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Abstract
We present a silicon photonic integrated circuit (PIC) designed to generate high-dimensional (HD) entanglement across four discrete waveguide path modes. Our PIC is foundry fabricated and fully packaged with electrical wirebonds and an epoxy bonded fiber array. We performed a partial characterization of our system’s density matrix by demonstrating spectral indistinguishability of photon pairs, measuring the joint coincidence distribution of photon pairs across all pairs of output modes, and measuring the coherences of different mode pairs using two-photon interference. Using a quantitative entanglement witness, we have determined that our system generates at least 1.45 ± 0.15 ebits of HD photonic entanglement.
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1. Introduction
Entanglement is a resource [1] which enables many applications of quantum information systems (QIS) [2–4]. Recently, systems capable of generating entanglement across high-dimensional (HD) degrees of freedom have garnered aggressive research interest as they provide enhanced utility compared with two-level systems. Access to larger Hilbert spaces has led to novel quantum computing algorithms [5,6], and allows information to be encoded more densely, for example, using particles in a quantum communication channel [7–9].
Photonic QIS systems are an attractive platform for studying HD entanglement. Photons have access to several HD degrees of freedom such as temporal modes, spatial modes, and frequency modes [10]. Furthermore, optical QIS allow us to transmit data at the speed of light and with perfect security [4,11]. HD photonic entanglement has historically been investigated using bulk optical setups [12–14]; however, this approach is impractical due to large spatial footprints and the need to precisely align many components.
Photonic integrated circuits (PICs) are able to address these challenges. The ability to integrate entire optical setups onto microchips is a key ingredient for scaling “proof-of-concept” experiments into systems with real-world utility. The silicon photonic platform is particularly attractive due to foundry-level fabrication techniques and optimization for telecom wavelengths [11].
As PICs continue to scale for practical applications, with some circuits incorporating hundreds of components onto a single microchip [15,16], considering systems’ “ease-of-use” becomes critical. Advanced packaging techniques such as fiber-attach and wirebonding [17] streamline calibration and control of increasingly sophisticated PICs while also reducing mechanical noise at optical and electrical interfaces. Techniques for eliminating cross talk between thermal elements have also been investigated [17–19].
Here we report a silicon PIC designed to generate photon pairs entangled across four discrete waveguide modes. The integrated circuit consists of four microring photon pair sources, pumped in parallel from a single input waveguide, and spectrally tuned using a series of thermo-optic phase shifters. The device has been fully packaged, with an epoxy bonded fiber array and electrical wirebonds, to make operation and characterization of the system more efficient. By measuring the joint coincidence distribution of generated photon pairs along with two-photon interference curves for pairwise combinations of output modes, we have partially characterized our system’s density matrix. Finally, using an entanglement witness, we have calculated a lower bound for the entanglement of formation of the state generated by our system.
2. Theory
Entanglement is both a property of a system [20,21], and a quantifiable resource [1]. A unit of measure used to quantify the entanglement resource is the ebit, which represents the amount of entanglement observed in a Bell pair [20,22]. Such a system is described by Eq. (1), where particles $a$ and $b$ are entangled across modes $|0\rangle$ and $|1\rangle$:
One metric used to quantify the amount of entanglement produced by a HD system is the entanglement of formation (EoF). EoF represents the average number of two-qubit Bell states required to synthesize a given state [25]. References [25,26] provide Eqs. (4) and (5), which bound the EoF of a HD system described by the density operator $\hat {\rho }$:
When our model includes the frequency dependence of photon pairs, two-photon interference patterns can change because of discrepant joint spectral amplitude functions for photon pairs in different modes. Equation (8) describes $\langle {\hat {P}_c}\rangle$, the likelihood of a detecting a coincident pair of photons in separate output paths of a beam splitter, as a function of $\theta$, the relative phase delay inside the interferometer:
Fig. 1. (a) Interference visibility as a function of the normalized single mode pair rate $\rho _i$. Visibility is maximized when photon pair generation is balanced between interfering sources. (b) Interference visibility as a function of the spectral overlap integral $\Phi _{ij}$. Visibility is maximized when the spectra of interfering sources are identical. (c) Heatmap of theoretical interference visibilities as a function of both pair generation rate and spectral overlap.
