Microwave billiards and quantum chaos

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Hans-Jürgen Stöckmann (2010), Scholarpedia, 5(10):10243. doi:10.4249/scholarpedia.10243 revision #127194 [link to/cite this article]
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Curator: Hans-Jürgen Stöckmann

Up to about 1990 the quantum mechanics of classically chaotic systems, shortly termed "quantum chaos", was essentially a domain of theory (Haake 2001). Only two classes of experimental result had been available at that time. First, there were the spectra of compound nuclei giving rise to the development of random matrix theory in the sixties of the last century, and second the experiments with highly excited hydrogen and alkali atoms in strong magnetic or strong radio frequency fields. The situation changed with the appearance of experiments using classical waves, starting with microwave billiards . The distinction between classical waves and matter waves is not of relevance in the present context, since all features touched in this article are common to all types of waves. This is why some authors prefer the term "wave chaos" to describe this field of research.


From classical to quantum mechanics

Figure 1: Classical trajectories in a circular(a) and a stadium (b) billiard.

Billiards are particularly well suited to illustrate the difficulties one is facing with the concept of chaos in quantum mechanics. For a circular billiard the trajectory is regular ( Figure 1(a)). There are two constants of motion, the total energy \(E\ ,\) and the angular momentum \(L\ .\) Since there are two degrees of freedom as well, the system is integrable, and the distance between two nearby trajectories increases linearly with time. The situation is qualitatively different for the stadium billiard ( Figure 1(b)). There is only one constant of motion left, the total energy \(E\ ,\) and the distance between neighbouring trajectories increases exponentially with time. The stadium billiard thus is chaotic.

In quantum mechanics this distinction between integrable and chaotic systems does not work any longer. The initial conditions are defined only within the limits of the uncertainty relation \[ \Delta x\,\Delta p\ge \frac{1}{2}\hbar\,, \] and the concept of trajectories looses its significance. One may even ask whether quantum chaos does exist at all. Since the Schrödinger equation is linear, a quantum mechanical wave packet can be constructed from the eigenfunctions by the superposition principle. There is no room left for chaos. On the other hand the correspondence principle demands that there must be a relation between linear quantum mechanics and nonlinear classical mechanics at least in the regime of large quantum numbers. This defines the program of quantum chaos research, namely to look for the fingerprints of classical chaos in the quantum mechanical properties of the system.

Billiards are ideally suited systems for this purpose. The numerical calculation of the classical trajectories is elementary, and the stationary Schrödinger equation reduces to a simple wave equation

\[ -\frac{\hbar^2}{2m}\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2} \right)\psi_n=E_n\psi_n\,. \]

The potential appears only in the boundary condition, \( \left.\psi_n\right|_S=0\ ,\) where \(S\) is the surface of the billiard. In the absence of potentials the stationary Schrödinger equation is equivalent to the time-independent wave equation, the Helmholtz equation

\[\tag{1} -\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2} \right)\psi_n=k_n^2\psi_n\,, \]

where \(\psi_n\) now means the amplitude of the wave field.

This opens the opportunity to study questions and to test theories, originally initiated by quantum mechanics, by means of classical waves. The boundary conditions for the classical and the corresponding quantum mechanical systems may differ, but this is not of relevance for the questions to be treated in this article.

Microwave billiards

Figure 2: Chladni figures for plates with various shapes mounted in the centre (Stöckmann 1992, Nat.Wiss. 79: 443).

The first experiment of this type dates back already more then 200 years. At the end of the 18th century E. Chladni developed a technique "to make sound visible" by decorating the nodal lines of vibrating plates with grains of sand (Chladni 1802). Figure 2 shows Chladni figures for three typical situations. The plates are fixed in the centre and had been excited to vibrations by means of a loudspeaker. Figure 2(a) shows a typical pattern for a circular plate. In this case the integrability of the system is not perturbed by the mounting. One observes a regular pattern of nodal lines with many intersections. The next example in Figure 2(b) shows a rectangle. It is integrable, too, but now the integrability is slightly perturbed by the mounting resulting in a curvature of the nodal lines and a partial conversion of crossings into anti-crossings. The last example in Figure 2(c) belongs to the class of Sinai billiards, a rectangle with an excised quarter circle from one of the corners. Now all nodal line crossings have completely disappeared resulting in a meandering pattern of nodal lines. Two centuries after Chladni's discovery the study of nodal lines of chaotic plates has become a very active field of research again.

