ARTICLE OPEN
Controlling the photon number coherence of solid-state
quantum light sources for quantum cryptography
Yusuf Karli
1,9
✉, Daniel A. Vajner
2,9
, Florian Kappe
1,9
, Paul C. A. Hagen
3
, Lena M. Hansen
4,5
, René Schwarz
1
,
Thomas K. Bracht
6,7
, Christian Schimpf
8
, Saimon F. Covre da Silva
8
, Philip Walther
4,5
, Armando Rastelli
8
, Vollrath Martin Axt
3
,
Juan C. Loredo
4,5
, Vikas Remesh
1
, Tobias Heindel
2
, Doris E. Reiter
6
and Gregor Weihs
1
Quantum communication networks rely on quantum cryptographic protocols including quantum key distribution (QKD) based on
single photons. A critical element regarding the security of QKD protocols is the photon number coherence (PNC), i.e., the phase
relation between the vacuum and one-photon Fock state. To obtain single photons with the desired properties for QKD protocols,
optimal excitation schemes for quantum emitters need to be selected. As emitters, we consider semiconductor quantum dots, that
are known to generate on-demand single photons with high purity and indistinguishability. Exploiting two-photon excitation of a
quantum dot combined with a stimulation pulse, we demonstrate the generation of high-quality single photons with a controllable
degree of PNC. The main tuning knob is the pulse area giving full control from minimal to maximal PNC, while without the
stimulating pulse the PNC is negligible in our setup for all pulse areas. Our approach provides a viable route toward secure
communication in quantum networks.
npj Quantum Information (2024) 10:17 ; https://doi.org/10.1038/s41534-024-00811-2
INTRODUCTION
Single photons are an essential resource for future high-security
communication networks, with applications like measurement-
based or distributed quantum computing and quantum crypto-
graphy
1–3
. Every quantum information protocol has its unique set
of practical requirements
4
. While early quantum key distribution
(QKD) protocols
5,6
primarily relied on high single-photon purity,
more advanced schemes have further requirements such as high
indistinguishability, for example, in quantum repeaters or
measurement-device-independent (MDI)-QKD, which relies on
remote two-photon interference
7,8
. The search for efficient
single-photon sources has led to semiconductor quantum dots
9
,
thanks to their high single-photon purity
10
, brightness
11
, indis-
tinguishability
12
, scalability
13
, and above all, versatility in emission
wavelength selection.
Photon number coherence (PNC)
14,15
is another crucial quantity
relevant to the security of single-photon quantum cryptography
schemes. In the ideal case, a single photon can be represented by
the one-photon Fock state, but due to imperfections or coupling
to the environment, other Fock states can become occupied as
well. The unwanted occupation of the two-photon Fock state is
reflected by a reduced single-photon purity. In contrast, the
occupation of the vacuum state is not accounted for in a
correlation measurement. Instead, the PNC gives information on
the influence of the vacuum state, as it is defined as the phase
relation between vacuum and one-photon Fock state. This
deviation from the one-photon Fock state can compromise
security
16,17
, but might also be a resource for advanced QKD
protocols
18
.
For most protocols
18
, the PNC compromises the security
16,17
and therefore must vanish. Security is impaired by side-channel
attacks enabled by the fixed relative phase between different
photon number states
19,20
. While there are more general security
proofs that allow a non-zero PNC, they lead to lower key rates, as
some of the bits must be devoted to compensate for the
additional information leakage towards an eavesdropper
21
.To
achieve zero PNC in practice, actively phase-randomized single
photons can be used, as typically implemented for faint laser
pulses
22,23
. Otherwise, a suitable excitation scheme without PNC
must be chosen, which might however deteriorate other single-
photon properties
14,18
.
Due to its versatility, the excitation scheme presented here
covers the requirements of a broad range of quantum crypto-
graphic protocols. An overview of various applications in the
context of PNC and indistinguishability requirements is given in
Fig. 1. For example, established protocols like BB84
5
, decoy-
BB84
24
, 6-state-protocol
25
, SARG04
26
, LM05
27
and primitives like
strong quantum coin flipping
28,29
, unforgeable quantum
tokens
30,31
, quantum bit commitment
32
or quantum oblivious
transfer
33
require the absence of PNC to ensure, for instance,
security in QKD or fairness in coin-flipping protocols. On the other
hand, there exist protocols that benefit from a finite amount of
initial PNC like MDI-QKD when done with phase encoding
8
or
twin-field QKD protocols
34
to know and set the initial phase
18
.
Therefore, besides proper knowledge of the degree of PNC, one
also requires a way to tune it, thereby enabling a wide range of
quantum cryptographic applications.
In this work, we achieve tailored degrees of PNC from a
quantum dot photon source, on-demand, assuring high purity and
1
Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria.
2
Institute of Solid State Physics, Technische Universität Berlin, 10623 Berlin, Germany.
3
Theoretische Physik III, Universität Bayreuth, 95440 Bayreuth, Germany.
4
University of Vienna, Faculty of Physics, Vienna Center for Quantum Science and Technology
(VCQ), Vienna, Austria.
5
Christian Doppler Laboratory for Photonic Quantum Computer, Faculty of Physics, University of Vienna, Vienna, Austria.
6
Condensed Matter Theory,
Department of Physics, TU Dortmund, 44221 Dortmund, Germany.
7
Institut für Festkörpertheorie, Universität Münster, 48149 Münster, Germany.
8
Institute of Semiconductor and
Solid State Physics, Johannes Kepler University Linz, 4040 Linz, Austria.
9
These authors contributed equally: Yusuf Karli, Daniel A. Vajner, Florian Kappe.
✉email: yusuf.karli@uibk.ac.at
www.nature.com/npjqi
Published in partnership with The University of New South Wales
1234567890():,;
indistinguishability. Our optical excitation protocol helps gen-
erate single-photon states in a well-defined polarization basis,
and the photon counts are almost twice as large as achieved by
the resonant excitation. We therefore set the stage for our
quantum dot platform for its use in advanced cryptographic
implementations.
