7.4: Single Photon Interference - Fourth Version

This analysis of the operation of a Mach-Zehnder Interferometer (MZI) will use tensor algebra and the creation and annihilation operators.

An interferometer arm can be occupied or unoccupied. These states are represented by the following vectors.

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

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

After the first beam splitter the photon is in an even superposition of being in both arms of the interferometer. By convention a 90 degree phase shift is assigned to arm b to preserve probability. In terms of the concept of occupation, the superposition takes the following form in tensor algebra.

\[ |S \rangle \xrightarrow{i} \frac{1}{ \sqrt{2}} [|1 \rangle_a | 0 \rangle_b + i | 0 \rangle_a 1 | \rangle_b ] = \frac{1}{ \sqrt{2}} \bigg[ \begin{pmatrix}
0\\
1
\end{pmatrix}_a \otimes \begin{pmatrix}
1\\
0
\end{pmatrix}_b + i \begin{pmatrix}
1\\
0
\end{pmatrix}_a \otimes \begin{pmatrix}
0\\
1
\end{pmatrix}_b \bigg] = \frac{1}{ \sqrt{2}} \begin{pmatrix}
0\\
i\\
1\\
0
\end{pmatrix} \nonumber \]

\[ \Psi = \frac{1}{ \sqrt{2}} \begin{pmatrix}
0\\
i\\
1\\
0
\end{pmatrix} \nonumber \]

The matrix operators required for this analysis are as follows.

Creation:

\[ C = \begin{pmatrix}
0 & 0 \\
1 & 0
\end{pmatrix} \nonumber \]

Annihilation:

\[ A = \begin{pmatrix}
0 & 0 \\
1 & 0
\end{pmatrix} \nonumber \]

Number:

\[ N = \begin{pmatrix}
0 & 0 \\
1 & 0
\end{pmatrix} \nonumber \]

Identity:

\[ I = \begin{pmatrix}
0 & 0 \\
1 & 0
\end{pmatrix} \nonumber \]

The effect of the creation, annihilation and number operators on |0> and |1>:

\[ C \begin{pmatrix}
1\\
0
\end{pmatrix} = \begin{pmatrix}
0\\
1
\end{pmatrix} \nonumber \]

\[ A \begin{pmatrix}
0\\
1
\end{pmatrix} = \begin{pmatrix}
1\\
0
\end{pmatrix} \nonumber \]

\[ N \begin{pmatrix}
1\\
0
\end{pmatrix} = \begin{pmatrix}
0\\
0
\end{pmatrix} \nonumber \]

\[ N \begin{pmatrix}
0\\
1
\end{pmatrix} = \begin{pmatrix}
0\\
1
\end{pmatrix} \nonumber \]

The creation operator is the Hermitian adjoint of the annihilation operator and the annihilation operator is the Hermitian adjoint of the creation operator.

\[ \overline{(A^T)} = \begin{pmatrix}
0 & 0\\
1 & 0
\end{pmatrix} \nonumber \]

\[ \overline{(C^T)} = \begin{pmatrix}
0 & 1\\
0 & 0
\end{pmatrix} \nonumber \]

The number operator is the product of the creation and annihilation operators.

\[ CA = \begin{pmatrix}
0 & 0\\
0 & 1
\end{pmatrix} \nonumber \]

\[ \overline{(A^T)} A = \begin{pmatrix}
0 & 0\\
0 & 1
\end{pmatrix} \nonumber \]

\[ \overline{(C^T)} C = \begin{pmatrix}
0 & 0\\
0 & 1
\end{pmatrix} \nonumber \]

The eigenvectors of the number operator are |0> and |1> with eigenvalues 0 and 1, respectively:

\[ eigenvecs(N) = \begin{pmatrix}
1 & 0\\
0 & 1
\end{pmatrix} \nonumber \]

\[ eigenvals(N) = \begin{pmatrix}
0\\
1
\end{pmatrix} \nonumber \]

There are two paths to each detector. This provides the opportunity for constructive and destructive interference. To arrive at D₁ the a-arm photon state is reflected (90 degree phase shift) at BS₂ and the b-arm photon state is transmitted at BS₂. Therefore, photon detection requires the annihilation of the superposition of these paths to D₁. The annihilation is achieved with the following operator.

\[ \frac{iA_a + A_b}{ \sqrt{2}} \nonumber \]

The product of this operator with its Hermitian conjugate (see above) creates the number operator for photon detection at D1.

\[ N_{D1} = \frac{iC_a + C_b}{ \sqrt{2}} + \frac{iA_a + A_b}{ \sqrt{2}} \nonumber \]

The D₁ number operator is formed using Mathcad's kronecker command as follows:

N_D1 = \( \frac{1}{2}\) (-i kronecker (C, I) + kronecker (I, C)) (i kronecker (A,I) + kronecker (I, A))

To arrive at D₂ the a-arm photon state is transmitted at BS₂ and the b-arm photon state is reflected (90 degree phase shift) at BS₂. Photon detection at D₂ requires the annihilation of the superposition of these paths to the detector. The annihilation is represented the following operator.

\[ \frac{iA_a + A_b}{ \sqrt{2}} \nonumber \]

Therefore, the number operator for photon detection at D₂ is:

\[ N_{D2} = \frac{C_a + iC_b}{ \sqrt{2}} \frac{A_a + iA_b}{ \sqrt{2}} \nonumber \]

The D₂ number operator is formed using Mathcad's kronecker command as follows.

N_D2 = \( \frac{1}{2}\) (kronecker (C, I) - i kronecker (I, C)) (kronecker (A,I) + i kronecker (I, A))

We now show that the photon always arrives at D₁ and never at D₂ for an equal arm MZI.

Expectation value for photon detection at D₁:

\[ \overline{ \Psi ^T} N_{D1} \Psi = 1 \nonumber \]

Expectation value for photon detection at D₂:

\[ \overline{ \Psi ^T} N_{D2} \Psi = 1 \nonumber \]

Equivalent results can be obtained algebraically. Operating on \( \Psi\) with the D₁ number operator yields \( \Psi\). In other words, \( \Psi\) is an eigenfunction of N_D1 with eigenvalue 1.

\[ \bigg[ \frac{-C_a + C_b}{ \sqrt{2}} \frac{iA_a + A_b}{\sqrt{2}} \bigg] \frac{1}{ \sqrt{2}} \big[ |1 \rangle_a |0 \rangle_b + i|0 \rangle_a |1 \rangle_b \big] = \frac{1}{ \sqrt{2}} \big[ |1 \rangle_a |0 \rangle_b + i|0_a |1 \rangle_b \big] \nonumber \]

Operating on \( \Psi\) with the D₂ number operator yields 0.

\[ \bigg[ \frac{C_a - iC_b}{ \sqrt{2}} \frac{A_a + iA_b}{\sqrt{2}} \bigg] \frac{1}{ \sqrt{2}} \big[ |1 \rangle_a |0 \rangle_b + i|0 \rangle_a |1 \rangle_b \big] = 0 \nonumber \]