Oxide”
Supplementary
Information on ”Giant Rydberg Excitons in Cuprous
Oxide”
1 , S. Scheel2 , H. Stolz2 & M. Bayer1,3
T.Supplementary
Kazimierczuk1 , D. Fröhlich
Information
on ”Giant Rydberg Excitons in Cuprous
1
1
2
T.
Kazimierczuk , D. Fröhlich , S. Scheel , H. Stolz2 & M. Bayer1,3
1 Experimentelle
Oxide”
Physik 2, Technische Universität Dortmund, D-44221 Dortmund, Germany
SUPPLEMENTARY
INFORMATION
Institut für Physik, Universität Rostock, D-18051
Rostock, Germany
Experimentelle
2, Technische
Universität
Dortmund,
D-44221 Dortmund, Germany
T. KazimierczukPhysik
, D. Fröhlich
, S. Scheel
, H. Stolz
& M. Bayer
2
1
1
1
2
2
1,3
3 Ioffe Physical-Technical Institute of the Russian Academy of Sciences,
2 Institut für Physik, Universität Rostock, D-18051 Rostock, Germany
doi:10.1038/nature13832
St. Petersburg, 194021, Russia
3 Ioffe
1
Physical-Technical
Institute
of theUniversität
Russian Academy
of Sciences,
St. Petersburg,
194021, Russia
Experimentelle
Physik 2,
Technische
Dortmund,
D-44221 Dortmund,
Germany
für Physik,
Rostock,
D-18051
Rostock, Germany
First we present
detailsUniversität
of the material
Cu2 O
and its electronic
structure with emphasis on the lowest exciton
3 Ioffe Physical-Technical Institute of the Russian Academy of Sciences, St. Petersburg, 194021, Russia
series (yellow series). We then describe the experimental setup and give the corrections of the nP resonances
First we present details of the material Cu2 O and its electronic structure with emphasis on the lowest exciton
due to phonon background, which allows us to extract the P exciton energies. In the next section we discuss
series (yellow series). We then describe the experimental setup and give the corrections of the nP resonances
details of the theoretical estimate of the dipole blockade efficiency and the blockade volume.
due
to we
phonon
background,
which
allows
toand
extract
the P exciton
energies.
the nextonsection
we discuss
First
present
details of the
material
Cuus
its electronic
structure
with In
emphasis
the lowest
exciton
2O
details
of
the
theoretical
estimate
of
the
dipole
blockade
efficiency
and
the
blockade
volume.
series (yellow series). We then describe the experimental setup and give the corrections of the nP resonances
Cuprous
oxide background, which allows us to extract the P exciton energies. In the next section we discuss
due to phonon
details ofoxide
the theoretical estimate of the dipole blockade efficiency and the blockade volume.
Cuprous
Cu O crystallizes
in a cubic lattice (space group
2 Institut
2
b
Oh ). The crystal unit cell and the electronic band a
E
Cu
O
crystallizes
in
a
cubic
lattice
(space
group
2
Cuprousareoxide
structure
shown in Fig. S1. The two upmost va- a
b
unit
cell
and
the
electronic
band
Oh ). The crystal
CB
E
lence bands (Γ+
Γ+
stem from Cu
7 and
8 symmetry)
structure
are shown
in
S1. The
two(space
upmostgroup
vaΓ
Cu2 O crystallizes
in Fig.
a cubic
lattice
0.45eV
3d electrons, the
lowest+ conduction band from Cu
CBb
a
lence
(Γ+
and
Γ
symmetry)
stem
from
Cu
Γ
The crystal
unit
cell
and
the
electronic
band
Oh ).+bands
7
Γ
E
and 8the next higher conduction
4s (Γ6 symmetry)
0.45eV
3dstructure
electrons,
lowest
conduction
from Cu
are the
shown
in Fig.
S1.
two upmost
va− The band
Ett=t2.17eV
Γ
symmetry).
Exciband +from O 2p+electrons
(Γ
+ next
8 higher conduction
CB
symmetry)
andΓthe
4slence
(Γ bands
(Γ7 and
stem from Cu
8 symmetry)
ΓΓ
tation6 from the two
valence
bands
to the two conCu
−
VB
Ett=t2.17eV
0.45eV
Δttt=t0.13eV
Exciband
from O 2p
(Γ8 symmetry).
