# Microwave-optical spectroscopy of Rydberg excitons in the ultrastrong
driving regime

Alistair Brewin1, Liam A P Gallagher1, Jon D Pritchett1, Horatio Q X Wong1,
Robert M Potvliege1, Stewart J Clark1, Matthew P A Jones1
<EMAIL_ADDRESS>1Department of Physics, Durham University, Durham DH1
3LE, United Kingdom

(August 28, 2024)

###### Abstract

We study the ultrastrong driving of Rydberg excitons in Cu2O by a microwave
field. The effect of the microwaves was studied using optical absorption
specstroscopy, and via the observation of sidebands on the transmitted laser
light. A model based on Floquet theory was constructed to explore the system
beyond the rotating wave approximation. We obtain near quantitative agreement
between theory and experiment across a 16-fold range of microwave field
strength, from $43.5\pm 0.3$ V m-1 up to $688\pm 5$ V m-1, crossing from the
perturbative to the ultrastrong driving regime. Compared to Rydberg atoms, the
large non-radiative width of Rydberg exctions leads to new behaviour such as
the emergence of an absoprtion continuum without ionization.

††: New J. Phys.

##  1 Introduction

Strong coupling between light and matter is a significant topic in quantum
physics [1], with applications from quantum communication [2, 3, 4] and
simulation [5, 6] to quantum chemistry [7, 8]. Typically, high quality
cavities are used to couple single photons to two-level quantum systems, such
as atoms or artificial analogues [4, 9]. Closely related is the topic of
strong and ultrastrong driving, where large classical fields are used to drive
quantum systems, resulting in non-perturbative [10] optical nonlinearities
such as high harmonic generation [11].

The strong (coupling) driving regime is commonly defined to be when
$\nu_{0}>\Omega>\Gamma$ [8, 12], where $\hbar\nu_{0}$ is the bare energy gap
between the excited and ground state, $\Omega$ is the (vacuum) Rabi frequency
and $\Gamma$ is the relevant rate of energy loss to the environment. In this
limit, nonlinear effects such as saturation and power broadening are observed
[13], while in the cavity case, a photon may be exchanged many times between
the cavity and the two-level system before being lost [14].

In contrast, ultrastrong (also called deep strong) coupling/driving (USD)
occurs when the coupling to the electromagnetic field is increased further to
become the largest energy scale i.e when $\Omega>\nu_{0},\Gamma$ [8, 12]. New
effects occur in this extreme limit [1, 12]. In particular the commonly used
rotating-wave approximation (RWA) no longer holds, and complex dynamics take
over instead of Rabi oscillations [15]. This regime has been observed in
systems where large electric dipole moments coincide with small energy level
splittings, including Rydberg atoms [16], superconducting resonators coupled
to artificial atoms [17] and optomechanical devices [18]. As techniques for
coupling quantum systems together improve, understanding the USD regime holds
great promise including better quantum computation [19, 20], control of
material properties [7, 8, 21] and microwave-optical conversion [22, 23].

In Rydberg atoms, the electric dipole moment associated with transitions
between neighbouring opposite-parity states scales with principal quantum
number $n$ as $n^{2}$, while the separation scales as $n^{-3}$ providing an
excellent platform for realising ultrastrong driving [24, 25, 16]. More
recently, highly excited Rydberg states of excitons (bound states of an
electron and hole) have also been observed in semiconductor materials [26]. In
cuprous oxide, Cu2O, Rydberg excitons have been observed up to $n=30$ [27,
28], with wavefunctions spanning up to $1$ $\mu$m in real space, sparking a
great deal of research into the material [29, 30, 31]. Previous work by our
group has shown that strong microwave electric dipole transitions between
neighbouring states also exist for Rydberg excitons [32], culminating in the
observation of a record microwave-optical Kerr nonlinearity [33]. In these
experiments the microwave field strength was weak, enabling a quantitative
description using a perturbative approach [33].

In this paper we study the ultrastrong microwave driving of Rydberg excitons
in Cu2O. We measure the changes in the optical absorption spectrum and the
intensity of the microwave-induced sidebands over a range of $\sim 50$ in
microwave field strength, and develop a theoretical model to understand our
results. We use non-Hermitian Floquet theory [34] to go beyond the RWA, and
calculate the frequency-dependent susceptibility of the medium in the steady
state limit. The predictions of the model are compared successfully with the
experimental results up to $\Omega/\nu_{0}\sim 4$, deep in the ultrastrong
coupling regime (at $n=11$). The model allows us to gain insight into workings
of the system which are inaccessible to measurement. Crucially, the observed
behaviour is distinct from its atomic counterpart [16] because the non-
radiative broadening of the exciton states allows many excitonic states to be
strongly coupled to a single microwave frequency. The agreement between
experiment and theory opens exciting avenues for realising excitonic
microwave-to-optical conversion with strong fields.

Figure 1: (a) Schematic of the experiment. A laser at optical frequency
$\omega_{\text{p}}$ is incident on the sample, creating excitons in $n$P
states. A stripline antenna driven at microwave frequency $\omega_{\text{m}}$
generates a strong microwave field in the sample which is cooled to $\sim$ 4
K. A Fabry-Pérot etalon (FPE) may be inserted to resolve the spectrum of the
transmitted light. (b) Diagram showing the states involved in the model and
their couplings. Dashed arrows denote examples of the laser field coupling the
valence band to $n$P states, and solid arrows denote examples of the microwave
field coupling exciton states that differ in angular momentum by $1$. Here
$nl$ labels the states in the same way as in Hydrogen.

##  2 The Experiment

Details of the experimental setup are available in [33] and will be summarized
here. Figure 1 shows a diagram of the experiment. A thin slab of Cu2O ($55\pm
10\leavevmode\nobreak\ $\mathrm{\SIUnitSymbolMicro m}$$ thick) is mounted on a
CaF2 window inside a closed-cycle crysotat at $\sim 4$ K. The probe laser was
a narrowband diode laser, frequency doubled to $\lambda\approx 571$ nm
(details in [35]). The laser intensity on the sample was approximately 20
$\mathrm{\SIUnitSymbolMicro W}\text{\,}{\mathrm{mm}}^{-2}$ which is
approximately 2 times lower than the probe intensity used in [36], allowing
interactions between excitons to be neglected [27, 36]. A microwave field was
applied at a frequency of 7 GHz using a stripline antenna [32]. This leads to
two effects: a change in the absorption spectrum, and the generation of
sidebands on the laser [33]. The probe absorption data was taken with a
polarisation angle between the probe laser and the microwave field of
$90^{\circ}$, while the sideband data was taken with the polarisation of the
two fields parallel.

Absorption spectra were taken by measuring the light transmitted through the
sample, $T(\omega_{\mathrm{p}}$), and comparing it to a reference value,
$T_{0}$, to give the optical depth,
$-\mathrm{ln}(T(\omega_{\mathrm{p}})/T_{0})$, as a function of probe energy
(figure 2a). Experimentally the Rydberg states are observed to sit on a non-
resonant absorption background due to phonon-assisted processes [37]. As this
background is not affected by the microwave field, it was not included in the
model and was subtracted from the experimental data (see A for details). To
extract the size of the sidebands, a temperature tuned Fabry-Perot etalon with
a finesse of $44.5\pm 0.7$ and a free spectral range of $60.1\pm 0.2$ was
inserted after the sample. Details of how the sideband amplitude was extracted
from this data can be found in chapter 5 of [38].

Figure 2: Plots of the (a) experimental and (b) theoretical absorption
spectrum for the excitons at different microwave field strengths,
$\mathcal{E}_{\text{m}}$. Each spectrum has been offset in optical depth by
$\mathcal{E}_{\text{m}}/100$ to display the microwave field strength it was
taken at, shown on the righthand y-axis. The dashed vertical lines show the
positions of the zero-field $n$P resonances.

