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Chemistry LibreTexts

277: A Quantum Circuit for a Michelson Interferometer

  • Page ID
    141691
  • Schematic diagram of a Mach‐Zehnder interferometer (MZI).

    Screen Shot 2019-02-28 at 5.35.32 PM.png

    The following quantum circuit simulates the MZI.

    Screen Shot 2019-02-28 at 5.36.19 PM.png

    The arms of the MZI are represented by the following orthonormal basis.

    \[ | 0 \rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}\]

    \[ |1 \rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}\]

    The matrices representing the Hadamard and phase shift gates are:

    \[ H = \frac{1}{ \sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}\]

    \[ A ( \theta) = \begin{pmatrix} 1 & 0 \\ 0 & e^{i \phi} \end{pmatrix}\]

    Step‐by‐step through the circuit. The first Hadamard gate creates a superposition of the |0 > and |1 > states. The phase shifter operates on the lower arm of the MZI. The final Hadamard gate allows interference between the two arms of the MZI.

    \[ H \begin{pmatrix} 1 \\ 0 \end{pmatrix} \rightarrow \begin{pmatrix} \frac{ \sqrt{2}}{2} \\ \frac{ \sqrt{2}}{2} \end{pmatrix}\]

    \[ A ( \phi) H \begin{pmatrix} 1 \\ 0 \end{pmatrix} \rightarrow \begin{pmatrix} \frac{ \sqrt{2}}{2} \\ \frac{ \sqrt{2} e^{ \phi i}}{2} \end{pmatrix}\]

    \[ H A ( \phi) H \begin{pmatrix} 1 \\ 0 \end{pmatrix} \rightarrow \begin{pmatrix} \frac{ e^{ \phi i}}{2} + \frac{1}{2} \\ \frac{1}{2} - \frac{ e^{ \phi i}}{2} \end{pmatrix}\]

    Probability of detection at the |0 > port:

    \[ P_0 ( \phi) = \left[ \left| \begin{pmatrix} 1 & 0 \end{pmatrix} H A( \phi) H \begin{pmatrix} 1 \\ 0 \end{pmatrix} \right| \right]^2 |_{simplify}^{assume,~ \phi = real} \rightarrow \frac{ \cos \phi}{2} + \frac{1}{2}\]

    Probability of detection at the |1 > port:

    \[ P_1 ( \phi) = \left[ \left| \begin{pmatrix} 0 & 1 \end{pmatrix} H A( \phi) H \begin{pmatrix} 1 \\ 0 \end{pmatrix} \right| \right]^2 |_{simplify}^{assume,~ \phi = real} \rightarrow \frac{1}{2} - \frac{ \cos \phi}{2}\]

    A graphical representation of the above calculations shows the interference effects as a function of ϕ.

    Screen Shot 2019-02-28 at 5.48.34 PM.png