General Introduction and Foundational Experiments

1. Introduction

The Stern-Gerlach experiments forced us to abandon the picture of a spin that would simply be a small classical vector. They also led us to write relations such as

But what does this addition concretely mean? Why add amplitudes, and then take the square of their modulus, instead of adding probabilities directly? And what use can be made of the complex phase that appears in a general superposition?

The Mach—Zehnder interferometer provides a particularly clear answer to these questions. It requires only two paths, two beam splitters and two detectors. Unlike Young's slits, the two possibilities remain spatially distinct during almost the whole experiment: one can act on one of the paths without touching the other, and then decide whether or not to recombine them. This separation makes the reasoning easier to follow.

We shall begin with the classical experiment using a light wave. We shall then reduce the source down to the regime where detections occur one by one. Finally, we shall attach a different mark to each of the two paths. This will lead us to the quantum eraser and to its essential point, often badly presented: interference reappears only in subsets obtained by post-selection. It never reappears in the raw data.

2. The Mach—Zehnder interferometer

2.1. Two paths that separate and rejoin

A Mach—Zehnder interferometer is composed of two semi-reflecting plates, written $\mathrm{BS}_1$ and $\mathrm{BS}_2$ for beam splitters, of two mirrors and of two detectors $D_1,D_2$.

The first plate receives a wave and produces two of equal intensity. Half of the amplitude follows arm $a$, the other arm $b$. The mirrors redirect them towards the second plate, where the two waves recombine.

The important word here is amplitude. If the incident amplitude equals $A$, a balanced plate does not produce two amplitudes $A/2$. The intensity is proportional to the square of the modulus of the amplitude. To share the intensity into two equal parts, one must therefore produce two amplitudes of modulus $|A|/\sqrt2$.

During their propagation, the two waves accumulate phases. If the optical lengths of the two arms are not exactly the same, they arrive on $\mathrm{BS}_2$ with a phase difference $\varphi$. One can vary $\varphi$ continuously by slightly displacing a mirror or by inserting a glass plate on one arm.

The second plate adds the contributions coming from the two paths. Depending on the output port, they add with the same sign or with an opposite sign. With a suitable choice of phase conventions, the output amplitudes are proportional to

After normalization, the intensities equal

When $\varphi=0$, all the contributions interfere constructively towards $D_1$ and destructively towards $D_2$. All the light exits through $D_1$, while $D_2$ remains dark. For $\varphi=\pi$, the roles are reversed. Between the two, the intensity passes continuously from one output to the other.

Nothing quantum has yet come into play. We have just described a classical experiment in wave optics. A water wave too can separate, travel along two paths and interfere with itself when they are brought together.

2.2. Why ordinary probabilities are not enough

Let us now imagine, for comparison, a purely corpuscular model. At the first plate, each particle would choose at random arm $a$ or arm $b$. At the second, it would again choose one of the two outputs. If the plates are balanced, this model predicts

whatever the length of the arms.

The reason is simple: the probabilities corresponding to two exclusive possibilities add. They cannot cancel. Two positive probabilities will never produce a perfectly dark detector.

A wave, on the other hand, can cancel because its amplitudes possess a sign, or more generally a phase. In expression (2), the two contributions towards $D_2$ cancel when $e^{i\varphi}=1$. It is only after this addition that one computes the intensity by taking the squared modulus.

3. One particle at a time

3.1. Indivisible impacts, wave-like statistics

Let us now reduce the light flux and use a source capable of preparing single photons. One ensures that there is, with very high probability, only one photon in the interferometer at a given instant.

The detectors no longer receive a continuous intensity. They produce individual events: a click at $D_1$, or a click at $D_2$. A photon is not observed in the form of two simultaneous half-clicks. In an ideal experiment, it is detected on one side or on the other.

One might think that this indivisibility restores the previous corpuscular model: the photon would have chosen an arm, and then chosen an output. But when the experiment is repeated and $\varphi$ is varied, the frequencies of the clicks follow

For $\varphi=0$, each photon is detected at $D_1$ and none at $D_2$. The statistics of indivisible particles obey exactly the interference law of the classical wave.

We are therefore faced with a combination that classical categories cannot describe:

• each detection is localized and indivisible;
• the probability of this detection depends on a phase acquired between two paths.

A purely wave-like theory explains the interference, but not the indivisible clicks. A purely corpuscular theory explains the clicks, but not their dependence on $\varphi$. Quantum mechanics preserves both facts by attributing an amplitude to each possibility, while attributing only one result to each measurement.

