Direct experimental test of Feynman's path integral postulates with single photons
Data files
Jun 30, 2026 version files 962.14 KB
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Fig3.xlsx
50.58 KB
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Fig4A.xlsx
10.70 KB
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Fig4B.xlsx
11.62 KB
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Fig4C.xlsx
15.06 KB
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FigS1B-L.xlsx
142.42 KB
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FigS1B-Minus.xlsx
142.79 KB
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FigS1B-Plus.xlsx
142.86 KB
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FigS1B-PSI.xlsx
144.70 KB
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FigS1B-R.xlsx
142.74 KB
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FigS1C.xlsx
20.02 KB
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FigS2A.xlsx
16.30 KB
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FigS2B.xlsx
14 KB
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FigS2C.xlsx
15.94 KB
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FigS2D.xlsx
13.78 KB
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FigS3A.xlsx
11.68 KB
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FigS3B.xlsx
16.79 KB
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FigS4A.xlsx
13.50 KB
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FigS4B.xlsx
25.20 KB
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README.md
11.48 KB
Abstract
The experimental validation of fundamental thought experiments in quantum mechanics has profoundly advanced quantum science and technology while deepening our understanding of quantum mechanics. However, experimental studies of the path integral formulation, a cornerstone of quantum physics, remain scarce, especially regarding the two fundamental postulates proposed by Feynman in 1948, neither of which has been directly tested. Here, we present a novel theoretical proposal for the direct experimental test of Feynman's postulates, achieved through the development of a rigorous propagator-based approach for the first time. Furthermore, we perform comprehensive measurements of single photon's probability amplitudes for more than 1.4 million (175) paths, achieving unprecedented fidelity in propagator measurements and enabling complete reconstruction of the path probability amplitudes. The results confirm both postulates: (i) that quantum probabilities emerge from the coherent superposition of all possible paths, and (ii) that all possible paths have equal-magnitude amplitudes, whereas each path’s phase is determined by the classical action (in units of ℏ). This work not only resolves a longstanding foundational gap but also establishes a general experimental framework for investigating path integrals in contemporary quantum systems. This dataset supports the study "Direct experimental test of Feynman’s path integral postulates with single photons ". It provides the experimental and processed data that enable full reproduction of the figures in the associated manuscript.
Dataset DOI: 10.5061/dryad.x0k6djj14
Description of the data and file structure
Files and variables
File: Fig3.xlsx
Description: Experimental result for the test of Postulate I.
Variables
- x_f/δx-theor. : Spatial positions of theoretical results (horizontal axis).
- P_q(x_f)-theor. : Theoretical predictions of probabilities Pq(xf), in arbitrary units (a.u.).
- P_c(x_f)-theor. : Theoretical predictions of probabilities Pc(xf), in arbitrary units (a.u.).
- x_f/δx-exp. : Spatial positions of experimental results (horizontal axis).
- P_e(x_f)-exp. : Experimental results of probabilities Pe(xf), in arbitrary units (a.u.).
- P_q(x_f)-exp. : Experimental results of probabilities Pq(xf), in arbitrary units (a.u.).
- S.E. of P_q(x_f)-exp. : Standard errors of the experimental probability Pq(xf).
- P_c(x_f)-exp. : Experimental results of probabilities Pc(xf), in arbitrary units (a.u.).
- S.E. of P_c(x_f)-exp. : Standard errors of the experimental probability Pc(xf).
File: Fig4A.xlsx
Description: Experimental result for the test of Postulate II. Path probability vs. length 𝐿.
Variables
- L/δx: The total length (L) of each path.
- P_L(a.u.): Mean probabilities for equal-length paths, in arbitrary units (a.u.).
- S.D. of P_L(a.u.): Standard deviations of P_L(a.u.).
File: Fig4B.xlsx
Description: Experimental result for the test of Postulate II. Path probability vs. action S.
Variables
- S(pi/\hbar): The action intervals Δ𝑆(𝑙), with 𝑙= 1 to 100.
- P_{(l)} (a.u.): Mean probabilities of paths as a function of Δ𝑆(𝑙), in arbitrary units (a.u.).
- S.D. of P_{(l)} (a.u.): Standard deviations of P_{(l)} (a.u.).
File: Fig4C.xlsx
Description: Experimental result for the test of Postulate II. Path phase angle vs. action S.
Variables
- S(\pi/\hbar): The action intervals, with l= 1 to 100.
- \theta(\pi)-exp. : Experimental results of mean phase angle 𝜃(𝑙) as a function of Δ𝑆(𝑙).
