Download Heisenberg microscope and which-way experiments

Heisenberg microscope and which-way experiments
Quantenoptik-Seminar
12.05.2004
1. Historical development of the wave-particle duality
2. Gedankenexperiments with Young’s double-slit experiment
2.1.Complementarity
2.2.Young’s double-slit experiment with electrons
2.3.Einstein’s recoiling slit
2.4.Heisenberg microscope
2.5.Feynman’s light microscope
3. Experimental realisation of a which-way experiment
3.1.Experimental setup
3.2.Results
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1. Historical development of the wave-particle duality
In 1802 Young performed an experiment, which is nowadays known as Young’s double-slit
experiment. It is one of the most important experiments of wave theory and a clear example of
the diffraction of light conducted with essentially basic scientific equipment. The double-slit
experiment consists of letting light diffract through two slits producing fringes on a screen.
These fringes or interference patterns have light and dark regions corresponding to where the
light waves have constructively and destructively interfered. The experiment can also be
performed with a beam of electrons or atoms, showing similar interference patterns; this is
taken as evidence of the "wave-particle duality explained by quantum physics. In Young's
original experiment, sunlight passed first through a single slit, and then through two thin
vertical slits in otherwise solid barriers, and was then viewed on a rear screen.
Young observed an interference pattern which is, of course, a direct consequence of the wave
nature of light.
For many years scientists believed that light behaved in the same way as a wave of water.
Behavior like refraction and interference was perfectly well explained in this way.
But three experiments, the photoeffect, compton-scatterin and the black-body spectra could
not be explained by the 19th century point of view.
In 1900 Planck was the first to provide a theory that explained the black-body radiation. He
reasoned that the energy was not emitted continously but in forms of “quanta”. So Planck was
now arguing that light could be thought of as a particle!
So there was a big problem: Does light have wave-like or particle-like behavior?
In 1905 Albert Einstein solved the problem. He explained that light should be thought of as a
stream of particles. Each one has a certain energy that depends on it’s wavelenght. These light
particles were called photons.
Albert Einstein had built on Planck’s theory and was now able to explain the photoelectric
effect. For this work he was awarded the Nobel prize in 1921.
In 1924 Louis Victor de Broglie introduced the idea of matter waves. He argued that just as
light could be thought of as a particle, particles could also be thought of as a wave.
In 1927 Werner Heisenberg worked out the measurement uncertainties in quantum mechanics
and described them in “Heisenberg’s uncertainty principle”.
At the Solvay Congress in Brussels in the same year, the famous Einstein-Bohr debate took
place. Einstein’s goal was to devise an experiment where it should be possible to obtain
which-path information without influencing the interference pattern, in which I will go in
detail later.
2. Gedankenexperiments with Young’s double-slit experiment
The wave-particle duality is the main point of demarcation between quantum and classical
physics. In this context Young’s double slit experiment is used as a testing ground of this
duality because quantum mechanics predicts that any detector capable of determing the path
taken by a particle though one or the other of a two-slit plate will destroy the interference
pattern.
Richard Feynman described these attemps in the following words:
”We choose to examine a phenomenon which is impossible,
absolutely impossible, to explain in any classical way, and
which has in it the heart of quantum mechanics. In reality,
it contains the only mystery.”
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2.1. Complementarity
In this context Niels Bohr introduced the concept of complementarity to explain the peculiar
situation that there exist mutually incompatible experiments in quantum mechanics.
The most striking example of the complementary behavior of nature is the wave-particle
dualism. This duality is often addressed in the context of which-way experiments.
A measurement device can yield only information about the wave or the particle aspect, but it
is not possible to observe both aspects simultaneously with perfect accuracy.
For instance, one can consider the double-slit situation: Any attempt to detect which way the
particle follows leads to a degradation of the fringe pattern on the screen. A perfect
determination of the path by a so called which-way detector results in a complete
disappearance of the interference.
2.2. Young’s double slit experiment with electrons
Quantum mechanics is the description of the behavior of matter and light in all its details.
Things on a very small scale do not behave like waves and they do not behave like particles.
