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Journal of Physics B Atomic Molecular and Optical Physics

W-state generation and verification in linearly coupled waveguide arrays

Kolja Bugarski, Aleksandra Maluckov, Jovana S. Petrovic

Published Dec 3, 2024• 0 citations
Publisher

Abstract

Abstract W states are maximally entangled states with excellent robustness to loss. They have been studied in ion, photon and waveguide-mode bases, however the entanglement verification has remained challenging. Here, we theoretically study W-state generation by the quantum walk of a single photon through a linearly coupled waveguide array (WGA) that supports self-imaging of light. We design W states in symmetric and asymmetric arrays with even and odd number of waveguides. Bipartite entanglement is formally proven using the von Neumann entropy of the reduced density matrix. We further use the self-imaging to construct an entanglement witness based on the exclusion principle. A key to verification is the interferometer which extends the length of a W-state generator to the revival length where the photon is fully recombined into a single waveguide mode. In addition, W-state coherence is proven by numerically demonstrating the far-field interference. The sensitivities of the proposed W-state generation and verification protocols to the fabrication tolerances are numerically evaluated in glass, silicon nitride and silicon-on-insulator WGAs, indicating the feasibility of their realisation by high-precision e-beam lithography and laser writing.

DOI
10.1088/1361-6455/ad9a2d

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Abstract W states are maximally entangled states with excellent robustness to loss. They have been studied in ion, photon and waveguide-mode bases, however the entanglement verification has remained challenging. Here, we theoretically study W-state generation by the quantum walk of a single photon through a linearly coupled waveguide array (WGA) that supports self-imaging of light. We design W states in symmetric and asymmetric arrays with even and odd number of waveguides. Bipartite entanglement is formally proven using the von Neumann entropy of the reduced density matrix. We further use the self-imaging to construct an entanglement witness based on the exclusion principle. A key to verification is the interferometer which extends the length of a W-state generator to the revival length where the photon is fully recombined into a single waveguide mode. In addition, W-state coherence is proven by numerically demonstrating the far-field interference. The sensitivities of the proposed W-state generation and verification protocols to the fabrication tolerances are numerically evaluated in glass, silicon nitride and silicon-on-insulator WGAs, indicating the feasibility of their realisation by high-precision e-beam lithography and laser writing.
Abstract We study the generation of a single photon perfect W-state. An important aspect of this perfect W-state is that it can be used for perfect teleportation and superdense coding, which are not achievable with maximally entangled W-state. Our scheme for generation involves entanglement between various path degrees of freedom of a single photon in a compact and weakly coupled integrated waveguide system, which can be fabricated precisely with a femtosecond laser direct writing technique. These platforms are interferometrically stable, scalable, less sensitive to decoherence and ensure a very low loss factor of 0.1dBcm −1 during photon propagation, and hence are ideal for generation of a perfect W-state. In addition to generation of the single photon perfect W-state, we study its non-local properties using the theory of local elements of reality.
Recently, integrated optic solutions for generating single photon N-partite W states by using a one-dimensional array of N identical single-mode optical waveguides, which are coupled via nearest-neighbour interaction, have been proposed. Due to the particular design of the proposed structure, the tolerances with respect to the length of the device and the input wavelength of the single photon for the generation of the W state are very critical, thus making an experimental realization troublesome. Here, we propose an alternative design of a waveguide array consisting of (2N-1) waveguides to create a single-photon N-partite W state that exhibits much greater tolerance to the fabrication conditions as well as to the input wavelength and hence is much more realizable experimentally. The same design is also shown to lead to the generation of two-photon W-states when using the post selection technique.
The generation of multi-photon quantum states in integrated waveguiding structures provides a route to the creation of quantum optical chips. In particular, the simultaneous generation of photons through spontaneous parametric down-conversion (SPDC) and their quantum walks in nonlinear waveguide arrays (WGAs) can be used to produce strongly spatially entangled biphoton states [1] with applications for quantum search algorithms [2]. Whereas up to now only arrays of single-mode nonlinear waveguides have been considered for SPDC, in general the waveguides may support several guided modes, which were demonstrated to strongly affect nonlinear wave mixing in the classical regime [3]. Here we show that using higher order modes for SPDC has a significant influence on the generated two-photon states and gives additional means to control them. In particular, we employ the symmetry of the modes to engineer effectively nonlocal photon pair generation by SPDC.
