1. Introduction

Femtosecond laser direct writing (FLDW) technique has shown powerful capability to induce permanent refractive index change localizing at the focal volume in the transparent material through nonlinear processes [1]. Benefiting from this mechanism, an optical waveguide, whose core is formed via a positive refractive index change relative to the unirradiated material, can be fabricated in the bulk substrate in single step [2], by conveniently translating the substrate with respect to the laser beam in two-dimensional plane or even in three-dimensional (3D) configuration. Integrated waveguide circuits are compact and stable compared with those using traditional bulk optical elements at large space volume. In recent decades, this technique has been widely applied to various photonic quantum applications [3] ranging from integrated quantum logic gates [4–7], quantum sources [8], quantum walk [9] to boson sampling [10], quantum key distribution devices [11], and so on.

In the realm of linear photonic quantum computation [12–17], two-qubit logic gates together with one-qubit gates are sufficient to form circuits to enact any desired unitary transformation [18, 19]. Integrated one-qubit gates are straightforwardly implemented by using directional couplers (DCs) analogous to bulk beam splitters [20] or wave plates [21, 22]. One of the most important two-qubit gate is the controlled-NOT (CNOT) gate that flips the target qubit (T_q) when the control qubit (C_q) is in logic state |1 > . In 2008, Politi et al. demonstrated the first integrated linear optical CNOT quantum gate for path-encoded qubits in silica-on-silicon chip fabricated through conventional 2D lithography [23]. Path-encoding is scalable and flexible for large-scale circuits as demonstrated recently by Wang et al.: integration of more than 550 photonic components, including 16 identical photon-pair sources, on a large-scale silicon photonics path-encoded quantum circuit with dimensions up to 15 × 15. Another important advantage is that universal operations on path-encoded multidimensional quantum systems are possible in linear optics for any dimension [24, 25].

In 2011, Crespi et al. fabricated an integrated controlled-Phase gate using femtosecond laser writing for polarization-encoded optical qubits. For the input T_q, they change the basis from the simple states |H > and |V > to a superposition of state (|H>+|V>)/√2 and (|H>-|V>)/√2, so the CNOT operation was achieved. Since the difference of the evanescent coupling coefficients for H and V states is very small, the length of the coupling region is as long as ∼ 7 mm [6]. In 2018, Zeuner et al. also realized a laser-written polarization-encoded CNOT gate, which requires additional polarization-entangled ancillary photons to herald the successful operation of the circuit, and thus the unmeasured output control and target qubits can be used as inputs for subsequent logic operation. However, to obtain the desired CNOT output state, one of four Pauli rotations must be performed on the output control–target state, depending on the polarization measurement outcome for the output ancilla state [7]. Meany et al. took advantage of the 3D machining ability of FLDW to inscribe a 3D heralded path-encoded controlled-Phase gate, which is similar to CNOT gate. The four constituent DCs are asymmetric in phase, so additional phase control is demanded in the fabrication [5]. Up to now, integrated photonic quantum CNOT for path-encoded qubits based on FLDW technique has not been reported.

Here we demonstrate the fabrication of a linear path-encoded CNOT quantum gate inside glass using FLDW, for the first time to the best of our knowledge, which provides important significance for the development of large-scale 3D quantum computing circuits based on femtosecond laser writing in the future. The overall configuration, as shown in Fig. 1, is similar to the proposal of Ref. [26] originated from Ref. [12], which consists of five DCs with their power reflectivities (R) indicated in the figure. The power reflectivity R (transmission T) is defined by the ratio of the output power from the through/bar (cross/coupled) arm to the total output power of a DC where the laser is launched into either input arm [6]. Then the splitting ratio of the DC can be expressed by R or T or R : T [27]. For a DC with R = 1/2, the splitting ratio is 0.5 or 50: 50. Nevertheless, in Ref. [26], asymmetric phase change is required with a different symmetry for the various DCs composing the CNOT device. In our design, the five DCs are all symmetric in phase and produce a π/2 phase change, which matches the inherent symmetry of femtosecond laser written circuits. The quantum CNOT operation is performed with an average fidelity higher than 0.98.

 

Fig. 1. Schematic of a path-encoded CNOT logic gate. (a) The circuit consists of three DCs with power reflectivity R = 1/3 and two DCs with R = 1/2. The R is defined by the ratio of the output power from the through arm to the total output power of a DC where the laser is launched into either input arm. Reflection off any surface produces a π/2 phase change. The C_q (T_q) is encoded via spatial paths C₀ (T₀) and C₁ (T₁), representing logic states |0 > and |1>, respectively. The remaining two paths A_c and A_t represent ancillary vacuum modes. The uniform purple lines in the interaction straight regions indicate all the surfaces yielding a relative π/2 phase change upon reflection. (b) CAD layout corresponding to the actual fabricated size of the CNOT gate. The overall length is 2.5 cm. The coupling length is less than 0.4 mm.

