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Quantum Information Processing

Accurately quantifying the superposition state of two different Laguerre–Gaussian modes with single intensity distribution measurement

Ya Xiao, Haida Liu, Yue-Wei Song, Qizhi HanZhao, Fengheng Che, Xuan Fan, Xinhong Han, Hui-Chao Qu, Weichen Li, Wendong Li, Yongjian Gu

Published Feb 18, 2022• 5 citations
Publisher

Abstract

Abstract We develop a method to quantify the superposition state of two different Laguerre–Gaussian modes. By analyzing the characteristics of the intensity distribution obtained in a single measurement, including the petal number, the position and value of the extremum intensity, one can quantify the angular momentum index, the radial node index and the superposition coefficient simultaneously. Experimentally, we measure a series of superposition states, whose angular momentum index ranges from −47 to 53, radial node index from 0 to 3 and superposition weight from 0.1 to 0.9. The average trace distance and the mean fidelity of these states are lower than 0.053 ± 0.001 and higher than 0.982 ± 0.002, respectively. Our method can further obtain the superposition coefficient compared with previous mode verification ones and can reduce the number of measurement settings compared with the traditional quantum state tomography, thus more applicable in practice.

DOI
10.1007/s11128-022-03432-w

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Abstract We develop a method to quantify the superposition state of two different Laguerre–Gaussian modes. By analyzing the characteristics of the intensity distribution obtained in a single measurement, including the petal number, the position and value of the extremum intensity, one can quantify the angular momentum index, the radial node index and the superposition coefficient simultaneously. Experimentally, we measure a series of superposition states, whose angular momentum index ranges from −47 to 53, radial node index from 0 to 3 and superposition weight from 0.1 to 0.9. The average trace distance and the mean fidelity of these states are lower than 0.053 ± 0.001 and higher than 0.982 ± 0.002, respectively. Our method can further obtain the superposition coefficient compared with previous mode verification ones and can reduce the number of measurement settings compared with the traditional quantum state tomography, thus more applicable in practice.
Here, joint spatial weak measurements with higher-order Laguerre-Gaussian (LG) point states are investigated experimentally. From the intensity patterns of the final LG point states, two dimensional position operators 〈X〉 and 〈Y〉 as well as high-order position operators 〈XY〉, 〈X2 − Y2〉, 〈X3〉, and 〈Y3〉 are extracted simultaneously, from which both the complex weak values and joint weak values of two non-commuting observables can be obtained. The enhancement of joint weak values by the postselection state are analyzed. The simple relationship between the expectation values of position operators and the azimuthal and radial indexes of LG modes allows us to identify the mode indexes directly. A simple and robust scheme based on an optical window is demonstrated to monitor the LG mode indexes. These findings deepen the understanding of the weak measurement and provide an alternate and effective method for LG mode index monitoring.
Classical optic entanglement between the radial and angular degrees of freedom in Laguerre-Gaussian mode superpositions is explored within the framework of symmetric first-order optical systems. The Gouy phase picked by a Laguerre-Gaussian mode on free propagation is seen to be of consequence to the radial-angular entanglement in the mode superpositions. We illustrate examples of mode superpositions for which radial-angular entanglement is preserved on passage through symmetric first-order optical systems. An indicator of radial-angular entanglement in two-mode Laguerre-Gaussian superpositions is demonstrated to be a robust free space signaler in the presence of atmospheric turbulence, through examples.
One of the most widely used techniques for measuring the orbital angular momentum (OAM) components of a light beam is to flatten the spiral phase front of a mode, in order to couple it to a single-mode optical fiber (SMOF). This method, however, suffers from an efficiency that depends on the OAM of the initial mode and on the presence of higher-order radial modes. The reason is that once the phase has been flattened, the field retains its ringed intensity pattern and is therefore a nontrivial superposition of purely radial modes, of which only the fundamental one couples to a SMOF. In this paper, we study the efficiency of this technique both theoretically and experimentally. We find that even for low values of the OAM, a large amount of light can fall outside the fundamental mode of the fiber, and we quantify the losses as functions of the waist of the coupling beam of the OAM and radial indices. Our results can be used as a tool to remove the efficiency bias where fair-sampling loopholes are not a concern. However, we hope that our study will encourage the development of better detection methods of the OAM content of a beam of light.
The Laguerre-Gaussian (LG) modes constitute a complete basis set for representing the transverse structure of a paraxial photon field in free space. Earlier workers have shown how to construct a device for sorting a photon according to its azimuthal LG mode index, which describes the orbital angular momentum (OAM) carried by the field. In this paper we propose and demonstrate a mode sorter based on the fractional Fourier transform to efficiently decompose the optical field according to its radial profile. We experimentally characterize the performance of our implementation by separating individual radial modes as well as superposition states. The reported scheme can, in principle, achieve unit efficiency and thus can be suitable for applications that involve quantum states of light. This approach can be readily combined with existing OAM mode sorters to provide a complete characterization of the transverse profile of the optical field.
Accurate reconstruction of the complex spectrum of orbital angular momentum (OAM) superposition states is crucial for their application in high-capacity optical communications and quantum information, yet existing methods are often constrained by system complexity, extensive data requirements, or incomplete information retrieval. Here, we propose and experimentally demonstrate an iterative algorithm that enables high-fidelity recovery of the complex OAM spectrum of high-dimensional superpositions from only a few sparsely sampled patterns. The method integrates the Laguerre-Gaussian (LG) modal basis as an a priori constraint, which enables the direct retrieval of spectral coefficients by circumventing intermediate field reconstruction. To validate its performance, we reconstructed a test field composed of a random superposition of 21 LG modes, achieving fidelities of 99.3% in simulation and 97.2% in experiment using only four sparse sampling patterns. This work provides an efficient and robust tool for characterizing high-dimensional structured light, with significant potential in optical communication and quantum science.
High-dimensional (HD) entanglement enables an encoding of more bits than in the two-dimensional case and promises to increase communication capacity over quantum channels and to improve robustness to noise. In practice, however, one of the central challenges is to devise efficient methods to quantify the HD entanglement explicitly. Full quantum state tomography is a standard technology to obtain all the information about the quantum state, but it becomes impractical because the required measurements increase exponentially with the dimension in HD systems. Hence, it is highly anticipated that a new method will be found for characterizing the HD entanglement with as few measurements as possible and without introducing unwarranted assumptions. Here, we present and demonstrate a scan-free tomography method independent of dimension, which only requires two measurements for the characterization of two-photon HD orbital angular momentum (OAM) entanglement. Taking Laguerre-Gaussian modes of photons as an example, the density matrices of OAM entangled states are experimentally reconstructed with very high fidelity. Our method is also generalized to the mixed HD OAM entanglement. Our results provide realistic approaches for quantifying more complex OAM entanglement in many scientific and engineering fields such as multiphoton HD quantum systems and quantum process tomography.