Wave-Particle Duality

The present resource was developed at the Physics Institute of the Universidade Federal do Rio Grande do Sul by Michel Betz, Ismael de Lima, and Gabriel Mussatto.

Its purpose is to present a discussion of wave-particle duality, a fundamental concept of quantum physics. The Mach-Zehnder interferometer is used as an illustration.

This experimental device, which is represented here only schematically, was developed simultaneously and independently by Zehnder and Mach (Ludwig, Ernst's son) in 1892. It is composed of a laser source whose emitted light is split by a semireflective mirror, such that half of the light intensity is reflected and the other half is transmitted. Both resulting beams - reflected and transmitted - propagate in perpendicular directions till each one of them encounters a totally reflective mirror, so that the two beams meet again at a second semireflective mirror. At that point, each beam is partially reflected and partially transmitted, again in equal proportions. A light detector is located on the path of each final beam component.

Given this situation, how much do you think each detector will register?

Since each component is divided in two equal parts by a semireflective mirror, it seems reasonable a priori to expect that each detector will measure 50% of the total beam intensity. However, this is not what is observed. As you may confirm by executing the animation, the initial beam intensity is entirely registered by the red detector; the blue detector does not register anything!

The phenomenon observed in the previous animation is explained by Classical Wave Theory. In order to understand how, consider the light beam as a monochromatic wave propagating in a straight line in each part of the interferometer. In the animation, such a wave can be represented as a succession of wave fronts perpendicular to the beam direction.

When it reaches the first semireflective mirror, the light wave splits in two components, each with half the original intensity. According to classical wave theory, a wave's intensity I is proportional to the square of its amplitude ψ. Therefore, since the intensities of the reflected and transmitted waves are half the initial intensity each (I₀/2), the amplitude must be ψ₀/√2 for both.

Eventhough these components have the same amplitude, there is a difference between them: reflection results in a phase shift, whereas transmission does not. As a result, there is a phase difference between the reflected and transmitted waves. As an analogy, recall that a wave pulse in a string fixed at one of its extremities comes back inverted after reflection. In the case of the interferometer, it will be assumed that the mirrors and wave splitters are such as to provoke a shift by a quarter of a wave length (λ/4) in a reflection - see the Additional Informations for details.

When these two beams reach the second beam splitter, they are out of phase, for one of them suffered one more reflection, compared to the other. At that point, each beam is divided again, with each component carrying half the intensity, which corresponds to an amplitude half the amplitude of the original beam (ψ₀/2). These new components interfere constructively or destructively, depending on their relative phases.

The components reaching the blue detector have the same amplitude and are out a phase by half a wave length (λ/2). This follows because the beam travelling along path B suffered three reflections (at S1, E2 and S2), thus being shifted by 3λ/4, whereas the beam along path A underwent one reflection only (at E1) and was shifted by λ/4. The outcome is a relative phase shift of λ/2 between the two components, corresponding to a situation of completely destructive interference and, therefore, zero intensity.

In contrast, the components which reach the red detector, beside possessing equal amplitudes, are in phase since each one of them suffered two reflections. The component which followed path A was reflected at E1 and at S2. The component which travelled along path B was reflected at S1 and at E2. Thus, the interference between them is completely constructive and their amplitudes add up. Since the amplitude of each component, after being split twice, is ψ₀/2, the wave resulting from the superposition has amplitude ψ₀/2 + ψ₀/2 = ψ₀. In other words, the value registered by the red detector corresponds to the total intensity of the initial beam.

What would happen if, after separating the beam into two components with the first beam splitter, we inserted a transparent blade on the path of one of the components?

When light goes through a physical medium of density different from that of air, its velocity v changes but its frequency f remains the same. In order that the relationship v=f. λ be maintained, the light's wave length λ must also necessarily be modified. As a consequence, the blade's presence induces a phase shift in the beam that goes through it. Recall that the refraction index is defined as n=c/v, where c is the velocity of light in vacuum and v is the velocity of light in the medium.

In the visualization, you can change the refraction index of the blade and/or its thickness and find out the value of the corresponding phase shift. The phase shift and the thickness d are given in units of the wave length. The blade's thickness may be adjusted between 1 and 2 λ. The refraction index may be varied between 1 and 2. The phase shift is given by the relationship φ=(n-1)d, in units of λ.

You will be able to notice that, when the refraction index and/or the thickess of the blade is modified, the percentage of the initial-beam intensity registered by each detector also changes. This is due to the fact that one of the components suffers an additional phase shift, which affects the phase difference between the two beam components. As a result, when these components go through the second semireflective mirror, the superpositions forming the components in the directions of the two detectors will not, in general, be totally constructive or totally destructive. Each detector will register a part of the original beam intensity, the percentage amount received by each detector depending on the phase shift induced by the transparent blade.

The experimental situations investigated thus far correspond to typically wave-like phenomena and can be understood on the basis of Classical Wave Physics. However, if we lower sufficiently the intensity of the beam, particle aspects will become manifest. This is the case when the experiment becomes capable of identifying individual energy quanta, known as photons. When a physical system satisfies this condition, one may say that it operates in the quantum regime. Phenomena under such conditions lie beyond the explicative domain of Classical Wave Physics to be more adequately described by a theory grounded at a deeper level, namely Quantum Theory.

