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Optics Express

Joint spatial weak measurement with higher-order Laguerre-Gaussian point states

Wenguo Zhu, Shuang Zhang, Xinzhou Liang, Huadan Zheng, Yongchun Zhong, Jianhui Yu, Zhe Chen, Li Zhang

Published May 3, 2022• 3 citations
Publisher

Abstract

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.

DOI
10.1364/oe.457656

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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.
As a special experimental technique, weak measurements extract very little information from the measured system and does not cause the measured state to collapse. When coupling the Laguerre-Gaussian (LG) state with a well-defined pre- and post-selected system of a weak measurement process, there will be an indirect interconnection between the expected value of coordinate operators of the final state and the mode indices of the measured LG state. The mode of the light is impacted very slightly after the weak measurement. Based on this we propose an experiment scheme and have managed to experimentally measure the mode indices of LG beams spanning from l = -6 to l = +6, p = 0 to p = +8 accurately with the final intensity distributions approximatly at their origin.
Abstract We propose an efficient method to accurately identify two mode indices ( p and l ) of Laguerre–Gaussian (LG) beams simultaneously with weak value measurement (WVM). Concise relations between the spatial displacements from the WVM and two mode indices of LG beams have been obtained. Compared with the traditional measurements, the advantage of using our method is that LG beams with large p and l can be identified precisely. A joint spatial WVM with a looser approximate condition has been presented, which realizes better identification than the individual spatial WVM with correction terms or strict approximation. Furthermore, the signal-to-noise ratio (SNR) of the spatial WVM is also discussed.
The joint weak value is a counterfactual quantity related to quantum correlations and quantum dynamics, which can be retrieved via weak measurements, as initiated by Aharonov and colleagues. In this paper, we provide a full analytical extension of the method described by Puentes et al. [Phys. Rev. Lett. 109, 040401 (2012)], to extract the joint weak values of single-particle operators from two-dimensional spatial displacements of Laguerre-Gauss probe states, for the case of the azimuthal index ∣l∣>1|l|>1∣l∣>1. This method has a statistical advantage over previous ones since information about the conjugate observable, i.e., the momentum displacement of the probe, is not required. Moreover, we demonstrate that, under certain conditions, the joint weak value can be extracted directly from spatial displacements without any additional data processing.
High-dimensional spatial entanglement generated via spontaneous parametric down-conversion (SPDC) provides a powerful resource for quantum information processing, yet its mode structure becomes increasingly complex when the pump occupies a higher-order Laguerre--Gaussian (LG) mode. Here, we develop an algebraic framework for characterizing biphoton spatial-mode entanglement by factorizing LG-pumped SPDC into a spatial-mode beam-splitter transformation followed by spatial two-mode squeezing. By choosing the biphoton LG basis radius as the geometric mean of the pump-beam radius and the crystal-induced correlation width, we obtain a natural modal basis in which the two operations can be treated separately. When the pump-beam radius matches the correlation width, spatial squeezing vanishes, and the two circular-mode components of the pump are independently conserved across the signal and idler photons, leading to conservation of both orbital angular momentum and the total spatial mode number. We further derive an analytical expression for the Schmidt number for arbitrary LG pump modes and squeezing strengths, revealing how the radial and azimuthal pump indices govern the dimensionality of the spatial entanglement. This algebraic characterization provides a unified description of the mode structure, conservation laws, and entanglement dimensionality of higher-order-LG-pumped SPDC.
Through the von Neumann interaction followed by post-selection, we can extract not only the eigenvalue of an observable of the measured system but also the weak value. In this post-selected von Neumann measurement, the initial pointer state of the measuring device is assumed to be a fundamental Gaussian wave function. By considering the optical implementation of the post-selected von Neumann measurement, higher-order Gaussian modes can be used. In this paper, we consider the Hermite–Gaussian (HG) and Laguerre–Gaussian (LG) modes as pointer states and calculate the average shift of the pointer states of the post-selected von Neumann measurement by assuming the system observable with and for an arbitrary interaction strength, where represents the identity operator. Our results show that the HG and LG pointer states for a given coupling direction have advantages and disadvantages over the fundamental Gaussian mode in improving the signal-to-noise ratio. We expect that our general treatment of the weak values will be helpful for understanding the connection between weak- and strong-measurement regimes and may be used to propose new experimental setups with higher-order Gaussian beams to investigate further the applications of weak measurement in optical systems such as the optical vortex.
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.