Quantum chaos

The foundations of modern quantum mechanics were laid in that period, essentially leaving aside the issue of the quantum-classical correspondence in systems whose classical limit exhibit chaos.

[5] Important observations often associated with classically chaotic quantum systems are spectral level repulsion, 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).

Other phenomena show up in the time evolution 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 distributions.

Quantum chaos typically deals with systems whose properties need to be calculated using either numerical techniques or approximation schemes (see e.g. Dyson series).

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.

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

Statistical measures of quantum chaos were born out of a desire to quantify spectral features of complex systems.

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.

The statistical tests mentioned here are universal, at least to systems with few degrees of freedom (Berry and Tabor[6] have put forward strong arguments for a Poisson distribution in the case of regular motion and Heusler et al.[7] 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.

Similarly, many systems which exhibit classical chaos have been found with quantum solutions yielding a Wigner-Dyson distribution, thus supporting the ideas above.

In contrast to the Einstein–Brillouin–Keller method 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: Recently there was a generalization of this formula for arbitrary matrix Hamiltonians that involves a Berry phase-like term stemming from spin or other internal degrees of freedom.

This presents several difficulties for chaotic systems: 1) The number of periodic orbits proliferates exponentially as a function of action.

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

Rather than dealing with the computational difficulties surrounding long-period orbits to try to find the density of states (energy levels), one can use standard quantum mechanical perturbation theory 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.

Note: Taking the trace tells you that only closed orbits contribute, the stationary phase approximation gives you restrictive conditions each time you make it.

Physically, these are associated with the outgoing waves that are generated when a tightly bound electron is excited to a high-lying state.

For scaling systems such as Rydberg atoms in strong fields, the Fourier transform of an oscillator strength spectrum computed at fixed

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

the density of states obtained from the Gutzwiller formula is related to the inverse of the potential of the classical system by

One open question remains understanding quantum chaos in systems that have finite-dimensional local Hilbert spaces for which standard semiclassical limits do not apply.

[10][11] The traditional topics in quantum chaos concerns spectral statistics (universal and non-universal features), and the study of eigenfunctions of various chaotic Hamiltonian.

For example, before the existence of scars was reported, eigenstates of a classically chaotic system were conjectured to fill the available phase space evenly, up to random fluctuations and energy conservation (Quantum ergodicity).

In particular, scars are both a striking visual example of classical-quantum correspondence away from the usual classical limit, and a useful example of a quantum suppression of chaos.

There is vast literature on wavepacket dynamics, including the study of fluctuations, recurrences, quantum irreversibility issues etc.

[19] In 1977, Berry and Tabor made a still open "generic" mathematical conjecture which, stated roughly, is: In the "generic" case for the quantum dynamics of a geodesic flow on a compact Riemann surface, the quantum energy eigenvalues behave like a sequence of independent random variables provided that the underlying classical dynamics is completely integrable.

Quantum chaos is the field of physics attempting to bridge the theories of quantum mechanics and classical mechanics . The figure shows the main ideas running in each direction.
Experimental recurrence spectra of lithium in an electric field showing birth of quantum recurrences corresponding to bifurcations of classical orbits. [ 3 ]
Comparison of experimental and theoretical recurrence spectra of lithium in an electric field at a scaled energy of . [ 4 ]
Computed regular (non-chaotic) Rydberg atom energy level spectra of hydrogen in an electric field near n=15. Note that energy levels can cross due to underlying symmetries of dynamical motion. [ 4 ]
Computed chaotic Rydberg atom energy level spectra of lithium in an electric field near n=15. Note that energy levels cannot cross due to the ionic core (and resulting quantum defect) breaking symmetries of dynamical motion. [ 4 ]
Nearest neighbour distribution for Rydberg atom energy level spectra in an electric field as quantum defect is increased from 0.04 (a) to 0.32 (h). The system becomes more chaotic as dynamical symmetries are broken by increasing the quantum defect; consequently, the distribution evolves from nearly a Poisson distribution (a) to that of Wigner's surmise (h).
Even parity recurrence spectrum ( Fourier transform of the density of states ) of diamagnetic hydrogen showing peaks corresponding to periodic orbits of the classical system. Spectrum is at a scaled energy of −0.6. Peaks labeled R and V are repetitions of the closed orbit perpendicular and parallel to the field, respectively. Peaks labeled O correspond to the near circular periodic orbit that goes around the nucleus.
Relative recurrence amplitudes of even and odd recurrences of the near circular orbit. Diamonds and plus signs are for odd and even quarter periods, respectively. Solid line is A/cosh(nX/8). Dashed line is A/sinh(nX/8) where A = 14.75 and X = 1.18.
Experimental recurrence spectrum (circles) is compared with the results of the closed orbit theory of John Delos and Jing Gao for lithium Rydberg atoms in an electric field. The peaks labeled 1–5 are repetitions of the electron orbit parallel to the field going from the nucleus to the classical turning point in the uphill direction.