## Abstract

Triggered sources of entangled photon pairs are key components in most quantum
communication protocols. For practical quantum applications, electrical triggering
would allow the realization of compact and deterministic sources of entangled
photons. Entangled-light-emitting-diodes based on semiconductor quantum dots are
among the most promising sources that can potentially address this task. However,
entangled-light-emitting-diodes are plagued by a source of randomness, which results
in a very low probability of finding quantum dots with sufficiently small fine
structure splitting for entangled-photon generation
(∼10^{−2}). Here we introduce strain-tunable
entangled-light-emitting-diodes that exploit piezoelectric-induced strains to tune
quantum dots for entangled-photon generation. We demonstrate that up to
30% of the quantum dots in strain-tunable entangled-light-emitting-diodes
emit polarization-entangled photons. An entanglement fidelity as high as 0.83 is
achieved with fast temporal post selection. Driven at high speed, that is
400 MHz, strain-tunable entangled-light-emitting-diodes emerge as
promising devices for high data-rate quantum applications.

## Introduction

A source of triggered entangled photon pairs is a fundamental element in quantum
information science and plays a key role in a number of photonic quantum technologies
such as linear quantum computation^{1}, quantum teleportation^{2} and quantum relays^{3}. So far, generation of polarization-entangled
photon pairs is mostly obtained through spontaneous parametric downconversion^{4} and four-wave mixing^{5} in nonlinear optical media. At
present, these sources are optically driven with lasers, which increases the complexity
of the systems. In addition, the nonlinear optical process occurs randomly so that the
emission of entangled photon pairs is probabilistic. This results in the generation of
zero or multiple entangled-photon pairs in most excitation cycles and unavoidably limits
the success of realizing deterministic photonic quantum technologies.

These complications could be alleviated by employing solid-state quantum systems such as
colour centres in diamond^{6}, intrinsic defects in silicon carbide^{7} and semiconductor quantum dots (QDs)^{8}. They exhibit
atomic-like optical transitions and therefore allow the generation of deterministic
single photons. Most importantly, they can be easily embedded in a light-emitting diode
so that electrically driven single-photon emission from these systems can be
realized^{9,10,11}. However, to date, colour centres in diamond and
intrinsic defects in silicon carbide are limited to emit single photons and the
generation of entangled-photon pairs from these two systems has not been demonstrated
yet. In contrast, semiconductor QDs are proven candidates for generation of
deterministic entangled-photon pairs. In the last decade we have witnessed tremendous
advancements in the field, for instance, ultrabright^{12,13} and highly
indistinguishable polarization-entangled photons^{14} and time-bin entangled
photons^{15} have been demonstrated successfully with semiconductor QDs.
Remarkably, a robust and compact entangled-light-emitting-diode (ELED)^{16}
based on semiconductor QDs has been realized, which represents a significant progress in
the field of entangled-photon sources. Furthermore, the recent seminal realization of
quantum teleportation with ELEDs^{17,18} has indicated the desirability and
the ultimate feasibility of using such ELEDs for various future quantum
applications.

In analogy to the cascade emission in atomic systems^{19}, electrical
injection of electrons and holes into an ELED containing QDs switches on the radiative
decay of biexciton (*XX*) to the exciton (*X*) and finally to the ground state
(*0*)^{20}. The two ideally degenerate intermediate *X* states
(spin±1) ensure emission of polarization-entangled photon pairs and the
two-photon quantum-mechanical state can be expressed with the Bell state (H and V denote the orthogonally horizontal and vertical
polarizations). In real QDs, however, a reduced structural symmetry due to the
anisotropy in strain, composition and shape results in the appearance of an energetic
splitting between the two bright *X* states, the so-called fine structure splitting
(FSS)^{21}. In the presence of a FSS, the entangled state evolves over
the *X* lifetime and the time-averaged fidelity to the Bell state reveals classical
correlations among the emitted photons. High fidelity to the state
|Ψ^{+}〉 can be only observed by temporal post
selection of the emitted photons that, however, results in a strong reduction of the
brightness of the quantum source. Therefore, the FSS in an ELED containing QDs is the
key parameter determining the quality of the entangled-photon pairs. Recent work has
shown that in standard self-assembled QDs^{22}, the probability of finding
QDs with a FSS smaller than the radiative linewidth of the *X* emission
(1 μeV) is lower than 10^{−2}. This finding
implies that as-grown QDs are still impractical for scalable quantum networks. For
example, it was reported that only one QD per ELED shows an FSS small enough for
entangled-photon generation^{16}. In this context, the real potential of
ELEDs for entangled-photon generation can be harnessed only when a tight control over
the FSS is achieved.

