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1.49: A Stern‐Gerlach Quantum ʺEraserʺ

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    155902
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    This tutorial provides a brief mathematical analysis of a proposed quantum eraser experiment involving spin‐1/2 particles which is available at arXiv:quant‐ph/0501010v2. Please see the two immediately preceding tutorials for another example of the quantum eraser and additional mathematical detail.

    clipboard_e1aa69294918e5962b94bbe0241bf8a13.png

    The first magnet attaches which‐way information such that the spin‐1/2 particles leaving the double‐slit screen are described by the following entangled wave function,

    \[
    |\Psi\rangle=\frac{1}{\sqrt{2}}\left[\left|\uparrow_{z}\right\rangle\left|z_{1}\right\rangle+\left|\downarrow_{z}\right\rangle\left|z_{2}\right\rangle\right]
    \nonumber \]

    where z1 and z2 represent the positions of the horizontal slits on the z‐axis and the spin eigenstates in the z‐direction are given below.

    \[
    \Psi_{\mathrm{zup}} :=\left(\begin{array}{l}{1} \\ {0}\end{array}\right) \qquad \Psi_{\mathrm{zdown}}=\left(\begin{array}{l}{0} \\ {1}\end{array}\right)
    \nonumber \]

    Recognizing that a diffraction pattern is actually a momentum distribution function, we project \(\Psi\) onto momentum space as follows (in atomic units, h = 2\(\pi\)).

    \[
    \langle p | \Psi\rangle=\frac{1}{\sqrt{2}}\left[\left|\uparrow_{z}\right\rangle\left\langle p | z_{1}\right\rangle+\left|\downarrow_{z}\right\rangle\left\langle p | z_{2}\right\rangle\right]=\frac{1}{\sqrt{2}}\left[\left|\uparrow_{z}\right\rangle \frac{\exp \left(-i p z_{1}\right)}{\sqrt{2 \pi}}+\left|\downarrow_{z}\right\rangle \frac{\exp \left(-i p z_{2}\right)}{\sqrt{2 \pi}}\right]
    \nonumber \]

    Here the exponential terms are the position eigenfunctions in momentum space for infinetesimally thin slits located at z1 and z2. For slits of finite width \(<p | \Psi>\) is written as shown below. Again see the previous tutorials in this series for further mathematical detail. The slit positions and slit width chosen are arbitrary.

    Slit positions: \(\mathrm{z}_{1} :=1 \qquad \mathrm{z}_{2} :=2\) Slit width: \(\delta :=0.2\)

    \[
    \Psi(\mathrm{p}) :=\frac{1}{\sqrt{2}}\cdot \left(\Psi_{\mathrm{zup}} \cdot \int_{\mathrm{z}_{1}- \frac{\delta}{2}}^{\mathrm{z}_{1}+\frac{\delta}{2}} \frac{1}{\sqrt{2 \cdot \pi}} \cdot \exp (-\mathrm{i} \cdot \mathrm{p} \cdot \mathrm{z}) \cdot \frac{1}{\sqrt{\delta}} \mathrm{d} \mathrm{z} +\Psi_{\text { zdown }}\cdot \int_{z_{2}-\frac{\delta}{2}}^{z_{2}+\frac{\delta}{2}} \frac{1}{\sqrt{2 \cdot \pi}}\cdot \exp(-\mathrm{i} \cdot p \cdot z) \cdot \frac{1}{\sqrt{\delta}} \mathrm{d} z\right)
    \nonumber \]

    Because of the addition of path information there are no interference fringes associated with this two‐slit wave function; the encoded orthogonal z‐direction eigenstates destroy the interference cross terms as shown graphically below.

    clipboard_e127c2ffefd9e76026c04b9ce58264740.png

    The second Stern‐Gerlach magnet oriented in the x‐direction, according to the conventional interpretation, ʺerasesʺ the which‐way information. This is shown by projecting the state after the first magnet and the slit screen, \(\Psi\)(p), onto the x‐direction spin eigenstates.

    \[
    \Psi_{\mathrm{xup}}=\frac{1}{\sqrt{2}} \cdot(1 \quad 1) \qquad \Psi_{\mathrm{xdown}}=\frac{1}{\sqrt{2}} \cdot(1\quad -1)
    \nonumber \]

    \[
    \Psi_{\text { left }}(\mathrm{p}) :=\Psi_{\mathrm{xup}} \cdot \Psi(\mathrm{p}) \qquad \Psi_{\mathrm{right}}(\mathrm{p}) :=\Psi_{\mathrm{xdown}} \cdot \Psi(\mathrm{p})
    \nonumber \]

    clipboard_e1bc80f9beded889e43ccc38712e10376.png
    Figure 2b in the reference cited and shown below.