3d electrons,
theelectrons
lowest conduction
band from
Cu
Γ
O
duction+bands results in four exciton series, which
Γ
Γ
tation
the two valence
to theconduction
two conk
Cu
VB
and the bands
next higher
4s (Γfrom
Δttt=t0.13eV
6 symmetry)
are named
yellow, green, blue and
violet series, as
−
Γ
Ett=t2.17eV
O
duction
bandsO results
in four(Γ
exciton
series, which
Exciband from
2p electrons
8 symmetry).
shown in Fig. S1. We are concerned
with exci- Fig. S1: Structure of Cu2 O. (a)Γ Elementaryk cell of
are
named
yellow,
green,
bluebands
and violet
as Cu2 O: The Cu
tation
from
the two
valence
to theseries,
two conO-atoms (red VB
spheres) are arranged
in a
Δttt=t0.13eV
tons from the lowest energy series (Γ+
to Γ+
tran- Fig.
7
6
S1: Structure
of Cu(yellow
Elementary
of
2 O. (a)Γspheres)
shown
in bands
Fig. results
S1. Wein are
with exciO Cu-atoms
duction
fourconcerned
exciton series,
which bcc lattice,
the
formcell
a fcc
sition). The exciton motion can be+divided
k in a
+ into Cu2 O: The O-atoms (red spheres) are arranged
tons
theyellow,
lowestgreen,
energyblue
series
to Γseries,
are from
named
and(Γviolet
6 tran-as lattice. The sub-lattices are shifted relative to each
the centre of mass motion and the 7relative
mo- bcc
the
Cu-atoms
(yellow
spheres)
form cell
acorfccof
by
quarter
ofofthe
body
diagonal,
which
Fig.lattice,
S1:one
Structure
Cu
Elementary
sition).
The
exciton
motion
can
be
divided
2 O. (a)
shown in Fig. S1. We are concerned with into
exci- other
lattice.
The
sub-lattices
are
shifted
relative
to
each
tion of electron and hole. The relative
motion
is
+
to cubic
symmetry
Oh . (b) are
Schematic
elecO-atoms
(red spheres)
arranged
in a
Cu2 O: The
the
the(Γrelative
mo- responds
tonscentre
from of
the mass
lowestmotion
energyand
series
to Γ+
7 number
6 tranby
one structure
quarter
ofaround
the (yellow
body
diagonal,
which
corcharacterized by the principal quantum
n other
tronic
band
the
center
of
Brillouin
bcc
lattice,
the
Cu-atoms
spheres)
form
a
fcc
tion
of electron
and hole.
Thecan
relative
motioninto
is
sition).
The exciton
motion
be divided
to cubic
symmetry
. (b) Schematic
h
and the orbital angular momentum quantum number responds
ΓO
irreducible
repzone
(Γ-point)
in Cu
lattice.
The
sub-lattices
shifted
relative toeleceach
2 O. Theare
i denote
characterized
the principal
quantum
numbermon tronic band structure around the center of Brillouin
the centre ofbymass
motion and
the relative
23
the band
symmetry
Oh .corother by oneofquarter
of functions
the body for
diagonal,
which
l = 0, 1, 2, . . . (labeled S, P, D, as in hydrogen). Be- resentations
and
angular
numberis Transitions
O.
The
Γ
denote
irreducible
repzone
(Γ-point)
in
Cu
2
i
tiontheoforbital
electron
and momentum
hole. The quantum
relative motion
between
the two valence
(VB) and
responds to
cubic symmetry
Oh . (b)bands
Schematic
eleccause of its band structure, Cu2 O has rather unique resentations
ofstructure
the band
functions
forcenter
symmetry
Oh .23
l characterized
= 0, 1, 2, . . . (labeled
S, P, D, asquantum
in hydrogen).
Be-n the
by the principal
number
two
conduction
bands
(CB)
lead
to
four
exciton
tronic
band
around
the
of
Brillouin
properties as compared to other direct-gap semicon- Transitions between the two valence bands (VB) and
cause
of orbital
its bandangular
structure,
Cu2 O has
rather unique
green,
and violet.
and the
momentum
quantum
number series
The Γblue
irreducible repzone denoted
(Γ-point) yellow,
in Cu2 O.
i denote
ductors like e.g. GaAs. One unusual property is that the two conduction bands
(CB) lead to four exciton
properties
as.compared
semiconl = 0, 1, 2,
. . (labeledtoS,other
P, D,direct-gap
as in hydrogen).