The FSS of semiconductor QDs can be suppressed or tuned to zero via the application of
either a vertical electric field^{23,24} or a combination of strain and
electric field^{25}. The main drawback of these approaches is the difficulty
of using the electric field to control the FSS and to inject carriers simultaneously, as
in ELEDs. Other techniques such as thermal annealing^{26} and in-plane
magnetic field^{27} would be potentially compatible with the ELEDs, but the
former requires a lengthy procedure and the latter bulky setups, thus rendering a
practical implementation inconvenient. Recent theoretical works (although not
experimentally realized so far) suggest that the FSS can be eliminated using solely a
well-controlled strain field^{28,29}. Using piezoelectric-induced strains
to engineer the properties of ELEDs would be highly desirable, because this fully
electro-mechanically controlled tuning knob would allow the problems related to the FSS
in ELEDs to be overcome.

Here we experimentally demonstrate such a quantum device by integrating ELEDs onto a
piezoelectric actuator featuring giant piezo-electric response and capable of delivering
well-defined anisotropic strain fields. With this device—which we call
strain-tunable ELED (ST-ELED)—we show that the FSS of QDs can be tuned
effectively with the elastic strain fields without affecting the electrical injection of
the operation of the ELEDs. Up to 30% of the QDs are tuned to be suitable for
the generation of entangled-photon pairs (more than an order of magnitude more than in
previous devices^{16}) and a high operation speed for an entangled-photon
source is achieved, that is 400 MHz. This set of properties paves the way
towards the exploitation of ELEDs in high data-rate entangled-photon applications
involving a large number of quantum emitters.

## Results

## Self-assembled QDs in strain-tunable diode devices

The ST-ELED studied in this work is schematically shown in Fig.
1a. A 440-nm-thick *n-i-p* nanomembrane containing InGaAs QDs is
integrated onto a 0.3-mm-thick
[Pb(Mg_{1/3}Nb_{2/3})O_{3}]_{0.72}[PbTiO_{3}]_{0.28}
(PMN-PT) single piezoelectric crystal. The detailed fabrication process is
described in Methods. Different from previous works dealing with strain tuning
of QDs via PMN-PTs^{30,31}, the actuator used here has pseudo-cubic
cut directions [100], [0-11] and
[011], denoted by *x*, *y* and *z* axis,
respectively. When the PMN-PT is poled along the *z* axis, in-plane strain
fields with normal components *ɛ*_{xx} along the *x*
axis and *ɛ*_{yy} along the *y* axis with opposite
sign can be transferred to the nanomembrane. Accounting for its relevant
piezoelectric coefficients
*d*_{31}∼+420 pC N^{−1}
along the *x* axis and
*d*_{32}∼−1,140 pC N^{−1}
along the *y* axis^{32}, the in-plane anisotropy is estimated
to be
*ɛ*_{xx}≈−0.37*ɛ*_{yy}.
The large and well-controlled strain anisotropy and the broad range of
attainable strain magnitudes are unique and turned out to be vital in our
work.

## Tunable electroluminescence under applied strain fields

Apart from the strain fields, electrical contacts are arranged in such a way that
electrical fields can be independently applied across the diode and the PMN-PT
actuator. By biasing the diode and applying a variable electric field
(*F*_{p}) to the PMN-PT simultaneously, energy-tunable
electroluminescence (EL) from a single QD is produced, as shown in Fig. 1b. According to the power and polarization-resolved
measurements the observed EL lines are ascribed to exciton (*X*), biexciton
(*XX*) and charged exciton emission
(*X*^{+}), respectively. As the magnitude of
*F*_{p} is varied, all the emission lines shift in energy. In a
first approximation, this shift is due to the strain-induced change of the
energy bandgap of the material, which in turn is proportional to the volumetric
strain
*ɛ*_{tot}=*ɛ*_{xx}+*ɛ*_{yy}+*ɛ*_{zz}
at the QD position. For in-plane stress and cubic materials,
*ɛ*_{tot} is given by which, in the present case, is (with
*S*_{ij} the compliance coefficients of the host
material, see Supplementary Note
1). As *ɛ*_{tot} has the same sign as
*d*_{32}, which has a relatively large magnitude and is
negative, we expect a positive *F*_{p} to induce a compressive
strain, which results in a blue shift of the EL, whereas a negative
*F*_{p} induces a tensile strain, which results in a red shift.
We also note that a total energy shift of about 2.5 meV is achieved
as *F*_{p} is varied from −6.7 to
20 kV cm^{−1}.