##  3 The Model

The calculations exploit the general non-Hermitian Floquet theory of linear
absorption spectroscopy recently outlined in [34]. The Hamiltonian of our
model can be thought of as the sum of a field-free Hamiltonian (figure 1b:
solid horizontal lines), a coupling term between the valence band and the
exciton states (figure 1b: dashed arrows), and another between pairs of
exciton states (figure 1b: solid arrows). This can be written, respectively,
as

$\displaystyle\hat{H}(t)=$
$\displaystyle\,\hbar\sum_{j=1}^{\mathcal{N}}\omega^{(j)}\ket{j}\bra{j}$
$\displaystyle-\frac{\hbar}{2}\sum_{k=1}^{\mathcal{N}}\Big{[}\Big{(}\Omega_{0k}\exp{-i\omega_{\text{p}}t}+\Omega_{k0}^{*}\exp{i\omega_{\text{p}}t}\Big{)}\ket{0}\bra{k}+\text{h.c.}\Big{]}$
$\displaystyle-\frac{\hbar}{2}\sum_{j,k=1}^{\mathcal{N}}\Big{[}\Big{(}\Omega_{jk}\exp{-i\omega_{\text{m}}t}+\Omega_{kj}^{*}\exp{i\omega_{\text{m}}t}\Big{)}\ket{j}\bra{k}\Big{]},$
(1)

or, making the rotating wave approximation for the probe field after
transformation to slowly rotating coordinates by the unitary transformation
$\hat{U}=\ket{0}\bra{0}+\sum_{j\not=0}\exp{i\omega_{\text{p}}t}\ket{j}\bra{j}$,

$\displaystyle\hat{H}_{\rm tr}(t)=$
$\displaystyle-\hbar\sum_{j=1}^{\mathcal{N}}\Big{(}\omega_{\rm
P}-\omega^{(j)}\Big{)}\ket{j}\bra{j}$
$\displaystyle-\frac{\hbar}{2}\sum_{k=1}^{\mathcal{N}}\Big{[}\Omega_{k0}^{*}\ket{0}\bra{k}+\text{h.c.}\Big{]}$
$\displaystyle-\frac{\hbar}{2}\sum_{j,k=1}^{\mathcal{N}}\Big{[}\Big{(}\Omega_{jk}\exp{-i\omega_{\text{m}}t}+\Omega_{kj}^{*}\exp{i\omega_{\text{m}}t}\Big{)}\ket{j}\bra{k}\Big{]}.$
(2)

Here, $\ket{j}$ is an excitonic state with energy $\hbar\omega^{(j)}$
($\ket{0}$ being the ground state representing the ‘valence band’, with
$\hbar\omega^{(0)}=0$) and the abbreviation ‘h.c.’ stands for Hermitian
conjugate. $\mathcal{N}$ is the number of excitonic states we include in the
calculation. The Rabi frequencies between states $\ket{j}$ and $\ket{k}$ are

$\displaystyle\Omega_{0k}=\mathcal{E}_{\text{p}}\bra{0}\hat{D}^{\text{p}}_{z}\ket{k}/\hbar,$
(3)
$\displaystyle\Omega_{jk>0}=\mathcal{E}_{\text{m}}\bra{j}\hat{D}^{\text{m}}_{z}\ket{k}/\hbar.$
(4)

The amplitudes of the laser and microwave electric fields are given by
$\mathcal{E}_{\text{p}}$ and $\mathcal{E}_{\text{m}}$ respectively, and their
frequencies are $\omega_{\text{p}}$ and $\omega_{\text{m}}$, respectively.
Since the linearly polarized probe laser and microwave fields define two
different z-axes, $\hat{D}^{\text{p}}_{z}$ and $\hat{D}^{\text{m}}_{z}$ denote
the $z$-component of the dipole operator in each coordinate system.

The couplings to the valence band, $\Omega_{0k}$, are derived from experiment,
detailed in B. The uppermost valence band of Cu2O has even parity, so due to
dipole selection rules, only odd parity states (and in particular, we assume
only those with $m=0$ in the probe basis) directly couple to it. Since the F
states’ coupling to the valence band is several orders of magnitude weaker
than that of the P states [39], they are difficult to measure accurately, and
so are not included in this model. To evaluate the dipole matrix elements
$\bra{j}\hat{D}^{\text{m}}_{z}\ket{k}$, we need wavefunctions for the exciton
states. Rydberg excitons are commonly approximated by the Mott-Wannier model
[40], a hydrogen-like model, with the effective masses of the electron and the
hole derived from the dispersion of the conduction and valence bands as
calculated with density functional theory [41]. As detailed in C, the
dependence on the relative polarisation between the probe laser and microwave
field predicted by this model differs significantly from experiment. Therefore
to keep a self-consistent theory that is compatible with experiment, we make
the following approximation:

$\Omega_{jk>0}=\begin{cases}-e\mathcal{E}_{\text{m}}\bra{R^{n_{j}}_{l_{j}}}r\ket{R^{n_{k}}_{l_{k}}},&\quad\text{if
}|l_{j}-l_{k}|=1,\\\ 0,&\quad\text{otherwise,}\end{cases}$ (5)

which gives good agreement between the model and data. To keep with
convention, we still label our states $\ket{n,l}$, even they are not the
eigenstates of the $L^{2}$ operator. In addition, to explain the
experimentally observed breaking of degeneracy of the $l$ states, we include
an $l$-dependent quantum defect [42] in the exciton energies. Each different
set of angular momenta $l$ then follows its own Rydberg series,
$E_{g}-R_{y}/(n-\delta_{l})^{2}$, sharing a Rydberg constant $R_{y}$ but
offset by a quantum defect $\delta_{l}$ [35].

The excitonic states have a non-negligible energy width arising from their
non-radiative decay to a state outside our basis. Taking this decay into
account would normally require solving the Lindblad master equation for the
relevant populations and coherences. However, since our probe is weak, we can
take the population of the valence band (state $\ket{0}$) to be 1 at all
times, and the populations of all the excited states to remain essentially 0,
and calculate the coherences only to first order in $\mathcal{E}_{\text{p}}$.
It can be shown [34] that within this weak probe approximation, the full
Lindblad master equation [43] simplifies to the following set of linear
equations for the coherences $\rho_{j0}(t)$:

$\frac{d\rho_{j0}}{dt}=\Big{(}i\omega_{\text{p}}-i\omega^{(j)}+\frac{\Gamma^{(j)}}{2}\Big{)}\rho_{j0}+\frac{i}{2}\Omega_{j0}+\frac{i}{2}\sum_{k=1}^{\mathcal{N}}\Big{(}\Omega_{jk}\exp{-i\omega_{\text{m}}t}+\Omega^{*}_{kj}\exp{i\omega_{\text{m}}t}\Big{)}\rho_{k0},$
(6)

where $\Gamma^{(j)}$ is the decay rate of state $\ket{j}$. The same result
would also be obtained within the weak probe approximation if these coherences
were calculated by replacing the real energies $\hbar\omega^{(j)}$ by the
complex energies $\hbar[\omega^{(j)}-i\Gamma^{(j)}/2]$ in the Hamiltonian
$H_{\rm tr}(t)$ [34] and solving the von Neumann equation rather than the
Lindblad master equation.

