Direct experimental test of Feynman’s path integral postulates with single photons

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 theoretical proposal for the direct experimental test of Feynman’s postulates, achieved through the development of a rigorous propagator-based approach. Furthermore, we perform comprehensive measurements of single photon’s probability amplitudes for more than 1.4 million (17⁵) paths, achieving high 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.

INTRODUCTION

The fundamental thought experiments in quantum mechanics—such as Schrödinger’s cat (1), the Einstein-Podolsky-Rosen paradox (2), and the quantum double-slit experiment (3)—revealed counterintuitive phenomena such as quantum superposition and nonlocality through idealized frameworks, shaping the theory’s conceptual bedrock. These Gedanken experiments not only deepened our understanding of quantum principles but also laid the groundwork for experimental quantum science by identifying key phenomena to test (4–9). The experimental investigation of these Gedanken experiments has since catalyzed the emergence of quantum science and technology as a cutting-edge disciplines. Pioneering work includes the realization of Schrödinger cat states (9–11), the tests of Bell’s inequalities (12–15), the multiphoton Greenberger-Horne-Zeilinger states for nonlocality studies (7, 8, 16), and quantum eraser and delayed-choice experiments (6, 17–28), showcasing optics as a powerful tool for probing quantum foundations (4–7, 12–18, 29–34). However, while these experiments validate the superposition and nonlocal nature of quantum mechanics, they do not address another fundamental question: How do quantum systems evolve between initial and final states? This question lies at the heart of Feynman’s path integral formulation, which remains experimentally untested despite its foundational role in modern physics.

Among the unverified pillars of quantum theory, Feynman’s two fundamental postulates for his famous propagator equation (35) stand out as a critical gap. Feynman’s propagator equation not only is equivalent to the Schrödinger equation but also elegantly entangles three fundamental concepts in physics: quantization, symmetry, and phase factor (36) (see the “Feynman’s propagator formula and his two postulates in path integrals” section in the Supplementary Materials). The path-integral formulation has proven crucial to numerous theoretical developments in fields ranging from quantum field theory to cosmology, providing a unified framework that connects quantum mechanics with classical physics through the principle of least action (35, 37, 38). However, its core assumptions (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), have never been directly tested. Furthermore, whether these paths constitute physical realities or are merely computational tools requires further experimental investigation and awaits elucidation. This lack of experimental exploration is notable, given that the path integral’s predictions underpin modern quantum technologies, from condensed matter physics to quantum field theory to quantum statistics physics. Recent advances in measuring quantum wave functions and quantum propagators (39–44) based on weak values (33, 45–47) have begun to bridge this gap, but a direct test of Feynman’s postulates requires overcoming the challenge of probing large-scale propagators and the global structure of path probability amplitudes with sufficiently high fidelity.

In this work, we bridge this critical gap by developing a combined theoretical and experimental framework to directly test Feynman’s two fundamental postulates. Central to our approach is the measurement of individual path probability amplitudes, which, according to the original path integral formulation, can be decomposed into products of propagators. This propagator-based methodology enables a rigorous test of both postulates. We measure thousands of propagators for single photons with high fidelity to construct the probability amplitudes of 1,419,857 (17⁵) possible paths. With these data, we establish three key results: (i) validation of the coherent superposition postulate governing final-state probabilities, (ii) experimental confirmation of the equal-magnitude postulate for path amplitudes, and (iii) direct evidence that each path’s phase is determined by the classical action (in units of ħ). We quantify the agreement between experimental data and the postulates using the mean absolute percentage error (MAPE) and fidelity. The MAPE for postulate I [II] is 4.45% [17.4%], while the fidelity of postulate I [II] is 94.9% [94.7%]. While the path integral formulation has long served as a profound theoretical framework, our approach offers an experimental platform to directly visualize its fundamental components.

RESULTS

Theoretical model

The fundamental idea of path integrals is illustrated in Fig. 1A. At time $t_{\mathrm{i}}$, a particle is localized at $x_{\mathrm{i}}$ with its wave function denoted as $\mathrm{\psi }(x_{\mathrm{i}},t_{\mathrm{i}})$. The aim is to find the probability of the particle reaching $x_{\mathrm{f}}$ at time $t_{\mathrm{f}}$. Feynman derived that $\mathrm{\psi }(x_{\mathrm{f}},t_{\mathrm{f}})=K(x_{\mathrm{f}},t_{\mathrm{f}};x_{\mathrm{i}},t_{\mathrm{i}})\mathrm{\psi }(x_{\mathrm{i}},t_{\mathrm{i}})$, where the propagator is given by

$$K(x_{\mathrm{f}},t_{\mathrm{f}};x_{\mathrm{i}},t_{\mathrm{i}})=\int \mathcal{D}\left[x\left(t\right)\right]e^{\frac{i}{\mathrm{\hslash }}S\left[x\left(t\right)\right]}$$ (1)

Fig. 1. Schematic of path integrals and the measurement of propagators.

(A) Particle can traverse any possible paths from initial point a to final point b. Three paths represented with ${\mathrm{\varphi }}_j (j=\mathrm{1,2,3})$ are exampled. (B) Discretized paths: squares mark discrete positions; shaded regions represent propagation between time intervals. Line segments depict evolution, with propagator $K(x_{p_k},t_k;x_{p_{k-1}},t_{k-1})$ as the probability amplitude for transitions $(x_{p_{k-1}},t_{k-1})\to (x_{p_k},t_k)$. (C) Measurement of propagator $K(x_{p_3},t_3;x_{p_2},t_2)$.

Equation 1 is the famous Feynman’s propagator equation, where $S=\int \mathcal{L}\mathit{dt}$ with $\mathcal{L}$ being the Lagrangian represents the classical action. The probability of the particle reaching $x_{\mathrm{f}}$ at time $t_{\mathrm{f}}$ is determined through all possible paths connecting points A at $(x_{\mathrm{i}},t_{\mathrm{i}})$ and B at $(x_{\mathrm{f}},t_{\mathrm{f}})$. The j-th path contributes a probability amplitude ${\mathrm{\varphi }}_j$. Feynman derived Eq. 1 by proposing two fundamental postulates in his seminal paper (35):

Postulate I: The probability P of the particle reaching $x_{\mathrm{f}}$ at time $t_f$ equals the absolute square of the sum of probability amplitudes ${\mathrm{\varphi }}_j$ for all paths, i.e.

$$P={\mid \mathrm{\psi }(x_{\mathrm{f}},t_{\mathrm{f}})\mid }^2={\mid {\displaystyle \sum _j}{\mathrm{\varphi }}_j\mid }^2$$ (2)

which generally differs from the classical probability sum $P_{\mathrm{c}}={\sum }_j{\mid {\mathrm{\varphi }}_j\mid }^2$.