RESULTS
Two-photon excitation of quantum dots
Being excellent photon sources, we choose to work with
semiconductor quantum dots. The quantum dot can be modeled
as a four-level system with the ground state g
ji
, two linearly
polarized exciton states xV=H
and a biexciton state xxjias
depicted in Fig. 2a. Among the different excitation protocols
available
35
, we choose the ones that are based on resonant two-
photon excitation (TPE) of the quantum dot from the ground state
g
jiinto the biexciton state xx
ji
36,37
with the experimental setup
shown in Fig. 2d.
The TPE excitation leads to Rabi rotations between the ground
and the biexciton state. The TPE pulse area πis defined such that
the biexciton state occupation is maximum. A simulation of TPE
Rabi rotations, i.e., the biexciton occupation as a function of pulse
area, is shown in Fig. 2b. The oscillatory behavior of the Rabi
rotations is clearly visible also in the measurement (Fig. 2c). The
TPE excitation is a coherent excitation method, in contrast to the
incoherent excitation methods like phonon-assisted state pre-
paration
38
. Thus, the coherence between ground and exciton state
builds up, whenever a superpositon state is reached, e.g., for a π/2
pulse. In contrast to the ideal case (dashed lines), in reality, the
coupling to the environment, i.e., to phonons and radiative losses
into the photonic environment, disturbs the Rabi rotations. The
simulations for an environment at T=1.5 K (solid lines) show that
the maximal occupation is reduced, but more importantly for later,
the oscillation of the coherence is less pronounced. In particular, it
does not reach zero anymore at the π-pulse.
From the biexciton state, the system relaxes into either
horizontally xH
jior vertically xV
jipolarized exciton state, from
which we collect only horizontally (H) polarized photons. We call
this scheme relaxation into the exciton (reX). The reX scheme is
advantageous over the direct, resonant excitation of the exciton,
due to the suppressed re-excitation and therefore provides high-
purity photon states
10
. Because the exciting laser energy is
different from the emitted photon energy, a challenging cross-
polarization filtering is avoided which is also achieved by several
other recently proposed excitation schemes
39–45
. However, the
indistinguishability of the single photons via the reX scheme
suffers greatly from the spontaneous decay of the biexciton
46
and
if a specific polarization is required, the photon output is reduced
due to the two available decay channels.
An improved protocol to overcome these problems uses an
additional stimulation laser pulse following the TPE pulse
47
. This
stimulated preparation of the exciton (stiX) scheme can generate
higher indistinguishability exciton photons due to the reduced
time jitter
48–50
. Because the stimulation pulse determines the
polarization of the emitted photon, the photon counts in that
polarization state are also enhanced up to a factor of two (see also
Fig. 2c). Although the presence of PNC under resonant excitation
and reX has been investigated before
14,18
, it remains to be seen if
PNC exists in the stiX scheme. Additionally, assessing the
controllability of PNC is essential for advancing optical preparation
schemes of quantum dot states for quantum cryptography
applications.
Definition of photon number coherence
Following the optical excitation, the quantum dot will emit a
specific photon state, which can either be a pure state or a
statistical mixture. In a pure photon state Ψji¼
P1
n¼0cnnji,
written in the photon number Fock basis with eigenstates n
ji
and the complex coefficients c
n
,wedefine PNC as the absolute
value of the coherence between the Fock states. For QKD based
on single photons, as considered in this paper, the PNC refers to
the coherence between the Fock states 0
ji
and 1
ji
. More generally
and accounting also for statistical mixtures, we employ a density
matrix description using
ρ¼ρ0;0ρ0;1
ρ1;0ρ1;1
!
with PNC ¼jρ0;1j;(1)
ρ
1,1
(ρ
0,0
) being the occupation of the one (zero)-photon state and
ρ
0,1
being the coherence. We recall that it holds that
∣ρ
0,1
∣
2
≤ρ
1,1
ρ
0,0
with equality in the case of a pure state. The
inequality implies that for ρ
1,1
=1orρ
0,0
=1 the PNC vanishes,
while for ρ
0,0
=ρ
1,1
=1/2 it is maximal.
There are several factors that affect the PNC. One aspect is the
imperfect control of the electronic system, from which the photon
properties are inherited. The controllability of the electronic
system of a quantum dot is known to be strongly influenced by
the interaction with the lattice vibration, i.e., the phonons
35,51
.In
addition, phonons can degrade the coherence properties of the
photons
52
and therefore also the PNC. In addition to phonons,
there are other loss channels that can further affect the photon
properties, for example, losses of photons into other undetected
modes.
It is likewise important to consider the measurement process.
To detect PNC, a phase-evolving Mach-Zehnder Interferometer
(MZI) is employed
14
. The outputs of the MZI are simultaneously
recorded with two avalanche photodiodes (APDs). The count rates
N
1
,N
2
in the APD result in the visibility
vi¼Nmax
iNmin
i
Nmax
iþNmin
i
:(2)
Photon number coherence
Indistinguishablity
BB84, 6-state,
SARG04, LM05,
QBC, UT, CF, OT
MDI-QKD,
quantum repeater,
teleportation
MDI-QKD with
phase encoding
TF-QKD with
single photons
Fig. 1 Overview of various quantum information protocols. The
protocols are sorted by their requirements on indistinguishability I
and PNC, specifically focusing on discrete variable cryptographic
protocols using polarization or time-bin encoding. Protocols that
require low PNC and need no high Iare BB84
5
, decoy-BB84
24
,6-
state-protocol
25
, SARG04
26
, LM05
27
, QBC: quantum bit commit-
ment
32
, UT: unforgeable quantum tokens
30,31
, CF: quantum coin
flipping
28,29
, OT: Oblivious Transfer
59
; low PNC and high Iis
required by MDI-QKD
8
, quantum repeaters and entanglement
swapping for QKD
7,60
, quantum teleportation for QKD
61
, DI-QKD
with single photons
62
; high PNC and variable Iare needed in TF:
twin-field QKD with single photons
34
. Note that carrying out
protocols from the low PNC column with phase encoding would
require an initially defined phase, before randomizing it in a
reversible way, which requires PNC in the beginning. To illustrate
this we have also added MDI-QKD with phase encoding to the
diagram.