## Changes in polarization and FSS with the strain fields

To control the FSS of the QDs embedded in the diode, the crystal axes
[1-10] and [110] of the GaAs nanomembrane
were carefully aligned along the *x* and *y* axes of the PMN-PT
actuator, respectively (see Fig. 1a). Representative plots
of the FSS (*s*) of different QDs as a function of *F*_{p} are
shown in Fig. 2a. Although the emission energy shift is
only about 2.5 meV as *F*_{p} is varied from
−6.7 to 28 kV cm^{−1} as a
consequence of the strong strain anisotropy (and thus relatively small
*ɛ*_{tot}), the FSS is tuned over a broad range from
30 to 0 μeV. Away from the minimum FSS, all studied QDs
exhibit an approximately linear change in the FSS with *F*_{p} at a
rate of about
2.0 μeV kV^{−1} cm^{−1},
which is about nine times larger than what has been reported for vertical
electric fields^{23}. In addition to this drastic change in FSS, we
also observe rotations of the exciton polarization angle *θ*,
that is, the polarization direction of the high-energy line of the exciton with
respect to the [110] direction of the GaAs nanomembrane (see
Fig. 2b). At the largest available tensile
(compressive) strain, *θ* tends to be directed along the
[1-10] ([110]) direction for all QDs.
Furthermore, we note that the above tuning behaviour is mainly determined by the
exciton polarization angle at zero strain fields
(*θ*_{0}) with respect to the predefined direction of the
strain^{29,33}. Experimentally, this polarization angle can be
extracted from Δ*E*=|*E*(*ϕ*,
*F*_{p}=0)−*E*_{min}|, where
*ϕ* is the polarization direction selected by our polarization
analyser (which varies from 0° to 360°),
*E*(*ϕ*, *F*_{p}=0) is the
relative position between *X* and *XX* as a function of
*ϕ* and it can be directly extracted from a
polarization-resolved EL measurement, and *E*_{min} is the minimum
energy of *E*(*ϕ*, *F*_{p}=0; see
Supplementary Fig. 1). Figure 2c–g show
Δ*E*(*ϕ*, *F*_{p}=0)
for the five studied QDs. The initial FSS *s*_{0} and
*θ*_{0} are represented by the magnitude and
orientation of the lobes in each polar plot. For the dot A and B,
*θ*_{0} are 102.0°±0.4°
and 92.7°±0.2° (the numbers quoted are the
mean±s.d. and the same definition is applied below), respectively,
suggesting positive deviations of *θ*_{0} from the strain
*x* axis. Consequently, as *F*_{p} is increased, the
polarization angle *θ* rotates counterclockwise and the FSS
experiences a finite lower bound
*s*_{min}=7.2±0.2 and
2.2±0.1 μeV (the green and wine curves in Fig. 2a,b). For dot C, *θ*_{0} is
found to be 86.1°±0.4°, which suggests a negative
deviation from the strain *x* axis. Thus, a clockwise rotation of
*θ* over *F*_{p} is observed, together with an
*s*_{min}=4.2±0.2 μeV
(the purple curve). For the QDs D and E, *θ*_{0} is found
to be 90.4°±0.3° and
90.6°±0.3° respectively, which indicate exact
alignment of *θ*_{0} with the strain *x* axis. The
polarization angles for both QDs rotate clockwise as the strain is varied and
this implies that their polarization angles at zero strain fields are oriented
at angles slightly <90°. It should be noted that this difference
from the observed values (larger than 90°) is ascribed to the limited
alignment precision of the polarizer (within a few degrees)^{29}.
Most importantly, owing to this exact alignment between the exciton polarization
angle *θ*_{0} and the strain axes for dot D and E, their
FSS can be reduced well below 1 μeV and
*s*_{min} is found to be 0.30±0.25 and
0.60±0.20 μeV, respectively. By simply treating
the anisotropic strain as an effective uniaxial strain^{28} (see
Supplementary Note 2), the
behaviour of *s* and *θ* are well fitted (solid lines in
Fig. 2a,b). Noticeably, the minimum reachable FSS
value, following the form
*s*_{min}=*s*_{0}|sin(2*θ*_{0})|,
is determined by *s*_{0} and *θ*_{0} (see
Supplementary Table 1). Even
more, the *s*_{min} predicted before varying the strain fields are
8.18±0.20, 2.34±0.21, 3.3±0.18,
0.24±0.32 and 0.76±0.15 μeV for dot
A, B, C, D and E, respectively, which show excellent agreement with our
experimental data. From our experimental observations and theoretical analysis,
it is clear that, to cancel the FSS with our external strain fields, the strain
principal axes should be as close as possible to the polarization angle of QDs
at zero applied stress.