Postulate II: All paths contribute equally in magnitude, with each contribution’s phase given by the classical action (in units of $\mathrm{\hslash }$)

$$\mid {\mathrm{\varphi }}_1\mid =\mid {\mathrm{\varphi }}_2\mid =\dots =\mid {\mathrm{\varphi }}_j\mid$$ (3)

where

$${\mathrm{\varphi }}_j=A{\mathrm{e}}^{{\mathit{\text{iS}}}_j/\mathrm{\hslash }}$$ (4)

Here, A is a path-independent real constant. This postulate constitutes the “equal probability hypothesis.”

While path integral theory treats space time as continuous, practical implementation requires discretization. As shown in Fig. 1B, we decompose continuous paths into sequences of discrete propagation events, each described by a propagator. The total path probability amplitude is then expressed as a product of these propagators

$${\mathrm{\varphi }}_j={\displaystyle \prod _{k=1}^M}K(x_{p_k},t_k;x_{p_{k-1}},t_{k-1})$$ (5)

where $K(x_{p_k},t_k;x_{p_{k-1}},t_{k-1})$ represents the probability amplitude for propagation from $(x_{p_{k-1}},t_{k-1})$ to $(x_{p_k},t_k)$. This decomposition reduces measuring ${\mathrm{\varphi }}_j$ to successive measurements of individual propagators.

We now propose an optical experiment to directly measure the probability amplitude ${\mathrm{\varphi }}_j$, thereby enabling test of the fundamental postulates described in Eqs. 2 to 4. We assume photons propagate along the z direction and consider a one-dimensional system with x as the position coordinate. The Hamiltonian of single photons can be written as $H=c\mid \overrightarrow{p}\mid =c\sqrt{{\hat{p}}_x^2+{\hat{p}}_y^2+{\hat{p}}_z^2}$, where ${\hat{p}}_{x,y,z}$ are momentum operators and c is the velocity of light. In our experiment, the transverse spatial mode of the photons is a Gaussian wave packet in x-y plane, with a waist of $a\approx 0.57 \text{mm}$. The momentum of a photon in the z direction is $p_z=\mathrm{\hslash }{k}_z=2\mathrm{\pi \hslash }/\mathrm{\lambda }$, with a wavelength of $\mathrm{\lambda }=795 \text{nm}$. The resulting transverse momentum uncertainties are $\mathrm{\Delta }{p}_x=\mathrm{\Delta }{p}_y=\mathrm{\hslash }/\left(2a\right)$, which satisfies $p_z\gg p_x,p_y$. Under this condition, the Hamiltonian can be rewritten as an approximation

$$H\approx \frac{c{\hat{p}}_x^2}{2p_z}+\frac{c{\hat{p}}_y^2}{2p_z}+{\mathit{cp}}_z$$ (6)

For simplicity and without loss of generality, we restrict our analysis to the x-z plane. Given that $p_z$ is nearly constant, the propagation distance z is proportional to time $t=z/c$. Consequently, the evolution of the spatial wave function $\mathrm{\psi }(x,t)$ is governed by a Schrödinger-like equation (48, 49)

$$i\mathrm{\hslash }\frac{\mathrm{\partial }}{\mathrm{\partial }t}\mathrm{\psi }(x,t)=\frac{c{\hat{p}}_x^2}{2p_z}\mathrm{\psi }(x,t)$$ (7)

Hence, the evolution of a photon along the x direction can be separated and described by a nonrelativistic particle of effective mass $m=p_z/c=2\mathrm{\pi \hslash }/\left(\mathrm{\lambda }c\right)$. Accordingly, the characteristic de Broglie wavelength along x is given by ${\mathrm{\lambda }}_x=2\mathrm{\pi \hslash }/\left(\mathrm{\Delta }{p}_x\right)\approx 7.16 \text{mm}$. Although a rigorous treatment of photons requires quantum field theory, this first quantization approach is a standard and reliable approximation for paraxial propagation. Notably, this system has been used to detect the quantum wave functions (39, 50, 51).

We use photon polarization as the measurement pointer (39). We describes in Materials and Methods a detailed scheme for detecting a general propagator $K(x_{p_k},t_k;x_{p_{k-1}},t_{k-1})$, with each propagator measurement involving four steps: (i) State preparation: Initializing a polarized state and coupling the spatial mode to a polarization pointer. We apply a unitary coupling operation $\hat{U}$ to prepare the initial state $\mid {\mathrm{\Psi }}_{k-1}\rangle$ at time $t_{k-1}$. (ii) System evolution: The spatial mode evolves under the free-space propagation operator $\hat{T}(t_k-t_{k-1})$ between $t_{k-1}$ and $t_k$. (iii) Postselection: Projecting the system to a position state $\mid x_{p_k}\rangle$ at time $t_k$. (iv) Pointer measurement: Measuring the pointer state at $x_{p_k}$. In Materials and Methods, we demonstrate that the propagator $K(x_{p_k},t_k;x_{p_{k-1}},t_{k-1})$ can be obtained through

$$K(x_{p_k},t_k;x_{p_{k-1}},t_{k-1})=\frac{\langle f_k\mid {\hat{\mathrm{\sigma }}}_x\mid f_k\rangle +i\langle f_k\mid {\hat{\mathrm{\sigma }}}_y\mid f_k\rangle }{\mathrm{\Phi }\left(t_{k-1}\right)\mid {\mathrm{\psi }}^{\ast }(x_{p_k},t_k)\mid }$$ (8)

where $\mathrm{\Phi }\left(t_{k-1}\right)$ represents the zero-momentum wave function and can be eliminated through normalization. The numerator in Eq. 8 is obtained through pointer state measurements, while the probability $\mid {\mathrm{\psi }}^{\ast }(x_{p_k},t_k)\mid$ at space-time point $(x_{p_k},t_k)$ is directly detectable using a camera.

By sequentially measuring propagators from $t_{\mathrm{i}}$ to $t_{\mathrm{f}}$, we reconstruct all path probability amplitudes within a specified space-time region. Consequently, the probability amplitude for the j-th path is given by

$${\mathrm{\varphi }}_j=\frac{{\displaystyle \prod _{k=1}^M}[\langle f_k\mid {\hat{\mathrm{\sigma }}}_x\mid f_k\rangle +i\langle f_k\mid {\hat{\mathrm{\sigma }}}_y\mid f_k\rangle ]}{\mathcal{N}{\displaystyle \prod _{k=1}^M}\mid {\mathrm{\psi }}^{\ast }(x_{p_k},t_k)\mid }$$ (9)

where $\mathcal{N}={\prod }_{k=1}^M\mathrm{\Phi }\left(t_{k-1}\right)$.

Experiment

The experiment was performed with single photons and the experimental setup is depicted in Fig. 2. The single photons are generated via spontaneous parametric down conversion (SPDC) (52, 53). The single-photon source exhibited a second-order correlation function $g_{\mathrm{c}}^{\left(2\right)}\approx 0.234\pm 0.015$ with a center wavelength of $\mathrm{\lambda }=795 \text{nm}$.

