# Scaling laws of Rydberg excitons

J. Heckötter  Experimentelle Physik 2, Technische Universität Dortmund,
D-44221 Dortmund, Germany  M. Freitag  Experimentelle Physik 2, Technische
Universität Dortmund, D-44221 Dortmund, Germany  D. Fröhlich  Experimentelle
Physik 2, Technische Universität Dortmund, D-44221 Dortmund, Germany  M.
Aßmann  Experimentelle Physik 2, Technische Universität Dortmund, D-44221
Dortmund, Germany  M. Bayer  Experimentelle Physik 2, Technische Universität
Dortmund, D-44221 Dortmund, Germany  Ioffe Institute, Russian Academy of
Sciences, 194021, St.-Petersburg, Russia  M. A. Semina  Ioffe Institute,
Russian Academy of Sciences, 194021, St.-Petersburg, Russia  M. M. Glazov
Ioffe Institute, Russian Academy of Sciences, 194021, St.-Petersburg, Russia

###### Abstract

Rydberg atoms have attracted considerable interest due to their huge
interaction among each other and with external fields. They demonstrate
characteristic scaling laws in dependence on the principal quantum number $n$
for features such as the magnetic field for level crossing or the electric
field of dissociation. Recently, the observation of excitons in highly excited
states has allowed studying Rydberg physics in cuprous oxide crystals. While
bearing striking similarities to Rydberg atoms, also fundamentally new
insights may be obtained for Rydberg excitons, as the crystal environment and
the associated symmetry reduction compared to vacuum give easy optical access
to many states within an exciton multiplet. Here we study experimentally and
theoretically the scaling of several characteristic parameters of Rydberg
excitons with $n$ providing also insight into interaction effects. From
absorption spectra in magnetic field we find for the first crossing of levels
with adjacent principal quantum numbers a $B_{r}\propto n^{-4}$ dependence of
the resonance field strength, $B_{r}$, due to the dominant paramagnetic term
unlike in the atomic case where the diamagnetic contribution is decisive
resulting in a $B_{r}\propto n^{-6}$ dependence. By contrast, in electric
field we find scaling laws just like for Rydberg atoms. The resonance electric
field strength shows a scaling as $E_{r}\propto n^{-5}$. Similar to Rydberg
atoms with the exception of hydrogen we observe anticrossings of the states
belonging to multiplets with different principal quantum numbers. The energy
splittings at the avoided crossings scale as $n^{-4}$ which we relate to the
crystal specific deviation of the exciton Hamiltonian from the hydrogen model.
The data also allow us to assess the susceptibility of Rydberg exciton to the
external fields: We observe the exciton polarizability in the electric field
to scale as $n^{7}$. In magnetic field, on the other hand, the crossover field
strength from a hydrogen-like exciton to a magnetoexciton dominated by
electron and hole Landau level quantization scales as $n^{-3}$. Also the
exciton ionization can be studied with ionization voltages that demonstrate a
$n^{-4}$ scaling law as for atoms. Particularly interesting is the field
dependence of the width of the absorption lines which remains constant before
dissociation for high enough $n$, while for small $n\lesssim 12$ an
exponential increase is found. These results are in excellent agreement with
theoretical calculations.

##  I Introduction

The recent observation of highly excited excitons in bulk cuprous oxide
crystals up to principal quantum numbers of $n=25$ has opened appealing
perspectives for studying Rydberg physics in the solid state Kazimierczuk  _et
al._ (2014). The exciton Rydberg energy, $\mathcal{R}$, of about $90$ meV is
reduced by more than two orders of magnitude as compared to the atomic Rydberg
energy. This reduction makes excitons quite susceptible to external fields and
enables reaching the strong field regime where the characteristic interaction
energies with an electric or magnetic field prevail over the Coulomb
interaction. Furthermore, the different states of the hydrogen-like excitonic
series are spaced rather closely facilitating systematic high resolution
studies because the absorption across the whole series can be recorded by
tuning a single light source such as a frequency stabilized laser with neV-
resolution.

These opportunities have been exploited already to elaborate similarities and
differences of Rydberg systems of atoms and excitons. A particular example is
the quantum defect, which for atoms describes the deviation of the energy
levels of the valence electron from a pure hydrogen series. Similar deviations
were observed for Rydberg excitons in cuprous oxide whose binding energies can
be also approximately cast into the form Schöne  _et al._ (2016):

$E_{n,l}=\frac{{\mathcal{R}}}{n^{2}\left(1-\frac{\delta_{n,l}}{n}\right)^{2}},$
(1)

where $\delta_{n,l}$ is the quantum defect of the exciton in the state with
principal quantum number $n$ and orbital angular momentum $l$ of the envelope
wave function. Despite of this similarity the origin of the quantum defect is
very different: For atoms it accounts for the shielding of the Coulomb
potential of the nucleus by the closed shells of inner electrons, leading to
an effective deviation from the $1/r$-dependence. In semiconductors like Cu2O,
it results predominantly from the details of the valence band structure, which
has a dispersion deviating strongly from a parabolic law.

This exemplary finding asks for a more detailed study of the two different
classes of Rydberg objects. Particularly useful in this respect is the
application of external electric and magnetic fields, which cause
characteristic shifts of the energy levels and also lift level degeneracies.
For atoms, e.g., there is still a degeneracy in the magnetic quantum number,
$m$, caused by the spherical symmetry of the system. This degeneracy can be
lifted by a magnetic field. Unlike for atoms, the anisotropic crystal
structure makes this description approximative only for excitons as
demonstrated by the observation of a triplet splitting of the $F$-excitons,
$l=3$, in absence of an external field Thewes  _et al._ (2015).

While the effects of external fields were investigated already right after the
first observation of excitons in cuprous oxide Gross (1962); Agekyan  _et al._
(1974); Agekyan (1977), recently corresponding studies were performed in both
electric Heckötter  _et al._ (2017) and magnetic bfield  field with
unprecedented resolution. Elaborated calculations accounting for the details
of the band structure allowed a quantitative description of the observed field
dispersions in the regime of low $n\leq 6$, where the exciton lines can be
reliably assigned. For higher $n$ the field induced splittings become so
diverse that statistical methods have to be applied for the level analysis. In
magnetic field, this analysis revealed quantum chaotic behavior Assmann  _et
al._ (2016).

The focus here is placed on the high-$n$ Rydberg regime ($n>6$) addressing
quantities that can be uniquely determined and are therefore characteristic
for the exciton states, in particular in an external electric or magnetic
field. These quantities are the field strengths of the first resonance of
states belonging to adjacent manifolds with principal quantum numbers $n$ and
$n+1$, and in electric field the polarizability, the spectral linewidth, and
the dissociation field strength. A systematic picture is developed by scanning
the excited state resonances from low $n$ up to the band gap, from which
similarities to atoms can be extracted such as the $n^{-5}$ scaling of the
first resonance on voltage in electric field Gallagher (2005). At these
resonances anticrossings are observed which we attribute mainly to non-
parabolic terms in the kinetic energy of the valence band hole, which
represents a crystal specific feature. At higher field strengths, exciton
dissociation takes place at voltages that scale as $n^{-4}$ like in the atomic
case. On the other hand, in magnetic field we find a scaling different from
the $n^{-6}$-law in atoms, namely a $n^{-4}$ law of the resonance field
strength. This difference arises from the modification of the selection rules
for excitons as compared with atoms which basically makes the whole exciton
multiplet accessible in optical absorption. As a result, the states with large
paramagnetic energy shift are involved in the resonances in contrast to the
dominant diamagnetic behavior for atoms. We also address the multiplet width
at zero field as a function of $n$.

After presenting the experimental details in the next Section II, we discuss
the scaling law behavior in Section III, considering first the zero-field
situation and then turning to the behavior in external fields. In each part we
also give the corresponding theoretical analysis. The manuscript is concluded
in Sec. IV.

