Near-infrared Hong-Ou-Mandel interference on a silicon quantum photonic circuit

Near-infrared Hong-Ou-Mandel quantum interference is observed in silicon nanophotonic directional couplers with raw visibilities on-chip at 90.5%. Spectrally-bright 1557-nm two-photon states are generated in a periodically-poled KTiOPO4 waveguide chip, serving as the entangled photon source and pumped with a self-injection locked laser, for the photon statistical measurements. Efficient four-port coupling in the communications C-band and in the high-index-contrast silicon photonics platform is demonstrated, with matching theoretical predictions of the quantum interference visibility. Constituents for the residual quantum visibility imperfection are examined, supported with theoretical analysis of the sequentially-triggered multipair biphoton contribution and techniques for visibility compensation, towards scalable high-bitrate quantum information processing and communications.


Introduction
In recent years, quantum information has been popular for its robust applications on cryptography [ thermally-tuned and stabilized by self-injection locking to 778.9-nm, which is exactly half of the center working wavelength of the PPKTP waveguide. The temperature of the PPKTP waveguide is typically controlled to ~ 25C for optimal phase matching. A long-pass-filter with cutoff at 1064-nm (Semrock BLP01-1064R-25) blocks pump photons after the SPDC process, and a band-pass filter with 3-nm (Semrock NIR01-1570/3-25) passes the non-degenerate biphoton states. The polarization controller right before the fiber-based PBS is used to tune the polarization so that the fiber-based polarization beamsplitter (PBS) spatially separates the correlated photons. In one branch, a tunable delay is realized by a retroreflector (Thorlabs PS971-C) and a picomotor stage with loss less than 1-dB. In both branches, polarization controllers are introduced to respectively change the polarization of each channel to match the transverse magnetic (TM) mode for coupling into the chip waveguides ( Figure 1b).
The chip coupling setup is built with six aspheric lenses, each mounted on individual three-axis precision stages. The two input and output beams are separated by a D-shaped mirror after 60 cm divergence to avoid crosstalk. Single and coincidence measurements are performed by two InGaAs single photon Geiger-mode avalanches detectors D 1 and D 2 from Princeton Lightwave, with ~ 300 ps gate widths and ~ 20% detection efficiencies. The clock of D 1 is set to 15 MHz, and its output signal triggers D 2 . This allows the coincidence rate to be read directly from the D 2 counting rate, with the optical delay calibrated to compensate the electronic delay.

Design and fabrication of silicon chip-scale two-photon interference directional coupler
To ensure good quantum interference on-chip, we examined the design space of the directional couplers, in both transverse electric (TE) and TM polarization states as shown in Figure 2. Differential gap widths (g), cross-over coupling lengths (l c ) and waveguide widths (w) are illustrated for the optimal coupling length and splitting ratios. The silicon waveguides are designed with a 250-nm thickness and for operation at 1550-nm wavelengths.
in which l eff is the effective coupler length for the incoming and outgoing bend regions, which can be estimated by an integral of coupling length as a function of gap size along the bending region and computed to be 3-um in our designs ( Figure 1b). In addition to the MPB and integral computations, the designs were examined with both rigorous finite-difference time-domain computations and semi-vectorial BeamPROP method from RSoft. With the birefringent character of the directional coupler, we work with the TM mode rather than the TE mode due to its shorter coupling length and greater length control sensitivity. Furthermore, our simulation models and experimental measurements confirm lower loss in the TM mode for straight waveguide as well as the directional coupler regime due to lower electromagnetic field amplitude at the sidewalls (typically rougher than the top and bottom surfaces) [48-51]. The lower loss helps to increase the coincidences count rates and reduce the internal phase shift fluctuations of directional coupler. A quantitative calculation suggests the loss of TE mode is 7.4 times higher than TM mode for a consistent sidewall roughness. In one optimized instance, the waveguide width and coupler length for TM symmetric splitting is chosen to be 400-nm and 15-um, respectively, as illustrated in Figure   2 (Design 1). In this design, the corresponding TE-polarization splitting ratio imbalance was numerically computed to be 9-dB. The excess loss at the optimized directional coupler of Design 1 is estimated to be 0.1-dB by finite-difference time-domain computations.
Further increasing the coupler length will change the splitting ratio imbalance (SR), which could be determined by:   Supported by these designs, the devices were next fabricated at the Institute of Microelectronics.
Silicon-on-insulator wafers were used, with 248-nm deep-ultraviolet lithography for resist patterning. With our sequential triggering approach (detector D 2 triggered by D 1 ), instead of time-tagging, the coincident dark counts are negligible. An example coincidence versus the relative optical delay is illustrated in Figure 3a, with the observed near-infrared Hong-Ou-Mandel quantum interference on-chip.