Accordingly, we can write the coherences $\rho_{j0}(t)$ as $c_{j}(t)c_{0}^{*}$
with $c_{0}\equiv 1$, define the state
$\ket{\Psi(t)}=\sum_{j}c_{j}(t)\ket{j}$, and write down the equations of
motion of the coefficients $c_{j}(t)$ using the Schrödinger equation for the
transformed Hamiltonian. They read

$\frac{dc_{j}}{dt}=\Big{(}i\omega_{\text{p}}-i\omega^{(j)}+\frac{\Gamma^{(j)}}{2}\Big{)}c_{j}+\frac{i}{2}\Omega_{j0}+\frac{i}{2}\sum_{k=1}^{\mathcal{N}}\Big{(}\Omega_{jk}\exp{-i\omega_{\text{m}}t}+\Omega^{*}_{kj}\exp{i\omega_{\text{m}}t}\Big{)}c_{k}.$
(7)

We do not include an equation for the population of the valence state,
$c_{0}$, here, since we assume that $c_{0}\equiv 1$. Solving this system of
equations shows that in the long time limit the system evolves into a steady
state where the coefficients $c_{j}(t)$ oscillate a multiples of the microwave
frequency. Accordingly, we look for the Floquet-like solutions

$c_{j}=\sum_{N=-\infty}^{\infty}c_{j;N}\exp{-iN\omega_{\text{m}}t}.$ (8)

where the coefficients $c_{j;N}$ are constant in time and $N$ is the Floquet
number. Equating terms with equal frequency in equation 7 we find

$\Big{(}\omega_{\text{p}}-\omega^{(j)}+i\frac{\Gamma_{j}}{2}+N\omega_{c}\Big{)}c_{j;N}+\frac{1}{2}\sum_{k=1}^{\mathcal{N}}\big{[}\Omega^{*}_{kj}c_{k;N+1}+\Omega_{jk}c_{k;N-1}\big{]}=-\frac{1}{2}\Omega_{j0}\delta_{N,0},\text{
}j=1,...,\mathcal{N}.$ (9)

These equations can be solved numerically, with finite $N$, for the steady-
state Floquet components of the wavefunction. All the parameters in these
equations can be derived from experiment or the Mott-Wannier theory, as
detailed in B.

To extract the susceptibility of the medium, we need the polarisation,

$P(x,t)=n_{d}\bra{\Psi}\hat{D}^{\text{p}}_{z}\ket{\Psi},$ (10)

where $n_{d}$ is the effective density of exciton states. Expanding equation
10 in the manner of equation 8, considering that only terms oscillating at
frequency $\omega$ contribute significantly to $\chi(\omega)$, the
susceptibility can be shown to take the form

$\chi(\omega_{\text{p}}+N\omega_{M})=\frac{n_{d}}{\epsilon_{0}\mathcal{E}_{\text{p}}}\sum_{j=1}^{\mathcal{N}}\bra{0}\hat{D}^{\text{p}}_{z}\ket{j}c_{j;N}.$
(11)

For comparison with experiment, in the weak probe limit, the Beer-Lambert law,
$I(x)=\exp{-\alpha x}I(x=0)$, gives the absorption of the laser at a distance
$x$ into the medium for absorption coefficient $\alpha$. This is calculated
from the susceptibility by

$\alpha(\omega_{\text{p}})=2\frac{\omega_{\text{p}}}{c}\text{Im}\Big{[}\sqrt{1+\chi(\omega_{\text{p}})}\Big{]},$
(12)

where $c$ is the speed of light in vacuum.

At the probe frequency, we study the absorption of the probe laser, but the
medium also generates fields at frequencies not present in the incident light.
Consequently, sidebands on the laser are generated with intensity

$I(\omega_{\text{p}}+N\omega_{\text{m}})=\eta\big{|}\chi(\omega_{\text{p}}+N\omega_{M})\big{|}^{2},$
(13)

for $N\neq 0$, and $\eta$ is a constant of proportionality which depends on
the length of the crystal, the frequency of the light, the refractive index of
Cu2O, and the collection efficiency of the sideband light. Due to a lack of
knowledge about collection efficiencies, we do not have access to $\eta$, so
it is fit from data (see results).

To calculate the contribution from individual Floquet states, we first recast
equation 9 in terms of vectors and matrices. With $F$ as the Floquet
Hamiltonian (to within a factor of $\hbar$), one can see that

$(\omega_{\text{p}}I-F)\boldsymbol{c}=\boldsymbol{d},$ (14)

where the components of the vectors $\boldsymbol{c}$ and $\boldsymbol{d}$ are
$c_{j;N}$ and $-\Omega_{j0}\delta_{N,0}/2$, respectively. Assuming that $F$
has a complete set of eigenvectors (i.e., that $F$ is not defective),

$F=\sum_{q}\frac{E_{q}}{\hbar}\boldsymbol{R}_{q}\boldsymbol{L}_{q}^{\text{\textdagger}}$
(15)

and

$I=\sum_{q}\boldsymbol{R}_{q}\boldsymbol{L}_{q}^{\text{\textdagger}},$ (16)

where $E_{q}$ are the (complex) eigenenergies of $F$ and $\boldsymbol{L}_{q}$
and $\boldsymbol{R}_{q}$ are the associated left and right eigenvectors of
$F$, normalised such that
$\boldsymbol{L}_{q}^{\dagger}\boldsymbol{R}_{q^{\prime}}=\delta_{qq^{\prime}}$.
($\boldsymbol{L}_{q}^{\dagger}\boldsymbol{R}_{q^{\prime}}$ is the inner
product of the column vectors $\boldsymbol{L}_{q}$ and $\boldsymbol{R}_{q}$,
i.e., the matrix product of the row vector $\boldsymbol{L}_{q}^{\dagger}$ with
the column vector $\boldsymbol{R}_{q}$, which can also be written as the dot
product $\boldsymbol{L}_{q}^{*}\cdot\boldsymbol{R}_{q^{\prime}}$;
$\boldsymbol{R}_{q}\boldsymbol{L}_{q}^{\text{\textdagger}}$ is their outer
product.) The left and right eigenvectors of $F$ need not be the identical
since $F$ is not Hermitian, but they do share eigenvalues. The assumption that
$F$ is not defective is in general reasonable given that this matrix is
Hermitian when the widths $\Gamma^{(j)}$ are all zero. We have verified
numerically that the coefficients obtained from equation 18 below are
identical to those obtained by solving equation 9, which justifies this
assumption in the present case. Now, substituting into equation 14 and left-
multiplying by $\boldsymbol{L}_{q^{\prime}}^{\text{\textdagger}}$ we find

$\boldsymbol{L}_{q^{\prime}}^{\dagger}\boldsymbol{c}=\frac{\boldsymbol{L}_{q^{\prime}}^{\dagger}\boldsymbol{d}}{(\omega_{\text{p}}-E_{q^{\prime}}/\hbar)}.$
(17)

Finally, left-multiplying by $\boldsymbol{R}_{q^{\prime}}$ and summing over
$q^{\prime}$, we see that the steady-state coefficients decompose into Floquet
eigenstates as

$\boldsymbol{c}=\sum_{q}\boldsymbol{R}_{q}\frac{\boldsymbol{L}_{q}^{\dagger}\boldsymbol{d}}{(\omega_{\text{p}}-E_{q}/\hbar)},$
(18)

where we relabel $q^{\prime}\rightarrow q$. Substituting this result into
equation 11 and reordering the sums reveals that

$\chi(\omega_{\text{p}}+N\omega_{\text{m}})=\frac{-n_{d}}{2\epsilon_{0}\hbar}\sum_{q}\frac{1}{(\omega_{\text{p}}-E_{q}/\hbar)}\sum_{j,k=1}^{\mathcal{N}}R_{j;N}^{(q)}\big{[}L_{k;0}^{(q)}\big{]}^{*}\bra{0}\hat{D}^{\text{p}}_{z}\ket{j}\bra{k}\hat{D}^{\text{p}}_{z}\ket{0},$
(19)

where $L_{j;N}^{(q)}$ and $R_{j;N}^{(q)}$ are the components of the left and
right eigenvectors of $F$ labelled in the $\ket{j,N}$ basis.

