On-Demand Generation of Entangled Photon Pairs in the Telecom C-Band with InAs Quantum Dots

Entangled photons are an integral part in quantum optics experiments and a key resource in quantum imaging, quantum communication, and photonic quantum information processing. Making this resource available on-demand has been an ongoing scientific challenge with enormous progress in recent years. Of particular interest is the potential to transmit quantum information over long distances, making photons the only reliable flying qubit. Entangled photons at the telecom C-band could be directly launched into single-mode optical fibers, enabling worldwide quantum communication via existing telecommunication infrastructure. However, the on-demand generation of entangled photons at this desired wavelength window has been elusive. Here, we show a photon pair generation efficiency of 69.9 ± 3.6% in the telecom C-band by an InAs/GaAs semiconductor quantum dot on a metamorphic buffer layer. Using a robust phonon-assisted two-photon excitation scheme we measure a maximum concurrence of 91.4 ± 3.8% and a peak fidelity to the Φ+ state of 95.2 ± 1.1%, verifying on-demand generation of strongly entangled photon pairs and marking an important milestone for interfacing quantum light sources with our classical fiber networks.


List of Figures
S7 Laser state tomography measured for vertical input polarization into our analysis setup, (a) real part and (b) imaginary part of the resulting density matrix.S11 S8 Density matrices of the raw data (a, real part) and (b, imaginary part) and of the compensated data (c, real part) and (d, imaginary part). . . . . . . . S14 S9 Fidelities to Φ + for different measurement bases. The open circles represent the data, the bold solid lines correspond to fits to the data. (a) Fidelity of Φ meas to Φ + with fit (blue) and Φ meas with fit (turquoise) (b) Fidelity of Φ comp to Φ + with fit (green open circles) andΦ meas to Φ max (gray stars). . . . . . . S15 S10 Concurrence as a function of coincidence percentage in the center peak. . . . S16

Sample growth
The sample was grown by metal-organic vapor-phase epitaxy (MOVPE) on Si-doped GaAs (001)-oriented substrates in an Aixtron 200/4 low-pressure (100 mbar) horizontal reactor with H 2 as carrier gas and trimethylgallium (TMGa), trimethylaluminium (TMAl), trimethylindium (TMIn), and arsine (AsH 3 ) as precursors. The epitaxial layer structure is given Table   1. The distributed Bragg reflector (DBR) and compositionally graded InGaAs metamorphic buffer layer (MMBL) were first grown at 670°C (calibrated wafer surface temperature) after which the growth was stopped and the temperature was reduced to 515°C for quantum dot growth. Next, a 10 s ripening step was used and the low-temperature part of the capping layer was grown. Finally, the temperature was increased to 670°C and the structure was completed with the high-temperature part of the capping layer. A three-lambda cavity is formed between the DBR and the semiconductor-air interface with the MMBL and capping layer thicknesses chosen to optimize the extraction efficiency. The lattice relaxation of the MMBL layer allows for the growth of large QDs with an emission wavelength of around 1550 nm, which is significantly longer than what can be obtained from the coherent growth on the GaAs substrate (typically <1300 nm) 1 . In our previous work, using similar growth conditions, we estimated the QD density to be in the 1 × 10 7 cm −2 range 2 .

Rabi oscillations of QD1
It has been reported in several articles in the literature that additional white light (or in our case above-band laser) can help to stabilize the charge environment 3,4 by saturating charges in the vicinity of the quantum dot. Unsaturated charges can in turn lead to fluctuating electric fields resulting in the quantum dot jumping in and out of resonance again and not yielding highest occupation probability in the π-pulse, visible in the power-dependence in The population of the excited state is determined via the following fit to the data 5 : Here, c 2 (t) is the probability amplitude to find the quantum system in the excited state.
The following relations apply: . Ω 0 is the Rabi frequency and Γ 1 corresponds to the decay rate of the excited state. Ω 0 = | − eE 0 µ 12 /h| is the Rabi frequency and E 0 the electrical field. The dipole matrix element µ 12 is proportional to the excitation power Ω 0 ∝ E 0 ∝ √ P . This allows us to express the population of the excited S7 state depending on the pulse area via: ∞ −∞ E 0 (t)dt is the excitation pulse envelope. Via fits we obtain a population of 82.6 ± 1.6 % for the biexciton and 84.6 ± 2.7 % for the exciton. By multiplying these probabilities, we obtain a photon pair generation efficiency of 69.9 ± 3.6 %. Another way of determining the photon pair generation efficiency is discussed in the supplementary material of reference 6 .
Here, the authors suggest calculating the pair generation efficiency by comparing the peak areas of the center peak A center in co-polarized cross correlation measurements (e.g. HH) and the corresponding side peak areas A side . By taking the ratio of both quantities, the result is no longer dependent on the optical path efficiencies for biexciton and exciton. The pair generation efficiency p is related to the peak areas as follows: Here, theĀ indicates that we average over all 6 co-polarized measurements. An excited-state preparation and radiation probability of 66.4 ± 1.8 % is found, which is in good agreement S8 with the value obtained via fitting the Rabi oscillations.

