Quantum chaos

Quantum chaos

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Quantum chaos is a branch of physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

 which studies how chaotic
Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...

 classical dynamical systems can be described in terms of quantum theory. The primary question that quantum chaos seeks to answer is, "What is the relationship between quantum mechanics and classical chaos?" The correspondence principle
Correspondence principle
In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics reproduces classical physics in the limit of large quantum numbers....

 states that classical mechanics is the classical limit
Classical limit
The classical limit or correspondence limit is the ability of a physical theory to approximate or "recover" classical mechanics when considered over special values of its parameters. The classical limit is used with physical theories that predict non-classical behavior...

 of quantum mechanics. If this is true, then there must be quantum mechanisms underlying classical chaos; although this may not be a fruitful way of examining classical chaos. If quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

 does not demonstrate an exponential sensitivity to initial conditions, how can exponential sensitivity to initial conditions arise in classical chaos, which must be the correspondence principle
Correspondence principle
In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics reproduces classical physics in the limit of large quantum numbers....

 limit of quantum mechanics? In seeking to address the basic question of quantum chaos, several approaches have been employed:
  1. Development of methods for solving quantum problems where the perturbation cannot be considered small in perturbation theory
    Perturbation theory (quantum mechanics)
    In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for which a mathematical solution is known, and add an...

     and where quantum numbers are large.
  2. Correlating statistical descriptions of eigenvalues (energy levels) with the classical behavior of the same Hamiltonian
    Hamiltonian (quantum mechanics)
    In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

  3. Semiclassical methods such as periodic-orbit theory connecting the classical trajectories of the dynamical system
    Dynamical system
    A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

     with quantum features.
  4. Direct application of the correspondence principle
    Correspondence principle
    In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics reproduces classical physics in the limit of large quantum numbers....



During the first half of the twentieth century, chaotic behavior in mechanics was recognized (as in the three-body problem
Three-body problem
Three-body problem has two distinguishable meanings in physics and classical mechanics:# In its traditional sense the three-body problem is the problem of taking an initial set of data that specifies the positions, masses and velocities of three bodies for some particular point in time and then...

 in celestial mechanics
Celestial mechanics
Celestial mechanics is the branch of astronomy that deals with the motions of celestial objects. The field applies principles of physics, historically classical mechanics, to astronomical objects such as stars and planets to produce ephemeris data. Orbital mechanics is a subfield which focuses on...

), but not well-understood. The foundations of modern quantum mechanics were laid in that period, essentially leaving aside the issue of the quantum-classical correspondence
Correspondence principle
In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics reproduces classical physics in the limit of large quantum numbers....

 in systems whose classical limit exhibit chaos.


Questions related to the correspondence principle
Correspondence principle
In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics reproduces classical physics in the limit of large quantum numbers....

 arise in many different branches of physics, ranging from nuclear
Nuclear physics
Nuclear physics is the field of physics that studies the building blocks and interactions of atomic nuclei. The most commonly known applications of nuclear physics are nuclear power generation and nuclear weapons technology, but the research has provided application in many fields, including those...

 to atomic
Atomic physics
Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. It is primarily concerned with the arrangement of electrons around the nucleus and...

, molecular
Molecular physics
Molecular physics is the study of the physical properties of molecules, the chemical bonds between atoms as well as the molecular dynamics. Its most important experimental techniques are the various types of spectroscopy...

 and solid-state physics
Condensed matter physics
Condensed matter physics deals with the physical properties of condensed phases of matter. These properties appear when a number of atoms at the supramolecular and macromolecular scale interact strongly and adhere to each other or are otherwise highly concentrated in a system. The most familiar...

, and even to acoustics
Acoustics is the interdisciplinary science that deals with the study of all mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician while someone working in the field of acoustics...

, microwave
Microwaves, a subset of radio waves, have wavelengths ranging from as long as one meter to as short as one millimeter, or equivalently, with frequencies between 300 MHz and 300 GHz. This broad definition includes both UHF and EHF , and various sources use different boundaries...

s and optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...

. Important observations often associated with classically chaotic quantum systems are spectral level repulsion
Level repulsion
A system of two coupled oscillators has two natural frequencies. As the coupling strength between the oscillators increases, the lower frequency decreases and the higher increases. This effect can be viewed as a 'repulsion' between the frequencies...

