Elitzur-Vaidman bomb tester

Victor Chausse, Diego Monserrat and Jordi Sastre-PellicerEngineering Physics, Univesitat Politècnica de Catalunya

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TABLE OF CONTENTS

Introduction! 2

Contextualization! 4

Elitzur-Vaidman thought experiment! 6

Set-up of the thought experiment! 6

Mach-Zehnder interferometer! 6

Detecting the bombs! 11

Conclusions of the qualitative analysis! 13

Formalizing the thought experiment! 14

Experimental realizations of interaction-free measurements! 17

Introduction and first experiments! 17

The Quantum Zeno Effect (QZE)! 19

Combining the interferometer with QZE! 20

Conclusions! 23

Bibliography ! 25

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INTRODUCTION

Is it possible to detect something without “seeing” it? That’s to say, could we know the properties of an object without any kind of interaction? This is known as “nonlocal measurements” and from this question arises an interesting and complex field of quantum physics. But not only it is relevant for scientists, it even has implications in the current philosophical inquiry.

In physics, locality is a concept that reflects the fact that an object can only be influenced by its surroundings and thus, information (for instance, an electromagnetic wave or a gravitational wave) cannot travel across the space faster than the speed of light. That means that if we have separate objects, it is impossible that one of them causes an effect in the other in less time than that enforced by the speed of light. Although this concept seems very intuitive, quantum mechanics comes up with cases in which locality does not apply. This is what we will call nonlocality.

When speaking about measurements, a local measurement implies a direct interaction with the object. For instance, we can identify the location of an object by the collision with it of a beam of photons or by its gravitational field. These examples require a direct interaction between the observer and the object. However, is it possible to detect an object without this kind of interactions? In other words, is it possible to do nonlocal measurements?

We can come up with simple examples of this. Let’s think for example about a particle which is located in one of two boxes. If we do a measurement inside one of the boxes and we do not detect it, we will know right away that it is located in the other box. Then, we will have succeeded in doing a nonlocal measurement. But the fact is that we had a prior knowledge about the location of the particle: we knew that it was either in one box or in the other. A more sophisticated example would be having two entangled particles. If we knew that they are entangled, measuring a value of one of them (e.g., the spin) would let us know immediately the one of the other, no matter the distance between them.

Nevertheless, as we have stated, all of this nonlocal measurements can be carried out given a prior knowledge of the object. But, could we succeed in detecting the existence of an object, located in a certain place, without any kind of interaction with it and without prior information about it? Avshalom C. Elitzur and Lev Vaidman from the Tel-Aviv University concluded in a paper published in 1993 that indeed this was possible and came up with a thought experiment that suggested a way of carrying out this kind of nonlocal measurements.

The basics of the thought experiment, known as the Elitzur-Vaidman bomb tester, are theories of quantum mechanics such as the wave-particle duality, the Schrödinger equation, quantum superposition, etc. The thought experiment revolves around a set of bombs that can be triggered by a photon sensor. The problem lies in the fact that there are bombs which are dud, and those are undistinguishable from the good ones. So, how can some of the usable bombs

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be spotted out without destroying them? The answer is given in the Elitzur-Vaidman’s experiment and we will approach it along this paper.

In this work, we will not only focus on the characteristics and results of this thought experiment. We will begin with a contextualization of the experiment in the current knowledge of quantum physics and in the ideas of the physicists that have developed this modern field of physics. We will also present the real-life implications of it by going over the real experiments that have been made to prove that the ideas of Elitzur and Vaidman were right. And to conclude, we will expose the consequences of this thought experiment in quantum physics and in the way the universe is conceived nowadays.

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CONTEXTUALIZATION

The experiment proposed by Avshalom C. Elitzur and Lev Vaidman, from the Tel-Aviv University, in 1993, shows that it is possible to ascertain the existence of an object in a given region of space without interacting with it.

To be able to approach this mental experiment from the right point of view, it is absolutely fundamental to understand in depth the duality wave-particle of the light. In that sense, prior to the explanation of the Elitzur-Vaidman bomb tester, it might be a good approach to present first the double-slit experiment, sometimes called Young’s experiment.

