1. science
2. physics

QUANTUM CHAOS
DOMINIQUE DELANDE
Laboratoire Kastler-Brossel
Chaos is a well defined concept for classical systems. In these lectures, I study the
manifestations of chaos for microscopic objects, for which a quantum description
must be used. Various examples, mainly but not exclusively coming from atomic
physics, are used to illustrate our current understanding of the problem.
1
1.1
What is Quantum Chaos?
Classical Chaos
Chaos is usually defined for classical systems, i.e. systems whose dynamics
can be described by deterministic equations of evolution in some phase space.
The general form of these equations is:
dX
= f (X)
dt
(1)
where X is a vector (in phase space) representing the relevant physical properties of the system [1] – in the simplest case, it can be the position and
momentum of a single particle. In this case, the number of components of
the vector X, i.e. the dimension of phase space, is twice the number d of
degrees of freedom of the system. In the following, we will be interested in
systems with a small number of degrees of freedom, typically d ≤ 3. The function f depends only on the position X in phase space, which expresses the
deterministic character of the dynamics.
In the specific case of a time-independent Hamiltonian system for a single
particle, the phase space coordinates are the position q and momentum p,
and the equations of motion can be expressed using the Hamilton function
H(q, p) as [2]:
dqi
∂H
=
dt
∂pi
dpi
∂H
=−
.
dt
∂qi
(2)
(3)
Basically, classical chaos is exponential sensitivity on initial conditions:
two neighbouring trajectories diverge exponentially with time, i.e. the distance between the two trajectories generically increases as exp(λt) where λ is
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000
1
called the Lyapounov exponent of the system[1]. Sensitivity on initial conditions is responsible for the the decrease of correlations over long times, loss
of memory of the initial conditions and ultimately for deterministic unpredictibility of the long time behaviour of the system. Most often, when the
system is sensitive on initial conditions, it is also mixing and ergodic [1], i.e.
a typical trajectory uniformly fills up the entire phase space at long time.
For low-dimensional systems we are interested in, the dynamics is often a
mixed regular-chaotic one, depending on the initial conditions; also, when a
parameter is changed in the Hamilton function, the transition from regularity
to chaos is usually smooth with intermediate mixed dynamics. Such mixed
systems are rather complicated and not too well understood – at least for
quantum effects to be discussed in these lectures – and we will here restrict to
the two extreme simple situations where the motion is almost fully integrable
or almost fully chaotic.
1.2
Quantum dynamics
In quantum mechanics, there is neither any phase space, nor anything looking like a trajectory. Hence, the notion of classical chaos cannot be simply
extended to quantum physics. Quantum mechanics uses completely different
notions, like the state vector |ψi belonging to some Hilbert space, which describes all the physical properties of the system. Its evolution is given by the
Schr¨
odinger equation:
d|ψ(t)i
= H(t) |ψ(t)i
(4)
dt
where h
¯ is the Planck’s constant.
The linear Hamiltonian operator H(t) is acting in the Hilbert space. The
connection between this operator and the classical Hamilton function is far
from obvious. The usual rule is that the quantum Hamiltonian is obtained
from the classical one through replacement of the classical position by the
position operator (which is diagonal in the standard position representation
of the state by its wavefunction ψ(q) = hq|ψi) and replacement of the classical momentum by −i¯
h∂/∂q. There is a difficulty because the position and
momentum operators do not commute, which is solved by using symmetrized
combinations ensuring the hermiticity of H [3].
ψ(q) is not directly observable in quantum mechanics. In general – according to the standard Copenhagen interpretation of quantum mechanics –
the result of a measure is some diagonal element of an Hermitean operator,
something like hψ|O|ψi [3]. The physical processes involved in an experimental measurement are quite subtle, difficult and interesting, but beyond the
i¯
h
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000
2
subject of these lectures. It is also the subject of a vast litterature [4]. I will
not consider this problem and restrict to a purely Hamiltonian evolution.
The time-evolution operator U (t0 , t) is by definition the linear operator
mapping the state |ψ(t)i onto the state |ψ(t0 )i. It obeys the following equation
(which is equivalent to Schr¨
odinger equation):
i¯
h
∂U (t0 , t)
= H(t0 ) U (t0 , t).
∂t0
(5)
U (t0 , t) is the major object for studying the quantum dynamics. Because
H is an Hermitean operator, U (t0 , t) is a linear unitary operator. An immediate consequence is that the overlap between two states is preserved during
the time evolution. Indeed, one has:
hψ1 (t0 )|ψ2 (t0 )i = hψ1 (t)|U † (t0 , t)U (t0 , t)|ψ2 (t)i = hψ1 (t)|ψ2 (t)i
(6)
which implies that two “neighbouring” states will remain neighbors forever.
Because of linearity and unitarity, quantum mechanics cannot display any
sensitivity on initial conditions, hence cannot be chaotic in the ordinary sense!
However, the previous statement must be considered with care. Indeed,
classical mechanics can also be seen as a linear theory if one considers the
evolution of a classical phase space density ρ(q, p, t) given by the Liouville
equation [2]:
∂ρ
= {ρ, H}
∂t
(7)
where {, } denotes the Poisson bracket. The fact that we obtain both in
classical and quantum mechanics a linear equation of evolution in some space
just implies that the above argument on linearity in quantum mechanics is
irrelevant. Discussions on subjects like “Is there any quantum chaos?” are in
my opinion completely uninteresting because they focus on the formal aspects
of the mathematical apparatus used.
We will here define quantum chaos as the study of quantum systems whose
classical dynamics is chaotic. The questions we would like to answer are thus:
• What are the appropriate observables to detect the regular or chaotic
classical behavior of the system?
• More precisely, how the chaotic or regular behaviour expresses in the
energy levels and eigenstates of the quantum system?
• What kind of semiclassical approximations can be used?
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000
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These are the questions discussed in these lectures. I will only present
selected topics, forgetting lots of interesting questions and relevant references.
Thes questions of course go towards an intrinsic definition of quantum
chaos not refering to the classical dynamics [5]. Thus, the problem of quantum chaos is essentially related to the correspondance between classical and
quantum dynamics, the subject of semiclassical physics.
1.3
Semiclassical dynamics
The whole idea of a semiclassical analysis is to obtain approximate solutions of
the quantum equation of motion (the Schr¨
odinger equation) using only classical ingredients (trajectories...) and the Planck’s constant h
¯ . For a macroscopic
object, our common knowledge is that an approximate semiclassical solution
should be very accurate. Technically, this is true because ¯h is much smaller
than any classical quantity of interest (such as the classical action of the particle). One often refers to the “correspondance principle” as an explanation.
However, this is a very vague concept which is usually not clearly stated,
not proved and whose conditions of validity are not discussed. Actually, it
is so vague and qualitative that it should be rejected. Part of these lectures
are devoted to a serious scientific discussion of this issue, using the modern
knowledge on classical chaos.
In order to make the connection between classical and quantum quantities,
it is useful to define the Wigner representation defined as [6]:
Z p.x x ∗ x
1
ψ
q
−
ψ
q
+
exp
i
dx
(8)
W (q, p) =
(2π¯
h)N
2
2
¯h
This is a real phase space density probability, or rather quasi-probability
because it can be either positive or negative. Its evolution equation is simple
to compute [6]:
∂W
2
¯hΛ
= − H(q, p) sin
W (q, p)
(9)
∂t
¯h
2
with:
Λ=
−−−−→
X ←∂
∂
i
∂pi ∂qi
−
←−−−−→
∂ ∂
∂qi ∂pi
(10)
where the left (resp. right) arrow indicates action on the quantity on the left
(resp. right) side.
An explicit power expansion of the sine function is possible. This is in fact
a power expansion in ¯h, hence well suited for a semiclassical approximation.
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000
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At lowest (zeroth order), one finds exactly the classical Liouville equation,
thus establishing a link between the quantum and classical dynamics. At next
order in ¯h (actually ¯h2 ), one finds terms involving third partial derivatives of
the Hamiltonian. For harmonic systems, these terms vanish, proving that the
classical and quantum phase space dynamics completely coincide.
For non-harmonic systems, the corrective terms produce an additional
spreading of initially localized wavepackets. For chaotic systems, the classical
solutions of the Liouville equations tend to stretch and fold along (exponentially) unstable directions and – because of conservation of volume in phase
space – to shrink along (exponentially) attractive directions. This rapidly
creates “whorls” and “tendrils” in the classical phase space density, which
in turn implies more and more rapid spatial changes of the density. Thus,
as time goes on, one expects some higher order partial derivatives to grow
exponentially. Although the corresponding terms in the quantum equation
of evolution are multiplied by ¯h2 , they will unavoidably grow and overcome
the classical Liouville term a . Hence, after some “break time”, the detailed
quantum evolution will differ from the classical one. The estimation of this
break time is a very difficult questions, and different answers are possible,
depending on which aspect of the dynamics is under study (local, global...).
