# 260: Single Photon Interference - Mathcad version

- Page ID
- 137795

The schematic diagram below shows a Mach-Zehnder interferometer for photons. When the experiment is run so that there is only one photon in the apparatus at any time, the photon is always detected at D_{1} and never at D_{2}.

This surprising phenomenon will be analyzed using matrix mechanics. State vectors for photon motion in the x- and y-direction, plus matrix operators for beam splitters and mirrors are defined below. For background and references to the primary literature see: V. Scarani and A. Suarez, "Introducing Quantum Mechanics: One-particle Interferences," Am. J. Phys. 66, 718-721 (1998).

**Orthonormal basis states**:

Photon moving in x-direction: \[ x = \begin{bmatrix}

1\\

0

\end{bmatrix} ~~~x^T x = 1\]

Photon moving in y-direction: \[ y = \begin{bmatrix}

0\\

1

\end{bmatrix} ~~~y^T y = 1~~~ x^T y = 0 ~~~ y^T x = 0\]

**Operators**:

Operator for interaction with the mirror:

\[ M = \begin{bmatrix}

0 & 1\\

1 & 0

\end{bmatrix}\]

Operator for interaction with a 50/50 beam splitter:

\[ BS = \frac{1}{ \sqrt{2}} \begin{bmatrix}

1 & i\\

i & 1

\end{bmatrix}\]

**Operations**:

\[ M (x) = \begin{bmatrix}

0\\

1

\end{bmatrix} ~~~ M (y) = \begin{bmatrix}

1\\

0

\end{bmatrix} ~~~ BS (x) = \begin{bmatrix}

0.707\\

0.707i

\end{bmatrix} ~~~ BS (y) = \begin{bmatrix}

0.707i\\

0.707

\end{bmatrix} ~~~ BS (M) BS (x) = \begin{bmatrix}

i\\

0

\end{bmatrix}\]

Quantum Mechanical Calculation of Experimental Results:

To be detected at D_{1} the photon must be moving in the x-direction (photon state = |x>). To be detected at D_{2} the photon must be moving in the y-direction (photon state = |y>). It is shown below that the probability the photon is moving in the x-direction is 1 and the probability it is moving in the y-direction is 0. In its course from source to detector the photon encounters a beam splitter, a mirror, and another beam splitter. Thus,

Probability photon will arrive at detector D_{1}:

\[ (| x^T BS(M)BS(x)|)^2 = 1\]

Probability photon will arrive at detector D_{2}:

\[ (| x^T BS(M)BS(x)|)^2 = 1\]

**Further analysis**:

The photon leaves the source traveling in the x-direction. Interaction with the beam splitter puts the photon in an even linear superposition of traveling in the x- and y-directions with a 90^{o} phase shift ( \( \frac{ \pi}{2}\) or i) assigned by convention to motion in the y-direction (see note below).

\[ BS (x) = \begin{bmatrix}

0.707\\

0.707i

\end{bmatrix} ~~~ \frac{x+iy}{ \sqrt{2}} = \begin{bmatrix}

0.707\\

0.707i

\end{bmatrix}\]

The interaction of this state with the mirrors transfers the 90^{o} phase shift to motion in the x-direction.

\[ (M) BS (x) = \begin{bmatrix}

0.707i\\

0.707

\end{bmatrix} = \frac{ix+y}{ \sqrt{2}} = \begin{bmatrix}

0.707i\\

0.707

\end{bmatrix}\]

Finally the interaction of this state with the second beam splitter yields a 90^{o} phase-shifted photon travelling in the x-direction.

\[ BS(M)BS(x) = \begin{bmatrix}

i\\

0

\end{bmatrix} ~~~ ix = \begin{bmatrix}

i\\

0

\end{bmatrix}\]

Thus, the probability amplitude that it will be detected at D_{1} is i and the probability amplitude that it will be detected at D_{2} is 0.

\[ x^T BS(M)BS(x) = (i) ~~~ y^T BS(M)BS(x) = (0)\]

The probability for an event is the square of the absolute magnitude of the probability amplitude, so the probability that the photon will be detected at D_{1} is 1.

**Note**: The justification for the 90 (\( \frac{ \pi}{2}\) or i) phase shift between transmission and reflection at the beam splitter is conservation of energy. Assuming there is no phase shift requires that the BS operator be defined as:

\[ BS = \frac{1}{ \sqrt{2}} \begin{bmatrix}

1 & 1\\

1 & 1

\end{bmatrix}\]

This leads to the following calculation for the probability of the detection events:

Probability photon will arrive at detector D_{1}: \( (| x^T BS(M)BS(x)|)^2 = 1\)

Probability photon will arrive at detector D_{2}: \( (| y^T BS(M)BS(x)|)^2 = 1\)

According to this analysis, the single photon leaving the source has arrived at both detectors. One photon has become two, an obvious violation of the energy conservation principle.