## 1. Introduction

- introduction and operator theory modeling of two novel quantum resources, i.e., QPE and QPI, denoting temporal correlations and the interference among quantum trajectories, respectively, in MPD while utilizing the tensor product structure for future quantum computing and communication architectures and foundational QM studies;
- theoretical modeling and numerical analysis of MPD setup for the violation of LGI, with the ambiguous and no-signaling forms recently proposed by Emary in Reference [16], reaching $>1.2$ of correlation amplitude numerically obtained for three-time formulation while leaving the maximization of the violation to the boundary levels as an open issue;
- a novel setup, i.e., MPD, violating the ambiguous form of LGI with classical light sources complementing the recent experiment utilizing linear polarization degree of freedom of the classical light [19] while MPD setup with remarkably low complexity design utilizing classical light sources and photon-counting intensity detection;
- theoretical modeling and numerical analysis of counterintuitive properties and examples of the interference among MPD-based Feynman paths denoted as QPI promising to be easily verified experimentally in future studies;
- the modeling and numerical analysis of the coherence properties of the light sources in terms of spatial and temporal dimensions while discussing design issues for MPD setup with coherent light sources; and
- discussion for future applications of QPE and QPI as quantum resources and experimental implementations.

## 2. Results

#### 2.1. MPD Setup for Quantum Temporal Correlations

#### 2.2. Diffractive Projection and Measurement Operators

#### 2.3. History State Modeling of QPE

#### Event Probabilities

#### 2.4. Modeling of the Violation of LGI in MPD

#### 2.5. Modeling of QPI

#### 2.6. Numerical Results

#### 2.6.1. Violation of LGI

#### 2.6.2. QPI Analysis

## 3. Discussion and Conclusions

## 4. Methods

#### 4.1. Parameters for FPI Modeling of the Violation of LGI

#### 4.2. Temporal and Spatial Coherence of the Light Sources

## Funding

## Conflicts of Interest

## Abbreviations

MPD | Multiplane diffraction |

QC | Quantum computing |

QM | Quantum mechanical |

QPE | Quantum path entanglement |

QPI | Quantum path interference |

FPI | Feynman’s path integral |

LGI | Leggett-Garg Inequality |

MR | Macroscopic realism |

NIM | Non-invasive measurability |

SIT | Signaling-in-time |

GHZ | Greenberger-Horne-Zeilinger |

FWHM | Full width half maximum |

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**Figure 1.**(

**a**) System model of the free propagating light with velocity c in the z-direction and MPD through N planes, where jth plane includes ${S}_{j}$ slits at positions ${X}_{j,i}$ for $i\in [1,{S}_{j}]$ and interplane distance of ${L}_{j,j+1}$. (

**b**) Example of three plane diffractions ($N=3$) with two slits for the first and second planes showing all the possible seven types of histories composed of diffractions or projections ${P}_{1,1}$, ${P}_{1,2}$, ${P}_{2,1}$, and ${P}_{2,2}$ through slits and measurements ${M}_{1}$, ${M}_{2}$, and ${M}_{3}$ on the planes. There are ${N}_{p}\equiv {\prod}_{j=1}^{N-1}{S}_{j}=2\times 2=4$ paths detected on the third plane.

**Figure 2.**(

**a**) The violation of Leggett–Garg Inequality (LGI) with the setup of two planes with triple slits where the event set at time ${t}_{1}$ is $\left[{P}_{1,1}\right]$, $\left[{P}_{1,2}\right]$, and $\left[{P}_{1,3}\right]$ and, at time ${t}_{2}$, are $\left[{P}_{2,1}\right]$, $\left[{P}_{2,1}\right]$, $\left[{P}_{2,3}\right]$, and $\left[{M}_{2}\right]$ and ambiguous measurement setups by closing (

**b**) the third, (

**c**) the second, and (

**d**) the first slits on the first plane.

**Figure 3.**Setup for constructive and destructive interferences in time for the probabilities to diffract through each plane showing the history states (

**a**) $|{\Psi}_{3}^{a})\equiv \left[{\mathbf{P}}_{3,1}\right]\odot \left[{\mathbf{P}}_{2,1}\right]\odot \left(\left[{\mathbf{P}}_{1,1}\right]+\left[{\mathbf{P}}_{1,2}\right]\right)\odot \left[{\rho}_{0}\right]$ as the superposition of $|{\Psi}_{3}^{b})$ and $|{\Psi}_{3}^{c})$, (