I will not discuss this important point here, see [4,5].
Of course, for smaller h
¯ , the higher order terms are smaller and it requires
a longer time for them to perturb the dynamics. Hence, the break time has
to tend to infinity in the semiclassical limit h
¯ → 0. For a fixed time interval,
one can always find a sufficiently small ¯h such that the quantum and classical
dynamics are almost identical. In other words, over a finite time range, the
quantum dynamics tends to the classical one as ¯h → 0. However, this limit is
not uniform. For fixed ¯h, there is always a finite time after which the quantum
dynamics differs from the classical one. In other words, the two limits t → ∞
and ¯h → 0 do not commute. Taking first h
¯ → 0, then t → ∞ is studying
the long time classical dynamics, i.e. classical chaos. The other limit t → ∞,
then ¯h → 0, is what we are interested in, namely quantum (and semiclassical)
chaos. In fact, the semiclassical limit ¯h → 0 is highly singular and quantum
chaos is essentially the problem of understanding correctly this limit.
aA
rather similar conclusion can be obtained using the so-called Ehrenfest theorem, which
gives the time evolution of average values of the position and momentum [3]. Provided the
wavefunction is a localized wavepacket, these equations coincide with the classical equations
of motion. However, the unavoidable spreading of the wavepacket destroys its localized
character and breaks this simple correspondance. Again, the problem with this approach
is to estimate precisely how the spreading affects the global dynamics.
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000
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1.4
Physical situations of interest
Simple equations of motion may produce a chaotic behaviour. A rather nonintuitive result is that chaos may take place in low dimensional systems. On
the other hand, classical chaos can only exist in systems where different degrees of freedom are strongly coupled (this is a consequence of the KAM theorem [1]). This implies that a small perturbation added to a regular system
cannot make it chaotic.
The simplest possible chaotic systems are thus time-independent 2dimensional systems. It is also simpler to consider bound systems with a
discrete energy spectrum. Various model systems have been studied, among
which billiards are the simplest ones. A billiard is a compact area in the plane
containing a point particle bouncing elastically on the walls. Depending on
the shape of the boundary, the motion may be regular or chaotic. From the
quantum point of view, one has to find the eigenstates of the Laplace operator
whose wavefunction vanishes on the boundary [7].
Open (i.e. not bound) systems have also been studied, mainly because
the classical phase space structure is usually simpler in such systems. The
simplest example is the “three disks system” which is an open billiard with
three identical circular obstacles centered on a equilateral triangle. This is an
example of “chaotic scattering” [8], where the chaotic behaviour comes from
the existence of arbitrarily long and complex trajectories bouncing off the 3
disks without escaping. From the quantum point of view, there are no longer
discrete bound states, but rather resonances with complex energies which are
poles of the S-matrix or of the Green’s function.
If we now turn to “experimental” systems, it is obvious that quantum
effects are likely to be noticeable only for microscopic systems. The dynamics
of nucleons in an atomic nucleus might be chaotic – at least at sufficiently
large energy – and the experimental results on highly excited states played a
major role in the early development of quantum chaos [7]. The drawback is
the existence of complex collective effects and the fact that the interaction is
not perfectly well known.
Atoms are among the best available prototypes for studying quantum
chaos and I will use them as examples in these lectures. Compared with other
microscopic complex systems (nuclei, atomic clusters, mesoscopic devices...),
atoms have the great advantage that all the basic components are well understood : these are essentially point particles (electrons and nucleus) interacting through a Coulomb static field, and interacting with the external world
through electromagnetic forces. Hence, it is possible to write down an explicit
expression of the Hamiltonian. Another crucial advantage of atomic systems
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000
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is that they can be studied theoretically and experimentally. The word “experiment” must here be understood as traditional laboratory experiments, but
also as “numerical experiments”. Indeed, currently available computers make
it possible to numerically compute properties of complex systems described by
simple Hamiltonians. During the last fifteen years, the constant interaction
between the experimental results and the numerical simulations led to major
advances in the field of quantum chaos.
Depending on the energy scale involved, different parts of the atomic dynamics are relevant. At “large” energy – of the order of 1eV – it is the internal
dynamics of the atomic electrons (their motion around the nucleus) which may
be chaotic. At much lower energy – 1 µeV – it is the external dynamics of the
center of mass of the atom (considered as a single particle) which may display
a chaotic behavior under the influence of an external electromagnetic field
[9]. The latter case has been made possible because of the impressive recent
improvements on the control of ultra-cold atomic gases using quasi-resonant
laser beams [10].
Let us illustrate the first case by considering simple isolated atoms with
few electrons. The simplest atom – hydrogen – can be exactly solved both in
classical and quantum mechanics and is thus not chaotic at all. The helium
atom brings the three-body problem described by the following Hamiltonian
(Z is the charge of the nucleus and m the mass of the electrons):
H=
Ze2
Ze2
e2
p21 + p22
−
−
+
2m
r1
r2
r12
(11)
which is known to be classically essentially chaotic [45]. From the chaos point
of view, the interesting situation is when the two electrons have comparable
excitations. Strong dynamical correlations between the two electrons are expected, leading to a breakdown of the independent electron picture for highly
doubly excited states. Indeed, the most recent experimental results close to
the double ionization threshold display extremely complex structures in the
ionization cross-section, which have been shown to be related with the onset
of (quantum) chaos [12,13]. The helium atom is briefly discussed in section
4.6.
In molecules, the dynamics of the electrons may also be chaotic. In some
cases, the motion of the nuclei in the effective potential created by the electrons (which follow the nuclei adiabatically) is chaotic. Some interesting results on the N O2 molecules have been obtained [14].
At the microscopic level, the dynamics of electrons in a solid state sample
may present a chaotic dynamics in, for example, suitable combinations of
external fields. This has lead to dramatic results showing very clearly the
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000
7
relevance of periodic orbits for understanding the quantum chaotic dynamics
[15].
Another possibility exists for experimental study of quantum chaos. One
can consider wave equations describing some other physical phenomena, which
have a structure very similar to the Schr¨
odinger equation. As what we are
interested in is in fact “wave chaos” (properties of eigenmodes for example)
whatever the waves themselves are, this opens a wide variety of possible experiments. The best example is provided by flat microwave cavities where
solving the Maxwell equations is equivalent to calculating the eigenstates of
the corresponding two-dimensional Schr¨
odinger billiard [16]. The advantage
is that a measure of the “wavefunction” is possible.
I know work out in some detail the simplest atomic prototype, which will
be discussed as an example in the rest of these lectures.
1.5
A simple example: the hydrogen atom in a magnetic field
We consider the simplest atom – hydrogen – exposed to a strong external
uniform magnetic field directed along the z-axis.
Using the symmetric gauge A = 12 r × B for the vector potential, the
Hamiltonian is given by (q is the charge of the electron):
p2
q2
qB
q 2 B 2 ρ2
−
−
Lz +
(12)
2m 4π0 r 2m
8m
where Lz is the z-component of the angular momentum. In atomic units
(¯
h = m = |q| = 4π0 = 1), it reads:
H=
p2
1 γ
γ 2 ρ2
− + Lz +
(13)
2
r
2
8
where γ denotes the magnetic field in atomic units of 2.35 × 105 T.
Because of the azimuthal symmetry around the magnetic field axis, the
paramagnetic term γLz /2, responsible for the usual Zeeman effect, is just a
constant. The diamagnetic term, γ 2 ρ2 /8, is directly responsible for the onset
of chaos in the system. The competition between the Coulomb potential with
spherical symmetry and the diamagnetic potential with cylindrical symmetry
governs the dynamics. As a crude criterion, chaos is most developped when
these two terms have the same order of magnitude. This can be realized in a
laboratory experiment with Rydberg states n ' 40 − 150 [17,18,19].
When written in cylindrical coordinates, the Hamiltonian (13) describes
a time-independent two-dimensional system belonging to the class of the simplest possibly chaotic systems [1]. This makes this system an almost ideal
prototype for the study of quantum chaos [20].
H=
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000
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One of the main difficulties in the study of the semiclassical limit ¯h → 0
is that the value of the Planck’s constant is fixed in laboratory experiments.