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2. Design of the path-encoded CNOT quantum gate

Figure 1 shows the schematic configuration of the CNOT gate. The C_q (T_q) is encoded via spatial paths C₀ (T₀) and C₁ (T₁), representing logic state |0 > and |1>, respectively. The remaining two paths A_c and A_t represent ancillary vacuum modes to complete the network. It is noteworthy that the T₀ and T₁ mode labels are exchanged with respect to those in Refs. [26]. This is because the two top DCs obtain a π phase shift only when the reflection occurs through the upper surface of the DC while the other three DCs obtain the phase shift only when the reflection occurs through the lower surface in Refs. [26]. However, in our design, all the five DCs produce a π/2 phase shift through both the upper and lower surfaces reflections. The ideal CNOT operation is: T_q flips from the initial state |1> (|0>) to the opposite state |0> (|1>) if and only if C_q is in the state |1 > . The key effects in this linear optical circuit are two kinds of interferences: Mach-Zehnder Interference (MZI) and Hong-Ou-Mandel Interference (HOMI) [28]. They together play a decisive role in the fidelity of the CNOT gate. High fidelity of the CNOT gate requires high interference visibilities of these two effects.

The MZ interferometer consists of two balanced DCs (R = 1/2) connected by paths ${T_1}$-$T_0^{\prime}$ and ${T_0}$-$T_1^{\prime}$ . An excellent performance of the MZI depends on two strict conditions: (1) accurate R, i.e., the splitting ratio, of the two 50:50 DCs, which requires precise control of evanescent coupling modes and highly repeatable fabrication process; (2) completely symmetrical two arms with exact and stable phase difference between them. The ideal outcome for single photons injected from T₁ (T₀) is that the photons appear in $T_1^{\prime}$ ($T_0^{\prime}$), $C_1^{\prime}$ and $A_t^{\prime}$ equiprobably, but never appear in $T_0^{\prime}$ ($T_1^{\prime}$). Therefore, when two photons enter C₀ and T₁ (T₀) simultaneously, corresponding to |C_qT_q>_input = |01> (|00>), the output should be |C_qT_q>_output = |01> (|00>), which means that the target qubit has no change on condition that the control qubit is in state |0 > .

The two-photon HOMI occurs when two indistinguishable single photons coming from C₁ and T₁ (T₀) mix in the central DC with R = 1/3 simultaneously. Given the fact that $R \ne \textrm{1/2}$ the two photons undergo a partial bunching effect and have some probability of taking different paths after the DC. This is indeed the situation required for the CNOT proper operation since in this case a $\Pi$-phase shift is generated [29]. Combining with MZI, the case that one photon outputs from $C_1^{\prime}$ and the other from $T_1^{\prime}$ ($T_0^{\prime}$) at the same time never happens. Hence, the two-photon input-output relations are |C_qT_q>_input = |11> → |C_qT_q>_output = |10>, or |C_qT_q>_input = |10> → |C_qT_q>_output = |11>, which means the target qubit flips on condition that the control qubit is in state |1 > .

3. Experimental and results

The CNOT gates are inscribed in borosilicate glass (EAGLE2000, Corning) by a regeneratively amplified Yb: KGW femtosecond laser system (Pharos-20W-1 MHz, Light conversion) producing ∼ 240 fs pulses at 1030 nm at a repetition rate of 1 MHz. In the fabrication, laser pulses are focused 170 μm beneath the glass surface using a 0.5 numerical aperture microscope objective (RMS20X-PF, Olympus).

We first fabricate straight waveguides to optimize fabrication parameters. The optimal pulse energy is ∼ 296 nJ and the sample translation speed is 40 mm s⁻¹ executed by a computer-controlled high-precision three-axis air-bearing stage (FG1000-150-5-25-LN, Aerotech). The 808- nm guided mode of a 2.5-cm-long straight waveguide for the vertical polarized light shown in Fig. 2(a) is slightly elliptical with a mode-field diameter (MFD) of 14.0 µm × 16.4 µm, which deviates from that of the classical 808-nm single mode fiber (SMF) butt-coupled to the waveguide leading to high coupling loss. Experimental results indicate that the output guided mode of the waveguide for the horizontal polarized light is almost the same. The insertion loss is estimated to be ∼ 4.1 dB by measuring both the input power of the incident beam and the output power from the waveguide. This value is relatively high due to the high coupling loss. The measured coupling loss is ∼ 1.9 dB/facet via numerically evaluating mode overlapping integral [30], which can be further reduced by beam shaping [31] or adiabatic thermal methods [32, 33] to improve the mode-overlap between fiber and waveguide. The Fresnel reflection loss is estimated to be ∼ 0.2 dB/facet. The measured propagation loss is ∼ 0.7 dB/cm by subtracting the coupling loss and Fresnel reflection losses from the total insertion loss, which is a little smaller than that in Ref. [6].

