Local realistic simulation of a Bell type experiment with two beams of equally polarized light.

Note: Light is considered a moving spread disturbance of a background medium. The disturbance can implement all of the features of  waves (intensity, wavelength, frequency, phase, polarization, phase speed, signal speed, etc) in such a way that within some local domain (of space and of time), well defined values of those wave magnitudes can be assigned to the local disturbance. Nothing new with that. It has been very well known for centuries.  But to understand what follows, the idea that light is made by localized particles (photons) must be rejected. The corpuscular behavior of light (in its interactions with matter) admits other explanations (not studied here) without the need to suppose that light itself is made of particles of specific energy, momentum, spin, etc.

Consider a source that at any given time emits light of equal linear polarization along two different directions (arms of the apparatus). The polarization direction itself of the emitted light varies with time in a random way.  

Suppose that there is at each arm of the apparatus a detecting device. The detecting device of each arm consists of a calcite (or more generally a polarization analyzer) and two identical photo-detectors. It will be admitted that the calcite splits its incident light in two beams of mutually perpendicular plane polarized light that go to their corresponding photo-detector.

It will be admitted that the intensities of both beams emerging from a calcite satisfy exactly Malus law. More precisely:

Let the calcite be oriented along direction z.

Let q be the angle that the direction of polarization of an incident light makes with the axis z. Then the intensity of light emerging from channel "up" of the calcite is I_o Cos²(q) while the intensity emerging from channel "down" of the calcite is I_o Cos²(q - p/2). Notice also that I_o Cos²(q) + I_o Cos²(q - p/2) = I_o

In each arm of the apparatus its photo-detectors will be labeled detector "up" and detector "down" (or more simply u and d) according to the beam (channel) of the splitter in which they are placed.

To simplify the analysis of the experiment it can be supposed that the light reaching the four photo-detectors is weak enough so that during the short time intervals in which these detectors are "open" they produce at most one click (i.e. only one ejected photo-electron starts the cascade). In these circumstances, it also seems reasonable to assume that there will be cases in which a photo-detector makes no click at all during some of the time intervals in which it is "open".

The name "count" will be used for a set of the 4 photo-detectors outputs (click or no-click) during a corresponding time interval (coincidence window) in which they are open to receive light. The four outputs of a count must correspond to observation intervals during which the four detectors are receiving light emitted by the source at the same epoch.

In respect to the output of the 4 detectors during a corresponding "count", the following 16 possibilities can therefore take place:

uu   dd   ud   du     u0    d0   0d   0u   00     u2    d2   2d   2u   22    02    20

where the first character of the pair refers to the output of the detectors of the first arm and the second character to the output of those of the second arm. The characters indicate the following:

u = the detector "up" of the corresponding arm makes a click (but not the "down" detector).

d = the detector "down" of the corresponding arm makes a click (but not the "up" detector).

0 = neither detector of the corresponding arm makes a click.

2 = both detectors of the corresponding arm make a click.

Counts (observations) are made at "all" possible relative angles of the two calcites. The angle that the "significant" axis of the second calcite makes with the "significant" axis of the first will be called F. The fact that the lights reaching the calcites during a corresponding count are supposed to have the same plane polarization direction implies therefore that if such light makes an angle q with (the significant axis of) the calcite of the first arm, it makes an angle q-F with the significant axis of the calcite of the second arm.

A computer simulation has been made in which it has been supposed (for simplicity and without loss of generality) that the first calcite is in all observations oriented along the direction z while the second calcite is oriented relative to the first (and therefore relative to the direction z) at an angle F that can be chosen at will. As said above, in "correspondence" with Malus law, it has been supposed that, for an incoming light polarized along direction q (relative to z), the probabilities of click of the 4 detectors are respectively:

Detector up of the first arm Cos²(q)                                                       [BE-1u]

Detector down of the first arm Cos²(q - p/2)                                         [BE-1d]

Detector up of the second arm Cos²(q - F)                                           [BE-2u]

Detector down of the second arm Cos²(q - F -p/2)                              [BE-2d]

For example, a simulation has been made making 20,000 counts for each of the 33 following relative orientations of the calcites: F = {0, p/64, 2p/64, 3p/64, … , p/2}. In each of the 33 x 20,000 counts the direction of polarization q of the light reaching both calcites has been assumed to take a fresh random value (giving an equal probability to all the directions between q = 0 and q = p).

For any given "count", the click occurrences of the four detectors have been implemented by the program as follows:
- As just said, first a fresh direction q (the same for both arms) is randomly obtained.
- Next, four random numbers between 0 and 1, (i.e. a fresh random number for each detector) are obtained that are compared with the classic (Malus) light intensity reaching the corresponding photodetector at its particular orientation relative to the direction of polarization q of the incident light:

Suppose for example that the four detector’s random numbers of a given count are respectively {0.987,  0.022,  0.561,  0.842}

The detector up of the first arm makes a click if and only if              0.987 £ Cos²(q)

The detector down of the first arm makes a click if and only if         0.022 £ Cos²(q - p/2)

The detector up of the second arm makes a click if and only if         0.561 £ Cos²(q - F)

The detector down of the second arm makes a click if and only if     0.842 £ Cos²(q - F -p/2)

-----------------------

The following table collects the results of a simulation made with Visual-Basic 6.0 and its Rnd (random) function.