One can get around this difficulty in atomic systems thanks to the existence
of scaling laws. There is a convenient scaling of all variables and external
fields which leave the equations of motion invariant [21]:

r → λ−1 r,



p → λ1/2 p,
H → λH,



γ → λ3/2 γ,
(14)
where λ is any positive real number.
This means that different initial conditions with different external fields
may have exactly the same classical dynamics. This is no longer true in
quantum mechanics, since there is an absolute scale imposed by the Planck’s
constant h
¯ . Different scaled situations observed experimentally correspond
to the same classical dynamics with different effective values of the Planck’s
constant.
The scaled energy
= Eγ −2/3
(15)
measures the energy of the electron in units of magnetic field. Because of
the scaling law, the classical dynamics, instead of depending both on E and
γ, actually depends only on , whereas quantum properties depend a priori
on both quantities. Hence, in a real (or numerical) experiment, the semiclassical limit h
¯ → 0 can be studied, just by tuning simultaneously the energy
and the characteristics of the external fields according to Eq. (14) towards
higher excited states. This possibility has revealed extremely important for
understanding the classical-quantal correspondance [18].
At low scaled energy (roughly < −0.5) the classical dynamics is mainly
regular (this is the low field limit where the magnetic field is a small perturbation). Increasing from −0.5, the system smoothly evolves to a fully chaotic
situation reached above = −0.13. Finally, the phase space opens to infinity
at = 0.
¿From the theoretical point of view, the use of group theory allows extremely efficient numerical experiments [21,20,22], making the computation of
very accurately highly excited energy levels and wavefunctions possible. The
calculated quatitites are found in exact agreement with the (less accurate)
experimental measurement (see [17])!
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000
9
Figure 1. The important time and energy scales for a chaotic quantum system. The shortest
relevant time scale is TMin , the period of the shortest periodic orbit. The most important
quantum time scale is THeisenberg , associated the mean energy level spacing. In the semiclassical limit, THeisenberg is much larger than TMin . One expects a universal classical
behaviour at long times, thus universal statistical properties of the energy levels, described
in section 3.4. At short times (long energy range), the specificities of the system appear to
be related to the periodic orbits of the system, as explained in section 4.
2
Time scales - Energy scales
For a correct undertsanding of the connections between the quantum and the
classical properties of a chaotic system, it is crucial to know the relevant time
scales (and the corresponding energy scales) of the problem.
The shortest time scale is simply the typical time scale for the simplest
evolution of the system. It is conveniently taken as the period of the shortest
periodic orbit TMin . A slightly longer time scale is given by the time taken
for chaos to manifest, that is the inverse of the typical Lyapounov exponent.
The larger the sensitivity on initial conditions, the shorter this time scale.
These two time scales have of course nothing to do with h
¯ . The corresponding
energy scale, 2π¯
h/TMin , see Fig. 1, is the largest energy scale of interest in
the problem.
There is also a basic quantum time scale. To understand its origin, let us
consider a time-independent bound quantum system with Hamiltonian H, in
an arbitrary initial state |ψ(t = 0)i. Its evolution can be expressed using the
discrete eigenstates and eigenvalues of the Hamiltonian H
H|φi i = Ei |φi i
(16)
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000 10
with the following expression:
|ψ(t)i =
X
i
Ei t
ci exp −i
¯h
|φi i,
(17)
where the constant coefficients ci are computed from the initial state using:
ci = hφi |ψ(0)i.
(18)
The autocorrelation function of the quantum system is a diagonal element
of the time-evolution operator:
X
Ei t
2
C(t) = hψ(0)|ψ(t)i = hψ(0)|U (t, 0)|ψ(0)i =
|ci | exp −i
.
(19)
¯h
i
It is a discrete sum of oscillating terms, and, consequently, a quasi-periodic
function of time. This is extremely different from a classical autocorrelation
function for a chaotic system which is decreasing on the characteristic time
scale TMin and does not show any revival at longer times [1].
The Fourier transform of the autocorrelation function is:
Z ∞
X
1
˜
C(E) =
eiEt/¯h C(t)dt =
|ci |2 δ(E − Ei )
(20)
2π¯
h −∞
i
that is a sum of δ-peaks at the positions of the energy levels.
If we now consider the Fourier transform not over the whole range of time
from −∞ to +∞, but over a finite time interval, we obtain a smoothed version
of the quantum spectrum:
1
C˜T (E) =
T
Z
T /2
−T /2
eiEt/¯h C(t)dt =
X
i
|ci |2
i)
sin T (E−E
2¯
h
T (E−Ei )
2¯
h
,
(21)
where all the peaks are smoothed δ-peaks of width 2π¯h/T. For short T , the
different broadened peaks centered at the energy levels Ei overlap, and C˜T (E)
is a globally smooth function, like its classical counterpart. In such a situation,
it is possible (although nothing proves that is is always the case) that the
quantum C˜T (E) mimics the classical chaotic behaviour. The important point
is that, for large T , the different peaks do not overlap and the discrete nature
of the energy spectrum must appear in C˜T (E), whatever the initial state.
The typical time needed for resolving individual quantum energy levels is
called the Heisenberg time and is simply related to the mean level spacing ∆
through:
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000 11
2π¯h
.
(22)
∆
After this time, the quantum system cannot mimic the classical chaotic
behaviour which has a continuous spectrum. Since THeisenberg depends on h
¯,
one can understand how quantum tends to classical dynamics as ¯h goes to
zero. The mean level spacing is given by the Weyl’s rule and scales as ¯hd ,
with d the number of degrees of freedom (see Eq. (25)). For two- (or higher)
dimensional systems, THeisenberg tends to infinity as ¯h → 0, see Fig. 1.
In some sense, after the Heisenberg time, the quantum system “knows”
that the energy spectrum is discrete, it has resolved all individual peaks and
the future evolution cannot bring any essential new information. As a consequence, the system cannot explore a new part of the phase space, it freezes
its evolution, repeating forever the same type of dynamics.
Other time scales may exist in specific systems. For example, in an open
Hamiltonian system, the typical time scale for escaping the chaotic region is
obviously important. Also, in mixed chaotic-regular systems, different time
scales coexist in the different regions of phase space (and at their boundaries)
making general statements extremely difficult. For systems coupled to their
environments, dissipation and decoherence of the quantum wavefunction is
known to play a very important role [4] and these effects may be dominant
over chaotic effects. For the internal motion of electrons in atoms, the most
important dissipative effect is spontaneous emission of photons, a process usually rather small, acting significantly only after thousands of classical periods
[23]. For the sake of simplicity, we will restrict to the case where TMin and
THeisenberg are the only relevant time scales. In the semiclassical limit ¯h → 0,
the corresponding energy scales 2π¯h/TMin and 2π¯h/THeisenberg = ∆ are both
small compared to the energy itself. This means that we will always look at
relative small changes in the energy, such that the classical dynamics does
not substantially changes over the energy range considered. This is of course
possible in the semiclassical regime thanks to the large density of states. For
low excited states, such a local approach lacks any relevance.
THeisenberg =
3
3.1
Statistical Properties of Energy Levels – Random Matrix
Theory
Level Dynamics
The goal of traditional spectroscopy is to assign quantum numbers to the
different energy levels in order to obtain a complete classification of the spec-
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000 12
trum. When little is known about the system, it is difficult to identify the
good quantum numbers and their physical interpretation, or even to know
whether they exist or not. A simple tool is to look at the level dynamics,
that is the evolution of the various energy levels as a function of a parameter.
As good quantum numbers are associated with conserved quantities, i.e. operators commuting with the Hamiltonian, energy levels with different sets of
good quantum numbers are not coupled and thus generically cross each other
[24]. On the contrary, if two states are coupled, the energy levels will repel
each other, producing an avoided crossing. The width of the avoided crossing,
i.e. the minimum energy difference between the two energy curves, is a direct
measure of the strength of the coupling.
Thus, looking at the level dynamics gives some qualitative information
on the properties of the systems. This is illustrated in Fig. 2 which shows
the evolution of the energy levels of a hydrogen atom as a function of the
magnetic field strength. At low magnetic field, Fig. 2a, there are only level
crossings. A given eigenstate can be unambiguously followed in a wide range
of field strength, since it crosses (or has very small avoided crossings with)
the other energy levels, which proves that there are at least approximate good
quantum numbers. At higher magnetic field, Fig. 2b, the sizes of the avoided
crossings increase and individual states progressively loose their identities. In
other words, the good quantum numbers are destroyed.