## Quantum state tomography measurments

The ability to tune the FSS of the QDs to zero allows us to investigate the
capability of the ST-ELED to generate polarization-entangled photon pairs
without the aid of post-filtering techniques^{34,35}. Figure 3a shows the polarization resolved co- and
cross-polarization correlation between the *XX* and *X* photons
emitted by dot E under electrically pulsed excitation and tuned to a FSS of
0.60±0.20 μeV (at
*F*_{p}=18.7 kV cm^{−1}).
The periodic correlation peaks with well-separated temporal distance of
5.4 ns arise from the chosen repetition rate of 185.2 MHz.
Importantly, for co-polarized two-photon, strong correlations are observed in
linear (HV) and diagonal (DA) bases, while strong anti-correlations are observed
in circular basis (RL), as expected for the photon pairs emitted in the
maximally entangled Bell state |Ψ^{+}〉. To
quantify the degree of polarization entanglement, we have reconstructed the
two-photon density matrix by performing quantum state tomography measurements,
as described in ref. 36. Sixteen polarization
correlation measurements were performed and the density matrix is reconstructed
using a maximum likelihood estimation. The imaginary part and the real part of
the density matrix are displayed in Fig. 3b,c. The outer
off-diagonal elements in the real part of the density matrix reveal a high
probability for a superposition of the two-photon wave function, being a clear
signature of polarization entanglement^{36}. Specifically, the
density matrix can be used to quantify the degree of entanglement by extracting
the tangle *T*, the concurrence *C*, the largest eigenvalue
*λ* and the Peres criterion *P*. We find
*C*=0.688±0.040 (>0),
*T*=0.474±0.055 (>0),
*λ*=0.795 (>0.5) and
*P*=−0.30±0.02 (<0). All these tests
exceed the classical limit, proving that the quantum state obtained in our
experiment is highly entangled. Using the largest eigenvalue, we are able to
determine the most probable state of the system: ,
in which the presence of the phase
*φ*_{0}=−0.11*π* is
likely to be due to the reflection at the beam splitter^{37} and the
residual FSS ^{38}. As a consequence, the fidelity to the maximally
entangled Bell state |Ψ^{+}〉 is found to
be *f*^{+}=0.766±0.051.

