7.3: Single-photon Interference - Third Version

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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 D₂ and never at D₁.(1,2,3) The quantum mechanical analysis of this striking phenomenon is outlined below. There are two paths (upper and lower) to each detector, and they both contain a beam splitter, a mirror, and another beam splitter before the detectors are reached. At the beam splitters the the probability amplitude for transmission is \( \left( \frac{1}{2} \right)^{ \frac{1}{2}}\), while for reflection it is \( \left( \frac{i}{2} \right)^{ \frac{1}{2}}\). The origin of the 90° phase difference between transmission and reflection is found in the principle of energy conservation.

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Because there are two paths to each detector the probability amplitudes for these paths may interfere constructively or destructively when added. For detector D₂ the probability amplitudes for the two paths interfere constructively, while for detector D₁ they interfere destructively.

For example, the probability for the photon being detected at \(D_2\) is calculated as follows:

\[ P(D_2) = |< D_2 | S >|^2 = |< D_2 | T >< T | S > + < D_2 | R >< R | S >|^2 = \bigg| \left( \frac{i}{2} \right)^{ \frac{1}{2}} \left( \frac{1}{2} \right)^{ \frac{1}{2}} + \left( \frac{1}{2} \right)^{ \frac{1}{2}} \left( \frac{i}{2} \right)^{ \frac{1}{2}} \bigg|^2 = 1 \label{258.1} \]

The probability that the photon will be detected at \(D_1\) is:

\[ P(D_1) = |< D_1 | S >|^2 = |< D_1 | T >< T | S > + < D_1 | R >< R | S >|^2 = \left| \left( \frac{1}{2} \right)^{ \frac{1}{2}} \left( \frac{1}{2} \right)^{ \frac{1}{2}} + \left( \frac{i}{2} \right)^{ \frac{1}{2}} \left( \frac{i}{2} \right)^{ \frac{1}{2}} \right|^2 = 0 \label{258.2} \]

energy conservation

Suppose there is no phase difference between transmission and reflection. Then the probability amplitudes for transmission and reflection are both \( \left( \frac{1}{2} \right)^{ \frac{1}{2}}\). Under these circumstances Equations \ref{258.1} and \ref{258.2} become

\[ P(D_2) = \left| \left( \frac{1}{2} \right)^{ \frac{1}{2}} \left( \frac{1}{2} \right)^{ \frac{1}{2}} + \left( \frac{1}{2} \right)^{ \frac{1}{2}} \left( \frac{1}{2} \right)^{ \frac{1}{2}} \right|^2 = 1 \nonumber \]

\[ P(D_1) = \left| \left( \frac{1}{2} \right)^{ \frac{1}{2}} \left( \frac{1}{2} \right)^{ \frac{1}{2}} + \left( \frac{1}{2} \right)^{ \frac{1}{2}} \left( \frac{1}{2} \right)^{ \frac{1}{2}} \right|^2 = 1 \nonumber \]

This result violates the principle of conservation of energy because the original photon has a probability of 1 of being detected at D₁ and also a probability of 1 of being detected at \(D_2\). In other words, the number of photons has doubled. Thus, there must be a phase difference between transmission and reflection, and a 90^o phase difference, as shown above, conserves energy.

References

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).
V. Scarani and A. Suarez, "Introducing Quantum Mechanics: One-particle Interferences," Am. J. Phys. 66, 718-721 (1998).
Kwiat, P, Weinfurter, H., and Zeilinger, A, "Quantum Seeing in the Dark," Sci. Amer. Nov. 1996, pp 72-78.