Quantum projectors implemented with optical directional couplers in ion-exchanged glasses

7676 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 40, NO. 23, DECEMBER 1, 2022 Quantum Projectors Implemented With Optical Directional Couplers in Ion-Exchanged Glasses Xesús Prieto-Blanco , Carlos Montero-Orille , Jesús Liñares , Héctor González-Núñez , and Daniel Balado (Integrated Photonics 2022) Abstract—Ion-exchanged glass as a platform for quantum photonics is proposed. Quantum projectors are implemented with integrated optical directional couplers fabricated by ion-exchange K+/Na+in soda-lime glass. We consider devices composed of concatenated directional couplers which implement N-dimensional quantum projective measurements, and concomitantly the production of 1-qudit states. The fundamental units of these devices are 2 ×2 directional couplers that are experimentally studied in order to obtain, through an optical characterization, empiric relationships between fabrication and optical parameters of such couplers. Likewise, a two-dimensional quantum projector is demonstrated so that projective measurements are obtained for the states of bases X(diagonal) and Y(circular). Index Terms—Integrated quantum passive elements, integrated quantum projective meters, ion-exchanged glass platform. I. INTRODUCTION INTEGRATED quantum photonic devices can implement different quantum operations for quantum communications, quantum computation, quantum metrology, and so on, with the inherent advantages associated to integrated optics. Different platforms have been proposed and used to develop these integrated quantum devices, including silicon-on-insulator (as SiO2), Lithium Niobate, and Gallium Arsenide [1]–[3]. However, to our knowledge, ion-exchanged glass (IExG) platform has not been used as an integration technology for quantum photonics, despite the fact that it is present in many classical optical communications systems and optical sensors, and has been Manuscript received 1 March 2022; revised 11 May 2022 and 17 June 2022; accepted 21 June 2022. Date of publication 7 July 2022; date of current version 16 December 2022. This work was supported in part by the MICIN, European Union NextGenerationEU under Grant PRTR-C17.I1, and in part by the Galician Regional Government through Planes Complementarios de I + D + I con las Comunidades Autónomas in quantum communication. (Corresponding author: Jesús Liñares.) Xesús Prieto-Blanco and Jesús Liñares are with the Quantum Materials and Photonics Research Group, Optics Area, Applied Physics Department, Faculty of Physics/Faculty of Optics and Optometry, Universidade de Santiago de Compostela, E-15782 Santiago de Compostela, Galicia, Spain, and also with the iMA- TUS, Institute of Materials, Universidade de Santiago de Compostela, E-15782 Santiago de Compostela, Galicia, Spain (e-mail: [email protected]; [email protected]). Carlos Montero-Orille, Héctor González-Núñez, and Daniel Balado are with the Quantum Materials and Photonics Research Group, Optics Area, Applied Physics Department, Faculty of Physics/Faculty of Optics and Optometry, Universidade de Santiago de Compostela, E-15782 Santiago de Compostela, Galicia, Spain (e-mail: [email protected]; hector[email protected]; [email protected]). Color versions of one or more figures in this article are available at https://doi.org/10.1109/JLT.2022.3189206. Digital Object Identifier 10.1109/JLT.2022.3189206 suggested several years ago for this task [4]. It is well known that this platform has advantages with respect to cost, technological requirements, and fibre optical compatibility [5]. Moreover, it can also be reconfigurable in a thermo-optical way [6]. In this work we show the implementation of passive quantum optical projectors with integrated optical elements in IExG platform. We must stress that passive integrated optical elements for certain applications present clear advantages respect to, for instance, electro-optical elements, such as energy saving and no complex geometries of electrodes required. Projective measurements are the most common way to detect quantum states, in general, and quantum light states, in particular. It is an important task in most of quantum optical circuits intended for quantum computation, quantum cryptography, and quantum processing [7], [8]. Likewise, generalized measurements as POVMs (Positive operator-valued measurements) are required to obtain an unambiguous quantum non-orthogonal state discrimination. Projective measurements are a particular case of POVMs [9]. In the case of quantum communications, quantum light states propagate in optical fibres, or free space, and can be excited in codirectional optical modes, such as those of Multi-Core optical Fibres (MCF) [10]–[12], or in collinear optical modes, such as those of Few Mode optical Fibres (FMF) [13], [14]. Moreover, codirectional modes coming from MCFs can be easily coupled to an integrated optical circuit. Therefore, such circuits are ideal candidates to perform quantum transformations and, in particular, the quantum projective measurements mentioned above. We call