the coherent state:

$\displaystyle\langle\xi|\hat{\textbf{A}}|\xi\rangle$
$\displaystyle=\sum_{\alpha=1}^{M}\sqrt{\frac{\hbar}{2\epsilon_{0}\omega_{\alpha}V}}\,2|\xi|\cos\phi\,\textbf{e}_{\alpha}\,.$
(S6)

The coherent state has the form
$|\xi\rangle=e^{-|\xi|^{2}/2}\sum_{n=0}^{\infty}\left(\xi^{n}/\sqrt{n!}\right)|n\rangle$
where $\xi=|\xi|e^{i\phi}$ with $|\xi|$ being the amplitude, $\phi$ is the
phase and $|n\rangle$ the Fock number state of the photon mode. At the initial
time, there are $n=\langle\xi|\hat{a}^{\dagger}\hat{a}|\xi\rangle=|\xi|^{2}$
number of photons in the pump field.

Figure S3: The time-evolved Mandel $Q$ parameter for (a) SHG and (b) FHG for
an amplitude $\xi=2$. The photon statistics of SHG varies between a bunching
and anti-bunching character. FHG shows mainly super-Poissonian statistics.

Since the pump field is treated as a quantized field, we can investigate its
photon statistics. We determined this property by computing the Mandel
$Q_{\alpha}$ parameter Mandel (1979) defined as

$\displaystyle
Q_{\alpha}=\frac{\langle\hat{a}_{\alpha}^{\dagger}\hat{a}_{\alpha}^{\dagger}\hat{a}_{\alpha}\hat{a}_{\alpha}\rangle-\langle\hat{a}_{\alpha}^{\dagger}\hat{a}_{\alpha}\rangle^{2}}{\langle\hat{a}_{\alpha}^{\dagger}\hat{a}_{\alpha}\rangle}\,,$
(S7)

where $\hat{a}_{\alpha}^{\dagger}$ and $\hat{a}_{\alpha}$ are the creation and
annilation operators of the $\alpha$ mode. The parameter $Q_{\alpha}$ measures
the deviation of the photon statistics from a Poisson distribution and thus is
a measure for the non-classicality. A field with non-classical properties have
a range of values between $-1\leq Q_{\alpha}<0$ which corresponds to sub-
Poissonian statistics (anti-bunching behavior). Fields with super-Poissonian
statistics (bunching behavior) have $Q_{\alpha}>0$ and for a coherent state
with Poissonian statistics we have $Q_{\alpha}=0$ Loudon (2000). Figure (S3)
shows super-Poissonian statistics for FHG since it couples weakly and for SHG,
it shows statistics that vary between a bunched and antibunhed field. This is
due to the coherent exchange of energy between the pump mode and the matter
system. From the non-classical characteristic of the coherent pump field, we
infer that the generated second-harmonic will have similar features as the
coupled system coherently interacts and exchange energy. Our strongly coupled
light-matter treatment brings to light new observables that are not usually
considered in SHG or FHG since the pump field is treated classically.
Furthermore, it highlights the possibility where the generated harmonics have
non-classical signatures which is important for applications in quantum-
information-processing protocols O’Brien  _et al._ (2009).

##  III Strong fields versus strong coupling

In this section, we investigate the outcome of the harmonic generation process
when the system is excited with a strong intense field versus a few-photon
field in the strong coupling regime. To do this, we perform two simulations
where the case involving the few-photon field is treated as describe above and
the case involving a strong intense field is treated with a semi-classical
description. This means that in Eq. (1) the contribution of the
electromagnetic energy (last term) is dropped out and the vector potential is
simply a time-dependent external field that perturbs the system which has the
following form