##  4 Microwave Field Strength Calibration

To quantitatively compare the model to experimental data we must calibrate the
microwave field strength inside the medium. There is no way to independently
probe the strength of the microwave field without disturbing the system.
Therefore we obtain the calibration between applied microwave power end
electric field strength by fitting the model to the data at low field
strengths, where the model presented in section 3 reduces to the perturbative
model used in [33]. We fit across the energy spectrum the change in absorption

$\Delta\alpha
L=\alpha(\mathcal{E}_{\text{m}})L-\alpha(\mathcal{E}_{\text{m}}=0)L,$ (20)

where $L$ is the thickness of the crystal. Examples of this data can be seen
in figure 3. For 8 different input powers to the antenna between $0.3$ and
$4.2$ mW, the model gave a calibration factor of $43.5\pm 0.3$ V
m${}^{-1}(\text{mW})^{-1/2}$. This is within the errorbar of the calibration
in Pritchett et al.[33], but differs slightly from that work because more
field strengths were used in the calibration here. It is also in reasonable
agreement with finite element analysis modelling of the field in the sample
[32]. Using the output port of the stripline antenna, we checked that the
transmitted microwave power was linear with the input power over the range
considered in this paper. Therefore this calibration is used for all microwave
powers. The approach of calibrating at low field enables us to independently
test the behaviour of the high-field (ultrastrong driving) regime.

##  5 Results & Discussion

###  5.1 Probe Laser Absorption

Figure 3: Comparison between the experimental (filled circles) and theoretical
(solid lines) values of the change in absorption $\Delta\alpha L$ as a
function of laser energy. Three different values of microwave field strength
are shown spanning the perturbative (a) strong (b) and ultrastrong (c) driving
regimes. The dashed vertical lines show the positions of the zero-field $n$P
resonances.

Figure 2a compares the experimental and theoretical results for the probe
laser absorption. Each curve shows the optical depth, $\alpha L$, over the
laser energy range. The curves are vertically offset in optical depth by
$\mathcal{E}_{\text{m}}/100$ to show the microwave field strength they were
taken at. As the field strength is increased, starting with the high-$n$
peaks, absorption on the peaks is reduced and absorption between the peaks is
increased, resulting in a broad continuum of absorption across much of the
spectrum. Qualitatively, our model reproduces these effects well. At all
microwave powers, the same peaks disappear in the model as in the experimental
data, with both showing a flattening of the spectrum above $9$P into an
absorption continuum at the maximum field strength. In contrast to atoms [44],
we do not observe clear splitting and shifting of peaks in the manner of
Autler-Townes in either the theoretical or experimental spectra. In
experiments on atoms, Autler-Townes splitting occurs when the Rabi frequency
exceeds the width of the peak, so that two peaks can be resolved. For Rydberg
excitons, the non-radiative broadening of the exciton states leads to widths
comparable to the state separation. Due to this, any given state is coupled to
many others at once, leading to an absorption continuum.

To quantitatively compare the experimental data to the model, $\Delta\alpha L$
(equation 20) was plotted in figure 3. Figure 3a shows the model in a region
of $\mathcal{E}_{\text{m}}$ where it reduces to the perturbative model used in
previous work [32, 33]. Here the agreement with data is excellent across the
range of $n$, which is to be expected since this is one of the field strengths
used for the microwave field calibration. Figures 3b and 3c show the theory
and experiment at the threshold of ultrastrong driving and at the maximum
$\mathcal{E}_{\text{m}}$ achieved in the experiment respectively. Again, both
show extremely good and almost quantitative agreement a significant result for
an increase in $\mathcal{E}_{\text{m}}$ by a factor of $16$. We stress that
once the electric field is calibrated at low $\mathcal{E}_{\text{m}}$ (figure
3a) there are no additional free parameters. Excellent agreement is maintained
through all the intermediary field strengths as well. The change in absorption
is most accurately reproduced at the $n$P peaks. For reference, at
$\hbar\omega_{\text{p}}=E_{11\text{P}}$, for $\mathcal{E}_{\text{m}}=688$ V
m-1, the perturbative model predicts $\Delta\alpha L=-6$, several times the
size of the absorption peak without the microwave field, indicating we are
well beyond the regime where the perturbative model is valid.

There are some discrepancies between the theory and the experiment. Note the
asymmetric lineshape of the excitons in the experimental data, the precise
origin of which is disputed [45, 46, 47]. This was not modelled in the theory
as no self-consistent method could be found. Weaker agreement between peaks
can be partially explained by this lack of asymmetry in the model. However, it
does not account for the spurious features in the model, for example between 8
& 9P, or the splitting of $10$P into two peaks. The response of the $9$P
resonance to the microwave field is also underestimated by the theory. These
problems are likely caused by inaccuracies in the state structure and coupling
constants between excitonic states. Nevertheless, the good qualitative
agreement strongly suggests the model captures the essential physics of the
system.

The model provides insight into the origin of the absoprtion continuum
observed at high microwave field strengths. In figure 4, the real part of the
eigenenergies of the Floquet Hamiltonian, Re$[E_{q}]$, are plotted against
field strength. Grey lines indicate how strongly the state overlaps with
states that can couple to the valence band, characterised by
$a=\sum_{n}R_{n\text{P};0}^{(q)}\big{[}L_{n\text{P};0}^{(q)}\big{]}^{*}$. As
field strength increases, states in between the zero-field $n$P peaks
(vertical lines) gain character that can couple to the valence band, so can
contribute to the absorption. Due to the scaling of the transition dipole
moments with $n$, and the widths of the eigenstates, one can see a forest of
states that couple to the valence band in the high $n$ region. This forest
leads to the absorption continuum observed in the experiment. As the field
strength is increased, the forest can be seen to spread to lower energy, such
that at the highest field strengths studied, the absorption continuum has
spread as low as $9$P. Shown in red are the states that contribute to the
absorption at the indicated probe laser energy, with their opacity showing how
much they contribute, characterised by the imaginary part of that states’
contribution to equation 19. Due to the widths of the states, at the highest
field strengths shown, a broad collection of states make up the absorption at
any given laser energy. The solid lines in figure 4 indicate the field at
which the $n$P to $n$D transitions enter strong driving (orange), where
$\nu_{\text{$n$P-$n$D}}>\Omega_{\text{$n$P-$n$D}}>\Gamma_{\text{$n$P-$n$D}}$,
and ultrastrong driving (green), where
$\Omega_{\text{$n$P-$n$D}}>\nu_{\text{$n$P-$n$D}},\Gamma_{\text{$n$P}}$. For
example, the $11$P to $11$D transition enters SD at
$\mathcal{E}_{\text{m}}=200\pm 1$ V m-1 and USD at
$\mathcal{E}_{\text{m}}=300\pm 2$ V m-1. At the highest field strengths
achieved, all states above $8$P have entered the USD regime, coinciding with
the absorption continuum.