Y. Karli et al.
2
npj Quantum Information (2024) 17 Published in partnership with The University of New South Wales
1234567890():,;
In the case of an ideal measurement and perfectly indistinguish-
able photons, the visibility is connected to the PNC via
v¼jρ0;1j2
ρ1;1
:(3)
In the MZI used in the experiment, the interference of subsequent
single photons takes place. Phase scrambling between subse-
quent emission events leads to a further reduction of the visibility
in addition to the aforementioned phonon and loss effects. It
should be kept in mind that a vanishing visibility in the
experiment can therefore result either from vanishing PNC, phase
scrambling, or a combined effect of both.
Theoretical expectations
Initially, we perform theoretical simulations to estimate the PNC
for both reX and stiX for a quantum dot modeled as a four-level
system driven by a (classical) laser field and coupled to two
discrete photon modes. The simulation considered the presence
of three-photon states, however, only the vacuum and one-
photon states were found to be noticeably populated. Note that
the frequencies of the photon modes were set in resonance to the
quantum dot transition from the ground to the exciton state and
therefore direct two-photon emission is off-resonant. We further
account for coupling with acoustic phonons within a numerically
exact path-integral formalism
53
. In addition, we include the
relaxation between the quantum dot states accounting for
photons not being emitted into the relevant modes (see Methods
section for details of the model and calculation). We assume an
ideal detection, i.e., no phase scrambling and perfect indistin-
guishability and model the visibility via Eq. (3).
The time evolution of the biexciton and the xH
ji
exciton
occupation together with the PNC ∣ρ
0,1
∣is shown in Fig. 3a, b. The
laser sequence is shown in Fig. 3c. Both schemes start with an
excitation from the ground into the biexciton state induced by a
Gaussian-shaped laser pulse (orange curve in Fig. 3c) with a TPE
pulse area of π/2. In the reX scheme, the biexciton state then
relaxes into the exciton state via the emission of photons.
However, these photons are at a different wavelength and
therefore ignored. The exciton state is transiently occupied
a
NF
time delay
pol
VOA
PBS
BS
BS
pol
X
HWP
QWP
VOA
QD @1.5K
APD2
APD1
FBS
φ
λTPE
λstim
4f pulse shapers PNC setup
laser
slit M
slit M
G
G
wavelength (nm)
X
XX
397597
d
0
0
5
2.5
2π
π
X counts (103 cps)
cb
TPE Pulse area
TPE
|g⟩
|xH⟩
|xV⟩
|xx⟩
single
photon
stim. pulse reX
stiX
xx Occup. (ideal)
xx Occup. (1.5K)
g-xx Coher. (ideal)
g-xx Coher. (1.5K)
TPE Pulse area
02ππ
0
0.5
1
π
Fig. 2 Generating single photons with variable PNC. a Level scheme of a quantum dot consisting of the ground state g
ji
, two linearly
polarized exciton states xV=H
and biexciton state xx
ji
. Straight lines indicate laser excitation, while dashed lines denote relaxation processes
with rate γ. Both schemes start with a two-photon excitation (TPE) from g
ji!xx
ji
. In stiX, an additional H-polarized laser pulse stimulates the
transition xx
ji
!xH
ji
. We only collect H-polarized photons. bTheoretically calculated biexciton ( xx
ji
) occupation showing Rabi rotations as a
function of the TPE pulse area (red curve) and the corresponding coherence between g
jiand xx
ji
(purple curve). cExciton photon counts
recorded under reX (blue dots) and stiX (red dots) manifest the enhancement of photon counts when the emission with a specific linear
polarization is required. dSketch of the experimental setup: a Ti:Sapphire laser source producing ≈2 ps-long laser pulses, with a spectral
FWHM of 0.5 nm, is used to spectrally shape TPE and stim. pulses at appropriate wavelengths λ
TPE
and λstim using two 4fpulse shapers. A fiber-
optic delay line enables the time control of the stim. pulse with respect to the TPE pulse. An electronic variable optical attenuator (VOA) helps
sweeping the laser power. The two pulses meet at a 10:90 beam splitter (BS) and propagate to the cryostat which holds the quantum dot at
1.5 K. Emitted single photons from the quantum dot are spectrally filtered by a notch filter (NF) and sent to an unbalanced Mach-Zehnder
interferometer with a freely evolving phase on one arm (labeled as PNC setup). Two single-photon sensitive avalanche photodiodes (APD1
and APD2) detect the single-photon counts at the output arms of the interferometer. pol: linear polarizer, HWP: half-wave plate, QWP: quarter-
wave plate, BS: beam splitter, PBS: polarizing beam splitter, FBS: fiber beam splitter.
Y. Karli et al.
3
Published in partnership with The University of New South Wales npj Quantum Information (2024) 17
because it rapidly generates the desired photon relaxing further
into the ground state. The corresponding PNC (blue curve in
Fig. 3a) is almost vanishing, too, because of the incoherent
biexciton-exciton relaxation destroying the electronic coherence.
The remaining PNC can be traced back to deviations from the
ideal case caused by phonon interaction, radiative losses, as well
as relaxation into other (undesired) states in the quantum dot.