##  II Experiment

The experiments were performed on bulk-like cuprous oxide crystal slabs that
were cut and polished from a natural rock. The samples were placed strain-free
in a sample holder that allowed application of an electric field Brandt
(2007). The holder was placed at $T=1.3$ K in the liquid helium insert of an
optical cryostat equipped with a magnet coil for fields up to 7 T. Both
electric and magnetic field were applied in longitudinal field configuration,
i.e. they were oriented along the optical axis normal to the crystal slab
plane. Further technical details can be found in Refs. Heckötter  _et al._
(2017); bfield .

The absorption was measured, depending on the required spectral resolution, by
using either a broadband white light source or a frequency stabilized dye
laser with neV-linewidth for excitation. For the white light source the
bandwidth was reduced to the wavelength range of interest by a double
monochromator. After transmission through the crystal the white light was
dispersed by another double monochromator and detected by a Si-charge coupled
device camera. The spectral resolution provided is less than 10 $\mu$eV which
is sufficient for exciton states with $n<20$ to study the effects of interest
here. When the laser was used to obtain a particularly high spectral
resolution, its output was stabilized by a noise eater. After transmission
through the sample it was detected by a avalanche photodiode. The power
density of the exciting light was chosen low enough that the excitation of
dressed states as discussed in Ref. Grünwald  _et al._ (2016) can be neglected
and the experimentally observed spectral lines correspond to resonant
absorption peaks.

Recently we showed that the absorption spectra of cuprous oxide crystals vary
with the light polarization in combination with the chosen crystal
orientation. For example, in electric field applied to a $[110]$ oriented
crystal, the quadrupole allowed transitions to the $S$\- and $D$-excitons
which can be observed for light polarized linearly along the $[001]$-direction
are absent for light polarized along the $[1\bar{1}0]$ direction, simplifying
the spectra significantly Heckötter  _et al._ (2017). Depending on the problem
to be studied, we chose the appropriate polarization configuration. If the
task was to address the whole multiplet belonging to a particular $n$ for
which as many states as possible need to be resolved in a manifold, we used
the $[001]$-polarization, while the $[1\bar{1}0]$-polarization was chosen
when, for example, the field dispersion of the $P$-excitons was to be measured
with high accuracy.

For the subsequent discussion several remarks are due. For atoms, in a
constant external field the spherical symmetry is reduced, but the rotational
symmetry about the field is maintained, so that the angular momentum component
along the field direction, $m$, remains a good quantum number. In a
longitudinal field configuration, where the optical axis is oriented along the
field, well defined selection rules hold in electric dipole approximation for
single photon absorption: Starting from an electron in a $s$-shell, only
transitions to $p$-shell states are possible due to the restriction $\Delta
l=\pm 1$, whereby all states in this state triplet can be excited leaving the
magnetic quantum number unchanged, $\Delta m=0$, or changing it by unity,
$\Delta m=\pm 1$, for linearly or circularly polarized light, respectively.
Other states of the Rydberg manifold with higher angular momenta $l>1$, for
example, cannot be excited out of a $s$-shell state.

The situation is different for the yellow excitons in cuprous oxide, where the
dominant dipole-allowed excitons are those with $P$-type envelope
wavefunction. The absorption of light takes place through excitation of an
exciton in a particular quantum state out of the ground state of the crystal.
Due to the reduction from continuous to discrete symmetry oscillator strength
is shuffled from the $P$-states to other odd exciton states, namely the $F$-,
$H$-, $\ldots$ excitons, even though it remains at least two orders of
magnitude smaller compared to the $P$-states. Furthermore, the $S$-exciton
transitions are allowed in quadrupole approximation, and, through the exchange
coupling, also the $D$-excitons become accessible. Thereby, if allowed by the
transition matrix elements, the observation of states is facilitated or
enhanced at the expense of the $P$-excitons, as described quantitatively in
recent studies of low lying excitons Heckötter  _et al._ (2017); bfield .

Figure 1: Contour plot of absorption spectra versus magnetic field applied in
the Faraday configuration (light propagation along the magnetic field),
recorded at $T$ = 1.3 K on a Cu2O crystal slab. The dashed line indicates the
band gap at zero field. Landau level quantum numbers that can be approximately
assigned to a bunch of clustered transitions are indicated at the right and
top axes. The black frame indicates the area from which a close-up is shown in
the Appendix A as Fig. 13. The scale on the right shows gives the strength of
the absorption features in arbitrary units.

In more detail, a magnetic field enhances the visibility of the weak states
that are accessible through the crystal symmetry-induced mixing of excitons
with odd parity envelope as this mixing is strongly increased by the field.
Quadrupole-allowed excitons with even parity remain weak bfield . By contrast,
in electric field the excitons with opposite parity become mixed from which
the even parity excitons profit in oscillator strength. Hence, the $S$-, $D$-,
…excitons become more pronounced in the absorption spectra. As a consequence
of this mixing by the crystal structure and by the external field, the
majority of states within an exciton multiplet becomes easily detectable, in
contrast to atoms. This allows for access also to high angular momentum states
by single photon absorption.

##  III Scaling of exciton properties

Before discussing the different scaling laws, we first present briefly the
comprehensive set of data from which the laws are extracted. These data
comprise spectra recorded in magnetic and electric field, from which also the
understanding of the zero field level spectrum can be deepened. Figure 1 shows
a contour plot of exciton absorption spectra versus the magnetic field applied
in the Faraday geometry. At zero field the hydrogen-like series of exciton
states is seen, out of which a complex Zeeman splitting pattern arises. Better
insight into the details of these spectra can be obtained by taking their
second derivatives from which weak features become well resolved. The
corresponding contour plot of the second derivatives is shown in Fig. 2: in
panel (a) for the states with $n>4$ up to $B$ = 7 T, while in panel (b) a
close-up of the low-field range up to 2 T is shown for $n>6$. At zero field
the levels belonging to a particular principal quantum number are spread over
an energy range which becomes the narrower, the higher $n$ is. However, in
magnetic field the splitting of a multiplet becomes larger for higher $n$,
because higher angular momentum states limited by $l\leq n-1$ contribute.

Due to the multitude of observed levels in particular at medium field
strengths it appears difficult to extract generally valid scaling laws for the
dependences on the principal quantum number there. For that purpose we have to
restrict our analysis to ranges, where states can be identified well. This is
the case at low fields up to the point, where resonances of states belonging
to exciton multiplets with different principal quantum numbers occur.
Similarly, states can be quite well identified in high magnetic fields, where
the dominant observed lines tend to cluster around transitions that correspond
to those between Landau levels, see Fig. 1. In fact, the close-up of Fig. 1
shown in the Appendix A as Fig. 13 reveals transitions which can be associated
with Landau level quantum numbers up to 79, which arise in magnetic field from
$P$-excitons with $n$=79 in zero field, see also discussion below in Sec.
III.2. The corresponding average radius of this exciton wavefunction would be
10.4 $\mu$m, which is squeezed in magnetic field, thereby enhancing the
oscillator strength so that these highly excited states become visible. The
$n$=79 state can be observed starting from 0.5 T, where the Landau level
extension is about 300 nm.

Figure 2: Top panel: Same data as in Fig. 1, but in form of a contour plot of
the second derivative of the absorption spectra versus magnetic field for
$n\geq 5$. Lower panel: Close-up of the states for $n\geq 7$ in $B$ up to 2 T.
$T$ = 1.3 K. The scales on the right give the strength of the features in
arbitrary units. The weak equidistant vertical stripes which are apparent
mostly at low energies in the upper panel, are artifacts of taking the 2nd
derivative.