this type of circuits as integrated quantum optical circuits (IQOCs). In addition, this solution can be extended to collinear modes. For that, a prior spatial demultiplexing process, along with the use of modal converters, is required in order to get a spatial separation of optical modes, which facilitates their coupling to an IQOC. Photonic lanterns can implement such operations in a compact form [15]. Accordingly, the implementation of the mentioned quantum transformations and/or projective (or POVMs) measurements can be made. As commented, these quantum circuits can perform different transformations, but projective measurements are usually required in the final part of quantum communications systems (Bob system). In this work we focus on these projective —and also generalized— measurements used for quantum cryptography in optical fibres [7], [11], [12], although other applications are also possible, in quantum computation (for specific purposes) [1], [3], [16], quantum simulations [8], [17], [18], QRW [16], This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ PRIETO-BLANCO et al.: QUANTUM PROJECTORS IMPLEMENTED WITH OPTICAL DIRECTIONAL COUPLERS IN IExG 7677 [19], [20], quantum metrology and sensors [21]–[23], and so on. IQOCs, for N-dimensional quantum spaces, can be realized with integrated optical elements as single mode channel waveguides (SMW), directional couplers and integrated phase shifters [24]–[27]. So, integrated quantum optical projectors (IQOPs) can be implemented with these integrated optical elements and, therefore, an experimental realization of these projectors is an important and relevant task. Here, we propose the experimental use of IExG platform for implementing IQOPs, although many other quantum components could be implemented, such as quantum simulators [28], quantum random walk, and so on. These implementations will be efficient as long as a few qubits are considered, which is related to the well-known scalability difficulty of quantum integrated photonics. In particular, we propose passive quantum projectors implemented with concatenated optical directional couplers and phase shifters fabricated by K+/Na+ion-exchange processes in soda-lime glass. Since these projectors are intended for quantum cryptography purposes, they are designed in such a way that a quantum random choice of measurement basis is done [12]. The fundamental or basic unit of these projectors is the 2x2 directional coupler, that is, N=2. Therefore, we present the results of fabrication as well as the optical tests of such couplers. In particular, we make a semiclassical test of projective measurements of the quantum states belonging to bases Xand Y. The results show that IExG is a platform with enough advantages to be used in quantum technology along with other platforms. The organization of the paper is as follows. In Section II we present the theoretical fundamentals on passive quantum projectors with random choice of measurement basis, as required in most of the quantum cryptography protocols. In particular, we introduce this kind of projectors with the case N=2, although their generalization to an arbitrary dimension is quite straightforward. Next, in Section III, the fabrication of a 2x2 projector by IExG is described, specifically, by a K+/Na+ion-exchange process in soda-lime glass, and paying special attention to the experimental curves obtained in order to determine the fabrication parameters of the circuit. In Section IV, an optical test of a basic projector 2x2 is shown. This includes the use of an external optical grating to change the basis of quantum states and shows, in a semiclassical way, that projective measurements can be realized. Finally, in Section V, conclusions are presented. II. PASSIVE QUANTUM PROJECTORS BASED ON DIRECTIONAL COUPLERS The quantum light states with a major interest are the single photon states —which are, for instance, used in quantum cryptography— and, in particular, 1-qubit (two optical modes) and 1-ququart states (four optical modes). In fact, 1-ququart states are the states which give rise to the beginning of the so-called high-dimensional quantum cryptography with 1-qudit, that is, quantum states of dimension d, which can be implemented with a number Nof optical modes (or optical paths), that is, d=N≥2: |L= d=N  j=1 cj|000...1j...000= d=N  j=1 cj|1j,(1) where, in addition, the states have to belong to the so called mutually unbiased bases (MUBs), that is, |c1|=... =|cj|=... =|cN|=1/√N. A source of single photons can be obtained from SPDC (spontaneous parametric down-conversion) sources [8] or even from lasers, that is, from multimode coherent states |α1...α Nwith a very low photonic excitation (weak coherent states), that is, |αj|1, therefore, |L≈|0+ N  j=1 αj|000...1j...000.