, dynamical localization in time evolution (e.g. ionization rates of atoms), and enhanced stationary wave intensities in regions of space where classical dynamics exhibits only unstable trajectories (as in scattering
Scattering is a general physical process where some forms of radiation, such as light, sound, or moving particles, are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which they pass. In conventional use, this also includes deviation of...


In the semiclassical approach of quantum chaos, phenomena are identified in spectroscopy
Spectroscopy is the study of the interaction between matter and radiated energy. Historically, spectroscopy originated through the study of visible light dispersed according to its wavelength, e.g., by a prism. Later the concept was expanded greatly to comprise any interaction with radiative...

 by analyzing the statistical distribution of spectral lines and by connecting spectral periodicities with classical orbits. Other phenomena show up in the time evolution
Time evolution
Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state . In this formulation, time is not required to be a continuous parameter, but may be discrete or even finite. In classical physics, time evolution of a collection of rigid bodies...

 of a quantum system, or in its response to various types of external forces. In some contexts, such as acoustics or microwaves, wave patterns are directly observable and exhibit irregular amplitude
Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...


Quantum chaos typically deals with systems whose properties need to be calculated using either numerical techniques or approximation schemes (see e.g. Dyson series
Dyson series
In scattering theory, the Dyson series, formulated by British-born American physicist Freeman Dyson, is a perturbative series, and each term is represented by Feynman diagrams. This series diverges asymptotically, but in quantum electrodynamics at the second order the difference from...

). Simple and exact solutions are precluded by the fact that the system's constituents either influence each other in a complex way, or depend on temporally varying external forces.

Quantum mechanics in non-perturbative regimes

For conservative systems, the goal of quantum mechanics in non-perturbative regimes is to find
the eigenvalues and eigenvectors of a Hamiltonian of the form

where is separable in some coordinate system, is non-separable in the coordinate system in which is separated, and is a parameter which cannot be considered small. Physicists have historically approached problems of this nature by trying to find the coordinate system in which the non-separable Hamiltonian is smallest and then treating the non-separable Hamiltonian as a perturbation.

Finding constants of motion so that this separation can be performed can be a difficult (sometimes impossible) analytical task. Solving the classical problem can give valuable insight into solving the quantum problem. If there are regular classical solutions of
the same Hamiltonian, then there are (at least) approximate constants of motion, and by solving the classical problem, we gain clues how to find them.

Other approaches have been developed in recent years. One is to express the Hamiltonian in
different coordinate systems in different regions of space, minimizing the non-separable part of the Hamiltonian in each region. Wavefunctions are obtained in these regions, and eigenvalues are obtained by matching boundary conditions.

Another approach is numerical matrix diagonalization. If the Hamiltonian matrix is computed in any complete basis, eigenvalues and eigenvectors are obtained by diagonalizing
the matrix. However, all complete basis sets are infinite, and we need to truncate the basis and still obtain accurate results. These techniques boil down to choosing a truncated basis from which accurate wavefunctions can be constructed. The computational time required to diagonalize a matrix scales as , where is the dimension of the matrix, so it is important to choose the smallest basis possible from which the relevant wavefunctions can be constructed. It is also convenient to choose a basis in which the matrix
is sparse and/or the matrix elements are given by simple algebraic expressions because computing matrix elements can also be a computational burden.

A given Hamiltonian shares the same constants of motion for both classical and quantum
dynamics. Quantum systems can also have additional quantum numbers corresponding to discrete symmetries (such as parity conservation from reflection symmetry). However, if we merely find quantum solutions of a Hamiltonian which is not approachable by perturbation theory, we may learn a great deal about quantum solutions, but we have learned little about quantum chaos. Nevertheless, learning how to solve such quantum problems is an important part of answering the question of quantum chaos.

Correlating statistical descriptions of quantum mechanics with classical behavior

Statistical measures of quantum chaos were born out of a desire to quantify spectral features of complex systems. Random matrix
Random matrix
In probability theory and mathematical physics, a random matrix is a matrix-valued random variable. Many important properties of physical systems can be represented mathematically as matrix problems...

 theory was developed in an attempt to characterize spectra of complex nuclei. The remarkable result is that the statistical properties of many systems with unknown Hamiltonians can be predicted using random matrices of the proper
symmetry class. Furthermore, random matrix theory also correctly predicts statistical properties
of the eigenvalues of many chaotic systems with known Hamiltonians. This makes it useful as a tool for characterizing spectra which require large numerical efforts to compute.

