Some Implications of Quantum Superposition

You have already seen an introduction to quantum mechanics and the ideas of wave functions.  This reading will build upon that introduction and illustrate some of the consequences of quantum superposition of states.
 

Wave-Particle Duality

The Double-Slit Experiment Revisited



Discussion Question: Think about what you have learned during this course about wave-particle duality.  When does light behave like a wave?  When does it behave like a particle?  When do electrons exhibit wave properties?  What would happen to an electron double-slit interference pattern if you put a detector at each of the slits to count electrons?

You should understand by now that all objects have both particle and wave properties. The wavelengths of macroscopic objects such as elephants, bullets, and computers are so small that they cannot be measured. Thus the wave nature of these macroscopic objects is never observed. Sub-atomic particles like the electron, however, can easily have wavelengths on the order of micrometers.  Electrons traveling in a narrow beam through a double slit will produce an interference pattern of maxima and minima rather than the single spot expected if electrons always behaved as particles.  This interference pattern is identical to the pattern produced by light passing through slits.  The first figure below illustrates what would occur if electrons always acted like particles; the second if they always acted like waves; and the third what is actually observed to occur.

THESE DIAGRAMS ARE NOT AT ALL TO SCALE

What the intensity pattern would look like for a double-slit experiment if electrons behaved completely like particles.  They would go through either one slit or another, creating two mounds where they hit the screen.  Notice that no electrons hit the outer regions of the screen
What the intensity pattern would look like for a double-slit experiment if electrons behaved completely like waves.  They would pass through both slits simultaneously, creating an interference pattern on the other side of the screen.  Notice that electrons produce a non-zero intensity in the outer regions of the screen
What the intensity pattern really looks like for a double-slit experiment with electrons.  They pass through both slits simultaneously, interfereing with each other on the opposite side of the slits, but when they hit the screen, they hit at one point only.  Notice that some of these hits occur in the outer regions of the screen.

As shown in the rightmost figure above, electrons travel through the slit like waves but do not strike the screen as waves. Instead, the same electron that exhibits wave properties traveling through the slits interacts with the screen as a particle, striking one and only one spot. If one were able to slow the electron emission and observe single electrons interacting with the screen, one would see the interference pattern build up one spot at a time.  As the number of electrons having hit the screen increases, the pattern of where individual electrons strike takes the form of an interference pattern.  Some of the "spots" produced by individual electrons striking the screen are located far away from the points directly in front of the slits that a particle would be expected to hit.  Thus each electron passes through both slits like a wave, interfering with itself, before interacting with the screen in one point like a particle.

As if this dual behavior is not confusing enough, trying to detect the electron as it passes through the slits changes the entire outcome of the experiment.  The electron is never detected simultaneously in both slits; instead, a detector will find it passes through only one opening.  But when this detector is in place and making this measurement, the interference pattern disappears.  We are left instead with the leftmost pattern on the screen above which was predicted if electrons are particles going through one opening or another:  two bright spots in front of the each individual slits.  Our measurement at the openings has forced the electron to behave like a classical particle.

Variation on a Theme

A variation of the double-slit experiment is presented in Gerard Milburn's book, Schrödinger's Machines:  The Quantum Technology Reshaping Everyday Life.  Consider the arrangement of laser beam, beam-splitters, mirrors, and detectors shown below.
 
(a) Light from a laser is incident on a beam splitter.  The beam splitter will transmit half of the light while reflecting the other half (b) The transmitted and reflected light is directed by completely reflecting mirrors back to the second beam splitter, where each beam is split once more. (c) Light that has been reflected twice (RR) or transmitted twice (TT) ends up at the U detector, while light that has been reflected once and transmitted once (RT and TR) ends up at the D detector.
(d) In a Mach-Zhender interferometer, one of the mirror can be adjusted (amount it moves is exaggerated here) to make the top path different from the bottom path. (e) Depending upon the location of the mirror, you can achieve complete destructive interference at the bottom detector and complete constructive interference at the top detector, or . . . (f) . . . complete constructive interference at the bottom detector and complete destructive interference at the top detector, or anything in between.  This occurs no matter how low the intensity of the input beam.

The only explanation for the behavior of figures (e) and (f) above is interference.  So far, this experiment, like the double-slit experiment, seems to verify the wave theory of light.  Even if you decrease the intensity of the input to the point that the particle theory predits only one photon would be in the detector at a time, you still see the interference effect.  But another interesting aspect of the experiment is illustrated below.
 
 

When a detector is placed to determine which path the light takes, the interference effects vanish!

If a detector is placed in one of the paths, the U and D detectors will once again each get half of the light.  This happens even if the detector is a "non-demolition" detector rather than the high-demolition bomb we discussed in class.  And if the intensity of the input beam is decreased to send one photon at a time, a dector sensitive enough to measure a single photon would only detect light half of the time.  This last behavior seems to indicate particle behavior, but neither the wave nor the particle theory can explain why the presence of an interference pattern depends on the type of measurements taken.
 