Let’s consider Young's two-slit interference experiment with light. As we can see in Figure 1, the original wave front is split into two wave fronts by the slits. These overlapping wave fronts produce the interference fringes on the screen that are so character is t ic o f wave phenomena. Nevertheless, we can also design a small variation of the experiment supposing that we replace the screen by a photoelectric surface. If we measure where the electrons are ejected from the surface, that gives us a pattern corresponding to the double-slit intensity pattern, so the wavelike aspects of the radiation seem to be present. But if the energy of the ejected electrons is measured, we can determine that light consist of a succession of photons, so the particle-like aspects will seem to be present. Each photon must pass through either one slit or the other; if this is the case, how can its motion beyond the slits be influenced by the interaction of its associated waves with a slit through which it did not pass? Are we facing an apparent paradox?

The fallacy in the paradox lies in the statement that each photon must pass through either one slit or the other. How can we actually determine experimentally whether a photon detected at the screen has gone through the upper or the lower of the two slits? To do this we would have to set up a detector at each slit, but the detector that interacts with the photon at a slit disturbs its momentum so much that the double-slit interference pattern is destroyed. In other words, if we do prove that each photon actually passes through one slit or the other, we shall no longer obtain the interference pattern. If we wish to observe the interference pattern, we must not try to observe them as particles. We can observe either the wave or the particle behavior of radiation; but the uncertainty principle prevents us from observing both together.

So, in conclusion, when we allow the photon to go through both slits, we can observe the interference, so light is behaving as a wave. However, if at some point we introduce an element (the detector) to determine which of the two paths has the photon followed, then there’s no longer interference and light behaves as a particle. That means it is impossible to design an experiment where both behaviors (wave and particle) can be observed at the same time.

Figure 1

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It is important to keep in mind the conclusions on the Young’s experiment when reading the part concerning to the discussion on the Mach-Zehnder interferometer.

Let’s go back to the goal of Elitzur and Vaidman, that’s it, to be able to get information from an object without interacting directly with it. Throughout the twentieth century, since the most important postulates of quantum physics were established, there were different points of view in the way of how measurements interact with reality. What we are going to explain is under the Copenhagen interpretation.

The essential concepts of the Copenhagen interpretation were devised by Niels Bohr, Werner Heisenberg and others in the decade of the twenties. It is one of the earliest and most commonly taught interpretations of quantum mechanics. It holds that quantum mechanics does not give a description of an objective reality but deals only with probabilities of observing, or measuring, various aspects of energy quanta, entities which fit neither the classical idea of particles nor the classical idea of waves. According to the interpretation, the act of measurement causes the set of probabilities to immediately and randomly assume only one of the possible values. This feature of the mathematics is known as wave function collapse.

This interpretation has been criticized by many physicists and philosophers, mainly on the ground that it is non-deterministic. Einstein's comments "I, at any rate, am convinced that He (God) does not throw dice" and "Do you really think the moon isn't there if you aren't looking at it?" exemplify this. Bohr, in response, said "Einstein, don't tell God what to do".

After all these discussions, we can understand in more depth the importance of making a measurement without interacting with the object that we want to measure. That object, when we are not trying to get information from it and, consequently, we are not interfering with it, is described by a superposition of different quantum states. However, as soon as we interfere with the object, we oblige him to choose randomly one of the possible states that could describe him. That’s it, the wave function collapses and it is no longer a distribution of probabilities. One of the best explanations to understand the impact that ourselves have towards reality is the mental experiment of Schrödinger’s cat, proposed in 1935, with the cat dead and alive simultaneously until we observe the inside of the box, and impose that reality can only take one of the two opposite states.

To conclude this contextualization, we hope that it is crystal clear for the reader the crucial paper that measurements have in quantum physics, and how jaw-dropping it is to get information from a system without interacting with it. That is exactly what Elitzur and Vaidman proved with their mental experiment, so let’s go into details and try to comprehend this stunning result of vital importance for quantum physics and our future.

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ELITZUR-VAIDMAN THOUGHT EXPERIMENT

Set-up of the thought experiment

Even though the results of the Elitzur-Vaidman’s thought experiment can be presented in different ways, the authors first conceived it as a bomb detector. The bombs they imagined have a peculiar behavior: they are triggered by a detector sensible to light. The detector of this kind of bombs is supposed to be highly sensible. Indeed, even when a single photon reaches one of these bombs, the detector absorbs it and the bomb explodes. So, to prevent them from exploding they should be kept in a completely dark room.