A number of statistical measures are available for quantifying spectral features in a simple way. It is of great interest whether or not there are universal statistical behaviors of classically chaotic systems. The statistical tests mentioned here are universal, at least to systems with few degrees of freedom (Berry and Tabor have put forward strong arguments for a Poisson distribution in the case of regular motion and Heusler et al. present a semiclassical explanation of the so-called Bohigas-Giannoni-Schmit conjecture which asserts universality of spectral fluctuations in chaotic dynamics). The nearest-neighbor distribution (NND) of energy levels is relatively simple to interpret and it has been widely used to describe quantum chaos.

Qualitative observations of level repulsions can be quantified and related to the classical dynamics
using the NND, which is believed to be an important signature of classical dynamics in quantum systems. It is thought that regular classical dynamics is manifested by a Poisson distribution
Poisson distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since...

 of energy levels:

In addition, systems which display chaotic classical motion are expected to be characterized by the statistics of random matrix eigenvalue ensembles. For systems invariant under time reversal, the energy-level statistics of a number of chaotic systems have been shown to be in good agreement with the predictions of the Gaussian orthogonal ensemble (GOE) of random matrices, and it has been suggested that this phenomenon is generic for all chaotic systems with this symmetry. If the normalized spacing between two energy levels is , the normalized distribution of spacings is well approximated by
which is the Wigner distribution
Wigner distribution
The Wigner distribution is either of two things:* Wigner semicircle distribution - A probability function used in mathematics * Wigner quasi-probability distribution - A distribution in phase space encoding, in a convenient representation, properties of quantum-mechanical wave-functions...


Many Hamiltonian systems which are classically integrable (non-chaotic) have been found to have quantum solutions that yield nearest neighbor distributions which follow the Poisson distributions. Similarly, many systems which exhibit classical chaos have been found with quantum solutions yielding a Wigner distribution
Wigner distribution
The Wigner distribution is either of two things:* Wigner semicircle distribution - A probability function used in mathematics * Wigner quasi-probability distribution - A distribution in phase space encoding, in a convenient representation, properties of quantum-mechanical wave-functions...

, thus supporting the ideas above. One notable exception is diamagnetic lithium which, though exhibiting classical chaos, demonstrates Wigner (chaotic) statistics for the even-parity energy levels and nearly Poisson (regular) statistics for the odd-parity energy level distribution.

Periodic orbit theory

Periodic-orbit theory gives a recipe for computing spectra from the periodic orbits of a system. In contrast to the Einstein-Brillouin-Keller method
Einstein-Brillouin-Keller method
The Einstein–Brillouin–Keller method is a semiclassical method to compute eigenvalues in quantum mechanical systems.-See also:*Quantum mechanics*WKB approximation*Albert Einstein*Léon Brillouin*Joseph B. Keller...

 of action quantization, which applies only to integrable or near-integrable systems and computes individual eigenvalues from each trajectory, periodic-orbit theory is applicable to both integrable and non-integrable systems and asserts that each periodic orbit produces a sinusoidal fluctuation in the density of states.

The principal result of this development is an expression for the density of states which is the trace of the semiclassical Green's function and is given by the Gutzwiller trace formula:

The index distinguishes the primitive periodic orbits: the shortest period orbits of a given set of initial conditions. is the period of the primitive periodic orbit and is its classical action. Each primitive orbit retraces itself, leading to a new orbit with action and a period which is an integral multiple of the primitive period. Hence, every repetition of a periodic orbit is another periodic orbit. These repetitions are separately classified by the intermediate sum over the indices . is the orbit's Maslov index.
The amplitude factor, , represents the square root of the density of neighboring orbits. Neighboring trajectories of an unstable periodic orbit diverge exponentially in time from the periodic orbit. The quantity characterizes the instability of the orbit. A stable orbit moves on a torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...

 in phase space, and neighboring trajectories wind around it. For stable orbits, becomes , where is the winding
number of the periodic orbit. , where is the number of times that neighboring orbits intersect the periodic orbit in one period. This presents a difficulty because at a classical bifurcation
Bifurcation means the splitting of a main body into two parts.Bifurcation or Bifurcated may refer to:*Bifurcation , the division of issues in a trial for example the division of a page into two parts....

. This causes that orbit's contribution to the energy density to diverge. This also occurs in the context of photo-absorption spectrum.

Using the trace formula to compute a spectrum requires summing over all of the periodic orbits of a system. This presents several difficulties for chaotic systems: 1) The number of periodic orbits proliferates exponentially as a function of action. 2) There are an infinite number of periodic orbits, and the convergence properties of periodic-orbit theory are unknown. This difficulty is also present when applying periodic-orbit theory to regular systems. 3) Long-period orbits are difficult to compute because most trajectories are unstable and sensitive to roundoff errors and details of the numerical integration.

Gutzwiller applied the trace formula to approach the anisotropic Kepler problem (a single particle in a potential with an anisotropic mass tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

semiclassically. He found agreement with quantum computations for low lying (up to ) states for small anisotropies by using only a small set of easily computed periodic orbits, but the agreement was poor for large anisotropies.

The figures above use an inverted approach to testing periodic-orbit theory. The trace formula asserts that each periodic orbit contributes a sinusoidal term to the spectrum. Rather than dealing with the computational difficulties surrounding long-period orbits to try and find the density of states
Density of states
In solid-state and condensed matter physics, the density of states of a system describes the number of states per interval of energy at each energy level that are available to be occupied by electrons. Unlike isolated systems, like atoms or molecules in gas phase, the density distributions are not...

 (energy levels), one can use standard quantum mechanical perturbation theory
Perturbation theory
Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...

 to compute eigenvalues (energy levels) and use the Fourier transform to look for the periodic modulations of the spectrum which are the signature of periodic orbits. Interpreting the spectrum then amounts to finding the orbits which correspond to peaks in the Fourier transform.

Closed orbit theory

Closed-orbit theory was developed by J.B. Delos, M.L. Du, J. Gao, and J. Shaw. It is similar to
periodic-orbit theory, except that closed-orbit theory is applicable only to atomic and molecular spectra and yields the oscillator strength density (observable photo-absorption spectrum) from a specified initial state whereas periodic-orbit theory yields the density of states.

Only orbits that begin and end at the nucleus are important in closed-orbit theory. Physically, these are associated with the outgoing waves that are generated when a tightly bound electron is excited to a high-lying state. For Rydberg atoms and molecules, every orbit which is closed at the nucleus is also a periodic orbit whose period is equal to either the closure time or twice the closure time.

According to closed-orbit theory, the average oscillator strength density at constant is given by a smooth background plus an oscillatory sum of the form

is a phase that depends on the Maslov index and other details of the orbits. is the recurrence amplitude of a closed orbit for a given initial state (labeled ). It contains information about the stability of the orbit, its initial and final directions, and the matrix element of the dipole operator between the initial state and a zero-energy Coulomb wave. For scaling systems such as Rydberg atoms in strong fields, the Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

 of an oscillator strength spectrum computed at fixed as a function of is called a recurrence spectrum, because it gives peaks which correspond to the scaled action of closed orbits and whose heights correspond to .

Closed-orbit theory has found broad agreement with a number of chaotic systems, including diamagnetic hydrogen, hydrogen in parallel electric and magnetic fields, diamagnetic lithium, lithium in an electric field, the ion in crossed and parallel electric and magnetic fields, barium in an electric field, and helium in an electric field.

Recent directions in quantum chaos

The traditional topics in quantum chaos concerns spectral statistics (universal and non-universal features), and the study of eigenfunctions (Quantum ergodicity
Quantum ergodicity
In quantum chaos, a branch of mathematical physics, quantum ergodicity is a property of the quantization of classical mechanical systems that are chaotic in the sense of exponential sensitivity to initial conditions...

, scars) of various chaotic Hamiltonian .

Further studies concern the parametric () dependence of the Hamiltonian, as reflected in e.g. the statistics of avoided crossings, and the associated mixing as reflected in the (parametric) local density of states (LDOS). There is vast literature on wavepacket dynamics, including the study of fluctuations, recurrences, quantum irreversibility issues etc. Special place is reserved to the study of the dynamics of quantized maps: The Standard map and The Kicked Rotator
The Kicked Rotator
The kicked rotator, also spelled as kicked rotor, is a prototype model for chaos and quantum chaos studies. It describes a particle that is constrained to move on a ring . The particle is kicked periodically by an homogeneous field...

 are considered to be prototype problems.

Recent works are also focused in the study of driven chaotic systems, where the Hamiltonian is time dependent, in particular in the adiabatic and in the linear response regimes.

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