A crucial observation is that the transition from crossings (or tiny avoided
crossings) to large crossings takes place where the classical dynamics evolves
from regular to chaotic. The transition is smooth – with the proportion of
large avoided crossings progressively increasing – and there is a large intermediate region where crossings and large avoided crossings coexist. This corresponds to the range of scaled energies ∈ [−0.5, −0.13], in complete agreement
with the classical transition from regularity to chaos, see section 1.5. From
a pure quantum point of view, this phenomenon is extremely difficult to understand: when the magnetic field strength increases, the only change in the
matrix representing the Hamiltonian, Eq. (12), in any basis, is a global multiplication of all the matrix elements of ρ2 by a constant factor. The dramatic
effect on the energy level dynamics is a direct manifestation of chaos in the
quantum properties of the system.
In section 4, I will give an explanation of this transition from the regular
region where good quantum numbers, i.e. conserved quantities, exist to the
chaotic region where they are destroyed.
In the fully chaotic regime, the energy levels and the eigenstates strongly
fluctuate when the magnetic field is changed. In that sense, the quantum system shows a high sensitivity on a small perturbation, like its classical equiva-
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000 13
Figure 2. Map of the energy levels of a hydrogen atom versus magnetic field for typical
Rydberg states of the (Lz = 0, even parity) series. At low energy (a), the classical dynamics
is regular and the energy levels (quasi-) cross. The quantum eigenstates are defined by a
set of good quantum numbers. At high energy (b), the classical dynamics is chaotic, the
good quantum numbers are lost and the energy levels strongly repell each other. The strong
fluctuations in the energy levels are characteristic of a chaotic behaviour.
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000 14
lent. The energy spectrum of a classically chaotic system displays an extreme
intrisic complication, which means the death of traditional spectroscopy. Such
extremely complex spectra have been observed experimentally in atomic systems in external fields [25], on the eigenmodes of microwave billiards (when
a parameter of the billiard shape is varied) [16] and numerically on virtually
all chaotic systems [24,7]. It should be emphasized that level dynamics in the
chaotic regime looks extremely similar whatever the system is, as long as its
classical dynamics is chaotic. It is probably the simplest and most universal
property.
3.2
Statistical analysis of the spectral fluctuations
This qualitative property has been put on a firm ground by the study of the
statistical properties of energy levels [7,24,26]. The idea is the following: there
are far too many levels and their evolution is far too complicated to deserve
a detailed explanation, level by level. In complete similarity with a gas of
interacting particles where the detailed positions of the various particles do
not really carry the relevant information which is rather contained in some
statistical properties, we must use a statistical approach for the description of
the energy levels of a chaotic quantum system. In order to compare different
systems and characterize the spectral fluctuations, we must first define proper
quantities. For a complete description, see [7,24].
Density of states
The density of states is:
d(E) =
X
δ(E − Ei )
(23)
i
where the Ei are the energy levels of the system. The cumulative density of
states counts the number of energy levels below energy E. It is thus:
Z E
X
n(E) =
d() d =
Θ(E − Ei )
(24)
−∞
i
This is a step function with unit steps at each energy level. When there is a
large number of levels, one can define the averaged cumulative density of states
n
¯ (E), a function interpolating n(E) by smoothing the steps. Its derivative
¯
is the averaged density of states d(E).
There are several cases where this
quantity contains the only relevant quantity for the physics of the system.
For example, in a large semiconductor sample, the averaged density of states
at the Fermi level is what determines the contribution of electrons to the
specific heat at low temperature [27].
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000 15
The averaged density of states can be determined in the semiclassical
approximation (see section 4) by the Weyl’s rule (also known as the ThomasFermi approximation):
Z
1
¯
d(E)
=
dp dq δ(H(q, p) − E)
(25)
(2π¯h)d
It only depends on the classical Hamilton function H and not on the regular
or chaotic nature of the dynamics.
Unfolding the spectrum
The next step is to eliminate the slow changes in the averaged density of states
by defining an unfolded spectrum through the following quantity:
ˆ (x) = n(¯
N
n−1 (x))
(26)
which is nothing but the cumulative density of states represented as a function
of a rescaled variable such that the “energy levels” now appear equally spaced
by one unit. These rescaled energy levels xi = n
¯ (Ei ) have by construction
density unity. It allows to compare spectra got for different parameters or
even for completely different systems.
Nearest Neighbor Spacing Distribution
The simplest quantity is the distribution of nearest neighbour spacings, i.e.
of energy difference between two consecutive levels si = xi+1 − xi . This distribution is traditionaly denoted P (s). By virtue of the unfolding procedure,
the average spacing is one. Its behaviour near s = 0 measures the fraction of
very small spacings (quasi-degeneracies), hence the degree of level repulsion.
Number Variance
The use of the nearest neighbor spacing distribution is simple, but not very
logical from the statistical physics point of view. Indeed, P (s) involves all
correlation functions among the energy levels. It is simpler to consider separately the two-point, three-point, etc... correlation functions. The two-point
correlation function R2 depends only on the energy difference if the spectrum
is stationary (i.e. statistically invariant by a global translation, which is likely
for a large unfolded spectrum). Near 0, it again measures the degree of level
repulsion. A more global quantity is the number variance Σ2 (L) which measures the variance of the number of levels contained in an energy interval of
length L. It is related to the two-point correlation by [7]:
2
Σ (L) = L + 2
Z
L
(L − x)(R2 (x) − 1) dx
(27)
0
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000 16
It is a measure of the rigidity of the spectrum, that is, it measures how the
spectrum deviates from a uniform spectrum of equally spaced levels.
Spectral Rigidity
A related quantity is the so-called spectral rigidity ∆3 (L) which measures
how much the cumulative density of states differs from its best linear fit on
an energy interval of length L. The relation is:
∆3 (L) =
2
L4
Z
L
(L3 − 2L2 x + x3 )Σ2 (L) dx
(28)
0
It is again an alternative to the two-point correlation function. Its advantage is
that it is very robust against imperfections such as spurious or missing energy
levels and can be determined rather safely from a limited number of energy
levels. This is of major importance for example in analyzing experimental
atomic [28,29] or nuclear spectra [7].
3.3
Regular Regime
In the regular regime (see Fig. 2a), consecutive energy levels generally do not
interact. Thus, from the statistical point of view, they can be considered
as independent random variables. The distribution of spacings is the one of
uncorrelated levels, that is a Poisson distribution:
P (s) = e−s ,
(29)
which nicely reproduces the numerical results obtained on different systems
(see Fig. 3a) and also several experimental results [30,16]. Note that the maximum of the distribution is near s = 0 which shows that quasi-degeneracies
are very probable and that level repulsion is absent.
This is a universal result which applies generically to regular systems.
Other statistical quantities can be described as well. The two-point correlation
function is simply R2 (x) = 1 leading to the number variance
Σ2 (L) = L.
(30)
Fig. 3b shows the numerical result for the hydrogen atom in a magnetic
field in the regular regime. The agreement with the prediction is good, at
least for low L. The saturation at large L can be quantitatively understood
using periodic orbit theory (see section 4). In simple words, it is due to long
range correlations in the spectrum induced by periodic orbits.
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000 17
Figure 3. Statistical properties of energy levels for the hydrogen atom in a magnetic field,
obtained from numerical diagonalization of the Hamiltonian in the regular regime. (a)
Nearest neighbor spacing distribution. The distribution is maximum at 0 and well fitted by
a Poisson distribution (dashed line). (b) Number variance. Again, the Poisson prediction
(dashed line) works quite well. The saturation at large L is due to the residual effects of
periodic orbits and is well understood.
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000 18
Figure 4. Same as figure 3, but in the classically chaotic regime. (a) The probability
of finding almost degenerate levels is very small (level repulsion). The results are well
reproduced by the Wigner distribution (dashed line) and Random Matrix Theory. (b)
The number variance is much smaller than in the regular case, showing the rigidity of the
energy spectrum. The results agree perfectly with the prediction of Random Matrix Theory
(dashed line).
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000 19
3.4
Chaotic Regime – Random Matrix Theory
In the chaotic regime, the strong level repulsion induces a completely different
result for the spacing distribution – see Fig. 4a – with practically no small
spacing, and also a lack of large spacings.
A simple model is able to predict the statistical properties of energy levels.
