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A Means to Purify an Entangled Source
Remi Cornwall
Future Energy Research Group
Queen Mary, University of London, Mile End Road, London E1 4NS
Abstract
A technique is presented for improving the ratio of entangled photons to un-entangled photons for any
means of generation. The approach takes advantage of the entangled nature of the photons of interest
and their concomitant temporal coherence length, to separate that component by a combination of beam
convergence, destructive interference, Faraday-Rotators, polarising filters and then beam divergence.
The method applies to energy-time entangled photons and matter waves too.
1. Introduction
Entanglement is a multi-body phenomenon
peculiar to quantum mechanics. It is related to
classical correlation if two or more particles
are constrained by a rule. For instance (this
example due to J. S. Bell[1]), if we had a sock
with a red or green ball and two people could
only chose a ball blindfolded, the first person
on eventually looking at their colour would
immediately know the other person’s ball
colour. This rule is a toy model of
conservation laws and we could say too that if
the rule was that the balls had equal and
opposite angular momentum, person 1 on
receiving a left spinning ball would
immediately know that person 2 had a right
spinning ball.
The situation in Quantum Mechanics is
intriguing due to the indeterminacy of the
measurement process. In the contrived
example with the sock and the balls, in a
quantum setting it wouldn’t be correct to say
before measurement that we know we must
have a red or green ball before measurement
and looking at the result only reveals what had
been predetermined; the balls are in a
superposition of red and green and the act of
measurement sets the state, not just locally but
over space-like separation[2].
There is no “spooky action at distance”
involving forces moving faster than light but
what we can ascertain and tabulate by
experiment number, after the fact, by
coincidence testing[3] if many such
experiments are performed. Any unitary[4]
(i.e. non-measurement) operation performed
on one particle of the set only changes the
correlations perceived at the end. For instance,
in our contrived example with balls, imagine
there was a unitary operation that substituted
(or “rotated”) our “colour states” from red-
green to yellow-blue respectively and this was
only applied to one ball of the pair (say the
first), coincidence counting after measurement
would then find, 1:yellow/2:green or
1:blue/2:red instead of 1:red/2:green or
1:green/2:red randomly. We don’t influence
the distant system even though it is subject to
the same conservation law (we made 1’s balls
yellow or blue, it doesn’t follow that 2 will see
yellow or blue balls too). Remote operations
cannot influence physical quantities over
space-like intervals, they can only collapse the
remote wavefunction and change the statistics
from being indeterminate to being fixed[5].
As a slight digression, it is believed that the
“No-communication theorem” saves Special
Relativity[6] but the author believes that,
somehow, only information is passed because
it has no mass-energy and so it not speed
constrained[7, 8] by showing an omission in
the no-communication theorem (it doesn’t take
account of phase and this can be ascertained by
an interferometer).
Thus on the level of pure science, if not
philosophical, work with entangled systems is
en vogue, though more prosaic reasons of
engineering a communications device by the
measure/no-measure protocol[7, 8] are greatly
assisted by better entangled sources.
2. Entangled sources
Entangled photons can be generated by a
variety of means: spontaneous parametric
down-conversion in non-linear material[9],
radiative decay of electron-hole pairs in a
quantum dot[10-12] or energy-time
entanglement from ions in potential traps[13].
An example of the first case, figure 1 shows
the spatial layout of the single photon downconversion system, where a high frequency
© Remi Cornwall 2014
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laser source is incident on a crystal. The
majority of the pumping high frequency
radiation passes through the crystal (some
1012:1) and various non-linear processes occur
producing uncorrelated beams of different
frequencies. Of most interest is the process that
leads to down conversion to photons of half
the energy and correlation/entanglement. At
the intersection of the two middle cones one
finds with a ratio of some 1:10,000, entangled
photon pairs occasionally created by
spontaneous emission and constrained by
energy and momentum conservation. Higher
powered pumping of the crystal generates
more entangled photons though the ratio
suffers. Cryogenic temperature can boost the
ratio of the desirable entangled photons to unentangled.
In this paper will we present a method of
improving this ratio of entangled to unentangled ratio by using the sine-qua-non of
entangled systems – their correlation.