Nevertheless, not all of these bombs work correctly. Some of them are dud and thus, when a photon reaches their detector, nothing happens. In fact, the photon is not even absorbed. It passes through the detector unaltered. So, given an assortment of bombs of this kind, we would find that some of them would be broken and the rest would work correctly.

The problem is that both types of bombs are completely undistinguishable by inspection. Apparently, and according to classical physics, the only possible way to check whether they are real or they are fake is to expose them to a beam of light and let the good ones go off. But by doing this, we would lose all the usable bombs and only keep the dud ones, which are useless.

On the basis of wave-particle duality theory, Elitzur and Vaidman came up with a way to detect some of the real bombs. To do so, they proposed the use of a Mach-Zehnder interferometer.

Mach-Zehnder interferometer

The solution to our problem consists basically in the use of a device known as Mach-Zehnder interferometer. Now we will explain what it is and how it works.First of all, we consider a light source, specifically a high precision laser capable of emitting photons individually. The beam emitted from the source collides with a beam splitter, in particular, a half-silvered mirror. This optical device is simply a glass with a very thin layer of aluminium (usually deposited from aluminium vapour). When the light beam meets the splitter at an angle of

Figure 2

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45º, half of it is reflected by the surface and half of it is transmitted through it. In other words, a photon incident on the half-silvered mirror has a 50% probability of going through and 50% of being reflected.

As shown in Figure 3, due to the incident angle of 45º, two perpendicular beams will be obtained with half of the original intensity each. The reflected light beam will be called the sample beam and the transmitted one, the reference beam.

How is the wave function of the photon modified when interacting with the beam splitter? The sample beam’s wave function undergoes a phase shift of π (denoted as INV in the pictures) relative to the original beam when it is reflected by the aluminium. As we see, it also changes from moving to the right to do it upwards. Meanwhile the reference beam undergoes certain modifications because of the refraction through the aluminium and the glass. This unknown modifications are due to the nature of the material, its thickness, etc. In the pictures, this kind of modification will be denoted as G.S. (glass shift).

The good news are that these changes are unimportant because they do not affect the final results. Therefore, we will neglect these changes and we will consider that the reference beam is a wave that moves to the right in the same way as the original.

Note that it is possible to build an interferometer in which the half-silvered mirrors act as we are considering, that’s to say, producing a lag of 2π with respect to the incident wave. This would not produce any modification in the refracted beam of light. However, as we have already said, this is not necessary for our experiment because these modifications do not affect the final results.

Continuing with our interferometer, now we have that each of the two beams is reflected in a regular mirror, a fully-silvered surface mirror. As shown in Figure 4, the sample beam now travels to the right and because it undergoes a phase shift of π once again, the reflected beam will be equal to the original one.

Figure 3

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On the other hand, the reference beam is reflected upwards and it undergoes a phase shift of π as well (see Figure 5).

Finally, we put another half-silvered mirror (with the same characteristics as the initial one) in the place where the two beams will meet, as shown in Figure 6. We will have to analyse now what happens to each of these beams.

On the one hand, we see that the reference beam collides with the aluminum part of the splitter. As we already know, it will be divided into two new beams with half of the incident intensity. One will be reflected by the aluminum and

Figure 4

Figure 5

Figure 6

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will travel to the right with a new phase shift of π, which added to the shift it already had makes it equal to the original beam (the one emitted by the source). But of course, with less intensity. The other one goes through the aluminum and is refracted by the glass. As we have seen, its associated wave function will undergo certain modifications that do not affect the results. So, we will have a beam traveling upwards with a phase lag of π relative to the original beam. The reference beam had already this phase difference before colliding.

On the other hand, the sample beam collides with the glass part of the splitter. This beam is divided once again in two new beams. One goes through the half-silvered mirror and because it only suffers the modifications due to the glass, that we are not taking into consideration, it continues traveling to the right with a wave function equal to the original. The other beam produced is reflected by the mirror. In this case, it goes first through the glass and then it is reflected in the aluminum layer. After crossing the glass again, it continues its way upwards.