Be- resentations of the band functions for symmetry Oh .23
denoted yellow, green, blue and violet.
the upmost valence and the lowest conduction band series
ductors
like
GaAs.
One unusual
property
that Transitions between the two valence bands (VB) and
cause of
itse.g.
band
structure,
Cu2 O has
rather is
unique
are derived from states of the same
ion (see above). Because
both bands have the same parity, dipole transithe
upmost valence
and the
lowest
conduction
band the two conduction bands (CB) lead to four exciton
properties
as compared
to other
direct-gap
semicontions between these band states are forbidden. Transitions
from
the ground
(fullblue
valence
band, empty
series
denoted
yellow,state
green,
and violet.
are
derived
the unusual
same ionproperty
(see above).
ductors
likefrom
e.g. states
GaAs.ofOne
is thatBecause both bands have the same parity, dipole transiconduction band) to excitons, however, obtain the additional quantum numbers of the relative motion of
tions
betweenvalence
these band
states
are forbidden.
from the ground state (full valence band, empty
the upmost
and the
lowest
conduction Transitions
band
electron and hole. The total symmetry of the exciton is thus given by the direct product of the symmetries
conduction
to excitons,
however,
obtain
the additional
quantum
the parity,
relativedipole
motion
of
are derivedband)
from states
of the23same
ion (see
above).
Because both
bandsnumbers
have theof
same
transiof the bands and the envelope :
electron
and
hole.
The
total
symmetry
of
the
exciton
is
thus
given
by
the
direct
product
of
the
symmetries
tions between these band states are forbidden. Transitions from the ground state (full valence band, empty
Γex = Γv ⊗ Γc ⊗ Γenv
23
ofconduction
the bands and
thetoenvelope
band)
excitons,: however, obtain the additional quantum numbers of the relative motion of
Γv ⊗ Γisc thus
⊗ Γenv
electron and hole. The total symmetry ofΓthe
exciton
given by the direct product of the symmetries
ex =
23
of the bands and the envelope :
1
Γex = Γv ⊗ Γc ⊗ Γenv
1
+
7
yellow
green
blue
violet
+
6
yellow
green
blue
violet
8
+
6
8
8
yellow
green
blue
violet
+
8 +
+6
7
+
8
g
g
SO
SO
g
+
7
+
8
SO
1
W W W. N A T U R E . C O M / N A T U R E | 1
RESEARCH SUPPLEMENTARY INFORMATION
Fig. S2: Scheme of the experimental setup. CCD, charge-coupled device camera; P1-P3, Glan polarizers; λ/2, half wave plates; cryostat, Oxford Variox or Oxford Spectromag; VA, variable attenuators.
+
Γ+
7 ⊗ Γ6 ⊗











Γ+
for S envelope
1
Γ−
for P envelope
4
+
+
Γ3 ⊕ Γ5 for D envelope .
..
.
Because of parity, S excitons are dipole forbidden but quadrupole allowed, whereas P excitons are
dipole allowed24 . Because of their strong coupling to light, the excitons discussed so far (free excitons)
in general have to be considered as polaritons. Polaritons are exciton–photon mixed states25 that move
through the crystal with their group velocity vg , which can be derived from their dispersion relation. For the
quadrupole allowed 1S excitons of the yellow series the polariton character was clearly demonstrated26–28 .
Although our results are well described within the exciton picture, it will be interesting to analyse the P
series within a multi-resonant polariton concept.
Experimental setup
In Fig. S2 we present the schematic setup of our experiment. The setup serves two purposes:
(i) classical transmission spectroscopy by use of a white light source and a monochromator,
(ii) high resolution spectroscopy by use of single frequency dye lasers.