It assumes a maximum disorder in the system and that – from a statistical
point of view – all basis sets are equivalently good (or bad). It therefore models
the Hamiltonian by a set of random matrices which couple any basis state to
all the other ones. Depending on the symmetry properties of the Hamiltonian
(especially with respect to time reversal, see section 3.5), different ensembles
of random matrices have to be considered. Let us assume for the moment
that the system is time-reversal invariant and can be represented by a real
symmetric matrix Hij in some basis. If the matrix size is N (not to be
confused with the number of degrees of freedom), this leaves N (N + 1)/2 real
independent random variables. The natural (normalized) measure over the
matrix space is:
Y
Y
dH ∝
dHii
dHij
(31)
i=1..N
i,j=1..N ;i<j
which is invariant by any orthogonal transformation and thus puts all the
orthonormal basis on the same footing. As a consequence, the probability
density P (H) itself must be invariant by any orthogonal transformation.
For simplicity, we will assume that the various matrix elements are independent random variablesb . With these basic assumptions, it is tedious but
rather easy to show that the probability density can be written as [24]:
Tr(H 2 )
(32)
P (H) ∝ exp −
4σ 2
where σ is the only remaining free parameter.
¿From this equation and expanding the trace of H 2 as a function of the
matrix elements Hij , one obtains easily that all matrix elements have a Gaussian distribution with zero average and variance:
2
< Hij
>= (1 + δij )σ 2
(33)
These properties define the Gaussian Orthogonal Ensemble (GOE) of random matricesc .
b This
hypothesis is not at all crucial. It can be easily relaxed, generating other ensemble
of random matrices with similar statistical properties.
c An alternate derivation of the GOE is based on information theory. If we look for the prob-
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000 20
Knowing the probability density, we have to extract the statistical properties of the eigenvalues. The ensemble being invariant by any orthogonal
transformation, it is simple to use as random variables the N eigenvalues
and the N (N − 1)/2 angles which characterize the orthogonal transformation
bringing H to its diagonal form. The joint probability distribution is then
obtained by tracing over the N (N − 1)/2 angles. This is rather straightforward, because of the orthogonal invariance. The angles appear neither in the
probability distribution itself, nor in the Jacobian of the transformation. The
calculation of the Jacobian is the only tricky point. For a 2X2 matrix, it is
straightforward (reader, you should do the calculation by yourself!) to show
that is is |E1 − E2 | where E1,2 are the two eigenvalues. For a N XN matrix,
it is simply the product of all |Ei − Ej | terms [24]. One finally obtains the
joint probability density:


!
PN
2
Y
E
i
i=1
P (E1 , .., EN ) ∝ 
|Ei − Ej | exp −
(34)
4σ 2
i,j=1..N ;i<j
This formula already contains a lot of information. Level repulsion is due
to the |Ei − Ej | factors which exclude level degeneracies. This factor is purely
geometrical: it comes from the Jacobian of the transformation from matrix
elements to eigenvalues.
Although it looks simple, it is quite difficult to extract from the joint probability density the various statistical quantities of interest. It is easy for N = 2
and also feasible in the limit N → ∞, but involves the use of either beautiful
old-fashioned mathematics [26] or almost incomprehensible supersymmetry
techinques [31]. Most formulas are explicit but not very illuminating; they
can be found in [7,26].
The spacing distribution cannot be calculated in closed form, but it happens to be very close to the result got for N = 2, known as the Wigner
distribution:
πs − πs2
P (s) =
e 4
(35)
2
This distribution, shown in Fig. 4a, agrees extremely well with the numerical results got on the hydrogen atom in a magnetic field. Similar results
have been obtained on dozens of quantum chaotic systems, both numerically
R
ability density which maximizes the entropy S = − P (H) ln P (H)dH with the constraint
that the average value of Tr(H 2 ) is fixed, one rediscovers immediately (using Lagrange
multipliers) the GOE. The idea behind this derivation is that we know basically nothing
about the distribution and have to take it as general as possible.
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000 21
and experimentally. Experimental examples are the energy levels of highly
excited nuclei [7], rovibrational levels of the N O2 molecules [14], energy levels
of the hydrogen atom in a magnetic field [29] and electromagnetic eigenmodes
of microwave cavities [16].
The transition from a Poisson distribution in the classically regular regime
to a Wigner distribution in the chaotic regime gives a characterization of
quantum chaos, at least for highly excited states.
Other statistical properties have been studied and are found in good agreement with the predictions of Random Matrix Theory [28]. For example, the
number variance, shown in Fig.4b, is in perfect agreement with the GOE
prediction which, for large L, is
1
ln 2πL
(36)
π2
Note that the number variance is much smaller here than in the regular case.
The spectrum is extremely rigid, as for L = 106 ,Σ2 is only of the order of
3. This means that the typical fluctuation of the number of levels is 1 or 2
additional or missing levels over a range of√one million level. In the Poisson
model, the typical fluctuation would be L = 1000 levels! This extraordinary large rigidity is due to the strong couplings existing between all the
states in the model. If a fluctuation makes the level repulsion abnormally
large between two states, they cannot repell too strongly because they are
themselves strongly pushed by the other levels. From maximum disorder at
the microscopic level, a globally rigid structure is born.
Finding universal properties in the local statistical properties of energy
levels for chaotic systems is not a real surprise. As discussed in the preceding
section, this range of energy (mean level spacing ∆) corresponds to a long
time behaviour (h/∆ = THeisenberg TMin ), where chaos is classically fully
developped with its universal properties. Universality is also observed in the
corresponding quantum dynamics. On the other hand, at shorter times of
the order of TMin , non-universal properties exist in the classical behaviour.
This implies also a deviation from the predictions of Random Matrix Theory
on a large energy scale, as has been numerically and experimentally observed
[21,30,29].
Σ2 (L) '
3.5
Random Matrix Theory – Continued
Random Matrix Theory can also predict the behaviour of quantities beyond
the energy levels. For example, it can predict the distribution of the wavefunction amplitude [32], the lifetimes of resonances in open systems [33,34] or
the distributions of transition matrix elements [22].
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000 22
I now come back to the time-reversal symmetry which is necessary to
obtain the GOE. If time-reversal symmetry (or more generally all anti-unitary
symmetry) is broken, the Hamiltonian cannot be written as a real symmetric
matrix, but rather as a complex Hermitian matrix. One has to change the
ensemble of random matrices to use and define the Gaussian Unitary Ensemble
(GUE). The natural measure is now:
Y
Y
dH ∝
dHii
dReHij dImHij
(37)
i=1..N
i,j=1..N ;i<j
which is invariant by any unitary transformation and thus puts again all the
orthonormal basis on the same footing. The probability distribution for H
is found again to be given by Eq. (32): both ReHij and ImHij are Gaussian
distributed. This adds more level repulsion because two arbitrary states have
two chances to be coupled and to repell. Not surprisingly, this is visible in
the joint probability distribution which takes the form:

P (E1 , .., EN ) ∝ 
Y
i,j=1..N ;i<j

2
|Ei − Ej |
exp −
PN
2
i=1 Ei
2
4σ
!
(38)
The calculations are similar to the GOE case (although sometimes simpler) and the predicted distributions agree very well with numerical results
[7,35]. As far as I know, there is no convincing experimental result obtained
in this regime.
One also has to consider the special case of half-integer spin systems with
time-reversal invariance: there, all levels are doubly degenerate (Kramers degeneracy). If some rotational invariance exists, this degeneracy is hidden and
the GOE should be used in each rotational series. If the rotational invariance
is broken, every level will be exactly doubly degenerate and the Gaussian
Symplectic Ensemble (GSE) of random matrices has to be used [7,24]. It is
essentially identical to the GUE, with an exponent 4 instead of 2 in the joint
probability density, Eq. (38).
It is important to notice the role of symmetries for level statistics. If a
good quantum number survives in a system (for example a discrete two-fold
symmetry), the states with the same good quantum number will interact,
but they will ignore the other states. Thus, even if each series with a fixed
quantum number obeys the GOE statistics, the total spectrum will appear as
the superposition of several uncorrelated GOE spectra, which has completely
different statistical properties. It is very important to be sure that one has
a pure sequence of levels before analyzing it. This may be difficult in a real
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000 23
experiment because of stray mixing between series, usually much easier in
numerical experiments.
Finally, intermediate regimes have been studied, for example between the
regular Poisson and the chaotic GOE regimes. In general, this transition is
not universal.