$\displaystyle\textbf{A}(t)$
$\displaystyle=\frac{\text{E}_{0}}{\omega}\cos(\omega
t)\sin^{2}\left(\pi\frac{t}{\tau}\right)\textbf{e}_{x}\,,\quad 0\leq
t\leq\tau\,,$ (S8)

where $\omega$ is the fundamental frequency, $E_{0}$ is the amplitude. As we
work in effective atomic units, it is important to present the rescaled
quantities. The electric field is
$E^{*}=E_{h}^{*}/e\,a_{0}^{*}=((0.067)^{2}/(12.7)^{3})(E_{h}/e\,a_{0})$ which
gives an intensity $I^{*}=1/2\,c\,\epsilon E^{*2}=((0.067)^{4}/(12.7)^{5})\,I$
where $(E_{h}/e\,a_{0})=5.1422\times 10^{11}$ V/m and $I=3.51\times 10^{16}$
W/cm2. For a strong field, we use an intensity $I_{0}=2.74\times 10^{4}$ W/cm2
($E_{0}=1.27\times 10^{5}$ V/m) which corresponds to a typical intensity
needed for pronounced dipole oscillations for quantum rings Fomin (2018). For
the few-photon case for strong coupling, the electric field is
$E_{0}=7.41\times 10^{3}$ V/m where $\xi=2$.

Figure S4: Strong field versus strong coupling for the case of THG which
yields the same harmonic intensity. The strong field case has an electric
field $E_{0}=1.27\times 10^{5}$ V/m while the strong coupling (with
$g_{\alpha}/\hbar\omega_{\alpha}=0.0017$) case has an electric field
$E_{0}=7.41\times 10^{3}$ V/m where $\xi=2$.

To make a concrete comparison for both cases, we investigate the third-
harmonic generation (THG) which does not depend on the symmetry of the
material system. This at the same time shows that our method is applicable for
noncentrosymmetric systems where THG is present. To simulate this nonlinear
optical process, the pump energy is one-third the transition energy (24.65
meV). In Fig. (S4), we show the strong field versus strong coupling for the
case of THG. By comparing the two approaches, we find that for the strong
field case with electric field $E_{0}=1.27\times 10^{5}$ V/m and strong
coupling (with $g_{\alpha}/\hbar\omega_{\alpha}=0.0017$) case with electric
field $E_{0}=7.41\times 10^{3}$ V/m where $\xi=2$, we obtain the same harmonic
intensity for both approaches. Yet, the electric field (and intensity) for the
strong field case is larger than that for the strong coupling case with just 4
photons in the pump field at the initial time. This highlights that strong
coupling in the few-photon limit is a suitable approach to achieve an
efficient nonlinear optical process.

##  IV Numerical Details

The numerical details to treat the matter-photon coupled system is outline
here. The matter Hamiltonian of Eq. (S1) is represented on a two-dimensional
uniform real-space grid of $N_{x}=N_{y}=127$ grid points (implying $127^{2}$
states are considered) with spacing $\Delta x=\Delta y=0.7052$ nm while
applying an eighth-order finite-difference scheme for the momentum operator
and Laplacian. The photon mode that represent the pump field is represented in
a basis of Fock number states. To represent the coherent state with
$n(t_{0})=|5|^{2}$ photons, we include 65 Fock number states. The combined
electron-photon space has dimensions $127\times 127\times 65$ and we
explicitly construct matrix representations for all operators in the
Hamiltonian and observables of interest. To solve the time-dependent
Schrödinger equation of the coupled system, we use a Lanczos propagation
scheme Hochbruck and Lubich (1997). We compute expectation values of the
observables for every time step $\Delta t=0.0012$ ps of the time-evolved wave
function.

The harmonic spectrum of the coupled system is calculated from the accelerated
dipole response. For a laser-driven system, the experimentally measured
harmonic spectrum is proportional to the modulus squared of the Fourier
transformed laser-induced dipole acceleration Schafer and Kulander (1997);
Sundaram and Milonni (1990):

$\displaystyle
H(\omega)=\left|\int_{0}^{T}dt\frac{d^{2}}{dt^{2}}\textbf{r}(t)\,e^{-i\omega
t}\right|^{2}.$ (S9)