Newton thought that light was made up of particles, but then it was discovered that it behaves
like a wave. In the beginning of the twentieth century it was found that light did indeed
sometimes behave like a particle.
We now want to study the quantum behavior of atomic objects (electrons, protons, neutrons,
photons, and so on). So what we learn about the properties of electrons (which we shall use
for the following experiment) will apply also to all "particles," including photons of light.
What you can see here is the experimental setup of a double-slit experiment with electrons.
Fig. 2.1: Setup of a double
slit-experiment with
electrons
electron
source
double-slit
screen
The result which is expected in this experimental setup is an interference pattern, shown in
Fig. 2.2, because of the wave like properties of electrons, which de Broglie postulated.
Fig. 2.2:
Interference
pattern caused by
electrons passing
through a double
slit
electron
source
double-slit
screen
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Now we want to analyse the curve to see, weather we can understand the behavior of the
electrons. The first thing is that we now consider the electron as a particle, which goes
through the upper or the lower slit. This can be written in a form of a proposition:
“Each electron either goes through the upper
or through the lower slit”
So what we want to get in principle is which-way information.
Assuming to the proposition, all electrons that arrive at the detector can be divided into two
classes:
• those that came though the lower slit:
Fig 2.3: Distribution
of electrons passing
through the lower slit
Distribution of particles,
which went through the
lower slit
electron
source
double-slit
•
screen
and those that came through the upper slit:
Distribution of particles,
which went through the
upper slit
electron
source
Fig 2.4: Distribution
of electrons passing
through the upper slit
double-slit
screen
So the result as the sum over these two measured distribution seams quite reasonable. But if
we now compare the interference pattern we had measured before with the sum of these two
distribution, which is shown in Fig 2.5, we clearly see that they’re not the same:
electron
source
double-slit
Fig 2.5: Comparison of the
interference pattern and the sum
over the two single distributions
of Fig. 2.3 and Fig. 2.4
screen
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So the question is now: What has happened?
The attempt to measure the trace of the path of the electron altered the experimental setup and
destroyed the interference pattern. That means that the proposition we made is not true: It is
not true that each electron either goes through the upper or the lower slit, because if they did,
the probabilities should just add!
So we have seen with this experiment that in some cases the electron must be considered as a
wave!
2.3. Einstein’s recoiling slit
Einstein attempted to disprove the concept of complementarity which was introduced by
Bohr. His goal was to devise an experiment where it should be possible to obtain which-way
information without influencing the interference pattern.
This lead to the famous debate between Einstein and Bohr in the late 1920’s. In this context,
the double-slit experiment was a pivotal point of contention at the fifth Solvay Congress in
Brussels in 1927. There exist three versions of the recoil gedanken experiment which are all
attributed to Einstein, Fig 2.6 shows of one of them:
Fig. 2.6: The recoiling slit
gedanken experiment: By
measuring the recoil of the
movable slit one can
deduce the path of the
interfering particle.
Light
source
The which path detector consists of a freely moveable slit between the light source and the
double slit. Each particle has to traverse this slit before it reaches the double slit.
We can now try to measure the recoil momentum of the movable slit after scattering a
particle: If the slit moves downwards, one can conclude that the particle follows the way to
the upper one of the double slits. If the slit goes upwards the photon chooses the path to the
lower one.
A precise measurement of this momentum transfer permits to decide through which of the two
slits the particle has passed before arriving at the screen. Einstein argued that the interference
pattern should not be altered due to this detection process.
But a argument given by Bohr showed, why the reasoning of Einstein is not correct. He
pointed out that it is necessary to describe the whole experiment, including the which-path
detector, with the laws of quantum mechanics. The reason is that the momentum transfer due
the interaction of the particle with the movable slit is so small, that quantum uncertainties start
to play a major role. The uncertainty in the slit position is transferred to an equal uncertainty
in the position of the fringe pattern at the screen. Consequently the interference pattern is
washed out.