Eventhough these two theories differ considerably in their mathematical formalism and conceptual basis, they possess some analogous features that are worth pointing out. In wave physics, as has already been seen, the intensity of a light beam is proportional to the square of the wave amplitude (I is proportional to Ψ²). In the quantum regime, intensity is conceived as the number of photons detected (per unit time). The wave function Ψ represents the wave packet associated to the photon. The square of the modulus of the Ψ function is proportional to the probability for the photon to be detected in a given region of space at a given time. (Probability proportional to │Ψ│²).

Let us now consider the behavior of the Mach-Zehnder interferometer in the case of monophotonic states, that is, when photons are injected in the apparatus one at a time. Will the interference phenomenon still manifest itself or will the photons behave like classical particles, each one of them travelling through one arm of the apparatus or the other?

In the visualization, the little ball represents the wave packet associated to a photon. Notice that, when it reaches the first semireflective mirror, it separates into two parts (represented by the same symbol in a weaker tone). But when they reach the second semireflective mirror, these two components recombine, restauring the packet to its initial state again. As one may observe, this packet propagates always in the direction of the red detector.

In fact, what happens to the wave packet is the same as with the components of the light wave in the classical regime. When a wave-packet component is reflected, it gets phase-shifted by λ/4. When it is transmitted, its phase remains unchanged. When they reach the second semireflective mirror, the two components interfere constructively and destructively, depending on their relative phases. Just as in the classical case, the interference is constructive in the direction of the red detector only, in which the components recombine to form a wave packet with amplitude Ψ equal to the initial one. In the direction of the blue detector, the interference is completely destructive and the amplitude vanishes.

According to the quantum interpretation mentioned above, the wave function is associated to the probability of detecting a photon in a given region. Therefore, where the interference of the wave-packet components is totally constructive, the presence of the photon is certain and the red detector has 100% chance to register it. Since the wave-packet components in the direction of the blue detector cancel each other, the probability of detecting the photon in that region is zero.

Suppose now that a transparent blade is inserted on the path of one of the beam components, as was done previously in the discussion based on classical wave theory. The difference is that now it is supposed that the beam intensity is so low and the detector sensitivity so good that each photon is detected individually.

As before, the wave packet associated to each photon is divided by the first semireflective mirror and the two resulting components interfere at the second semireflective mirror. Also as before, the blade provokes a phase shift in the wave-packet component that goes through it. Just as for the classical wave, this phase change depends on the thickness and on the refraction index of the blade.

In the visualization, you may alter these properties of the blade, thus inducing different phase shifts. Depending on the relative phases between the components, the interference may be of an intermediate type, neither totally constructive nor totally destructive. In such a case, the amplitude of the packet component moving in the direction of each detector is a fraction of the initial amplitude. The probability of detecting the photon in each detector shall depend on the amplitude of the packet component which reaches it. More precisely, this probability is proportional to the square of the modulus of the corresponding amplitude, as was already mentioned.

The table indicates the values predicted by theory for the probabilities of photon detection by each detector. The table also registers what is measured experimentally, namely the relative photon counts by each detector. Observe that, as the number of photons increases, the experimental values tend to get closer to the theoretical ones, although with considerable oscillations. You may find it interesting to let the animation run while you take a coffee break. When you come back, the experimental values should be close to the theoretical predictions.

Another feature you may notice if you watch the animation carefully is that, when a detector registers a photon, the associated wave-packet component at that detector briefly acquires its full intensity, while the wave-packet component at the other detector disappears. This illustrates the concept of state collapse, which will be discussed next.

As we have seen, the probability of encountering a photon is proportional to the square of (the modulus of) the function that represents the wave packet associated to that photon. In the interferometer, the packet is divided in two components by the first semireflective mirror, in such a way that the probability of detecting the photon in one of the arms of the apparatus is 50% for each arm.

What happens if we try to find out experimentally which path the photon actually followed? For this purpose, we need to insert a third detector in one of the arms, taking care of choosing a type of detector capable of observing the photon without absorbing it. In this way, if this detector is in arm A and gets triggered, we may infer that the photon went through arm A. If this same detector is not triggered, we may conclude that the photon followed path B.

If the photon is registered by a detector observing path A, what happens to the probability associated to the wave-packet component which follows path B? And vice-versa: if the photon presence is not registered at A, so that it must have followed path B, does the packet component at A still carry a probability of registering the photon?

In order to describe this situation, Quantum Physics makes use of a very peculiar concept: the concept of “collapse”. This vocable refers to a discontinuous alteration of the state of a quantum system when it is measured. In the example of the interferometer, it may be illustated as follows. Previous to being observed in one of the arms, the photon had a 50% probability of being in each one of them. After having performed the observation, the trajectory followed by the photon is known and the probability of it being found in the other arm abruptly becomes nil.

The rendering of the collapse is seen at the instant when one of the divided wave-packet components suddenly vanishes, whereas the other component, which until then possessed the relatively weak color corresponding to a 50% probability, suddenly acquires the same intense color as the initial wave packet, corresponding to a probability of 100%.

Notice also that when the path followed by the photon is observed, the characteristic interference phenomenon at the second beam splitter gets destroyed and both detectors now have the same probability of registering the photon, independently of the presence and the properties of the transparent blade. This is due to the fact that, when an observation is made, collapse occurs and from then on, the wave packet propagates along one of the paths only. In these conditions, there will be no interference between components at S2 and the presence of the blade on path A will have no effect. At S2 will occur the split of the wave packet into two new components of equal amplitudes. Therefore, to each one is associated a 50% probability of detecting the photon. And, furthermore, when one of the detectors, be it the red or the blue one, finally registers the photon, another collapse takes place.