Fig. 2. Experimental setup schematic.

(A) State initialization: Single photons with Gaussian transverse mode emerge from a single-mode (SM) fiber 1. A half-wave plate (HWP1) and PBS (PBS1) prepare horizontal polarization $\mid 0\rangle$. A beam splitter (BS) reflects 10% for photon number normalization. (B) Spatial-pointer coupling: PBS2 splits light into $\mid 0\rangle$ and $\mid 1\rangle$ polarization branches. The $\mid 0\rangle$ branch passes through CYL and slit, projecting onto position state $\mid x_{k-1}\rangle$; the $\mid 1\rangle$ branch remains unchanged. Recombination occurs at another PBS3. Piezo-driven prism stabilizes optical path difference. An 852-nm reference laser stabilizes Mach-Zehnder interferometer (MZI), with optical power detector (PD) providing feedback. (C) State evolution: 4f-system1 (equivalents to two convex lenses) maps slit exit wavefront to evolution region (see Materials and Methods). Initial time $t_{k-1}$ adjusted via slit-to-4f-system1 distance; position $x_{k-1}$ controlled by slit and CYL translation. (D) Position postselection and readout: Second 4f-system2 relays wavefront at measurement positions ($z={\mathit{ct}}_k$) to ICMOS camera (CISS, 2DSPC). The camera gated by SPDC idler triggers (see Materials and Methods). Longitudinal camera adjustment accesses different $t_k$ planes. Q/HWP and PBS4 enable pointer expectation measurements.

We initialized the photon spatial mode using a single-mode fiber. Under the paraxial approximation, the propagation time corresponds to the longitudinal distance traveled by the light. The initial polarization state was prepared using a Mach-Zehnder (MZ) interferometer configuration, where a polarizing beam splitter (PBS) divided the light into two arms with orthogonal polarization states $\mid 0\rangle$ and $\mid 1\rangle$. The $\mid 0\rangle$-polarized arm was compressed along the y direction using a cylindrical lens (CYL) and passed through a 15-μm slit, while the $\mid 1\rangle$-polarized arm remained unmodified. A second PBS recombined both arms.

A 4f-system (4f-system1) projected the recombined state into the free-space evolution region. The joint photon state at the slit’s image plane is denoted by $\mid \mathrm{\Psi }\rangle =\mid \mathrm{\psi }\left(t_{k-1}\right)\rangle \mid \mathrm{\sigma }\rangle$. The initial time $t_{k-1}$ of propagator $K(x_{p_k},t_k;x_{p_{k-1}},t_{k-1})$ can be adjusted by varying the slit-to-4f-system1 distance, with the time interval set to $\mathrm{ϵ}=t_k-t_{k-1}=15 \text{mm}/c$.

A second 4f-system (4f-system2) relayed the spatial mode at time $t_k$ to an intensified complementary metal-oxide semiconductor (ICMOS) camera, enabling x-basis projection through spatial distribution measurements. Camera pixel positions corresponded to projection locations. Before detection, polarization measurements were performed in four bases: two diagonal bases [$\mid +\rangle =1/\sqrt{2}(\mid 0\rangle +\mid 1\rangle )$ and $\mid -\rangle =1/\sqrt{2}(\mid 0\rangle -\mid 1\rangle )$] and two circular bases [$\mid R\rangle =1/\sqrt{2}(\mid 0\rangle +i\mid 1\rangle )$ and $\mid L\rangle =1/\sqrt{2}(\mid 0\rangle -i\mid 1\rangle )$]. Camera readouts in these bases provided the expectation values.

The propagator was measured in (44) but with a fidelity of only 87.6%, which is insufficient for verifying Feynman’s postulates. We implemented several experimental optimizations: First, as detailed in Materials and Methods, we significantly increased the pointer coupling strength, improving our measurement signal-to-noise ratio. Second, we enhanced the 4f imaging system’s precision. In addition, we improved optical path length stability in the MZ interferometer arms and reduced mechanical vibrations throughout the setup. These optimizations increased single-photon propagator measurement fidelity from 87.6 to 98.5%.

We further estimated the error of the propagator’s measurements by adopting the MAPE, expressed as $R_K=\frac{1}{17}{\displaystyle \sum _{x_{p_k}}}\frac{2\mid K_{\mathrm{e}}(x_{p_k},t_k;x_{p_{k-1}},t_{k-1})-K_{\text{th}}(x_{p_k},t_k;x_{p_{k-1}},t_{k-1})\mid }{\mid K_{\mathrm{e}}(x_{p_k},t_k;x_{p_{k-1}},t_{k-1})\mid +\mid K_{\text{th}}(x_{p_k},t_k;x_{p_{k-1}},t_{k-1})\mid }$, where $K_{\mathrm{e}}(x_{p_k},t_k;x_{p_{k-1}},t_{k-1})$ and $K_{\text{th}}(x_{p_k},t_k;x_{p_{k-1}},t_{k-1})$ are the measured and theoretical propagators, respectively. The mean value of $R_{\mathrm{K}}$ for all $17\times 5=85$ combinations of $x_{{\mathrm{p}}_{k-1}}$ is ${\overline{R}}_{\mathrm{K}}=(8.17\pm 3.50)\%$. In comparison, the corresponding values from prior work (44) were ${\overline{R}}_{\mathrm{K}}=24.3\%$. These results demonstrate a substantial improvement in the measurement accuracy of propagators compared to previous studies.

For a discretized space-time region with N spatial positions and M temporal intervals, the number of possible paths scales as N^M. Our experiments covered $x\in [-48.72,\text{μm},48.72,\text{μm}]$ (N = 17) and time domain $\mathrm{\tau }=5\mathrm{ϵ}=75 \text{mm}/c$ (M = 5). We measured all individual propagators in this region and the spatial distribution $P_{\mathrm{e}}(x_{{\mathrm{p}}_5},t_5)$ for initial space-time point $(x_{\mathrm{i}}=x_0=0,t_{\mathrm{i}}=t_0=0)$, without disturbing photons at intermediate times.

Test of postulate 1

To test postulate 1, we compared three probability distributions: $P_{\mathrm{q}}(x_{\mathrm{f}},t_{\mathrm{f}})={\mid {\sum }_j{\mathrm{\varphi }}_j\mid }^2$ is the quantum prediction derived from Eq. 2, and $P_{\mathrm{c}}(x_{\mathrm{f}},t_{\mathrm{f}})={\sum }_j{\mid {\mathrm{\varphi }}_j\mid }^2$ represents the classical probability sum, where ${\mathrm{\varphi }}_j$ can be derived from Eq. 9 after measuring the propagators; $P_{\mathrm{e}}(x_{\mathrm{f}},t_{\mathrm{f}})$ can be read directly from ICMOS camera. Postulate 1 states that $P_{\mathrm{e}}(x_{\mathrm{f}},t_{\mathrm{f}})$ should match $P_{\mathrm{q}}(x_{\mathrm{f}},t_{\mathrm{f}})$ while differing from $P_c(x_f,t_f)$.

Figure 3 demonstrates excellent agreement between $P_{\mathrm{e}}\left(x_{\mathrm{f}}\right)$ and $P_{\mathrm{q}}\left(x_{\mathrm{f}}\right)$, with $P_{\mathrm{c}}\left(x_{\mathrm{f}}\right)$ showing significant deviation. Each dataset involves 83,521 (17⁴) paths for fixed initial and final space-time points. We computed the MAPE between the experimental distribution $P_{\mathrm{q}}\left(x_{\mathrm{f}}\right)$ and its theoretical prediction $P_{\mathrm{q}}^{\text{th}}\left(x_{\mathrm{f}}\right)$, defined as $R_{\mathrm{q}}=\frac{1}{17}{\sum }_{x_{\mathrm{f}}}\mid \frac{2[P_{\mathrm{q}}\left(x_{\mathrm{f}}\right)-P_{\mathrm{q}}^{\text{th}}\left(x_{\mathrm{f}}\right)]}{P_{\mathrm{q}}\left(x_{\mathrm{f}}\right)+P_{\mathrm{q}}^{\text{th}}\left(x_{\mathrm{f}}\right)}\mid$.The results yield $R_{\mathrm{q}}=4.45\%$, which indicates agreement between the experimentally measured $P_{\mathrm{q}}\left(x_{\mathrm{f}}\right)$ and the theoretical probability distribution $P_{\mathrm{q}}^{\text{th}}\left(x_{\mathrm{f}}\right)$.

Fig. 3. Experimental test of postulate I.

Triangle symbols (Δ): Measured probability distribution $P_{\mathrm{e}}(x_{\mathrm{f}},t_{\mathrm{f}})$ (ICMOS direct measurement); Circles (∘): Quantum prediction $P_{\mathrm{q}}(x_{\mathrm{f}},t_{\mathrm{f}})={\mid {\sum }_j{\mathrm{\varphi }}_j\mid }^2$; Squares ( ): Classical sum $P_{\mathrm{c}}(x_{\mathrm{f}},t_{\mathrm{f}})={\sum }_j{\mid {\mathrm{\varphi }}_j\mid }^2$. Error bars are SE of 10 measurements. The curves are theoretical predictions (solid = $P_{\mathrm{q}}^{\text{th}}$, dashed = $P_{\mathrm{c}}^{\text{th}}$). All probabilities are in arbitrary units (a.u.)

We also quantify the agreement between the experimental data and postulate I with the fidelity between $P_{\mathrm{q}}\left(x_{\mathrm{f}}\right)$ and $P_{\mathrm{q}}^{\text{th}}\left(x_{\mathrm{f}}\right)$. We define the state vector of final position $x_{\mathrm{f}}$ as $\mid \mathrm{\psi }\left(x_{\mathrm{f}}\right)\rangle ={\sum }_{j\in Z_{x_{\mathrm{f}}}}{\mathrm{\varphi }}_j\mid j\rangle$, where $Z_{x_{\mathrm{f}}}$ is the ensemble of paths moving from $x_{\mathrm{i}}$ to $x_{\mathrm{f}}$. The hypothetical state vector (for postulate I) $\mid {\mathrm{\psi }}_{\mathrm{q}}\left(x_{\mathrm{f}}\right)\rangle$ and the theoretical state vector $\mid {\mathrm{\psi }}_{\text{th}}\left(x_{\mathrm{f}}\right)\rangle$ were constructed using the experimentally measured and theoretically predicated amplitudes ${\mathrm{\varphi }}_j$, respectively. Then the fidelity is expressed as ${\mathcal{F}}_{p1}=\frac{1}{17}{\sum }_{x_{\mathrm{f}}}\frac{\mid \langle {\mathrm{\psi }}_{\mathrm{q}}\left(x_{\mathrm{f}}\right){\mid {\mathrm{\psi }}_{\text{th}}\left(x_{\mathrm{f}}\right)\rangle \mid }^2}{\mid \langle {\mathrm{\psi }}_{\mathrm{q}}\left(x_{\mathrm{f}}\right){\mid {\mathrm{\psi }}_{\mathrm{q}}\left(x_{\mathrm{f}}\right)\rangle \mid }^2\cdot \mid \langle {\mathrm{\psi }}_{\text{th}}\left(x_{\mathrm{f}}\right){\mid {\mathrm{\psi }}_{\text{th}}\left(x_{\mathrm{f}}\right)\rangle \mid }^2}$. The results show ${\mathcal{F}}_{p1}=94.4\%$. These results provide a direct experimental illustration of the concepts encapsulated in postulate 1, establishing that quantum superposition of paths requires complex coefficients, in contrast to classical probability summation.

Test of postulate 2

To verify postulate 2, we first categorized all paths by their total length $L=\sum \mid x_k-x_{k-1}\mid$, where L = 0 represents the trivial case with the photon remaining at its initial position throughout evolution. The paths are chosen with fixed initial point (0,0) but variable final points at t_f = 5, yielding 1,419,857 (17⁵) total paths. Figure 4A shows the probability distribution as a function of L, with squares representing mean probabilities for equal-length paths. Shaded regions indicate one SD.

Fig. 4. Test of postulate II.

(A) Path probability vs. length L. Squares ( ) indicate mean probability of paths. (B) Probability versus action S. Circles (∘) indicate mean probability of paths. (C) Circles (∘) indicate phase angle $\text{arg}\left(\mathrm{\varphi }\right)$. The shaded bands in (A) to (C) indicate SDs.

We further analyzed paths by grouping them according to their action. Using the classical action S as our metric (with the classical path’s action as reference zero), we computed each path’s theoretical action $S_{\mathrm{j}}$ via the discrete free-particle Lagrangian $\mathrm{\pi }\hslash {(x_{p_k}-x_{p_{k-1}})}^2/\left(\mathrm{\lambda }c{\mathrm{ϵ}}^2\right)$, summed over all segments as $S_{\mathrm{j}}={\sum }_{k=1}^M{\mathcal{L}}_k\mathrm{ϵ}$. The action range $[0,2\mathrm{\pi \hslash })$ was divided into 100 intervals $\mathrm{\Delta }{S}_{\left(l\right)}=[2\mathrm{\pi \hslash }(l-1)/\mathrm{100,2}\mathrm{\pi \hslash }l/100)$ for $l=1,\dots ,100$. For each interval we calculated the mean path probability ${\overline{P}}_{\left(l\right)}={\langle {\mid {\mathrm{\varphi }}_j\mid }^2\rangle }_{S_j\in \mathrm{\Delta }{S}_{\left(l\right)}}$ and mean phase angle ${\overline{\mathrm{\theta }}}_{\left(l\right)}={\langle \text{arg}\left[{\mathrm{\varphi }}_j\right]\rangle }_{S_j\in \mathrm{\Delta }{S}_{\left(l\right)}}$. Figure 4 (B and C) presents these results, where blue circles in Fig. 4B show the mean probabilities and purple circles in Fig. 4C are the mean phase angles.

