## Using Dirac Notation to Analyze Single Particle Interference

### Frank Rioux

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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 1/21/2, while for reflection it is i/21/2.
The origin of the 90o phase difference between transmission and reflection
is found in the principle of energy conservation as is shown in the appendix.

Upper Path

\ BS              \
(S)- -> - -\- - - - T - - - -\ M
|\                |\
|                 |
|                 |
R                 |
|                 |
|                 |
\|             BS \|
M \- - - - - - - - -\- - - -> D2
\                |\                 BS = Beam splitter (50/50)
Lower Path    |                  D  = Detector
|                  M  = Mirror
v                  S  = Source
D1

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

For example, the probability for the photon being detected at D2 is calculated
as follows:

(1)  P(D2) = |< D2 | S >|2 = |< D2 | T >< T | S > + < D2 | R >< R | S >|2

= |(i/21/2)(1/21/2) + (1/21/2)(i/21/2)|2 = 1

The probability that the photon will be detected at D1 is:

(2)  P(D1) = |< D1 | S >|2 = |< D1 | T >< T | S > + < D1 | R >< R | S >|2

= |(1/21/2)(1/21/2) + (i/21/2)(i/21/2)|2 = 0

Appendix:

Suppose there is no phase difference between transmission and reflection.
Then the probability amplitudes for transmission and reflection are both 1/21/2.
Under these circumstances equations (1) and (2) become

(1')  P(D2) = |(1/21/2)(1/21/2) + (1/21/2)(1/21/2)|2 = 1

(2')  P(D1) = |(1/21/2)(1/21/2) + (1/21/2)(1/21/2)|2 = 1

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 D2. In other words, the number
of photons has doubled. Thus, there must be a phase difference between
transmission and reflection, and a 90o 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.
```

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