## High yield of QDs for entangled-photon generation

Having demonstrated generation of entangled-photon pairs from our ST-ELED, we
present one of the most important results of our work, that is, the capability
of anisotropic strain fields to tune about 30% of the QDs for
entangled-photon emission. A statistical study from 82 randomly selected QDs
revealed that the majority of QDs in our ST-ELED device have
*θ*_{0} oriented close to the
[1-10] crystal axis (see Supplementary Fig. 2). Therefore, the
alignment of the strain axes parallel (or perpendicular) to the
[1-10] crystal axis of the GaAs ensures frequent observation
of QDs with low value of *s*_{min}. Figure 4a
shows the statistical investigation of *s*_{min}, falling into a
range of 0 to 40 μeV. Remarkably, nine QDs have
*s*_{min} <1 μeV, which is of the
order of the homogenous linewidth of the *X* emission. Therefore,
11% of dots have sufficiently small FSS for strong entanglement in
our device (see Supplementary Note
3, Supplementary Fig. 3
and Supplementary Table 2 for the
entanglement results measured for other dots at
*s*_{min}<1 μeV). In addition, recent
works have suggested that for InGaAs QDs entanglement is robust and the
violation of the classical limit can be achieved for FSS smaller than
3–4 μeV (refs 37,
39). To quantify this probability in our
ST-ELEDs, we use the following approach: we study the evolution of the fidelity
to the maximally entangled Bell state
|Ψ^{+}〉 as a function of the FSS for one
single QD and we use the obtained result to estimate at which value of the FSS
it is possible to overcome the classical limit. To determine the entanglement
fidelity, polarization correlations were performed for each value of the FSS and
entanglement was equivalently quantified by measuring the degree of correlation
(see Methods). As shown in Fig. 4b, the maximum fidelity
*f*^{+}=0.75±0.02 is
achieved when the FSS is tuned close to zero. For FSS values larger than
3 μeV, the fidelity drastically decreases below the
classical limit (see the dashed line). Taking into account that the exciton
lifetime of the InGaAs QDs in our ST-ELED device has typical values of about
1 ns, this FSS of about 3 μeV provides an upper
limit to observe entanglement, consistent with previous reports for InGaAs
QDs^{12,27,37,39}. From the statistical investigation we find
that 27 QDs can be tuned below 3 μeV, which indicates a
probability as high as 33% of QDs that can be exploited as
entangled-light emitters in our ST-ELEDs. Compared with the only work on ELEDs
present in the literature^{16}, the yield demonstrated in our work
is more than an order of magnitude higher (a factor of ∼30). This
probability is higher than what was reported for highly symmetric pyramidal QDs
where, however, electrical injection has not been realized yet^{40}.
We note that the yield of dots tuned for entangled-photon generation in our
ST-ELED can be further improved by optimizing the alignment of the strain
principal axes to the statistical mean value of the initial polarization
direction. According to our statistical measurement, the mean value of the
initial polarization direction is in fact 92.25° instead of 90°
(see Supplementary Fig. 2).

## High speed generation of entangled-photon pairs

In addition, we can increase the pulsed excitation rate to achieve fast
generation rate of the entangled-photon pairs. This feature is highly desirable
for high data-rate quantum information processing. Figure
5 shows the results of polarization correlation measurements at
400 MHz for the dot E. Similar to the case of 185.2 MHz,
we observe correlations in the HV and DA bases and anti-correlation in the RL
basis for co-polarized two photons. In Fig. 5b the degrees
of correlation in given bases are reported. We find a state fidelity as high as
0.66±0.02, which exceeds the classical limit of 0.5 and thus proves
generation of entangled-photon pairs at 400 MHz. We observe that the
fidelity at 400 MHz is smaller than the fidelity at
185.2 MHz reported above. This is likely ascribed to the contribution
of a small amount of uncorrelated photon pairs due to the time-dependent
re-excitation process, residual FSS and background emission^{16,38,39}. By temporal post selection of the emitted photons we
can alleviate these effects for the entanglement degradation. The curves in
Fig. 5c show the degrees of correlation for a temporal
gate Δ*τ*=0.8 ns at which
∼20% of the coincidence counts are discarded. We measure
*C*_{HV}=0.67±0.06,
*C*_{DA}=0.63±0.04 and
*C*_{RL}=−0.78±0.07, corresponding
to *f*^{+}=0.77±0.03. The degree
of correlation and entanglement fidelity can be improved by further shortening
Δ*τ*^{16,13,38,39}. With the narrowest
available gate width of 0.1 ns applied, ∼80%
coincidence counts are discarded and the degrees of correlation increase
significantly: *C*_{HV}=0.74±0.12,
*C*_{DA}=0.74±0.09 and
*C*_{RL}=−0.84±0.12, which
provides the highest fidelity of 0.83±0.05. It is interesting to
investigate whether such a high level of entanglement is sufficient to violate
Bell’s inequality. Using the measured values of the degree of
polarization correlation, it is possible to determine Bell parameters
*S*_{RD}, *S*_{RC} and *S*_{DC},
which are related to three different planes of the Poincaré sphere
(see Methods). Our results show that the Bell parameters increase as the gate
width is decreased, and the Bell inequality is found to be violated starting
from the gate width of 0.8 ns. From the degrees of correlation at
Δ*τ*=0.8 ns, we calculate
*S*_{RD}=1.83±0.07,
*S*_{RC}=2.04±0.09 and
*S*_{DC}=2.00±0.08. Two of these values are
>2 and they indicate violations of the Bell inequality. In particular,
*S*_{RD} is found to be less than *S*_{RC} and
*S*_{DC}, and this is due to the weaker degree of correlation
observed in the HV and DA bases (*C*_{HV},
*C*_{DA}<|*C*_{RL}|) (refs 16, 37, 39). This common feature is usually observed for the QDs
entangled-photon sources and is most probably ascribed to the weak coupling
between the two bright exciton states^{41}. Therefore, for the
actual entangled state *S*_{RD} is not optimally chosen to inspect
violation of the Bell inequality. In addition, for the gate width of
0.1 ns, we find
*S*_{RD}=2.09±0.21,
*S*_{RC}=2.23±0.21 and
*S*_{DC}=2.23±0.24 (see Supplementary Fig. 4). All these three
parameters are above the threshold of 2, thus proving that our ST-ELED is
capable of generating non-local states of light in response to an electrical
trigger.

