Robust excitation of C-band quantum dots for quantum communication

The emergence of practical quantum technology paves the way to a quantum internet—a network of connected quantum computers capable of reaching computational speed-ups in various tasks such as prime factoring,¹ machine learning,² and the verification of NP-complete problems with limited information.³ Although such schemes are appealing, most are technologically challenging, while the security advantages provided by quantum cryptography are more tangible.^4–6 A broad range of quantum-cryptographic primitives including quantum key distribution (QKD),^4–6 quantum coin flipping,^7–9 unforgeable quantum tokens,^10–12 and quantum bit commitment^13–15 have been developed to demonstrate some security advantage over their classical counterparts. The success of a future quantum internet then relies on the development of fundamental quantum hardware (sources, repeaters, and detectors), which should adhere to these primitives' security standards, provide high communication rates, and operate reliably in a real-world environment.¹⁶ 

Non-classical light sources such as spontaneous parametric downconversion,^17,18 nitrogen-vacancy centers,¹⁹ and trapped atoms²⁰ have been used as hardware for the first quantum networks. In recent years, semiconductor quantum dots (QDs) have materialized as highly versatile and quality single-photon sources,^21–25 with outstanding end-to-end efficiencies overcoming 57% and the potential to reach repetition rates of tens of GHz.²⁴ Such emission properties of QDs have led to the implementation of complex network building blocks relying on quantum teleportation^26,27 and quantum entanglement swapping.^22,23,28 Regarding the emission wavelength, the spectral regime of the telecom C-band (1530–1565 nm) is highly appealing, due to its global absorption minimum in standard silica fibers, the possibility to implement daylight satellite communication,²⁹ and the compatibility with the mature silicon photonic platforms.³⁰ QDs with emission wavelengths in and around the C-band are available on indium phosphide (InP)^31–33 and gallium arsenide (GaAs) material system,^34–36 and circumvent the technical overhead and losses of quantum frequency conversion.³⁷ Embedded in circular Bragg cavities, QDs based on the well-established GaAs platform have simultaneously demonstrated high brightness and high purity values recently.³⁶ 

Previous works have investigated the advantages and drawbacks of various optical pumping schemes (resonant, phonon-assisted, and two-photon excitation) in terms of efficiency, single-photon purity, and indistinguishability.^38–40 On the other hand, it was recently shown that such schemes must be carefully tuned to satisfy the security assumptions of each quantum-cryptographic application.⁴¹ Crucially, quantum coherences between the emitted photon-number components must be scrambled for optimal performance, which is inherently provided by some optical pumping schemes such as longitudinal phonon-assisted (LA) excitation and two-photon excitation (TPE).⁴¹ On top of their intrinsic security benefits, LA schemes are fairly insensitive to pump instabilities like power or polarization fluctuations, making them suitable for real-life communication networks.^38,39 These excitation schemes are also beneficial for QDs with a complex charge environment, while other pumping schemes such as TPE can typically only address charge-neutral transitions. Moreover, unlike for neutral transitions, charged excitons can enhance polarized emission in polarized cavities, an important feature for most applications.²⁴ Finally, LA schemes do not require challenging single-photon polarization filtering (contrary to the resonant counterpart), and, thus, promise an experimentally straightforward way to obtain simultaneously high brightness and purity with high reproducibility in quantum dot fabrication and experimental setups.³⁸ 

In this work, we combine all aforementioned advantages of LA excitation in the C-band and exploit its tunable parameters to investigate the complex dependence of brightness and purity on pump power and spectral detuning. We illustrate how this non-trivial behavior affects the security of quantum-cryptographic primitives with the example of single-photon QKD, and how the optimal operation conditions depend on the communication distance. In agreement with theoretical findings,⁴² our results show that the characteristics of LA excitation can be tuned to achieve the ideal photon-number statistics. This optimization is reminiscent of the mean-photon adjustment required in weak coherent state (WCS) implementations.⁴³ 