4
Semiclassical Approximation
The previous section has shown the existence of universal fluctuation properties associated with chaos for quantum systems. These properties take place
at short energy range, of the order of the mean level spacing, that is for times
of the order of the Heisenberg time, much longer that the period of the shortest periodic orbit. This also implies that a detailed analysis of all energy
levels and eigenstates does not make sense: no interesting information can
be brought to the physics of the chaotic phenomenon, beyond the statistical
aspects. On the other hand, this does not mean that these enegy levels do
not carry any information; it is just that this information has to be extracted
in a different way. More precisely, as the individual specificity of a chaotic
system manifests at relatively short times, before universal chaotic features
dominate, it has to be found in the long energy range characteristics of the
quantum spectra.
For such a short time scale, as discussed in section 2, a semiclassical
approximation might be used. It is the goal of this section to show how this can
be implemented and eventually used to make some quantitative predictions
on quantum chaotic systems which go beyond simple statistical statements.
4.1
Regular Systems – EBK/WKB Quantization
For completely integrable systems, where there exist as many independent
constants of motion as the number of degree of freedoms, there is a standard
semiclassical theory which is a simple extension of the well known WKB theory
for time-independent one-dimensional systems [36].
We assume the integrability of the system [1], which implies the existence
of d pairs of canonically conjugate action-angle variables (θi , Ii ) for 1 ≤ i ≤ d
such that the Hamiltonian depends only on the actions:
H = H(I1 ..Id )
(39)
For a given trajectory, the actions Ii are constants of motion which define
a invariant torus, a d-dimensional manifold embedded in the 2d-dimensional
phase space, and the angles θi (defined modulo 2π) are evolving linearly with
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000 24
time. A generic trajectory densely and uniformly fills the invariant torus,
which implies that invariant tori are stationary structures during the time
evolution. Hence, they are the relevant structures for building – in the semiclassical approximation – the eigenstates of the system.
Let us now turn to the technicalities. One writes the wavefunction as:
S(q)
(40)
ψ(q) = A(q) exp i
¯h
where A(q) and S(q) are real functions. We also assume that the Hamiltonian
is of the form:
p2
H=
+ V (q)
(41)
2m
An elementary manipulation of the time-independent Schr¨
odinger equation shows that it leads without any approximation to the two following real
equations:
∇(A2 (q)∇S(q)) = 0
(42)
2
2
(∇S(q))
¯h 4A(q)
+ V (q) − E = −
2m
2m A(q)
(43)
The EBK approximation amounts to neglect the right-hand side term in
the second equation, because it is multiplied by ¯h2 and thus likely to be small
in the semiclassical limit. With this approximation, the equation becomes the
classical Hamilton-Jacobi equation for the action [2]:
H(q, ∇S(q)) − E = 0
Hence, its solutions are known and can be written, at least locally:
Z
S(q) = p.dq
(44)
(45)
where the integral is calculated along a trajectoryd .
As equation (44) is a purely classical one, we can perform a canonical
change of coordinates to action-angle variables in order to solve it. As the
actions are constant, we get the trivial solutions:
X
S(θ1 ..θd ) =
Ii θi
(46)
i=1..d
This is a locally uniquely defined function of the coordinates, i.e. a singlevalued solution of the Hamilton-Jacobi equation, and provides us with an
d The
first equation (42) is nothing but a continuity equation which allows to compute the
amplitude A of the wavefunction once the phase S is known.
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000 25
approximate solution of the Schr¨odinger equation. However, this solution
is not defined everywhere in configuration space, because the projection of
the invariant torus over configuration space is only a finite region of it. At
the boundary of this region, there are caustics. The simplest example is for
a one-dimensional potential well: the oscillatory motion covers only a finite
position range. The two extreme positions are turning points of the classical
motion where the velocity changes sign and the particle traces back from
where it came. There, ∇S(q) vanishes which produces a divergence of the
amplitude A(q). This is turn makes the rhs of Eq. (43) tending to infinity and
invalidates the semiclassical approximation. Each caustic requires a careful
specific treatment is order to overcome this problem. Such a treatment goes
beyond the scope of these lectures, but the result is simple: the solution,
Eq. (46), can be continued through the caustics, provided a −π/2 phase factor
is added to the wavefunction for each caustic crossed.
When the angle θi is smoothly increased by 2π, (with other angles fixed),
one follows a closed loop on the invariant torus and comes back to the initial
point. In order for the wavefunction to be single-valued, the total phase
accumulated on such a closed loop must be an integer multiple of 2π. There
are two contributions to this phase: the first one is the change in the action
S divided by h
¯ , that is 2πIi /¯h, the second one is −π/2 multiplied by the
number of caustics crossed. The single-valued character of the wavefunction
thus implies:
αi ¯h
(47)
Ii = ni +
4
where ni is a non-negative integer number and αi the Maslov index counting
the number of caustics.
Alternatively, this quantization condition can be rewritten as a function
of the original coordinates as:
I
αi 1
p.dq = ni +
¯h
(48)
2π γi
4
where the integral is evaluated along a closed loop γi at the surface of the
invariant torus.
As there are d independent irreducible closed loops at the surface of the
invariant torus (or equivalently d actions Ii ), this provides us with a set of
d quantization conditions and d quantum numbers. These quantization rules
are known as the EBK (Einstein, Brillouin, Keller) quantization conditions or
invariant torus quantization [37].
The important point is that they do not use the classical trajectories, but
the classical invariant tori. For a one-dimensional system, the trajectories
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000 26
coincide with the tori and one rediscovers the standard WKB quantization.
This is also true for a degenerate multi-dimensional system like the hydrogen
atom where all trajectories are closed.
The EBK quantization rules can be used in a practical calculation. For
example, for the hydrogen atom in a weak magnetic field, the classical dynamics is mainly regular with a phase space full of invariant tori and the EBK
scheme can be used. The semiclassical prediction for the energy levels is very
accurate and practically indistinguishable from the exact quantum result in
Fig. 2a.
Another consequence of the EBK semiclassical quantization is that the
eigenstates are localized on the invariant tori and that d good quntum numbers
exist. As discussed in section 3.1, this implies that energy levels cross and
that the statistical properties of the energy spectrum are well described by
a Poisson law. In other words, the EBK quantization rules correctly predict
the observed statistical properties of energy levels, see section 3.3.
4.2
Semiclassical Propagator
For a chaotic system, the invariant tori do not exist and the preceding analysis
totally breaks down. There is no longer any structure which can be used to
build global solutions of the Hamilton-Jacobi equation with a single-valued
wavefunction. A completely different approach has to be used. As a direct
solution of the time-independent Schr¨
odinger equation seems out of reach,
one tries to calculate a semiclassical approximation of the unitary evolution
operator. This is also more convenient if one wants to compare to the classical
dynamics, as the regular or chaotic character expresses in the time domain.
The propagator is defined as a matrix element of the evolution operator
in the configuration space representation:
K(q0 , t0 ; q, t) = hq0 |U (t0 , t)|qi
(49)
The semiclassical approximation for the propagator is very similar to the
one already discussed for the time-independent Schr¨
odinger equation is section
4.1. It relies on a separation of phase and amplitude and neglection of higher
order terms in h
¯ . One then finds the time-dependent Hamilton-Jacobi equation
for the action [2], which can be locally solved along trajectories. The result is
known as the Van Vleck propagator [38,39,40]:
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000 27
0
0
K(q , t ; q, t) =
X
Clas. Traj.
1
2iπ¯h
1/2
d/2 2
0 0
Det ∂ R(q , t ; q, t) ×
∂q0 ∂q
0 0
iR(q , t ; q, t)
πν
exp
−i
¯h
2
(50)
where the sum is over all the classical trajectories going from (q, t) to (q0 , t0 ).
The function R(q0 , t0 ; q, t) is called the classical action, although it is different
from the action used previously which, according to [2], should be called
reduced action. The difference is that R is suitable when the time interval
(t, t0 ) is fixed while S is used at fixed energy. Altogether, the two functions
differ by E(t0 − t).
R is just the integral of the Lagrangian along the trajectory:
Z t0
0 0
˙ τ ) dτ
R(q , t ; q, t) =
L(q, q,
(51)
t
The non-negative integer ν counts the number of caustics encountered
along the trajectory and is called a Morse index e .
Few remarks should be made on this formula:
• The structure of this formula is completely analogous to the one used in
the energy domain, with a phase expressed as a purely classical quantity evaluated along a trajectory, divided by h
¯ , and a smoothly varying
amplitude.
• The fact that the same quantity R appears in the phase and the amplitude
is not surprising. It ensures the unitarity of the time evolution. It is the
counterpart – in time domain – of the continuity
equation in the energy
√
domain, Eq. (42). In fact, the prefactor Det is of purely classical origin.