Exciton level splitting can be also induced by applying an electric field and
exploiting the Stark effect. Corresponding spectra are shown in Fig. 3 where
their second derivatives are plotted as function of the voltage applied to the
sample for the excitons with $n\geq 5$. Overall, the number of lines observed
there remains smaller than in magnetic field because already for pretty low
voltages below 10 V the exciton states with $n>$10 are subject to field-
induced dissociation. Still, for identifying scaling laws, one has to restrict
also here to the low field strength regime or to lines with dominant
oscillator strength compared to the other features. As discussed above,
variation of the linear polarization of the exciting light allows one to vary
the number of detected spectral lines. In the left panel, the light
polarization was $\hat{\bm{e}}\parallel[1\bar{1}0]$ and in the right panel the
polarization was $\hat{\bm{e}}\parallel[001]$. In the former case, the
quadrupolar-active, even exciton states (e.g., $S$\- and $D$-excitons) are
forbidden, while in the latter case they are allowed, giving the spectra a
more complex appearance.

###  III.1 Zero-field behavior

Figure 3: Contour plot of the second derivatives of absorption spectra versus
applied voltage. The spectra were recorded by white light excitation at
$T=1.3$ K on a Cu2O crystal slab with $[110]$ orientation. The light was
polarized along the $[1\bar{1}0]$ direction in the left panel and along the
$[001]$ direction in the right panel. The mid panel shows a close-up of the
section of the contour plot in the left panel, marked there by the solid box,
in order to highlight the anticrossing at the first resonance involving the
state of the $n$ = 6 multiplet showing the strongest field dispersion to
higher energies and the state of the $n$ = 7 multiplet with the strongest
dispersion to lower energies. The weak equidistant vertical stripes in the
left and right panel are artifacts of taking the 2nd derivative.

The exciton energies in zero external field can be described well by the
quantum defect formalism of Eq. (1) Schöne  _et al._ (2016). We recall that in
Cu2O the origin of the quantum defect is quite different from that in many-
electron Rydberg atoms and is related with the complex valence band structure:
The separation between the topmost $\Gamma_{7}^{+}$ and the closest
$\Gamma_{8}^{+}$ valence subbands is on the same order of magnitude as the
exciton Rydberg energy $\mathcal{R}$. Hence, the mixing of these bands via the
off-diagonal elements of the Luttinger Hamiltonian gives rise to the deviation
of the excitonic series from a hydrogenic one and to the fine structure of the
exciton energy spectrum. Moreover, the exchange interaction between the
electron and the hole also contributes significantly to the exciton state
splitting and mixing Uihlein  _et al._ (1981); Thewes  _et al._ (2015);
Schweiner  _et al._ (2016); Heckötter  _et al._ (2017). For excitons with
principal quantum numbers $n\gtrsim 4$ estimates show that the mixing of the
valence subbands can be treated perturbatively and the effective Hamiltonian
can be recast in the following form [cf. Schöne  _et al._ (2016); Heckötter
_et al._ (2017)]:

$\mathcal{H}=\frac{p^{2}}{2\mu}-\frac{e^{2}}{\varepsilon
r}+\mathcal{A}p^{4}+\\\ \frac{e^{2}}{\varepsilon
r^{3}}\left[\mathcal{B}_{e}(\bm{l}\cdot\bm{s}_{e})+\mathcal{B}_{h}(\bm{l}\cdot\bm{s}_{h})\right]+\\\
\mathcal{C}\delta(\bm{r})(\bm{s}_{e}\cdot\bm{s}_{h})+\mathcal{H}_{cubic}+\mathcal{H}_{sd}.$
(2)

Here and through the rest of the manuscript, we consider excitons with center
of mass wavevector $\bm{K}=0$. In Eq. (2) the first two terms, subsequently
denoted in short by $\mathcal{H}_{0}$, describe the standard hydrogen-like
problem with the relative motion momentum $\bm{p}$, the relative motion
coordinate $\bm{r}$, the electron charge $e$ and the dielectric constant
$\varepsilon$. The quartic term $\mathcal{A}p^{4}$ describes the
nonparabolicity of the kinetic energy with $\mathcal{A}$ being a constant. The
parameter $\mathcal{A}$ can be estimated in the spherical approximation of the
Hamiltonian via the Luttinger parameter $\gamma_{2}$ as $\mathcal{A}\approx
2\gamma_{2}^{2}/(m_{0}^{2}\Delta)$, where $m_{0}$ is the free electron mass
and $\Delta$ is the splitting between the $\Gamma_{8}^{+}$ and
$\Gamma_{7}^{+}$ bands. The second line of Eq. (2) describes the spin-orbit
interaction, where $\bm{l}=\hbar^{-1}[\bm{r}\times\bm{p}]$ is the angular
momentum operator, $\bm{s}_{e}$ ($\bm{s}_{h}$) are the electron (hole)
spin-$1/2$ operators acting on the basis functions of the $\Gamma_{6}^{+}$ and
$\Gamma_{7}^{+}$ representations, respectively, and $\mathcal{B}_{e}$
($\mathcal{B}_{h}$) are constants. In the third line, the term
$\mathcal{C}\delta(\bm{r})(\bm{s}_{e}\cdot\bm{s}_{h})$ describes the short-
range electron-hole exchange interaction with the exchange constant $C$, and
the additional terms $\mathcal{H}_{cubic}$ and $\mathcal{H}_{sd}$ account for
the cubic anisotropy and the $S-D$-exciton mixing, respectively. In the
following the contributions $\mathcal{H}_{cubic}$ and $\mathcal{H}_{sd}$ are
neglected. The Hamiltonian (2) produces quantum defects that are in reasonable
agreement with the experiment. To that end we first solve the hydrogenic
problem and obtain $\Psi_{nlm}(\bm{r})$, which are the eigenfunctions of
$\mathcal{H}_{0}$. The remaining terms are treated perturbatively. To that
end, we introduce the Hamiltonian that is within the applied approximations
the extension of the Hamiltonian beyond the hydrogen model:

$\mathcal{H}_{d}=\mathcal{A}p^{4}+\frac{e^{2}}{\varepsilon
r^{3}}\left[\mathcal{B}_{e}(\bm{l}\cdot\bm{s}_{e})+\mathcal{B}_{h}(\bm{l}\cdot\bm{s}_{h})\right]+\mathcal{C}\delta(\bm{r})(\bm{s}_{e}\cdot\bm{s}_{h}),$
(3)

and is responsible for the quantum defect appearance. Retaining first-order
perturbation theory in $\mathcal{H}_{d}$ the contributions leading in the
inverse principal quantum number we arrive at Berestetskii  _et al._ (1999)

$\displaystyle\langle
p^{4}\rangle_{nlm}=\left(\frac{\hbar}{a_{B}}\right)^{4}\frac{4}{n^{3}(l+1)},$
(4a)
$\displaystyle\left\langle\frac{1}{r^{3}}\right\rangle_{nlm}=\frac{1}{a_{B}^{3}}\frac{1}{n^{3}l(l+1/2)(l+1)},$
(4b)
$\displaystyle\langle\delta(\bm{r})\rangle_{nlm}=\frac{\delta^{K}_{l,0}}{\pi}\frac{1}{a_{B}n^{3}},$
(4c)

where $\langle\ldots\rangle_{nlm}$ denotes the quantum-mechanical average of
the corresponding quantity over the exciton state $\Psi_{nlm}(\bm{r})$,
$a_{B}=h^{2}\varepsilon/(\mu e^{2})\approx 1.11$ nm is the exciton Bohr radius
for the $P$-excitons, and $\delta^{K}_{a,b}$ is the Kronecker $\delta$-symbol.

The analysis of the experimental data shows that the exciton quantum defects
for large $n$ quickly converge to constant values for fixed angular momentum
$l$. This observation is supported by Eqs. (4), which show that each
contribution to the energy deviation of the levels within a multiplet from the
simple hydrogen formula scales as $1/n^{3}$, in agreement with the quantum
defect description, Eq. (1). Furthermore, we found experimentally that with
increasing $l$ the quantum defects drop continuously from finite positive
values towards zero, again, in full agreement with Eqs. (4). The $S$-excitons
demonstrate the largest quantum defect $\delta_{n,l=0}\approx 0.65$, for the
$P$-excitons it is about $0.34$ and for the $D$\- and $F$-excitons it is
reduced to $0.18$ and $0.12$, respectively, in the large-$n$ limit.
Theoretical estimates of the non-parabolicity contribution to the quantum
defects, Eq. (4a), for $\gamma_{2}=0.8$ Thewes  _et al._ (2015); Schweiner
_et al._ (2016) yield $\delta_{n,l=0}=0.87$, $\delta_{n,l=1}=0.43$,
$\delta_{n,l=2}=0.28$ and $\delta_{n,l=3}=0.21$ in reasonably good agreement
with experiment. It supports our conjecture that the dominant contribution to
quantum defects arises from the $p^{4}$ nonparabolic term in the valence band
dispersion.