Until now the reflection of a beam had always taken place when the beam was coming from the air and impacted on the aluminum. Because of that the beams suffered a phase lag of π. But now, due to the fact that the material from which the beam comes is glass and not air, the beam does not undergo any phase shift of its wave function. The reason of this phenomenon is that the refractive index of air is lower than that of aluminum, while the index of glass is greater. Then, we have to consider that this beam is only affected by the modifications due to the glass, that we already know that are negligible. So we will have a beam with a wave function identical to the original one (considering the glass shift is being neglected) and traveling upwards.

Note that the glass shift is the same in the exiting beams. The beams moving to the right will have suffered one glass shift and the ones moving upwards will have suffered two glass shifts. When considering the interactions between the beams, these glass shifts will not affect the results at all because they are the same in both cases. That is why we neglected them along the analysis. This can be observed clearly in Figure 8.

Figure 7

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We have almost completed our interferometer, we have to analyze now the interactions of the exiting beams. We place two detectors: one on the right side of the last beam splitter, which we will call detector 1, and another above the splitter, detector 2.

From the beam splitter, two beams with an identical wave function will emerge towards the right detector. They are in phase so they will undergo a constructive overlap and consequently the detector 1 will detect the wave and it will light. However, the two beams that are directed upwards have the same wave equation but they have a phase difference of π. Therefore they will suffer a destructive overlap and detector 2 will never light because no wave will collide with it.

We should highlight that the same results are obtained even when a single photon is emitted by the source. This is due to the wavelike behavior that light shows in this experiment.

Figure 8

Figure 9

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Detecting the bombs

Now it is the moment of introducing the famous bombs in scene. We put a bomb in the interferometer, at the position that we can see below in Figure 10, between the two upper mirrors. So, what will happen when a single photon is emitted from the source?

If the bomb is dud, it will not alter at all the photon’s wave function. So it will happen exactly the same as before: detector 1 will always light. In Figure 11, we see that the bomb does not disturb the photon’s wave function and it behaves as if there was nothing in the path.

However, if we put a usable bomb, everything changes. Until now we have considered the photon as a wave that travels through the two different ways at the same time and at the end, it interacts with itself creating destructive or constructive overlaps. Nevertheless, when the usable bomb is put in the interferometer, it behaves as an observer and makes the wave function collapse. Thus, we have to consider the photon as a particle now. To

Figure 10

Figure 11

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understand this we should recall from the wave-particle duality theory that it is not possible to observe both behaviors at the same time in an experiment. Then, when the photon (acting as a particle) collides with the first half-silvered mirror it has fifty percent possibilities of taking the upper way and another fifty percent of taking the lower way.

If it goes through the upper path, it will collide with the bomb and it will explode. We will have discovered that it is a usable bomb but we will have lost it. Indeed, this does not suppose any improvement with respect to the classical procedure of pointing a beam of light at the bomb. Anyway, let’s see what happens in the other cases.

If the photon takes the lower path, two new possibilities with the same probability arise. One of these possibilities is going to the right and colliding with detector 1. What we will observe is exactly the same as if the bomb was dud: detector 1 lighting.

Figure 12

Figure 13

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The other possibility for the photon is going upwards and colliding with detector 2. Now we have achieved our goal: when detector 2 lights we are completely sure that the bomb is usable and it will have not exploded!

Conclusions of the qualitative analysis

To sum up, let’s make a brief compendium of the results that we can obtain when we place either a dud bomb or a usable one.

When the bomb is dud detector 1 always lights. However, when we put usable bombs we have three results. Half of the bombs will explode. A quarter of them will not explode but the nature of them will be unknown (this will happen when detector 1 lights). And the other quarter will not explode either and we will be completely sure that they are usable (this will happen when detector 2 lights). This means than we can identify a twenty five per cent of the usable bombs.

This procedure has two big inconveniences: half of the usable bombs will explode, and when detector 1 lights, we will not know what kind of bomb it is.

Apparently, the first one has not solution (although we will see in a subsequent section that introducing some modifications in this experiment we can achieve

Figure 14

! ! ! 0% explodeDud bombs! ! 100% detector 1! ! ! 0% detector 2

! ! ! 50% explodeUsable bombs! 25% detector 1! ! ! 25% detector 2

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better results). But regarding the second problem, we can do something to get better results. If we repeat the experiment with the bombs of which we do not know the nature, we will increase the percentage of usable bombs detected without explosion. When the repetitions tend to infinity, this percentage becomes thirty three per cent:

So we see that in the limit, it is possible to detect 33,33% of the usable bombs without making them explode. The rest will explode, though.