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2.4. Heisenberg microscope
screen
The Heisenberg microscope is an experiment whose
purpose is to measure the position of an electron with
very high resolution.
microscope
The incident photon is coming from the left. If the
objective
electron is assumed to be initially stationary, it’s
momentum is thus known exactly and we can then try
p sin
to determine it’s position simultaneously.
To observe where the electron is, one of the incident
p
photons must hit the electron and then be scattered
into the microscope’s objecive. When the photon
bounces against the electron, it transfers momentum to
photon
the electron.
electron
What we want to do is to measure the position of the
electron accurately with a very high resolution. Due to
Fig. 2.7: Setup of a Heisenberg microscope
the wave nature of light, there is a limitation on how
close two spots can be and still be seen as two separate spots. This distance is of the order of
the wavelenght of light.
In Feynman’s analysis of the following experiment, the loss of visibility of the interference
pattern occurs when the light wavelenght is shorter than the separation between the two slits,
allowing which path information to be obtained from a single scattered photon.
That means using photons with a high energy and a short wavelength. But a high energetic
photon imparts a large kick to the electron.
The aim is nearly not to disturb the electron by observing them. So to determine the
momentum of the electron accurately, the electron only must be given a small kick. This
means using a photon with a long wavelength and a low energy.
We can conclude that there is a certain trade off between an accurate measurement of the
position and an accurate measurement of the momentum which refers to Heisenberg’s
uncertainty relation.
2.5 Feynman’s light microscope
In 1960 Feynman proposed a gedankenexperiment which is now known as Feynman’s light
microscope. In this gedankenexperiment a perfect light microscope is used to determine
which-way information in a two slit experiment with electrons by analysing single scattered
photons.
Heisenberg
The setup is shown in Fig. 2.8:
microscope
Fig. 2.8: Experimental
setup of Feynman’s light
microscope
electron
source
light
source
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In addition to the double-slit experiment of Fig. 2.1 a light source is placed behind and
between the double slits.
When an electron passes the double slit on its way to the screen, it will scatter some light
which can be detected by an Heisenberg microscope.
Every time a photon is scattered by an electron, we see a flash of light near the upper or near
the lower slit, but never both at once. The result is always the same, no matter where the
microscope is. We allways observe the sum of the two probabilities of the which-way
measurement (blue curve).
But when we don’t look at the position of the electrons, the distribution is different to the one
when we look at them (red curve).
Werner Heisenberg formulated the attemps of observing an interference pattern and determine
which-way information as follows:
“ It is impossible to design an apparatur to determine which slit the electron passes through
that will not at the same time disturb the elecrons enough to destroy the interference pattern.”
3. Experimental realisation of an which way experiment
The first experimental realisation of a which way experiment by Michael Chapman et al.
succeeded in 1995.
The idea was to scatter single photons from interfering de Broglie waves in an atom
interferometer. What they observed was not only a a loss of coherence expected from
complementarity but also several surprising revivals of the fringe contrast. The question we
are now interested in to answer is:
Where is the coherence lost and how may it be regained?
3.1 Experimental setup
The experimental setup is shown here:
Fig.3.1: Experimental setup of the realisation of a which way experimentby M. Chalpamn et al. in 1995
At the beginning, a beam of atomic sodium with a narrow velocitiy distribution is optically
pumped with a +-polarized laser. Then the beam is collimated by two slits, separated by 85
cm.
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The atom interferometer uses three 200 nm period nanofabricated diffraction gratings which
are separated by 65 cm. The interference fringes are recorded by the measurement of the
atomic flux transmitted through the gratings while varying their relative positions.
By using a +-polarized laser beam to reasonately excite the atoms, single photons are
scattered from the atoms within the interferometer between the first and the second grating.
The atoms must decay back from the excited state to the ground state via spontaneous
scattering.
To study the effects of photon scattering on atomic coherence as a function of the separation
d, the excitation laser beam is translated along the atomic beam axis.
3.2 Results
The relative fringe contrast is recorded versus the
realative displacement of the two different paths (Fig.
3.2).
A beam separation d smaller than 0 indicates scattering
before the first grating.
