1.47: Terse Analysis of Triple-slit Diffraction with a Quantum Eraser
- Page ID
- 155898
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.
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.
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.