- S.D. of \theta(\pi): Standard deviations of \theta(\pi)-exp.
- \theta(\pi)-theor. : Theoretical results of mean phase angle 𝜃(𝑙) as a function of Δ𝑆(𝑙).
File: FigS1B-L.xlsx
Description: The first column and the first row of the table represent the coordinates x_{p3} and y_{p3}, respectively, in units of δx. The remaining values are the grayscale values of the image acquired by the ICMOS camera when projected onto the polarization basis |L⟩.
File: FigS1B-Minus.xlsx
Description: The first column and the first row of the table represent the coordinates x_{p3} and y_{p3}, respectively, in units of δx. The remaining values are the grayscale values of the image acquired by the ICMOS camera when projected onto the polarization basis |-⟩.
File: FigS1B-Plus.xlsx
Description: The first column and the first row of the table represent the coordinates x_{p3} and y_{p3}, respectively, in units of δx. The remaining values are the grayscale values of the image acquired by the ICMOS camera when projected onto the polarization basis |+⟩.
File: FigS1B-R.xlsx
Description: The first column and the first row of the table represent the coordinates x_{p3} and y_{p3}, respectively, in units of δx. The remaining values are the grayscale values of the image acquired by the ICMOS camera when projected onto the polarization basis |R⟩.
File: FigS1B-PSI.xlsx
Description: The first column and the first row of the table represent the coordinates x_{p3} and y_{p3}, respectively, in units of δx. The remaining values are the grayscale values of the image acquired by the ICMOS camera when measuring the probability distribution.
File: FigS1C.xlsx
Description: Sample results from the propagator measurements.
Variables
- x_{p3}/dx-theor.: Spatial positions of theoretical results (horizontal axis).
- Im[K]-theor. : Theoretical predictions for the imaginary components of propagator 𝐾(𝑥𝑝3, 𝑡3; 𝑥𝑝2, 𝑡2).
- Re[K]-theor. : Theoretical predictions for the real components of propagator 𝐾(𝑥𝑝3, 𝑡3; 𝑥𝑝2, 𝑡2).
- x_{p3}/dx-exp. : Spatial positions of experimental results (horizontal axis).
- Im[K]-exp. : Experimental results for the imaginary components of propagator 𝐾(𝑥𝑝3, 𝑡3; 𝑥𝑝2, 𝑡2).
- S.E. of Im[K]-exp. : Standard errors of Im[K]-exp.
- Re[K]-exp. : Experimental results for the real components of propagator 𝐾(𝑥𝑝3, 𝑡3; 𝑥𝑝2, 𝑡2).
- S.E. of Re[K]-exp. : Standard errors of Re[K]-exp.
File: FigS2A.xlsx
Description: Distribution of the phase differences 𝜗 between the measured propagator phases in various position difference Δ = |𝑥𝑝𝑘 - 𝑥𝑝𝑘-1 |. The first column are the 𝜗 values across 120 bins within the range [-0.2𝜋, 𝜋]. The first row position difference Δ = |𝑥𝑝𝑘 - 𝑥𝑝𝑘-1 | (in unit of step size 𝛿𝑥 ). The remaining values are the frequency counts of 𝜗 values across 120 bins.
File: FigS2B.xlsx
Description: Statistical analysis of the measured phase of the propagators.
Variables
- Δ/δx-exp. : Position difference Δ (in units of step size δx), used as the horizontal axis for experimental results.
- ϑ(π)-exp. : Experimental results for phase differences 𝜗 between the measured propagator phases in various position difference Δ.
- S.D. of ϑ(π)-exp. : Standard deviations of ϑ(π)-exp.
- Δ/δx-theor. : Position difference Δ (in units of step size δx), used as the horizontal axis for theoretical results.
- ϑ(π)-theor. : Theoretical predictions for phase differences 𝜗 between the measured propagator phases in various position difference Δ.
File: FigS2C.xlsx
Description: Distribution of the amplitude 𝜂 in unit of mean amplitude \bar{𝜂}. The first column are the 𝜂 values across 100 bins within the range [0.5, 1.5]. The first row position difference Δ = |𝑥𝑝𝑘 - 𝑥𝑝𝑘-1 | (in unit of step size 𝛿𝑥 ). The remaining values are the frequency counts of 𝜂 values across 100 bins.
File: FigS2D.xlsx
Description: Statistical analysis of measured amplitudes of propagator.
Variables
- Δ/δx-exp. : Position difference Δ (in units of step size δx), used as the horizontal axis for experimental results.