3. Method
“Signal”
Polariser
or
polarising beam splitter
“Idler”
Mirror block or lens
bringing beams into
divergence
separates unentangled photons
Rotator
Mirror block or lens
bringing beams into
convergence
“Signal”
Polarisation
and phase
“Idler”
Crystal
At the intersection of the
two cones, entangled
photons are collected
Image copyright of the European
Space Agency
Figure 1 – Entanglement by down conversion
Obviously physical filtering by restricting
most of the gathered photons to the
intersection points (we can also use colour
filters) increases our chances of recovering the
entangled photons and we arrive at a density
matrix (without the unwanted element)
below[14].
Figure 2 – Density Matrix for
Entangled Photons[14]
Figure 3 – The Apparatus
The apparatus depicted above makes use of the
correlation (temporal, spatial) of entangled
photons to make them interfere. The “idler”
beam can be brought into the same polarisation
and anti-phase with the “signal” beam. They
then are made convergent in a region that has a
Faraday Rotator then a polarizing filter or
polarizing beam splitter (PBS). Since the
electrical fields of the entangled photons are
coherent and made to destructively interfere,
the rotator, responding to the electric field
strength will preferentially rotate the unwanted
un-entangled photons in the statistical blend of
photons travelling in the signal and idler
beams. The polariser or PBS then removes
these photons (figure 4). After this, the beams
can be made divergent again to recover the
signal and idler beams.
Improvements to the one stage scheme can be
made by sending the signal and idler beams
through multiple stages of the process.
Figure 1 depicts a narrow beam incident on the
crystal and though making the beam wider will
blur the collection points at the intersection of
© Remi Cornwall 2014
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the middle cones, the convergence region in
figure 3 would become wider and longer
allowing the collection of more entangled
photons. The rotation or rejection of the
rotator, polariser or PBS is a function of the
field strength squared and this very selectively
favours the near zero field strength from the
destructively interfering entangled photon
field.
arandom iθrandom
e
H A or B
2
(
)
6.
Hall, M.J.W., Imprecise
Measurements and Non-Locality in
Quantum Mechanics. Physics Letters
A, 1987. 125(2,3): p. 89,91.
7.
Cornwall, R.O., Secure Quantum
Communication and Superluminal
Signalling on the Bell Channel.
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Burnham D. C., Weinberg.D.L.,
Observation of simultaneity in
parametric production of optical
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25(84).
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Young, R.J.et.al., Improved fidelity of
triggered entangled photons from
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Akopian, N.et.al., Entangled photon
pairs from semiconductor quantum
dots. Phys. Rev. Lett., 2006. 96: p.
130501–130504.
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Muller, A., Fang, W., Lawall, J. &
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polarization-entangled photon pairs
from a semiconductor quantum dot
using the optical Stark effect. Phys.
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Franson, J.D., Bell Inequality for
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Dousse, A.S., et al, Ultrabright source
of entangled photon pairs. Nature,
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Blocked by
polarisers
ae
( H A ⊗ VB + VA ⊗ H B
2
)
Unaffected
Figure 4 – How separation is affected
4. Conclusion
This paper will be updated with experimental
results in due course. The method is general to
all sources of entangled photon production.
References
1.
Bell, J.S., Foundations of Quantum
Mechanics. New York: Academic,
1971. 171.
2.
Einstein, A.; Podolsky.B; Rosen.N.,
Can Quantum-Mechanical Description
of Physical Reality Be Considered
Complete? Phys. Rev., 1935. 47(777).
3.
Aspect, A.; Grangier, P; Roger, G.,
Experimental Realization of EinsteinPodolsky-Rosen-Bohm
Gedankenexperiment: A New
Violation of Bell's Inequalities. Phys.
Rev. Lett., 1982. 49(91).
4.
Peleg, Y.P., Reuven; Zaarur Elyahu,
Schaum's Outlines, Quantum
Mechanics. 1998: McGraw-Hill.
5.
Dopfer, B., Two experiments on the
interference of two-photon states. PhD
Thesis, University of Innsbruck, 1998.
© Remi Cornwall 2014