Natural Science Forum / Physics / Research / September 2007

    Strange Aspects of the Franson Experiment
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    This is a long post, so I will first state
its gist so that you can decide if you want to invest
the time to read it.  Franson proposed an experiment
which has since been performed by others.
Certain results were predicted which were confirmed
by the experiment.  Later it was discovered
that the predicted results were mathematically impossible.
What went wrong?

__________________________________________________________________

    J.D. Franson, "Bell inequality for position and time",
Phys. Rev. Letters 19 (1989), 2205-8, proposed an experiment
to test aspects of "quantum interference".  
The experiment involves generating two photons at the same time,
which travel in opposite directions, passing through interferometers
on the way, to eventually reach detectors which report their polarizations.
In this experiment, "polarization" may be imagined as a binary variable
which can take on just the two values +1 and -1.

    detector 1 <---<  MZI <---< PS >---> MZI >---> detector 2

Here PS stands for a photon source, which produces a pair of photons.
    [In an idealized situation, we can imagine that there
    is just one pair of photons in the diagram during one run
    of the experiment.]
The arrows indicate the direction of travel of the photons.

    MZI represents an unbalanced Mach-Zehnder interferometer,
where "unbalanced" means that its arms are unequal.  There are two paths
through the interferometer, a long path and a short path.  The
length of the long path can be varied, which is thought to change
the "phase" of a photon passing through it.  For our purposes,
it is sufficient to imagine the phase adjustment as a switch
which can slightly lengthen the long path.
I am deliberately refraining from diagramming the MZI in detail because
I want to think of it as a "black box" with a switch
with two possible settings, "l" (for "long") and "ll" (for "longer"):  
            ___________
            |         |
            |     .l  |
            | .-->.ll |
            |         |
            |_________|
               MZI   

Readers who want to know in detail what is inside the box
can find a clear schematic in one of the papers I want to discuss:

    Aerts, S. Kwiat, P., Larsson, J-A, and Zukowski, M.,
        "Two-photon Franson-type experiments and local
        realism:, Phys. Rev. Letters 83 (1999), 2872-2875.

The other paper to be discussed describes an actual performance
of the Franson experiment:  

    Kwiat, P. Steinberg, A., and Chiao, R., "High-visibility
        interference in a Bell-inequality experiment for
        energy and time", Phys. Rev. A 47 (1993),
        www.arxiv.org/quant-ph/9912064 v1.

    Quantum mechanics suggests that the polarizations observed
in the Franson experiment should be correlated in particular ways,
for which it is hard to think of a classical explanation.  

    It is easy to imagine how the polarizations
*could* be correlated classically.  A photon pair is
produced together at a particular place at a particular time.
We can imagine each of the two photons as carrying a piece of
information furnished by the source (usually called a "hidden variable")
which specifies whether its detector should read +1 or -1.
For example, if it were true
(as quantum mechanics predicts for some situations)
that the detectors on both sides always gave the same reading  
(both +1  or both -1),
this could be explained by assuming that both photons carry the same
hidden variable, which determines the detectors' responses.

    J.S. Bell discovered that this "hidden variable" model
cannot reproduce *all* correlations predicted by quantum mechanics.
He proved that the correlations must satisfy certain inequalities,
now called "Bell inequalities",
which can be violated by quantum-mechanical predictions.  
Subsequently, other authors derived many similar inequalities,
which are also commonly called  "Bell inequalities" even though
Bell may not have proposed them.

    Franson's paper predicted that the results of his proposed
experiment would violate a Bell inequality, which implies that
they cannot be explained by any classical "hidden variable" model.
The experiment has since been performed by Kwiat, Steinberg, and Chiao
(reported in the above-referenced paper) and by other groups.

    [References to similar experiments of two other groups
    can be found in Chiao, Kwiat, and Steinberg, ``Quantum nonlocality
    in two-photon experiments at Berkeley",
    www.arxiv.org/qunt-ph9501016v1.  One of these references
    reports experimental violation of a Bell inequality
    "in principle".  I have not seen the other, which
    appears in a journal to which I do not have access.]
   

The paper of Kwiat, Steinberg, and Chiao concludes:

    "Furthermore, contingent on reasonable extra assumptions,
    we can infer a violation of a Bell inequality
    by more than 16 standard deviations.  We interpret these
    results to rule out the possibility of any local realistic
    theory underlying the simultaneous energy and time correlations
    of down-converted photon pairs."

That seems reasonably straightforward,
but next the plot thickens.   

    A few years after the above paper of Kwiat, Steinberg, and Chiao,
the previously-referenced paper
"Two-photon Franson-type experiments and local realism" of
Aerts, Kwiat, Larsson, and Zukowski appeared.    
This paper  presents a "hidden-variable" model
of the type considered by Bell
which reproduces the predicted results of quantum mechanics
for the Franson experiment.  The paper states:

        "... the above construction implies
        that *the Franson experiment does not and cannot
        violate local realism if one disregards the fact
        that the unbalanced Mach-Zehnder interferometers
        are extended objects.* [emphasis theirs]"  

Here the context makes clear that "local realism" can be identified
with experimental confirmation of the Bell inequalities.

    At this point, a reader who has been following the discussion
should do a double-take.  It is an undisputed mathematical fact that
a hidden variable theory MUST satisfy all Bell inequalities.
Results of the Franson experiment can be explained by a hidden variable
theory (that of Aerts, Kwiat, Larsson, and Zukowski).  Therefore,
results of the Franson experiment MUST satisfy all Bell inequalities.  
But Kwiat, Steinberg, and Chiao report that
under "reasonable additional assumptions",
they VIOLATE a Bell inequality by 16 standard deviations.  

