7.22: A Quantum Circuit for a Michelson Interferometer
- Page ID
- 141691
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Schematic diagram of a Mach‐Zehnder interferometer (MZI).
The following quantum circuit simulates the MZI.
The arms of the MZI are represented by the following orthonormal basis.
\[ | 0 \rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \nonumber \]
\[ |1 \rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \nonumber \]
The matrices representing the Hadamard and phase shift gates are:
\[ H = \frac{1}{ \sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} \nonumber \]
\[ A ( \theta) = \begin{pmatrix} 1 & 0 \\ 0 & e^{i \phi} \end{pmatrix} \nonumber \]
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.
\[ H \begin{pmatrix} 1 \\ 0 \end{pmatrix} \rightarrow \begin{pmatrix} \frac{ \sqrt{2}}{2} \\ \frac{ \sqrt{2}}{2} \end{pmatrix} \nonumber \]
\[ A ( \phi) H \begin{pmatrix} 1 \\ 0 \end{pmatrix} \rightarrow \begin{pmatrix} \frac{ \sqrt{2}}{2} \\ \frac{ \sqrt{2} e^{ \phi i}}{2} \end{pmatrix} \nonumber \]
\[ H A ( \phi) H \begin{pmatrix} 1 \\ 0 \end{pmatrix} \rightarrow \begin{pmatrix} \frac{ e^{ \phi i}}{2} + \frac{1}{2} \\ \frac{1}{2} - \frac{ e^{ \phi i}}{2} \end{pmatrix} \nonumber \]
Probability of detection at the |0 > port:
\[ P_0 ( \phi) = \left[ \left| \begin{pmatrix} 1 & 0 \end{pmatrix} H A( \phi) H \begin{pmatrix} 1 \\ 0 \end{pmatrix} \right| \right]^2 |_{simplify}^{assume,~ \phi = real} \rightarrow \frac{ \cos \phi}{2} + \frac{1}{2} \nonumber \]
Probability of detection at the |1 > port:
\[ P_1 ( \phi) = \left[ \left| \begin{pmatrix} 0 & 1 \end{pmatrix} H A( \phi) H \begin{pmatrix} 1 \\ 0 \end{pmatrix} \right| \right]^2 |_{simplify}^{assume,~ \phi = real} \rightarrow \frac{1}{2} - \frac{ \cos \phi}{2} \nonumber \]
A graphical representation of the above calculations shows the interference effects as a function of ϕ.