**b**) $|{\Psi}_{3}^{b})\equiv \left[{\mathbf{P}}_{3,1}\right]\odot \left[{\mathbf{P}}_{2,1}\right]\odot \left[{\mathbf{P}}_{1,1}\right]\odot \left[{\rho}_{0}\right]$, and (

**c**) $|{\Psi}_{3}^{c})\equiv \left[{\mathbf{P}}_{3,1}\right]\odot \left[{\mathbf{P}}_{2,1}\right]\odot \left[{\mathbf{P}}_{1,2}\right]\odot \left[{\rho}_{0}\right]$. The targeted scenario with classically counterintuitive nature where a specific example of interference pattern (represented as the number of lambs denoting the number of photons for a practical counting experiment) for the cases of (

**d**) two slits on PL-1 both open and (

**e**) only the second slit open. The operation of closing the first slit decreases the number of photons diffracted through PL-2 while counterintuitively increases the number of photons through PL-3 since we classically expect a decrease. This scenario shows the interference of histories at two different time instants for PL-2 and PL-3 with firstly constructive and then destructive effects, respectively.

**Figure 4.**The layouts used in (

**a**) $Si{m}_{1}$ and (

**b**) $Si{m}_{2}$, where for $Si{m}_{2}$, the fixed values of the parameters are ${\sigma}_{0}=200$ ($\mathsf{\mu}$m), ${t}_{01}=0.5$ (ns), ${t}_{12}=0.2$ (ns), ${t}_{23}=0.1$ (ns), ${\beta}_{1}=25$ ($\mathsf{\mu}$m), ${\beta}_{2}=35$ ($\mathsf{\mu}$m), and ${\beta}_{3}=45$ ($\mathsf{\mu}$m) in addition to the fixed values of the slit positions on the first plane. The practical measurement setups to be utilized in future experiments are illustrated for the probabilities (

**c**) ${p}_{1}(\{1,2\})$ and (

**d**) ${p}_{1,2}(\{1,3\},2)$. The measurement planes count the detected number of photons compared with the number of photons emitted by the source in unit time.

**Figure 5.**(

**a**) LGI violation (${K}_{A}\phantom{\rule{0.166667em}{0ex}}-\phantom{\rule{0.166667em}{0ex}}{K}_{V}$) and signaling (${K}_{V}\phantom{\rule{0.166667em}{0ex}}-1\phantom{\rule{0.166667em}{0ex}}$) for varying ${D}_{s}$, where ${t}_{01}=0.2$ (ns), ${t}_{12}=0.1$ (ns), $\Delta x=7$, ${\beta}_{1}=15$ ($\mathsf{\mu}$m), ${\beta}_{2}=30$ ($\mathsf{\mu}$m), and ${\sigma}_{0}=130$ ($\mathsf{\mu}$m) and (

**b**) the corresponding dichotomic sign assignments for ambiguous measurements maximizing the violation for each ${D}_{s}$.

**Figure 6.**(

**a**) Maximum LGI violation (${K}_{A}-{K}_{V}$) and the corresponding amount of signaling (${K}_{V}-1$) for varying ${\sigma}_{0}$, $\Delta x$, and ${t}_{01}={t}_{12}$ and (

**b**) the corresponding values of ${\beta}_{1}$, ${\beta}_{2}$, and ${D}_{s}$ maximizing the violation for each ${\sigma}_{0}$ assuming fully coherent sources. Maximum violation for varying $({\beta}_{1},{\beta}_{2})$ pairs for fully coherent sources where (

**c**) $\Delta x=7$ and ${t}_{01}={t}_{12}=0.1$ (ns) at the maximizing ${\sigma}_{0}=30$ ($\mathsf{\mu}$m), (

**d**) $\Delta x=7$ and ${t}_{01}={t}_{12}=0.2$ (ns) at ${\sigma}_{0}=230$ ($\mathsf{\mu}$m), and (

**e**) $\Delta x=11$ and ${t}_{01}={t}_{12}=0.1$ (ns) at ${\sigma}_{0}=150$ ($\mathsf{\mu}$m). It is observed that there is a large set of slit pairs and beam width resulting in LGI violation reaching $\approx 0.4$ for $\Delta x=7$ and $\approx 0.23$ for $\Delta x=11$, respectively, while there are local peaks for $({\beta}_{1},{\beta}_{2})$ pairs for all cases. Increasing ${t}_{01},{t}_{12}$ values expands the $({\beta}_{1},{\beta}_{2})$ pairs for similar values of violations. (

**f**) The comparison of the spatial coherence diameters ${D}_{c}$ with the diffraction setup diameters ${D}_{1}$ and ${D}_{2}$ for the first and second planes, respectively, where the targeted case is $\Delta x=11$ and ${t}_{01}={t}_{12}=0.1$ (ns), i.e., analyzed as the red curve in Figure 6a, and (

**g**) the corresponding LGI violation curve plotted again by emphasizing the coherence including the peak points.

**Figure 7.**(

**a**) $|{\psi}_{2,0}({x}_{2})\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}{\psi}_{2,1}({x}_{2}){|}^{2}$ compared with $|{\psi}_{2,1}({x}_{2}){|}^{2}$ and $|{\psi}_{2,0}({x}_{2}){|}^{2}$ for diffraction through the layer PL-2, (

**b**) $\underset{{x}_{3}}{max}\left\{|{\psi}_{3,0}({x}_{3})\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}{\psi}_{3,1}({x}_{3}){|}^{2}-{\left|{\psi}_{3,1}({x}_{3})\right|}^{2}\right\}$ for varying ${X}_{2,1}$ on PL-3 such that destructive interference is maximized for each ${X}_{2,1}$ with respect to ${x}_{3}$ while ${X}_{2,1}\approx 140\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}$m maximizes the destructive interference, (

**c**) ${X}_{3,1}$ maximizing the destructive interference for varying ${X}_{2,1}$, (

**d**) the comparison of $|{\psi}_{3,0}({x}_{3})\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}{\psi}_{3,1}({x}_{3}){|}^{2}$ and $|{\psi}_{3,1}({x}_{3}){|}^{2}$ on PL-3 for specific ${X}_{2,1}\approx 140\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}$m showing the destructive interference maximized with ${X}_{3,1}\approx 143\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}$m, and (

**e**) the marked regions satisfy the counterintuitive scenario in (53)–(55) for varying ${X}_{2,1}$ with the corresponding ${X}_{3,1}$ pair in Figure 7c. Constructive and destructive interferences are observed for diffraction through PL-2 and PL-3, respectively, with different kinds of correlation of the paths at different times as a proof-of-concept numerical simulation of quantum path interference (QPI) in time between the two paths. (

**f**) The comparison of setup diameters on the second and third planes, i.e., ${D}_{2}$ and ${D}_{3}$, respectively, with the spatial coherence diameters ${D}_{c}({t}_{12},{\beta}_{1})$ and ${D}_{c}({t}_{23},{\beta}_{2})$, respectively, in the targeted range of ${X}_{2,1}\in [140,170]$ ($\mathsf{\mu}$m) in Figure 7e.

**Figure 8.**(

**a**) The conventional modeling for the spatial coherence of light sources based on double-slit diffraction [43], where $\Delta \theta \phantom{\rule{0.166667em}{0ex}}\Delta s\le \lambda $ is required for the fringes to be observed determining the spatial coherence diameter (${D}_{c}$); (

**b**) free-space propagation of Gaussian beam, where ${D}_{c}$ is approximated as the $1/{e}^{2}$ intensity beamwidth of $2\phantom{\rule{0.166667em}{0ex}}\sqrt{2}\phantom{\rule{0.166667em}{0ex}}{\sigma}_{0}$ with the standard deviation of ${\sigma}_{D}$. The descriptions of the calculation of the setup diameters on the planes to include the slits are denoted by ${D}_{j}$ for $j\in [1,3]$ with respect to the location and the standard deviation of the source on the previous plane (${\sigma}_{0}$ for the first plane and ${\beta}_{j-1}$ for the jth plane) for (

**c**) LGI violation numerical analysis $Si{m}_{1}$ with two planes of triple slits on each plane and (

**d**) interference in time scenario $Si{m}_{2}$ with three planes.

ID | Property | Value | ID | Property | Value |
---|---|---|---|---|---|

$Si{m}_{1}$ | ${\overrightarrow{X}}_{1}^{T}$ | ${D}_{s}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}\left[-\Delta x\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}0\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\Delta x\right]\times {\beta}_{1}$ | $Si{m}_{2}$ | ${\overrightarrow{X}}_{1}^{T}$ | $\left[-4\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}4\right]\times {\beta}_{1}$ |

${\overrightarrow{X}}_{2}^{T}$ | $\left[-\Delta x\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}0\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\Delta x\right]\times {\beta}_{2}$ | ${X}_{2,1}$ ($\mathsf{\mu}$m) | $\left[0,\phantom{\rule{0.166667em}{0ex}}500\right]$ | ||