Formalizing the thought experiment

As a continuation of the analysis of the thought experiment proposed by Elitzur and Vaidman, we will formalize what we have explained before. To do so, we will follow the same procedure and we will use the same notation that was used by Elitzur and Vaidman in their original papers. First of all, we have to define:

as the state of the photon moving to the right, as the state of the photon moving upwards.

We consider that these two states form the axes of the complex plane. The photon moves along the positive direction of the real axis, while moves along the positive direction of the imaginary axis. Clearly, we see that the angle between the relative orientations of these two states is π/2. When the photon changes from one state to the other, it means that a phase shift of π/2 is being added to its initial state. We can consider that its initial state is being

multiplied by the factor (note that ).

First, we will analyze the case in which there is not any bomb in the interferometer. Considering what we have just explained, it is easy to see that when the photon coming from the source, with an initial state , collides with the half-silvered mirror, it yields two superposed states:

sample beam, reference beam.

We can represent the change of the initial state as a combination of the two new superposed states:

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The factor that appears in the formula is the normalization constant.

The modifications produced by the fully-silvered mirror can be described as:

in the sample beam, in the reference beam.

Once again, we can express this modifications together as:

As we have seen at the beginning, when a photon in the state interacts with a half-silvered mirror, the result is:

Then, the result in the last half-silvered mirror is:

• Sample beam! ! !

• Reference beam !! ! !

Doing some calculations, we obtain the final state of the beam:

As we see, the final state of the beam is . The negative sign in the result corresponds to a global phase and it has no physical meaning. In quantum mechanics, the observables depend on the square of the arguments. Therefore, since (-1)2 = 1, the negative sign is insignificant and we can ! 15

consider the final state to be . Hence, at the end the photon will move to the right and with 100% chance, it will collide with detector 1.

Next, we will consider the case when the usable bomb is put in the interferometer. So now, we have to define:

as the state of the photon scattered by the object.

In this case, the original photon undergoes the following modifications:

Due to the final superposition of states, we can know the probabilities of the possible events:

Thus, this formal analysis yields the same results as the ones we obtained in the previous section, where we approached the thought experiment qualitatively. We observe that half of the usable bombs will explode, a quarter of them will be indefinite and the other quarter will be for sure usable and they will not have exploded.

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EXPERIMENTAL REALIZATIONS OF INTERACTION-FREE MEASUREMENTS

Introduction and first experiments

As we have explained in detail in the previous section, Elitzur and Vaidman have pointed out that it is possible to make interaction-free quantum mechanical measurements. They arrived to the conclusion that by doing the experiment once, we can verify in an interaction-free way the presence of the bomb only 25% of the time. However, by repeating the experiment or recycling the photon in cases where it leaves the Mach-Zehnder interferometer by detector 1, we can increase the total probability of an interaction-free measurement.

We can define the probability of making an interaction-free measurement as the following:

η = P(D2)P(D2)+ P(Boom) (1)

In this case, according to Elitzur and Vaidman conclusions, it yields 1/3. However, we can set another experiment that yields more than one third, actually one half. That is exactly what Paul Kwiat, Harald Weinfurter, Thomas Herzog and Anton Zelinger, from Universität Innsbruck, in collaboration with Mark Kasevich, from Stanford University, did in 1994.

Paul Kwiat et altri set an experiment with the production of pairs of correlated photons via the process of spontaneous down-conversioning a nonlinear crystal (LiIO3). They selected photon pairs at 702 nm: one member of each pair was directed to the “trigger” detector Dt and the other one to a Michelson interferometer, as it can be observed in Figure 15.

Figure 15

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By means of a removable mirror, it was possible to direct the photons from one of the arms to detector Db, thereby producing the “bomb in” configuration. In the absence of the bomb mirror the path difference in the interferometer was adjusted to produce a minimum at detector Difm. What they did was to record data periodically switching from a “bomb out” to a “bomb in” configuration. The results are shown in Figure 16. A careful observation of the data shows that even in the absence of the bomb, the data still display counts at Difm, falsely indicating the presence of the bomb mirror. These counts constitute the noise of their detection scheme, due to accidental coincidences and non-perfect destructive interference.

