There is one way around this problem. That is to create a coarse-grained probability distribution for the quantum state in phase space. Such a distribution is called a Husimi distribution and it is basically a probability distribution in which the probability has been "smeared out" on the scale of h-bar. This allows us to comply with the Uncertainty Principle and still get a pretty good idea of what the quantum state looks like in phase space. The small-scale details of the quantum state get washed out, but we can still see where the quantum state sits in the phase space. These Husimi distributions can then be compared to the classical phase space to determine whether or not a particular eigenstate is associated with chaotic or regular classical motion. Examples of Husimi distributions are shown below.
This figure shows Husimi distributions for a variety of
quantum eigenstates. These distributions should be compared with the
classical phase space shown in part (b) of the
figure in the section on classical chaos. It is clear that the Husimi
distribution shown in (a) is
associated with the region of regular motion near the top of the classical
phase space. The Husimi distributions in (b), (c), and (d) are associated
with the resonance region near J=16.
The states shown in (e) and (f) are associated with the chaotic region of
the classical phase space.
This figure shows Husimi distributions that should be compared to the
classical phase space shown in part (c) of the
figure in the section on classical chaos. These states are all
associated with the region of classical chaos. Note that these states,
particularly (c) and (d), are much more spread out than the "chaotic"
states shown in parts (e) and (f) of the previous figure. This corresponds
with the fact that the chaotic region in part (c) of the classical phase
space figure is much larger in part (b).
The answer lies in a phenomenon known as an avoided crossing. In a
time-periodic system (a system in which the forces are periodic in time),
the eigenvalues of the quantum system change as the strength of the forces
is increased and the system becomes more chaotic. It is observed that when
two eigenvalues come close to each other in value, they will often repel
each other. This phenomenon is called an avoided crossing. The figure
to the right shows the eigenvalues of a quantum system (y-axis) as a function of the
strength of the periodic forces (x-axis). Numerous avoided crossings can
be seen. Note that the number of avoided crossings increases as the
strength of the periodic forces is increased.
These avoided crossings are important because they are known to lead to changes in the statistical distribution of the eigenvalues. As the classical system becomes more and more chaotic, the quantum system has more and more avoided crossings and the statistical distribution of its eigenvalues changes. But what happens to the eigenstates when the quantum system has an avoided crossing? The answer depends on what type of avoided crossing occurs. Avoided crossings can be classified as sharp or broad, and the effect of an avoided crossing on the eigenstates of the quantum system depends on whether the avoided crossing is sharp or broad.
A sharp avoided crossing, like the one shown to the right, causes little change in
the overall structure of the quantum eigenstates. In fact, what happens
is that the two eigenstates whose eigenvalues are involved in the sharp
avoided crossing simply exchange their structure as they pass through the
avoided crossing. This phenomenon can be clearly seen in a movie that illustrates the changes that occur in the Husimi distributions of the two quantum states
involved in this avoided crossing. A different version of this movie shows the sum
of the two Husimi distributions plotted as a 3D surface plot. The colors
represent which portions of the sum come from state A or from state B. It
is clear that the effect of the avoided crossing is just to switch the
colors around without effecting the overall structure.
A broad avoided crossing, like the one shown to the right, does result in important
changes to the structure of the eigenstates that are involved. After the
system passes through the broad avoided crossing the eigenstates that were
involved in the avoided crossing tend to be more spread out. This fits
with our observation that Husimi distributions become more spread out as
the classical system becomes more chaotic. Since there are three states
involved in the broad avoided crossing shown here, we must follow the
changes in the Husimi distributions of all three states. The changes that
take place can be clearly seen in a movie
that shows the Husimi distributions of the three states as they pass
through the broad avoided crossing.
Clearly avoided crossings play an important role in determining the evolution of quantum eigenstates as the classical system becomes increasingly chaotic. More research is needed to determine exactly how to distinguish between a sharp avoided crossing and a broad one, and to determine why these avoided crossings occur in the first place.
One of the reasons we have focused our study of quantum chaos on systems
that have periodic driving forces is that these systems are the simplest
systems that can exhibit classical chaos. Another reason, though, is that
these systems exhibit a number of interesting phenomena that are associated
with quantum chaos. One such phenomenon is high harmonic generation.
High harmonic generation occurs because a charged particle will radiate
light when it is driven by periodic forces. Usually, the light that is
radiated is at a frequency equal to the frequency with which the particle
is being driven. However, chaotic systems can radiate at integer multiples
of the driving frequency, sometimes reaching frequencies over one hundred
times greater than the driving frequency. This is known as high harmonic
generation and it has many potential applications such as the development
of inexpensive x-ray lasers. We have observed high harmonic generation in
our simulations and we are seeking to understand the properties of a
quantum system that lead to this phenomenon.
This figure displays some radiation
spectra showing the radiation from various quantum states as a function of
frequency. The spectrum in part (a) is from a state that sits in the
regular region of the classical phase space and it produces radiation only
at the driving frequency. The spectrum in part (b) is
from a state that sits inside a classical resonance and it produces
radiation at a number of frequencies but with no clearly defined harmonic
peaks. The spectrum in
part (c) is from a state that sits in the classical chaotic region of phase
space. In this last spectrum there are well-defined harmonic peaks that
produce significant radiation out to frequencies that are 19 times the
driving frequency. This is an excellent example of high harmonic
generation from a "chaotic" quantum state.
The locations of periodic orbits are indicated by the black dots. The leftmost periodic orbit is stable and it is surrounded by a regular classical resonance. The other two periodic orbits, though, are unstable. Note that some of the Husimi distributions have peaks on or near the location of even these unstable periodic orbits. We have also found that the more strongly a quantum state is peaked on a periodic orbit, the less likely it is to ionize. We are currently exploring this phenomenon in greater detail.
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