Interference visibility (and therefore EoF) is at a maximum when photon pairs are equally likely to be found in any path mode, and when photon pairs in different output modes have identical spectra.
3. Experiments
A schematic of our circuit is shown in Fig. 2(a). It is designed to produce the biphoton NOON state $|\psi \rangle _{\text {NOON}}$ in Eq. (7). The circuit is driven by a continuous wave (CW) telecom laser which edge-couples to our PIC via SMF-28 fiber. Input light is then divided evenly by a series of cascaded Y-splitters and used to pump an array of four photon pair sources in parallel.
Fig. 2. (a) The circuit that was designed to generate a biphoton NOON state entangled in path. Photon pairs can be created by each of the four DMZIMRR sources with equal probability. Each source is equipped with three thermal phase shifters for precise control of its transmission spectra. Metal pads are placed over each source to reduce cross talk between neighboring heating elements. (b) A microscope image of a single DMZIMRR photon pair source. This image was taken from a PIC that was a part of the 2019 AFRL AIM Photonics multi-project wafer (MPW). (c) A microscope image highlighting the on-chip portions of fiber attach and wirebond packaging techniques. (d) Our fully packaged PIC on a benchtop. Optical signal is transmitted and received through a fiber array. Electrical signal is provided by a programmable current source which interfaces with our PIC’s printed circuit board (PCB) carrier.
To generate entanglement, we require photon pair sources which are both bright and tunable. Tunability is critical, as it allows us to ensure photon pairs produced by different sources have indistinguishable joint-spectra. We have chosen to implement interferometrically coupled dual Mach–Zehnder microring resonators (MRRs) as our source of photon pairs [27–29]. An image of a single source is shown in Fig. 2(b). Our photon pair sources were designed to the same specifications as those reported in Ref. [27]. These sources generate photon pairs by spontaneous four wave mixing (SFWM), whereby silicon’s $\chi ^{(3)}$ nonlinearity may convert two pump photons of frequency $\omega _p$ into non-degenerate signal and idler photons at frequencies $\omega _a$ and $\omega _b$ [30]. Integrating Mach–Zehnder interferometers (MZIs) at the coupling sections of microrings gives us control of the coupling coefficients at each bus waveguide. This allows us to enhance the SFWM process and tune the transmission spectra of generated photon pairs. In addition, the transmission spectra of MRR photon pair sources is a frequency comb [29]. This not only limits the bandwidth of generated photon pairs, but also provides an opportunity to study entanglement in the frequency domain [31].
Our integrated circuit has nine optical channels and 34 electrical connections in total, each of which must be calibrated and monitored. Because of its complexity, this PIC is an example of a photonic system which requires advanced packaging techniques. To that end we have implemented: (i) gold wirebonds connecting the electrical pads to a PCB carrier; (ii) an epoxy-bonded SMF28 fiber array which couples to the input/output waveguides; and (iii) “micro-ovens” to thermally isolate neighboring sources [19]. Together, these features reduce the effects of environmental noise, eliminate the need for probing and eliminate the need for micrometer stage alignment between the chip and the fiber array. An operator need only couple their optical power source, configure each photon pair source by sending electrical signals to the PCB carrier, and then interface with optical FC (ferrule connector) connections at the fiber array’s output. The chip packaging advances are shown in Fig. 2(c) and the complete system is shown in Fig. 2(d). We measured $84{\% }$ coupling efficiency between our PIC and the fiber array (see Supplement 1).
To calibrate our photon pair sources so that their spectra were identical and optimized for photon pair production, we conducted a series of frequency sweeps using our tunable laser. During each sweep we monitored the transmitted intensity at the through and drop ports of all sources. After each sweep, we manually adjusted the currents applied to the heaters of a single pair source until it directed signal and idler frequencies to its drop port, and pump light to its through port. This is the optimal configuration for photon pair generation [27]. Once a single source was configured properly the spectra of all other sources were tuned to match.