Michael Chapman et al. observed that for small beam
separations scattering the photons before and
immediately after the first grating does not affect the Fig. 3.2: Relative contrast C(laser on)/C(laser off)
contrast. So at this point of scattering we can observe a as a fction of d
good interference pattern.
With increasing beam separation the contrast, which is a direct measurement of the coherence,
decreases smoothly towards zero. At around d
it is not possible to observe an
2
interference pattern. At this point the separation between the two paths is equal to the
microscope resolution.
This is what complementarity suggests: The fringe contrast disappears when the scattered
photon can determine through which slit the atom passed.
What we now expect is that with increasing beam separation d we can observe an even better
resolution. But as the separation d increases further we get partial revivals of the contrast.
There are two different ways to explain this behavior:
We consider one atom in the absence of scattering. Then the wavefunction at the third grating
can be written as:
ik x
i 0
2
u 2 ( x )e g
with k g =
0 ( x ) = u1 ( x ) + e
g
where u1/2(x) are the real amplitudes of the upper and lower beam,
0 is the relative phase which we may take to be zero and
g is the period of the grating.
Fig. 3.3: Interference pattern of one single
atom
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The interference pattern which we observe when this single atom described by the
wavefunction above reaches the third grating is shown in Fig.3.3.
If we now consider an atom elastically scattering a photon with a well defined incident and
final momentum we receive the following wavefunction:
i (k x+ )
u1 ( x
x) + u 2 ( x
x )e g
s ( x)
= k d = k x d which is shown in Fig. 3.4 (red curve) and just
(2 L z ) k x
which may be neglected.
a small spatial shift of the fringe envelope x =
k atom
We observe a spatial shift
By summing the two curves (blue and red one) up, we observe again an interference pattern,
shown in Fig. 3.5.
Fig. 3.4
Fig. 3.5
But in the process of scattering, to some atoms there is a small and to some there is a great
momentum transferred. This is the reason for the change of the phase shift at different beam
separations which can be seen in the following pictures:
At a beam separation of about zero the product of
= k d = k x d is small for k=const.
and small d. The result is only a small phase shift of the interference pattern and the sum over
the single distributions of the two atoms forms again an interference pattern (Fig. 3.6).
d
d
0
0 ,5
Photon
Photon
Fig. 3.7
Fig. 3.6
With increasing beam separation we observe smearing of the interference pattern and finally
around d =
there is a loss of the pattern (Fig. 3.7).
2
As the beam separation increases further, a periodic rephasing of the interference pattern
gives rise to partial revivals of the contrast, because for certain values of ei , s can show an
interference pattern.
We have now considered only two atoms. But to describe the effects that happen in the
interferometer in the correct way we have to sum up all atoms of the beam.
The interference pattern of each single atom can be described by its original contrast C0
) which refers to the different momentum
multiplied with an oscillation term cos(k g x +
transfers given to the atoms by scattering a photon.
The sum of the single waves is then given by the following integral:
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d ( k x ) P ( k x )C 0 cos(k g x +
)
where P( k) is the probability distribution of the momentum transfer,
where the average transfer is 1 k . A momentum transfer of 0 k occurs
for forward scattering of the incoming photon and 2 k occurs for
backward scattering (Fig. 3.8).
Because of the average of over the angular distribution of the scattered
photons we observe a loss of contrast C’ and a phase shift ’ of the
interference pattern:
d ( k x ) P ( k x )C 0 cos(k g x +
) = C ' cos(k g x + ' )
Fig, 3.8
The integral has the form of a Fourier transform. By regarding the Fourier transform of the
probability distribution which can be approximated as a rectangle function we get indeed the
observed distribution.
C ' (laser on)
C0 (laser of )
FT
beam separation d
Fig. 3.9: Fourier transform of the probability distribution of the momnetum transfer
The second way to explain why the coherence is lost and why it can be regained is to find the
answer to the question: What does the result tell us about which-way information?
By looking at the radiation characteristics of one atom in each interfering arm
we can explain the loss of coherence as a result of receiving which-way
information in the context of complementarity.
Fig. 3.10 shows with which probability the scatterd photons can be seen at the
different positions by varying the detector position.