Rabi oscillations of QD2
Power-dependent measurements for QD2 are shown in Fig. S4. During the measurement, we suffered from spectrally impure laser pulses after our pulse slicer, which did not allow us to perform pure two-photon resonant excitation. While we attempted to excite via the two-photon resonance, the effect due to the spectral profile of the laser can be described more as a two-photon excitation scheme with additional phonon contribution. As the laser pulses were not spectrally pure, there were additional spectral components partially detuned towards the phonon resonance energy. This results in the power-dependent measurement shown in Fig. S4 (a) that is not displaying the damped Rabi oscillations expected for two-photon resonant excitation. The additional and unwanted spectral components were visible in the spectrum and much broader than the bandwidth of our notch filters. This again highlights the robustness of the phonon-assisted scheme that is not relying on precise overlap of a laser spectrum with the quantum dot resonance. To be able to estimate the state population, we subtract the phonon background (see Fig. 2 (c) in the main text), obtaining the data points shown in Fig. S4 (b). A fit to the data yields a state population of 80.1 ± 9.5 % in the π-pulse.

Lifetimes of QD2
We examine the influence of the excitation scheme on the radiative lifetimes of the biexciton of QD2. A decay time measurement under above-band excitation with 2 ps pulses at 1200 nm is shown in Fig. S5 (a). We find a decay time of τ above = 794±3 ps. The used fitting function is similar to the one in the supplementary material of Ref. 7 for the charged exciton. We would like to note that this is not the biexciton lifetime, as discussed in the appendix of Ref. 8 .
In comparison, by using the phonon-assisted two-photon resonance excitation scheme the

Exciton autocorrelation
In Figure S6 we show the autocorrelation measurements performed on the exciton of QD1 under two-photon resonant excitation in (a) and QD2 under phonon-assisted two-photon excitation in (b). We determine g (2) TPE (0) = 0.07 ± 0.004 for two-photon excitation and g (2) Phonon (0) = 0.068 ± 0.004 for phonon-assisted excitation. To make sure our quantum state tomography setup is carefully aligned, we perform a state tomography of the excitation laser, which is coupled into the analysis setup (see main text Fig. 1(e)), without passing the cryostat. The polarization of the laser is set to vertical using a half and a quarter waveplate, which is verified with a polarimeter. We perform all 36 measurements of the quantum state tomography with the vertically polarized laser, yielding the density matrix shown in Fig. S7 (a) and (b). From the matrix, we can infer that the waveplate angles are well calibrated, since there is only a peak for the |VV VV| state. The density matrix element of the vertical states amounts to 0.99, while the absolute values of all other elements are no larger than 0.03. From the imaginary part shown in Fig. S7 (b) we can infer that our analysis setup is not introducing significant additional phases.

Two-photon quantum state reconstruction
As mentioned in the main text, we perform a transformation from our birefringent mea- The applied transformation is keeping the orthogonality of the polarizations. Furthermore, the fidelity of our state to |HH + |ṼṼ in theHṼ-basis is the same as the fidelity to |HH + |VV in the HV-coordinate system. In Fig. S9 (a) we show how the fidelity to Φ + is evolving over time for different initial states. The fidelity of the stateΦ meas in the   Figure S8: Density matrices of the raw data (a, real part) and (b, imaginary part) and of the compensated data (c, real part) and (d, imaginary part).
the oscillation of the emitted quantum state between Φ + and Φ − due to the finestructure splitting. After applying the transformation to the HV-coordinate system, the maximum fidelity of our state compared to Φ + is increased to 95.2 ± 1.1 % shown in turquoise. Now an almost perfect visibility in the oscillation between the two states is achieved. Finally, we calculate the fidelity of the measured state to Φ + after applying an optimal waveplate to each individual time bin compensating for the FSS. This yields a nearly flat fidelity to Φ + , with a maximum fidelity of 95.4 %, which is shown in Fig. S9 (b) with green open circles.
In comparison, we plot the fidelity ofΦ meas to a maximally entangled state in theHṼ-basis,

S14
showing full overlap within measurement uncertainty. igure S9: Fidelities to Φ + for different measurement bases. The open circles represent the data, the bold solid lines correspond to fits to the data. (a) Fidelity ofΦ meas to Φ + with fit (blue) and Φ meas with fit (turquoise) (b) Fidelity of Φ comp to Φ + with fit (green open circles) andΦ meas to Φ max (gray stars).

Concurrence
To demonstrate that the concurrence is only decreasing over time due to an increased noise level for time delays larger than 3 ns, we add a different way of plotting the data. In Figure   S10, we plot the concurrence not as a function of time but instead as a function of already detected coincidences of the center peak. To do so, we sum up all coincidences in the center peak and normalize to the total amount of coincidences. In this representation it is apparent S15 that the concurrence is only decreasing distinctly after more than 90 % of the coincidences in the center peak have been measured, emphasizing that the decrease is only related to the noise level at this point in the measurement and not dephasing effects.