    clipboard_e646e28582ac0520a15f828725717a4bf.png

    The horizontal blue line marks p = 0 on the z‐axis. On the left is the interference pattern of the part of the beam emerging from the x‐up magnet direction with spin state \(\Psi_{xup}\), and on the right is the interference pattern of the part of the beam emerging from the x‐down magnet direction with spin state \(\Psi_{xdown}\). As shown in the Summary, \(\Psi\)(p) can be rewritten in terms of the x‐direction spin states clearly show in the superpositions responsible for the interference fringes on the left and right.

    \[
    \langle p | \Psi\rangle=\frac{1}{2}\left[\left|\uparrow_{x}\right\rangle\left(\left\langle p | z_{1}\right\rangle+\left\langle p | z_{2}\right\rangle\right)+\left|\downarrow_{x}\right\rangle\left(\left\langle p | z_{1}\right\rangle-\left\langle p | z_{2}\right\rangle\right)\right]
    \nonumber \]

    In the absence of both Stern‐Gerlach magnets the usual double‐slit interference pattern is observed.

    \[
    \Psi(p)=\frac{1}{\sqrt{2}}\cdot\left(\int_{\mathrm{z}_{1}-\frac{\delta}{2}}^{\mathrm{z}_{1}+\frac{\delta}{2}} \frac{1}{\sqrt{2\cdot\pi}}\cdot\exp(-\mathrm{i}\cdot\mathrm{p}\cdot\mathrm{z})\cdot\frac{1}{\sqrt{\delta}} \mathrm{d} \mathrm{z}+\int_{z_{2}-\frac{\delta}{2}}^{z_{2}+\frac{\delta}{2}}\frac{1}{\sqrt{2 \cdot \pi}}\cdot\exp(-i \cdot p \cdot z)\cdot\frac{1}{\sqrt{\delta}} d z\right)
    \nonumber \]

    clipboard_eecc879599b0d8bbcfe8033a656dc9c96.png

    Alternative Analysis

    It is possible to express the mathematics in an alternative but equivalent form. The first wave function,

    \[
    \frac{1}{\sqrt{2}}\left[\left|\uparrow_{z}\right\rangle\left|z_{1}\right\rangle+\left|\downarrow_{z}\right\rangle\left|z_{2}\right\rangle\right]
    \nonumber \]

    can be expressed explicitly in vector format in the momentum representation. This analysis will be based on infinitesimally thin slits as introduced earlier.

    \[
    \frac{1}{\sqrt{2}}\left[\left(\begin{array}{l}{1} \\ {0}\end{array}\right)\left|z_{1}\right\rangle+\left(\begin{array}{l}{0} \\ {1}\end{array}\right)\left|z_{2}\right\rangle\right]=\frac{1}{\sqrt{2}}\left(\begin{array}{c}{\left|z_{1}\right\rangle} \\ {\left|z_{2}\right\rangle}\end{array}\right) \xrightarrow{\langle p |} \frac{1}{\sqrt{2}}\left(\begin{array}{c}{\left\langle p | z_{1}\right\rangle} \\ {\left\langle p | z_{2}\right\rangle}\end{array}\right)
    \nonumber \]

    It is easily shown that this wave function does not lead to interference fringes at the detection screen by calculating the square of its absolute magnitude.

    \[
    \frac{1}{2}\left(\left\langle z_{1} | p\right\rangle\quad\left\langle z_{2} | p\right\rangle\right)\left(\begin{array}{c}{\left\langle p | z_{1}\right\rangle} \\ {\left\langle p | z_{2}\right\rangle}\end{array}\right)=\frac{1}{2}\left[\left|\left\langle p | z_{1}\right\rangle\right|^{2}+\left|\left\langle p | z_{2}\right\rangle\right|^{2}\right]
    \nonumber \]

    \[
    \Psi(\mathrm{p}) :=\frac{1}{2 \cdot \sqrt{\pi}} \cdot\left(\begin{array}{c}{\exp \left(-\mathrm{i} \cdot \mathrm{p} \cdot \mathrm{z}_{1}\right)} \\ {\exp \left(-\mathrm{i} \cdot \mathrm{p} \cdot \mathrm{z}_{2}\right)}\end{array}\right)
    \nonumber \]

    clipboard_e71e6afd8e62b966431acdb6284805caf.png

    However, in the presence of the second Stern‐Gerlach magnet vector B is projected onto the two output channels of the magnet.