(2) Let us start by considering the case of a two-dimensional quantum problem, that is, N=2, so 1-qubit states will be used. This case will help us to explain the implementation of integrated quantum optical projectors. As commented, the fundamental units of these devices are the synchronous directional couplers along with integrated phase shifters. It is well known that synchronous directional couplers can be represented by the following matrix: Xθ=cos θisin θ isin θcos θ,(3) where the coupling phase is θ=κl, with κthe coupling coefficient and lthe length of the coupler. For a two-dimensional IQOP the required couplers are, by one hand, a Xπ/4,thatis,a 3 dB (or 1:1) coupler, which corresponds to the logic gate √X and, on the other hand, a Xπ/2corresponding to the usual logic gate X. Moreover, phase shifters are required, that is, ZΦ=10 0eiΦ.(4) Phase shifters Zπ/2and Zπare usually needed. So, by concatenating Xθand ZΦelements, we can implement complex and fully operational IQOPs. For example, let us consider the bases Xand Ywith the following single photon states excited in two channel waveguides jand j, X={|1D,|1A} =1 √2(|1j+|1j),1 √2(|1j−|1j), (5) Y={|1L,|1R} =1 √2(|1j+i|1j),1 √2(|1j−i|1j). (6) If these channel waveguides are coupled by a coupler Xπ/4then a IQOP is implemented for the basis Y, that is, the input states |1Land |1Rbecome the states |1jand |1j, respectively, at the output. Likewise, the IQOC defined by Xπ/4Zπ/2implements an IQOP for the basis X, that is, the input states |1Dand |1A also become the states |1jand |1j, respectively, at the output. On the other hand, a random choice of the measurement basis is required in quantum cryptography. For that, a more complex integrated circuit is needed. In Fig. 1 we draw an IQOC with four paths, j=1–4, for making projective measurements of states in the basis Xor Yin a random way. Indeed, let us consider 1-qubit input states, in a vector representation (c1,c 3), coming from a transmission line where a Quantum Key Distribution (QKD) is distributed, as for example by the BB84 protocol (see, 7678 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 40, NO. 23, DECEMBER 1, 2022 Fig. 1. Drawing of an integrated quantum optical projector for the twodimensional bases Xand Y. for instance, [29]). Likewise, let us consider that these 1-qubit states are excited in paths (optical modes) 1 and 3, that is, |L=c1|11+c3|13.(7) The first part of the IQOC (S in Fig. 1) is formed by two couplers Xπ/4and one coupler Xπ/2. They implement the following operation: S|L=1 √2⎛ ⎜ ⎜ ⎜ ⎝ 1i00 00i−1 −1i00 00i1 ⎞ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎝ c1 0 c3 0 ⎞ ⎟ ⎟ ⎟ ⎠ =1 √2⎛ ⎜ ⎜ ⎜ ⎝ c1 ic3 −c1 ic3 ⎞ ⎟ ⎟ ⎟ ⎠ (8) which is a state division that ensures a random choice of the measurement basis. Note that this operation generates a relative phase π/2 between paths 1 and 2, and −π/2 between paths 3 and 4. These phases must be taken into account in the projective measurement. In particular, the former is equivalent to a gate Zπ/2between paths 1 and 2. So, a projective operation in the basis Xis performed after the top right Xπ/4coupler of Fig. 1: PX=Xπ/4Zπ/2. Similarly, a projection in the basis Y is obtained in the subspace formed by path modes 3 and 4, by cancelling their relative phase -π/2byagateZπ/2in path 4, before the bottom right Xπ/4coupler: PY=Xπ/4. The matrix associated with this second part of the IQOC (C in Fig. 1) is given by C=1 √2⎛ ⎜ ⎜ ⎜ ⎝ 1i00 i10 0 001−1 00ii ⎞ ⎟ ⎟ ⎟ ⎠ .(9) Therefore, the whole circuit implements the following projective transformation: P=CS =1 2⎛ ⎜ ⎜ ⎜ ⎝ 1i−1−i i−1i−1 −1i−i−1 −i−1−1i ⎞ ⎟ ⎟ ⎟ ⎠ .(10) As an example, let us consider the input state |L=|1D= (1/√2)(|11+|13), then the state division operation implemented by Sgives the state S|1D=(1/√2)(|11+i|12− |13+i|14)and, next, the action of Cgives the output state Fig. 2. Drawing of an integrated quantum optical circuit for implementing a POVM for states |L±=cosα|11±isin α|12. P|1D=(i/√2)|12−(1/2)eiπ/4(|13+|14). That is, the state |1Dis projected on exit 2. Note that a detection on exits 3 and 4 will be discarded by Bob’s system after Alice communicates the measurement basis (X, in this case) [29]. Likewise, the state |1Abecomes P|1A=(1/√2)|11− (1/2)e−iπ/4(|13−|14), and is projected on exit 1. Therefore the states of basis Xare measured at output channel guides 2 and 1. Similarly, the states |1Land |1Rof the Y-basis are measured at output channel guides 4 and 3, respectively. In short, we formally have a projective operation PXforstatesofthebasis Xin the subspace defined by modes 1 and 2, and another one PY for the states of basis Yin the subspace defined by modes 3 and 4 (see Fig. 1). Therefore this IQOC works as a passive IQOP with random choice of measurement basis, where the Xπ/4and Xπ/2couplers can be regarded as its basic functional units. Their fabrication in an IExG platform will be shown in the next section. On the other hand, it is worth showing that generalized measurements can be also implemented with these circuits based on directional couplers. The projective measurement described above are useful for orthogonal states. However, in some cases, non orthogonal states are produced and, therefore, a different measurement approach has to be used. Positive operator-valued measurement (POVM) is one of the generalized measurement approach for non orthogonal states. Let us consider, for example, the following non-orthogonal states in the subspace of two path modes (two integrated channel guides): |L±=cosα|11±isin α|12.