Figure 4: Bottom: a map of the real part of the eigenenergies of the Floquet
Hamiltonian, Re$[E_{q}]$, shaded in grey by $a$ to demonstrate their overlap
with states that couple to the valence band. The shading in red shows each
states’ contribution to the absorption of the laser at the indicated probe
laser energy (the imaginary part of that states contribution to equation 19).
Above the orange curve, the $n$P to $n$D transitions enter the strong driving
regime. Above the green curve, they enter the ultrastrong driving regime. Top:
the theoretical absorption spectrum at $\mathcal{E}_{\text{m}}=688$ V m-1.
Only the absorption at the indicated probe laser energy is composed of the
states in red. Different sets of grey states contribute to different parts of
the absorption spectrum. Figure 5: Contours showing the fraction of the
wavefunction that is rotating at most at $|N|\omega_{\text{m}}$ plotted
against microwave field strength, for probe laser energy
$\hbar\omega_{\text{p}}=E_{12\text{P}}$. This provides a proxy for the minimum
number of microwave photons involved in the system since a transition like
$+\omega_{\mathrm{M}}-\omega_{\mathrm{M}}$ is a two-photon microwave
transition which contributes to the $N=0$ Floquet component, but only a
transition involving at least $N\omega_{\text{m}}$ can excite the $N$-th
component.

As we enter the ultrastrong driving regime, one might suspect the presence of
multi-microwave photon transitions. To characterize the extent of multiphoton
transitions in the system, we construct the fraction

$X=\frac{\sum_{-N<N^{{}^{\prime}}<N}\sum_{j\neq\ket{0}}|c_{j;N^{{}^{\prime}}}|^{2}}{\sum_{N^{{}^{\prime}}}\sum_{j\neq\ket{0}}|c_{j;N^{{}^{\prime}}}|^{2}},$
(21)

which captures the proportion of the wavefunction rotating at a maximum
frequency of $|N|\omega_{\text{m}}$. In figure 5 contours of $X$ are plotted
through Floquet $N$ vs $\mathcal{E}_{\text{m}}$ space. We see clearly that the
Floquet approach is justified, as higher than second order components become
significant above field strengths of $200\pm 1$ V m-1, around where we enter
the ultrastrong driving regime. One can also see there is a significant
fraction of the wavefunction oscillating at $|N|>5$ at the high end of
considered $\mathcal{E}_{\text{m}}$, well into the ultrastrong driving regime.

###  5.2 Sideband Generation

Figure 6: Second (colour) and fourth (black) order red- and blue-detuned
sidebands for theory (lines) and experiment (points) at 4 different probe
laser detunings. They are taken at probe laser energies of (a) $2.170606$ eV,
(b) $2.170912$ eV, (c) $2.171252$ eV, (d) $2.171366$ eV. Each theory curve is
a different, randomly-chosen, set of phases for the valence band to $n$P
dipole matrix elements, $\bra{0}\hat{D}_{z}\ket{j}$, since they could not be
determined from experiment.

In figure 6, the sidebands predicted by the model are compared to experiment.
As mentioned in B, we do not have access to the relative phases of different
dipole elements $\bra{0}\hat{D}_{z}^{\text{p}}\ket{j}$, so to investigate
their relevance, $100$ sets of uniformly random phases were generated and the
resulting carrier and sideband information plotted. Largely independent of the
choices of phase, the model accurately reproduces the shape and onset of the
rollover of both the second- and fourth-order sidebands over almost two orders
of magnitude in field strength and nearly three orders of magnitude in
sideband amplitude. For any given energy, there are even some choices that
quantitatively reproduce the sidebands, providing good evidence that the model
is caputuring the essential physics of these processes. The quantitative
agreement is better at lower laser energies, but weakens as the dependence on
phase increases for higher energies. This is to be expected, since the phase
dependence is introduced through cross-terms between different pairs of matrix
elements in the susceptibility. These processes occur much more strongly at
higher $n$, where the states are closer together. There is no consistent
choice of phase that works for all energies, suggesting an energy-dependent
phase, possibly coming from the phonon background maybe be present. It was
considered beyond the scope of this work to add a large set of free parameters
to the model in an attempt to control for this phenomenon. The choice of
phases made little meaningful impact on the carrier absorption spectrum, so
were taken to all be positive real for that analysis.

The $y$-axis scaling of the sidebands is a free parameter, due to the lack of
access to photon collection efficiency in the experiment. It was fitted from
the $\hbar\omega_{\text{p}}\approx\text{E}_{8\text{P}}$ 2nd order sidebands
(figure 6a), since the $8$P exciton is relatively far away from neighbouring
states. Here, the sum in equation 11 is dominated by one term, i.e. that
sidebands are generated principally by decay from just the $8$P state,
removing dependency on cross terms between these matrix elements, and so
dependence on their relative phases.

###  5.3 Discussion

The ultrastrong driving regime is defined as when the Rabi frequency exceeds
the width of and energy spacing between the two considered levels. Although we
cannot treat this system as a two-level one, it is still useful to calculate
the ultrastrong driving regime condition. For example, we find that for the
$11$P to $11$D transition we enter the ultrastrong driving regime for field
strengths above $300\pm 2$ V m-1. The point where all $n$P to $n$D transitions
reach the strong and ultrastrong driving regimes can be seen in figure 4. At
the highest fields applied, we can reach an ultrastrong driving parameter of
up to $U=\Omega_{\text{$13$P,$13$D}}/\nu_{\text{$13$P-$13$D}}\sim 4$ for the
$13$P to $13$D transition, significantly exceeding parameters of $U=1$ [16]
and $U=2.1$ [15] in other works. It is worth noting that the strong driving
regime, where the Rabi frequency is larger than the state linewidth, is
reached at a similar microwave field strength, as the state linewidth (FWHM of
$40\leavevmode\nobreak\ $\mathrm{\SIUnitSymbolMicro eV}$$ for 11P) is
comparable to the energy separation
($E_{11\text{D}}-E_{11\text{P}}=58\leavevmode\nobreak\
$\mathrm{\SIUnitSymbolMicro eV}$$).

The system presented here is qualitatively different to the analogous case of
atoms. We see rapid hybridization and the creation of a quasi-resonant
continuum of Floquet states, rather than Autler-Townes splitting and shifts.
This continuum maintains a strong coupling to the microwave field, as shown by
the measurements of the microwave-induced sidebands. The near-quantitative
agreement means that this model can be used to predict the behaviour of
devices used for microwave-optical conversion well beyond the linear regime
and deep into the ultrastrong driving regime, maximising the exploitation of
the Rydberg mediated microwave-optical nonlinearity.

One thing which is not considered by the model presented here is the effect of
exciton-exciton interactions. Previous work on interactions between Rydberg
states has studied the van der Waals interactions [41] which scale with
exciton separation $R$ as $R^{-6}$. These experiments looked for exciton
density dependent effects by varying the intensity of the optical excitation.
Here, our intensity is $20$ $\mathrm{\SIUnitSymbolMicro W}$mm-2 which is $50$
time smaller than the intensity at which interactions have been seen at $n=12$
[27] and so van der Waals interactions can be neglected. However, when the
microwave field is applied, the excitons states become hybridized and resonant
dipole-dipole interactions can occur. These interactions are longer range than
van der Waals, scaling as $R^{-3}$ and so they may be present at lower exciton
densities. Some of the disagreement between the experiment and the model at
high field strength may be due to these interactions.

At the largest field strengths applied, we must consider the relevance of
microwave field induced ionization, though it isn’t included in the model.
This is analogous to ionization in atoms, since the electron is being promoted
to a continuum state, which in the case of excitons is in the conduction band.
In static fields, the ionization field threshold has been observed to follow
an $n^{-4}$ trend [48, 49, 50]. These results predict that the static field
required for ionization would be $\sim 6000$ V m-1 at $n=11$, which is almost
an order of magnitude larger than the highest field strength used in this
work. However, the case of microwave field induced ionization is more
complicated than the static field case as multi-photon absorption processes
can also lead to ionization [51]. In atomic Rydberg systems, hydrogen-like
atoms follow an $n^{-5}$ scaling when the microwave frequency is less than the
level spacing, and ionize at lower microwave field strengths than
corresponding static fields [52].