We computed the MAPE for all path probabilities, defined as $R_{\mathrm{P}}=\frac{1}{N^M}{\sum }_{j=1}^{N^M}\mid \frac{2(P_j-\overline{P})}{P_j+\overline{P}}\mid$, where N = 17, M = 5, $P_j$ is the probability of the j-th path, and $\overline{P}$ is the mean probability. The results yield $R_{\mathrm{P}}=17.4\%$. To further analyze the error in testing postulate II, we performed a numerical simulation by constructing noisy path probability amplitudes for all N^M paths. The noise level was determined from the measured propagator errors, and the MAPE derived in the simulation was $R_{\mathrm{P}}=18.0\%$. With these conditions, the simulated probabilities were plotted in fig. S3 (Supplementary Materials), which aligns well with the experimental data in Fig. 4.

To quantify the agreement between the experimental data and postulate II, we calculated the fidelity between the theoretically derived state vector and the experimentally measured state vector encompassing all possible paths. We define a state vector composed of all possible paths as $\mid \mathrm{\Phi }\rangle ={\sum }_{j=1}^{N^M}{\mathrm{\varphi }}_j\mid j\rangle$, where ${\mathrm{\varphi }}_j$ represents the amplitude of the j-th path. The hypothetical state vector (for postulate II) $\mid {\mathrm{\Phi }}_{\mathrm{P}2}\rangle$ was constructed using the experimentally measured amplitudes ${\mathrm{\varphi }}_j$, while the theoretical state vector $\mid {\mathrm{\Phi }}_{\mathrm{T}}\rangle$ was generated from the theoretically predicated amplitudes. The fidelity between these two states is given by ${\mathcal{F}}_{\mathrm{p}2}=\frac{{\mid \langle {\mathrm{\Phi }}_{\mathrm{P}2}\mid {\mathrm{\Phi }}_{\mathrm{T}}\rangle \mid }^2}{{\mid \langle {\mathrm{\Phi }}_{\mathrm{P}2}\mid {\mathrm{\Phi }}_{\mathrm{P}2}\rangle \mid }^2\cdot \mid \langle {\mathrm{\Phi }}_{\mathrm{T}}{\mid {\mathrm{\Phi }}_{\mathrm{T}}\rangle \mid }^2}$. The results show ${\mathcal{F}}_{\mathrm{p}2}=94.7\%$, demonstrating agreement with postulate II.

Our results confirm two key predictions of postulate 2: First, path probabilities remain constant across all L and S values. Second, phase angles $\mathrm{\theta }=\text{arg}\left[\mathrm{\varphi }\right]$ scale precisely with $S/\mathrm{\hslash }$. Our work offers a method to test both the equiprobability of paths and the relationship between quantum phase and classical action.

DISCUSSION

We make some remarks here to compare with previous work (44), in which an initial method was demonstrated for measuring propagators and testing the principle of least action (PLA). Our present work yields three key distinguished results: (i) The accuracy of propagator measurements has been significantly improved after overcoming various technical challenges. The methodology itself is significantly important; (ii) we report a direct experimental test of Feynman’s two postulates for path integrals; (iii) our experiment can help resolve a debate regarding path integral theory. We now address these results in more detail as follows.

First, we have significantly improved the accuracy of propagator measurements, which enables high-fidelity reconstruction of path amplitudes at a large scale. When we used the same setup as in (44) to test Feynman’s path integral postulates, we encountered a significant bottleneck: The performance of our original propagator measurement setup was insufficient for this new task. For the PLA demonstration, which involved only the amplitudes of two paths, the scheme was barely sufficient to identify the classical path. However, testing the path integral postulates necessitates the reconstruction of probability amplitudes for millions of possible evolution paths. In such a regime, even marginal errors in individual propagator measurements accumulate multiplicatively, leading to a near-random reconstructed probability distribution and the loss of phase information within the noise. To overcome this, in this paper, we have optimized our experimental framework through four critical advancements addressed in Materials and Methods, and we have successfully enhanced the single-photon propagator fidelity to meet the prerequisite for the current study.

The path integral formulation is an expansive and profound framework in which the propagator serves as the central physical quantity. Consequently, any experimental investigation of path integrals naturally relies on propagator measurement techniques. We can say that the position of the propagator in the path integral formulation is equivalent to the time evolution operator in the Schrödinger picture. Therefore, the propagator is a basic tool widely used in various studies in quantum mechanics. In the present work, we present a more accurate method to improve the measurement of this fundamental tool. It holds the potential to unlock exploration into a broad spectrum of complex quantum phenomena, serving as a useful tool for future advancements in quantum physics.

Second, our work provides a direct experimental test of Feynman’s two postulates. We test both postulates by measuring the probability amplitudes of 1.4 million paths, demonstrating that quantum probabilities emerge from path interference (postulate I) and that path amplitudes are equiprobable with action-dependent phases (postulate II). We might all think that postulate I is a natural idea in quantum theory, but the two points in postulate II—the equal-magnitude postulate for all (millions of) paths and the fundamental relationship between the quantum phase and classical action—are not natural consequences directly derived from the Schrödinger picture. Therefore, illustrating these fundamental ideas in an experiment is valuable.

Third, our work provides experimental evidence to address the important question of whether the paths in the path integral correspond to physical reality or are merely computational tools. Modern techniques such as weak measurement enable us to construct the trajectories of millions of single photons. The observation that each photon’s behavior is consistent with a superposition of all these paths provides strong experimental insight into this debate, supporting the view that the paths are more than mere mathematical artifacts—they reflect a physical aspect of quantum reality.

In summary, we have experimentally tested Feynman’s two foundational postulates of path integrals, the equal probability postulate and the coherent superposition postulate, by measuring the probability amplitudes of more than 1.4 million single photon’s paths in a high-fidelity optical system. These findings provide a direct experiment to visualizing the Feynman’s original ideas of path integrals, bridging a long-standing gap between theoretical insights and empirical evidences. While our experiments provide evidence for the physical reality of all paths, more refined methodologies are required to definitively answer this fundamental question.