    Something is seriously wrong here!  
What can it be?  I hope that some expert can clear this up.

    I haven't carefully checked all details
of the hidden variable model of Aerts, Kwiat, Larsson, and Zukowski,
but it looks OK.  And if there were an error in it,
it seems likely that Kwiat would have discovered it, since
he is an author of both papers, and the second paper casts
serious doubts on the first.

    Can the "reasonable additional assumptions" of
Kwiat, Steinberg, and Chiao be wrong?  It is impossible to say
because their paper doesn't state these assumptions.
It also doesn't state which Bell inequality is violated,
though I get the impression that it is probably the so-called
"CHSH" inequality.  Such important omissions make it very
difficult to evaluate the paper.

    Assuming that these assumptions are as reasonable as the authors
claim, it seems to me that there must be something badly wrong
with the experiment itself.  If so, this would be disturbing.
If one can't trust experimental results with a statistical significance
of 16 standard deviations as reported by experimenters of good repute,
how can one trust any reported experimental results?

    E. Santos has argued in a long paper

        ["The failure to perform a loophole-free test of
        Bell's inequality supports local realism",
        Found. Phys. 34 (2004), 1643-1673]

that the failure, over 40 years, to produce an unequivocal refutation
of "local realism" suggests that nature respects local realism,
and that this may herald a new principle of nature,
yet to be discovered.  
He thinks this may be analogous to the fact that
failures to produce a perpetual motion machine
led to the discovery of the Second Law of Thermodynamics.

   
    The Santos paper makes a number of thought-provoking points.
One is that when an experiment fails to reproduce results predicted
by quantum mechanics, it is generally assumed that
there must be something wrong with the experiment.
But experiments which confirm quantum mechanics
are accepted without question.  

    I take no position on these issues.
I am deeply puzzled and uncertain how to evaluate
the available experimental evidence.

    [Before concluding, I want to digress to address
    a possible misconception.  The above referenced paper of
    Aerts, Kwiat, Larsson, and Zukowski contains much more
    than I have stated so far.  

        The above discussion treated the MZI's as
    "black boxes" with irrelevant internal structure.
    That was appropriate for the experiment reported
    by Kwiat, Steinberg, and Chiao, but it is not
    necessarily appropriate for similar experiments.

    Aerts, et al., consider a similar experiment
    (a "thought" experiment which has not been performed)
    in which the MZI "switches" are thrown rapidly enough
    that the time it takes light to cross the interferometer
    becomes significant.  They conclude that
    the CHSH "Bell" inequality is not applicable to this case,
    so that its violation has no significance.
    A casual reader might well come away with the impression
    that this resolves the contradiction of the experimental
    results of Kwiat, et al., with the existence of the
    hidden variable model of Aerts, et al. (which treated
    the MZI's as "black boxes" of irrelevant internal structure).

    It may be true that the Bell inequality is inapplicable
    to an experiment in which the switching is so fast that
    the internal structure of the MZI has to be taken into account.
    But there is no suggestion in the above-cited report
    of Kwiat, Steinberg, and Chiao, that this is the case
    for the reported experiment.  
    Switching speed is never mentioned.
    The impression given is that in a given run of the experiment,
    the switches were set and the correlation data
    gathered for that setting.  Then the process was presumably
    repeated.  In other words, in a given run of the experiment,
    the switches were presumably not moved at all.

    I have to keep saying "presumably" because the paper does
    specifically address these points.  The above is my best
    guess.
   
    It is understandable that the original Kwiat, et al., paper
    did not clarify this point because the authors might well
    not have thought of it.  They probably believed that they
    were merely confirming an unequivocal prediction of quantum
    mechanics.  (It is disturbing that this prediction, which
    *was* presented as essentially unequivocal in both
    the original Franson paper and the Kwiat, et al. paper, is
    now known to be mathematically impossible.)
    But it is truly disappointing that the later Aerts, et al.,
    paper did not address this obviously important issue.]
   
    I have long lamented the unreliability of the theoretical
physics literature.  A surprising proportion of papers contain
serious mathematical errors which not unfrequently
affect some of their conclusions.  
It is often virtually impossible to figure out
what a paper claims to establish due to unstated assumptions or
undefined notation.  Known errors are seldom corrected.
Every tedious calculation has to be repeated by every reader
who wants to be sure of its correctness.

    I guess there is no reason to expect the experimental
literature to be any more reliable, but I nevertheless
am deeply disturbed by the above contradictions.  
I find it very difficult to give up the idea of "local realism",
but even more difficult to ignore the evidence against it.
If the evidence can be neither trusted nor ignored,
what is one to think?

    Am I missing something?
Can the above contradictions be resolved?
Or have experts long known that the results of the experiment
of Kwiat, Steinberg, and Chiao are not to be trusted,
but no one will say so in print?

    Or is the experimental evidence reliable, but
the "reasonable extra assumptions" of Kwiat, et al. (which
imply that the experimental results violate a Bell inequality)
are not so reasonable after all?  
If so, shouldn't these assumptions be clearly stated so that
other authors may look at them more skeptically in the future?

    Another disturbing issue is that the general quality
of the above-discussed papers seems much higher than is  
typical in the literature.  In particular, I have always
admired the clarity of Kwiat's papers.  
If the *best* turns out to be questionable,
maybe one *should* demand
unequivocal, loophole-free, repeatable evidence
before drawing conclusions.