Table [BE-3]

(The last row, labeled S, presents the sums of the respective columns).

The following graphic plots the number of counts with "coincidences" as a function of the angle F between the calcites.

1) considering coincidences only those counts in which the output is either uu or dd (i.e. only the first two columns of the table):

                            Fig [BE-1]

Defining the correlation function (at a given relative angle F between the calcites) as:

                        [BE-3]

the above simulation gives the following plot:

                         Fig [BE-2]

2) considering coincidences, in a non strict way, all those counts in which the output is either uu, dd, u2, d2, 2u, 2d, 22 (i.e. in which detectors of the same kind (u or d) click in both arms including those cases in which both detectors of the same arm simultaneously click), the above simulation gave the following plot:

            Fig [BE-3]

Note: if in the above "non strict coincidences", those cases/counts (labeled 22) in which the four detectors click simultaneously are counted twice, a very similar plot to Fig(BE-3) is obtained.

Comments:

1) - I did not had access to any of the famous articles (e.g. those of J.F. Clauser or A. Aspect) describing the Bell type experiments with singlet "paired photons" showing the violation of Bell’s inequalities and I only have indirect information about their results. (An important part of my information about the experiments and controversies related with EPR and Bell tests comes from reading the interesting critical articles of Caroline H. Thompson http://users.aber.ac.uk/cat/ )

If I understood correctly, when the Bell testers plot their experimentally obtained coincidences and/or correlations they obtain "cosine type" curves that seem to be a signature of violation of Bell’s inequalities. The above plots of this article are also "cosine type" curves. Furthermore, when the so called CHSH69 test is applied to the correlations obtained in the above simulation it gives:

Calling two specific calcite orientations of the first arm of the apparatus: a and a’

Calling two specific calcite orientations of the second arm of the apparatus: b and b’

For example, choosing for a, a’, b, b’ the following values (commonly used in these kinds of Bell tests):

                        Table [BE-4]

Calling C(p,q) the correlation function measured (or in this case predicted by the simulation) when the 1^st calcite is oriented along direction p and the 2nd along direction q, the CHSH test requires to measure the value of

S = C(a, b) - C(a, b’) + C(a’, b) + C(a’, b’)

And considering that for any pair of orientations p, q of the calcites the correlation results with polarized light do only depend on their relative orientation F=|q-p| , using the settings of the above Table [BE-4]:

S = C(a, b) - C(a, b’) + C(a’, b) + C(a’, b’) = C(0, p/8) - C(0, 3p/8) + C(p/4, p/8) + C(p/4, 3p/8) =

= C(p/8) - C(3p/8) + C(p/8) + C(p/8) =

and substituting the values obtained in the above simulation (that can also be read from Fig[BE-2]):

= 0.6373 - (-0.6138) + 0.6373 + 0.6373   =   2.5257

But the CHSH form of Bell’s inequalities states that (whatever orientations a, a’, b, b’) the correlations obtainable from local realistic models must obey

-2 £  C(a, b) - C(a, b’) + C(a’, b) + C(a’, b’) £  2

but the model of light simulated above is local and realistic and violates such relation. A local (hidden variables) reality can under some reasonable suppositions violate Bell’s inequalities. Therefore "Bell theory of non-locality (and therefore experimental Bell-type tests) does not have general validity because the theory makes wrong assumptions about the necessary features of local realistic models".

I (and many others) consider for example a wrong assumption to suppose that light is made of photons (i.e. localized particles). If this wrong assumption is made it must also be assumed that when a photon (one of the "entangled" pair emitted by the source) reaches a calcite it must take either one or the other of the two possible paths leading to the u or d detectors. It can not, according to the orthodox concept of photon, take both paths and trigger both detectors. It can neither disappear without triggering any of the detectors. Making that wrong assumption it can indeed be reasoned (in the line of Bell's theoretical reasonements) that a local realistic photon can not violate Bell's inequalities. (Simulation 2, below, reinforces such assertion). But instead, as the above simulation shows, it can also be reasoned that with a non-photonic but otherwise local realistic model of light, the Bell inequalities can be violated.

Simulation 2.

