## Using Dirac Notation to Analyze Single Particle Interference

### Frank RiouxDepartment of ChemistrySaint John's University College of Saint Benedict

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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.  The photon leaves the source, S, and whether it takes the upper
or lower path it interacts with a beam splitter, a mirror, and another
beam splitter before reaching the detectors. At the beam splitters there
is a 50% chance that the photon will be transmitted and a 50% chance that
it will be reflected.

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

After the first beam splitter the photon is in an even linear superposition
of being transmitted and reflected. Reflection involves a 90o (p/2)
phase change which is represented by exp(ip/2) = i, where i = (-1)1/2. (See
the appendix for a simple justification of the 90o phase difference
between transmission and reflection.) Thus the state after the first beam splitter
is given by equation (1).

(1)        |y> = [|T> + i|R>]/21/2

Now |T> and |R> will be written in terms of |D1> and |D2> the states they
evolve to at detection. |T> reaches |D1> by transmission and |D2> by
relection.

(2)        |T> = [|D1> + i|D2>]/21/2

|R> reaches |D1> by reflection and |D2> by transmission.

(3)        |R> = [i|D1> + |D2>]/21/2

Equations (2) and (3) are substituted into equation (1).

(4)        |y> = [|D1> + i|D2> + i2|D1> + i|D2>]/2

It is clear (i2 = -1) that the first and third terms cancel (the amplitudes
are 180o out of phase), so that we end up with a final state given
by equation 5.

(5)        |y> = i|D2>

The probability of an event is the square of the absolute magnitude of the
probability amplitude.

(6)        P(D2) = |i|2 = 1

Thus this analysis is in agreement with the experimental outcome that no
photons are ever detected at D1.

Appendix:

Suppose there is no phase difference between transmission and reflection.
Then equations (1), (2), and (3) become

(1')        |y> = [|T> + |R>]/21/2

(2')        |T> = [|D1> + |D2>]/21/2

(3')        |R> = [|D1> + |D2>]/21/2

Substitution of equations (2') and (3') into equation (1') yields

(4')        |y> = |D1> + |D2>

Thus, the detection probabilities at the two detectors are:

(5')        P(D1) = 1  and  P(D2) = 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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