## Two Photon Interference:The Creation of an Entangled State

#### 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 the upper path and the
other the lower path. The results of this experiment are that both photons
are detected at either A or B.  One photon is never detected at A while
the other is detected at B. A quantum mechanical analysis of this phenomena
is provided below.

UP

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

DOWN

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

(1)        |Source> ® [|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, |Source> would have to be
antisymmetric with respect interchange of coordinates and the positive

A photon that takes the upper path has a 50% chance of being detected
at A or B.  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> the photon taking
the upper path evolves into the state shown in equation (2).

(2)        |up> ® [i|A> + |B>]/21/2

Similar arguments show that the photon taking the lower path will
evolve to 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 (A and B represent the detectors,
while 1 and 2 designate the photons):

(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 recorded at A or both photons will be recorded
at B 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 equation (1)
would become,

(6)        |Source> ® [|up>1|down>2 - |down>1|up>2]/21/2

After substitution of equations (2) and (3) into (6) we find

(7)        |Source> ® [|B>1|A>2 - |A>1|B>2]/21/2

In other words the fermions are always detected at different detectors
and are never found at the same detector at the same time. In summary,
the sociology of bosons and fermions can be briefly stated: bosons are
gregarious and enjoy company; fermions are antisocial and prefer
solitude.

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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