Another computer simulation with these "photon" assumptions has been made:

For any given "count", the click occurrences of the four detectors have now been implemented by the program as follows:

- First, a fresh polarization direction q (the same for both arms) is randomly obtained.
- Next, only two (instead of four) random numbers between 0 and 1, (i.e. a fresh random number for each arm) are obtained that are compared with the classic (Malus) probability that the photon takes the path "up" of the corresponding calcite at its particular orientation (relative to the direction of polarization q of the incident light):

Suppose for example that the two calcite’s random numbers of a given photon pair are respectively {0.874, 0.369}

The detector up of the first arm makes a click if and only if              0.874 £ Cos²(q)

The detector down of the first arm makes a click if and only if       0.874 > Cos²(q)

The detector up of the second arm makes a click if and only if         0.369 £ Cos²(q - F)

The detector down of the second arm makes a click if and only if  0.369 > Cos²(q - F)

With the same definition of the correlation function  

a typical photon model local realistic simulation gives instead the following plot:

                      Fig [BE-5]

The CHSH test now gives

S = C(a, b) - C(a, b’) + C(a’, b) + C(a’, b’) =

   = C(p/8) - C(3p/8) + C(p/8) + C(p/8) =

   =  0.3512 - (-0.3597) + 0.3512 + 0.3512   =   1.4133

that now obeys    -2 £  S £  2  therefore contradicting the experimental facts.

Summarizing:

A wave type disturbance (spread in space) model of light can be assumed to have a strictly locally defined polarization and can be assumed to trigger or not the photo-detectors according only to local conditions. (See the end of Section 6 of my @ether model http://personales.ya.com/carlosla/model/EVA6/Eva6.htm where I try to explain that the photo-detectors are triggered if the intensity of the light disturbance is "bigger" than that of the local @ether noise). This local realistic model of light can (in principle) explain the experimental Bell's inequalities violations without the weirdness of non-locality

If instead a photon type (localized in space) model of light is assumed to have a strictly locally defined polarization it can be reasoned and predicted (as Bell does) that it will obey Bell's inequalities and therefore will be unable to explain the experimental facts.

The theoretical analysis (and conclusions about non-locality) of Bell (and followers) do not have a general validity.

 Simulation 3. With reduced photo-detector’s efficiencies.

28-Mar-2003. Having posted a summary of the above results to the Yahoo group qm2 http://groups.yahoo.com/group/qm2/ , Caroline H. Thompson made several comments and suggestions one of which was:

Before going much further, how about doing the following:

(a) Run the model simulating imperfect detectors.  To make it possible (in
theory) to check the results, do this by first using your Malus' Law method
then throwing away the count with probability (1 - e), where e is the
"efficiency".  You should, if I'm right, find that when you have very low
efficiency (as in real experiments) you can no longer violate your test …

My first thought was to answer:

Well, …, if I suppose in my simulations that the "efficiency" e of the photo-detectors is independent of the intensity of the radiation that they receive this should have no influence in the violation or not of the test that I have so far simulated, because such supposition would only have the effect to multiply the number of counts of the types uu, dd, ud and du by the same constant value e². Therefore the correlation function that I have been using (uu+dd-ud-du)/(uu+dd+ud+du) would be the same as before (at any given relative angle) whatever the efficiency e.

But, just in case my intuition was wrong I made the simulation suggested by Caroline and: Caroline is right. Here is the result of the simulation suggested by Caroline:

I chose an "efficiency" e = 0.4 (the same of course for the four photo-detectors). As above the program samples 20,000 times (counts) each relative angle of the calcites. For each count:

- first a fresh direction q (the same for both arms) is randomly obtained.
- next, four random numbers between 0 and 1, (i.e. a fresh random number for each detector) are obtained that are compared with the classic (Malus) light intensity reaching the corresponding photo-detector at its particular orientation relative to the direction of polarization q of the incident light.

Suppose for example that the four detector’s random numbers of a given count are respectively {0.987, 0.022, 0.561, 0.842}

The detector up of the first arm makes a pre- click if and only if              0.987 £ Cos²(q) . e

The detector down of the first arm makes a click if and only if         0.022 £ Cos²(q - p/2) . e

The detector up of the second arm makes a click if and only if         0.561 £ Cos²(q - F) . e

The detector down of the second arm makes a click if and only if     0.842 £ Cos²(q - F -p/2) . e

Note: this way of deciding the click/no-click multiplying the Malus intensities by the efficiency e and comparing the product with the random number has the same effect (i.e. produces the same distribution of counts) as the routine suggested by C.H.T.   that consists in "throwing away the count with probability (1 - e)" doing the following : (for example with the detector u of the first arm): If   0.987 > Cos²(q) consider it straight away a no-click and go to deduce the event of the next detector. But if 0.987 £ Cos²(q) consider it a candidate for a click and find another random number r (again between 0 and 1) . If r > (1-e) then consider it a click and otherwise a no-click, and go to deduce the event of the next detector. (In spite, I insist, that as I have checked, this second routine produces statistically identical results, I feel more comfortable with the first that depends on only one random "event").

                Fig [BE-6]

Using the same definition of the correlation function     

                    Fig [BE-7]

The CHSH test now gives

S = C(a, b) - C(a, b’) + C(a’, b) + C(a’, b’) =

   = C(p/8) - C(3p/8) + C(p/8) + C(p/8) = 1.767 < 2

and therefore, as Caroline predicted, the test is no longer violated (for this example detector efficiency e=0.4)

Trying to understand where failed my intuition I compare the absolute number of counts of the types uu,dd,ud,du of the first test with those of this latter   "reduced efficiency test" that I label with upper case letters UU, DD, UD, DU in the following table: