# Theory of Quantum Path Entanglement and Interference with Multiplane Diffraction of Classical Light Sources

## Abstract

**:**

## 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 |

## References

- Cotler, J.; Wilczek, F. Entangled histories. Phys. Scr.
**2016**, 2016, 014004. [Google Scholar] [CrossRef] [Green Version] - Feynman, R.P.; Hibbs, A.R.; Styer, D.F. Quantum Mechanics and Path Integrals; Dover: Mineola, NY, USA, 2010. [Google Scholar]
- Griffiths, R.B. Consistent histories and the interpretation of quantum mechanics. J. Stat. Phys.
**1984**, 36, 219–272. [Google Scholar] [CrossRef] - Griffiths, R.B. Consistent interpretation of quantum mechanics using quantum trajectories. Phys. Rev. Lett.
**1993**, 70, 2201. [Google Scholar] [CrossRef] [PubMed] - Griffiths, R.B. Consistent Quantum Theory; Cambridge University Press: Cambridge, UK, 2003. [Google Scholar]
- Aharonov, Y.; Vaidman, L. The two-state vector formalism: An updated review. In Time in Quantum Mechanics, 2nd ed.; Muga, G., Sala Mayato, R., Egusquiza, I., Eds.; Springer: Berlin/Heidelberg, Germany, 2008; Volume 1, pp. 399–447. [Google Scholar]
- Nowakowski, M.; Cohen, E.; Horodecki, P. Entangled histories vs. the two-state-vector formalism-towards a better understanding of quantum temporal correlations. arXiv
**2018**, arXiv:1803.11267. [Google Scholar] - Gulbahar, B. Quantum path computing: Computing architecture with propagation paths in multiple plane diffraction of classical sources of fermion and boson particles. Quantum Inf. Process.
**2019**, 18, 167. [Google Scholar] [CrossRef] [Green Version] - Gulbahar, B. Quantum path computing and communications with Fourier optics. arXiv
**2019**, arXiv:1908.02274. [Google Scholar] - Gulbahar, B.; Memisoglu, G. Quantum spatial modulation of optical channels: Quantum boosting in spectral efficiency. IEEE Comm. Lett.
**2019**, 23, 2026–2030. [Google Scholar] [CrossRef] - Cotler, J.; Duan, L.M.; Hou, P.Y.; Wilczek, F.; Xu, D.; Yin, Z.Q.; Zu, C. Experimental test of entangled histories. Ann. Phys.
**2017**, 387, 334–347. [Google Scholar] [CrossRef] [Green Version] - Leggett, A.J.; Garg, A. Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks? Phys. Rev. Lett.
**1985**, 54, 857. [Google Scholar] [CrossRef] [Green Version] - Emary, C.; Lambert, N.; Nori, F. Leggett–Garg inequalities. Rep. Prog. Phys.
**2013**, 77, 016001. [Google Scholar] [CrossRef] - Emary, C.; Lambert, N.; Nori, F. Leggett-Garg inequality in electron interferometers. Phys. Rev. B
**2012**, 86, 235447. [Google Scholar] [CrossRef] [Green Version] - Wilde, M.M.; Mizel, A. Addressing the clumsiness loophole in a Leggett-Garg test of macrorealism. Found. Phys.
**2012**, 42, 256–265. [Google Scholar] [CrossRef] [Green Version] - Emary, C. Ambiguous measurements, signaling, and violations of Leggett-Garg inequalities. Phys. Rev. A
**2017**, 96, 042102. [Google Scholar] [CrossRef] [Green Version] - Katiyar, H.; Brodutch, A.; Lu, D.; Laflamme, R. Experimental violation of the Leggett–Garg inequality in a three-level system. New J. Phys.
**2017**, 19, 023033. [Google Scholar] [CrossRef] [Green Version] - Morikoshi, F. Information-theoretic temporal Bell inequality and quantum computation. Phys. Rev. A
**2006**, 73, 052308. [Google Scholar] [CrossRef] [Green Version] - Zhang, X.; Li, T.; Yang, Z.; Zhang, X. Experimental observation of the Leggett-Garg inequality violation in classical light. J. Opt.
**2018**, 21, 015605. [Google Scholar] [CrossRef] - Wang, K.; Emary, C.; Zhan, X.; Bian, Z.; Li, J.; Xue, P. Enhanced violations of Leggett-Garg inequalities in an experimental three-level system. Opt. Express
**2017**, 25, 31462–31470. [Google Scholar] [CrossRef] [Green Version] - Xu, J.-S.; Li, C.F.; Zou, X.B.; Guo, G.C. Experimental violation of the Leggett-Garg inequality under decoherence. Sci. Rep.
**2011**, 1, 101. [Google Scholar] [CrossRef] [Green Version] - Asano, M.; Hashimoto, T.; Khrennikov, A.; Ohya, M.; Tanaka, Y. Violation of contextual generalization of the Leggett-Garg inequality for recognition of ambiguous figures. Phys. Scr.
**2014**, T163, 014006. [Google Scholar] [CrossRef] - Losada, M.; Laura, R. Generalized contexts and consistent histories in quantum mechanics. Ann. Phys.
**2014**, 344, 263–274. [Google Scholar] [CrossRef] - Howard, M.; Wallman, J.; Veitch, V.; Emerson, J. Contextuality supplies the ‘magic’ for quantum computation. Nature
**2014**, 510, 351–355. [Google Scholar] [CrossRef] [Green Version] - Zavatta, A.; Viciani, S.; Bellini, M. Quantum-to-classical transition with single-photon-added coherent states of light. Science
**2004**, 306, 660–662. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Glauber, R.J. Quantum Theory of Optical Coherence: Selected Papers and Lectures; John Wiley & Sons: Weinhaim, Germany, 2007. [Google Scholar]
- Chen, Z.; Beierle, P.; Batelaan, H. Spatial correlation in matter-wave interference as a measure of decoherence, dephasing, and entropy. Phys. Rev. A
