Testing Born’s rule with massice molecules

Classical vs. quantum

Classical physics explain the physical phenomena in our everyday world very well. On the other hand, quantum mechanics as formulated around 100 years ago describe how atoms and molecules behave. Now, imagine you start within something very small like a single atom and you start making that system larger and larger. In the beginning, the object will surely follow the rules of quantum mechanics, but at some point, the system will be so large that it counts as a classical object. The big question is when this transition will happen and what kind of rules govern the transition region.

A number of theories discuss extensions of quantum mechanics, which only show an effect for objects with a high mass. Hence, conducting tests with massive particles is the way to go if you want to learn something about the transition region.

Born’s rule

In this study, we take a look at a rule stated by the physicist Max Born in the early days of quantum mechanics. The rule says that the probability P to find a quantum-mechanical particle at a certain position can be inferred from the squared absolute wavefunction ψ.

P=|ψ|2 (1)

What does that mean? Say the wavefunction looks like depicted below. These are the two energetically lowest wavefunctions of a particle in a box, an easy model system. You can understand their shape when you think about a string of a guitar: When you play a note, the string vibrates in its fundamental tone. When you strike it harder you get a higher note, the first overtone. When we compute the probability to find the quantum mechanical particle, we observe that it does not depend on the relative sign of ψ. The wavefunction tells you where to find the particle in analogy to where the guitar string vibrates.

Double slit and more

In a diffraction experiment, the matter-wave traverses several slits at the same time and essentially, you get a wavefunction for each slit. Let’s consider a double slit with the individual slits A and B. The respective wavefunctions are ψA and ψB. To compute the probabilities, we insert them into equation (1) and get

P=|ψAB|2=|ψA|2+|ψB|2+2|ψAψB|. (2)

The first two terms give the probability behind the single slits A and B. The term 2|ψAψB|, however, describes the interference of the waves: The wave goes through two slits at the same time.

If we increase the number of slits to three the probability is as follows

P=|ψABC|2=|ψA|2+|ψB|2+|ψC|2+2|ψAψB|+2|ψAψC|+2|ψBψC| (3)

From our previous discussion, we can identify the components easily. The first three correspond to the probabilities behind three single slits A, B, and C. The last three describe the interference of the wave going through all possible combinations of double slits: AB, AC, and BC. Interestingly, the wave goes never through three slits at the same time, only through one and two according to Born. This does not change when you increase the number of slits; the maximum number of slits contributing to one term is two. However, must this be true? The only method to find that out is an experiment.

How it’s done

To test this rule we recorded the pattern behind a triple slit and compare it to the signal from all possible single and double slits which contribute to the pattern. Essentially, we tried to measure all the terms of equation 3 individually. In Figure 2 you can see the mask that is needed for such a test.

However, we simplified things a little. The signal through all three single slits should look identical. The only difference is that the patterns will be displaced by 100 nm. However, we cannot resolve such a small shift. Hence, we measured only one. The same is true for the combination of double slits AB and BC, where the spacing between the slits is identical. This means that the minimum set to conduct the test is the triple slit and those in the red rectangle. You just have to multiply the intensity of the single slit by 3 and the intensity of the BC slit by 2 to account for the masks A, B, and AB.

If Born’s rule is correct, the difference between the pattern behind the triple slit |ψABC|2 and all other patterns should be zero and nothing should remain. If Born’s rule is incomplete, a pattern should emerge.

After doing the analysis we indeed saw no pattern which points towards an additional term, namely higher-order interference. In the end, we saw that less than one molecule in a hundred deviates from Born’s rule. This is not a terribly small number. However, this experiment is limited by statistics, that is the count rate of the experiment. The recording of each diffraction pattern took around 8 hours and the experiment was conducted 5 times. The patterns themselves are quite beautiful as you see below.

The study is the first dedicated test of Born’s rule with massive particles. Although the accuracy is three orders of magnitude lower than in current experiments with light [1], it is an important step to test quantum mechanics with massive particles. Interestingly, it is quite easy to generate a violation of Born’s rule. However, this is not done with massive matter-waves but with microwaves that have a much longer wavelength [2]. These experiments do not search for a violation of quantum mechanics with massive particles but take a look at the mathematical subtleties in the formulation. In our experiment, the difference in wavelength predicts a deviation of one molecule in a billion. Hence, there is quite a bit of room left to search for new effects.


[1] T. Kauten, R. Keil, T. Kaufmann, B. Pressl, C. Brukner, and G.Weihs, “Obtaining tight bounds on higher-order interferences with a 5-path interferometer“, ArXiv 1508.03253v3 (2016)

[2] G.Rengaraj, U.Prathwiraj, Surya Narayan Sahoo, R.Somashekhar, and Urbasi Sinha, “Measuring the deviation from the superposition principle in interference experiments“, ArXiv 1610.09143 (2016)