3.2. The path state

Just after the first plate, we shall represent the state by

After the propagation, a phase difference appears:

The symbols $\ket a$ and $\ket b$ do not denote two different particles. They represent the two path possibilities for one and the same preparation. The second plate transforms these possibilities in such a way that their amplitudes contribute together to the same events $D_1$ and $D_2$.

One must also distinguish global phase from relative phase. Multiplying the whole state by the same number $e^{i\alpha}$ changes no probability:

describe the same physical state. By contrast, the factor $e^{i\varphi}$ placed on a single path modifies the relation between the two amplitudes. It is this relative phase that the interferometer measures.

Complex numbers provide the most natural representation of a phase that varies continuously while preserving the modulus of the amplitude. The experiment therefore shows the physical necessity of a phase variable. It does not however constitute, by itself, a proof that any real reformulation of larger dimension is impossible; this finer question concerns the foundations of the theory.

4. Looking for which path the particle took

4.1. An absorbing detector would answer too brutally

What happens if one seeks to determine the path followed? The most direct means would consist in placing a detector on arm $a$. If it clicks, one knows that the photon went through $a$, but the photon is then absorbed and never reaches the second plate. If it does not click, one attributes it to arm $b$.

The interference disappears, but the explanation seems immediate: the detector stopped the particle. This version does not allow one to distinguish a properly quantum phenomenon from a crude mechanical disturbance.

Let us therefore use a much gentler marking. Suppose that the photon initially possesses a polarization $\ket d$. Let us place on the two arms optical elements that act without ideal absorption and produce two orthogonal polarizations:

The letters $H$ and $V$ denote horizontal and vertical polarizations. By then measuring the polarization, one could know through which arm the photon passed: $H$ marks $a$, $V$ marks $b$.

After the marking, the complete state is written

The first label indicates the path, the second the polarization. We shall learn later that the corresponding mathematical space is a tensor product. For the moment, it suffices to read $\ket a\ket H$ as “path $a$ accompanied by the mark $H$”.

After the second plate, the two paths recombine spatially. But the two contributions are no longer completely identical: one carries the mark $H$, the other the mark $V$. Since these polarizations are orthogonal, their amplitudes can no longer cancel one another in a detector that ignores polarization. One then finds

independently of $\varphi$. The interference has disappeared.

4.2. It is not the knowledge of an observer

Nobody is obliged actually to measure the polarization. One can record only the clicks of $D_1$ and $D_2$, throw the polarization measuring apparatus into a cupboard, or even never install one. The interference remains absent as soon as the two paths have left orthogonal marks in another degree of freedom.

One must therefore avoid a misleading formulation, often heard: “the fringes disappear because a human could know the path”. The physical role is played neither by consciousness nor by abstract knowledge. It is played by the correlation between path and polarization. The two alternatives no longer lead to the same complete final state and become distinguishable.

If the two marks are neither identical nor perfectly orthogonal, the interference is neither perfect nor totally absent. Its visibility is governed by their overlap $|\langle m_a|m_b\rangle|$. Almost identical marks leave the fringes almost intact; orthogonal marks suppress them. Complementarity is therefore not a mysterious switch, but a continuous relation between distinguishability of the paths and visibility of the interference.

5. The quantum eraser

5.1. Changing the question put to the mark

The expression “quantum eraser” suggests that one destroys already recorded information. That is not quite what happens. Instead of measuring the polarization in the basis $\{\ket H,\ket V\}$, which reveals the path, let us measure it in the diagonal basis

Conversely,

Substituting these expressions into the marked state (10), one obtains

This simple rewriting contains the whole phenomenon.

If the polarization measurement gives $+$, the corresponding subset is associated with the superposition

The two paths interfere.

If the measurement gives $-$, the corresponding subset is associated with

There is interference as well, but with an opposite sign. The maxima of the first subset coincide with the minima of the second: one sometimes speaks of fringes and antifringes.

With a convention adapted to the outputs, one obtains for example

5.2. Why post-selection is indispensable

Let us now look at the data without taking the polarization result into account. The events $+$ and $-$ each have probability $1/2$. For the detector $D_1$,

The same holds for $D_2$. The fringes and the antifringes compensate exactly.

The quantum eraser therefore does not make an interference pattern reappear in the raw set of impacts. It allows one to sort after the fact the events into two subsets, each of which exhibits an opposite modulation.

A polarizer oriented at $45^\circ$ placed before the detectors realizes a partial version of this sorting: it transmits only one of the two diagonal polarizations and absorbs the other. Fringes reappear among the transmitted photons, but half of the events have been rejected. A diagonal polarization splitter allows one, on the contrary, to record both outputs: one then observes the fringes in one channel, the antifringes in the other, and no fringes when they are added.