$\Delta x$; ${D}_{s}$ | $\{7,11\}$; $\left[0,3000\right]$ ($\mathsf{\mu}$m) | ${X}_{3,1}$ ($\mathsf{\mu}$m) | $\left[-600,\phantom{\rule{0.166667em}{0ex}}800\right]$ | ||

${t}_{01}={t}_{12}$ (ns) | $\{0.1,\phantom{\rule{0.166667em}{0ex}}0.2\}$ | ${t}_{01},{t}_{12},{t}_{23}$ (ns) | $0.5$, $0.2$, $0.1$ | ||

${\beta}_{1}$, ${\beta}_{2}$ ($\mathsf{\mu}$m) | $\left[1,\phantom{\rule{0.166667em}{0ex}}50\right]$, $\left[1,\phantom{\rule{0.166667em}{0ex}}100\right]$ | ${\beta}_{1}$, ${\beta}_{2}$, ${\beta}_{3}$ ($\mathsf{\mu}$m) | 25, 35, 45 | ||

${\sigma}_{0}$ ($\mathsf{\mu}$m) | $\left[10,\phantom{\rule{0.166667em}{0ex}}800\right]$ | ${\sigma}_{0}$ ($\mathsf{\mu}$m) | 200 |

**Table 2.**Parameters for modeling LGI and path integrals $({\psi}_{2,i}({x}_{2})$ for $i\in [0,2]:$ three paths).

Formula | Formula | Formula | |||
---|---|---|---|---|---|

${k}_{1}$ | $\frac{-\hslash \phantom{\rule{0.166667em}{0ex}}m\phantom{\rule{0.166667em}{0ex}}{t}_{12}({a}_{t,\sigma}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}{\hslash}^{2}\phantom{\rule{0.166667em}{0ex}}{t}_{01}\phantom{\rule{0.166667em}{0ex}}{t}_{12})}{2\phantom{\rule{0.166667em}{0ex}}{k}_{11}}$ | ${k}_{8}$ | $-\frac{1}{4}\phantom{\rule{0.166667em}{0ex}}\left(\frac{1}{{\beta}_{1}^{2}}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}\frac{1}{{\beta}_{1}^{2}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}{d}_{t,\sigma}}\right)$ | ${\xi}_{1}$ | $\frac{{\beta}_{1}^{2}\phantom{\rule{0.166667em}{0ex}}m\phantom{\rule{0.166667em}{0ex}}{\vartheta}_{t}}{\left({\beta}_{1}^{2}\phantom{\rule{0.166667em}{0ex}}m\phantom{\rule{0.166667em}{0ex}}(m\phantom{\rule{0.166667em}{0ex}}{\sigma}_{0}^{2}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}\u0131\phantom{\rule{0.166667em}{0ex}}\sqrt{{b}_{t}})\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}\u0131\phantom{\rule{0.166667em}{0ex}}\hslash \phantom{\rule{0.166667em}{0ex}}{t}_{1,2}\phantom{\rule{0.166667em}{0ex}}{\vartheta}_{t}\right)}$ |

${k}_{2}$ | $\hslash \phantom{\rule{0.166667em}{0ex}}{m}^{3}\phantom{\rule{0.166667em}{0ex}}{\sigma}_{0}^{2}{t}_{12}\left({\beta}_{1}^{2}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}{d}_{t,\sigma}\right)\phantom{\rule{0.166667em}{0ex}}/\phantom{\rule{0.166667em}{0ex}}{k}_{11}$ | ${k}_{9}$ | $\frac{1}{2}\phantom{\rule{0.166667em}{0ex}}\left(\frac{1}{{\beta}_{1}^{2}}\phantom{\rule{0.166667em}{0ex}}-\phantom{\rule{0.166667em}{0ex}}\frac{1}{{\beta}_{1}^{2}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}{d}_{t,\sigma}}\right)$ | ${A}_{1}$ | $\frac{-{\beta}_{1}^{2}\phantom{\rule{0.166667em}{0ex}}{m}^{2}\left({\hslash}^{2}\phantom{\rule{0.166667em}{0ex}}{t}_{0,1}^{2}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}{m}^{2}\phantom{\rule{0.166667em}{0ex}}{\sigma}_{0}^{2}\phantom{\rule{0.166667em}{0ex}}{\Xi}_{1}\right)}{(2\phantom{\rule{0.166667em}{0ex}}{\alpha}_{t,\sigma ,\beta})}$ |

