Correlated photon pair generation in AlGaAs nanowaveguides via spontaneous four-wave mixing

We demonstrate a source of correlated photon pairs which will have applications in future integrated quantum photonic circuits. The source utilizes spontaneous four-wave mixing (SFWM) in a dispersion-engineered nanowaveguide made of AlGaAs, which has merits of negligible two-photon absorption and low spontaneous Raman scattering (SpRS). We observe a coincidence-to-accidental (CAR) ratio up to 177, mainly limited by propagation losses. Experimental results agree well with theoretical predictions of the SFWM photon pair generation and the SpRS noise photon generation. We also study the effects from the SpRS, propagation losses, and waveguide lengths on the quality of our source.


Introduction
Correlated photon pairs have potential applications in future communications and information processing systems. Recent demonstrations utilizing entangled photons have been reported including secure communications [1], teleportation [2], and quantum computing [3].
Despite these impressive results, they rely on free-space optics, which hinders scaling and applying the technology beyond the lab. In addition, bulk systems are prone to external interference and are sensitive to perturbations. Recently, there has been a growing interest in the use of guided-wave integrated photonics to realize a compact, robust quantum circuit that contains all necessary components from photon sources to routing elements, interferometers, and photon detectors. The integrated platform offers inherent stability in interfering quantum states, which is the crux of several quantum algorithms.
The III-V semiconductor material AlGaAs/GaAs is a promising candidate for the implementation of complete integrated quantum photonic circuits. The integration of an optical arXiv:1602.03096v1 [physics.optics] 9 Feb 2016 pump source can be achieved significantly easier in this material system than in silicon-based materials [4]. An active interferometer circuit [5] and a GaAs-based single photon detector [6] have also been demonstrated. Integrated sources of correlated photon pairs in AlGaAs have been rigoursly studied with device structures including superlattice waveguides [7,8] and Bragg-reflection waveguides [4,9,10]. These structures rely on the second-order susceptibility. To the best of our knowledge, there is no study in the performance of the pair production in AlGaAs-based devices via the third-order nonlinearity, regardless of a vast body of the literature on the classical counterpart. Therefore, in this paper, we focus on the pair generation via the third-order nonlinearity in AlGaAs waveguides.
The generation of correlated photon pairs can be based on either spontaneous parametric down-conversion (SPDC) which makes use of the second-order susceptibility χ (2) or spontaneous four-wave mixing (SFWM) which harnesses the third-order susceptilibity χ (3) . In SPDC, one pump photon is annihilated and two correlated photons are created with efficiency linearly dependent on the pump power. In general, the pump photon and the resulting photon pair are typically spectrally far apart, leading to (post-filtering) high photon pair purity; however, this approach is difficult to phase-match and susceptible to temporal walk-off.
The poor phase matching limits the pair generation bandwidth, and the walk-off degrades temporal indistinguishability of pairs (which would degrade pair entanglement properties) generated at different positions along the nonlinear medium. These difficulties can be mitigated by using a shorter medium but with a compromise on the pair production efficiency.
Pair generation in AlGaAs has mostly relied upon the SPDC process [4,7,9], and hence faces these challenges.
On the other hand, SFWM provides correlated photon pair generation with easier phase matching requirements. During the SFWM process, two pump photons are annihilated and two correlated photons are created with an efficiency that depends quadratically on the pump power. Because the pump photons and the pair are spectrally close to one another, the phase matching can be maintained over a broad spectral range and results in a wide pump photon acceptance bandwidth. The photons also travel with an approximately similar group velocity that leads to a minimal walk-off problem. Additionally, SFWM could create simultaneous, independent photon pairs into different wavelength channels over a broad spectral range given a fixed pump photon wavelength. This feature is very attractive in quantum channel multiplexing of the future quantum network [11,12].
The AlGaAs alloy has several advantageous properties that can exploit benefits from the SFWM process in making an efficient and bright photon pair source. AlGaAs can make high index contrast waveguides and has high χ (3) coefficients, which in combination results in a large effective nonlinear coefficient (γ ≈ 500 m −1 W −1 [13]). Inefficiencies from two-photon absorption (TPA) and free carrier absorption (FCA) are negligible by a proper choice of the alloy compositions [14,15]. Its crystalline structure leads to narrow spontaneous Raman scattering (SpRS), which can be filtered easily. In addition, control in AlGaAs waveguide dispersion allows a wide-band pump photon acceptance and a broad correlated photon pair generation bandwidth. These characteristics, in conjunction with on-chip integrateability, makes AlGaAs a very attractive platform to realize a quantum photonic circuit.
In this work, we demonstrated an integrated correlated photon pair source in a dispersionengineered AlGaAs nanowaveguide via SFWM. We characterize the pair generation rate, the signal-to-noise ratio (i.e. CAR), and the SpRS noise in our device and analyze the measurement results, which agree well with theoretical results. The interplay between SFWMgenerated photon pairs, SpRS noise, propagation loss, and waveguide length is also studied.