Our work opens several promising avenues for future research. First, the proposed propagator-based methodology can be applied to investigate fundamental quantum phenomena, including tests of entangled histories (30, 34, 54, 55) and indefinite causal order (56, 57), from several space-time points to a large number of spacetime points. Second, our experimental framework can be extended to study more complex quantum systems, particularly those with interactions or in curved spacetimes, potentially providing insights into relativistic quantum mechanics and quantum gravity. Third, the achieved propagator measurement precision enables deeper exploration of quantum-to-classical transitions, decoherence mechanisms, and semiclassical approximations. Last, our approach may facilitate previously unexplored quantum simulation techniques that use path integrals to examine condensed matter systems and quantum field phenomena, such as instantons and magnetic monopoles.

MATERIALS AND METHODS

Measurement of probability amplitude

We take the propagator $K(x_{p_k},t_k;x_{p_{k-1}},t_{k-1})$ with $x_{p_k}$ being the p-th point at time $t_k$ as an example to describe our scheme for measuring propagators. We first divide the transverse position into N slits, so the wave function of transverse position at time $t_{k-1}$ can be written as $\mid \mathrm{\psi }\left(t_{k-1}\right)\rangle ={\sum }_{p=1}^N\mathrm{\psi }(x_{p_{k-1}},t_{k-1})\mid x_p\rangle$. We choose the photon’s polarization as the measuring pointer, which is a two-dimensional qubit space with eigenstates $\mid 0\rangle$ and $\mid 1\rangle$. Our initial state $\mid {\mathrm{\Psi }}_{\text{in}}\rangle$ at $t_{k-1}$ is prepared as

$$\mid {\mathrm{\Psi }}_{\text{in}}\rangle =\mid \mathrm{\psi }\left(t_{k-1}\right)\rangle \mid 0\rangle$$ (10)

At this moment, we perform a Hadamard gate on the pointer, yielding

$$\mid \mathrm{\Psi }\rangle =\frac{1}{\sqrt{2}}\mid \mathrm{\psi }\left(t_{k-1}\right)\rangle (\mid 0\rangle +\mid 1\rangle )$$ (11)

We then let the $\mid 0\rangle$ component remain unchanged while performing a Fourier-like operation on the $\mid 0\rangle$ component. This operation can be represented as

$$\hat{U}=\hat{T}(f/c)\otimes \mid 1\rangle \langle 1\mid +{\hat{\mathrm{\pi }}}_p\hat{F}\otimes \mid 0\rangle \langle 0\mid$$ (12)

where ${\hat{\mathrm{\pi }}}_p$ is the projection operator $\mid x_{p_{k-1}}\rangle \langle x_{p_{k-1}}\mid$ for position $x_{p_{k-1}}$ and c is the speed of light. The parameter f is experimentally determined. In an optical system, f represents the focal length of a convex lens. The free evolution operator is $\hat{T}(f/c)=\text{exp}[-i\frac{{\hat{p}}_x^2}{2m}\frac{f}{c\mathrm{\hslash }}]$, and the Fourier transform operator is

$$\hat{F}=\text{exp}[-i\frac{{\hat{p}}_x^2}{2m}\frac{f}{c\mathrm{\hslash }}]\times \int \text{exp}[-i\frac{\mathit{mc}{(x_{p_{k-1}}-x)}^2}{2\mathrm{\hslash }f}]\mid x\rangle \langle x\mid \mathit{dx}$$ (13)

where $m=2\mathrm{\pi \hslash }/\left(\mathrm{\lambda }c\right)$ is the equivalent mass of a photon with wavelength $\mathrm{\lambda }$. After applying $\hat{U}$, the joint state becomes

$$\mid {\mathrm{\Psi }}_{k-1}\rangle =\frac{1}{\sqrt{2}}\hat{T}(f/c)\mid \mathrm{\psi }\left(t_{k-1}\right)\rangle \mid 1\rangle +\frac{1}{\sqrt{2}}\mathrm{\Phi }\left(t_{k-1}\right)\mid x_{p_{k-1}}\rangle \mid 0\rangle$$ (14)

where $\mathrm{\Phi }\left(t_{k-1}\right)={\mathcal{N}}_{\mathrm{F}}\int \mathrm{\psi }(x_{k-1},t_{k-1}){\mathit{dx}}_{k-1}$. Here, ${\mathcal{N}}_{\mathrm{F}}$ is a proportional constant that accounts for the normalization of the optical Fourier transform. Notably, $\mathrm{\Phi }\left(t_{k-1}\right)$ is proportional to the wave function at zero momentum. The momentum wave function with $p_0=0$ at $t_{k-1}$ is expressed as

$$\begin{array}{c}\mathrm{\phi }(p_0,t_{k-1})=\frac{1}{\sqrt{2\mathrm{\pi \hslash }}}\int \mathrm{\psi }(x_{k-1},t_{k-1}){\mathrm{e}}^{{\mathit{ip}}_0{x}_{k-1}/\mathrm{\hslash }}{\mathit{dx}}_{k-1}\\ =\frac{1}{\sqrt{2\mathrm{\pi \hslash }}}\int \mathrm{\psi }(x_{k-1},t_{k-1}){\mathit{dx}}_{k-1}\end{array}$$ (15)

Here, $\mathrm{\Phi }\left(t_{k-1}\right)={\mathcal{N}}_{\mathrm{F}}\sqrt{2\mathrm{\pi \hslash }}\mathrm{\phi }(p_0,t_{k-1})$.

We choose a Gaussian wave as the spatial wave function

$$\mathrm{\psi }(x_{k-1},t_{k-1})=\mathcal{A}\text{exp}[-\frac{x_{k-1}^2}{2(a^2+i\frac{\mathrm{\hslash }{t}_{k-1}}{m})}]$$ (16)

where $\mathcal{A}$ is the normalization factor and a is the waist of the wave packet. When $a\gg \mathrm{\lambda }$, we have $\hat{T}(f/c)\mid \mathrm{\psi }\left(t_{k-1}\right)\rangle \approx \mid \mathrm{\psi }\left(t_{k-1}\right)\rangle$. In our experiment, $x_k\in [-0.0487 \text{mm},0.0487 \mathrm{mm}]$, $\mathrm{\lambda }=795 \text{nm}$, $a=0.57 \text{mm}$, and $f=25 \text{mm}$. Under these conditions, the trace distance between quantum states satisfies $\mathcal{T}=\sqrt{1-\mid \langle \mathrm{\psi }\left(t_{k-1}\right)\mid \hat{T}(f/c){\mid \mathrm{\psi }\left(t_{k-1}\right)\rangle \mid }^2}<3.41\times {10}^{-6}$. This negligible $\mathcal{T}$ indicates $\mid \mathrm{\psi }\left(t_{k-1}\right)\rangle$ remains essentially unchanged after $\hat{T}(f/c)$ evolution. The joint state can thus be approximated as

$$\mid {\mathrm{\Psi }}_{k-1}\rangle =\frac{1}{\sqrt{2}}\mid \mathrm{\psi }\left(t_{k-1}\right)\rangle \mid 1\rangle +\frac{1}{\sqrt{2}}\mathrm{\Phi }\left(t_{k-1}\right)\mid x_{p_{k-1}}\rangle \mid 0\rangle$$ (17)