**2018**, 97, 043608. [Google Scholar] [CrossRef] [Green Version] - DiVincenzo, D.; Terhal, B. Decoherence: The obstacle to quantum computation. Phys. World
**1998**, 11, 53. [Google Scholar] [CrossRef] - da Paz, I.; Vieira, C.H.S.; Ducharme, R.; Cabral, L.A.; Alexander, H.; Sampaio, M.D.R. Gouy phase in nonclassical paths in a triple-slit interference experiment. Phys. Rev. A
**2016**, 93, 033621. [Google Scholar] [CrossRef] [Green Version] - Santos, E.A.; Castro, F.; Torres, R. Huygens-Fresnel principle: Analyzing consistency at the photon level. Phys. Rev. A
**2018**, 97, 043853. [Google Scholar] [CrossRef] [Green Version] - Sawant, R.; Samuel, J.; Sinha, A.; Sinha, S.; Sinha, U. Nonclassical paths in quantum interference experiments. Phys. Rev. Lett.
**2014**, 113, 120406. [Google Scholar] [CrossRef] [Green Version] - Ozaktas, H.M.; Zalevsky, Z.; Kutay, M.A. The Fractional Fourier Transform with Applications in Optics and Signal Processing; John Wiley and Sons: Chichester, UK, 2001. [Google Scholar]
- Dowker, H.; Halliwell, J.J. Quantum mechanics of history: The decoherence functional in quantum mechanics. Phys. Rev. D
**1992**, 46, 1580. [Google Scholar] [CrossRef] - Meng, Z.; Stewart, G.; Whitenett, G. Stable single-mode operation of a narrow-linewidth, linearly polarized, erbium-fiber ring laser using a saturable absorber. J. Lightw. Technol.
**2006**, 24, 2179. [Google Scholar] [CrossRef] - Sinha, A.; Vijay, A.H.; Sinha, U. On the superposition principle in interference experiments. Sci. Rep.
**2015**, 5, 10304. [Google Scholar] [CrossRef] [Green Version] - Magana-Loaiza, O.S.; De Leon, I.; Mirhosseini, M.; Fickler, R.; Safari, A.; Mick, U.; McIntyre, B.; Banzer, P.; Rodenburg, B.; Leuchs, G.; et al. Exotic looped trajectories of photons in three-slit interference. Nat. Commun.
**2016**, 7, 13987. [Google Scholar] [CrossRef] [Green Version] - Gagnon, E.; Brown, C.D.; Lytle, A.L. Effects of detector size and position on a test of Born’s rule using a three-slit experiment. Phys. Rev. A
**2014**, 90, 013832. [Google Scholar] [CrossRef] - Brukner, C.; Taylor, S.; Cheung, S.; Vedral, V. Quantum entanglement in time. arXiv
**2004**, arXiv:1701.08116. [Google Scholar] - Aharonov, Y.; Popescu, S.; Tollaksen, J.; Vaidman, L. Multiple-time states and multiple-time measurements in quantum mechanics. Phys. Rev. A
**2009**, 79, 052110. [Google Scholar] [CrossRef] [Green Version] - Gell-Mann, M.; Hartle, J.B. Alternative decohering histories in quantum mechanics. In Proceedings of the 25th International Conference on High Energy Physics, Singapore, 2–8 August 1990. [Google Scholar]
- Omnès, R. Interpretation of quantum mechanics. Phys. Lett. A
**1987**, 125, 169–172. [Google Scholar] [CrossRef] - Isham, C.J. Quantum logic and the histories approach to quantum theory. J. Math. Phys.
**1994**, 35, 2157–2185. [Google Scholar] [CrossRef] [Green Version] - Mandel, L.; Wolf, E. Coherence properties of optical fields. Rev. Mod. Phys.
**1965**, 37, 231. [Google Scholar] [CrossRef] - Akcay, C.; Parrein, P.; Rolland, J.P. Estimation of longitudinal resolution in optical coherence imaging. Appl. Opt.
**2002**, 41, 5256–5262. [Google Scholar] [CrossRef] [PubMed] - Latychevskaia, T. Spatial coherence of electron beams from field emitters and its effect on the resolution of imaged objects. Ultramicroscopy
**2017**, 175, 121–129. [Google Scholar] [CrossRef] [Green Version] - Zernike, F. The concept of degree of coherence and its application to optical problems. Physica
**1938**, 5, 785–795. [Google Scholar] [CrossRef]

**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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Gulbahar, B.
Theory of Quantum Path Entanglement and Interference with Multiplane Diffraction of Classical Light Sources. *Entropy* **2020**, *22*, 246.
https://doi.org/10.3390/e22020246

**AMA Style**

Gulbahar B.
Theory of Quantum Path Entanglement and Interference with Multiplane Diffraction of Classical Light Sources. *Entropy*. 2020; 22(2):246.
https://doi.org/10.3390/e22020246

**Chicago/Turabian Style**

Gulbahar, Burhan.
2020. "Theory of Quantum Path Entanglement and Interference with Multiplane Diffraction of Classical Light Sources" *Entropy* 22, no. 2: 246.
https://doi.org/10.3390/e22020246