Theoretical background and device design
As described briefly in the previous section, SFWM requires the participating photons to satisfy the conservation of energy. For a frequency-degenerate pumping scheme, two pump photons at ω p are are annihilated to yield two correlated photons at the signal frequency ω s and the idler frequency ω i , such that For efficient and broadband SFWM, the dispersion of the nonlinear waveguide must be carefully engineered such that the phase matching condition is satisfied. A phase matching equation for the degenerate-pump co-polarized SFWM is where β p,s,i is the propagation constant of the pump, the signal, and the idler modes, γ is the nonlinear coefficient, and P is the input pump power. In the low-dispersion regime, Eq.
(2) can be further simplified to where β (2) ωp = d 2 β(ω)/dω 2 is the second-order dispersion coefficient calculated at the pump frequency and ∆ω = ω p − ω i is the angular frequency detuning. It is evident from Eq. (3)  shows the simulated β (2) curves of the TE mode for different waveguide widths, and it shows that the 700-nm-wide waveguide becomes anomalous at wavelengths beyond 1550 nm [16]. For a TE mode, the electric field is oriented horizontally and experiences a high index contrast between AlGaAs and air, which results in tight mode confinement. This effect leads to a significant waveguide dispersion that can outweigh the material dispersion; therefore, anomalous dispersion is achieved.
In the lossless limit, the idler wave that is generated through the SFWM processes has an optical power of [17] where B is the filter passband (in Hz), L is the waveguide length, and the sinc function is defined as sinc = sin x/x. The (squared) sinc function term in Eq. (4) dictates the efficiency and the bandwidth of the idler generation (or frequency conversion) process, and it is regarded as a phase matching function.
Another benefit of tight mode confinement is a small effective mode area, A eff , that boosts the nonlinear coefficient according to γ = βn 2 /A eff , where n 2 is the nonlinear refractive index.
In effect, the nano-scale waveguides deliver two-fold benefits: broadband phase matching and enhanced nonlinear interaction. However, it is well-known that a large n 2 is accompanied by a significant TPA coefficient β 2 , as both scale with χ (3) . Fortunately, this problem is suppressed in Al x Ga 1−x As as its bandgap is tunable via aluminum concentrations [14]. With x ∼ 0.20, the bandgap energy corresponds to a photon energy at 720 nm [18], hence, larger than a two-photon energy at 1550 nm, rendering a two-photon electronic transition forbidden. Even though a three-photon transition is still allowed, it requires a larger pump power rarely attained in typical SFWM operation. Another benefit of using AlGaAs (which is a direct bandgap material) is that its free carriers have a short lifetime, yielding low FCA and plasma dispersion effects.

Fabrication and classical characterization
The AlGaAs waveguide structure, whose compositional details can be found in [19], was grown via Metal Organic Vapour Phase Epitaxy (MOVPE) on a GaAs substrate. The nonlinear AlGaAs waveguides were then fabricated using electron-beam lithography patterning of HSQ resist followed by a SiCl 4 -based reactive ion etching step to an etch depth of 3 micrometers. The resulting nonlinear nanowaveguide was 700-nm wide and 4.5-mm long, matching its designed dimensions. The ends of the nanowaveguide were tapered out to 2µm-wide waveguides to facilitate coupling. The propagation loss of the fundamental mode was α = 10 dB/cm. which was measured using the Fabry-Perot technique. Therefore, the propagation loss contribution from the nanowaveguide was 4.5 dB. The total loss from the taper sections was approximately 1 dB.
In order to gauge the possible SFWM bandwidth of our device, we characterize the device using classical pump-degenerate four-wave mixing (FWM), as its bandwidth characteristics are related to that of the SFWM process [20]. In the classical domain, the generated idler power is expressed as [21] P cls where P s,in is the input signal power, L eff = [1 − exp(−αL)]/α is the nonlinear effective length, and ζ is regarded as the loss-included phase matching function and is defined by The shapes of ζ for the TE and TM (transverse magnetic) modes are plotted in Fig. 2(a) versus signal-pump wavelength detunings at pump peak powers of 0, 1, and 2 W, assuming the pump wavelength λ p = 1555 nm, γ = 150 m −1 W −1 , α = 10 dB/cm, and L = 4.5 mm, corresponding to L eff = 2.8 mm. Plots of conversion efficiency, CE, from classical cw-FWM measurements and from theoretical predictions for both TE (blue circles for data and curve for theory) and TM (green triangles and curve). Note the discrepancy between the measured data and the theoretical plot for the TM mode is due to our OSA's limited sensitivity.
The FWM measurement was performed by introducing a linearly-polarized strong pump (at a fixed wavelength of 1555 nm) and a weak co-polarized signal, whose wavelength was swept. Both the pump and the signal waves are continuous-wave (cw). The idler wavelength and power were monitored with a calibrated optical spectrum analyzer (OSA). The FWM conversion efficiency is defined as a ratio of the output idler power to the input signal power (CE = P i,out /P s,in ). In Fig. 2 In the actual SFWM experiment, we only focus on co-polarized TE-mode SFWM where both the signal and the idler photons are created in the TE modes of the waveguide. Therefore, the difference in the spectral profiles of ζ of the TE and TM modes is not an issue.
However, we would consider the difference in detail in the Discussion section.