After applying $\hat{U}$, the system propagates from $t_{k-1}$ to $t_k$ in free space

$$\begin{array}{c}\mid {\mathrm{\Psi }}_k\rangle =\frac{1}{\sqrt{2}}\hat{T}(t_k-t_{k-1})\mid \mathrm{\psi }\left(t_{k-1}\right)\rangle \mid 1\rangle \\ +\frac{1}{\sqrt{2}}\mathrm{\Phi }\left(t_{k-1}\right)\hat{T}(t_k-t_{k-1})\mid x_{p_{k-1}}\rangle \mid 0\rangle \end{array}$$ (18)

At $t_k$, we perform postselection at position $x_{p_k}$, yielding the final pointer state

$$\begin{array}{cc}\hfill \mid f_k\rangle & =\frac{1}{\sqrt{2}}\langle x_k\mid \hat{T}(t_k-t_{k-1})\mid \mathrm{\psi }\left(t_{k-1}\right)\rangle \mid 1\rangle \hfill \\ & +\frac{1}{\sqrt{2}}\mathrm{\Phi }\left(t_{k-1}\right)\langle x_k\mid \hat{T}(t_k-t_{k-1})\mid x_{p_{k-1}}\rangle \mid 0\rangle \hfill \\ & =\frac{1}{\sqrt{2}}\mathrm{\psi }(x_{p_k},t_k)\mid 1\rangle +\frac{1}{\sqrt{2}}\mathrm{\Phi }\left(t_{k-1}\right)K(x_{p_k},t_k;x_{p_{k-1}},t_{k-1})\mid 0\rangle \hfill \end{array}$$ (19)

Therefore, the propagator $K(x_b,t_b;x_a,t_a)$ is effectively mapped onto the pointer, allowing for its extraction via a direct measurement approach (39). Specifically, Calculating the expectation values $\langle f_k\mid {\hat{\mathrm{\sigma }}}_x\mid f_k\rangle$ and $\langle f_k\mid {\hat{\mathrm{\sigma }}}_y\mid f_k\rangle$ in Eq. 19 yields the real and imaginary components of the propagator, respectively

$$\begin{array}{c}{\mathrm{\psi }}^{\ast }(x_{p_k},t_k)\mathrm{\Phi }\left(t_{k-1}\right)K(x_{p_k},t_k;x_{p_{k-1}},t_{k-1})\\ =[\langle f_k\mid {\hat{\mathrm{\sigma }}}_x\mid f_k\rangle +i\langle f_k\mid {\hat{\mathrm{\sigma }}}_y\mid f_k\rangle ]\end{array}$$ (20)

We define the nonreduced propagator as

$$\begin{array}{c}{K}^{\prime }(x_{p_k},t_k;x_{p_{k-1}},t_{k-1})\\ ={\mathrm{\psi }}^{\ast }(x_{p_k},t_k)\mathrm{\Phi }\left(t_{k-1}\right)K(x_{p_k},t_k;x_{p_{k-1}},t_{k-1})\end{array}$$ (21)

Then, the path probability amplitude is given by

$${\mathrm{\varphi }}_j=\frac{{\displaystyle \prod _{k=1}^M}{K}^{\prime }(x_{p_k},t_k;x_{p_{k-1}},t_{k-1})}{{\displaystyle \prod _{k=1}^M}{\mathrm{\psi }}^{\ast }(x_{p_k},t_k)\mathrm{\Phi }\left(t_{k-1}\right)}$$ (22)

Since $\mathrm{\Phi }\left(t_{k-1}\right)$ is spatially independent and the time interval $\mathrm{ϵ}$ is uniform, their product can be normalized as a global factor $\mathcal{N}={\displaystyle \prod _{k=1}^M}\mathrm{\Phi }\left(t_{k-1}\right)$. Generally, obtaining ${\mathrm{\varphi }}_j$ requires measuring the wave function $\mathrm{\psi }(x_{p_k},t_k)$. However, measuring the wave function inevitably introduces measurement errors. The product of wave functions can be expressed as ${\displaystyle \prod _{k=1}^M}{\mathrm{\psi }}^{\ast }(x_{p_k},t_k)={\displaystyle \prod _{k=1}^M}{\mathrm{e}}^{i{\mathrm{\phi }}_k}\mid {\mathrm{\psi }}^{\ast }(x_{p_k},t_k)\mid$. In our experiment, the Gaussian beam with large waist results in a nearly collimated beam, implying a uniform phase distribution of $\mathrm{\psi }(x_{p_k},t_k)$ on the x axis. Under this condition, we have an approximation of ${\displaystyle \prod _{k=1}^M}{\mathrm{\psi }}^{\ast }(x_{p_k},t_k)\approx {\mathrm{e}}^{\mathrm{i\gamma }}{\displaystyle \prod _{k=1}^M}\mid {\mathrm{\psi }}^{\ast }(x_{p_k},t_k)\mid$, where ${\mathrm{e}}^{\mathrm{i\gamma }}$ is the global phase factor accumulated from the individual wave function phases. Since ${\mid {\mathrm{\psi }}^{\ast }(x_{p_k},t_k)\mid }^2$ is the probability distribution of the system at $t_k$, $\mid {\mathrm{\psi }}^{\ast }(x_{p_k},t_k)\mid$ can be directly obtained from the ICMOS. We evaluated the accuracy of this approximation within our experimental conditions. The SD of $\mathrm{\gamma }$ across all $\mathrm{\varphi }$ is $<1.63\times {10}^{-4}\text{rad}$. Typically, this approximation error is significantly smaller than the measurement error of the wave function. Therefore, using a quasi-collimated Gaussian beam with direct measurement of $\mid {\mathrm{\psi }}^{\ast }(x_{p_k},t_k)\mid$ proves significantly more accurate than wave function measurement approaches. The propagator is then obtained by

$$K(x_{p_k},t_k;x_{p_{k-1}},t_{k-1})=\frac{\langle f_k\mid {\hat{\mathrm{\sigma }}}_x\mid f_k\rangle +i\langle f_k\mid {\hat{\mathrm{\sigma }}}_y\mid f_k\rangle }{\mathrm{\Phi }\left(t_{k-1}\right)\mid {\mathrm{\psi }}^{\ast }(x_{p_k},t_k)\mid }$$ (23)

The path probability amplitude is given by

$${\mathrm{\varphi }}_j=\frac{{\displaystyle \prod _{k=1}^M}[\langle f_k\mid {\hat{\mathrm{\sigma }}}_x\mid f_k\rangle +i\langle f_k\mid {\hat{\mathrm{\sigma }}}_y\mid f_k\rangle ]}{\mathcal{N}{\displaystyle \prod _{k=1}^M}\mid {\mathrm{\psi }}^{\ast }(x_{p_k},t_k)\mid }$$ (24)