(11) Note that |L+|L−| =cos2α=0.Ifα=π/4 then we obtain the orthogonal states |1(L R). By following the POVM proposed in [30] for polarization modes, we can implement a POVM for quantum states given by (11), which are excited in path modes. A sketch of a circuit to implement this kind of measurement is shown in Fig. 2. The circuit is made of a directional coupler Xθ, in the space of path modes 0-1, followed by a directional coupler Xπ/4. Again, the calculation of the total action of the circuit gives the following matrix: M=⎛ ⎜ ⎝ cos θisin θ0 isin θcos θ/√2i/√2 −sin θ/√2icos θ/√21/√2 ⎞ ⎟ ⎠.(12) Next, the action on the states |L±has to discriminate between them, that is, cos θ=tanα. Then, the directional coupler Xθ must meet θ=cos −1(tan α)=κL, which can be achieved experimentally as we will see in Section IV. Therefore, the PRIETO-BLANCO et al.: QUANTUM PROJECTORS IMPLEMENTED WITH OPTICAL DIRECTIONAL COUPLERS IN IExG 7679 following transformation is obtained M=1 √2cosα⎛ ⎜ ⎝ √2sinαi √2cos2α0 i√2cos2αsin αicos α −√2cos2αisin αcos α ⎞ ⎟ ⎠.(13) Note that the vectors defined by (M22,M23) and (M32,M33) are the vector representation of the bras L⊥ ±|. Therefore, in the subspace of path modes 1-2, the following POVM operators are implemented ˆπ±=|L⊥ ±L⊥ ±| 1+|L+|L−| (14) and the measurement is obtained in detectors D+and D−. Indeed, for the state |L+the output state is M|L+=i√2cos2α|10+i(sin 2α/√2cosα)|12. So, if detector D−clicks then the state |L+is measured, with a probability 2sin 2α, such as the general theory predicts. That is, L+|ˆπ+|L+=0 and L−|ˆπ+|L−=2sin 2α. The detector D0 corresponds to the case of non conclusive outcome, that is, the outcome governed by the operator ˆπ0=I−ˆπ+−ˆπ−. In short, an additional path mode, corresponding to the channel guide 0 shown in Fig. 2, together with the coupler Xθwith cos θ=tanα, implements a POVM for states |L±. A similar POVM can be implemented for the states |L ±=cosα|11±sin α|12. III. FABRICATION OF A 2x2PROJECTOR BY ION-EXCHANGE IN GLASS This section focuses on the fabrication of the Xπ/4and Xπ/2basic units composed of 2x2 directional couplers. These units were fabricated by thermal ion-exchange K+/Na+in glass together with photolithographic procedures to generate the suitable patterns. As it is well known, ion exchange modifies the chemical composition of appropriate substrates and, as a consequence, waveguides can be created using this technique. The ion-exchange procedure that we have used is similar to that presented in [31], where the authors carried out a selective ion-exchange followed by a planar one. However, we used a burial process as the second step. This choice presents some advantages which we will comment below. With respect to the photolithographic procedure, we produced a metal mask with several basic units, each having a different geometry. So, modal coupling empiric curves —and the required directional coupler— could be obtained, once a particular set of parameters of the ion-exchange process was selected. Note that, usually, photolithographic parameters are selected and the ion-exchange parameters are changed. Next, we explain the details of the fabrication process. At first, let us consider a non-masked glass substrate containing sodium ions. If this substrate is dipped into a salt (KNO3,for example) at a high temperature T,theNa +ions of the glass are exchanged for the K+ions of the salt. As a result, the refractive index of the glass increases in a micrometric layer beneath the glass surface and an integrated waveguide can be obtained. In addition, some birefringence arises from the mechanical stresses due to the difference in size of K+and Na+cations [32]. A subsequent thermal diffusion can be used to carry out a burial Fig. 3. Sketch of a selective ion-exchange process. process of this waveguide by immersing the substrate into a molten salt containing sodium ions (NaNO3, for example). So, the Na+ions present in the salt replace the most surface doping cations K+and the refractive index at the glass surface drops to its original value. This second step, i.e. the burial process, is not mandatory but presents two great advantages. First, the modal field is located further from the glass surface, which can present irregularities, and so losses are reduced [33]. Second, it reduces the typical anisotropy of K+/Na+ion-exchange waveguides since the accumulated mechanical stresses are diminished [34]. On the other hand, if a selective ion-exchange is required to produce, for example, channel waveguides, a suitable mask can be patterned on the glass surface by some lithographic method (see Fig. 3). This is the case of directional couplers, which are usually formed by single-mode channel waveguides. The dimensions of these waveguides are determined by the penetration of ions, both in the x-direction (in-depth diffusion) and in the y-direction (side diffusion), as well as the size of the openings of the metal mask. Fig. 3 shows a sketch of a selective ion-exchange process where the mask, the molten salt, and