In contrast, in the stiX scheme, the stimulating pulse (red curve
in Fig. 3c) brings the biexciton coherently into xH
ji
by the
application of a π-pulse resonant to the xxji!xH
jitransition as
evidenced in Fig. 3b. The small oscillations on top of the
population-exchange result from the off-resonant driving of the
complementary transition xH
ji!gji. Because the transition to
the exciton state xH
ji
is coherent, the electronic coherence, which
translates to the PNC, is preserved. Accordingly, in Fig. 3a (red
curve), we see that as soon as the stimulating pulse sets in, the
PNC becomes very high. In other words, a timed stimulation
preparation of the exciton state recovers the PNC that is lost in the
reX scheme.
By controlling the electronic coherence through the pulse areas
of the exciting pulses, we can thus manipulate the PNC. To ensure
the best comparability, we fix the stimulating pulse to a πpulse
and vary the pulse area of the TPE pulse. We remind that TPE is a
coherent excitation and that the corresponding coherence
oscillates as a function of pulse area (cf. Fig. 2b). The time-
integrated occupation of the one-photon state occ
calc
follows the
Rabi rotations of the biexciton (cf. Fig. 2b). We have checked that
under the present conditions, the higher Fock states always have
negligible occupations.
To estimate the PNC as a function of pulse area, we consider the
time-integrated variable PNC
calc
=∣ρ
0,1
∣in Fig. 3e. The PNC
calc
should follow the coherence oscillations of the electronic system,
i.e., the coherence between ground and biexciton state. Thus, the
highest PNC is expected for pulse areas (2n+1)π/2, where also
the electronic coherence is maximal. However, due to the
incoherent relaxation process from the biexciton into the exciton,
in reX the PNC
calc
is close to zero for all pulse areas as expected.
This is confirmed by the numerical results in Fig. 3e. Only for large
pulse areas, detrimental processes due to phonons or losses lead
to some residual PNC
calc
.
The behavior changes totally for stiX, where the coherence is
preserved due to the stimulating pulse. Hence we find an
oscillating behavior of PNC
calc
as a function of the TPE pulse area.
The maxima of the PNC
calc
coincide with the maxima of the
electronic coherence as a function of the TPE pulse area, while the
minima of the electronic coherence should lead to vanishing
PNC
calc
. In Fig. 3e maxima of PNC
calc
occur for TPE pulse areas
(2n+1)π/2 and minima for pulse areas nπ. Ideally, PNC should be
zero for pulse areas nπ. In the full simulation including finite pulse
lengths and losses, the TPE pulse does not fully invert the system
and the electronic coherence is reduced (cf. Fig. 2b). Additionally,
the PNC itself is affected by environmental losses. This leads to a
residual PNC even for a TPE π-pulse.
To compare with the experiment, we also compute the visibility
v
calc
in Fig. 3d. The visibility behaves differently compared to
PNC
calc
. While a clear minimum at πis recovered, the PNC is not
maximal at π/2. Instead, due to its definition, v
calc
increases for the
even smaller TPE pulse areas. Still it remains true that, compared
to reX, the visibility for stiX shows a strong dependence on the TPE
pulse area.
Experimental data
We perform the reX and stiX experiments to test the theoretical
prediction on a single quantum dot in our setup displayed in
Fig. 2d. We note that in stiX we fix the time delay between the TPE
pulse and stimulating pulse to 7 ps, where the photon count is
maximal. A detailed description of the experiment is provided in
the Methods Section.
We start by quantifying the photon properties for reX and stiX,
at various powers (see Supplementary Table 1) measuring the
single-photon purity in a Hanbury Brown and Twiss (HBT) setup
and the indistinguishability via Hong-Ou-Mandel (HOM) measure-
ments. At πpower of both reX and stiX we validate that the
generated photons have high purity with gð2Þ
reXð0Þ¼0:0004ð1Þand
gð2Þ
stiXð0Þ¼0:0009ð1Þ. For the indistinguishability, the HOM visibility
reaches only 58(3)% under reX, while for stiX it increases to
95(6)%, in line with previous observations
48–50
. These results
already underline that stiX is advantageous over reX.
We then sweep the TPE pulse area under reX and stiX yielding
Rabi rotations for the exciton (X) photon counts (blue and red
0.0 0 20 40 60 80 100 120 140
laser
time (ps)
0.0
0.5
occupation
biexciton
exciton
10−5
10−3
10−1
ρ01 (log scale)
stiX
reX
b
c
a
calc calc
0
2
4
6
8
0.5 π π
PNC (arb. u)
TPE pulse area
one photon occ.
reX x 10
stiX
0
5
v (arb. u)
stiX
reX x1000
d
e
||
reX stiX
TPE Pulse
Stim. Pulse
Fig. 3 Theoretical predictions. a,b,cDynamics of the four-level system coupled to two-photon modes including phonons and losses
calculated via a numerically exact path-integral formalism. The exciting laser pulses with the TPE pulse (orange) and stimulating pulse (red) are
shown in c. The occupation of the biexciton and exciton state for stiX are displayed in bwith the dashed line indicating the behavior for reX.
The PNC for stiX (red) and reX (blue) is displayed in a. Note the logarithmic scale. During the stimulating pulse, the exciton becomes occupied
resulting in a rise of the PNC. d,eTime-integrated coherence PNC
calc.
(e) and visiblity v
calc
(d) as a function of TPE pulse area for both reX (blue,
magnified) and stiX (red). The TPE pulse areas of πand π/2 are marked by vertical lines. The time-integrated occupation of the one-photon
Fock state occ.
calc
is shown as a green dashed line. Due to the relaxation process, the PNC is almost lost in the reX case. In stiX, we find that the
PNC is controlled via the TPE pulse area.