2
2 | W W W. N A T U R E . C O M / N A T U R E
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We started our experiments by use of a broadband white light source (Energetiq LDLS) in front of the
sample and a double monochromator in second order with a CCD camera as a detector (spectral resolution
of about 10 µeV). In our first try we detected resonances only up to n = 12. We discovered that broadband
white light leads to a quenching of higher excitons, probably due to the fact that, besides the simultaneous
excitation of all P excitons from the yellow series, also excitons from the other series (green, blue and
violet series, see Fig. S1) and even free carriers beyond the band gap are excited, which leads to multiple
interactions, potentially destroying high-n excitons. Inserting cut-off and interference filters in front of the
sample brought some improvement (resonances up to n = 13 were detected). After inserting a small double
monochromator (Solar MSA-130, bandwidth about 0.1 nm ∼ 300 µeV), we were able to detect resonances
up to n = 17 (Fig. S3). Because of the limited spectral resolution of the detection monochromator (about
10 µeV) these resonances are broader than the resonances determined with the laser. The limitation of the
bandwidth of the illumination monochromator to about 300 µeV was a possible reason, why only resonances
up to n = 17 were resolved.
Optical density
We then used a single frequency dye laser.
Most of the measurements were done using a Co1.5
n=8
white-light
herent 899-21 laser with a linewidth of about 5 neV
n=9
scanned laser
corresponding to 1.2 MHz. We first focused the
laser down to about 50 µm spot size. In that way
1.0
we obtained resonances up to n = 25, if the laser
power was cut down to about 500 nW. A single
frequency laser has a broad band background due
0.5
to ASE (amplified spontaneous emission). Sending
the laser through the Solar-monochromator did not
2.1705
2.1710
2.1715
bring any further improvement. We improved the
Energy (eV)
uniformity of the laser cross-section by inserting a
fibre and obtained the best results by use of a 2 mm Fig. S3: Absorption spectra of P excitons of Cu2 O
spot size of the laser on the crystal. This rather large from a 34 µm thick sample at 1.2K. Upper trace:
spot size has the advantage that laser power levels Spectrum measured using a white light source with
of several µW could be used without running into a resolution of 10 µeV. The highest observed resonance n = 17 is marked with an arrow. Lower trace:
any detectable exciton blockade effects. For the balSpectrum measured with a single frequency dye laser
anced detection system (NewFocus Nirvana 2007) with a line width of 5 neV, n = 25 is the highest resopower levels of > 5 µW were necessary to guaran- nance resolved. The spectra are shifted vertically for
tee linearity and low noise operation. The balanced clarity.
detection led to a suppression of laser fluctuations
to less than 0.1%. A careful alignment of sample and reference beam on the detector was important to
achieve sufficient resolution of the small absorption signals. We tried several different beam splitter setups.
We finally decided to use a Glan prism as a beam splitter. The laser system allows an automatic tuning of
20 GHz (83 µeV) only. The spectral range from 2P up to 25P, however, corresponds to 27 meV. Besides
the electronically controlled thick etalon there are two other rougher tuning elements (Lyot-filter and thin
etalon). The tuning of the laser was automated with use of a stepper motor for the micrometer screw of
the Lyot-filter and also for the potentiometer of the thin etalon. The wavelength setting was controlled by a
high resolution wavemeter (HighFinesse WSU). All tuning and diagnostic elements were implemented in a
LabView program, which allowed us to control spectral range, scan speed, etc. For the measurements of the
intensity dependence (Fig. S2), it was of importance to stabilize the laser power. For that purpose we used
a liquid crystal-based noise eater BEOC LPC (Laser Power Controller in Fig. S2).
3
W W W. N A T U R E . C O M / N A T U R E | 3
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The bulk of experience gained in optimizing the single color transmission studies could be exploited
in setting up the two-color pump-probe experiment. For these experiments we additionally used a second
tunable single-frequency dye laser Matisse DS with a line width of 1 neV, corresponding to about 250
kHz band width. One of the two lasers (Matisse) was used to measure transmission through the sample as
described above, while the other was focused at the same spot but not detected after passing through the
sample. The overlap of the two beams was controlled by use of a 50µm pin-hole.
The samples were strain-free mounted in an Oxford Variox cryostat. For the magnetic field measurement we used an Oxford Spectromag 4000 cryostat. The cryostat is fitted with a split-coil superconductive
magnet. In the reported measurements a magnetic field of 0.8T was applied in the Faraday configuration
(the field is oriented parallel to the optical axis, along which laser beams propagate).