Figure 4: Energy spread of the exciton states within a multiplet with fixed
principal quantum number versus $n$ in a double logarithmic representation.
The line gives a fit to the data after a $n^{-3}$ scaling law, following the
theoretical considerations in the text, see Eq. (4).

The consequence of the quantum defect series for a particular principal
quantum number is that the associated states are spread over a finite energy
range with the low (high) angular momentum states on the low (high) energy
side, as clearly seen in the spectra in Fig. 2 and in Fig. 3. The widths of
these multiplets were determined from the data in zero field and also in weak
external fields, applied to optically activate excitons that are dark without
field or to enhance their visibility so that their energies could be
extrapolated to zero field with high accuracy. The widths of the multiplets
are plotted in Fig. 4 as function of $n$ in double-logarithmic representation
and show a strong drop with increasing principal quantum number from $3$ meV
for $n=3$ to less than 0.1 meV for $n$ exceeding 10. The data follow the
$1/n^{3}$ scaling law expected from the quantum defect description and
corroborated by the calculations according to Eqs. (4), as shown by the red
line fit.

The definition of the Rydberg exciton regime is to some extent arbitrary. We
take it here as the regime in which the principal quantum number exceeds
$n=5$, because the exciton extension given by twice the average exciton radius
exceeds then 100 nm, which is roughly two orders of magnitude larger than the
ground state wave function extension. Correspondingly the data in Fig. 4 and
also in all subsequent scaling plots were fitted for $n\geq 6$, but the plots
of the fits were extended also towards lower $n$ in case corresponding data
were available like for the multiplet splitting just discussed.

###  III.2 Scaling in magnetic field

The magnetic field behavior of the excitons in the Rydberg regime has been
assessed in detail in Refs. bfield ; Assmann  _et al._ (2016). Entering this
regime or $n\gtrsim 6$, the density of states becomes so large that a precise
assignment of the observed states is difficult, in particular, because here
avoided crossings and therefore state mixings dominate. Hence statistical
methods have been applied by analyzing the distribution of energy separations
between nearest exciton levels at fixed transition energies. For $n>6$ clear
signatures of quantum chaos were found due to the dominance of anticrossings
as can be seen also from the spectra in Fig. 2 in the energy range above about
$2.170$ eV. Here we focus on different features: (i) an estimate of the size
of the excitons, and (ii) the magnetic field induced crossing of states with
adjacent $n$.

Exciton size evaluation. So far, the Rydberg exciton size has been determined
somewhat ‘‘indirectly’’ from their principal quantum number $n$ using the
formula for the average radius of an orbital in the hydrogen model. This
average radius is given by Landau and Lifshitz (1977)

$\langle
r\rangle_{nlm}=\frac{a_{B}}{2}\left[3n^{2}-l\left(l+1\right)\right]\approx\frac{3a_{B}}{2}n^{2}$
(5)

with the approximation being well suited for $n\gtrsim 6$ for the
$P$-excitons. The magnetic field introduces an independent length scale given
by the magnetic length, $\ell_{c}=\sqrt{\hbar c/eB}$ characterizing the
extension of the magnetic confinement potential, which competes with the
Coulomb interaction: At low magnetic fields, the main $P$-exciton lines in
Fig. 1 show a rather weak dependence on magnetic field, corresponding to a
diamagnetic shift $\propto B^{2}$, which changes with increasing field to a
stronger shift that can be roughly approximated by a $B$-linear dependence. In
order to assess this transition theoretically, it is sufficient to present the
effective Hamiltonian of the exciton in the magnetic field $\bm{B}\parallel z$
in the form:

$\mathcal{H}_{B}=\mathcal{H}_{0}+\frac{\hbar eB}{2\mu
c}l_{z}+\frac{\hbar^{2}e^{2}B^{2}}{8\mu c^{2}}(x^{2}+y^{2}).$ (6)

The terms responsible for the quantum defect are not of importance here. We
also disregard the electron and hole Zeeman spin splittings. For strong
magnetic fields one can neglect in first approximation the Coulomb interaction
and approximate the states by the electron-hole Landau levels. More precisely,
even at strong magnetic fields the Coulomb interaction provides bound
magnetoexciton states with the binding energy scaling as
${\mathcal{R}}\ln(a_{B}/\ell_{c})$ Landau and Lifshitz (1977). With increasing
field the changeover from an exciton behavior dominated by the Coulomb
interaction at low fields to a behavior with dominant magnetic confinement at
high fields, where Landau levels are formed, occurs.

Correspondingly, an exciton resonance in the optical spectrum transforms to a
good approximation into a transition between electron and hole Landau levels.
Roughly, $P$-excitons with principal quantum number $n$ transform into a
transition between Landau levels with quantum number $n$ Yafet  _et al._
(1956); Zhilich  _et al._ (1969). The extension of these Landau levels in real
space is given by

$\ell_{c,n}\approx\ell_{c}n^{1/2}=\sqrt{\frac{\hbar cn}{eB}}$ (7)

for not too small $n$. The transition takes place for

$\ell_{c,n}\sim\langle r\rangle_{nlm},$

i.e., where the Landau level extension is about equal to the Coulomb
extension, which can be achieved by increasing the magnetic field to a
particular crossover field strength $B_{c,n}$. From this crossover field one
obtains $\ell_{c,n}=25.6\leavevmode\nobreak\
$nm$\sqrt{n}/\sqrt{B_{c,n}[\mathrm{T}]}$, where $B_{c,n}$ should be inserted
in units of Tesla. In the experiment we can determine $B_{c,n}$ as the field
strength at which the quadratic field dependence of the $P$-exciton energies
changes into a linear one. Note that the crossover fields and the resulting
Landau level extensions can be only approximately determined in that way, but
they are nevertheless sufficient to test their scaling with $n$. From the
considerations above we expect a scaling with $n^{-3}$ for $B_{c,n}$.

From the spectra it is confirmed that the transition to Landau-level like
behavior occurs at continuously decreasing magnetic fields with increasing
principal quantum number. In the high field regime, where also quantum chaos
was detected for $n>6$ no single strong transition is observed for a
particular $n$, but bunches of lines which cluster around the Landau level
transitions appear. Technically, to determine the crossover field we have
taken the center of the line multiplet around a Landau transition and have
extrapolated its energy linearly to lower fields, so that we could determine
the crossing point $B_{c,n}$ with the quadratic diamagnetic shift of the
$P$-exciton line. The $B_{c,n}$ determined in that way are shown by the red
circles in Fig. 5, demonstrating the strong drop with increasing $n$. The
dotted line gives a $n^{-3}$ fit to the data from which excellent agreement is
seen.