Y. Karli et al.
4
npj Quantum Information (2024) 17 Published in partnership with The University of New South Wales
curves in Fig. 4c, g), and investigate the PNC. For each TPE pulse
power, we analyze the spectrally filtered X photons using a phase-
evolving MZI
14
. Its outputs are simultaneously recorded with two
avalanche photodiodes (APDs) for 20 s each. In Fig. 4d, h, we
display exemplary time traces (denoted by green and magenta
curves, representing the two detector outputs of the MZI, see also
Fig. 2d) at TPE pulse areas 0.5π,1π, 1.5π. From the time traces, we
compute the visibility according to Eq. (2) from the normalized
detector counts taking the average of the two detectors as
vexp ¼ðv1þv2Þ=2.
The visibility vexp as a function of pulse area is displayed
alongside the respective Rabi rotations in Fig. 4c, g as black dots.
Under reX, the visibility vexp is vanishing for all TPE pulse areas and
no clear dependence is found. This is in agreement with the
exemplary time traces (displayed in Fig. 4d), where indeed no
oscillations are seen for different pulse areas.
In contrast, for stiX, the PNC shows a more interesting behavior:
in the representative time traces of the MZI outputs (displayed in
Fig. 4h), we observe clear oscillations for TPE powers 0.5π, and 1.5π
and almost no oscillations at π. Accordingly, the visibilities vary
from ≈0.6 at TPE power 0.5πto being minimal at πand then rise
again until 1.5π.
From the visibilities, using the formalism from ref.
14
, we extract
the PNC
exp.
shown as the yellow line in Fig. 4c, g. In the formalism
PNC
exp.
is reconstructed via the so-called λ-parameter, which is
gained from the visibilities. Hence, a larger parameter range can
be gained for PNC
exp.
then for the visibilities (cf. SI).
The data clearly confirms the trend expected from the theory:
We find minima of PNC when exciting with TPE pulses of pulse
area nπand maxima at (2n+1)π/2. This behavior is evident in stiX,
while in reX only a small modulation is found.
Hence, we conclude that the PNC is negligible in reX, while in
stiX we have tuneable PNC controlled via the TPE pulse area.
DISCUSSION
We now set our results in the context of finding the optimal
photon source for high-security quantum networks. As indicated
before, purity, indistinguishability, and PNC are the key para-
meters that must be known when choosing an excitation scheme.
We have shown that reX generates high-purity photons, while
indistinguishability and PNC are low, and also, if filtering only a
single polarization, the photon output is reduced. Looking back at
Fig. 1,wefind that reX produces photons in the bottom left corner
with low PNC and low indistinguishability, which limits the range
of applicable protocols.
With the stiX method, photons with high purity and high
indistinguishability are generated. More importantly, the PNC can
reX stiX
-25 0 25
Time (ns)
100
103
Coinc.
Time (ns)
HOM
stiX
= 95%
52-520
0
0.5
1
Coinc.
g(2)(0) = 0.0009
-25 0 25
Time (ns)
100
103
Coinc.
Time (ns)
HOM
reX
= 58%
52-520
0
0.5
1
Coinc.
g(2)(0) = 0.0004
time time
ab
c
d
ef
g
h
Fig. 4 Controlled generation of PNC on a single quantum dot for reX and stiX. a,eMeasured purity of the generated single photons.
agð2Þ
reXð0Þ(blue), the TPE pulse power is kept at π-power, and the stim pulse is absent. egð2Þ
stiXð0Þ(red), TPE and stim pulses are kept at π-power.
b,fHOM
reX
, blue and gray shaded curves represent HOM coincidences recorded for parallel and orthogonal polarizations respectively, bfor π
TPE pulse area. HOM
stiX
, red and gray shaded curves represent HOM coincidences recorded for parallel and orthogonal polarizations
respectively, fTPE and stim pulses are kept at π-power. c,gExtracted visibilities vexp (dark-blue dots) and the reconstructed PNCexp (yellow) at
different TPE pulse areas alongside the measured X counts (blue curve for X
reX
and red curve for X
stiX
). The X counts are normalized to their
respective values at π-power. d,hExemplary time traces were recorded at the two detector outputs of the PNC setup at TPE pulse areas of
0.5π,πand 1.5π.
Y. Karli et al.
5
Published in partnership with The University of New South Wales npj Quantum Information (2024) 17
be controlled via the TPE pulse area. If the TPE power is set to
(2n+1)π/2, one obtains high PNC, enabling protocols in the top
right corner of the diagram in Fig. 1. By changing the pulse area to
nπ, the PNC is minimal while indistinguishability is maintained
high, enabling protocols in the top left corner of the diagram in
Fig. 1. For all TPE powers, stiX is the superior method for
generating single photons for protocols that require high
indistinguishability.
So far, we have discussed the TPE pulse area as the main control
parameter for the time-integrated PNC. But stiX offers a much
richer parameter space for tuning the PNC. As the stiX scheme is
composed of two pulses, there is a wide playground of control
options. We display additional control parameters in Supplemen-
tary Fig. 4. By varying the time delay between the TPE and the
stimulating pulse, first a relaxation and then a stimulation occurs.
This leads to a varying PNC as a function of time delay with loss of
PNC for increasing time delays. More remarkably, the time delay
control offers a different type of control compared to the pulse
area control, because changing the time delay also affects the
indistinguishability as indicated in Supplementary Fig. 5a.
In addition, we explored the polarization of the stimulating laser
pulse showing high PNC and high indistinguishability in the
chosen polarization basis (see Supplementary Fig. 5).
In summary, we showed the generation of single photons with
variable degrees of PNC as well as high purity, high indistinguish-
ability, and high brightness via a two-photon excitation combined
with a stimulating pulse. This is a significant step forward towards the
realization of secure quantum networks based on single photons.