As the beam splitter in their interferometer was a 1 mm thick glass plate, they were able to choose between reflectivities of 54%, 43%, 33%, 19% and 11%. The expected probability of making an interaction-free measurement (η) can be generalized to a function of beam splitter reflectivity. In the absence of losses, the probability of an incident photon going towards bomb is R, while the probability of it going towards Db is RT. Assuming a lossless beam splitter, T=1-R, we have:

η = RTRT + R

= 1− R2 − R ! (2)

Experimental and theoretical values for η in a Michelson interferometer scheme, as a function of the beam splitter reflectivity, can be observed in Figure 17.

Note from equation (2) that as the reflectivity becomes smaller, the fraction of interaction-free measurements tends to the value one half, that’s it, 50%. The

Figure 16

Figure 17

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question immediately arises: is 50% the best we can do? Are we destined to be blown up half the time?

The Quantum Zeno Effect (QZE)

To be able to understand the next improved experiment, we need to consider another rather peculiar quantum mechanical phenomenon, the so-called "Quantum Zeno effect". First discussed in 1977 by Misra and Sudarshan, the QZE involves using repeated quantum measurements to inhibit the evolution of a quantum system. It relies on the quantum "projection postulate", which basically states that, for any measurement made on a quantum system, only certain answers are possible, and that the resulting quantum system is then in a state determined by the obtained results. This is easiest to understand with a particular example.

Let’s consider a series of N polarization rotators, each of which rotates the polarization of light by an angle 90°/N, as in Figure 18. Therefore, after passing through all N of them, an initially horizontally-polarized photon will be vertically polarized. That is, it will have zero chance of passing through a horizontal polarizer and being detected.

However, if we put horizontal polarizers between every two polarization rotators, then the outcome is quite different (Figure 19). For concreteness, consider the case of six cycles, so that the rotation angle at each stage is 15°. At the first polarizer, the photon has only a small chance of being absorbed: sin2 (15º ) = 6.7% .

Figure 18

Figure 19

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If it is not absorbed, then by the projection postulate, the photon must be horizontally polarized. The identical process happens at every stage. For the case of N=6, the chance that the photon was transmitted through all 6 polarizers is simply 6 ⋅cos2(15°) , which is about 2/3.

Note that without the interspersed polarizers, we never saw light at the detector. Hence, whenever we get a "click" at the detector, we know that the extra polarizers were inserted. Moreover, if we perform the experiment with a single photon input, and it shows up at the final detector, then of course it could not have been absorbed by any of the extra polarizers. And as we let the number of stages N become larger (at the same time reducing the angle of polarization rotation accordingly), then the probability that the photon is absorbed vanishes. The photon is always transmitted!

To demonstrate this phenomenon, Paul Kwiat et altri did not use six rotators and six polarizers, mostly because “we did not have so many elements”, as Paul Kwiat admitted. Instead, they used a single rotator and a single polarizing beam splitter, but arranged the light to pass through these six times, as shown below in Figure 20.

What they found, was that when the polarizing beam splitter (which transmits horizontal polarized light and reflects vertical polarized light) was not in place, the detector D never fired; whereas when the polarizing beam splitter was in place, the detector D fired 2/3's of the time. So these experimental results agree completely with the theoretical values.

Combining the interferometer with QZE

Using the Elitzur-Vaidman simple interferometer, we can detect the presence of any object (i.e., an opaque object), but only up to 50% of the time in an interaction-free way. And with the quantum Zeno technique we have shown that we can detect the presence of a polarizing object better than 50% of the time. By making a hybrid of the two schemes, one can achieve the detection of an opaque object with an arbitrarily small chance of it absorbing a photon.

Figure 20

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While there are many methods to do this, we will present here only the one most closely connected with the previous discussions.

Consider the system shown below (Figure 21). We again have a cycled photon, as in the previous demonstration of the Zeno effect. It makes N cycles in the system before being let out, and its polarization analyzed. At each cycle, again there is a rotation of the polarization by π/2N, so that at the end, the initially horizontally-polarized photon is vertically polarized. The key difference to the previous experiment is the inclusion of the polarization Mach-Zehnder interferometer. Instead of normal beam splitters, it uses two polarizing beam splitters, which transmit horizontal and reflect vertical polarized light. The two arms of the interferometer are set up to have equal lengths. Therefore, any incident light is split into the horizontal and vertical components (the former taking the "high road", the latter taking the "low road"), which are then added up again to reform the original polarization state.