From these fields $\ell_{c,n}(B_{c,n})$ can be calculated. The Landau level
extensions determined in that way are shown in Fig. 5 by the black squares and
compared to the average exciton radius $r_{n,l}$ according to the hydrogenic
formula. For example, for $n=6$ the hydrogenic formula (5) gives an average
radius of 60 nm. The changeover field strength, on the other hand, is $1.3\pm
0.3$ T, from which we obtain $\ell_{c,6}(B_{c,6})\approx 55\pm 8$ nm which are
surprisingly close values. For all other $n$ we find similarly good agreement
between $\ell_{c,n}$ and $\left(r_{n,l}\right)$, confirming the expected
$n^{2}$ scaling of the wave function extension:
$\ell_{c,n}\propto(n/B_{c,n})^{1/2}\propto\sqrt{n/n^{-3}}$. Note that the
inaccuracy in determining $B_{c,n}$ translates through the square-root
connection in moderate inaccuracies of the Landau level extension $\ell_{c,n}$
that are indicated by the error bars. For high $n$ sufficient accuracy was
achieved by recording the spectra in 5 mT steps which in combination with the
steeper slope of the Landau level transitions facilitates the separation from
the low-field behavior. While the observed agreement with the hydrogenic
formula may be expected, it also validates the applied exciton description
using, for example, a uniform dielectric screening over large length scales.
It also reassures the huge extension of the highly excited Rydberg exciton,
for which we estimate for $n=20$ an average radius of about 0.7 $\mu$m.

Figure 5: Red circles: crossover field strength $B_{c,n}$ from diamagnetic to
Landau level-like behavior as function of principal quantum number. The red
dotted line is a fit to the data using a $n^{-3}$ dependence. Black squares:
Landau level radius $\ell_{c,n}$, Eq. (7), estimated from $B_{c,n}$ as
function of $n$. The black solid line gives the average radius of the orbital
according to the hydrogenic formula, Eq. (5).

_Magnetic field-induced crossings._ For the field-induced resonances we focus
on the first one between exciton states belonging to the multiplets with
principal quantum numbers $n$ and $n+1$, as shown in detail in the bottom
panel of Fig. 2 for field strengths up to $2$ T. At these resonances we
observe systematic crossings, in contrast to the general trend of
anticrossings in the spectra. From lower to higher energies we clearly see
that the field strength, $B_{r}$, at which these crossings occur shifts
strongly to lower values. This can be expected from (i) the reduced splitting
between the exciton states at zero field, (ii) the enhanced field-induced
splitting of each of them involving larger angular momenta, and (iii) the
stronger diamagnetic shift of each multiplet for higher $n$.

From the data we can determine the resonant field strengths $B_{r}$ versus the
principal quantum number. This dependence is shown in Fig. 6 using a double-
logarithmic representation. The resonant fields decrease from $B_{r}=2$ T for
$n=6$ to $B_{r}=0.04$ T for $n=16$. Fitting the observed data for the Rydberg
exciton regime with a power law form reveals a dependence as $n^{-4}$, unlike
for Rydberg atoms, where one finds a $n^{-6}$-dependence Gallagher (2005).

Figure 6: Resonant magnetic field strength, $B_{r}$, at which the first
crossing between states belonging to the exciton multiplets $n$ and $n+1$
occurs, as function of the principal quantum number $n$ in a double
logarithmic presentation. Symbols give the experimental data, the line is a
fit to these data following a $n^{-4}$ law, Eq. (10).

This scaling can be well understood by applying the simplified hydrogen-like
model, which is to a good approximation justified because of the quite small
impact of the quantum defect on the large-$n$ Rydberg states, as already
demonstrated by the $n^{-3}$ scaling of the multiplet width in Fig. 4. Using
atom-like excitonic units, i.e., $2\mathcal{R}=\mu
e^{4}/(\varepsilon^{2}\hbar^{2})$ as the unit of energy, the Bohr radius
$a_{B}$ as the unit of length, and $\hbar c/(ea_{B}^{2})$ as the unit of
magnetic field, and

$\langle r^{2}\rangle_{nlm}=\frac{n^{2}}{2}[5n^{2}+1-3l(l+1)]\propto n^{4}$
(8)

for sufficiently large $n$, we obtain for the level energies to a good
approximation:

$E_{nl}\approx-\frac{1}{2n^{2}}+\frac{mB}{2}+A_{l}n^{4}B^{2}.$ (9)

Here $A_{l}$ is a constant that depends on the angular momentum $l$ of the
studied level. Neglecting the diamagnetic shift $\propto B^{2}$, we obtain for
the resonance field of the sublevels $n,l=n-1,m=n-1$ and $n+1,l=n,m=-n$, which
are involved in the crossing (corresponding to the states with the maximal $m$
of the $n$th multiplet and with the minimal $m$ of the $n+1$st multiplet), in
perturbation approach:

$B_{r}(n)\propto\frac{1}{n^{4}}.$ (10)

It is noteworthy that for the crossing of these states the diamagnetic
contribution is vanishingly small, because $A_{l}n^{4}B_{r}^{2}(n)\propto
1/n^{4}\ll mB\propto 1/n^{3}$ in Eq. (9). This is different from the atomic
physics case where one, as a rule, considers states with magnetic quantum
numbers, $m=0$ or $1$, as only these states can be observed in single photon
absorption out of an $s$-state. Correspondingly, in the atomic case the
paramagnetic term $\propto mB/2$ in Eq. (9) is not important and the
diamagnetic term $\propto A_{l}n^{4}B^{2}$ dominates Gallagher (2005). For
atoms one finds therefore a $n^{-6}$ dependence of the resonance field
strength.

###  III.3 Electric field

Let us turn now to the scaling of exciton properties in electric field. As
outlined in Sec. II, we recorded spectra for two different configurations,
taken on a $[110]$ oriented crystal. For the exciting light propagating along
the same $[110]$-direction the linear light polarization was chosen either
along $[1\bar{1}0]$ (left panel of Fig. 3) or along $[001]$ (right panel of
the same figure). In the first case the quadrupolar transitions are forbidden
so that it is easy to follow the dispersion of the $P$-excitons, which is the
first major point of this subsection. In the latter case, they are allowed, so
that $S$\- and $D$-excitons appear, allowing a more comprehensive insight into
the different states within an exciton multiplet Heckötter  _et al._ (2017).
This is exploited here to determine the fields at which states from different
multiplets come in resonance, representing the second major point of this
part. The same configuration was also used to study the polarizability of the
$S$-excitons which are well separated on the low energy flanks of the
$P$-excitons. Finally we also address exciton ionization processes, for which
we use again the first configuration (without quadrupolar transitions) to
determine the ionization field strength as well as the linewidth of
$P$-excitons.

_Scaling of $S$\- and $P$-exciton polarizability._ For the $P$-exciton
dispersion, we have to distinguish between the low-$n$ and high-$n$ states.
For low $n$ the effect of the quantum defect, namely the lifting of level
degeneracy, is particularly relevant. In electric field, this leads to the
observation of a quadratic Stark effect for the non-degenerate states, because
in centrosymmetric crystals any non-degenerate excitonic state has no electric
dipole moment without electric field application. The dipole moment is induced
by the electric field which subsequently orients it, leading to the quadratic
energy shift. This is in contrast to the linear Stark effect obtained by
degenerate perturbation theory for a multiplet of levels each having the same
energy. In this case, the degenerate states in the multiplet that become
coupled by the electric field are linearly combined such that an electric
dipole moment is established. This dipole moment only has to be oriented by
the field leading to linear energy shifts with increasing field. This
situation is to a good approximation relevant for the high $n$-range. Strictly
speaking, the exciton differentiation between small and large values of the
principal quantum number as well as between non-degenerate and degenerate
states depends on the experimental resolution: For each $n$, independent of
its value, there is a range of small fields in which a quadratic Stark shift
occurs due to level splitting in the crystal. Roughly, this is the range in
which the associated quantum defect exceeds the field induced energy shift.
This range of quadratic Stark effect decreases, however, strongly with
increasing $n$.

Figure 7: Polarizability $\alpha_{n,l}$ of the $S$\- (open triangles) and
$P$-excitons (closed squares) versus principal quantum number $n$. The line is
a fit to the data following the expected $n^{7}$ scaling law, Eq. (12).