- η/(\bar{η})-exp. : Experimental results of the amplitude 𝜂 in unit of mean amplitude \bar{𝜂}, in various position difference Δ.
- S.D. of η/(\bar{η})-exp. : Standard deviations of η/(\bar{η})-exp.
- Δ/δx-theor. : Position difference Δ (in units of step size δx), used as the horizontal axis for theoretical results.
- η/(\bar{η})-theor. : Theoretical predictions of the amplitude 𝜂 in unit of mean amplitude \bar{𝜂}, in various position difference Δ.
File: FigS3A.xlsx
Description: Numerical simulation for the test of Postulate II. Path probability vs. length 𝐿.
Variables
- L/δx: The total length (L) of each path, expressed in units of δx.
- P_L(a.u.): Mean probabilities for equal-length paths, in arbitrary units (a.u.).
- S.D. of P_L(a.u.): Standard deviations of P_L(a.u.).
File: FigS3B.xlsx
Description: Numerical simulation for the test of Postulate II. The action range [0, 2𝜋ℏ) was divided into 100 intervals Δ𝑆(𝑙) , with 𝑙=1,2,3,...,100.
- S(pi/\hbar): The action intervals Δ𝑆(𝑙) , with 𝑙= 1 to 100.
- P_{(l)} (a.u.): Mean probabilities of paths as a function of Δ𝑆(𝑙), in arbitrary units (a.u.).
- S.D. of P_{(l)} (a.u.): Standard deviations of P_{(l)} (a.u.).
- \theta(\pi)-sim. : Numerical simulation of mean phase angle 𝜃(𝑙) as a function of Δ𝑆(𝑙).
- S.D. of \theta(\pi): Standard deviations of \theta(\pi)-sim.
- \theta(\pi)-theor. : Theoretical calculation of mean phase angle 𝜃(𝑙) as a function of Δ𝑆(𝑙).
File: FigS4A.xlsx
Description: Experimental results for both fixed and varying final positions.
Variables
- L/δx: The total length (L) of each path, expressed in units of δx.
- P_L(a.u.)-varying_final_position: Mean probabilities for mean probability values for varying final positions, in arbitrary units (a.u.).
- S.D. of P_L(a.u.)-fix_final_position: Standard deviations of P_L(a.u.)-varying_final_position.
- P_L(a.u.)-fix_final_position: Mean probabilities for mean probability values for fixed final positions, in arbitrary units (a.u.).
- S.D. of P_L(a.u.)-fix_final_position: Standard deviations of P_L(a.u.)-varying_final_position.
File: FigS4B.xlsx
Description: Experimental results for both fixed (Sheet2 in the .xlsx file) and varying (Sheet1 in the .xlsx file) final positions.
Variables in Sheet1
- S(pi/\hbar): The action intervals Δ𝑆(𝑙) , with 𝑙= 1 to 100.
- P_{(l)} (a.u.)_varying final positions: Mean probabilities of paths as a function of Δ𝑆(𝑙) for varying final positions, in arbitrary units (a.u.).
- S.D. of P_{(l)} (a.u.)_varying final positions:: Standard deviations of P_{(l)} (a.u.)_varying final positions.
- \theta(\pi)-exp._varying final positions : Experimental results of mean phase angle 𝜃(𝑙) as a function of Δ𝑆(𝑙) for varying final positions.
- S.D. of \theta(\pi)-exp._varying final positions: Standard deviations of \theta(\pi)-exp._varying final positions.
- \theta(\pi)-theor._varying final positions : Theoretical calculation of mean phase angle 𝜃(𝑙) as a function of Δ𝑆(𝑙) for varying final positions.
Variables in Sheet2
- S(pi/\hbar): The action intervals Δ𝑆(𝑙), with 𝑙= 1 to 100.
- P_{(l)} (a.u.)_fixed final positions: Mean probabilities of paths as a function of Δ𝑆(𝑙) for fixed final positions, in arbitrary units (a.u.).
- S.D. of P_{(l)} (a.u.)_fixed final positions: Standard deviations of P_{(l)} (a.u.)_fixed final positions.
- \theta(\pi)-exp._fixed final positions : Experimental results of mean phase angle 𝜃(𝑙) as a function of Δ𝑆(𝑙) for fixed final positions.
- S.D. of \theta(\pi)-exp._fixed final positions : Standard deviations of \theta(\pi)-exp._fixed final positions.
- \theta(\pi)-theor._fixed final positions : Theoretical calculation of mean phase angle 𝜃(𝑙) as a function of Δ𝑆(𝑙) for varying final positions.