Figure 3: Microwave set-up to study spectra and wave functions (top), and a typical microwave reflection spectrum for a quarter-stadium (b= 20 cm, l=36 cm) shaped resonator (bottom) (Stöckmann and Stein 1990, Phys. Rev. Lett. 64: 2215).

First modern experimental billiard studies started with microwave resonators. Meanwhile the technique is used by several groups worldwide (see Section Further reading for more details).Figure 3(top) shows a typical set-up. The cavity is formed by a bottom plate supporting the entrance antenna, and by an upper part whose position can be moved with respect to the lower one. As long as a maximum frequency \(\nu_{\rm max} = c/2d\) is not exceeded, where \(d\) is the height of the resonator and \(c\) the velocity of light, the system can be considered as quasi-two-dimensional. In this situation the electro-magnetic wave equations reduce to the scalar Helmholtz equation (1), where \(\psi_n\) corresponds to the electric field pointing perpendicularly from the bottom to the top plate. Since the electric field component parallel to the wall must vanish, we have the condition \(\psi_n|_S = 0\) on the outer circumference \(S\) of the resonator. We have thus arrived at a complete equivalence between a two-dimensional quantum billiard and the corresponding quasi-two-dimensional microwave resonator, including the boundary conditions. As an example Figure 3(bottom) shows the reflection spectrum of a microwave resonator of the shape of a quarter stadium. Each minimum in the reflection corresponds to an eigenfrequency of the resonator, and the depth of the resonance corresponds to the modulus square \(|\psi_n(r)|^2\) of the wave function at the antenna position.

Figure 4: Wave functions in a stadium-shaped microwave resonator (Stein et al. 1992, Phys. Rev. Lett. 68: 2867). The figure shows \(\left|\psi_n(\vec{r})\right|^2\) in a colour plot.

By scanning with the antenna through the billiard \(|\psi_n(\vec{r})|^2\) may thus be spatially resolved. To get the sign as well, a transmission measurement to an additional fixed antenna has to be performed. Figure 4 shows a number of stadium wave functions obtained in this way. All wave functions show the phenomenon of scarring, meaning that the wave function amplitudes are not distributed more or less homogeneously over the area, but concentrate along classical periodic orbits. One could get the impression that scarred wave functions are dominating, but this is only true for the lowest eigenvalues. With increasing energy the fraction of scarred wave functions tends to zero.

For a quantitative description of the experiments scattering theory has to be applied, developed half a century ago in nuclear physics. Compared to nuclei microwave billiards have a number of advantages: wave lengths are of the order of mm to cm, resulting in very convenient sizes for the used resonators, and all relevant parameters can be perfectly controlled. This is why a number of predictions of scattering theory have been tested not in nuclei but microwave billiards (Mitchell et al. 2010).

Random matrices

In the midst of the last century little was known on the origin of the nuclear forces. Here one idea showed up to be extremely successful, notwithstanding its obviously oversimplifying nature: If the details of the nuclear Hamiltonian \(H\) are not known, just let us take its matrix elements in some basis as random numbers, with only some global constraints, e. g. by taking the matrix \(H\) symmetric for systems with, or non-symmetric Hermitian for systems without time-reversal symmetry, and by fixing the variance of the matrix elements. Assuming basis invariance the matrix elements can be shown to be uncorrelated and Gaussian distributed (Mehta 1991). The classical Gaussian ensembles are the orthogonal one (GOE) for time-reversal invariant systems with integer spin, the unitary one (GUE) for systems with broken time-reversal symmetry, and the symplectic one (GSE) for time-reversal invariant systems with half-integer spin. Here "orthogonal" etc. refers to the invariance properties of the respective ensembles.

Figure 5: Level spacing distribution for a Sinai billiard (a), a hydrogen atom in a strong magnetic field (b), the excitation spectrum of a NO2$ molecule (c), the acoustic resonance spectrum of a Sinai-shaped quartz block (d), the microwave spectrum of a three-dimensional chaotic cavity (e), and the vibration spectrum of a quarter-stadium shaped plate (f) (taken from Stöckmann 1999). In all cases a Wigner distribution is found though only in the first three cases the spectra are quantum mechanically in origin.

The quantity most often studied in this context is the distribution of level spacings \(p(s)\) normalised to a mean level spacing of one. For \( 2\times 2\) matrices this quantity can be easily calculated, yielding for the GOE the famous Wigner surmise

\(\tag{2} p(s)=\frac{\pi}{2}s\exp\left(-\frac{\pi}{4}s^2\right)\,. \)

For large matrices Eq. (2) is still a good approximation with errors on the percent level (Haake 2001). Figure 5 shows level spacings distributions for a variety of chaotic systems, all exhibiting the same behaviour. Such observations had been the motivation for the famous conjecture by Bohigas, Giannoni and Schmitt (1984) that the spectra of completely chaotic time-reversal-invariant systems should show the same fluctuation properties as the GOE.