SFWM experimentation and results
A schematic of the SFWM experiment is shown in Fig. 3. We also analyze matching of the frequency-conjugate filters by comparing their transmission spectra. In an ideal case, we expect that if a signal photon is transmitted by the signal filter, so should the correlated idler photon be transmitted by the idler filter. In practice, the two filters might not be perfectly frequency-conjugate. The degree of conjugation is defined here as where T s,i is the transmission spectrum of the signal or of idler filters. We measured the transmission spectra of our filters and calculated that the conjugation is at f = 0.95, almost perfect.
Due to the detector deadtime, the measured detection rate must be normalized appropriately by the duty cycle by using the following relation [22] where r m is the measured rate, r actual is the actual rate, and τ dt is the detector deadtime.
Onward we normalize all measured rates to their corresponding actual rates unless explicitly specified.
In order to analyze our waveguide behavior, we account for both SFWM pairs and Raman generated noise photons. The SFWM pair generation rate, the Raman photon generation rate, and the singles count rates at the signal and the idler bands are expressed as where the subscript j = s, i denotes the signal or the idler, µ is the number of SFWM photon pairs per pump pulse, P is the coupled input pump peak power, r j is the number of Raman noise photons per pump pulse, which scales linearly as the pump power [23], and η s,i is the effective detection efficiency. The coefficients a and b j determine the efficiency of photon generation through the SFWM and the Raman process respectively. Note that both µ and r j are rates at the end of the waveguide before out-coupling. As a result of dark counts and SpRS noise, the registered coincidence rate contains accidental coincidence counts as well.
The coincidence rate (per pulse) expected from the measurement, with fiber-based Raman noise safely neglected, can be described as In Eq. (12), only the first term accounts for true coincidence. The term d j = τ w D j is the dark count rate within the coincidence window. The rate C is therefore a 3 rd -order polynomial function of the pump power P . An accidental coincidence rate due to the adjacent pump pulse is expressed as Experimentally we calculate C and A from the detection coincidence histogram ( Fig. 4(a)).
The main coincidence peak appears at the zero time delay between the signal and the idler.
The total measured coincidence rate per second C m at this peak is determined from the total registered counts within the 1.5-ns coincidence window, divided by the total integration time, and corrected for the detection deadtime. However, C m also contains the accidental rate A.
Therefore, C m = C + A. The accidental rate is calculated from counts registered from the next coming pulse, corresponding to a 100-ns delay (not shown in Fig. 4(a)). The measured coincidence-to-accidental ratio, CAR, can be defined as We then seek to extract values for the photon pair generation efficiency a, the Raman noise generation rate b j , and the detection efficiencies η j by fitting with the theoretical models in Eq. (12)- (14). The optimization fit procedure yields η s = 0.024, η i = 0.016, a = 7.6 × 10 −3 pair/(pulse · W 2 ), b s = 2.8 × 10 −3 photon/(pulse · W), and b i = 2.9 × 10 −3 photon/(pulse · W), and the CAR fit is plotted with the experimental data in Fig. 4(b). We observe a CAR of 177 when a pair generation rate is µ = 29, 000 pair/s/nm = 2.9 × 10 −3 pair/pulse/nm at a coupled pump peak power of 0.54 W. The vertical error bar in Fig. 4(b) is calculated from the square root of the coincidence counts √ C m by employing the Poissonian nature of the detection. Since the observed rate is µ < 0.1, we can safely assume that multipair generation is negligible. Note that at high pump powers, CAR decreases due to the fact that the accidental rate increases. In this regime, the dark count rates d j are negligible such that CAR varies with ∼ 1/µ.