Enter the Wavefunction

Wave Functions Exposed

In the highly-successful quantum theory of physics, the behavior of objects is described by a wavefunction, Y(x,t). The evolution (how it changes in time and space) of the wavefunction is determined by the Schrödinger equation in the same manner that the evolution of water waves is determined by Newton's laws. So as the electron moves through the crystal, the interference effects are caused by interference of the wavefunction.  You have seen in the previous reading how we separate the wavefunction Y(x,t) into an exponential time component and a position-dependent wavefunction c(x): Y(x,t)=c(x)exp(-i2pt/T).  The Schrödinger equation can be written in a manner that depends only on the position-dependent part:

In our electron double-slit experiment, the electron has two possible paths: take the top slit, or the bottom slit.  Rather than try to figure out an exact way to describe those options, we will use a notation developed by Dirac:  going through the top slit will be denoted |top>, while going through the bottom slit will be denoted |bot>.  Some portion of our electron wave has passed through the top, and some portion through the bottom, so the wavefunction is

c(x) = a |top> + b |bot>

Unlike the amplitude y(x,t) describing a water wave, the wavefunction is NOT measurable. When the position of an electron is measured (such as when we put a detector at the slit), the electron will appear in one point, not as a wave spread out over the screen. The probability of finding the electron at a given point is found from the square of the wavefunction:

Prob = ò|c(x)|2dx = |a|2 <top|top> + |b|2 <bot|bot>

 So our electron has probablility of |a|2 of going through the top slit and |b|2 of going through the bottom slit.  Since the total probability must be one,

|a|2 + |b|2 =1

If the slits are equally likely, then |a|2 = |b|2 = 0.5.  This equation may appear to give us a and b, and thereby tell us what the wavefunction of the electron is, but it does not.  a and b can be (and generally are) imaginary and/or negative.  We can measure the magnitude of the wavefunction amplitudes, but we cannot directly determine the entire amplitude.

If a measurement is made to determine which slit the electron goes through, it will turn up that |a|2 of measurements find the electron in the top slit, and the remaining |b|2 of measurements find it in the bottom slit.  After this measurement, the electron is no longer a superposition of states, but it is entirely the state corresponding to the slit in which it was measured:

c(x) =  |top>, or c(x) =  |bot>.

We say that the measurement collapses the wavefunction into a single state.  Since the wavefunction no longer contains both possibilities, the interference between different possible paths doesn't occur.  The act of measurement alters the results of the experiment.

Let me restate that since it is a key concept in quantum computing applications: measuring a system changes the system in an irreversible manner.  This aspect of the quantum theory could be utilized to indicate whether a message has been intercepted and read by someone on its way to a recipient, as explained later in this reading.
 

Wave Function Explanation of the MZ Interferometer

Let us take another look at the Mach-Zhender interferometer described above.  In this experiment, we have four possibilities:  |RR>, |TT>, |TR>, and |RT>, where R is reflected, T is transmitted, the first letter refers to what occurs at the first beamsplitter on the left, and the second letter refers to what occurs at the second, righthand, beamsplitter.  We can therefore write the wavefunction as

c(x) = a |RR> + b |TT> + c |TR> + d |RT>.

Since each path is equally likely when the mirrors are truly half-mirrored and splite the beam 50/50, the probability of a photon taking a given path must be 1/4:

|a|2 = |b|2 = |c|2 = |d|2 = 1/4.

It is tempting to just make each amplitude equal to 1/2, since |1/2|2 is 1/4.  We must, however, not give way to this temptation.  The wavefunction at the upper detector contains the |RR> and the |TT> paths, while the wavefunction at the lower detector contains the other two paths:

Prob (U detector) = |a + b|2, and
Prob (D detector) = |c + d|2,

where we add the amplitudes of the two indistinguishable paths before squaring to find the probability.  If each amplitude is 1/2, the probability at the U detector is 1, as is the probability of the photon reaching the D detector.  The total probability is therefore 2, which is impossible.  If no interference occurs, the probability at each detector is 1/2.  The only way to meet this condition while also requiring that each amplitude squared is 1/4 is to use complex amplitudes.  One (but certainly not the only) solution is
 
 
 

ac
b = d
|a + b|2 = |c + d|2
1/2

 

Applications of Quantum Effects

Randomness

Current computers cannot find a truly random number since every step is determined. Many routines come close, using user-chosen seeds or clock times to start the algorithm, but the outcome will always be the same if the seed is the same. Quantum computers, however, offer a source of truly random numbers.