The quadratic Stark effect is described by

$\Delta E_{nl}^{(2)}=-\alpha_{nl}F^{2},$ (11)

where $\alpha_{nl}$ is the polarizability of the exciton state $(n,l)$.
Although a quadratic Stark effect is already expected for the hydrogenic model
with $\alpha_{nl}\propto n^{6}$ due to the field induced mixing of the
multiplets with different principal quantum numbers Landau and Lifshitz
(1977), the presence of the quantum defect leads to an important change of the
polarizability scaling with $n$: Mixing becomes possible for states within a
multiplet of particular $n$, which are split due to the $\mathcal{H}_{d}$ part
of the Hamiltonian, Eq. (3). The scaling relation of the polarizability can be
understood making use of second order perturbation theory to evaluate the
quadratic Stark shift: Taking into account the dipole moment operator matrix
elements between neighbouring states with the same $n$ that scale as $e\langle
r\rangle_{nlm}\propto n^{2}$, Eq. (5), and the scaling of the energy gap
between these states $\propto 1/n^{3}$, see Fig. 4 and Sec. III.1, the
polarizability scales as

$\alpha_{nl}\propto n^{7},$ (12)

in accordance with the behavior typically observed for Rydberg atoms Hahn
(2000); Saffman  _et al._ (2010).

Here, we observe for the $P$-excitons in Fig. 3 a quadratic Stark effect for
$n\leq$ 13, in agreement with Eq. (11), while the splitting pattern for the
higher-$n$ excitons approaches the linear Stark fan of hydrogen within the
experimental resolution. In Fig. 3(a) the shift of the $P$-excitons to lower
energies with increasing field can be well resolved and increases drastically
with increasing principal quantum number, see the blue-colored feature of
lowest energy in each $n$-manifold. From the data we can assess the
polarizability $\alpha_{n,P}$ ($l=1$) which is shown by the solid squares in
Fig. 7 as function of $n$ in a double-logarithmic plot. The polarizability
increases from about 5 $\mu$eV/V2 for $n$ = 5 to about 2000 $\mu$eV/V2 for $n$
= 12. The solid lines give the fit to these data by a power law scaling with
the seventh power of $n$, in accordance with the theoretical expectations, Eq.
(12), from which a good description of the data is obtained. In Fig. 3(b), due
to the chosen polarization configuration, also the $S$-excitons can be
observed, which show also a quadratic Stark effect. Deriving also for them the
polarizability, which is possible for $n\leq 10$, one obtains the data shown
by the open triangles in Fig. 7. Within the experimental error no difference
in polarizability between $S$\- and $P$-excitons can be resolved.

_Scaling of electric field induced resonances._ Next we turn to the first
resonances occurring with increasing electric field between states of adjacent
principal quantum numbers. Similar to the magnetic field case, we find also
here a strong shift of the resonance voltage $U_{r}$ to lower values with
increasing $n$. Looking at the resonances in closer detail, we find, however,
that at these resonances levels do not cross but systematically avoid each
other, as shown in the mid panel of Fig. 3. Extrapolating the dispersions, one
can determine the resonance voltages which are shown in Fig. 8 in a double
logarithmic representation versus $n$. $U_{r}$ decreases from 8 V for $n=5$ to
about 40 mV for $n=13$. The data are in reasonably good accord with a power
law scaling like $n^{-5}$ which is in line with the results on Rydberg atoms.
Resonance voltage and field strength $F_{r}$ are connected by a simple
proportionality relation for the chosen capacitor geometry by which the field
is applied. However, care needs to be exercised in doing this conversion using
just the nominal geometric and dielectric parameters, because surface charges,
charged defects etc. can lead to depolarization effects in the crystal, so
that there may be some discrepancy between the nominally calculated and the
actually present field strength by a factor of $3\ldots 5$, as discussed in
detail in Ref. Heckötter  _et al._ (2017).

Figure 8: Principal quantum number $n$-dependence of the resonance voltage
$U_{r}$ at which the first resonance of levels belonging to the multiplets
with principal quantum numbers $n$ and $n+1$ occurs. At these resonances
avoided crossings are observed. The red line shows a fit according to a
$n^{-3}$ scaling law, Eq. (15).

This behavior can be understood again using the hydrogen-like model, because
the quantum defect is not essential for the evaluation of the resonant fields.
In the presence of an electric field the simplified Hamiltonian of the exciton
reads:

$\mathcal{H}_{E}=H_{0}+eFz,$ (13)

where we assumed that the field $\bm{F}$ is applied along the $z$-direction.
In first-order perturbation theory:

$\frac{E_{nl}}{2\mathcal{R}}=-\frac{1}{2n^{2}}+\frac{3}{2}\tilde{F}n(n_{1}-n_{2}),$
(14)

where $\tilde{F}=Fa_{B}/(2\mathcal{R})$ is the reduced electric field. Here
$n_{1}$ and $n_{2}$ are the parabolic quantum numbers being non-negative
integers with $n=n_{1}+n_{2}+|m|+1$. The resonance field strength, $F_{r}$,
can be evaluated from Eq. (14) putting $n_{1}-n_{2}\approx\pm n$ for $n\gg 1$
and for the states with highest and lowest energy of the manifold with the
result Gallagher (2005):

$F_{r}\propto\frac{1}{3n^{5}}.$ (15)

Note that the power-law dependence is different from the magnetic field case.
This is because the electric dipole moment increases as $n^{2}$ Landau and
Lifshitz (1977), whereas the magnetic field induced splitting is determined by
the paramagnetic contribution, which scales linearly with $n$ for the lowest
and highest energy states in a multiplet.

We have also determined the energy splittings at these anticrossings as
function of $n$ which are shown in Fig. 9. The data show a strong drop with
$n$, even though one has to emphasize the splitting magnitude is only slightly
larger than the linewidths of the involved transitions, resulting in
considerable error bars, despite of which the drop from about 80 $\mu$eV
splitting for $n$ = 5 to about 10 $\mu$eV for $n$ = 10 can be determined. The
data can be reasonably well described by a $n^{-4}$ fit following also from
perturbation theory. In order to estimate the anticrossing energy we follow
Refs. Gallagher (2005); popov  and evaluate the matrix elements of the
‘‘quantum defect’’ Hamiltonian (3) on the eigenfunctions of the hydrogen atom
in the parabolic coordinates $\Psi_{nn_{1}n_{2}m}(\bm{r})$, where $n_{1}$,
$n_{2}$, and $m$ are the parabolic quantum numbers. The avoided crossing
energy reads

$\delta E_{anti}=2\left|\int
d\bm{r}\Psi_{n^{\prime}n_{1}^{\prime}n_{2}^{\prime}m}^{*}(\bm{r})\mathcal{H}_{d}\Psi_{nn_{1}n_{2}m}(\bm{r})\right|.$
(16)

For the hydrogenic states with adjacent principal quantum numbers
$n^{\prime}=n\pm 1$ we have Gallagher (2005)

$\int
d\bm{r}\Psi_{n^{\prime}lm}^{*}(\bm{r})\mathcal{H}_{d}\Psi_{nlm}(\bm{r})\sim\frac{\mathcal{R}\delta_{nl}}{n^{3}}.$

Finally, taking into account that

$\sum_{l}\int
d\bm{r}\Psi_{n^{\prime}n_{1}^{\prime}n_{2}^{\prime}m}^{*}(\bm{r})\Psi_{nlm}(\bm{r})\propto\frac{1}{n-m},$

we obtain the scaling law

$\delta E_{anti}\sim\frac{\mathcal{R}\delta_{nl}}{n^{3}(n-m)}\propto 1/n^{4}.$
(17)

Using for a rough estimate $\delta_{nl}=0.5$ (Sec. III.1) we obtain for the
$n=5$ and $n=6$ multiplets an anticrossing energy of about $70$ $\mu$eV in
reasonable agreement with the data in Fig. 9.

Figure 9: Black squares: Dependence of the energy splittings due to
anticrossings at the first resonances of states from the exciton multiplets
$n$ and $n+1$, induced by the electric field, on the principal quantum number.
The red line gives a fit to the data according to a $n^{-4}$ dependence, Eq.
(17).