METHODS
Theoretical model
For the theoretical modeling, we set up the Hamiltonian
consisting of the quantum dot system ^
HQD, the out-coupling to
two-photon modes ^
Hphoton, the excitation of the TPE ^
HTPE and the
stimulating laser pulse ^
Hstim, as well as the coupling to phonons
^
H¼^
HQD þ^
Hphoton þ^
HTPE þ^
Hstim þ^
Hphonon :(4)
In addition, we consider radiative decay and losses by a Lindblad
operator L. In the following, we describe the individual terms in
detail.
The quantum dot is modeled using four states (see also Fig. 2b)
denoted by g
jias the ground state, xH
ji
and xV
ji
as the two
excitons and xx
ji
as the biexciton. The ground-state energy is set
to zero, while both excitons have the same energy ℏω
x
, i.e., no
fine-structure splitting is assumed. The biexciton has a binding
energy E
B
such that its energy is given by ℏω
xx
=2ℏω
x
−E
B
.
^
HQD ¼_ωxxH
jih
xHjþjxVixV
hj
ðÞþ_ωxx xx
ji
xx
hj (5)
The quantum dot is coupled to two-photon modes with
polarizations Vand Hfor the out-coupling of the photons, similar
to positioning the quantum dot in a photonic cavity. We model
the photon modes by the Fock states nH
ji
and nV
ji
with the
frequency ω
c
via the annihilation (creation) operators ^
aH=V(^
ay
H=V).
The photonic modes are coupled to the quantum dot transitions
with the same strength via the coupling constant ℏg=0.05meV,
yielding
^
Hphoton ¼_ωc^
ay
H^
aHþ^
ay
V^
aV
þ_g^
aHxH
jihgjþjxxixH
hj
ðÞ
þh:c:
þ_g^
aVxV
jih
gjþjxxixV
hj
ðÞþh:c:
¼^
Hphoton
0þ^
Hphoton
coupl::
(6)
We use the Hamiltonian in a rotating frame with ω=ω
l
=ω
x
−E
B
/
(2ℏ), which corresponds to the frequency of the TPE laser pulse.
With this, the QD-photon Hamiltonian has the form
^
HQDphoton ¼_ΔωxlxH
jih
xHjþjxVixV
hj
ðÞ
þð_2ΔωxlEBÞxx
ji
xx
hj
þ_Δωcl^
ay
H^
aHþ^
ay
V^
aV
þ^
Hphoton
coupl::
(7)
The index convention of frequency differences is chosen such
that the second index is subtracted from the first, e.g.,
Δω
x−l
=ω
x
−ω
l
. We choose the photon mode to be resonant
with the quantum dot transition from the ground to the
excited state, i.e., ℏΔω
c−x
=0meV.
The TPE is modeled by an external classical laser field with
diagonal polarization in dipole and rotating wave approximation.
We consider a resonant TPE process and accordingly set the
detuning Δω
x−l
=E
B
/(2ℏ). With this, the Hamiltonian reads
^
HTPEðtÞ¼
_
2fTPEðtÞg
jih
xHjþjgixV
hj
ð
þxH
jih
xxjþjxVixx
hj
þh:c:Þ:(8)
Here, f
TPE
(t) denotes the instantaneous Rabi frequency as given by
the product of dipole moment and electric field. We use Gaussian
pulses
fTPEðtÞ¼ ΘTPE
ffiffiffiffiffiffi
2π
pσTPE
et2
2σ2
TPE ;(9)
with the pulse area Θ
TPE
and the pulse width σ
TPE
. We assign the
TPE pulse area πin the plot (cf. Fig. 3to the one which results in
the first maximum of the biexciton occupation and the TPE pulse
area of π/2 to the first maximum of the electronic coherence. In
the calculations, these values were determined numerically.
We describe the stimulating laser with the same approxima-
tions, but assume it to be horizontally polarized. Its frequency is
set to match the xx
ji
!xH
ji
transition, such that
^
HstimðtÞ¼_
2fstimðtÞeiΔωstim
ltg
jixH
hj
ð
h
þxH
ji
xx
h jÞþh:c::
(10)
Here, Δωstim
l¼ωstim
lωl¼EB=ð2_Þ. The stimulating laser’s
envelope function fstim is delayed by a time Δtcompared to the
TPE laser. We also assume a Gaussian envelope for the stimulating
pulse
fstimðtÞ¼ Θstim
ffiffiffiffiffiffi
2π
pσstim
eðtΔtÞ2
2σ2
stim :(11)
with the pulse area Θstim and the pulse length σstim. Here, a “π-
pulse”refers to a full inversion of the resonantly driven transition
for ideal conditions (without losses/phonons).
In addition we consider the coupling to longitudinal-acoustic
(LA) phonons via the deformation potential coupling. Here, ^
bk(^
by
k)
annihilates (creates) a phonon of mode kwith energy ω
k
.We
consider the typical pure-dephasing type coupling in the standard
Hamiltonian
54,55
^
Hphonon ¼_P
k
ωk^
by
k
^
bk
þ_P
k;S
γS
k
^
by
kþγS
k
^
bk
S
jiS
hj
;
(12)
coupling each mode kto the quantum dot state S
ji
, where
S∈{x
H
,x
V
,xx}. The coupling constant γS
kand the material
parameters are taken to be the same as in ref.
53
.
Both, the cavity and quantum dot, are subject to losses into the
free photonic field outside of the cavity. These losses are
described by Lindblad-superoperators, affecting the density
Y. Karli et al.
6
npj Quantum Information (2024) 17 Published in partnership with The University of New South Wales
operator ^
ρ
L^
O;δ½^
ρ¼δ^
O^
ρ^
Oy1
2
^
ρ;^
Oy^
O
hi
þ
;(13)
where ^
Ois an operator, δa rate and [. , . ]
+
the anti-commutator.