Again consider the simple double slit experiment. Let's say electrons are sent one at a time toward the double slit. A counter is placed in the top slit. As the electron approaches the slits, it is described by a wavefunction that spreads out in space encompassing both slits. Once the wavefunction hits the counter, however, it collapses. If the collapse of the wavefunction leaves the electron in the top slit, the counter is triggered, and a "1" bit is sent to the computer. If the collapse leaves the electron in the bottom slit, the counter does not trigger and a "0" bit is sent. This procedure could be repeated for each bit of an arbitrarily-long random number. Assuming the electron beam is symmetric around the slits, half of the time an electron will be found in the top slit. But you cannot determine which slit the electron will be found in.  Quantum behavior is truly unpredictable and governed by probabilities of random events.
 

Paradoxes

Quantum theory has many aspects that have troubled even the greatest scientific thinkers.  Schrödinger, one of the early developers of the wave function representation, posed a paradox that we still wrestle with, called Schrödinger's cat.  Suppose one put a cat in a box with a nuclear sample, a gieger counter and an apparatus that would result in poison being released to kill the cat if the geiger counter detected a nuclear decay.  The halflife of the sample is picked so there is a 50% chance of the decay occurring in an hour.  If states really exist in superpositions until an observation is made to collapse the wavefunction, then the sample is both decayed and undecayed (with equal amplitudes) after that hour.  The geiger counter is both triggered and untriggered, the poison is both released and unreleased, and the cat is both alive and dead.  This state of superposition lasts until an observer opens the box and checks to see whether the cat is dead.  Until then, the cat is both alive and dead.  Animal rights' activists can relax; to my knowledge this experiment has never been actually performed.  Yet it exemplifies the difficulties with wave functions.  Perhaps the cat is an observer and therefore the wavefunction collapses immediately, but one could imagine putting a rat, an amoeba, or a plant that is killed if the nuclear decay occurs and survives if it does not occur.  What defines a conscious observer?  When exactly does the state of superposition end?

Einstein also wrestled with the implications of quantum theory, never coming to accept it. He co-authored a paper putting forth an objection now known as the Einstein-Podolsky-Rosen (EPR) paradox.  While quantum mechanics may involve uncertainty and superposition of states, it does not involve breaking physics' conservation laws (such as energy, charge, or momentum) in any experiment.  And it does involve Heisenberg's uncertainty principle (you can't know both position and momentum to arbitrary accuracy).  The EPR paradox considers two electrons emitted by a system at rest.  If the net spin is zero to start with, the two electrons must have opposite spins. But until a measurement is taken, each electron could have either spin, so the wavefunction of each must contain both possibilites. Once you measure the spin of one elctron, however, the spin of the other is immediately determined.  But how could the electron that is not measured "know" that its partner electron has been detected with a particular spin?  It would seem that some sort of information is sent from the first electron to the second instantaneously.  Such instantaneous information transfer is at odds with relativity and violates causality.
 

Quantum Cryptography

Consider two co-workers, Alice and Bob, who want to transmit company secrets, such as the software used in the latest retinal scan security system, without the information falling into the hands of their competitors.  Eve is an employee of the competing company assigned with the task of finding out how the program works.  Traditional cryptography uses sophisticated encryption methods that could require more computing power to break than is currently available in the world.  But if Eve manages to obtain a copy of the key to the code, she can decipher Alice's transmission to Bob just as easily as Bob can.  And while Alice and Bob take many precautions, such as ownind the only two copies of the key, sending data only over secure, dedicated lines, and carefully screening all employees who would come near either of their computers, Eve might still find a way to infiltrate their company and/or hack into their computers.  She could then steal the code, tap their dedicated transmission line, and find out how the retinal scan software works.  So Alice and Bob turn to quantum cryptography.

Quantum cryptography provides the ability to determine whether a message has been intercepted, since Eve's measurement would change the system.  It relies upon the very behavior that Einstein, Podolsky, and Rosen claimed was impossible in their EPR paradox.  This behavior, in which measuring a property of a photon in one location immediately determines that property for its partner photon that could be miles away, is commonly known as "entangled states."  Alice produces pairs of photons and sends one of each pair to Bob while keeping the others for herself.  They each take measurements, and convey what they have measured to create a key.  If Eve has intercepted the signal, and thus discovered the key, Bob's photons will have inconsistencies when compared to Alices' photons.
 

Like it or not, understand it or not, quantum mechanics works.  The properties that the paradoxes take issue with, the superposition of states and the uncertainty principle, have been tested and re-tested, showing that they are indeed the best way we have developed so far to describe the world around us.  You will see in class how this quantum behavior can be used to perform parallel computation and for cryptography.
 
 

Bibliography

Schrödinger's Machines:  The Quantum Technology Reshaping Everyday Life, by Gerard Milburn.  This 1997 publication appears to be a revision of his previous book, the Feynman Processor.
 
 

Copyright © 2000 Doris Jeanne Wagner.  All Rights Reserved.