This is only true if the arms of the interferometer are unblocked. If, instead, we have an object in the lower path, then the evolution is totally different. Now with each cycle there is only a small chance that the photon chooses the lower path and triggers the bomb, with probability P = sin2 π / 2N( ) . If this does not happen, then the photon’s wave function is "collapsed" into the upper path – the photon is again completely horizontally polarized with a probability P = cos2 π / 2N( ) . The same thing happens at every cycle, until on the Nth cycle the photon is allowed to leave. If it has successfully survived every cycle, then the photon is

Figure 21

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definitely horizontally polarized. That means that probability of finding the photon after N cycles to be horizontally-polarized is just the probability for it to have taken the upper path during each cycle:

P = cos2 π2N

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟N

!(3)

Remember that without the object, the photon was definitely vertically polarized. By measuring the final state of the photon's polarization, we can tell whether or not an object was blocking the lower path.

If we take the probability determined by (3), in the limit for large N it becomes P = 1−π 2 / 4N +O(1 / N 2 ) . Of course, the probability that the bomb explodes is just the complementary of (3).

(3) is plotted in Figure 22. We can see immediately that as long as N ≥ 4, there exists a greater than 50% probability of making an interaction-free measurement, surpassing the limit of the original Elitzur-Vaidman research. Moreover, in these figure we can also see the latest data from Innsbruck Universität that shows that measurements around 70% are now possible, and theoretically it is possible to get to a 100% accuracy in interaction-free measurements.

Figure 21

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CONCLUSIONS

When we face the problem introduced by the bombs of the Elitzur and Vaidman’s thought experiment, intuition and classical physics tell us that the only possible way to recognize the intrinsic nature of the bomb is by triggering it with a photon. Of course, that leads to the loss of all usable bombs. When generalizing, it seems that given an unstable system the only possible way to measure it or detect it is by the interaction with, for example, a photon. This may lead to the modification of the quantum state of the system and the loss of the initial configuration. Therefore, if an unstable quantic system is required with a certain state, when a measurement is carried out in the classical way, the interaction needed to make the measurement will modify its state and the final state may not be the same as the one measured. So, the selection of a quantum system with a fixed state may become impossible in certain cases.

However, Elitzur-Vaidman’s thought experiment shows that it is possible to carry out detections without interacting at all with the system and thus, not affecting the state of the system measured. This procedure is based in the principles of quantum physics and contradicts both classical physics and intuition, though that does not mean it is not right. As it has been exposed previously, several real experiments have been done and all of them have shown that the ideas introduced in the thought experiment are right.

This experiment leads to a lot of questions about the nature of the universe. How can something be detected without any kind of interaction? How can the behavior of a photon be modified by something with which it does not interact? And even, how can the photon know the nature of an object that is located far away without interacting with it at all?

We discussed at the beginning of this work the concept of locality. This experiment dismisses this concept and shows that our universe has a nonlocal nature. Somehow, this entails that actions can be conducted instantly from one point of the universe to another with no apparent mechanism of interaction. This was a quite controversial concept. Einstein and other important physicists questioned quantum mechanics because of aspects like this. Indeed, he described it as the “spooky” action at a distance.

In favor of the concept of nonlocality, Bell’s theorem (proposed by the Irish physicist John Stewart Bell in 1964) rejects locality and states that the predictions of quantum mechanics (which have been proved to be right) can never be reproduced by theories of local hidden variables, that’s to say, theories that maintain the concept of locality.

Another approach to the concepts introduced in the Elitzur-Vaidman’s thought experiment is conducted by the many worlds interpretation. Even though this interpretation has not a great acceptance among scientists, it can help to understand the questions that have come up. According to this theory, when the experiment is performed and the bomb is usable, the universe splits in three. In one of the universes, the photon takes the upper path, hits the bomb and it explodes. In the other two, it takes the lower path and either detector 1 or detector 2 clicks. The three copies of the photon are connected and interact

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with each other. So, when in a universe the photon interacts with the bomb’s detector, the other two photons will get the information about the interaction. Therefore, we cannot say that there was no physical interaction between the photon and the object.