6. Delayed choice and the absence of retrocausality

One can keep the mark in a second system and choose its measurement basis only after the detection of the first photon. In the so-called delayed-choice versions, the decision to read the mark in the $H/V$ basis or in the $+/-$ basis can therefore be taken when the main event has already been recorded.

The vocabulary easily gives the impression that this late decision changes what happened beforehand. This is not the case. Before knowing the result of the measurement of the mark, the data of the main detector are still devoid of interference:

The late choice modifies no click already recorded and allows no message to be sent into the past.

If one chooses the $H/V$ basis, one can classify the events according to the path with which their mark was correlated, but neither of the two subsets interferes. If one chooses the $+/-$ basis, one can classify the same types of events into fringes and antifringes. In both cases, the structure appears only after comparison of the two lists of results.

The temporal order of the two readings is therefore not what produces the correlations. These are already contained in the joint state (10). The choice of basis simply determines the way in which one organizes them.

7. Where are Young's slits hiding?

Young's double-slit experiment rests on exactly the same structure. The two slits play the role of the two arms $a$ and $b$. At a point of coordinate $x$ on the screen, the two paths do not have the same length; their difference produces a relative phase $\varphi(x)$. Instead of varying a phase in time by displacing a mirror, one observes simultaneously several values of the phase along the screen:

Young's bright and dark fringes are therefore the spatially deployed version of the oscillations between $D_1$ and $D_2$ in a Mach—Zehnder.

If the particles arrive one by one, each event is a point-like impact. The first impacts appear randomly distributed; as they accumulate, the interference pattern takes shape. This phenomenon is not specific to light: it is observed with electrons, neutrons, atoms and molecules. Quantum mechanics therefore does not describe a particular oddity of the photon, but a general structure of physical probabilities.

The two-slit quantum eraser consists in attaching a different mark to each slit. The raw pattern disappears. If one then measures the mark in a complementary basis, the impacts can be separated into two conditional patterns shifted with respect to one another. Their sum remains without fringes. This is exactly the computation of equations [eq:etat-gomme] to (17), with the position $x$ on the screen in place of the two detectors.

We shall therefore not need to redo the whole reasoning with Young's slits. They offer a spectacular image, but the Mach—Zehnder exposes its logic more cleanly.

8. What these experiments have taught us

8.1. A superposition is not a mixture

In a classical mixture, the particle takes arm $a$ with a certain probability or arm $b$ with another. The probabilities add. In a superposition, the two possibilities possess a relative phase and their amplitudes can reinforce or cancel one another when one and the same measurement recombines them.

The difference is not a preference of vocabulary: it is measured by opening or closing the possibility of interference.

8.2. The phase is physical when it is relative

The global phase of a state has no observable effect. A relative phase between two components changes the probabilities after recombination. It is this phase that interferometers turn into fringe displacements or into variations of the click rates.

8.3. A system cannot always be described in isolation

After the marking, it is no longer enough to describe the path without the polarization. The two degrees of freedom are correlated. The interference lost by the “path” subsystem has not been destroyed without leaving a trace: it subsists in the correlations of the complete state and can be revealed by an appropriate sorting.

This idea will become central when we study entanglement and decoherence. A system that interacts with its environment leaves in it marks of its different alternatives. Since we generally do not control the billions of degrees of freedom of the environment, we add their contributions without post-selecting them, and the interference becomes unobservable.

8.4. Measurement produces conditional probabilities

Quantum probabilities do not describe only the raw frequency of an event. They relate a preparation, a choice of measurement and possibly the result of another measurement. The quantum eraser shows why one must always specify whether a statistic is marginal, like $P(D_1)$, or conditional, like $P(D_1\mid +)$.

9. Summary

The single-particle Mach—Zehnder brings together two properties that classical physics kept separate. Each detection is localized and indivisible, but the detection frequencies result from the interference of amplitudes associated with several paths. Superposition is therefore not a mere draw between already exclusive possibilities.

Marking the paths by two orthogonal states suppresses the interference, even if nobody consults the mark. What acts is not the information in the mind of an observer, but the physical correlation with another degree of freedom. Measuring this mark in a complementary basis allows one to distribute the events into two subsets that exhibit opposite interference. Their sum, for its part, still does not interfere.

Stern—Gerlach had led us to states, incompatible measurements and probabilities. Interferometry adds amplitudes, their relative phase and the necessity of describing correlated systems together. We now have enough experimental clues to introduce the postulates of quantum mechanics properly. They will not fall from the sky: each will answer a precise difficulty encountered in the course of these two lessons.

10. References