We assume that the decay processes of the quantum dot take
place with rate γand losses of the photonic modes go with the
rate κ, such that Lindblad-superoperators are
L½^
ρ:¼L
^
aH;κ½^
ρþL^
aV;κ½^
ρ
þLgjixH
hj;γ½^
ρþLgjixV
hj;γ½^
ρ
þLxH
jixxhj;γ½^
ρþLxV
jixxhj;γ½^
ρ:
(14)
The rates are chosen such that we are in the weak coupling
regime.
With the Hamiltonian and the Lindbladian terms we calculate
the dynamics of the system states via the Liouville-von Neumann
equation
d
dt
^
ρ¼i
_
^
HðtÞ;^
ρ
þL½^
ρ:(15)
As the initial state, we assume that the quantum dot is in its
ground state and no photonic excitation exists. For the numerical
integration, we use a numerically complete path-integral method,
which is described in refs.
53,56
and the parameters from Table 1,
to solve Eq. (15).
We obtain results for the full density matrix, from which we can
obtain the reduced density matrices for the quantum dots ρQD
S;S0
with S∈{g,x
H
,x
V
,xx} and for the photons ρphoton
ni;n0
i
with i∈{H,V}, by
tracing out the other degrees of freedom. We are interested in the
coherence ρ0;1¼ρphoton
0H;1H. The absolute value of ρ0;1¼ρphoton
0H;1His
referred to as PNC.
As a measure for the overall PNC at a given pulse area, we
introduce the time-integrated absolute value of the instantaneous
PNC
PNCcalc /~
ρ0;1¼Zjρphoton
0H;1Hjdt :(16)
PNC
calc
is the calculated quantity that corresponds with the
experimental quantity PNCexp below in Eq. (20).
Analogously, we define the time-integrated occupation of the
one-photon number states as
occcalc /~
ρ1;1¼Zρphoton
1H;1Hdt :(17)
We assume that the photonic space can be reduced to a two-level
system consisting of 0H
ji
and 1H
ji
. This is reasonable because the
higher-order Fock states are not occupied. We then follow ref.
14
to calculate the visibility vas measured in an MZI for a mixed
state as
vcalc ¼
~
ρ2
0;1
~
ρ1;1
:(18)
We stress that this is an estimate of the visibility, which does not
account for the imperfection of the beam splitter, higher photon
states, phase scrambling, or reduced indistinguishability. None-
theless, we expect the qualitative behavior to agree with the
experiment.
Experimental setup
Our setup (Fig. 2d) consists of a Ti:Sapphire laser source (Tsunami
3950, SpectraPhysics) producing 2.7 ps pulses (measured as
intensity autocorrelation FWHM), that is tuned to 793 nm,
enabling spectral shaping of both the TPE and stimulating (stim).
pulses via two independent 4fpulse shapers. The intensities of the
TPE and stim pulses are individually controlled via electronic
variable optical attenuators (VOA, V800PA, Thorlabs) and the
arrival time of the stimulating pulse is precisely controlled via a
fiber-optic delay line (ODL-300, OZ Optics). The two beams are
combined at a 10:90 beam splitter near the optical window of a
closed-cycle cryostat (base temperature 1.5 K, ICEOxford) where
the quantum dot sample is mounted on a three-axis piezoelectric
stage (ANPx101/ANPz102, attocube systems AG). The two beams
are focused on a single quantum dot with a cold objective
(numerical aperture 0.81, attocube systems AG).
Our sample consists of GaAs/AlGaAs quantum dots with exciton
emission centered around 790 nm grown by the Al-droplet
etching method
57,58
. The dots are embedded in the center of a
lambda-cavity placed between a bottom (top) distributed Bragg
reflector consisting of 9 (2) pairs of λ/4 thick Al
0.95
Ga
0.05
As/
Al
0.2
Ga
0.8
As layers.
The quantum dot emission is collected via the same path as the
excitation, where the exciton (X) photons are spectrally separated
from the scattered laser light and phonon side bands using a
home-built monochromator equipped with two narrow-band
notch filters (BNF-805-OD3, FWHM 0.3 nm, Optigrate). To improve
the suppression of the reflected TPE pulse we employ a cross-
polarized configuration in which two orthogonal linear polarizers
on excitation and collection paths block any residual laser
scattering. In fact, this would not be necessary for a sufficiently
narrow laser spectrum, as the TPE energy is detuned from the
exciton energy.
To measure the spectra, collected photons are routed to a
single-photon sensitive spectrometer (Acton SP-2750, Roper
Scientific) equipped with a liquid Nitrogen-cooled charge-coupled
device camera (Spec10 CCD, Princeton Instruments). For lifetime
measurements, we use an avalanche photodiode (SPAD, Micro
Photon Device) together with time-tagging electronics.
To measure the indistinguishability, the filtered X photons are
sent through a Mach-Zehnder Interferometer (MZI) with a path-
length difference of 12.5 ns, to interfere with successively emitted
photons from the quantum dot in a 50:50 fiber beam splitter
(TW805R5A2, Thorlabs) for HOM measurement. The two output
ports of the fiber beam splitter are monitored by avalanche
photodiodes (SPCM-NIR, Excelitas). The arrival times of the
photons are recorded using a time tagger (Time Tagger Ultra,
Swabian Instruments), and coincidence counting is employed to
determine the correlation between the photons. In the HOM
measurement, the polarization in both MZI arms is controlled
individually, enabling a comparison between the co-polarized
scenario with maximum indistinguishability and the cross-
polarized situation with distinguishable photons to obtain the
HOM visibility.
For PNC measurements, a phase shifter is placed into one of the
arms of the unbalanced MZI. The phase shifter consists of a
Table 1. Parameters used in the simulation.
QD-cavity detuning ℏΔω
c−x
0 meV
QD-laser detuning ℏΔω
x−l
2 meV
detuning stim. pulse _Δωstim
l2 meV
duration stim. pulse FWHMstim 3ps
duration TPE pulse FWHM
TPE
4.5 ps
delay between pulses Δt15 ps
QD-cavity coupling ℏg0.05 meV
Binding energy E
B
4 meV
cavity loss rate κ0.577 ps
−1
QD loss rate γ0.001 ps
−1
QD size a3nm
temperature T1.5 K
Material parameters are taken as in ref.