We demonstrate experimentally simultaneous photon-pair generation and quantum walks in a lithium niobate waveguide array, where the output photon correlations can be controlled by varying the temperature-dependent phase-matching, switching from classical to quantum statistics. Integrated optical quantum circuits are increasingly gaining attention as a solution for scalable quantum technologies with important application to quantum simulations [1]. A key mechanism for such quantum simulations can be provided by the process of quantum walks, which in quantum optics is enabled in an array of coupled optical waveguides. However, in all integrated quantum walk schemes to date, the correlated photon pairs have been generated externally to the waveguide array (WGA), making it only a passive circuit. Here we experimentally demonstrate an active quantum circuit, where the WGA is used for both photonpair generation and quantum walks of the generated photons. We show that the output photon statistics can be controlled classically by changing the phase matching conditions for photon-pair generation, which allows for controllable engineering of the spatial quantum correlations exhibited through a quantum walk [4]. Importantly, this integrated scheme also allows avoiding input losses from the state generation to the waveguides. This is crucial as input losses in devices with linear coupled waveguides currently limit feasible experimental data acquisition times and hinder possible applications. In our experiments we use an x-cut LiNbO3 WGA [see Fig. 1(a)] fabricated by titanium indiffusion [5], which consists of 40 waveguides with a spacing of 14mm. A single waveguide of the WGA is excited by a continuous wave (CW) solid state laser at a wavelength of 671nm [see Fig. 1(b)]. The WGA is kept at a temperature of ~360C (with an accuracy of 0.1C) to phase-match the Type-I (oo-e) spontaneous parametric down-conversion (SPDC) process. At the pump wavelength the mode is well confined to the central waveguide and does not show evanescent coupling to the neighboring waveguides [see Fig. 1(a)]. However, for the generated low-frequency SPDC photons the waveguide modes extend towards the neighboring waveguides and can couple through the array experiencing quantum walks. Thus the generated output photon pairs demonstrate nontrivial photon statistics. Experimentally, we measure the photon-pair correlations at a wavelength of 1342 ± 6nm (filtered by an interference filter) on two cooled InGaAs single photon detectors (IDQuantique 210) in free-running mode [see Fig. 1(b)]. The non-polarizing beamsplitter is used to separate two photons of a pair with 50% efficiency, while irises allow for individual waveguide output selection. The resulting correlation measurements are shown in Fig. 2(a), where the coincidence counts are normalized with respect to the coupling efficiency. We observe that for a sample temperature of T = 364.5C close to degenerate SPDC phase-matching, photons from down-converted pairs leave the WGA from the same waveguide {-1,-1}, {0,0} or {1,1} (here two numbers correspond to two SPDC photons), defined as bunching, or from the opposite waveguides {-1,1} or {1,-1} defined as antibunching. The highest coincidence rate is achieved in the {0,0} case, since photon pairs are generated in waveguide 0 along the whole length of the WGA. Photon pairs coming from the other waveguides, namely {-1,0}, {1,0}, {0,-1} and {0,1} lead to less coincidences. This means that probabilities of photon bunching and antibunching are simultaneously dominant, which implies a highly non-trivial Paper code: Fig. 1. (a) Schematic representation of LiNbO3 WGA with pump (blue) coupled to a central waveguide and SPDC photon pairs (red) generated in the central waveguide and propagating in the WGA in the regime of quantum walks. (b) Schematic of the experimental setup. entangled state [3]. The theoretical quantum simulations of the process based on Ref. [4] [Fig. 2(b)] are in good agreement with our experiment. To demonstrate the reconfigurability of the device we also measure the correlation of the antibunched photons in waveguide {-1,1} as the temperature is varied, shown in Fig. 2(c). The theory developed for WGAs (blue curve), taking into account structural and temperature-related inhomogeneities in the sample, fits well the experimental data (red circles with error bars). This demonstrates that changing the temperature affects the coincidence counts from various waveguides, allowing control of the output photon statistics and revealing a route to reconfigurable quantum simulations based on walks. To conclude, our work opens a path for implementing combined photon-pair generation and quantum walks for engineering of spatial quantum states in novel active quantum integrated circuits.