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Demonstration of the quantum principle of least action with single photons

Nature Photonics volume 17, pages 717–722 (2023)Cite this article

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Abstract

The principle of least action is arguably the most fundamental principle in physics as it can be used to derive the equations of motion in various branches of physics. However, this principle has not been experimentally demonstrated at the quantum level because the propagators for Feynman’s path integrals have never been observed. The propagator is a fundamental concept and contains various significant properties of a quantum system in the path integral formulation, so its experimental observation is itself essential in quantum mechanics. Here we theoretically propose and experimentally observe the propagators of single photons based on the method of directly measuring quantum wave functions. Furthermore, we obtain the classical trajectories of single photons in free space and in a harmonic trap based on the extremum of the observed propagators, thereby experimentally demonstrating the quantum principle of least action. Our work paves the way for experimentally exploring the fundamental problems of quantum theory in the formulation of path integrals.

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Fig. 1: Schematic of path integrals and measuring method.
Fig. 2: Schematic of the experiment.
Fig. 3: Measured propagators of single photons in free space.
Fig. 4: Classical trajectories of single photons determined by the PLA.
Fig. 5: Theoretical comparison of \({{{{\mathcal{M}}}}}^{{\prime\prime} }(x,t)\) and \({{{{\mathcal{M}}}}}_{\epsilon }^{{\prime\prime} }(x,t-\epsilon )\) in the harmonic potential, where t = 1.1π/ω and ϵ = 0.003(tb − ta).

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Data availability

The data supporting the results of this study are available within the Article and its Supplementary Information. The raw datasets generated during the study are too large to be publicly shared, but they are available from the corresponding authors upon reasonable request.

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Acknowledgements

We thank Z. Y. Zhou for helpful discussions. This work was supported by the Key Area Research and Development Program of Guangdong Province (grant nos. 2019B030330001 (S.-L.Z. and H.Y.) and 2020A1515110848 (S.Z.)), the National Key Research and Development Program of China (grant nos. 2020YFA0309500 (S.Z.) and 2022YFA1405300 (S.-L.Z.)), the National Natural Science Foundation of China (grant nos. 12225405 (H.Y.), 12004120 (Y.W.), 62005082 (J.L.), 12074180 (S.-L.Z.) and U20A2074 (H.Y.)) and the Innovation Program for Quantum Science and Technology (grant no. 2021ZD0301705 (H.Y. and S.-L.Z.)).

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Author notes

  1. These authors contributed equally: Yong-Li Wen, Yunfei Wang, Li-Man Tian.

Authors and Affiliations

  1. Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou, China

    Yong-Li Wen, Yunfei Wang, Li-Man Tian, Shanchao Zhang, Jianfeng Li, Jing-Song Du, Hui Yan & Shi-Liang Zhu

  2. Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Frontier Research Institute for Physics, South China Normal University, Guangzhou, China

    Yong-Li Wen, Yunfei Wang, Li-Man Tian, Shanchao Zhang, Jianfeng Li, Jing-Song Du, Hui Yan & Shi-Liang Zhu

  3. National Laboratory of Solid State Microstructures and School of Physics, Nanjing University, Nanjing, China

    Yong-Li Wen

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Contributions

Y.-L.W. and S.-L.Z. developed the theory. Y.-L.W., Y.W., S.Z., H.Y. and S.-L.Z. designed the experiment. Y.-L.W., Y.W., L.-M.T., S.Z., J.L. and J.-S.D. carried out the experiments. Y.-L.W., Y.W., H.Y. and S.-L.Z. conducted the raw data analysis. Y.-L.W., H.Y. and S.-L.Z. wrote the paper, and all authors discussed the content of the paper. H.Y. and S.-L.Z. supervised the project.

Corresponding authors

Correspondence to Hui Yan or Shi-Liang Zhu.

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Supplementary Information

Supplementary sections 1–7, figs. 1–6 and discussion.

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Wen, YL., Wang, Y., Tian, LM. et al. Demonstration of the quantum principle of least action with single photons. Nat. Photon. 17, 717–722 (2023). https://doi.org/10.1038/s41566-023-01212-1

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