Corrections of nP resonances due to phonon background
The measured absorption lines were fitted using the formula10–12 :
Γn
2
αn (E) = Cn + 2qn (E − En )
Γn
2
2
+ (E − En )2
,
(S1)
where the amplitude Cn is proportional to the oscillator strength, the width Γn is related to the exciton
lifetime through τn = πh̄/Γn , the exciton resonance energy En is taken without the phonon background,
and the parameter qn describes the line asymmetry. In Table S1 we present the experimentally measured
positions of the absorption lines and the exciton energies En after unfolding using Eq. S1. For n = 2 the
relative shift from the unfolding amounts to more than 1 meV and for higher-n excitons it strongly decreases
similar to the linewidth (i.e., roughly proportional to n−3 ).
Description of exciton series by Rydberg formula including quantum defects
As shown in the manuscript for the P-excitons, the exciton energies is well described by the Rydberg formula
.
− Ry
n2
In hydrogen, levels of the same principal quantum number n are degenerate with respect to the orbital
angular momentum l (neglecting the Lamb shift). In cuprous oxide, however, we find in combination with
data recorded earlier29 for a fixed n small, yet systematic splittings between states of different angular
momentum (Fig. S4), which arise from a deviation of the electron-hole interaction potential from a pure
1/r-dependence due to the crystal environment.
The deviation arises from effects such as the central-cell correction due to band non-parabolicity, the
electron-phonon interaction etc. Deviations from the Rydberg formula are known also from the physics of
Rydberg atoms: The spectrum of hydrogen-like alkali atoms differs from the spectrum of hydrogen for low
angular momentum states because the valence electron interacts with the ionic core. The Pauli repulsion
and the dipole polarization of the rump ion’s electron cloud leads to a larger binding energy compared to
hydrogen.
From atomic physics it is established that the deviations of the energy levels from the Rydberg formula
can be accounted for by the concept of quantum defects in which in the Rydberg formula the following
replacement is made n → n − δl , where the δl is the quantum defect8 . Also in the context of semiconductors
4
4 | W W W. N A T U R E . C O M / N A T U R E
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Table S1: Energies of yellow excitons in Cu2 O. Data for the P excitons were measured on 34 µm
sample at 1.2K. Difference between the positions of the absorption maxima and exciton energies originate
from phonon background according to Eq. S1. Note, that since the experimental data are taken at a
photon momentum h̄K, the band gap energy at K = 0 is lower from the given value by the kinetic energy
∆E = h̄2 K 2 /2M ≈ 24 µeV, where the exciton mass M = 1.56m0 . Data for S and D series is taken from
Ref. 29.
5
W W W. N A T U R E . C O M / N A T U R E | 5
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the quantum defect concept has been already introduced theoretically30 . Hence, the resonance energies are
given by
Ry
.
(S2)
En,l = Eg −
(n − δl )2
By fitting Eq. S2 to our data for the P excitons we find a quantum defect for the P series δP = 0.23, which
describes the resonance energies of states with n > 10 extremely well.
Estimation of dipole blockade radius
A rough estimate of the exciton blockade volume
can be obtained from the absorbed laser energy,
given by the absorbed laser power times the exciton lifetime for which we use an upper boundary of
1 ns, based on the linewidth data. A pronounced
reduction of the absorption is obtained already for
100 µW absorbed power, resulting in an absorbed
energy of about 100 femtoJoule. With an exciton
energy of more than 2 eV we obtain a number of
about 300 000 excitons within the laser excitation
spot, having a diameter of about 2 mm, corresponding to one exciton in an volume of 300 µm3 . From
this we derive a blockade radius of about 5 µm.
2
Energy difference (meV)
We stress that this estimation of the quantum
defect is rather crude due to limited data about other
exciton series. Further studies are required to assess
whether in the case of excitons quantum defect can
be proven to be independent on n.
S
D
F
1
0
-1
3
4
5
6
7
n
Fig. S4: Effect of quantum defect on exciton series. The plot presents the differences between
known energies of S, D, as well as F excitons and
P excitons of the same principal number n. Data
for S and D series were taken from Ref. 29. Solid
lines represent n−3 fits revealing the dependences of
the quantum defects on exciton angular momentum:
This estimate of the dipole blockade vol- δS −δP = 0.17, δD −δP = −0.23, and δF −δP = −0.16.
ume can be done more quantitatively either based From the independent fit to the P exciton energy seon the experimentally measured saturation under ries δP ≈ 0.23.