To start investigating and optimizing our excitation parameters, it is important to note that most quantum-secured applications rely on few trusted parameters that are typically not all experimentally accessible. Here, we infer the photon-number probabilities ${ p k }$ from two measurements, the brightness $B = ∑ k = 1 ∞ p k$ and the single-photon purity $P = 1 − g ( 2 ) ( 0 )$⁠, where $g ( 2 ) ( 0 )$ is the second-order auto-correlation measurement evaluated at zero time delay. We use an InAs QD based on an InGaAs metamorphic buffer layer enabling emission in the telecommunication C-band.³⁵ The tunable excitation is provided by a mode-locked fiber laser with a pulse length of 17(1) ps and a FWHM spectral width of 210(20) pm. The QD transition line is filtered by a set of volume Bragg grating filters (FWHM = 0.2 nm). The total setup efficiency is determined to be 13%. For more experimental details, see supplementary material Note 1. The highest single-photon purity under pulsed LA excitation was measured as P = 0.982, the corresponding second-order auto-correlation measurement is shown in Fig. 1.

While scanning both the power and wavelength of the pump laser, we simultaneously measure the brightness B [experimentally evaluated according to Eq. (S2) and corresponding to first-lens brightness] and single-photon purity P. We then compile the results in 2D maps as shown in Figs. 2(a) and 2(b), respectively. Due to the low phonon density at a sample temperature of $∼$ 4 K, we excite the QD only with positive detunings $Δ = ℏ ( ω laser − ω dot ) > 0$⁠. The brightness map features a single, broad maximum around $Δ ≈ 0.8 meV ( Δ λ ≈ 1.5 nm$⁠) agreeing with similar experimental findings^44,45 and theoretical studies.^46,47 LA excitation with sufficiently smooth pulses⁴⁸ achieves a population inversion of the QD ground and excited state if the effective Rabi splitting of the laser-dressed states, $ℏ Ω eff = ( ℏ Ω ) 2 + Δ 2$⁠, ensures an efficient exciton-phonon coupling that is characterized by the spectral phonon density $J ( ω )$⁠.^44,46 The robustness of this scheme against power and wavelength fluctuations of the excitation laser is demonstrated by the broad maximum of the brightness in Fig. 2(a) and stems from the spectral width of $J ( ω )$⁠. Thus, the large bandwidth of the phonon interaction directly benefits a stable operation of the QD source. Only for large detunings and weak fields, the phonon-induced relaxation to the exciton level fails and the brightness drops significantly. Similarly, for high powers, the effective Rabi splitting is no longer in resonance with the phonon interaction resulting in reduced brightness.

In addition to emission efficiency, the single-photon purity of the quantum-light source is crucial to the performance of cryptographic protocols.⁴¹ Therefore, we analyze the purity P, depicted in Fig. 2(b), for the same parameter range as the brightness. We identify a broad region of high purity at similar detunings but shifted toward lower powers. At large detunings, the purity degrades because the exciton state preparation via LA phonons becomes less efficient [evident by the low brightness in the same area of Fig. 2(a)] and spurious contributions to the emission, including neighboring QDs or a quasi-continuum of transitions, are no longer negligible. Considering a perfect two-level system, Ref. 42 predicts an enhanced purity for increasing excitation field strength because the phonon-induced level inversion is delayed until the end of the pulse. As a consequence, the chance of a re-excitation event during the same pulse, as it is known for resonant pumping,^38,50 would be reduced. In our experiment, however, this process competes with, and is eventually out-weighed by, the aforementioned unintended emission decreasing the purity at high powers significantly.

Interestingly, our experimental findings imply the absence of a trivial set of optimal parameters (simultaneously maximizing brightness and single-photon purity), which confirms some of the complex behaviors predicted in previous theory works.^41,42 Instead, a careful tuning of the excitation parameters is required for each quantum-cryptographic application. Depending on the desired security of merit, the correct weighting of the photon-number populations ${ p k }$ used for the optimization⁴¹ must be defined. At the same time, fluctuations in the excitation parameters produce only small changes in photon-number populations. Furthermore, optimal brightness and near-optimal purity are achieved for a pump pulse detuned by $≈$ 1.5 nm from the QD transition that can be readily separated from the single-photon emission using efficient, off-the-shelf spectral filters. This simplifies source operation and optimizes brightness by removing the need for a cross-polarization setup, further underlining the practicality of LA excitation for network applications.^{4–9,11–15,51}