_Crossing vs. anticrossing_. To understand the origin of crossing or
anticrossing at the first resonance of states arising from the multiplets with
principal quantum number $n$ and $n+1$ in magnetic and electric field,
respectively, we have to consider the problem in more detail: Both electric
and magnetic field preserve the axial symmetry of the ‘‘hydrogenic’’ problem,
therefore the $z$-component of the angular momentum or ‘‘magnetic’’ quantum
number $m$ is a good quantum number. In magnetic field the first crossing
occurs between the states with $m=n-1$ (the maximal $m$ in the manifold with
$n$) and $m=-n$ (the minimal $m$ in the manifold with $n+1$). The difference
of the angular momentum components of these states is

$\Delta l_{z}=2n+1.$ (18)

This is an odd number $1$, $3$, …, so that the parity of excitonic states
which cross in magnetic field is different. In the centrosymmetric point group
$O_{h}$ the states are characterized by a definite parity as well and,
therefore, states with odd $\Delta l_{z}$ do not mix. Hence, one may expect
that the first crossing in magnetic field is indeed an _allowed_ crossing. In
electric field the situation is different. One can readily check by
perturbative calculations that the first resonance occurs for states with
$m=0$, e.g., $|n=2S\rangle+|n=2P_{z}\rangle$ and $|3S\rangle-|3P_{z}\rangle$.
These states have the same symmetry and, hence, the crossing of these states
is _avoided_.

However, one has to be careful when using this line of argumentation to
discuss the crossings/anticrossings of _observed_ states in the transmission
spectra. In fact, the consideration above neglects the symmetry of the two-
particle Bloch function of the exciton Thewes  _et al._ (2015). All observed
excitonic states (in a given geometry and polarization) interact with light
and, hence, have the _same_ symmetry, $\Gamma_{4}^{-}$. For example, the
states observed in the geometry where light with $[1\bar{1}0]\parallel y$
polarization propagates along the $[110]\parallel z$ direction, couple to the
$y$-component of the electric field and therefore transform as this
$y$-coordinate. Hence, at least the light-matter interaction, i.e., the
polariton effect Rashba and Sturge (1982); Fröhlich  _et al._ (1992);
Zielińska-Raczyńska  _et al._ (2016), can convert allowed crossings into
avoided ones. However, the polariton effect is relatively weak in cuprous
oxide crystals where the direct interband transitions are forbidden at the
$\Gamma$-point of the Brillouin zone, leading to a small oscillator strength
of allowed exciton transitions Nikitine  _et al._ (1961); Fröhlich  _et al._
(1992). The quantitative measure for the strength of the polariton effect is
the longitudinal-transverse splitting, $\hbar\omega_{nlm,LT}$, of the exciton
state with quantum numbers $(n,l,m)$. Provided that the longitudinal-
transverse splitting is smaller than the non-radiative damping of the
excitonic states,

$\hbar\omega_{nlm,LT}<\hbar\Gamma_{nlm},$

the light-matter interaction is in the weak coupling regime so that polariton
modes are not observable, and also the anticrossings are hidden in the optical
spectra.

_Ionization in electric field._ A particularly appealing feature of Rydberg
excitons in comparison to Rydberg atoms is the fact that the true high
electric field regime, in which the interaction strength with the field
exceeds the Coulomb interaction, can be reached easily in the laboratory,
while for atoms it is much harder to enter this limit. This is mostly due to
the different Rydberg energy in cuprous oxide, $\sim 90$ meV, which is more
than two orders smaller than the hydrogen atom Rydberg energy. From Fig. 3 one
sees that for elevated electric field strengths the excitons dissociate: The
absorption lines fade away and finally disappear with increasing field. The
fields required for dissociation are the smaller, the higher the principal
quantum number of the involved exciton state is. A series of spectra recorded
for the $n=11$ multiplet is shown in Fig. 10, from which the drop of the
$P$-exciton resonance with increasing applied voltage can be quantitatively
assessed. Simultaneously with the line drop the width of the resonance
increases.

On the high energy side of the $P$-exciton another peak can be seen, which can
be associated with the $F$-excitons that can hardly be identified without
applied voltage, but become more and more pronounced with increasing voltage.
At higher energies even further weak features appear. This increase of
visibility is partly related to the transfer of oscillator strength from the
$P$ to the $F$ and other higher angular momenta excitons. It is somewhat
surprising that the $F$-exciton absorption is still growing even when the
$P$-exciton absorption has basically completely vanished. A potential
explanation is the action of the centrifugal barrier in the kinetic energy
which varies with the angular momentum of the involved state. Due to its
action the particles are kept away from the field axis $z$, making these
states more robust with respect to field application with increasing $l$.
However, the complex phenomenology associated with these excitons has not yet
fully been developed so that we will not discuss it here in further detail.

Figure 10: High resolution absorption spectra of the $n=11$ exciton multiplet
recorded with a frequency stabilized laser for different applied voltages.

To assess the $P$-exciton behavior more quantitatively, we have to consider
the impact of the electric field in more detail. The associated potential
leads to a tilting of the Coulomb potential between electron and hole along
the field direction. Due to this tilting eventually tunneling of carriers
through the potential can occur for sufficiently high applied voltages.
Ultimately in the high field regime the exciton state is moved into the state
continuum so that it can no longer be observed. This behavior determines both
the absorption strength (given by the area below a resonance) and its
linewidth (given by the lifetime of the state): At low fields where tunneling
is still not relevant, both quantities should stay constant, while in the
tunneling regime we expect an exponential dependence for both the spectrum
area and linewidth.

For determining the ionization field strength we have calculated the area
below each resonance. For that purpose we have fitted each resonance with the
Toyozawa formula Toyozawa (1964); Ueno (1969); Jolk (1998)

$\alpha_{n}(E)=C_{n}\frac{\frac{\Gamma_{n}}{2}+2q_{n}(E-E_{n})}{\left(\frac{\Gamma_{n}}{2}\right)^{2}+(E-E_{n})^{2}}$
(19)

from which we obtain also the linewidth. Here $E_{n}$ and $\Gamma_{n}$ are the
energy and the damping of the state $n$. $q_{n}$ describes the asymmetry of
the observed resonance and $C_{n}$ gives its amplitude. Here we have
restricted to the voltage regime in which the $P$-exciton line represents a
well isolated line that is not influenced by adjacent absorption features.
This is the regime in which the emerging $F$-excitons are still much weaker in
comparison. Once higher angular momentum excitons become prominent, the
spectra on the high energy flank of each $P$-exciton show pronounced
modulations which we attribute to interactions with the electron-hole
continuum that are not fully understood so far. This modulation also
aggravates a lineshape analysis so that we refrain from a line analysis in
this regime. In Fig. 10 this regime corresponds to voltages higher than 0.25 V
and is decreasing for higher $n$. Then the area below each absorption peak
indeed shows to a good approximation an exponential drop with applied voltage
as soon as effects of the applied voltage set in, up to the point where the
line disappears.

From this dependence we have determined the effective ionization voltage taken
as the voltage at which the area below the resonance has dropped to $1/e$. The
drop has in fact two origins: (1) the redistribution of oscillator strength to
other excitons due to state mixing; (2) the actual dissociation of the
excitons. As all states are affected similarly by these two factors, we still
use the voltage determined in the described way as characteristic for
ionization. This voltage is plotted in Fig. 11 as function of the principal
quantum number. The data can be well described by a dependence proportional to
$n^{-4}$. This is the dependence that is also expected by setting Eq. (14) for
the exciton energy $E_{n,l}$ in electric field to zero.

Figure 11: Dependence of the dissociation voltage, determined by taking the
$1/e$ value of the area below a resonance, on the principal quantum number.
The solid line shows a fit to the data by a $n^{-4}$ dependence.