${k}_{3}$ | $\left(-{\beta}_{1}^{2}{m}^{2}({a}_{t,\sigma}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}{\hslash}^{2}\phantom{\rule{0.166667em}{0ex}}{t}_{01}\phantom{\rule{0.166667em}{0ex}}{t}_{12})\right)\phantom{\rule{0.166667em}{0ex}}/\phantom{\rule{0.166667em}{0ex}}{k}_{11}$ | ${k}_{10}$ | $-1\phantom{\rule{0.166667em}{0ex}}/\phantom{\rule{0.166667em}{0ex}}({\beta}_{1}^{2}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}{d}_{t,\sigma})$ | ${B}_{1}$ | $\frac{\left({\beta}_{1}^{4}\phantom{\rule{0.166667em}{0ex}}{m}^{3}\phantom{\rule{0.166667em}{0ex}}\hslash \phantom{\rule{0.166667em}{0ex}}{t}_{0,1}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}m\phantom{\rule{0.166667em}{0ex}}\hslash \phantom{\rule{0.166667em}{0ex}}{t}_{1,2}({\hslash}^{2}\phantom{\rule{0.166667em}{0ex}}{t}_{0,1}^{2}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}{m}^{2}\phantom{\rule{0.166667em}{0ex}}{\Xi}_{1}^{2})\right)}{(2\phantom{\rule{0.166667em}{0ex}}{\alpha}_{t,\sigma ,\beta})}$ |

${k}_{4}$ | ${\beta}_{1}^{2}\phantom{\rule{0.166667em}{0ex}}{m}^{4}\phantom{\rule{0.166667em}{0ex}}{\sigma}_{0}^{2}({\beta}_{1}^{2}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}{d}_{t,\sigma})\phantom{\rule{0.166667em}{0ex}}/\phantom{\rule{0.166667em}{0ex}}{k}_{11}$ | ${k}_{11}$ | $\begin{array}{l}{\beta}_{1}^{4}\phantom{\rule{0.166667em}{0ex}}{m}^{2}\phantom{\rule{0.166667em}{0ex}}\left({m}^{2}\phantom{\rule{0.166667em}{0ex}}{\sigma}_{0}^{2}\phantom{\rule{0.166667em}{0ex}}{\Xi}_{2}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}{b}_{t}\phantom{\rule{0.166667em}{0ex}}\right)\\ +\phantom{\rule{0.166667em}{0ex}}{\beta}_{1}^{2}{m}^{2}\left({\beta}_{2}^{2}\phantom{\rule{0.166667em}{0ex}}{a}_{t,\sigma}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}2\phantom{\rule{0.166667em}{0ex}}{c}_{t,\sigma}\right)\\ +\phantom{\rule{0.166667em}{0ex}}{\hslash}^{2}\phantom{\rule{0.166667em}{0ex}}{t}_{12}^{2}\phantom{\rule{0.166667em}{0ex}}{a}_{t,\sigma}\end{array}$ | ${\mathbf{H}}_{R,1}$ | $\frac{-{m}^{2}\left({\beta}_{1}^{2}({b}_{t}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}{m}^{2}{\sigma}_{0}^{4})\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}{c}_{t,\sigma}\right)}{(2\phantom{\rule{0.166667em}{0ex}}{\alpha}_{t,\sigma ,\beta})}$ |

${k}_{5}$ | $\frac{-{m}^{2}\left({\beta}_{1}^{2}({m}^{2}\phantom{\rule{0.166667em}{0ex}}{\sigma}_{0}^{2}\phantom{\rule{0.166667em}{0ex}}{\Xi}_{2}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}{b}_{t})\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}{c}_{t,\sigma}\right)}{{k}_{11}}$ | $\begin{array}{l}{a}_{t,\sigma}\\ {b}_{t}\\ {c}_{t,\sigma}\end{array}$ | $\begin{array}{l}{\hslash}^{2}\phantom{\rule{0.166667em}{0ex}}{t}_{01}^{2}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}{m}^{2}{\sigma}_{0}^{4}\\ {\hslash}^{2}\phantom{\rule{0.166667em}{0ex}}{({t}_{01}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}{t}_{12})}^{2}\\ {\hslash}^{2}\phantom{\rule{0.166667em}{0ex}}{\sigma}_{0}^{2}\phantom{\rule{0.166667em}{0ex}}{t}_{12}^{2}\end{array}$ | ${\mathbf{H}}_{I,1}$ | $\frac{m\phantom{\rule{0.166667em}{0ex}}\hslash \phantom{\rule{0.166667em}{0ex}}{t}_{1,2}\left({a}_{t,\sigma}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}{\hslash}^{2}\phantom{\rule{0.166667em}{0ex}}{t}_{0,1}\phantom{\rule{0.166667em}{0ex}}{t}_{1,2}\right)}{(2\phantom{\rule{0.166667em}{0ex}}{\alpha}_{t,\sigma ,\beta})}$ |