It just represents how a classical phase space density initially localized in
q and uniformly spread in p evolves according to the Liouville equation,
Eq. (7).
• At the caustics, the amplitude diverges and the semiclassical approximation breaks down. However, beyond the caustics, the semiclassical
approximation recovers its validity, provided the convenient −π/2 phase
factor is added (through the Morse index), in complete analogy with the
EBK approximation.
e The
Maslov and Morse indices are not necessarily equal as the first one deals with trajectories at fixed energy and the second at fixed t0 − t.
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000 28
• At short time difference t0 − t, there is only one classical trajectory connecting (q, t) to (q0 , t0 ) (more or less a straight line). The existence of
multiple trajectories connecting the starting and ending points is analog
to the existence of multiple paths at the surface of an invariant torus (see
section 4.1). For a chaotic system, at long times, the trajectories become
very complicated and their number grows exponentially, which renders
the use of the semiclassical propagator more and more difficult.
• A completely different derivation of the Van Vleck propagator is possible
using the Feynman path integral [41] formulation of quantum mechanics.
The propagator can be exactly written as a superposition of contributions
of all paths connecting the starting and ending points. The phase of each
contribution is the integral of the Lagrangian along the path divided by
h. In the semiclassical limit, the sum over paths can be calculated by
¯
the stationary phase approximation. The paths with stationary phase
are precisely the classical trajectories, and the prefactor in the stationary
phase integration exactly gives the Van Vleck amplitude. This approach
explains why the contributions of the different classical trajectories have
to be added coherently in the propagator. It also explains why the semiclassical propagator can cross the caustics safely.
4.3
Green’s function
The next step is to go from the time domain to the energy domain. The
Green’s function is:
Z
1 ∞
Eτ
1
=
U (τ, 0) exp i
dτ
(52)
G(E) ≡
E−H
i¯h 0
¯h
where the last equality is valid for E in the lower half complex plane.
In configuration space, this gives:
1
G(q , q, E) ≡ hq |G(E)|qi =
i¯h
0
0
Z
0
∞
Eτ
K(q , τ ; q, 0) exp i
¯h
0
dτ
(53)
In order to get a semiclassical approximation for the Green’s function,
one plugs the Van Vleck propagator, Eq. (50), into Eq. (53). The integral
over τ can be performed by stationary phase approximation. The stationary
phase condition writes:
∂R(q0 , τ ; q, 0)
+E =0
∂τ
(54)
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000 29
¿From the Lagrange-Hamilton equation [2,40], the partial derivative is minus
the Hamilton function. Hence, stationary phase selects trajectories going from
q to q0 with precisely energy E. This allows to write the semiclassical Green’s
function at energy E as a sum over classical trajectories with energy E, a
physicaly satisfactory result. The phase of each contribution is the sum of R
(calculated along the orbit) and Eτ, which gives the reduced action S. The
detailed calculation of the various prefactors is not very difficult, but rather
tedious; see [40] for details. It is convenient to distinguish the coordinate
qk chosen along the trajectory and the the coordinates q⊥ transverse to the
orbit. One finally obtains the semiclassical Green’s function [38,42,40]:
1/2
2
0
Det ∂ S(q, q , E) X 0
iS(q, q0 , E)
πν
1
∂q
∂q
⊥
⊥
q
exp
−i
G(q0 , q, E) =
i¯
h
¯h
2
h)(d−1)/2 |q˙k q˙0 k |
Clas. Traj. (2iπ¯
(55)
with
Z q0
0
S(q, q , E) =
p.dq
(56)
q
Again, the prefactor simply represents the classical evolution of a phase
space density with fixed energy, according to the Liouville equation. This
semiclassical approximation breaks down for very short trajectories. Indeed,
the integral over τ cannot be performed by stationary phase approximation
in such a case. A specific short time expansion is possible when S(q, q0 , E)
is not much larger than h
¯ . It basically consists in ignoring the effect of the
potential and using the free Green’s function [38].
The Green’s function by itself is not very illuminating. In order to obtain
some information on the energy spectrum and eigenstates, one needs a more
global quantity.
4.4
Trace Formula
The Green’s function, Eq. (52), has a singularity at each energy level, like the
density of states, Eq. (23). They are actually related by the simple equation:
Z
1
1
d(E) = − Im TrG(E) = − Im dq G(q, q, E)
(57)
π
π
If one uses the semiclassical Green’s function, Eq. (55), the density of
states is obtained as a sum over closed trajectories, starting and ending at
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000 30
position q. The last integral over position q can again be performed by
stationary phase. The Lagrange equations tell us that the partial derivative
of S(q, q0 , E) with respect to q0 is the final momentum p0 while its partial
derivative with respect to q is minus the initial momentum p. Stationary
phase thus selects closed orbits where the initial and final momenta are equal,
that is periodic orbits.
Putting everything together gives the celebrated trace formula (also
known as the Gutzwiller trace formula from one of its author), written here
for a two-dimensional system [42,39,38,40]:
S
π
k
X
−
ν
cos
r
T
k
k
h
¯
2
¯
p
(58)
d(E) = d(E)
+
π¯h
|det(1 − Mkr )|
p.p.o. k, repetitions r
where the sum is performed over all primitive periodic orbits (i.e. periodic
orbits which do not retrace the same path several times) and all their repetitions r > 0. Tk is the period of the orbit, Sk its action, νk its Maslov
index and Mk the 2X2 monodromy matrix describing the linear change of the
transverse coordinates after one period; for details, see [38,40].
¯
The term d(E)
whose expression is given in Eq. (25) comes from the
“zero-length” trajectories. Indeed, for such trajectories, the semiclassical approximation for the Green’s function breaks down and a repaired formula (see
previous section) has to be used, which produces this smooth term.
The trace formula deserves several comments:
• The trace formula is a central result in the area of quantum chaos, as it
expresses a purely quantum quantity (the density of states) as a function
of classical quantities (related to periodic orbits) and the constant h
¯.
• It uses only periodic orbits, which proves that they are the skeleton of
the chaotic phase space. In that sense, they replace the invariant tori
used for regular systems.
• Each periodic orbit contributes to the density of states with an oscillatory
contribution. The period of these modulations correspond to a change of
the argument of the cosine function Sk /¯h by 2π. As the derivative of the
action with respect to energy is the period Tk of the orbit, the corresponding characteristic energy scale is 2π¯h/Tk . In the semiclassical regime, this
is much larger than the mean level spacing ∆ = 2π¯h/THeisenberg . Hence,
the trace formula describes the long range correlations in the energy spectrum, not the short range fluctuations described by Random Matrix Theory, see section 3.4.
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000 31
• The trace formula breaks the simple connection between a given energy
level and a simple structure in phase space. An energy level is a δ singularity in the density of states while each orbit contributes to a modulation
of the density of states with finite amplitude. Thus, to build a δ peak
requires a coherent conspiration of an infinite number of periodic orbits.
• The present formula is restricted to isolated periodic orbits such that the
phase space distance to the closer periodic orbit is larger than ¯h.. For non
isolated periodic orbits, the simple stationary phase treatment fails. A
specific treatment is required and various similar formula can be written.
Especially, for integrable systems, the sum over periodic orbits can be
performed analytically using a Poisson sum formula [37]: the result is
exactly equivalent to the EBK quantization scheme exposed in section
4.1.
• The formula is valid only at lowest order in the Planck’s constant h
¯.
Including higher orders in the various stationary phase approximations
is tedious, but feasible [43].
• If one is not interested in the density of states, but in some other physical
quantity, it is often possible to get similar expressions. The general strategy is to express the quantity of interest using the Green’s function of the
system, then to use the semiclassical Green’s function. For example, the
photo-ionization cross-section of an hydrogen atom in a magnetic field
has been calculated in [44] as a sum over periodic orbits starting and
ending at the nucleus.
The practical use of the trace formula is difficult. Indeed, extracting individual energy levels by adding oscillatory contributions requires in principle
an infinite number of periodic orbits. In practice, it may be argued that it is
enough to sum up all orbits with periods up to the Heisenberg time. Indeed,
longer orbits will produce modulations on an energy scale smaller than the
mean level spacing, and are thus expected to cancel out and to be irrelevant
(only useful to make peaks narrower but not moving theirs positions). In the
semiclassical limit, THeisenberg is so much longer than TMin that the proliferation of long orbits makes the procedure unpractical. However, for relatively
low excited states, THeisenberg is not much larger than TMin and the trace
formula has been successfully used to compute several states [40,11].