    \[
    \frac{1}{2}\left(\begin{array}{ll}{1} & {1}\end{array}\right)\left(\begin{array}{l}{\left\langle p | z_{1}\right\rangle} \\ {\left\langle p | z_{2}\right\rangle}\end{array}\right)=\frac{1}{2}\left[\left\langle p | z_{1}\right\rangle+\left\langle p | z_{2}\right\rangle\right]
    \nonumber \]

    \[
    \frac{1}{2}\left(\begin{array}{ll}{1} & {-1}\end{array}\right)\left(\begin{array}{l}{\left\langle p | z_{1}\right\rangle} \\ {\left\langle p | z_{2}\right\rangle}\end{array}\right)=\frac{1}{2}\left[\left\langle p | z_{1}\right\rangle-\left\langle p | z_{2}\right\rangle\right]
    \nonumber \]

    The probability distributions of these states show interference fringes.

    \[
    \Psi_{\text { left }}(\mathrm{p}) :=\frac{1}{\sqrt{2}} \cdot(1\quad1) \cdot \frac{1}{2 \cdot \sqrt{\pi}} \cdot\left(\begin{array}{c}{\exp \left(-\mathrm{i} \cdot \mathrm{p} \cdot \mathrm{z}_{1}\right)} \\ {\exp \left(-\mathrm{i} \cdot \mathrm{p} \cdot \mathrm{z}_{2}\right)}\end{array}\right)
    \nonumber \]

    clipboard_ebdece3133a4c28c9b8725a2d0f96a2af.png

    \[
    \Psi_{\text { right }}(\mathrm{p}) :=\frac{1}{\sqrt{2}} \cdot(1\quad-1) \cdot \frac{1}{2\cdot\sqrt{\pi}} \cdot\left(\begin{array}{c}{\exp \left(-\mathrm{i} \cdot \mathrm{p} \cdot \mathrm{z}_{1}\right)} \\ {\exp \left(-\mathrm{i} \cdot \mathrm{p} \cdot \mathrm{z}_{2}\right)}\end{array}\right)
    \nonumber \]

    clipboard_ed7c81268604063532005f4e665ec98bd.png

    Summary

    The z‐direction Stern‐Gerlach magnet and the slit screen create the following entangled superposition which does not produce interference fringes due to the orthogonality of the spin states marking the slits.

    \[
    \langle p | \Psi\rangle=\frac{1}{\sqrt{2}}\left[\left|\uparrow_{z}\right\rangle\left\langle p | z_{1}\right\rangle+\left|\downarrow_{z}\right\rangle\left\langle p | z_{2}\right\rangle\right]
    \nonumber \]

    To understand what happens at the x‐direction magnet this state is rewritten in the x‐direction spin basis.

    \[
    \langle p | \Psi\rangle=\frac{1}{\sqrt{2}}\left[\frac{1}{\sqrt{2}}\left(\left|\uparrow_{x}\right\rangle+\left|\downarrow_{x}\right\rangle\right)\left\langle p | z_{1}\right\rangle+\frac{1}{\sqrt{2}}\left(\left|\uparrow_{x}\right\rangle-\left|\downarrow_{x}\right\rangle\right)\left\langle p | z_{2}\right\rangle\right]
    \nonumber \]

    Collecting terms on the x‐direction spin eigenstates yields,

    \[
    \langle p | \Psi\rangle=\frac{1}{2}\left[\left|\uparrow_{x}\right\rangle\left(\left\langle p | z_{1}\right\rangle+\left\langle p | z_{2}\right\rangle\right)+\left|\downarrow_{x}\right\rangle\left(\left\langle p | z_{1}\right\rangle-\left\langle p | z_{2}\right\rangle\right)\right]
    \nonumber \]

    The in‐phase and out‐of‐phase superpositions, highlighted in blue and red, exit the magnet in opposite directions. Because of this the superpositions become spatially separated which leads to two sets of interference fringes with a one‐fringe relative phase shift at the detection screen.

    Itʹs clear to me that erasure is not a satisfactory explanation for this process. Because fringes appear after the x‐direction magnet it might seem plausible, at first glance, to assume that the which‐way markers have been erased. But actually the x‐direction magnet sorts < p|\(\Psi\)> into two components in terms of the x‐direction spin eigenstates. Nothing has been erased.


    This page titled 1.49: A Stern‐Gerlach Quantum ʺEraserʺ 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.