Despite all of the philosophical questions arisen by this thought experiment, it also provides the basis for real applications, which can be very useful in the field of experimental and applied physics. As the authors proposed, this procedure can be used to select atoms in a specific metastable state. If an atom in a certain metastable state tends to absorb photons of a certain energy while the same atom behaves as transparent for those photons when it is not in this state, then the free interaction measurement procedure allows to select the atoms in this specific state, without modifying their state. If that kind of measurement could not be achieved, due to the absorption of a photon during the measurement, the state of the atom would be altered. The concepts of free interaction measurements proposed by Elitzur and Vaidman can also be found in quantum computation. In particular, in the development of the C-NOT gate conducted by A. A. Methot and Kai Wicker. Another example is the device proposed by Adonai S. Sant’Anna and Otavio Bueno that allows to detect an electric or magnetic field without interacting with it.

So then, we see that the thought experiment developed by Elitzur and Vaidman, that at first could seem quite simple, has a lot of implications in quantum mechanics and it leads a lot of new ideas around the concept of free interaction measurements to which it refers. We have seen that many philosophical questions arise from this thought experiment but also, many real experiments and applications have been developed from the basis provided by this work.

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BIBLIOGRAPHY

Introduction

- Locality and Quantum Mechanics: David M. Harrison (University of Toronto). Available online at: http://www.upscale.utoronto.ca/PVB/Harrison/Locality/Locality.html

Contextualization

- Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles: R. Eisberg, R. Resnick. 2nd Ed. (1985) Wiley.

- Figure 1 belongs to Forskning & Framsteg and has been obtained from this source: http://www.nobelprize.org/nobel_prizes/physics/articles/ekspong/.

Elitzur-Vaidman thought experiment

- How does a Mach-Zehnder interferometer work?: K. P. Zetie, S. F. Adams and R. M. Tocknell (Westminster School, London); 1999. Available online at: http://www.cs.princeton.edu/courses/archive/fall06/cos576/papers/zetie_et_al_mach_zehnder00.pdf.

- Quantum Mechanical Interaction-Free Measurements: Avshalom C. Elitzur and Lev Vaidman (Tel-Aviv University); 1993. Paper available online at: http://arxiv.org/pdf/hep-th/9305002v2.pdf.

- Conociendo el efecto Zenón cuántico en experimentos contrafácticos: una aproximación filosófica: Karim Gherab Martín (Harvard University) and Carmen Sánchez Ovcharov (Universidad Complutense de Madrid); 2009. Paper available online at: http://www.ontologia.net/studies/2010/gherab-sanchez_2010.pdf

- All the pictures that appear along this section are self-made.

Experimental realizations of interaction-free measurements

- Experimental realization of “interaction-free” measurements: Paul Kwiat, Harald Weinfurter, Thomas Herzog, Anton Zelinger (Universität Innsbruck); Mark Kasevich (Stanford University); 1st Ed. (1994), Editions Frontières. There’s a copy available online at: http://www.univie.ac.at/qfp/publications3/pdffiles/1994-08.pdf (Figure 16 has been obtained from this source).

- Paul Kwiat webpage: http://physics.illinois.edu/people/kwiat/interaction-free-measurements.asp (Figure 17 and Figure 21 have been obtained from this source).

- The rest of the pictures that appear along this section are self-made.

Conclusions

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- “Relative State” Formulation of Quantum Mechanics: Hugh Everett (Princeton University); 1957. Published in Review of Modern Physics 29, pp. 454-462.

- On the Einstein Podolsky Rosen Paradox: J. S. Bell; 1964. Published in Physics 1, 3, pp. 195-200. Original paper available online at: http://www.drchinese.com/David/Bell_Compact.pdf.

- Interaction-Free Measurement Applied To Quantum Computation: A New “CNOT” Gate: A. A. Méthot, Kai Wicker (Universität Heidelberg); 2008. Paper available online at: http://arxiv.org/pdf/quant-ph/0109105.pdf.

- Generalizing Elitzur-Vaidman interaction free measurements: Adonai S. Sant’Anna, Otávio Bueno (University of South Carolina); 2005. Paper available online at: http://arxiv.org/pdf/quant-ph/0503189.pdf.

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