Fig.3.11
dd
x
Fig.3.10
The red and the blue dashed curve in Fig. 3.11 show
the probability of scattering a photon at an atom of the
lower (blue) or the upper (red) one. So the range of
probable photon positions forms a single peak over
which the conditional visibility of the interference
pattern is high. At this postion which refers to the beam separation of zero
we observe no which-way information.
Fig.3.12
In Fig. 3.12 the opposite situation is shown. For a
0 ,5 we observe that the
beam separation of d
probabilities have
separated. There are
two distinct peaks, each corresponding to an atom in
one interfering arm.
The path is now sufficiently well separated and can be resolved by a measurement. This
behavior is what complementarity suggests: By observing which-way information, the
interference pattern is lost.
Photon
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Fig. 3.13. refers to further increasing beam
separation. For certain beam separations, however,
the position distribution consists of two fairly wellseparated peaks, but a secondary maximum of one
Fig.3.13
peak coincides with the maximum of the other one. So the path is in fact not
well determined and we get a partial revival of the contrast.
The next question we want to discuss is if we get a higher contrast by receiving less wich-way
information.
To find an answer we have to look at a variant of this experiment in order to reduce the
amount of which-way information. We now observe atoms which are correlated with photons
scattered into a restricted range of final directions. With this experiment it is now possible to
demonstrate that the coherence is indeed not truly lost and can be regained.
The observation of the correlated atoms could be in principle achieved by detecting the atom
in coincidence with photons scattered in a specific directions. But for some technical reasons
this is not feasible.
However, Michael Chapman et al. succeeded in realizing this type of correlation experiment,
because the deflection x of the atoms is a measurement of the final momentum projection
and includes the photon direction.
P( kx)
III
I
II
kx
Fig. 3.14: The different beam collimation positions with differetn momentum transfer distributions
By using a narrow beam collimation together with small detector acceptance it is possible to
selectively detect only those atoms correlated with photons scattered within a limited range.
The three different positions of the collimator refer to different atomic beam profiles
corresponding to different momentum transfer (Fig. 3.14).
The result is that the contrast
falls off more rapidly for the
faster beam (III) than for the
slower one (I and II) because
for the lower beam velocity
the momentum selectivity is
correspondingly lower. The
curves I and III also decrease
more slowly than the original
(dashed) one. And that leads
to the explanation with the
concept of complementarity:
It is possible to regain the
contrast by extracting less
which-way information.
Fig. 3.15: Results
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3. Summary and Outlook
The usual explanation for the loss of coherence is based on Heisenberg’s position-momentum
uncertainty relation. This is illustrated in the gedankenexperiments of Einstein’s recoiling slit
and Feynman’s light microscope.
The momentum uncertainty arises from momentum kicks transferred by scattered photons.
A new gedankenexperiment proposes that the loss of interference is not related to
Heisenberg’s uncertainty relation. Instead the correlation between the which-way detector and
the atomic beam are responsible for the loss of interference.
A controversial discussion started about the question:
Is complementarity more fundamental than the uncertainty principle?
S. Dürr, T. Nonn and G. Rempe showed that the momentum kicks are much to small to wash
out the interference pattern.
Instead correlations between the which-way detector and the atomic motion lead to the
concept of complementarity.
4. References
[1] M. S. Chapman, Phys. Rev. Lett. 75, 3783 (1995)
[2] D. E. Pritchard, Ann. Phys 10 (2000)
[3] R. Feynman, R. Leighton, and M. Sands, The Feynman Lectrues on Physics (AddisonWesley, Reading, MA, 1965), Vol. 3
[4] L. Mandel, J.Optics 10, pp. 51-64 (1979)
[5] S. Dürr, T. Nonn, G. Rempe, Phys. Rev. Lett. 395, 33-37 (1998)
[6] Yu Shi, Early Gedanken Experiments of Quantum Mechanics Revisited, Cavendish
Laboratory, University of Camebridge
[7] M. Rabinowitz, Examination of wave-particle duality via two-slit interference, Amor
Research, 715 Lakemead Way, Redwood City, CA
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