strong laser excitation or based on theoretical calculations of the dipole-dipole interaction. In case of
the experiment-based estimate we use an expression
for the blockade efficiency Sn derived in the same
way as Eq. 1 of the manuscript:
ηα0 Vblockade T1
,
Sn =
Aexc h̄ω
(S3)
where α0 ≈ 20 mm−1 is the peak absorption without the blockade, Vblockade is the volume around the
exciton unavailable for absorption, Aexc ≈ 3 mm2 is the laser spot area, h̄ω ≈ 2.1 eV is the energy of the
exciton and T1 is the n-dependent exciton lifetime, assuming that the linewidth is radiatively limited. The
lower bound for T1 is given by the exciton linewidth: T1 ≥ Γ2h̄n ≈ 7 · 10−14 s · n3 , using the experimental
data. The factor η = 0.5 accounts for the reflection losses on the cryostat windows and the crystal surface.
For simplicity we assume here a constant value of α0 . This simplification corresponds to the assumption
that the deviation from the power law for highest-n excitons (as shown in Fig. 2 in the manuscript) is due to
6
6 | W W W. N A T U R E . C O M / N A T U R E
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local inhomogeneity of the crystal. By solving the above equation for Vblockade we get:
Vblockade =
Sn Aexc h̄ω
1.3 · 106 mW · µm3
≤
Sn .
ηα0 T1
n3
(S4)
3
Blockade volume
The second approach to estimate the blockade volume is based on an analysis of the dipoledipole interaction between two excitons at a distance R. The dipole-dipole interaction may be
discussed either in the resonant or nonresonant
regime and can be computed either from perturbation theory or from an exact diagonalization of
the static dipole-dipole interaction potential. The
former method can be used to obtain the scaling of the interaction with the principal quantum
number n: At large spatial separations between
excitons, the dipole-dipole interaction contributes
to nondegenerate second-order perturbation theory
∝ |Vdd |2 /∆, where ∆ is the energy difference
between neighboring dipole-allowed states. The
strongest contribution comes from coupling with
the states of the same n, which would be degenerate
for a perfect 1/r potential. As mentioned, lifting
of the degeneracy between those states can be described using the quantum defect concept, according to which the resonance energy is given by Eq.
S2.
V blockade ( μm )
From inserting the fitted values for Sn (see manuscript) into this formula we obtain the blockade volumes
shown in Fig. S5 as function of the principal quantum number. The blocked volume increases drastically
with n by more than two orders of magnitude from n = 12 upwards.
10
3
10
2
10
1
experiment
theory
12
14
16
n
18
20
22
24
Fig. S5: Estimate of the blockade volume. Red
squares give experimental data for blockade volume
Vblockade calculated from the Sn values, solid line represents results of theoretical calculation. For n = 24
the corresponding critical radius Rc of the blockade
volume amounts to about 10 µm.
The energy difference between neighboring
dipole-coupled states scales as n−3 , both in case of the same n and different n states. The dipole moments
increase as n2 , similar to the size of a Rydberg state. The dipole-dipole interaction potential Vdd is quadratic
in the dipole moments and scales with the third power of the distance between interacting excitons: Vdd ∝
(p1 p2 ) /R3 , where p1 , p2 are the dipole moments of the interacting excitons. Hence, the nonresonant van
der Waals interaction takes the form:
|Vdd |2
n4 /R3
EvdW (n) ∝
∝
∆
n−3
2
∝
n11
.
R6
(S5)
The proportionality constant C6 (n = 1) can be calculated by adding up the contributions from all dipolecoupled states. On the other hand, for closer separation of the excitons, i.e., when the dipole-dipole interaction becomes comparable or larger than the energy spacing between the exciton levels, the interaction
contains also resonant contributions proportional to Vdd . By the above arguments, this means that this
Förster-type interaction takes the form:
EF (n) ∝ Vdd ∝ n4 /R3 .
(S6)
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In this case the proportionality constant C3 (n = 1) can be calculated by diagonalizing the dipole interaction
within the subspace of dipole states considered as degenerate.
As discussed in the manuscript, we define a blockade radius by comparing the dipole-dipole interaction strength with the linewidth of the excitonic line, which scale falls into the Förster regime. The C3
coefficient for the interaction of two particular P states can be computed from the eigenvalues of the (static)
dipole-dipole interaction potential in the two-exciton basis containing the nearest dipole-allowed states.