Our model provides insight into whether multi-photon ionization is
significant. As previously discussed, the Floquet components give contours for
the minimum number of microwave photons involved at a given field strength
(figure 5), which can be compared to the binding energy of the excitons. At
$\hbar\omega_{\text{p}}=E_{12\text{P}}$ the highest Floquet component within
the $95\%$ contour is $N=10$ which leads to a frequency of $70$ GHz compared
with the binding energy for $12$P of $117$ GHz implying that multi-photon
ionization is not significant. However, by
$\hbar\omega_{\text{p}}=E_{15\text{P}}$ the binding energy of $15$P has
dropped to 93 GHz and the maximum significant Floquet component has increased
to $N=17$ within $95\%$, a frequency of $119$ GHz, so microwave field induced
ionization starts to become important. A major difference from atomic Rydberg
systems is the lifetime of the states. The excitonic lifetime is $<1$ ns due
to phonon scattering, which is comparable to the oscillation period of the
microwave field, whereas in atomic Rydberg systems the state lifetime is much
larger, so it’s possible states could spontaneously decay before absorbing
enough photons for ionization. Extending the theoretical model to include a
continuum would allow microwave field induced ionization to be studied
further, however, the addition of ionization via an arbitrary number of
microwave photons is non-trivial [53]. For the field strengths considered
here, microwave field ionization is not expected to be playing a significant
role.

##  6 Conclusions

We have constructed a two-field model describing a probe laser field and a
microwave field acting on an ensemble of non-interacting Rydberg excitons in
cuprous oxide, and solved for its steady state using Floquet theory, something
not previously applied systems of this kind. Promising reproduction of the
changes in probe laser absorption led us to read into the internal workings of
the model. We are confident that, rather than ionize the excitons, the
microwave field hybridizes many exciton states together, creating a continuum
of dressed states. This does not occur in atomic Rydberg systems due to the
non-radiative broadening of the exciton lines. The excitons are ultrastrongly
driven by the microwave field, with Rabi frequencies greater than 4 times the
energy spacing of the respective levels, where processes involving in excess
of 6 microwave photons are significant. Sideband generation is well predicted
and understood near low energy peaks where states are far apart, but the
prediction is poorer where many states are involved, owing partly to unknown
phases but also hinting at missing physics.
The data and code used in this paper are available at [54].

##  7 Acknowledgements

The authors thank V. Walther for calculating dipole matrix elements for
transitions between Rydberg states and I. Chaplin and S. Edwards (Durham
University, Department of Earth Sciences) for preparing samples. This work was
supported by the Engineering and Physical Sciences Research Council (EPSRC),
United Kingdom, through research Grant Nos. EP/P012000/1 and EP/X038556/1. AB
and HQXW acknowledge funding from xx.

##  Appendix A Removal of absorption background

In the absorption experiment, the absorption due to excitons sits on top of a
large background of absorption due to the laser exciting a low lying exciton
state and an optical phonon. Modelling the microscopic origin of this
background is beyond the scope of this work, so it was removed from the
experimental data. The background function is based on the function used in
[50], consisting of a piece-wise function containing a below- and an above-
bandgap contribution:

$\alpha_{\mathrm{back}}=\begin{dcases}\alpha_{\mathrm{Ph}}+\alpha_{\mathrm{Ur}}&E<E_{\mathrm{g}}\\\
\alpha_{\mathrm{cont}}&E\geq E_{\mathrm{g}}\\\ \end{dcases}.$ (22)

Here, $\alpha_{\mathrm{Ph}}$ is the contribution to the background due to
phonon-assisted processes given by

$\alpha_{\mathrm{Ph}}=\alpha_{\mathrm{1Sy}}^{\Gamma_{3}^{-}}+\alpha_{\mathrm{1Sy}}^{\Gamma_{4}^{-}}+\alpha_{\mathrm{2Sy}}^{\Gamma_{3}^{-}}+\alpha_{\mathrm{1Sg}}^{\Gamma_{3}^{-}},$
(23)

where each term is a phonon branch consisting of a square-root dependence
starting at the energy of the phonon plus the labelled exciton state. The
Urbach tail, $\alpha_{\mathrm{Ur}}$ is a phenomenological term which describes
the exponential increase in absorption as the continuum is approached, given
by

$\alpha_{\mathrm{Ur}}=c_{\mathrm{U}}e^{(E-E_{\mathrm{g}})/E_{\mathrm{U}}},$
(24)

with amplitude $c_{\mathrm{U}}$ and Urbach energy $E_{\mathrm{U}}$. As the
experimental data in this paper does not extend far above the bandgap, the
absorption of the continuum is treated as constant. The energies and relative
amplitudes of the phonon absorption were taken from [50] while the parameters
in the Urbach tail were used as fitting parameters (as it is known the Urbach
tail depends on temperature and the presence of defects in the material [55]).
Figure 7 shows the absorption spectrum with the shaded region indicating the
background. The parameters used for this background function are available in
[54].

Figure 7: Removal of absorption background from experimental data. Shaded
region indicated fitted background which subtracted from the experimental data
in figure 2.

##  Appendix B Parameters for the model

The energies, widths, and peak heights of the $\ket{n\text{P}}$ states are
obtained for our sample from single-photon excitation [56]. Since the
$\mathcal{E}_{\text{m}}=0$ carrier spectrum reduces to a sum of Lorentzians,
the heights of these peaks were used to derive the magnitudes of
$\bra{0}\hat{D}_{z}\ket{j}$. However, it affords us no information about their
relative phases, so for the absorption spectrum they were all taken to be
positive real. The energies and widths of the S, D and F states come from two-
photon excitation [35], which is much less reliable due to additional
broadening in the SHG spectrum [57]. The energies were extended beyond what
was resolvable in the absorption spectrum (i.e. from $n=\\{6,\ ...,\ 15\\}$ to
$n=\\{6,\ ...,\ 20\\}$) by fitting. We included the Floquet states
$N={-15,...,15}$ in the calculation to achieve a convergence of less than
$0.01$ in optical depth across the spectrum.

The dipole matrix elements derived from experimental peak heights enter into
the model as the product

$n_{d}|\bra{0}\hat{D}^{\text{p}}_{z}\ket{n\text{P}}|^{2}=\frac{2A_{n\text{P}}\epsilon_{0}}{kL},$
(25)

where $A_{n\text{P}}$ are the experimental peak heights in optical depth, $k$
is the wavenumber of the laser at that peak, and $L$ is the thickness of the
sample. This makes the results of the model independent of $n_{d}$. It is also
obvious here that we do not have access to the phases of
$\bra{0}\hat{D}^{\text{p}}_{z}\ket{n\text{P}}$, only their magnitude, which is
very important for the calculation of the sidebands, but less so for the
absorption spectra, where they are all taken to be positive real.