both the in-depth and side diffusion of ions are drawn. Note that the burial step is carried out after mask removal. In order to select appropriate fabrication parameters (mainly the ion-exchange time, t), we adopted the effective index method (EIM) [24], [35], [36] together with a separability approximation, justified by the weak guidance. Thus, we obtained the single-mode condition in the y-direction, while in the xdirection, it was studied experimentally through the optical characterization of slab waveguides. With regard to the fabrication of the metal mask, a two-step photolithographic process was used [37]. In the first step, different coupler patterns were plotted on a sheet of heavy paper and then reduced -1/14.11 by using a lens Schneider XENON Sapphire 4.5/95. This lead to an aluminium master on a glass substrate that was then reduced by a Carl Zeiss S-planar lens working at magnification M=-1/5 and f/#=1.3. Despite this two-step process, we were not able to get minimum mask widths less than 3 μm, which prevented us from making channels narrower than this amount. Moreover, the side diffusion also widens the lateral size of the channel waveguide (see Fig. 3). Therefore, we made an estimate of this side diffusion and assumed that it is equal to the penetration depth. We used soda-lime glasses with refractive index ns=1.5104 for the working wavelength λo=632.8 nm. From previous studies [31], we set the temperature of both the first ion-exchange and the burial process to T=400 °C. Several thermal ion-exchanges in test glasses without metal masks were made and the effective 7680 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 40, NO. 23, DECEMBER 1, 2022 Fig. 4. Basic integrated optical element (BIOE) for optical characterization purposes. indices of the modes of the waveguides were measured by prism coupling (also known as M-lines method). This was done with an automated system (Metricon, Model 2010/M). From these measurements, and taking into account the numerical results provided by the EIM, we set a time of t1=30 min for the first ion-exchange and t2=10 min for the burial process. These parameters lead to waveguides with a single mode of effective index Neff=1.5117. As stated before, the design and fabrication of IQOPs need a prior optical characterization of the full fabrication process of the basic units Xπ/4and Xπ/2. For that we used a basic integrated optical element (BIOE) which is shown in Fig. 4. It consists of a set of three waveguides of width sforming a Y-junction and a directional coupler which will be object of optical characterization. The Y-junction begins in the waveguide 1 and has the purpose of providing a reference signal [38] in waveguide 5 after a 50%-50% split. Next, the waveguide 2 and one of the waveguides of the Y-junction are approached through a Minford curve [39], y=Δyz lm−1 2πsin 2πz lm,(15) with Δy=15.4 μm and lm=1520 μm, defining a directional coupler. Once the ion-exchange fabrication parameters are fixed (temperature, time, salt, ionic concentration, and glass substrate), the spatial parameters d(distance between coupled channel waveguides) and lc(coupling length), shown in Fig. 4, will determine the optical characteristics of the couplers. In order to determine dand lcfor a particular coupler, a calibration method is used, that is, the design of the mask includes a series of BIOEs with different dand lc. This has a double objective. By one hand, it ensures that some of the BIOEs can become equal or quite close to the remarkable couplers Xπ/4or Xπ/2. On the other hand, it makes possible to obtain the empiric curves of these parameters that in turn will allow us to design couplers with the desired coupling percentage in a subsequent fabrication process. So, we have designed a mask containing 16 BIOEs in 4 blocks. In each block, dis fixed while lcvaries. These values, after the photo-reduction process, are found in the interval d∈[3,7.5] μm and lc∈[0.5,2] mm. The results of the optical characterization of the BIOEs and the empiric curves of the fabrication parameters are analysed and shown in the next section. We must indicate that the side diffusion increases the width of the channel guides up to 4 μm with respect to the 3 μm of the mask openings. Furthermore, these BIOEs were designed to introduce light through end-fire coupling. Consequently, the waveguides have been ended in flat and polished faces, perpendicular to their propagation axis. IV. SEMICLASSICAL DEMONSTRATION AND TEST OF A 2x2 PROJECTOR A. Optical Characterization System The optical characterization of the BIOEs begins with a visual inspection, through a microscope, of the aluminium mask. Usually, some of the BIOEs have breaks or irregularities in the mask openings of a few micrometers in length. One would expect that they would have to be discarded. However, all of them worked well, showing that this technology has excellent tolerance to mask defects. Next, the end-fire coupling technique is used to couple light to the input channel waveguides and, after propagation through the BIOE, to observe it at the outputs. A sketch of the