Discussion
We verify that the extracted performance of our waveguide agrees well with theoretical predictions. First, we consider uncorrelated photon noise generated from the SpRS process.
Adapted from [23], the SpRS photon generation rate over a narrow filtered band is given by where g R is the Raman gain, ∆ν is the frequency detuning from the pump, τ = 6 ps is the pump pulse width, and n th is the phonon population evaluated at the ambient temperature.
In our case, ∆ν = 2.7 THz for the idler band. The filtered bandwidth B = 0.1 THz corresponds to a 1-nm passband. We found from the experiment that for the idler band b i = 2.9 × 10 −3 photon/(pulse·W) and equating to the factor b theory yields |g R (∆ν)| = 1×10 −10 cmW −1 (equivalent to 2 m −1 W −1 ), which is in good agreement with Raman gain values reported for AlGaAs [24].
For a theoretical prediction of the photon pair generation rate, we adapt Eq. (4) for pulsed excitation and a lossy waveguide. The effective photon pair rate is expressed as Since either of the photon pair members (signal or idler) can be lost due to the linear propagation loss, it is appropriate to capture this effect with the factor e −2αL for the survived photon pair rate in Eq. (18). The value of e −αL is taken from the linear loss measurement and it includes contributions from the nanowaveguide (4.5 dB; 10 dB/cm, 4.5-mm-long) and the taper sections (1 dB). The effective length is L eff = 2.8 mm. The nonlinear coefficient is assumed γ = 150 m −1 W −1 , which is consistent with a value observed in a waveguide of a similar structure fabricated using the same process [19]. This set of parameters yields a theory = 7.5 × 10 −3 pair/(pulse · W 2 ), and it matches really well with the experimental result.
The performance of our current waveguide is limited by the propagation loss since the pair is lost when one or both of the photons are lost. The pair generation rate per squared watts a theory is plotted against propagation losses and waveguide lengths in Fig. 5. A low propagation loss reduces the power requirement in generating a certain pair rate. It also boosts CAR as the correlated photon pairs are more sensitive (quadratically) to loss than other uncorrelated noise photons. This can be seen in Fig. 6 where, using the parameters extracted from the experiment, we plot CAR as a function of pump peak powers, waveguide lengths of 0.5, 1.0, and 1.5 cm, and propagation lossess of 1, 5, and 10 dB/cm. Evidently CAR degrades substantially at high propagation losses. It is worth noting that a propagation loss α ≤ 5 dB/cm is possible for deeply etched AlGaAs nanowaveguides [25].
Additionally, in the measurement, we neither observed power-dependent saturation in the single counts nor coincidence counts, which would otherwise indicate effects from TPA and FCA. Usually, in semiconductor-based nonlinear waveguides, TPA and FCA are major causes of saturation effects for parametric processes. However, it has been demonstrated that TPA in AlGaAs materials is very low and no saturation is observed in classical FWM idler generation measurements [15,25,26].
Regarding the difference in phase matching (ζ) profiles of the TE and TM modes as shown in Fig. 2, it would pose a problem to attempts in generating broadband polarizationentangled photon pairs or broadband applications making use of both polarizations. However, as evident from Fig. 2(b), both of the SFWM efficiency curves are quite flat over 40 nm and could produce a good polarization-entangled state (|HH + e iθ |V V ) if both modes are excited at the same pump frequency [27]. Therefore, the current waveguide functions well in a single-polarization or narrow-band but dual-polarization operation. It is possible to bring   performances of the two modes more similar, for instance, by altering waveguide geometries and compositions. In particular, a square waveguide provides identical dispersion properties for the two modes, and therefore identical conversion efficiency curves.
SFWM-based correlated photon pair sources have been demonstrated in a number of material platforms including silica [28,29], chalcogenide [30], crystalline silicon [31], and amorphous silicon (a-Si) [32]. However, silica fibers and chalcogenide glasses suffer from uncorrelated photons originating from the SpRS process, which has a broad gain profile.
In Table 1, we compare our source with several AlGaAs-based and silicon-based sources.
Comparing within the AlGaAs device family, our source, which utilizes χ (3) rather than χ (2) , has outperformed in the CAR measure and provides a competitive photon pair rate. Considering χ (3) -based devices, our source appears decent in generating photon pair with a high signal-to-noise ratio. Even though the performance we demonstrated here has not matched that of the best-reported c-Si source yet, there is a plenty of room for improvement including lower propagation loss and integrating pump sources. Our study proves the possibility of efficiently generating correlated photon pairs with the AlGaAs material system employing the third-order nonlinearity.

Conclusion
In summary, we have reported broadband (>80 nm) correlated photon pair generation in a dispersion-engineered AlGaAs nanowaveguide utilizing spontaneous four-wave mixing. The CAR of up to 177 is measured with a corresponding pair generation rate of 2.9 × 10 −3 pair/pulse/nm. The theoretical models incorporating both SFWM-generated photon pairs and SpRS noise agree well with the experimental results, revealing a feature of low SpRS noise (Raman gain of g R = 2 m −1 W −1 ). The nanowaveguide also showed no detrimental effects from two-photon absorption. We project that with lower propagation loss in future waveguides, it is possible to realize brighter correlated photon pair sources with CAR>300. This work shows that dispersion-engineered AlGaAs nanowaveguides can serve as a bright correlated photon pair source and offers promises for implementing scalable, integrated quantum photonic circuits.