## Discussion

We have presented ST-ELEDs in which anisotropic strain fields are used to tune QDs
for entangled-photon generation. We have shown that up to 30% of QDs
embedded in this device are capable of emitting polarization entangled-photon pairs.
This practically removes the tedious search for special QDs plaguing previous
ELEDs^{16}. Furthermore, we demonstrate triggered entangled-photon
emission at a repetition rate of 400 MHz. Our all electrically controlled
ST-ELEDs emerge as one of the most practical entangled-photon sources with fast
operation speed and great potential for large-scale quantum communication and
computation tasks.

Despite the considerable advances in our ST-ELED, it should be noted that the high
yield of QDs tuned for entangled-photon emission has been achieved by applying
different magnitudes of strain to different QDs in the single ST-ELED device due to
the structural randomness of the QDs. To achieve scalable on-chip integrative
entangled-photon applications, active engineering efforts are required to fabricate
well-defined microstructures on PMN-PT crystal so that different strain fields can
be exerted to different QDs simultaneously on one single chip. This will be of
particular interest for realizing QDs arrays of independently tunable electrically
triggered entangled-photon sources. Further improvements to the ST-ELED devices,
such as integration with microcavities^{42,43} or micro-lenses^{44}, to achieve bright entangled-photon emission, using III-nitride QDs
to develop entangled-photon sources operating at room temperature^{45,46}, constitute important steps towards realizing more practical electrically driven
entangled-photon sources for scalable quantum information applications.

## Methods

## Sample and device fabrication

The studied sample was grown on a (001) GaAs substrate by solid-source molecular
beam epitaxy. It consisted of a *p*-*i*-*n* heterostructure diode
composed of a 178-nm-thick *n*-type GaAs layer, a 160-nm-thick intrinsic
GaAs layer and a 96-nm-thick *p*-type GaAs layer from the bottom to the
top. A layer of low-density
(∼10^{6}–10^{7} cm^{−2}),
self-assembled InGaAs QDs was embedded in the middle of the intrinsic GaAs
layer. The entire diode structure was grown on a 100-nm-thick
Al_{0.75}Ga_{0.25}As sacrificial layer. As for the device
processing, first of all, standard ultraviolet photolithography and wet chemical
etching were used to fabricate mesa structures with size of 120 ×
160 μm^{2}. The longer edge of the GaAs
membrane was processed along [110] crystal axis of GaAs
and—during the transfer onto the piezoelectric actuator—was
carefully aligned along the *y* axis of the PMN-PT actuator. It is worth
noting that the bonded gold layer on the bottom formed a *p*-contact,
whereas the *n*-type contact was formed by depositing a gold pad with size
of 50 × 50 μm^{2} on the top of the
nanomembrane.

## Electrically pulsed excitation and spectroscopic measurements

The EL is observed when the diode is biased with a DC voltage
(*V*_{d}) above −1.7 V; however, it is
slightly different from one device to another due to the different bonding
conditions. The pulsed electrical excitation is accomplished by superimposing an
ultrafast electrical pulse stream onto a −1.6 V DC bias by
using a broad bandwidth bias-Tee. The pulse stream used in this work has nominal
duration of 300 ps and amplitude of
*V*_{pp}=−8.0 V. The large
magnitude of the pulse stream used here is caused by the low pulse injection
efficiency, which is probably ascribed to the imperfect electrical connections
and large impedance mismatching between the device and the external electronics.
Further improvements, including optimization of the electrical connections and
introducing an impedance matching network^{47}, are expected to
increase the pulse injection efficiency and consequently reduce the pulse
magnitude.