Sources of chip-scale interaction distinguishability
To further uncover the sources of distinguishability, we compare the on-chip Hong-Ou-Mandel visibility with that of a fiber beam splitter (without chip) as illustrated Figure 3b. We plot the visibility against different pump powers or the mean photon pair number to estimate the effects of the chip on the visibility. Since a higher pump power with more biphoton pairs will cause a higher probability of multiple biphoton pairs in one detector gate window, the visibility is inversely proportional to the pump power [43].
where the one half denotes the 50% probability that the biphotons will separate to two gates, and denotes the overall detection efficiency, including all losses and intrinsic detector efficiency. To calculate the probability of coincidence when two photons are indistinguishable, we consider only one and two photon pairs within the gate. Here we notice that even when there is only one photon pair within the detection gate of triggering detector D 1 , there are still some coincidences contributions ( Figure 4b): where the photon pair is considered uniformly distributed within the gate window, and the possible photon pair within the leak window due to gate time mismatch is considered (Figure 4b) Taking the first order approximation, we have that: Here, ,which denotes the probability distribution of the first arriving photon pair. We notice here the difference between the sequential triggering approach versus the time-tagging approach is that there is a situation that the second photon pair will be located within the gate window of one detector, but is cut off by the gate window of the other detector (Figure 4c). This portion equals to , which is exactly the same as the contribution of coincidence conditioning only one photon pair per gate (Equation 4) even when disregarding the detection efficiency distribution within the gate and timing jitter. As these two terms compensates each other, we conclude that, to first order, the visibility for the sequential triggering scenario is as same as time-tagging scenario: Fig. 4. Scenario of the timeline for the photon pairs. (a) The delay of two photon pairs is set to /2 to maximize the coincidences. (b) When there is only one photon pair in the gate window of D1, there is still possibility that D2 will record a photon event due to gate window time mismatch. (c) When there are two photon pairs within the gate window and separated to two detectors, there is possibility that the latter photon pair will be cut off due to the gate window time mismatch.
From fitting the chip result with the same slope as suggested by the above theory, we conclude that 6% of the imperfect visibility is therefore likely to be from the multiphoton pairs. The residual 3% is likely to be induced by processes on-chip. To further understand the chip mechanisms for visibility reduction, we next compared the visibility for different splitting ratios. We selected two devices with coupler lengths of 28-and 30-um, which has the TM mode splitting ratio imbalance of about 3-dB and 6-dB as measured. The comparison of the coincidence measurements between the three silicon chip devices is shown in Figure 5a (before normalization, with lower integral time of 120 seconds compared to Figure 3a). The inverse triangular fit is utilized to estimate the visibility and corresponding deviations. For the 28-um directional coupler, the visibility is measured to be 74 ± 8%, close to the theoretical estimate of 80%. For 30-um directional coupler, the visibility after fitting is 31 ± 11%, compared to the theoretical estimate of 47%, in similar ballpark. The deviations here from theory are due to on-chip directional coupler internal loss and high pump power. For our optimal 15-um directional coupler, the less than 1-dB splitting ratio imbalance (limited by precision of lens-chip coupling loss variations) with its 97% theoretical visibility can therefore account of a sizable portion of the residual 3% decrease in visibility.
Moreover, to understand the quantum interference effect with variation of polarization, we rotate the polarization for one branch of the input path before the chip using a half waveplate. The resulting visibility versus the linear polarization angle is depicted in Figure 5b. The result shows cosinusoidal behavior that reaches maximum visibility with no polarization rotation, and diminished visibility with orthogonal polarization. The maximum visibility in this set of measurements is 83% due to higher pump power of  Here we estimate that the 0.1-dB excess loss via vertical scattering from the chip even with ideal sidewalls, or 170˚ internal phase shift, computed by FDTD method as noted in the earlier design section, in the balanced directional coupler will reduce the visibility by 1.5%. This excess loss will be larger when including fabrication disorder-induced losses. For unbalanced directional coupler, the internal phase shift will be further away from 180˚ with corresponding reductions in the visibility. Formally, the output annihilation and creation operators of a lossy directional coupler have to include Langevin noise operators to maintain the commutation relation, while at the same time inducing additional phase shifts [60]. of the directional coupler. α denotes the birefringence at the directional coupler, β denotes half the internal phase shift for the TE polarization, and γ denotes half the internal phase shift for the TM polarization. In the waveguide region, the differential birefringence has a ϕ phase shift in the TM mode relative to the TE mode, and the differential loss from TM to TE is denoted as ld. The waveplate delays the TM mode with an additional ψ phase.

Compensation method for chip-scale two-photon interference
With the splitting ratio imbalance and excess loss of the directional coupler (due to fabrication imperfections or slight residual design), one of the co-authors has proposed an approach to compensate the imbalance and regain the indistinguishability [52], albeit for lossless and non-birefringent fiber beamsplitters. For the scalable chip implementation, a simple but lossy approach (shown in Figure 6a) would be to place half waveplates in one path before the chip and two polarizers after the chip, post-selecting the photons and removing the polarization information. When the splitting ratios are different for TE and TM modes, there exists a continuum of solutions for the angles of the polarizers to achieve the probability amplitude of rr A and tt A with the same amplitude and inverse phase. Here the distinguishability is removed as long as the following condition is met: (a) WP polarization information projection to the polarizer. In more complete scenarios where loss and loss-induced internal phase shifts are considered for TE and TM modes, this distinguishability could still be compensated in our approach since only two adjustable elements, for example the angles of the polarizer and of the waveplate, are needed to recover the two probability amplitudes inverse in phase and equal in amplitude. We define the directional coupler birefringence, internal phase shifts, waveplate phase, and waveguide differential birefringence and loss in Figure 6. In this complete case, we have

Conclusion
We have observed 1550-nm Hong-Ou-Mandel interference in silicon quantum photonic circuits, with raw quantum visibility up to 90.5% in near-symmetric directional couplers. With thermally-stabilized spectrally-bright PPKTP chip-scale waveguides as the entangled biphoton source, we examined the constituents of residual distinguishability through numerically-designed directional couplers, multiphoton pairs, polarization effects, excess loss, and imperfect phase shifts. With our sequential triggering approach for negligible coincidental dark counts, we present the theoretical analysis for multipair biphoton contribution to Hong-Ou-Mandel visibility reduction. Techniques for visibility compensation in chip-scale birefringent directional couplers in the presence of loss are described. The results presented here support the scalable realization of two-photon interaction elements on-chip, for quantum information processing and communications.