## Another Two Photon Interference Experiment

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

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In this experiment a down converter, DC, converts an incident photon
into two lower energy photons. One photon takes an upper path traveling
left or right, and the other photon takes one of the lower paths traveling
in the opposite direction. The results of this experiment are that both photons
are recorded at either the A detectors or B detectors.  One photon is never
observed at an A detector while the other is observed at a B detector. A
quantum mechanical analysis of this phenomena is provided below.

UP

M -----      ----- M        M = mirror
^          ^
/ \        / \
(detector) A     /   \      /   \     A (detector)
\   /     \    /     \   /
\ /       \  /       \ /
BS --x--       DC       --x-- BS    DC = down converter   BS = 50/50 beam splitter
/ \       /  \       / \
/   \     /    \     /   \
(detector) B     \   /      \   /     B (detector)
\ /        \ /
v          v
M -----      ----- M       M = mirror

DOWN

After the down converter the initial photon has evolved into a state which
is an entangled linear superposition.

(1)        |DC> ® [|up>1|down>2 + |down>1|up>2]/21/2

This is an entangled state (non-factorable) because it acknowledges
that it is unknown which photon takes which path. It also incorporates
the fact that photons are bosons and consequently the state function
must be symmetric with respect to interchange of the coordinates
(paths) of the photons. If photons were fermions, |DC> would have to be
antisymmetric with respect interchange of coordinates and the positive

A photon that takes one of the upper paths has a 50% chance of being recorded
at an A or a B detector.  To reach A it must be reflected at the beam splitter
and to reach B is must be transmitted. Conservation of energy requires a 90
degree phase difference between transmission and reflection, and by
convention this phase difference is assigned to reflection. To reach
detector A the upper photon must undergo a reflection at the beam
splitter and its phase shift is recorded by multiplying |A> by i [(-1)1/2].
Thus in terms of the detector states |A> and |B> a photon taking
an upper path evolves into the state shown in equation (2).

(2)        |up> ® [i|A> + |B>]/21/2

Similar arguments show that a photon taking one of the lower paths will
evolve into the state given by equation (3).

(3)        |down> ® [|A> + i|B>]/21/2

When equations (2) and (3) are substituted into equation (1) the
following final state results:

(4)  |Source> ® [i|A>1|A>2 + i2|A>1|B>2 + |B>1|A>2 + i|B>1|B>2

+ i|A>1|A>2 + |A>1|B>2 + i2|B>1|A>2 + i|B>1|B>2]/23/2

Thus there eight final probability amplitudes, and they come in four
pairs as can be seen above. However, two of the pairs destructively
interfer (see note below) with each other and the final state is,

(5)  |Source> ® [i|A>1|A>2 + i|B>1|B>2]/21/2

The probability of an outcome is found by taking the square of the
absolute magnitude of its probability amplitude. Thus the probability
that both photons will be observed at A detectors, or both photons will be observed
at B detectors is calculated as follows:

P(AA) = |i/21/2|^2 = 1/2         P(BB) = |i/21/2|^2 = 1/2

It was noted earlier that fermions require anti-symmetric state functions.
So if this experiment could be performed with fermions the results would
be P(AA) = P(BB) = 0, and P(AB) = P(BA) = 1/2.

Note: "The things that interfere in quantum mechanics are not particles.
They are probability amplitudes for certain events. It is the fact that
probability amplitudes add up like complex numbers that is responsible
for all quantum mechanical interferences."  Roy J. Glauber, American
Journal of Physics, 63(1), 12 (1995).

Reference: Greenberger, D. M.; Horne, M. A.; Zeilinger, A. Physics
Today, 1993, 44(8), 22.
Back to Frank Rioux's  homepage.

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