Following the parameter estimation, we calculate the attainable secure key bits per pulse (SK) in the asymptotic regime for standard and decoy-state BB84 QKD for each set of excitation parameters as shown for zero communication distance in Figs. 2(c) and 2(d), respectively. For the decoy protocol, we include a typical 3 dB loss for a high-bandwidth intensity modulator required to produce the decoys. While the qualitative dependence of the SK is very similar for both protocols, the performance gap is evident in the absolute values. Decoy states have been introduced to handle the risk of multi-photon contributions $p m$⁠, but since these are inherently small for QDs, introducing the constant loss of the intensity modulator outweighs the effect of an exact bounding of $p m$⁠. Furthermore, recalling that quantum cryptography with off-resonantly or two-photon excited QDs does not require any modulator for phase scrambling, adding an intensity modulator for decoy would increase the setup complexity. Comparing Fig. 2 finally shows that for zero distance the brightness (more accurately, p₁) dominates the SK map making a tight bounding of $p m$ even less relevant.

However, the impact of the multi-photon events on the SK comes into play for non-zero communication distances making the ideal set of ${ p k }$ no longer trivial but dependent on the channel loss. Computing SK maps at four distances, as depicted in Figs. 3(a)–3(d), visualizes the shift in source requirements. Short-distance transmissions benefit most from a bright source, whereas high-loss scenarios such as long-distance communication call for sources with high purity. Figure 3(e) then shows how these four ideal parameter sets behave over distances. The difference in performance underlines the potential of individually adjusting the excitation conditions with respect to the channel loss. Note that the joint optimization of ${ p k }$ by tuning the pump conditions is possible with resonant or two-photon excitation but less performant. In supplementary material Note 4, we also present the optimal finite-size SK for various block sizes.

Finally, we remark that implementing decoy states could be advantageous in the long-distance regime even for highly pure single-photon sources such as QDs, as reflected in Fig. 3(e). However, for low to moderate loss, standard BB84 outperforms the decoy-state protocol.

In conclusion, we have investigated the benefits that phonon-assisted excitation of a telecom C-band QD provides for quantum-secured applications. In addition to the convenient wavelength for communication applications, the InAs QDs provide a deep confinement potential, typically spanning over several hundred millielectron volts. As a consequence, their photon-number statistics are relatively insensitive to temperature fluctuations.⁵⁷ Moreover, the source can be operated at 25 K, which is feasible for a low-cost Stirling cryostat.³⁶ 

In addition to the previously simulated low photon-number coherence,⁴¹ the robustness to environmental fluctuations,³⁹ and the efficient single-photon filtering, we have shown that LA excitation allows to effectively optimize the photon-number statistics with respect to the desired application. This feature originates from interaction with the phonon environment and is, therefore, not common to resonant excitation schemes but can be exploited by tailoring the LA pumping conditions. The complex implications of phonon interactions for brightness and single-photon purity have also been theoretically predicted for idealized systems.⁴² Therefore, our observations can be generalized to other QD-based sources.

As a means of improving the emission statistics independently of the excitation mechanism, temporal filtering of the signal was proposed^58,59 but requires special hardware and comes at the price of additional loss. Moreover, we show in supplementary material Note 6 that optimizing ${ p k }$ using only the LA pump power is as efficient as temporal filtering with a fast and lossless modulator. Only at very large distances, temporal filtering performs better since it also reduces the source brightness and detector dark counts.

However, we note that two-photon excitation can simultaneously yield a higher brightness and purity than achievable for any parameter set using the LA scheme.⁵⁰ Nevertheless, two-photon excitation—being a resonant process—is sensitive to environmental fluctuations and is, thus, less suitable for real-world implementations.

Furthermore, we found that even for quantum light sources with inherently low multi-photon contribution, decoy states can push the maximum attainable distance in QKD. However, in consideration of the low SK at these distances and the experimental overhead involved, we believe that decoy states are not beneficial for QD implementations.

Finally, we would like to stress that we optimized the photon-number statistics in LA excitation with respect to QKD as an example, but the process is transferable to other quantum-secured applications^7–15 and prone to improve their performance.