In our calculation we include S and D states with the same and adjacent n. We neglect any retardation
effects that might contribute at larger separations. Taking into account that the excitons reside in a host
medium with a static permittivity of = 7.5 reduces the C3 coefficient from its nominal free-space value.
Thus, we estimate the C3 coefficient for the resonant interaction between two excitons in Rydberg states as
C3 /n4 ≈ 6 · 10−4 µeV µm3 . Consequently, the expected blockade volume is:
4 C3
≈ 3 · 10−7 µm3 · n7 .
Vblockade = π
3 Γn /2
(S7)
The results of these calculations are shown by the open symbols interpolated by a solid line in Fig. S5.
Here we take into account the influence of a single exciton only onto the absorption. For the van der Waals
form of the interaction the dependence looks similar. The calculations reproduce the increase of the blocked
volume with increasing principal quantum number quite well. The calculated volumes are comparable to
the experimentally estimated ones even though they tend to be somewhat smaller by a factor of 2 compared
to the estimates from experiment. This is still a reasonable agreement, given the simplicity of our estimates
done on purpose for transparency of the model, without trying to enforce agreement. For n = 24 the radius
of the blockade volume would be about 10 µm, almost an order of magnitude larger than the exciton Bohr
radius of this state, suggesting an enormous strength of the dipolar interactions.
Exclusion of alternate explanations for the weak absorption of high-n excitons
As discussed in the main text, the absorption strength for high-n excitons is weaker than the extrapolation
of the dependence for low-n excitons, which follows the n−3 scaling expected for isolated P excitons. We
have attributed this to the effect of the dipole blockade. For completeness, we discuss and exclude possible
alternative explanations.
At first sight, one may attribute the drop to thermal effects. The energy separation between the highest
P excitons (n = 25) and the ionization continuum amounts to only about 140 µeV, which is comparable
to the thermal energy at 1.2K. Yet their line width is below 3 µeV. This shows that scattering of these
Rydberg excitons by phonons and thermal excitation into the ionization continuum is not efficient. In any
case thermal scattering would only broaden an absorption line, but not reduce its area. Therefore it can be
safely ruled out as origin for the absorption drop.
The reduction may also point towards an invalidity of the electric dipole approximation, which is
based on the assumption that the product of the light wave number k and the extension of the state wave
function r is small compared to unity (kr 1). Usage of this approximation may be challenged due to
the large exciton size being comparable to or even considerably larger than the light wavelength. While
this issue is worth further investigation, any influence related to it would not depend on the laser intensity,
as observed in the experiment. The same argument applies to the temperature-based mechanism discussed
above, because the laser power causing bleaching of high-n excitons is well below the power level needed
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to induce any measurable heating effects. By monitoring the crystal transmission under modulated-in-time
laser excitation, we verified also that the excitation induced changes of the absorption do not arise from
long-lived exciton populations formed after relaxation of optically injected excitons. An example for such
a long-lived exciton is the paraexciton which is the spin-triplet configuration of the 1S ground state exciton
with lifetimes in the µs-range.
Supplementary references
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Groups (The MIT Press, Cambridge, 1963).
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phys. stat. sol. (a) 43, 11 (1977).
25. Andreani, C., L. Exciton-polaritons in bulk semiconductors and in confined electron and photon systems in Strong Light-Matter Coupling: from Atoms to Solid-State Systems (World Scientific, Singapore,
2014).
26. Fröhlich, D. et al. Coherent propagation and quantum beats of quadrupole polaritons in Cu2 O. Phys.
Rev. Lett. 67 (1991).
27. Fröhlich, D. et al. High resolution spectroscopy of yellow 1s excitons in Cu2 O. Sold State Comm. 134,
139 (2005).
28. Fröhlich, D., Brandt, J., Sandfort, C., Bayer, M. & Stolz, H. High resolution spectroscopy of excitons
in Cu2 O. phys. stat. sol. (b) 243, 2367 (2006).
29. Uihlein, C., Fröhlich, D. & Kenklies, R. Investigation of exciton fine structure in Cu2 O. Phys. Rev. B
23, 2731 (1981).
30. Wu, W. & Fisher, J., A. Exchange between deep donors in semiconductors: A quantum defect approach.
Phys. Rev. B 77, 045201 (2008).
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