optical setup used for this task is shown in Fig. 5. It is similar to the system described in reference [40] but, in this one, it is especially important the subsystem that includes, together with the microscope objectives, a light-emitting-diode (LED), two beam splitters (BS), and an eyepiece. It allows us to watch simultaneously the back-reflected beams on the input face of the BIOE as well as the channel waveguides illuminated by the LED. This makes the coupling process easier through a three-axis translation stage and, for fine positioning, a low-power positive lens L1 that can be moved up and down. If the Gaussian beam on the input face of the BIOE has the appropriate height and waist, it will match the fundamental mode of the channel waveguide and a high coupling to this mode will be obtained. In our case, the required waist of the beam was wider than the 10×objective resolution; then we arranged the system so that the beam illuminated only the central region of the objective aperture stop. On the other side of the BIOE, a 20× microscope objective images the exit face on a CMOS Thorlabs DCC1545 M camera. We have chosen a camera as the measuring instrument, instead of a power meter, because the extension of the output beams is comparable to their separation. With a power meter, it would be difficult to individualize the power measurements of each output beam. Likewise, the temporal evolution of the measurements can be studied easily with a camera and their values are not affected by the temporal fluctuations of the laser. Moreover, this CMOS camera exhibits a good linear response up to saturation, which ensures a true relative power measurement [41]. On the other hand, two polarisers were inserted in this setup to work with a specific polarization (TE or TM). First polariser ensures the polarization state at the entrance of the waveguides. The second one, placed in front of the CMOS camera, allows us to distinguish between polarization states at the output, since K+/Na+ion-exchanged waveguides in glass, after burial processes, can still have a certain anisotropy. B. Optical Characterization Results The characterization of the modal coupling in the BIOEs (16 units) was made by using the optical system described in the previous subsection. The light from the He-Ne laser is coupled to input waveguide 1 of each BIOE, that is, a coherent state |αis used. Then, as mentioned above, the Y-junction splits the light into two halves: one of them goes through the reference waveguide 5 and the other one through waveguide 4 which is a part of the coupler. So, three modal outputs 3, 4, and 5 are PRIETO-BLANCO et al.: QUANTUM PROJECTORS IMPLEMENTED WITH OPTICAL DIRECTIONAL COUPLERS IN IExG 7681 Fig. 5. Sketch of the optical setup used for the characterization of the BIOEs. Fig. 6. Images obtained by a CMOS camera of the output of different BIOEs after coupling TE polarized light into input 1 (Y-junction). The lower point corresponds to the output of the reference waveguide (5), while the two upper points are the outputs of the coupler waveguides (3 and 4). (a) Light exits by output 4 (θ=2nπ). (e) Light exists by output 3 (θ=(2n+1)π/2). (c) Light exits by both outputs 3 and 4 (θ=(2n+1)π/4). (b) and (d) are intermediate cases. TABLE I COUPLING RESULTS EXPRESSED AS POWER RATI OS (P4/(P3+P4)|P3/(P3+P4))IN PERCENTAGES AT OUTPUTS 4AND 3 observed on the CMOS camera. Next, by taking several pictures and processing them with a suitable software, the amount of light which comes out of each of the three waveguides can be evaluated. In particular, the relative percentage of power that leaves the waveguides (3 and 4) of the coupler can be calculated and compared to itself when light is coupled into input 2. We did this comparison for the sixteen different BIOEs of the sample and for TE polarization. Qualitatively, we saw that the results are similar for TM polarization. The presence of anisotropy in the waveguides made the measurements a bit difficult, detecting in some cases some energy transfer from TE to TM modes. Anyway, the energy transferred was small and its inclusion in the analysis did not change substantially the coupling results. In Fig. 6 some examples of images taken during the optical characterization process of the BIOEs are presented. Measurements of the powers that come out of outputs 4 and 3 are shown in Table I as percentages of power ratios, that is, P4/(P3+P4)and P3/(P3+P4). These values, according to Fig. 7. Coupling phase θas a function of the coupler length lcfor different waveguide separations d. (3), correspond to (cos2θ|sin2θ)and match the probability of detecting one photon at outputs 4 and 3. We must stress that at Table I there are two BIOEs that work approximately as a Xπ/4coupler and three as a Xπ/2coupler, which are the most interesting couplers for this work. These couplers, in (d, lc) notation, are (3.0,1.5) and (6.0,1.5) for Xπ/4, and (4.5,1.0), (4.5,1.5), and (3.0,2.0) for Xπ/2. Although these couplers are suitable for our testing purposes, quasi-exact couplers Xπ/4 and Xπ/2can be obtained by fitting a theoretical model to the