${k}_{6}$ | $\begin{array}{l}-{m}^{2}\left(2\phantom{\rule{0.166667em}{0ex}}{\beta}_{1}^{2}\phantom{\rule{0.166667em}{0ex}}({m}^{2}\phantom{\rule{0.166667em}{0ex}}{\sigma}_{0}^{2}\phantom{\rule{0.166667em}{0ex}}{\Xi}_{2}\phantom{\rule{0.166667em}{0ex}}+{b}_{t})\right)\phantom{\rule{0.166667em}{0ex}}/\phantom{\rule{0.166667em}{0ex}}(4\phantom{\rule{0.166667em}{0ex}}{k}_{11})\\ +\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{m}^{2}(-{\beta}_{2}^{2}\phantom{\rule{0.166667em}{0ex}}{a}_{t,\sigma}\phantom{\rule{0.166667em}{0ex}}-\phantom{\rule{0.166667em}{0ex}}2\phantom{\rule{0.166667em}{0ex}}{c}_{t,\sigma})\phantom{\rule{0.166667em}{0ex}}/\phantom{\rule{0.166667em}{0ex}}(4\phantom{\rule{0.166667em}{0ex}}{k}_{11})\end{array}$ | ${\alpha}_{t,\sigma ,\beta}$ | $\begin{array}{l}{\beta}_{1}^{4}{m}^{2}\left({b}_{t}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}{m}^{2}{\sigma}_{0}^{4}\right)\phantom{\rule{0.166667em}{0ex}}\\ +\phantom{\rule{0.166667em}{0ex}}2\phantom{\rule{0.166667em}{0ex}}{\beta}_{1}^{2}\phantom{\rule{0.166667em}{0ex}}{m}^{2}\phantom{\rule{0.166667em}{0ex}}{c}_{t,\sigma}\\ +\phantom{\rule{0.166667em}{0ex}}{\hslash}^{2}\phantom{\rule{0.166667em}{0ex}}{t}_{1,2}^{2}\phantom{\rule{0.166667em}{0ex}}{a}_{t,\sigma}\end{array}$ | ${c}_{1}$ | ${\beta}_{1}^{2}\phantom{\rule{0.166667em}{0ex}}{m}^{2}\left({a}_{t,\sigma}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}{\hslash}^{2}\phantom{\rule{0.166667em}{0ex}}{t}_{0,1}\phantom{\rule{0.166667em}{0ex}}{t}_{1,2}\right)\phantom{\rule{0.166667em}{0ex}}/\phantom{\rule{0.166667em}{0ex}}{\alpha}_{t,\sigma ,\beta}$ |

${k}_{7}$ | ${\beta}_{2}^{2}\phantom{\rule{0.166667em}{0ex}}{m}^{2}\phantom{\rule{0.166667em}{0ex}}{a}_{t,\sigma}\phantom{\rule{0.166667em}{0ex}}/\phantom{\rule{0.166667em}{0ex}}(2\phantom{\rule{0.166667em}{0ex}}{k}_{11})$ | ${\chi}_{0}$ | ${\pi}^{-1/4}\sqrt{\frac{m\phantom{\rule{0.166667em}{0ex}}{\sigma}_{0}}{m\phantom{\rule{0.166667em}{0ex}}{\sigma}_{0}^{2}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}\u0131\phantom{\rule{0.166667em}{0ex}}\hslash \phantom{\rule{0.166667em}{0ex}}{t}_{0,1}}}$ | ${d}_{1}$ | $\frac{-m\phantom{\rule{0.166667em}{0ex}}\hslash \phantom{\rule{0.166667em}{0ex}}{t}_{1,2}\left({\hslash}^{2}{t}_{0,1}^{2}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}{m}^{2}\phantom{\rule{0.166667em}{0ex}}{\sigma}_{0}^{2}\phantom{\rule{0.166667em}{0ex}}{\Xi}_{1}\right)}{{\alpha}_{t,\sigma ,\beta}}$ |

**Table 3.**Parameters for modeling the path integrals of QPI $({\psi}_{3,i}({x}_{3})$ for $i\in [0,1]:$ two paths).

Formula | Formula | $\mathit{j}\in [1,2]$ | Formula | ||
---|---|---|---|---|---|