##  Appendix C Angle Dependence

Mott-Wannier excitons can be separated into radial and angular parts,
$\ket{\Psi_{nlm}}_{\text{p}}=\ket{R^{n}_{l}}\ket{Y^{l}_{m}}_{\text{p}}$, where
the angular part is given by spherical harmonics, and the subscript p means
the $m$ label refers to the probe system of coordinates. Expressing the states
in the probe system as a sum of states in the microwave system,

$\ket{R^{n_{j}}_{l_{j}}}\ket{Y^{l_{j}}_{m_{j}}}_{\text{p}}=\ket{R^{n_{j}}_{l_{j}}}\sum_{m^{\prime}}d^{l_{j}}_{m_{j},m^{\prime}}(\beta)\ket{Y^{l_{j}}_{m^{\prime}}}_{\text{m}},$
(26)

where $d^{l}_{m,m^{\prime}}(\beta)$ the Wigner small-d rotation matrices [58],
$\beta$ is the angle between the probe laser and microwave field
polarizations, and the subscript m means the $m$ label refers to the microwave
system of coordinates. Thus

$\displaystyle\Omega_{jk>0}=$
$\displaystyle-e\mathcal{E}_{\text{m}}\bra{R^{n_{j}}_{l_{j}}}r\ket{R^{n_{k}}_{l_{k}}}$
$\displaystyle\times\sum_{m^{\prime}m^{\prime\prime}}[d^{l_{j}}_{m_{j},m^{\prime}}(\beta)]^{*}d^{l_{k}}_{m_{k},m^{\prime\prime}}(\beta)\bra{Y^{l_{j}}_{m^{\prime}}}_{\text{m}}\cos{\theta_{\text{m}}}\ket{Y^{l_{k}}_{m^{\prime\prime}}}_{\text{m}}$
(27)

where $\\{r,\theta_{\text{m}},\phi_{\text{m}}\\}$ are the spherical
coordinates in the microwave basis to be integrated over and $e$ is the charge
of the electron. However, as shown in figure 8, equation 27 predicts a strong
variation of the probe absorption with $\beta$ that is not observed
experimentally. The full cubic symmetry of the crystal lattice does allow for
the weaker dependence observed in figure 8. However the exciton wavefunction
is no longer separable, and the Mott-Wannier picture no longer applies, hence
the approximation made in equation 5.

In figure 8, we plot $\Delta\alpha L$ against polarisation angle for
$\Omega_{jk}$ given by equation 27. It can be seen that, while the
polarisation angle has little impact on the experimental data, this theory
predicts a very strong dependence. In the proper angle dependence approach,
the optically active S and D states both have $\Gamma_{5}^{+}$ symmetry,
implying the angle dependence will be the same for both states, but in the
atomic model S and D state have very different angle dependencies. This is
further evidence the spherical harmonics approach is wrong.

Figure 8: A plot of $\Delta\alpha L$ against the relative polarisation angle
between the probe laser and microwave field, $\beta$, at
$\hbar\omega_{\text{p}}=E_{11\text{P}}=2.171267$ eV and
$\mathcal{E}_{\text{m}}=688\pm 5$ V m-1. The dots show the angle dependence
observed in experiment, the red line shows the angle dependence predicted by
the model if we take the intra-excitonic dipole matrix elements,
$\Omega_{jk>0}$, from equation 27, and the black line shows the model given
$\Omega_{jk>0}$ from equation 5.