 

Fig. 2. (a) Near field image of the waveguide guided mode at 808 nm for the vertical polarized light. (b) The measured coupling coefficient k as a function of the interaction distance d. (c) Microscope image of part of the coupling region at d = 8 µm of the DC.

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Then we fabricate high quality DCs. A DC comprises two waveguides that interact with each other through evanescent coupling in the interaction region, where they are brought to a close interaction distance (d) for a certain coupling length (L) to facilitate optical power exchange between them. For the fabrication of DCs, we set bending radius to be 60 mm for the curved segments of the DCs, which gives < 1 dB of additional bending losses on the whole device. To determine an appropriate d for the gate fabrication, we need to understand how it affects the coupling coefficients (k). So we keep the L to be 1.5 mm and calculate the k of the two guided modes with different d as shown in Fig. 2(b), which well follows an exponential damping law [34]. In order to facilitate evanescent coupling, we select d = 8 µm with high enough k but without waveguide-overlapping geometrically. Figure 2(c) presents the microscopic top view of part of the interaction region of a DC with d = 8 µm.

To obtain the required splitting ratios of DCs for the CNOT gate indicated in Fig. 1, we vary L in the preselected range of 0-1 mm to tune reflectivities as well as transmissions at fixed d as shown in Fig. 3(a). As expected, we successfully find the satisfactory L₁ ≈ 0.13 mm and L₂ ≈ 0.40 mm for R₁ = 1/2 and R₁ = 1/3, respectively, which are much shorter than the coupling lengths used in Ref. [6]. According to the coupled mode theory [34], optical power exchange between two arms of DC should follow the square of sine (or cosine): T = sin²φ, R = cos²φ, where φ = kL, so the beating period depends upon k. The fittings for T and R really follow square of the sine and cosine curves as presented in the inset. Figure 3(b) shows that φ linearly varies with L. The short coupling length results from the advantage of the path-encoding that does not require tuning the propagation laws for two polarizations at the same time.

 

Fig. 3. Measured reflectivity R and transmission T (a) as well as phase φ (b) of the fabricated DCs as a function of coupling length L at interaction distance d = 8 µm. The fittings for T and R indicate a trend following the square sine and square cosine curves. φ linearly varies with L. Inset: fittings for a complete period showing square of the sine and cosine laws.

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Finally, we fabricate several CNOTs according to the schematic of Fig. 1 with slightly different coupling lengths around the values of L₁ and L₂ to take into account possible fabrication imperfections. The footprint of each integrated CNOT is about 640 μm × 2.5 cm. We choose the best one from these inscribed gates through classical characterization, and then measure the two photon coincidence using quantum sources. We input vertical-polarization and horizontal-polarization classical CW laser to each input port of the tested CNOT and record its corresponding output intensity distribution using a power meter. By carefully comparing the classical performance of each fabricated CNOT under different input polarizations, we select the one that best meets the calculated ideal intensity distribution. It’s worth noting that birefringence is present in these waveguides on the order of B = 7 × 10 ⁻⁵ [35], the coupling coefficient, and hence the beating period, can be different for the two polarizations. Therefore, the control of input polarization of the CW laser and single photons is very critical.

To characterize the performance of the quantum CNOT operation, we measure the HOMI visibilities and the truth table of the gate. The experimental setup is shown in Fig. 4(a). We generate 808-nm photon pairs by pumping a beta-barium borate (BBO) crystal through Type-I spontaneous parametric down-conversion (SPDC) with a 140-mW, 404-nm CW diode laser (ECL801, UniQuanta) and collect them into SMFs. We use 3-nm interference filters to ensure good spectral indistinguishability. A delay line is inserted to control the temporal distinguishability of the photons. We use wave plates and polarization controllers to compensate the rotation of the photon polarization in the SMFs. Single photons are launched into the waveguides inside the integrated CNOT chip and then collected at the outputs using two arrays of four SMFs with the same 127-µm spacing as that of waveguides. The SMF arrays and the chip are directly butt-coupled without index matching fluid. We use fiber-coupled commercial silicon avalanche photodiode single-photon counting modules (SPCMs) (Excelitas, SPCM-850-14-FC) connected with an 8-channel time to digital converter (ID800, IDQ) to conduct two-fold coincidence detections of different output-photon combinations.

