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1.47: Terse Analysis of Triple-slit Diffraction with a Quantum Eraser

  • Page ID
    155898
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    Slit positions, slit width and the wavefunction at the slit screen which is a superposition of the photon being simultaneously present at all three slits.

    \[
    x_{1} :=-\frac{1}{2} \quad x_{2}:=0 \quad \mathrm{x}_{3} :=\frac{1}{2} \qquad \delta :=0.1
    \nonumber \]

    \[
    |\Psi\rangle=\frac{1}{\sqrt{3}}\left[\left|x_{1}\right\rangle+\left|x_{2}\right\rangle+\left|x_{3}\right\rangle\right]
    \nonumber \]

    Calculate the diffraction pattern by a Fourier transform of the spatial wavefunction into momentum space.

    \[
    \langle p | \Psi\rangle=\frac{1}{\sqrt{3}}\left[\left\langle p | x_{1}\right\rangle+\left\langle p | x_{2}\right\rangle+\left\langle p | x_{3}\right\rangle\right]
    \nonumber \]

    \[
    \Psi(p)=\int_{x_{1}-\frac{\delta}{2}}^{x_{1}+\frac{\delta}{2}} \frac{1}{\sqrt{2 \cdot \pi}} \cdot \exp (-i \cdot p \cdot x) \cdot \frac{1}{\sqrt{\delta}} d x +\int_{x_{2} -\frac{\delta}{2}}^{x_{2}+\frac{\delta}{2}} \frac{1}{\sqrt{2 \cdot \pi}} \cdot \exp (-i \cdot p \cdot x) \cdot \frac{1}{\sqrt{\delta}} d x +\int_{x_{3} -\frac{\delta}{2}}^{x_{3}+\frac{\delta}{2}} \frac{1}{\sqrt{2 \cdot \pi}} \cdot \exp (-i \cdot p \cdot x) \cdot \frac{1}{\sqrt{\delta}} d x
    \nonumber \]

    Display the momentum distribution function which is the diffraction pattern.

    clipboard_e76b6f9d857d6c7d6ecf3915d5330b455.png

    Tag the slits with orthogonal states.

    \[
    \left\langle p | \Psi^{\prime}\right\rangle=\frac{1}{\sqrt{3}}\left[\left\langle p | x_{1}\right\rangle|\uparrow\rangle+\left\langle p | x_{2}\right\rangle|\rightarrow\rangle+\left\langle p | x_{3}\right\rangle|\downarrow\rangle\right]
    \nonumber \]

    Recalculate the momentum distribution.

    \[
    \Psi^{\prime}(p)=\int_{x_{1}-\frac{\delta}{2}}^{x_{1}+\frac{\delta}{2}} \frac{1}{\sqrt{2 \cdot \pi}} \cdot \exp (-i \cdot p \cdot x) \cdot \frac{1}{\sqrt{\delta}} d x \cdot\left(\begin{array}{l}{1} \\ {0} \\ {0}\end{array}\right) +\int_{x_{2}-\frac{\delta}{2}}^{x_{2}+\frac{\delta}{2}} \frac{1}{\sqrt{2 \cdot \pi}} \cdot \exp (-i \cdot p \cdot x) \cdot \frac{1}{\sqrt{\delta}} d x \cdot\left(\begin{array}{l}{0} \\ {1} \\ {0}\end{array}\right) +\int_{x_{3}- \frac{\delta}{2}}^{x_{3}+\frac{\delta}{2}} \frac{1}{\sqrt{2 \cdot \pi}} \cdot \exp (-i \cdot p \cdot x) \cdot \frac{1}{\sqrt{\delta}} d x \cdot\left(\begin{array}{l}{0} \\ {0} \\ {1}\end{array}\right)
    \nonumber \]

    Display the momentum distribution at the detection screen showing that the diffraction pattern has disappeared. The orthogonallity of the tags destroys the cross-terms in the momentum distribution, \(|\Psi^{'}(p)|^{2}\), which give rise to the interference effects shown in the original diffraction pattern.

    clipboard_eb2ccfcb04a87016602479a4af109349c.png

    Insert an "eraser" after the slit screen and before the detection screen.

    \[
    \Psi^{\prime \prime}(\mathrm{p}) :=\frac{1}{\sqrt{3}} \cdot\left(\begin{array}{l}{1} \\ {1} \\ {1}\end{array}\right)^{\mathrm{T}} \cdot \Psi^{\prime}(\mathrm{p})
    \nonumber \]

    The diffraction pattern is restored but attenuated because the so-called "eraser" filters out the orthogonal tags restoring the interference terms.

    clipboard_ede699f01d1be3de985222e75d92801cc.png


    This page titled 1.47: Terse Analysis of Triple-slit Diffraction with a Quantum Eraser is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Frank Rioux via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.