Figure 3 shows the transmission spectra at all sources’ through and drop ports. This data shows that signal and idler photons will be preferentially directed to the drop ports of each microring. We have computed the set of spectral correlation functions $\Phi _{ij}$ as a measure of spectral indistinguishability. These results are given in Table 1.
Table 1. Spectral Overlap Integrals $\Phi _{ij}$
The EoF of photon pairs generated by our PIC is determined by the density matrix of its output $\hat {\rho }$. The diagonal elements of $\hat {\rho }$ are the normalized joint coincidence distribution of signal and idler photons detected in different pairs of output modes [25]. Coherences are extracted from the visibility of two-photon interference curves generated between pairs of output modes [25,32]. Schematics of the experimental setups for these measurements are shown in Fig. 4.
Fig. 4. Block diagrams illustrating the following. (a) The experimental setup used to measure the rate of coincident photon pairs created in the same output mode. We normalized data from these experiments to produce the diagonal elements of $\hat {\rho }$. (b) The experimental setup used to measure the rate of coincident photon pairs created in different output modes. Since we expect our photon pairs to be path-correlated, we do not expect to observe large coincidence rates using these experiments. (c) The experimental setup used to mix signals from pairs of output modes. The visibilities of these interference curves are the off-diagonal elements of $\hat {\rho }$. Here $\hat {P}_c$ represents a coincidence detection with a single photon in each output arm of the interferometer.
Figure 4(a) illustrates the setup used to detect coincident photon pairs exiting the same path mode. Here 3.5 mW of CW 1549.85 nm light is sent through a 1 nm bandpass filter, and edge-coupled to the input waveguide of the PIC. The outputs of individual pair sources are sent to a $50:50$ fiber splitter, carrying signal and idler photons as well as residual pump light. In each arm, residual pump light is first removed using a two-channel wavelength-division multiplexer (WDM). Then, each arm selects either signal or idler photons using a 1544.01/1 nm or a 1555.78/1 nm bandpass filter. Both arms are sent to a superconducting nanowire single photon detector (SNSPD) and a time-correlated single photon counting (TCSPC) system where coincidence histograms are collected. We chose a 10 s integration time and a 1 ns coincidence window to maximize correlated count rates above the accidental noise floor (see Supplement 1).
A similar setup was used to detect coincident photon pairs created in pairwise combinations of path modes, illustrated by Fig. 4(b). Without the need to distill signal and idler photons from a single mode, we direct photon pairs traveling in modes $i$ and $j$ to the SNSPD/TCSPC. Each path selects either signal or idler photons using the combination of a pump removal WDM and a 1 nm bandpass filter centered at the appropriate wavelength. Coincidence histograms are collected using the same settings as before.
A diagram of the setup used for interference measurements is shown in Fig. 4(c). Photon pairs in modes $i$ and $j$ propagate through paths of equal length and a set of manual polarization controllers to arrive at a $50:50$ fiber splitter. The output arms of the interferometer select either signal or idler photons and are sent to the SNSPD/TCSPC system. The relative phase delay $\theta$ of photon pairs arriving at the beam splitter is allowed to drift slowly in time. We record this phase as a function of time using the pump light that is reflected by our WDMs. The arrival times of coincident photon pairs was recorded using the TCSPC system in time-tagged time-resolved (TTTR) mode. Photon pairs were then binned according to the relative phase in the interferometer at the time of their detection. Using this data we generated interference curves showing oscillations in coincidence probability $\hat {P}_c$ as a function of the relative phase $\theta$ choosing an integration time of 5 min.
