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Physical Review A

Complex structure and efficient characterization of multiphoton split states in integrated circuits

Jihua Zhang, Andrey A. Sukhorukov

Published Jun 30, 2023• 1 citation
Publisher

Abstract

Multiphoton split states, where each photon is in a different spatial mode, represent an essential resource for various quantum applications, yet their efficient characterization remains an open problem. Here, we formulate the general structure of their reduced spatial density matrices and identify the number of real and complex-valued independent coefficients, which in particular completely determine the distinguishability of all photons. Then, we show that this density matrix can be fully characterized by measuring correlations after photon interference in a static integrated circuit, where the required number of outputs scales subquadratically versus the number of photons. We present optimized circuit designs composed of segmented coupled waveguides, representing a linear optical neural network, which minimize the reconstruction error and facilitate robustness to fabrication deviations.

DOI
10.1103/physreva.107.062615

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Multiphoton split states, where each photon is in a different spatial mode, represent an essential resource for various quantum applications, yet their efficient characterization remains an open problem. Here, we formulate the general structure of their reduced spatial density matrices and identify the number of real and complex-valued independent coefficients, which in particular completely determine the distinguishability of all photons. Then, we show that this density matrix can be fully characterized by measuring correlations after photon interference in a static integrated circuit, where the required number of outputs scales subquadratically versus the number of photons. We present optimized circuit designs composed of segmented coupled waveguides, representing a linear optical neural network, which minimize the reconstruction error and facilitate robustness to fabrication deviations.
Multi-photon split states, where each photon is in a different spatial mode, represent an essential resource for various quantum applications, yet their efficient characterization remains an open problem. Here, we formulate the general structure of their reduced spatial density matrices and identify the number of real and complex-valued independent coefficients, which in particular completely determine the distinguishability of all photons. Then, we show that this density matrix can be fully characterized by measuring correlations after photon interference in a static integrated circuit, where the required outputs scale sub-quadratically versus the number of photons. We present optimized circuit designs composed of segmented coupled waveguides, representing a linear optical neural network, which minimize the reconstruction error and facilitate robustness to fabrication deviations.
Multi-photon split states, where each photon is in a different input mode, represent an essential resource for various quantum applications. Accordingly, characterisation of such states is important from the fundamental and practical perspectives. We propose a segmented coupled waveguide array as a new form of compact low-loss quantum circuit and apply it for the scalable multi-photon split state tomography with optimized performance and no need of reconfigurability. We develop an optimization algorithm to design the array with minimally necessary number of waveguides and smallest sensitivity to noise in the tomography. Such a novel platform can enable fast, scalable, and fully integrated quantum systems.
Summary form only given. Full characterization of quantum states is increasingly important as ever more complex quantum systems are realized. Quantum state tomography of these systems is a difficult task because individual measurements only give partial information about the underlying density matrix of the system. Thus, conventionally the measurement apparatus must be reconfigured to take multiple measurements in order to fully determine the density matrix. It is accepted that for an W-qubit system an exponentially increasing number of measurements are needed [1]. As a consequence, the complexity of typical quantum state tomography grows exponentially with increasing photon number, which makes characterizing large quantum systems highly challenging. We formulate a concept of on-chip quantum tomography where the complexity scales linearly with the number of photons. This is achieved by using a single specially designed non-reconfigurable linear optical circuit as in Fig. 1(a). Crucially, the number of waveguides in this circuit scales linearly with the number of photons in the state to be measured. Furthermore, our approach is simple and robust, and we demonstrate it experimentally with a specially designed photonic chip.
Multimode optical interferometers represent the most viable platforms for the successful implementation of several quantum information schemes that take advantage of optical processing. Examples range from quantum communication and sensing, to computation, including optical neural networks, optical reservoir computing, or simulation of complex physical systems. The realization of such routines requires high levels of control and tunability of the parameters that define the operations carried out by the device. This requirement becomes particularly crucial in light of recent technological improvements in integrated photonic technologies, which enable the implementation of progressively larger circuits embedding a considerable amount of tunable parameters. We formulate efficient procedures for the characterization of optical circuits in the presence of imperfections that typically occur in physical experiments, such as unbalanced losses and phase instabilities in the input and output collection stages. The algorithm aims at reconstructing the transfer matrix that represents the optical interferometer without making any strong assumptions about its internal structure and encoding. We show the viability of this approach in an experimentally relevant scenario, defined by a tunable integrated photonic circuit, and we demonstrate the effectiveness and robustness of our method. Our findings can find application in a wide range of optical setups, based on both bulk and integrated configurations.
Integrated single-photon detectors open new possibilities for monitoring inside quantum photonic circuits. We present a concept for the in-line measurement of spatially-encoded multi-photon quantum states, while keeping the transmitted ones undisturbed. We theoretically establish that by recording photon correlations from optimally positioned detectors on top of coupled waveguides with detuned propagation constants, one can perform robust reconstruction of the density matrix describing the amplitude, phase, coherence and quantum entanglement. We report proof-of-principle experiments using classical light, which emulates single-photon regime. Our method opens a pathway towards practical and fast in-line quantum measurements for diverse applications in quantum photonics.
Tunable linear photonic circuits are important elements in both classical and quantum information technologies. The scaling of such circuits is possible only in the integrated form, which complicates their characterization because of the impossibility of the reconstruction of each element individually. The existing methods of characterization of linear photonic circuits require multiple changes in the phases of transfer matrix elements at various values of control parameters, which provides significant experimental difficulties. In this work, a new approach is proposed: it is demonstrated that the measurement of only transmission coefficients for certain values of the control parameters of the photonic circuit allows the development of a mathematical model that can predicts transmission coefficients for arbitrary values of the parameters. This method has been successfully approved in a numerical experiment for a tunable four-channel interferometer implementing an arbitrary unitary transformation. The proposed method provides new possibilities to efficiently characterize and design tunable linear photonic circuits.