In optical measurements, the EL emitted from the diode is collected by a
× 50 microscope objective with numerical aperture of 0.42, which is
placed on the top of the nanomembrane and collects the photon emission from the
area close to the metal contact. By inserting a half-wave plate and a linear
polarizer directly after the collection lens, polarization-resolved measurements
were performed to obtain the FSS against *F*_{p}. The exciton
polarization is determined by aligning the fast optical axis of the polarizer
along [110] direction of the nanomembrane. The EL was directed
to a spectrometer with 750 mm focus length and the spectrum was
analysed using a nitrogen-cooled charge-coupled device. The FSS is determined
with an accuracy of sub-μeV by taking the experimental procedure in
refs 23, 25.

## Polarization-resolved photon correlation measurements

Regarding the polarization-resolved correlation measurements, a non-polarizing
50:50 beam splitter is placed directly after the collection objective, to divide
the optical paths between two spectrometers, which are used to detect *X*
and *XX* separately. After each spectrometer, a Hanbury–Brown
Twiss setup consisting of a polarizing beam splitter and two high-efficiency
single-photon avalanche detectors is placed. Half and quarter waves were used to
select the proper polarization basis. The temporal resolution of the system is
∼400 ps. The entanglement can be equivalently quantified by
measuring degree of correlation C, which is defined by

where *g*_{XX,X}^{(2)}(*τ*)
and are normalized second-order time correlations
for co-polarized and cross-polarized *XX* and *X* photons,
respectively. The fidelity *f*^{+} is calculated by
using the formula:
*f*^{+}=(1+*C*_{HV}+*C*_{DA}−*C*_{RL}),
in which *C*_{HV}, *C*_{DA} and *C*_{RL}
are degree of correlations in HV, DA and RL bases. The Bell parameters are
determined with the formulas: , and .

## Additional information

**How to cite this article:** Zhang, J. *et al.* High yield and ultrafast
sources of electrically triggered entangled-photon pairs based on strain-tunable
quantum dots. *Nat. Commun.* 6:10067 doi: 10.1038/ncomms10067 (2015).

## Change history

### Updated online 11 May 2016

A correction has been published and is appended to both the HTML and PDF versions of this paper. The error has not been fixed in the paper.

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## Acknowledgements

The work was supported financially by BMBF QuaHL-Rep (contract numbers 01BQ1032 and 01BQ1034), Q.Com-H (16KIS0106) and the European Union Seventh Framework Programme 209 (FP7/2007-2013) under grant agreement number 601126 210 (HANAS). J.X.Z. was supported by China Scholarship Council (CSC, number 2010601008). We thank B. Eichler, R. Engelhard, P. Atkinson, B. Höfer and S. Harazim for discussions and the technical support.

## Author information

## Author notes

- Eugenio Zallo

Present address: Paul-Drude-Institut für Festkörperelektronik, Hausvogteiplatz 5-7, 10117 Berlin, Germany.

## Affiliations

### Institute for Integrative Nanosciences, IFW Dresden, Helmholtzstrasse 20, 01069 Dresden, Germany

- Jiaxiang Zhang
- , Fei Ding
- , Yongheng Huo
- , Eugenio Zallo
- & Oliver G. Schmidt

### Institute of Semiconductor and Solid State Physics, Johannes Kepler University Linz, Altenbergerstrasse 69, 4040 Linz, Austria

- Johannes S. Wildmann
- , Rinaldo Trotta
- , Yongheng Huo
- , Daniel Huber
- & Armando Rastelli

### Material Systems for Nanoelectronics, TU Chemnitz, 09107 Chemnitz, Germany

- Oliver G. Schmidt

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## Contributions

The sample was grown by Y.H. and E.Z., and the device was fabricated by J.X.Z. The work was conceived and designed by J.X.Z., F.D. and R.T., and guided by A.R. and O.G.S. The optical measurements were performed by J.X.Z., J.W., R.T. and D.H. The results were discussed by all authors. J.X.Z. wrote the manuscript with help from all the other authors.

## Competing interests

The authors declare no competing financial interests.

## Corresponding authors

Correspondence to Jiaxiang Zhang or Fei Ding or Rinaldo Trotta.

## Supplementary information

## PDF files

- 1.
### Supplementary Information

Supplementary Figures 1-4, Supplementary Tables 1-2, Supplementary Notes 1-3 and Supplementary References.

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