 

Fig. 4. Experimental setup and two-photon HOMI in the central DC with R = 1/3 of the CNOT. (a) Photon pairs at 808 nm are generated through Type-I SPDC in a BBO crystal pumped by a 140-mW, 404-nm CW diode laser and are collected into SMFs. Long pass filters (LPFs) from 650 nm and interference filters (IFs) with Δλ = 3 nm are used to ensure good spectral indistinguishability. A delay line (DL) is inserted to control the temporal superposition of the photons. Half-wave plates (HWPs) and polarization controllers (PC) are used to compensate the rotation of the photon polarization in the SMFs. Single photons are launched into the waveguides inside the integrated CNOT chip and then collected at the outputs using two arrays of four SMFs. SPCMs and the connected 8-channel time to digital converter (TDC) are used to conduct two-fold coincidence detections of different output-photon combinations. (b), (c) The coincidence counts of detecting photons at $C_1^{\prime}$ and $T_1^{\prime}$ ($T_0^{\prime}$) when two photons are injected into C₁ and T₁ (T₀). The HOMI visibilities are 0.977 ± 0.007 and 0.971 ± 0.007, respectively.

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The HOMI visibilities are very high. As described in Section 2, identical single photons coming from C₁ and T₁ inputs will interact in the central DC with R = 1/3. By scanning their relative delay, as shown in Fig. 4(b), we find the interference area of HOMI reflected by the two-fold coincidence detection of two photons at $C_1^{\prime}$ and $T_1^{\prime}$ outputs, to derive the measured interference visibility to be V₁ = 0.977 ± 0.007. For the photons input from C₁ and T₀, two-fold coincidence detection conducted at $C_1^{\prime}$ and $T_0^{\prime}$ in Fig. 4(c) results in the visibility V₂ = 0.971 ± 0.007. The high visibilities strongly demonstrate occurrence of high quality HOMI and MZI. The coincidence counts are a little bit asymmetric with respect to the relative delay and there seems small difference between the experimental data and the fitting lines in Figs. 4(b, c). These fluctuations mainly attribute to the detection noise of the SPCMs and the positioning accuracy of the delay line, and may also be induced by the non-perfect indiscernibility of the two-photon spectrum in the quantum source setup [36].

To determine the truth table of the device operated as a CNOT logic gate, we inject into the chip the four computational basis states: |00>, |01>, |10 > and |11 > corresponding to the |C_qT_q> systems, which are different from the target qubits encoded into a superposition state of $({|0 \rangle \pm |1 \rangle } )\textrm{/}\sqrt 2 $ in Ref. [6]. We then conduct a two-fold coincidence detection of two output photons, exactly one from $C_0^{\prime}$ or $C_1^{\prime}$ and the other from $T_0^{\prime}$ or $T_1^{\prime}$ to identify whether the gate operation is successful. The obtained truth table is presented in Fig. 5. The probabilities of each computational-basis output for each computational-basis input (|00>, |01>, |10 > and |11>) represented by the red bars are 0.992, 0.982, 0.964, and 0.973, respectively, and those of others are all lower than 0.036. The experimentally observed fidelities of the output states [37] are all F_exp ≥ 0.980 ± 0.006, which is higher than that in Ref. [6]. The fidelity can be further improved by increasing the indistinguishability of photon wave packets and suppressing rotation of the photon polarization in the SMFs.

 

Fig. 5. Experimentally constructed CNOT logical truth table. The labels on the Input and Output axes identify the state | C_qT_q >. Ideally, a flip of the logical state of the target qubit T_q occurs only when the control qubit C_q is in the logical state |1 > . The average fidelity is higher than 0.98.

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4. Conclusion

We experimentally demonstrate an integrated path-encoded CNOT logic gate in glass fabricated by femtosecond laser direct writing, for the first time to the best of our knowledge. The five directional couplers constructing the gate are all symmetric in phase, only different in coupling length, which well matches the inherent symmetry of the FLDW technique. The inbuilt classical and quantum interferences, including MZI and HOMI, both achieve great performance. The measured quantum interference visibilities are all larger than 0.97 ± 0.007, which directly contributes to a near-perfect CNOT operation. The experimentally observed fidelities for each computational basis outputs are all higher than 0.980 ± 0.006. These results present the capability of femtosecond laser microfabrication to directly produce photonic quantum circuits on chip with high quality. Further efforts will be devoted towards more complex integrated multi-qubit logic gates composed of several two-qubit gates as well as one-qubit gates.