Improvements in the propagator measurement scheme

The propagator measurement scheme presented in this work builds upon the method established in (44), incorporating several key improvements. The primary enhancements manifest in the pointer coupling stage. In the scheme from (44), the pointer-system coupling is expressed as ${\hat{U}}^{\prime }={\mathrm{e}}^{-\mathrm{i}\frac{\mathrm{\pi }}{2}{\hat{\mathrm{\pi }}}_p{\hat{\mathrm{\sigma }}}_y}$. The final pointer state in that scheme becomes

$$\begin{array}{cc}\hfill \mid f^{\prime }\rangle & =\langle x_{p_k}\mid \hat{T}(t_k-t_{k-1}){\hat{U}}^{\prime }\mid {\mathrm{\Psi }}_{k-1}\rangle \hfill \\ & =\frac{1}{2}\mathrm{\psi }(x_{p_k},t_k)\mid 0\rangle \hfill \\ & -\frac{1}{\sqrt{2}}\mathrm{\psi }(x_{p_{k-1}},t_{k-1})K(x_{p_k},t;x_{p_{k-1}},t_{\mathrm{k}-1})\mid 1\rangle \hfill \end{array}$$ (25)

The nonreduced propagator is obtained through

$$\begin{array}{c}\frac{1}{\sqrt{2}}{\mathrm{\psi }}^{\ast }(x_{p_k},t_{k-1})\mathrm{\psi }(x_{p_{k-1}},t_{k-1})K(x_{p_k},t_k;x_{p_{k-1}},t_{k-1})\\ =\langle f^{\prime }\mid {\hat{\mathrm{\sigma }}}_x\mid f^{\prime }\rangle +i\langle f^{\prime }\mid {\hat{\mathrm{\sigma }}}_y\mid f^{\prime }\rangle \end{array}$$ (26)

The right-hand side represents experimental measurements. We define the amplification factor ${\mathcal{M}}^{\prime }=\frac{1}{\sqrt{2}}\mid {\mathrm{\psi }}^{\ast }(x_{p_k},t_{k-1})\mathrm{\psi }(x_{p_{k-1}},t_{k-1})\mid$, which quantifies detectable photon energy contribution. This factor directly determines signal intensity, with higher ${\mathcal{M}}^{\prime }$ values enhancing measurement sensitivity. Typically, $\mid {\mathrm{\psi }}^{\ast }(x_{p_k},t_{k-1})\mid$ and $\mid \mathrm{\psi }(x_{p_{k-1}},t_{k-1})\mid$ are small, yielding weak ${\mathcal{M}}^{\prime }$. Our improved scheme achieves $\mathcal{M}=\mid {\mathrm{\psi }}^{\ast }(x_{p_k},t_{k-1})\mathrm{\Phi }\left(t_{k-1}\right)\mid$.

Using Eq. 16 with experimental parameters $\mathrm{\omega }=0.57 \text{mm}$, $x_{k-1}\in [-48.72 \text{μm},48.72 \text{μm}]$, $t_k-t_{k-1}=15 \text{mm}/c$, and $\mathrm{\delta }x=5.73 \text{μm}$, we calculate ${\mathcal{M}}^{\prime }=(3.00\pm 0.0026)\times {10}^{-3}$. Momentum-space analysis with resolution $\mathrm{\delta }p=0.9061 \mathrm{\hslash }\cdot {\text{mm}}^{-1}$ (determined via $\mathrm{\delta }p=\mathrm{\pi }{d}_{\mathrm{s}}/\left(f_{\mathrm{c}}\mathrm{\lambda }\right)$ for slit width $d_{\mathrm{s}}=15 \text{μm}$ and CYL $f_{\mathrm{c}}=25 \text{mm}/c$) yields $\mathcal{M}=(3.95\pm 0.0024)\times {10}^{-2}$. This demonstrates signal amplification by $\mathrm{\alpha }=\mathcal{M}/{\mathcal{M}}^{\prime }>13$ in the enhanced scheme.

We denote $K_{\text{th}}(x_{p_k},t_k;x_{p_{k-1}},t_{k-1})$ as the ideal propagator derived from analytical expressions and $K_{\mathrm{e}}(x_{p_k},t_k;x_{p_{k-1}},t_{k-1})$ as the experimentally measured propagators. The fidelity of propagator measurement is quantified by

$$\mathcal{F}=\frac{{\mid {\displaystyle \underset{-\mathrm{\Delta }x}{\overset{\mathrm{\Delta }x}{\int }}}{K}_{\text{th}}^{\ast }(x_{p_k},t_k;x_{p_{k-1}},t_{k-1})K_{\mathrm{e}}(x_{p_k},t_k;x_{p_{k-1}},t_{k-1}){\mathit{dx}}_{p_k}\mid }^2}{{\displaystyle \underset{-\mathrm{\Delta }x}{\overset{\mathrm{\Delta }x}{\int }}}{\mid K_{\text{th}}(x_{p_k},t_k;x_{p_{k-1}},t_{k-1})\mid }^2{\mathit{dx}}_{p_k}\cdot {\displaystyle \underset{-\mathrm{\Delta }x}{\overset{\mathrm{\Delta }x}{\int }}}{\mid K_{\mathrm{e}}(x_{p_k},t_k;x_{p_{k-1}},t_{k-1})\mid }^2{\mathit{dx}}_{p_k}}$$ (27)

where $\mathrm{\Delta }x=48.72 \text{μm}$ corresponds to the experimental measurement range.

Besides increasing the pointer coupling strength, we have optimized our experimental framework through the following critical advancements: (i) Customized imaging system: A dedicated high-precision imaging system was designed to enhance the projection measurement accuracy of the photon’s spatial modes. (ii) Real-time normalization: By implementing a single-photon reference beam, we mitigated errors caused by fluctuations in photon generation rates, ensuring rigorous normalization for every measurement. (iii) Nanoscale mechanical stability: We achieved nanometer-level precision (far below the photon wavelength) in both the displacement accuracy and mechanical stability of the photon initial position scanning. This ensures the consistency required for million-scale photon path measurements. The synergy of these optimizations increased the single-photon propagator measurement fidelity from 87.6% in (44) to 98.5% in the present work. As also addressed in the main text, we achieved MAPE values of ${\overline{R}}_{\mathrm{K}}=(8.17\pm 3.50)\%$ in the present work. In comparison, the corresponding values from prior work (44) were ${\overline{R}}_{\mathrm{K}}=24.3\%$. These results demonstrate a significant improvement in propagator measurement accuracy compared to previous studies.

Supplementary Materials

This PDF file includes:

Supplementary Text

Figs. S1 to S6

References