Quantum chaos

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.

Periodic-orbit theory gives a recipe for computing spectra from the periodic orbits of a system. 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:

${\displaystyle g_{c}(E)=\sum _{k}T_{k}\sum _{n=1}^{\infty }{\frac {1}{2\sinh {(\chi _{nk}/2)}}}\,e^{i(nS_{k}-\alpha _{nk}\pi /2)}.}$

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.[9] The index ${\displaystyle k}$ distinguishes the primitive periodic orbits: the shortest period orbits of a given set of initial conditions. ${\displaystyle T_{k}}$ is the period of the primitive periodic orbit and ${\displaystyle S_{k}}$ is its classical action. Each primitive orbit retraces itself, leading to a new orbit with action ${\displaystyle nS_{k}}$ and a period which is an integral multiple ${\displaystyle n}$ 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 ${\displaystyle n}$. ${\displaystyle \alpha _{nk}}$ is the orbit's Maslov index. The amplitude factor, ${\displaystyle 1/\sinh {(\chi _{nk}/2)}}$, 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 ${\displaystyle \chi _{nk}}$ characterizes the instability of the orbit. A stable orbit moves on a torus in phase space, and neighboring trajectories wind around it. For stable orbits, ${\displaystyle \sinh {(\chi _{nk}/2)}}$ becomes ${\displaystyle \sin {(\chi _{nk}/2)}}$, where ${\displaystyle \chi _{nk}}$ is the winding number of the periodic orbit. ${\displaystyle \chi _{nk}=2\pi m}$, where ${\displaystyle m}$ is the number of times that neighboring orbits intersect the periodic orbit in one period. This presents a difficulty because ${\displaystyle \sin {(\chi _{nk}/2)}=0}$ at a classical bifurcation. 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 ${\displaystyle 1/r}$ potential with an anisotropic mass tensor) semiclassically. He found agreement with quantum computations for low lying (up to ${\displaystyle n=6}$) 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 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. Interpreting the spectrum then amounts to finding the orbits which correspond to peaks in the Fourier transform.

1. Start with the semiclassical approximation of the time-dependent Green's function (the Van Vleck propagator).
2. Realize that for caustics the description diverges and use the insight by Maslov (approximately Fourier transforming to momentum space (stationary phase approximation with h a small parameter) to avoid such points and afterwards transforming back to position space can cure such a divergence, however gives a phase factor).
3. Transform the Greens function to energy space to get the energy dependent Greens function (again approximate Fourier transform using the stationary phase approximation). New divergences might pop up that need to be cured using the same method as step 3
4. Use ${\displaystyle d(E)=-{\frac {1}{\pi }}\Im (\operatorname {Tr} (G(x,x^{\prime },E))}$ (tracing over positions) and calculate it again in stationary phase approximation to get an approximation for the density of states ${\displaystyle d(E)}$.

Note: Taking the trace tells you that only closed orbits contribute, the stationary phase approximation gives you restrictive conditions each time you make it. In step 4 it restricts you to orbits where initial and final momentum are the same i.e. periodic orbits. Often it is nice to choose a coordinate system parallel to the direction of movement, as it is done in many books.

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.

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 ${\displaystyle \epsilon }$ is given by a smooth background plus an oscillatory sum of the form

${\displaystyle f(w)=\sum _{k}\sum _{n=1}^{\infty }D_{\it {nk}}^{i}\sin(2\pi nw{\tilde {S_{k}}}-\phi _{\it {nk}}).}$

${\displaystyle \phi _{\it {nk}}}$ is a phase that depends on the Maslov index and other details of the orbits. ${\displaystyle D_{\it {nk}}^{i}}$ is the recurrence amplitude of a closed orbit for a given initial state (labeled ${\displaystyle i}$). 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 of an oscillator strength spectrum computed at fixed ${\displaystyle \epsilon }$ as a function of ${\displaystyle w}$ is called a recurrence spectrum, because it gives peaks which correspond to the scaled action of closed orbits and whose heights correspond to ${\displaystyle D_{\it {nk}}^{i}}$.

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 ${\displaystyle H^{-}}$ ion in crossed and parallel electric and magnetic fields, barium in an electric field, and helium in an electric field.

For the case of one-dimensional system with the boundary condition ${\displaystyle y(0)=0}$ the density of states obtained from the Gutzwiller formula is related to the inverse of the potential of the classical system by ${\displaystyle {\frac {d^{1/2}}{dx^{1/2}}}V^{-1}(x)=2{\sqrt {\pi }}{\frac {dN(x)}{dx}}}$ here ${\displaystyle {\frac {dN(x)}{dx}}}$ is the density of states and V(x) is the classical potential of the particle, the half derivative of the inverse of the potential is related to the density of states as in the Wu-Sprung potential.