experimental data shown in Table I. Indeed, the coupling phase (3) can be expressed as θ=κlc+κΔlc= arccos P4/(P3+P4),(16) where Δlcis an effective length corresponding to the interaction before and after the region defined by lc, that is, in the sections described by the Minford curve (15). We must stress that the experimental values of θ(right-hand side of (16)) belong to the first quadrant, therefore they have to be properly unwrapped. In Fig. 7 we represent these values with respect to the coupler length lcfor the different values of the waveguide separations d.The 7682 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 40, NO. 23, DECEMBER 1, 2022 Fig. 8. Coupling coefficient κas a function of the waveguide separation d. Fig. 9. Term κΔlcas a function of the waveguide separation d. best linear fit to the data is also shown. It is clearly seen how the coupling phase grows linearly with the coupler length. Besides, in the different curves we see that, the closer the waveguides are, the faster the coupling phase grows, as the coupling, given by the coefficient κ, is stronger. With respect to this coefficient, the coupled mode theory [24] shows that it is directly related to the overlap of the evanescent exponential fields of the optical modes outside the waveguide channels. Thus, the dependence between κand the waveguide separation dis also expected to be exponential, that is, κ=a·exp (−b·d),(17) where aand bare empiric fitting parameters. In Figs. 8 and 9 we represent the coupling coefficients κand the term κΔlc, taken from the best linear fits shown in Fig. 7, with respect to the waveguide separation d. The confidence intervals are also shown for a 95% confidence level. The best fit of (17) to these data gives rise to the following empiric function for (16): θ=3.372πexp (−0.564 d)lc+0.926πexp (−0.372 d), (18) where the fitting parameters has been obtained by minimizing the weighted root mean square error. This equation provides remarkable fabrication curves that enable the manufacture of any kind of coupler by K +/Na+ion-exchange with waveguides whose mask width is s=3μm and transitions defined by the Minford curve given by (15). With these curves, it is possible to fabricate different couplers Xθby varying the separation and coupling length. However, as mentioned above, among the different BIOEs used for calibration, there is a almost perfect Xπ/4coupler, that is, an element that can be used to test the projections of states belonging to the bases Xand Y. The scalability of the proposed IQOC is primarily limited by the length of the S-shaped waveguides (lm≃1.5 mm), which limits the number of BIOEs that can be concatenated in a single chip. For example, a length of 10 mm is needed to implement the IQOP of Fig. 1, even choosing the shortest distances between the waveguides of the couplers. We could improve this through a redesign that removes unnecessary curves. Furthermore, the length lmitself can be reduced by increasing the effective index, so that the waveguide admits greater curvatures. However, this also increases the length of the couplers for a given lithographic resolution, so a balance must be found. Although more studies are needed, a density of 5-10 couplers/cm seems possible, allowing for low and medium complexity IQOCs. C. Semiclassical Test of the Integrated Projector The projection test consists of introducing laser light with different relative phases into the two input channel waveguides of the directional coupler and observing the output as a function of these relative phases. This kind of illumination corresponds to a two-mode coherent state, that is, |L=|αjαj=|ajeijajeij,(19) where j=3 and j=4, because we turned around the BIOE and outputs 3 and 4 were used as inputs, and inputs 1 and 2 as outputs (see Fig. 4). This was done to prevent losses in the reference waveguide from distorting the results. Even so, some losses in output 1 due to the Y-junction were still observed. Thus, when light was coming entirely from output 1, its intensity was lower than when it was coming exclusively from output 2. Different quantum states can be obtained by introducing different relative phases =4-3. In our experiment, this was done by inserting a diffraction grating (G) between the auxiliary lens L3 and the first beam splitter (BS), as it is shown in Fig. 5. This grating was also fabricated by ion-exchange, following the method described in reference [42]. Its period is p=60 μm and it distributes light mostly between diffraction orders ±1, allowing us to obtain two light beams (coherent states) with thesamepower(|a3|2=|a4|2=|ao|2). The grating mount allows micrometric motions along the Yand Z-axes. Both diffraction orders remain focused on the input face of the BIOE regardless the grating position. By moving axially the grating, we can adjust the distance between the focused beams and couple accurately PRIETO-BLANCO et al.: QUANTUM PROJECTORS IMPLEMENTED WITH OPTICAL DIRECTIONAL COUPLERS IN IExG 7683 Fig. 10. Three channel waveguides (3, 4, and 5) of the BIOE (top) and the two focused diffracted beams (bottom), as observed through the eyepiece. Note that optical saturation of the camera produces a slightly distorted image. each beam to its respective waveguide (3 and 4). It can be easily checked through the eyepiece, while the BIOE is backlit with a LED to highlight the waveguides (see Fig. 10). On the other hand, by moving transversally the grating (Y-axis), a phase shift ±2πy/p is added in the respective order ±1. Particularly, a displacement of half a period (30 μm) corresponds to a relative phase shift =2π. Therefore, the two-mode coherent states produced by the grating are given, to good approximation, by |L=|aoe−i/2aoei/2. Now, let us consider a very weak coherent state, that is, ao1, in such a way that we obtain the single photon state |L≈|0304+aoe−i/2|1304+aoei/2|0314.