4. Results and discussion
Figure 5 shows the joint distribution of measured coincidence rates for signal and idler photons created in all possible pairs of output modes. We observe that coincidence rates are elevated along the diagonal, which is the expected behavior of a path-correlated system. Once normalized, we take the diagonal elements of this distribution to be the diagonal elements of $\hat {\rho }$. These, along with the spectral overlap integrals in Table 1 were used to generate theoretical interference curves based on our model in Eq. (8).
Fig. 5. Measured joint coincidence distribution for signal and idler photons over all possible pairs of output modes.
Figure 6 shows the two-photon interference curves generated by our model alongside measured two-photon interference curves produced by the PIC. Excellent agreement is found between predicted and measured values. The visibility of these curves can be used in conjunction with Eq. (6) to produce the coherence of pairwise combinations of output modes (see Supplement 1 for Monte Carlo error analysis).
Fig. 6. Expected interference curves generated by our model plotted along with measured interference curves produced by mixing pairwise combinations of output modes. The visibilities of these curves give us the coherences of our system's output signal.
Using Eq. (4) and the density matrix elements we have obtained from our experiments, we can estimate our PIC’s EoF. We estimate our PIC has generated at least $1.45 \pm 0.15$ ebits of HD entanglement which we compare with the upper limit for a four-dimensional system is $\log _2(4) = 2$ ebits.
We can improve the “quality” of generated entanglement, that is to say we can approach the upper limit of 2 ebits for our system without adding sources or modes. One way this can be achieved by balancing photon pair detection rates at the output ports of all sources. The imbalance in our measured coincidence rates can be attributed to an imperfect chain of integrated Y-splitters which distributes pump light to our array of photon pair sources with some prejudice. Discrepancies in the relative intensity of each source’s transmission spectra are also evidence of imbalanced pump power distribution. We can remedy this by replacing the Y-splitters with tunable MZIs, which can be configured to pump all sources identically.
Another approach to “higher-quality” entanglement is to increase the coherences of our system. We can achieve this both by improving the design of our circuit, and our experimental setup. Discrepancies in the relative intensity of each source’s transmission spectra would cause a decrease in the set of spectral correlation functions. Therefore, distributing pump power more evenly with tunable MZIs would also improve coherences. Considering our experimental setup, recall that we relied on an optical power meter and manual polarization control paddles in order to maximize coherence during two-photon interference measurements. If digital polarization controllers [33] were used instead, reduced potential for human error would increase the coherences of our system.
We can also generate more entanglement by adding more sources to increase the dimensionality of our system. Using fully packaged systems like the one reported here, the number of sources we can control is limited only by the dimensions of our microchip. Recently, 33-dimensional photonic entanglement has been generated and characterized in the frequency domain using 102 frequency modes [34]. The silicon photonic platform requires a larger spatial footprint to generate entanglement of that scale in the path basis. Because of this, we will likely need to design “modular” systems spanning multiple microchips [35] and implement novel packaging techniques such as photonic wirebonding in order to operate them [36].
We have presented a foundry-fabricated silicon photonic chip, designed to generate HD entanglement across four discrete waveguide path modes. We documented how this chip was packaged and how its advanced packaging enables us to efficiently calibrate our circuit/interface with other fiber-based systems. We have characterized our system’s density matrix using measured data and estimated the EoF of our system with an entanglement witness. The design of our circuit, its packaging, and the characterization procedure are all readily scalable to higher dimensions, providing a blueprint for generating more entanglement using larger circuits. Furthermore, as our system can initialize to a known state, we expect our device will be useful for characterizing the behavior of other systems that seek to manipulate quantum information encoded onto entangled states of light [37,38].
Funding
Air Force Research Laboratory (FA8750-21-2-0004); RIT-L3Harris Quantum Information Science and Technology Collaboration.
Disclosures
This work was supported in part by the Air Force Research Laboratory (FA8750-21-2-0004). The views/conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the United States Air Force, AFRL, or the U.S. Government.
E.M.S. and G.A.H. also gratefully acknowledge support from the RIT-L3Harris Quantum Information Science and Technology Collaboration.
Data availability
Data and code underlying the results presented in this paper are available upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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