_Exciton linewidth._ Finally, we turn to the dependence of the linewidth on
the electric field strength, for which one would expect – as described above –
a transition for increasing voltage from being constant for negligible
tunneling to strongly increasing as soon as tunneling can occur until
ultimately the potential barrier is lowered to an extent that the excitons
become unbound, which is given by the field strength at which the energy in
Eq. (14) is zero. The experimental data for the excitons from $n=10$ up to
$n=16$ are shown in Fig. 12. At zero electric field we observe the well known
decrease of the linewidth with increasing $n$ that has been described already
in Ref. Kazimierczuk  _et al._ (2014). For sufficiently low excitation powers
such that effects like power broadening can be disregarded, the linewidth
dependence on the principal quantum number can be described by a $n^{-3}$ law
due to the corresponding scaling of the transition rates for photon and phonon
emission.

When applying the electric field, we find that the low lying excitons in the
series indeed follow roughly the expected dependence (dashed line in Fig. 12)
which according to the theoretical calculations should be given by Merkulov
(1974); Rashba and Sturge (1982)

$\Gamma\propto\exp{\left(-\frac{2}{3n^{3}F}\right)},$ (20)

see Appendix B for details. Somewhat surprisingly, the linewidth shows a
different behavior for the high lying excitons in the shown set of states.
Their linewidth stays constant within the experimental accuracy over the range
of fields where they can be observed until they disappear.

Figure 12: Dependence of the linewidth of the $P$-exciton resonances on the
applied voltage for different principal quantum numbers from $n=10$ up to
$n=16$. The dashed lines show a fit to the data of the excitons $n=10$ and
$n=11$ according to Eq. (20). For clarity in the left (right) panel the even
(odd) $n$ states are shown.

This behavior can be understood as follows. The exponent becomes significant
at the critical field

$F_{\Gamma}\sim\frac{1}{n^{3}}.$ (21)

Since this field is parametrically larger than the ionization field $F_{i}$
for the states with large $n$, the increase in the linewidth is not seen,
because of faster ionization of the state.

Table 1: Comparison of scaling laws with principal quantum number $n$ for Rydberg atoms and Rydberg excitons. | Rydberg atoms | Rydberg excitons
---|---|---
_Zero field_ |  |
Multiplet splitting due to quantum defect |  $\propto n^{-3}$ (except of hydrogen) | $\propto n^{-3}$
_Electric field_ |  |
Polarizability |  $\propto n^{-7}$ ($\propto n^{-6}$ for hydrogen) | $\propto n^{-7}$
Resonance field of states from multiplets $n$ and $n+1$ | $\propto n^{-5}$ | $\propto n^{-5}$
Anticrossing energy at first resonance | $\propto n^{-4}$ | $\propto n^{-4}$
Ionization voltage | $\propto n^{-4}$ | $\propto n^{-4}$
_Magnetic field_ |  |
Crossover field to magnetoexciton | $\propto n^{-3}$ | $\propto n^{-3}$
Resonance field of states from multiplets $n$ and $n+1$ | $\propto n^{-6}$ | $\propto n^{-4}$

##  IV Conclusion

In summary, we have studied the scaling of several characteristic quantities
with the principal quantum number in the series of Rydberg excitons in cuprous
oxide and drawn the comparison to Rydberg atoms. The parameters considered are
related to the energy range covered by states at zero field due to finite
quantum defects and resonant field strengths in external field. They are
comprised in Table 1. The comparison shows that for most of the considered
parameters the scaling laws are identical, even though there are differences
in absolute magnitude due to the strikingly different Rydberg energy. Despite
of the same scaling, the origin of the parameter may be quite different like
for the zero field multiplet splitting: For atoms the deviation from a $1/r$
potential leads to this splitting described by the quantum defect, while for
excitons the splitting originates mostly from the complex valence band
structure deviating from a quadratic dispersion law.

When an external field is applied, identical scaling laws hold for atoms and
excitons for all parameters, but in magnetic field we find differences between
the two systems, mostly related to the different optical selection rules that
lead to a different scaling law for the resonance field strength of the states
from the multiplets with principal quantum number $n$ and $n+1$. However, even
when the same scaling law is found like for the energy splitting at the
resonance, the physics origin might be quite different as its size in the
exciton case is determined by the crystal specific of the Hamiltonian from the
hydrogen case.

From first sight the exciton level spectrum appears to be very complex,
particularly under field application. Here, the demonstrated scaling laws are
particularly appealing as they represent some generally valid rules for the
exciton spectrum which help to develop systematics in the diversely looking
spectra. The scaling laws may be transferred in a similar form also to other
semiconductors.

###### Acknowledgements.

We gratefully acknowledge the support of this project by Deutsche
Forschungsgemeinschft (DFG) and the Russian Foundation of Basic Research in
the frame of TRR 160(projects A1 and 15-52-12012) and by the DFG in the SPP
GiRyd. MMG is grateful to the Dynasty foundation and RF President grant
MD-1555.2017.2 for partial support. MB acknowledges support by RF Government
Grant No. 14.Z50.31.0021.

##  Appendix A Landau level transitions

Figure 13 zooms into the region of absorption spectra indicated by the black
box in Fig. 1, from which the resonances that can be in lowest approximation
assigned to transitions between electron and hole Landau levels with high
quantum number can be resolved in more detail. This assignment is only
approximate as for the corresponding energy range the system shows quantum
chaotic behavior with multiple anticrossings, so that no strict
$\bm{B}$-linear behavior of spectral line energies with some Coulomb related
modification can be identified. Still the transitions bunch around these pure
Landau level transitions so that a corresponding identification becomes
possible.

Figure 13: Close-up of the black boxed region of the magnetic field spectra in
Fig. 1, which allows one to resolve higher Landau level transitions. Zooming
in even further allows detection of transitions associated with Landau level
quantum numbers of more than 70.

From the representation in Fig.13 we can identify transitions up to Landau
level quantum number 57. Zooming in further reveals transitions up to quantum
number $n$=79. Transitions associated with a particular Landau level quantum
number $n$ arise with increasing magnetic field from the $B=0$ transitions
associated with $P$-excitons with the same principal quantum number $n$.
Experiments have shown that under the chosen experimental conditions it is
very hard to observe $P$-excitons with principal quantum number higher than 25
at $T$=1.2 K without magnetic field, not only because the energy spacing is
small well below 1 meV, but also because of the decreasing oscillator strength
with increasing $n$. Field application increases the splitting between the
levels and enhances the oscillator strength through squeezing the exciton wave
function. However, at $B$ = 0.5 T the extension of the $n$=73 Landau level
orbit is still about 300 nm.

##  Appendix B Exciton ionization in electric field

Equation (14) allows us also to estimate the ionization field strength from
the condition $E_{nl}=0$:

$F_{i}\sim\frac{1}{n^{4}},$ (22)

Note that the precise numerical coefficient depends on the angular momentum of
the state due to the centrifugal barrier Gallagher (2005).

The estimates (22) and (20) can be obtained in the quasi-classical picture
considering the trajectory corresponding to an energy below barrier. With
exponential accuracy it is sufficient to study the one-dimensional motion
along the field. Two turning points can be found from the equation

$\frac{1}{2n^{2}}=1/z+\tilde{F}z,$

which yields

$z_{1,2}=\frac{1/(2n^{2})\pm\sqrt{1/(2n^{2})^{2}-4\tilde{F}}}{2F}.$

In weak fields

$z_{1}\approx 2n^{2},\quad z_{2}\approx\frac{1}{2\tilde{F}n^{2}}.$

The barrier vanishes if $z_{1}=z_{2}$. Hence $\tilde{F}=1/(4n^{4})$ yielding
the ‘‘classical’’ ionization condition (the numerical coefficient can be
specified more accurately following Ref. Gallagher (2005)). The quasi-
classical motion under the barrier provides the following action ($z_{1}\to
0$, triangular barrier approximation):

$S=\int_{z_{1}}^{z_{2}}\sqrt{1/z+Fz-1/(2n^{2})}dz\sim
F^{1/2}z_{2}^{3/2}\sim\frac{1}{Fn^{3}},$

which gives the field and principal quantum number dependence of the exponent
in Eq. (20).

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