Another possibility is to use an open system with resonances instead of
bound states. There, the density of states has only bumps related to the
resonances and the use of a finite (and hopefully small) number of periodic
orbits may correctly reproduced the quantum properties. This is shown in
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000 32
Fig. 5 for the hydrogen atom in a magnetic field at scaled energy = 0.5.
With few hundred periodic orbits, we are able to reproduce the finest details
in the apparently random fluctuations of the photop-ionization cross-section.
This is a striking illustration of the strength of semiclassical methods.
4.5
Convergence Properties of the Trace Formula
Because of the proliferation of long periodic orbits, it is not clear whether
the sum in the trace formula converges or not. There is a competition between the exponential proliferation of periodic orbits and the decrease of the
individual amplitudes (long orbits tend to be very unstable hence creating
large denominators in the trace formula). For a generic bound system, the
proliferation overcomes the decrease of amplitudes and the sum does not converge [38]. This means that, depending on the order in which the various
periodic orbit contributions are added, the result can be anything! However,
it is rather clear that the periodic orbits are not independent from each other
and that the information that they contain is somewhat structured: most of
the information contained in the very long orbits can be essentially extracted
from shorter periodic orbits.
The idea is thus to use the structure of the classical orbits to make the
trace formula more convergent. The first step is to pass from the density of
states to the so-called spectral determinant defined by:
Y
f (E) =
(E − Ei )
(59)
i
which has a zero at each energy level. It can be rewritten as:
!
Z
1 E
Im TrG() d
f (E) = exp −
π
(60)
This quantity is of course highly divergent but can be regularized through
multiplication by a smooth quantity having no zero [38]. This corresponds to
removing in the Green’s function the smooth contribution of the zero length
trajectories.
One can thus define a “dynamical Zeta function” by:
1
Z(E) = exp −
π
Z
E
Im TrGreg () d
!
(61)
where Greg contains only the periodic orbit contributions. By inserting the
semiclassical Green’s function in Eq. (61), a rather simple manipulation allows
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000 33
Figure 5. The photo-ionization cross-section of the hydrogen atom in a magnetic field γ, at
constant scaled energy = Eγ −2/3 = 0.5 (plotted versus γ −1/3 ), where the classical motion
is fully chaotic. The smooth part of the cross-section is removed, in order to emphasize the
apparently erratic fluctuations. Upper panel: the “exact” quantum cross-section calculated
numerically. Lower panel: the semiclassical approximation of the cross-section, as calculated
using periodic orbit theory. All the fine details – which look like random fluctuations – are
well reproduced by periodic orbit theory.
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000 34
to sum over the repetitions and to obtain the infinite product [38]:
!
∞ Y
Y
exp i Sh¯k − iνk π2
Z(E) =
1−
|λk |1/2 λm
k
m=0 p.p.o.
(62)
where λk is the eigenvalue (larger than 1 in magnitude) of the monodromy
matrix Mk .
The transformation from an infinite sum to an infinite product does not
cure the lack of convergency. Of course, the zeros of the infinite product are
not the zeros of its individual terms (which do not have any for real energy,
because |λk | is always larger than 1). However, the larger m, the larger
the denominator and the more convergent the infinite product over primitive
periodic orbits. Hence, the most significant zeroes – the most important ones
for the physical properties – will come only from the m = 0 termf .
When expanding the infinite product, there are some crossed terms between orbits appearing with a positive sign which might cancel approximately
with negative terms from more complicated orbits. The idea of the cycle expansion is to group such terms so that maximum cancellation takes place.
Suppose that I have two simple orbits labelled 0 and 1 and a more complicated orbit labelled 01 which is roughly orbit 0 followed by orbit 1. If the
action (resp. Maslov index) of orbit 01 was exactly the sum of the actions
(resp. Maslov indices) of orbits 0 and 1 and its unstability eigenvalue λ the
product of the instability eigenvalues of orbits 0 and 1, complete cancellation
would take place. As these properties cannot be exact, only partial cancellation takes place. But, for longer and longer orbits, the cancellation is better
and better and the cycle expansion might be convergent (although there is no
proof).
This simple idea can be generalized to take into account all the orbits if
there exist a efficient coding scheme – also known as a good symbolic dynamics
– for the periodic orbits. Then, there are some cases like the 3-disks scattering
problem [38], where the cycle expansion can be made convergent and can be
used to efficiently calculate the quantum properties of the system.
4.6
An Example : the Helium Atom
The idea of the cycle expansion has been succesfully used by Wintgen and
coworkers [45] to calculate some energy levels of the helium atom. Although
the system is not fully chaotic, most of the dynamics is. Of special interest is
the eZe configuration where the electrons and the nucleus lie on a straight line,
f In
some cases, the m > 0 terms are absolutely convergent.
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000 35
State
1s1s
2s2s
2s3s
4s7s
5s5s
Periodic Orbit
-3.0984
-0.8044
-0.131
Cycle Expansion
-2.9248
-0.7727
-0.5902
-0.1426
-0.129
Exact quantum
-2.9037
-0.7779
-0.5899
-0.1426
-0.129
Table 1. Some energy levels (in atomic units) of the 1 S e series of the helium atom, compared
to the simple semiclassical quantization using only the simplest periodic orbit and the more
refined “cycle expansion” which includes a set of unstable periodic orbits. The agreement
is remarkable, which proves the efficiency of semiclassical methods for this chaotic system
(courtesy of D. Wintgen).
with the electrons on opposite sides of the nucleus. In such a configuration,
one can find a symbolic dynamics for the periodic orbits: any periodic orbit
can be uniquely labelled by the sequence in which the electrons hit the nucleus.
The motion transverse to the eZe configuration is stable and can be taken into
account. By calculating the zeros of the infinite product, Wintgen et al. have
been able to perform a fully semiclassical calculation of several energy levels of
the helium atom. Some results are displayed in Table 1. For the ground state,
it differs by only 0.7% from the exact quantum result. For excited states, it
is even better. Thus, these authors have been able to solve a problem open
since the beginning of the century, when pioneers of quantum mechanics tried
to quantize helium after having sucessfully quantized the hydrogen atom.
However, these pionners had no idea of the classically chaotic nature of phase
space, they were not even thinking of a trace formula – not to speak of cycle
expansion. They could not have the key idea that the correspondance between
a classical orbit and an energy level is not one-to-one but that an infinite
number of periodic orbits is needed to build a chaotic quantum eigenstate.
It is only after 70 years of work on classical and semiclassical dynamics that
their goal could be met.
5
Conclusion
In these lectures, I hope I could convince the audience that we have some
partial answers to the question raised in the introduction. Chaos manifest
itself in the quantum properties of the systems like the energy levels and the
eigenstates, in at least two ways:
• On a narrow energy interval — roughly at the level of individual eigen-
From: Peyresq Lectures in Nonlinear Phenomena, World Scientific, 2000 36
states — the quantum structures display strong, apparently random,
fluctuations and a high sensitivity on any small change of an external
parameter. This is the quantum counterpart of the classical sensitivity
on initial conditions.
• On a large energy scale, where spectral properties are averaged over several states, the specific features of the studied system become manifest
and are mainly related to the periodic orbits of the classical system.
For regular systems, efficient semiclassical methods exist. For chaotic
systems, we understand the role of periodic orbits. Yet, we are not often
able to compute individual highly excited states of a chaotic system from the
knowledge of its classical dynamics. Using periodic orbits, we can compute
low resolution spectra. Whether periodic orbit formulas are the end of the
game or just an intermediate step towards a more global understanding is
unknown.
Finally, several very important aspects of quantum chaos have not been
discussed in these lectures. The first one is the behaviour of the eigenstates
in the chaotic regime. As a first approximation – and in the spirit of Random
Matrix Theory – they are just unstructured random waves resulting from the
interferences between plenty of classical paths [7,26]. However, this is not
completely true and one often observes “scars” of periodic orbits, i.e. an
enhanced probability density in the immediate vicinity of a periodic orbit
[46,16,15]. Periodic orbit theory explains this phenomenon which has major
experimental consequences. The second major phenomenon not discussed
in these lectures is localization. At long times, it happens that – in sharp
contrast with the classical behaviour – the quantum behaviour is not ergodic
at all and the system remains localized [5,9]. This is related to the freezing
of the quantum dynamics after the Heisenberg time, see section 2 and is also
connected to the so-called Anderson localization in disordered systems.
I thank Christian Miniatura for providing me with his personal notes and
for careful reading of the manuscript, and Robin Kaiser for a constant kind
stimulation when writing these notes.
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