## References

## References

  * [1] Frisk Kockum A, Miranowicz A, De Liberato S, Savasta S and Nori F 2019 Nature Reviews Physics 1 19–40 ISSN 2522-5820 URL https://www.nature.com/articles/s42254-018-0006-2
  * [2] Hoi I C, Wilson C M, Johansson G, Palomaki T, Peropadre B and Delsing P 2011 Phys. Rev. Lett. 107(7) 073601 URL https://link.aps.org/doi/10.1103/PhysRevLett.107.073601
  * [3] Yan W B and Fan H 2014 Scientific Reports 4 4820 ISSN 2045-2322 URL https://www.nature.com/articles/srep04820
  * [4] Forn-Díaz P, García-Ripoll J J, Peropadre B, Orgiazzi J L, Yurtalan M A, Belyansky R, Wilson C M and Lupascu A 2017 Nature Physics 13 39–43 URL https://www.nature.com/articles/nphys3905
  * [5] García-Álvarez L, Casanova J, Mezzacapo A, Egusquiza I L, Lamata L, Romero G and Solano E 2015 Phys. Rev. Lett. 114(7) 070502 URL https://link.aps.org/doi/10.1103/PhysRevLett.114.070502
  * [6] Braumüller J, Marthaler M, Schneider A, Stehli A, Rotzinger H, Weides M and Ustinov A V 2017 Nature Communications 8 779 ISSN 2041-1723 URL https://www.nature.com/articles/s41467-017-00894-w
  * [7] Martínez-Martínez L A, Ribeiro R F, Campos-González-Angulo J and Yuen-Zhou J 2018 ACS Photonics 5 167–176 URL https://doi.org/10.1021/acsphotonics.7b00610
  * [8] Flick J, Rivera N and Narang P 2018 Nanophotonics 7 1479–1501 ISSN 2192-8614 URL https://www.degruyter.com/document/doi/10.1515/nanoph-2018-0067/html
  * [9] Liu Y, Wang Z, Yang P, Wang Q, Fan Q, Guan S, Li G, Zhang P and Zhang T 2023 Phys. Rev. Lett. 130(17) 173601 URL https://link.aps.org/doi/10.1103/PhysRevLett.130.173601
  * [10] Larson J and Mavrogordatos T 2021 The Jaynes–Cummings model and its descendants: modern research directions (IoP Publishing)
  * [11] Bhattacharya U, Lamprou T, Maxwell A S, Ordóñez A, Pisanty E, Rivera-Dean J, Stammer P, Ciappina M F, Lewenstein M and Tzallas P 2023 Reports on Progress in Physics 86 094401 URL https://dx.doi.org/10.1088/1361-6633/acea31
  * [12] Forn-Díaz P, Lamata L, Rico E, Kono J and Solano E 2019 Reviews of Modern Physics 91 ISSN 1539-0756 URL http://dx.doi.org/10.1103/RevModPhys.91.025005
  * [13] Loudon R 2000 The quantum theory of light (OUP Oxford)
  * [14] Kaluzny Y, Goy P, Gross M, Raimond J M and Haroche S 1983 Phys. Rev. Lett. 51(13) 1175–1178 URL https://link.aps.org/doi/10.1103/PhysRevLett.51.1175
  * [15] Deng C, Orgiazzi J L, Shen F, Ashhab S and Lupascu A 2015 Phys. Rev. Lett. 115(13) 133601 URL https://link.aps.org/doi/10.1103/PhysRevLett.115.133601
  * [16] Tommey J D R and Hogan S D 2019 Phys. Rev. A 100(5) 053417 URL https://link.aps.org/doi/10.1103/PhysRevA.100.053417
  * [17] Niemczyk T, Deppe F, Huebl H, Menzel E P, Hocke F, Schwarz M J, Garcia-Ripoll J J, Zueco D, Hümmer T, Solano E, Marx A and Gross R 2010 Nature Physics 6 772–776 ISSN 1745-2481 URL https://doi.org/10.1038/nphys1730
  * [18] Peterson G A, Kotler S, Lecocq F, Cicak K, Jin X Y, Simmonds R W, Aumentado J and Teufel J D 2019 Phys. Rev. Lett. 123(24) 247701 URL https://link.aps.org/doi/10.1103/PhysRevLett.123.247701
  * [19] Nataf P and Ciuti C 2011 Phys. Rev. Lett. 107(19) 190402 URL https://link.aps.org/doi/10.1103/PhysRevLett.107.190402
  * [20] Stassi R, Cirio M and Nori F 2020 npj Quantum Information 6 67
  * [21] Garcia-Vidal F J, Ciuti C and Ebbesen T W 2021 Science 373 eabd0336
  * [22] Lambert N J, Rueda A, Sedlmeir F and Schwefel H G L 2020 Advanced Quantum Technologies 3 1900077 URL https://onlinelibrary.wiley.com/doi/abs/10.1002/qute.201900077
  * [23] Han X, Fu W, Zou C L, Jiang L and Tang H X 2021 Optica 8 1050–1064 URL https://opg.optica.org/optica/abstract.cfm?URI=optica-8-8-1050
  * [24] Meschede D, Walther H and Müller G 1985 Phys. Rev. Lett. 54(6) 551–554 URL https://link.aps.org/doi/10.1103/PhysRevLett.54.551
  * [25] Raimond J M, Brune M and Haroche S 2001 Rev. Mod. Phys. 73(3) 565–582 URL https://link.aps.org/doi/10.1103/RevModPhys.73.565
  * [26] Aßmann M and Bayer M 2020 Advanced Quantum Technologies 3 1900134 ISSN 2511-9044 URL https://onlinelibrary.wiley.com/doi/abs/10.1002/qute.201900134
  * [27] Kazimierczuk T, Fröhlich D, Scheel S, Stolz H and Bayer M 2014 Nature 514 343–347 ISSN 14764687 URL https://www.nature.com/articles/nature13832
  * [28] Versteegh M A M, Steinhauer S, Bajo J, Lettner T, Soro A, Romanova A, Gyger S, Schweickert L, Mysyrowicz A and Zwiller V 2021 Phys. Rev. B 104(24) 245206 URL https://link.aps.org/doi/10.1103/PhysRevB.104.245206
  * [29] Heckötter J, Freitag M, Fröhlich D, Aßmann M, Bayer M, Grünwald P, Schöne F, Semkat D, Stolz H and Scheel S 2018 Phys. Rev. Lett. 121(9) 097401 URL https://link.aps.org/doi/10.1103/PhysRevLett.121.097401
  * [30] Ziemkiewicz D and Zielińska-Raczyńska S 2019 physica status solidi (b) 256 1800503 (Preprint https://onlinelibrary.wiley.com/doi/pdf/10.1002/pssb.201800503) URL https://onlinelibrary.wiley.com/doi/abs/10.1002/pssb.201800503
  * [31] Orfanakis K, Rajendran S K, Walther V, Volz T, Pohl T and Ohadi H 2022 Nature Materials 21 767–772 ISSN 1476-4660 URL https://www.nature.com/articles/s41563-022-01230-4
  * [32] Gallagher L A P, Rogers J P, Pritchett J D, Mistry R A, Pizzey D, Adams C S, Jones M P A, Grünwald P, Walther V, Hodges C, Langbein W and Lynch S A 2022 Phys. Rev. Res. 4(1) 013031 URL https://link.aps.org/doi/10.1103/PhysRevResearch.4.013031
  * [33] Pritchett J D, Gallagher L A P, Brewin A, Wong H Q X, Langbein W, Lynch S A, Adams C S and Jones M P A 2024 APL Photonics 9 031303 ISSN 2378-0967 URL https://doi.org/10.1063/5.0192710
  * [34] Potvliege R M 2024 arXiv:2406.16559
  * [35] Rogers J P, Gallagher L A P, Pizzey D, Pritchett J D, Adams C S, Jones M P A, Hodges C, Langbein W and Lynch S A 2022 Phys. Rev. B 105(11) 115206 URL https://link.aps.org/doi/10.1103/PhysRevB.105.115206
  * [36] Heckötter J, Walther V, Scheel S, Bayer M, Pohl T and Aßmann M 2021 Nature Communications 12 3556 ISSN 2041-1723 URL https://doi.org/10.1038/s41467-021-23852-z
  * [37] Baumeister P W 1961 Phys. Rev. 121(2) 359–362 URL https://link.aps.org/doi/10.1103/PhysRev.121.359
  * [38] Pritchett J 2023 Microwave induced optical nonlinearities in cuprous oxide Ph.D. thesis Durham University
  * [39] Thewes J, Heckötter J, Kazimierczuk T, Aßmann M, Fröhlich D, Bayer M, Semina M A and Glazov M M 2015 Phys. Rev. Lett. 115(2) 027402 URL https://link.aps.org/doi/10.1103/PhysRevLett.115.027402
  * [40] YU P and Cardona M 2010 Fundamentals of Semiconductors: Physics and Materials Properties Graduate Texts in Physics (Springer Berlin Heidelberg) ISBN 978-3-642-00710-1 URL https://books.google.co.uk/books?id=5aBuKYBT_hsC
  * [41] Walther V, Krüger S O, Scheel S and Pohl T 2018 Phys. Rev. B 98(16) 165201 URL https://link.aps.org/doi/10.1103/PhysRevB.98.165201
  * [42] Schöne F, Krüger S O, Grünwald P, Stolz H, Scheel S, Aßmann M, Heckötter J, Thewes J, Fröhlich D and Bayer M 2016 Phys. Rev. B 93(7) 075203 URL https://link.aps.org/doi/10.1103/PhysRevB.93.075203
  * [43] Manzano D 2020 AIP Advances 10 025106 ISSN 2158-3226 (Preprint https://pubs.aip.org/aip/adv/article-pdf/doi/10.1063/1.5115323/12881278/025106_1_online.pdf) URL https://doi.org/10.1063/1.5115323
  * [44] Delone N B and Krainov V P 1999 Physics-Uspekhi 42 669 ISSN 1063-7869 URL https://iopscience.iop.org/article/10.1070/PU1999v042n07ABEH000557/meta
  * [45] Toyozawa Y 1958 Progress of Theoretical Physics 20 53–81 ISSN 0033-068X URL https://doi.org/10.1143/PTP.20.53
  * [46] Schweiner F, Main J and Wunner G 2016 Physical Review B 93 085203 URL https://link.aps.org/doi/10.1103/PhysRevB.93.085203
  * [47] Stolz H, Schöne F and Semkat D 2018 New Journal of Physics 20 023019 ISSN 1367-2630 URL https://dx.doi.org/10.1088/1367-2630/aaa396
  * [48] Heckötter J, Freitag M, Fröhlich D, Aßmann M, Bayer M, Semina M A and Glazov M M 2018 Phys. Rev. B 98(3) 035150 URL https://link.aps.org/doi/10.1103/PhysRevB.98.035150
  * [49] Heckötter J, Freitag M, Fröhlich D, Aßmann M, Bayer M, Semina M A and Glazov M M 2018 Physics of the Solid State 60 1506–1509 ISSN 1090-6460 URL https://doi.org/10.1134/S1063783418080115
  * [50] Heckötter J 2020 Strongly interacting Rydberg excitons in Cu2O Ph.D. thesis TU Dortmund University
  * [51] Gallagher T F 1994 Microwave excitation and ionization (Cambridge University Press) p 162–194 Cambridge Monographs on Atomic, Molecular and Chemical Physics
  * [52] Gallagher T, Dalgarno A, Press C U, Read F and Zare R 1994 Rydberg Atoms Cambridge Monographs on Atomic, Molecular and Chemical Physics (Cambridge University Press) ISBN 9780521385312 URL https://books.google.co.uk/books?id=8JIpEhHWT-cC
  * [53] Shakeshaft R 1986 Phys. Rev. A 34(1) 244–252 URL https://link.aps.org/doi/10.1103/PhysRevA.34.244
  * [54] Microwave-optical spectroscopy of rydberg excitons in the ultrastrong driving regime
  * [55] Krüger S O, Stolz H and Scheel S 2020 Phys. Rev. B 101(23) 235204 URL https://link.aps.org/doi/10.1103/PhysRevB.101.235204
  * [56] Lynch S A, Hodges C, Mandal S, Langbein W, Singh R P, Gallagher L A P, Pritchett J D, Pizzey D, Rogers J P, Adams C S and Jones M P A 2021 Phys. Rev. Mater. 5(8) 084602 URL https://link.aps.org/doi/10.1103/PhysRevMaterials.5.084602
  * [57] Gallagher L 2022 Optical and microwave spectroscopy of Rydberg excitons in Cu2O Ph.D. thesis Durham University
  * [58] Wigner E 1959 Group theory: and its application to the quantum mechanics of atomic spectra (Academic Press, New York)