53
.
Y. Karli et al.
7
Published in partnership with The University of New South Wales npj Quantum Information (2024) 17
motorized rotation stage (ELL14K, Thorlabs) holding a half-wave
plate positioned between two quarter-wave plates that are
oriented orthogonally with respect to each other’s fast axis. This
arrangement effectively acts as a variable phase shifter for linearly
polarized input light since:
JðθÞ¼QWP π
4
HWP θðÞQWP π
4
¼i
2
1i
i1
cos2θsin2θ2 sin θcos θ
2 sin θcos θsin2θcos2θ
"#
1i
i1
¼0ei2θ
ei2θ0
"#
:
(19)
Here θis the orientation of the fast axis of the half-wave plate. By
rotating the half-wave plate at a fixed speed, the phase in one of
the arms is varied continuously without changing the polarization,
while the phase in the other arm remains constant on the
timescale of the rotation. The two arms are then recombined at
the fiber beam splitter, where the interference occurs and photons
are directed towards two separate single-photon detectors. The
matching of the timing and relative polarization of the two arms
was ensured by interfering the excitation laser with itself and
maximizing the contrast, which yielded a visibility of 98 %.
Extraction of the PNC from data
We follow ref.
14
to compute the PNC from the visibility. We
remind that we only consider the H-polarized photons and stay in
the approximation of the two-level system composed of the 0
ji
and 1jiFock state. From the detector counts, we obtain the
visibility vexp, which is proportional to the occupation ρ
0,0
. In the
next step, we decompose the density matrix ρ¼λρpure þð1
λÞρmixed into a part corresponding to a pure state and a part being
a statistical mixture with the off-diagonal elements being zero.
Note that we are only interested in the absolute value of the
coherence and not in its phase. Following ref.
14
, the visibility can
be approximated by vλ2ρ0;0ffiffiffiffiffiffiffiffiffiffiffi
VHOM
pwith 0 ≤λ≤1 and V
HOM
being the photon indistinguishability. Considering the slope of the
visibility as a function of ρ
0,0
=(1 −ρ
1,1
) allows us to extract λ(see
Supplementary Methods—Estimation of λfor PNC extraction from
visibility). Together with the knowledge of ρ
1,1
via the photon
counts, we can estimate the PNC as
PNCexp:¼λffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ρ1;1ð1ρ1;1Þ
q:(20)
DATA AVAILABILITY
The datasets generated and analyzed during the current study are available in the
Zenodo repository, accessible via the following https://doi.org/10.5281/
zenodo.7961599.
CODE AVAILABILITY
The code used for the simulation during the current study is available from the
corresponding author on reasonable request.
Received: 4 August 2023; Accepted: 11 January 2024;
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ACKNOWLEDGEMENTS
The authors gratefully acknowledge insightful discussions with Stefan Frick, Robert
Keil, Tommaso Faleo, Mathieu Bozzio and Serkan Ates. Nils Kewitz and Bhavana
Panchumarthi supported the early phases of the experiment. Y.K., F.K., R.S., V.R. and
G.W. acknowledge financial support through the Austrian Science Fund FWF projects
W1259 (DK-ALM Atoms, Light, and Molecules), FG 5, TAI-556N (DarkEneT), F 7114
(BeyondC) and I4380 (AEQuDot). DAV and TH acknowledge financial support by the
German Federal Ministry of Education and Research (BMBF) via projects 13N14876
(‘QuSecure’) and 16KISQ087K (tubLAN Q.0). TKB and DER acknowledge financial
support from the German Research Foundation DFG through project 428026575
(AEQuDot). A.R. and SFCdS acknowledge the FWF projects FG 5, P 30459, I 4320, the
Linz Institute of Technology (LIT) and the European Union’s Horizon 2020 research,
and innovation program under Grant Agreement Nos. 899814 (Qurope), 871130
(ASCENT+) and the QauntERA II Program (project QD-E-QKD). L.M.H., P.W. and J.C.L.
acknowledge financial support from the European Union’s Horizon 2020 and Horizon
Europe research and innovation program under grant agreement No 899368
(EPIQUS), the Marie Skłodowska-Curie grant agreement No 956071 (AppQInfo), and
the QuantERA II Program under Grant Agreement No 101017733 (PhoMemtor); FWF
through F7113 (BeyondC), and FG5 (Research Group 5); from the Austrian Federal
Ministry for Digital and Economic Affairs, the National Foundation for Research,
Technology and Development and the Christian Doppler Research Association. For
the purpose of open access, the author has applied a CC BY public copyright licence
to any Author Accepted Manuscript version arising from this submission.
AUTHOR CONTRIBUTIONS
The experimental setup was built by Y.K., F.K., R.S., V.R., J.C.L., D.A.V., L.M.H., and the
measurements were performed by Y.K., F.K., D.A.V. The main numerical calculations
were carried out by P.C.A.H., with supporting calculations provided by F.K and Y.K.
The sample was provided by C.S., S.F.CdS, A.R. The first draft of the manuscript was
written by D.A.V., F.K., Y.K., P.C.A.H., V.R., D.E.R. Conceptual work and supervision were
done by G.W., P.W., V.M.A, D.E.R., V.R., J.C.L., T.H., A.R. All authors discussed the results
and were involved in writing the manuscript.
COMPETING INTERESTS
The authors declare no competing interests.
ADDITIONAL INFORMATION
Supplementary information The online version contains supplementary material
available at https://doi.org/10.1038/s41534-024-00811-2.
Correspondence and requests for materials should be addressed to Yusuf Karli.
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Y. Karli et al.
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