${\mathbf{H}}_{2}$ | $\left(\begin{array}{cc}{\nu}_{2,2}{\left({\zeta}_{1,c}\phantom{\rule{0.166667em}{0ex}}+\u0131\phantom{\rule{0.166667em}{0ex}}{\zeta}_{1,d}\right)}^{2}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}{\nu}_{1,1}& 0\\ {\nu}_{3,2}\left({\zeta}_{1,c}\phantom{\rule{0.166667em}{0ex}}+\u0131\phantom{\rule{0.166667em}{0ex}}{\zeta}_{1,d}\right)& {\nu}_{1,2}\end{array}\right)$ | ${\nu}_{2,2}$ | $-\frac{{\beta}_{2}^{2}\phantom{\rule{0.166667em}{0ex}}\hslash \phantom{\rule{0.166667em}{0ex}}{t}_{2,3}}{2\phantom{\rule{0.166667em}{0ex}}\u0131\phantom{\rule{0.166667em}{0ex}}{\varsigma}_{2}}$ | ${\nu}_{1,j}$ | $-\frac{2\phantom{\rule{0.166667em}{0ex}}\hslash \phantom{\rule{0.166667em}{0ex}}{t}_{j,j+1}({A}_{j-1}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}\u0131\phantom{\rule{0.166667em}{0ex}}{B}_{j-1})\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}\u0131\phantom{\rule{0.166667em}{0ex}}m}{2\phantom{\rule{0.166667em}{0ex}}\u0131\phantom{\rule{0.166667em}{0ex}}{\varsigma}_{j}}$ |

${\nu}_{3,2}$ | $-\frac{\hslash \phantom{\rule{0.166667em}{0ex}}{t}_{2,3}}{\u0131\phantom{\rule{0.166667em}{0ex}}{\varsigma}_{2}}$ | ${\zeta}_{j}$ | $\begin{array}{l}4\phantom{\rule{0.166667em}{0ex}}{B}_{j-1}\phantom{\rule{0.166667em}{0ex}}{\beta}_{j}^{4}\phantom{\rule{0.166667em}{0ex}}\hslash \phantom{\rule{0.166667em}{0ex}}m\phantom{\rule{0.166667em}{0ex}}{t}_{j,j+1}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}{\beta}_{j}^{4}\phantom{\rule{0.166667em}{0ex}}{m}^{2}\phantom{\rule{0.166667em}{0ex}}\\ +\phantom{\rule{0.166667em}{0ex}}{\hslash}^{2}\phantom{\rule{0.166667em}{0ex}}{t}_{j,j+1}^{2}\phantom{\rule{0.166667em}{0ex}}{\varrho}_{j}\end{array}$ | ||

$\begin{array}{ll}{\overrightarrow{c}}_{2}& \left(\begin{array}{c}\phantom{\rule{0.166667em}{0ex}}{\nu}_{4,2}\phantom{\rule{0.166667em}{0ex}}{\zeta}_{1,c}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}{\nu}_{5,2}\phantom{\rule{0.166667em}{0ex}}{\zeta}_{1,d}\\ {\zeta}_{2,c}\end{array}\right)\\ {\overrightarrow{d}}_{2}& \left(\begin{array}{c}{\nu}_{4,2}\phantom{\rule{0.166667em}{0ex}}{\zeta}_{1,d}\phantom{\rule{0.166667em}{0ex}}-\phantom{\rule{0.166667em}{0ex}}{\nu}_{5,2}\phantom{\rule{0.166667em}{0ex}}{\zeta}_{1,c}\\ {\zeta}_{2,d}\end{array}\right)\end{array}$ | ${\nu}_{4,2}$ | ${\beta}_{2}^{2}\phantom{\rule{0.166667em}{0ex}}{\zeta}_{2,c}$ | ${\zeta}_{j,c}$ | $(2\phantom{\rule{0.166667em}{0ex}}{B}_{j-1}\phantom{\rule{0.166667em}{0ex}}\hslash \phantom{\rule{0.166667em}{0ex}}m\phantom{\rule{0.166667em}{0ex}}{t}_{j,j+1}\phantom{\rule{0.166667em}{0ex}}{\beta}_{j}^{2}+{\beta}_{j}^{2}\phantom{\rule{0.166667em}{0ex}}{m}^{2})\phantom{\rule{0.166667em}{0ex}}/\phantom{\rule{0.166667em}{0ex}}{\zeta}_{j}$ | |

