7.1: Single-Photon Interference - First Version

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 \(D_2\) and never at \(D_1\).(1,2,3) The qualitative explanation is that there are two paths to each detector and, therefore, the probability amplitudes for these paths may interfere constructively or destructively. For detector \(D_2\) the probability amplitudes for the two paths interfere constructively, while for detector \(D_1\) they interfere destructively.

A quantitative quantum mechanical analysis of this striking phenomenon is outlined below. The photon leaves the source, S, traveling in the y-direction. Whether the photon takes the upper or lower path it interacts with a beam splitter, a mirror, and another beam splitter before reaching the detectors.

Orthonormal basis states: (1 x 2 vectors)

Photon moving in the x-direction: \( | x > = \begin{vmatrix}
1\\
0
\end{vmatrix}~~~< x | = (1~~~0)~~~< x | x > = 1\)

Photon moving in the y-direction: \( | y > = \begin{vmatrix}
0 \\
1
\end{vmatrix}~~~< y | = (0~~~1) ~~~< y | y > = 1\)

< y | x > = < x | y > = 0

Operators: (2 x 2 matrices)

Operator for photon interaction with the mirror:

\[ \hat{M} = \( \begin{vmatrix} 0 & 1\\ 1 & 0 \end{vmatrix} \nonumber \]

Operator for photon interaction with the beam splitter:

\[\hat{BS} = \(\begin{vmatrix} T & iR\\ iR & T \end{vmatrix} \nonumber \]

\(T\) and \(R\) are the transmission and reflection amplitudes. For the half-silvered mirrors used in this example they are:

\[ T = R = \left( \dfrac{1}{2} \right) ^{ \frac{1}{2}} = 0.707 \nonumber \]

Operations:

After interacting with a beam splitter, a photon is in a linear superposition of |x> and |y> in which the components are 90 degrees out of phase.

\[\hat{BS}|x \rangle = \dfrac{[ |x + i|y]}{2}^{ \frac{1}{2}} \nonumber \]

\[\hat{BS}|y \rangle = \dfrac{[ i|x + i|y]}{2}^{ \frac{1}{2}} \nonumber \]

BS M BS|y > = i|y>

Interaction with the mirror merely changes the direction of the photon.

\[\hat{M} | x \rangle = | y \rangle \nonumber \]

\[\hat{M} | y \rangle = | x \rangle \nonumber \]

Matrix elements:

< x | M | x > = 0 < y | M | x > = 1 < x | M | y > = 1 < y | M | y > = 0

< x | BS | x > = < y | BS | y > = \( \frac{1}{2}^{ \frac{1}{2}}\) < y | BS | x > = < x | BS | y > = \( \frac{i}{2}^{ \frac{1}{2}}\)

Dirac brackets are read from right to left. In Dirac's notation < x | M | y > is the amplitude that a photon initially moving in the y-direction will be moving in the x-direction after interacting with the mirror. |< x | M | y >|² is the probability that a photon initially moving in the y-direction will be moving in the x-direction after interacting with the mirror. |< y | BS | y >|² is the probability that a photon initially moving in the y-direction will be found moving in the y-direction after interacting with the beam splitter.

(A) For the photon to be detected at D₁ it must be in the state |x> after interacting with two beam splitters and a mirror in the configuration shown above. The probability that a photon will be detected at D1:

< x | BS M BS | y > = 0 thus |< x | BS M BS |y>|2 = 0

(B) For the photon to be detected at D2 it must be in the state |y> after interacting with two beam splitters and a mirror in the configuration shown above. The probability that a photon will be detected at D₂:

< y |BS M BS| y > = i thus |< y |BS M BS| y >|2 = 1

It is also instructive to use Dirac's notation to examine upper and lower paths.

(A')

\[\begin{align} \langle D_1| y \rangle &= \langle D_1 | y \rangle_{upper} + \langle D_1 | y \rangle_{lower} \\[4pt] &= \langle x | \textbf{BS} | x \rangle\langle x | \textbf{M} | y \rangle\langle y | \textbf{BS} | y \rangle + \langle x | \textbf{BS} | y \rangle\langle y | \textbf{M} | x \rangle\langle x | \textbf{BS} | y > \\[4pt] &= \frac{i}{2}^{ \frac{1}{2}} \times 1 \times \frac{i}{2}^{ \frac{1}{2}} + \frac{i}{2}^{ \frac{1}{2}} \times 1 \times \frac{i}{2}^{ \frac{1}{2}} \\[4pt] &= \frac{i}{2} - \frac{i}{2} = i \end{align} \nonumber \]

This shows that upper and lower paths have the photon arriving 180 degrees out of phase. Thus the photon suffers destructive interference at D1.

(B') < D₂ | y > = < D₂ | y >_upper + < D₂ | Y >_lower

= < y | BS | x >< x | M | y >< y | BS | y > + < y | BS | y >< y | M | x >< x | BS | y >

= \( \frac{i}{2}^{ \frac{1}{2}} \times 1 \times \frac{i}{2}^{ \frac{1}{2}}\) + \( \frac{i}{2}^{ \frac{1}{2}} \times 1 \times \frac{i}{2}^{ \frac{1}{2}}\)

= \( \frac{i}{2} - \frac{i}{2} = i\) i

Thus, |< D₂ | y >|2 = 1

This calculation shows that the upper and lower paths have the photon arriving in phase at D₂.

If either path (upper or lower) is blocked the interference no longer occurs and the photon reaches D1 25% of the time and D2 25%. Of course, 50% of the time it is absorbed by the blocker.

Lower path blocked:

Probability photon reaches D₁: |< x | BS | x >< x | M | y >< y | BS | y >|2 = \( \frac{1}{4}\)

Probability photon reaches D₂: |< y | BS | x >< x | M | y >< y | BS | y >|2 = \( \frac{1}{4}\)

Upper path blocked:

Probability photon reaches D₁: |< x | BS | y >< y | M | x >< x | BS | y >|2 = \( \frac{1}{4}\)

Probability photon reaches D₂: |< y | BS | y >< y | M | x >< x | BS | y >|2 = \( \frac{1}{4}\)

References:

1. P. Grangier, G. Roger, and A. Aspect, "Experimental Evidence for Photon Anticorrelation Effects on a Beam Splitter: A New Light on Single Photon Interferences," Europhys. Lett. 1, 173-179 (1986).
2. V. Scarani and A. Suarez, "Introducing Quantum Mechanics: One-particle Interferences," Am. J. Phys. 66, 718-721 (1998).
3. Kwiat, P, Weinfurter, H., and Zeilinger, A, "Quantum Seeing in the Dark," Sci. Amer. Nov. 1996, pp 72-78.