(20) Then, by taking into account the transformations of the operators corresponding to a Xθcoupler, the normalized detection probability of a photon coming from outputs 1 or 2 of the coupler is given by P1,2=1 2(1 ±sin 2θsin ),(21) which, except constants, corresponds to theclassical light power. In short, a semiclassical analysis also provides a characterization of the IQOP for single photon states. So, a semiclassical test of the IQOP for analysing its capability for implementing projective measurements can be made. The basic unit selected for testing is the one labelled by (d, lc)=(6.0,1.5) in Table I, that is, a Xπ/4coupler. Once the two focused diffracted beams were coupled to its inputs 3 and 4, the diffraction grating was moved step by step along the Y-axis and, at the same time, light power was measured at both outputs 1 and 2. Moreover, due to losses in the Y-junction, the measured power at output 1 was corrected. This was done by taking into account that these losses are given by (1-P1m/P2m), being Pim the maximum measured power at outputs 1 and 2. Recall that the sample was turned around. The results of this optical characterization are represented in Fig. 11. They are compared to the theoretical curves given by (21). A sinusoidal behaviour of the light power, as a function of the grating displacement, is observed at both outputs. Maximum and minimum experimental values do not reach 1 and 0 because the coupler is not exactly a Xπ/4and there is some backlight that affects the measurements. We also observe that a full cycle is completed for a displacement of 30 μm. The most remarkable cases are those ones corresponding to phase shifts =π/2and 3π/2, because they simulate projective measurements of the quantum states belonging to bases Xand Y, Fig. 11. Normalized light power measured at both outputs (blue and red dots) as a function of both the grating displacement (upper X-axis) and the phase between beams (lower X-axis). The theoretical curves (21) for a Xπ/4coupler are also represented (blue line for output 1 and red line for output 2). Fig. 12. Output power after introducing two light beams with phase shift in both inputs. (a) =0,(b)=π/2,(c)=π,(d)=3π/2. The input states simulated with these light beams are indicated. which are commonly used in the BB84 protocol. Indeed, when the grating introduces phases =π/2and 3π/2, basis Yis implemented and the output states are i|11and |12, respectively. That is, the states of basis Yhave been projected on the outputs 1 and 2. Fig. 12(b) and (d) show the experimental results of these projections. If input states were |1Dand |1A, the use of an additional phase shifter Zπ/2would be required, as it was analysed in Section II. However, note that the above results, with the relative phases =π/2and 3π/2introduced by the grating, can be reinterpreted as if the input state were (1/√2)(|13±|14), but with an additional phase π/2introduced by the grating acting as a gate Zπ/2. Therefore, we can implement the phase shifter Zπ/2in this way. On the other hand, for phase shifts 0 and π, light comes out with almost the same intensity in both outputs (see Fig. 12(a) and (c)). This means that states |1Dand |1Aare eigenstates of the IQOP. In short, the directional coupler works as an IQOP for two-dimensional quantum states. In particular, for the states belonging to basis Y, that is, |1Land |1R.Forthe states |1Dand |1Aof basis Xa further relative phase π/2must be introduced. This can be achieved by a Zπ/2phase shifter, which can be easily implemented by changing, for instance, the length of the channel waveguides. 7684 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 40, NO. 23, DECEMBER 1, 2022 V. CONCLUSION Ion-exchanged glass as a platform for quantum photonics has been proposed. We have demonstrated in such a platform passive quantum projections for the states of bases X(diagonal) and Y(circular) by using directional couplers which are constituted by single-mode channel waveguides fabricated by a K+/Na+ion-exchange followed by a burial process. We have characterized these couplers and obtained empiric curves between both fabrication and optical parameters of such elements. In particular, we have modelled their coupling coefficient as an exponential function with respect to the separation of the channel waveguides. 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