${\nu}_{5,2}$ | $-\frac{2\phantom{\rule{0.166667em}{0ex}}\hslash \phantom{\rule{0.166667em}{0ex}}{t}_{2,3}\phantom{\rule{0.166667em}{0ex}}{A}_{2}}{m}$ | ${\zeta}_{j,d}$ | $\hslash \phantom{\rule{0.166667em}{0ex}}m\phantom{\rule{0.166667em}{0ex}}{t}_{j,j+1}\phantom{\rule{0.166667em}{0ex}}\left(2\phantom{\rule{0.166667em}{0ex}}{A}_{j-1}\phantom{\rule{0.166667em}{0ex}}{\beta}_{j}^{2}\phantom{\rule{0.166667em}{0ex}}-\phantom{\rule{0.166667em}{0ex}}1\right)\phantom{\rule{0.166667em}{0ex}}/\phantom{\rule{0.166667em}{0ex}}{\zeta}_{j}$ | ||

${A}_{0}$ | $-{m}^{2}{\sigma}_{0}^{2}\phantom{\rule{0.166667em}{0ex}}/\phantom{\rule{0.166667em}{0ex}}(2\phantom{\rule{0.166667em}{0ex}}{\hslash}^{2}{t}_{0,1}^{2}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}2\phantom{\rule{0.166667em}{0ex}}{m}^{2}{\sigma}_{0}^{4})$ | ${A}_{2}$ | $\frac{{\beta}_{2}^{2}\phantom{\rule{0.166667em}{0ex}}{m}^{2}\phantom{\rule{0.166667em}{0ex}}\left(2\phantom{\rule{0.166667em}{0ex}}{A}_{1}\phantom{\rule{0.166667em}{0ex}}{\beta}_{2}^{2}\phantom{\rule{0.166667em}{0ex}}-\phantom{\rule{0.166667em}{0ex}}1\right)}{2\phantom{\rule{0.166667em}{0ex}}{\zeta}_{2}}$ | $\begin{array}{l}{\varrho}_{j}\\ {\xi}_{j}\end{array}$ | $\begin{array}{l}4\phantom{\rule{0.166667em}{0ex}}{\beta}_{j}^{4}\left({A}_{j-1}^{2}+{B}_{j-1}^{2}\right)\phantom{\rule{0.166667em}{0ex}}-\phantom{\rule{0.166667em}{0ex}}4\phantom{\rule{0.166667em}{0ex}}{A}_{j-1}{\beta}_{j}^{2}+1\\ {\beta}_{j}^{2}\phantom{\rule{0.166667em}{0ex}}m\phantom{\rule{0.166667em}{0ex}}/\phantom{\rule{0.166667em}{0ex}}{\varsigma}_{j}\end{array}$ |

${B}_{0}$ | $\hslash \phantom{\rule{0.166667em}{0ex}}m\phantom{\rule{0.166667em}{0ex}}{t}_{0,1}\phantom{\rule{0.166667em}{0ex}}/\phantom{\rule{0.166667em}{0ex}}(2\phantom{\rule{0.166667em}{0ex}}{\hslash}^{2}{t}_{0,1}^{2}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}2{m}^{2}{\sigma}_{0}^{4})$ | ${B}_{2}$ | $\frac{2\phantom{\rule{0.166667em}{0ex}}{B}_{1}\phantom{\rule{0.166667em}{0ex}}{\beta}_{2}^{4}\phantom{\rule{0.166667em}{0ex}}{m}^{2}+\hslash \phantom{\rule{0.166667em}{0ex}}m\phantom{\rule{0.166667em}{0ex}}{t}_{2,3}\phantom{\rule{0.166667em}{0ex}}{\varrho}_{2}}{2\phantom{\rule{0.166667em}{0ex}}{\zeta}_{2}}$ | ${\varsigma}_{j}$ | $\begin{array}{l}\hslash \phantom{\rule{0.166667em}{0ex}}{t}_{j,j+1}\phantom{\rule{0.166667em}{0ex}}\left(2\phantom{\rule{0.166667em}{0ex}}{\beta}_{j}^{2}\phantom{\rule{0.166667em}{0ex}}({B}_{j-1}-\phantom{\rule{0.166667em}{0ex}}\u0131\phantom{\rule{0.166667em}{0ex}}{A}_{j-1})\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}\u0131\right)\\ \phantom{\rule{0.166667em}{0ex}}+{\beta}_{j}^{2}\phantom{\rule{0.166667em}{0ex}}m\end{array}$ |

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