The replacement of \(H\) by a random matrix means to abandon any hope to learn more about nuclei from the spectra but some average quantities such as the mean level spacings. But the loss of individual features in the spectra on the other hand suggests that it might be worthwhile to look for universal features being common to all chaotic systems. This approach showed up to be extremely fruitful. It allowed to apply results originally obtained for nuclei to many other systems, e. g. quantum-dot systems (Beenakker 1997) and microwave billiards (Stöckmann 1999).

Figure 6: Spectral form factor for the spectrum of a microwave hyperbola billiard (top), and for the subspectrum obtained by considering only every second resonance (bottom) (Alt et al. 1997, Phys. Rev E 55: 6674).

In addition to the level spacings distribution in particular spectral correlations related to the spectral auto-correlation function \(C(E)=\langle \rho(E_2)\rho(E_1)\rangle-\langle \rho(E_2)\rangle\langle\rho(E_1)\rangle\) are considered, where \(\rho(E)\) is the density of states, \(E=E_2-E_1\ ,\) and the brackets denote a spectral average. Quantities often studied in literature are number variance and spectral rigidity (Stöckmann 1999). Here another object shall be considered, the spectral form factor, which is obtained from the Fourier transform of the spectral auto-correlation function. Figure 6 shows an experimental illustration for a hyperbola microwave billiard. In the upper part of the figure the spectral form factor for the complete spectrum is shown. There is a good agreement with random matrix predictions from the GOE. This is consistent with the fact that microwave billiard systems are time-reversal invariant, and there is no spin. Spectra showing GSE statistics have not yet been studied experimentally, but there is the remarkable fact that GSE spectra can be generated by taking only every second level of a GOE spectrum (Mehta 1991). Exactly this had been done with the spectrum of the hyperbola billiard to obtain the spectral form factor in the lower part of the figure, being in perfect agreement with the expected GSE behaviour.

Semiclassical quantum mechanics

Before the final establishment of quantum mechanics Born and Sommerfeld developed a technique today known as semi-classical to calculate the spectrum of atomic hydrogen. At that time Einstein argued that this approach must be a dead end, since semi-classical quantisation needs invariant tori in phase space, preventing a semi-classical quantisation for non-integrable systems. This was one of the rare cases, where Einstein was wrong, though it needed half a century until Gutzwiller (1990) showed in a series of papers that chaotic systems, too, allow for a semi-classical quantisation.

Figure 7: Squared modulus \(|\hat{\rho}(t)|^2\) of the Fourier transform of the spectrum of a quarter Sinai billiard (a=56 cm, b=20 cm, r=7 cm). Each resonance can be associated with a classical periodic orbit~(Stöckmann and Stein 1990, Phys. Rev. Lett. 64, 2215).

For the density of states Gutzwiller's approach yields his famous trace formula. It becomes particularly simple in billiard systems, if the wavenumber \(k\) is taken as the variable. In terms of \(k\) the density of states reads

\(\tag{3} \rho(k)=\rho_0(k)+\sum\limits_n A_ne^{\imath kl_n}\,. \)

The first term varies smoothly with \(k\) and is given in its leading order by Weyl's formula

\( \rho_0(k)=\frac{A}{2\pi}k\,, \)

where \(A\) is the area of the billiard. The second term is heavily oscillating with \(k\ .\) The sum runs over all periodic orbits including repetitions. \(l_n\) is the length of the orbit, and \(A_n\) is a complex factor weighting the stability of the orbit.

The periodic orbit sum (3) is divergent for real \(k\ ,\) and resummation techniques are needed to calculate the spectrum from the periodic orbits. But the inverse procedure, namely to extract the contributions of the different periodic orbits out of the spectra, is straightforward. For billiards the Fourier transform of the fluctuating part of the density of states,

\(\tag{4} \hat{\rho}_\mathrm{osc}(l)=\int\rho_\mathrm{osc}(k)e^{-\imath kl}\,dk=\sum_n A_n\delta(l-l_n)\,, \)

directly yields the contributions of the orbits to the spectrum. Each orbit gives rise to a delta peak at an \(l\) value corresponding to its length, and a weight corresponding to the stability of the orbit.

For illustration Figure 5 shows the squared modulus of the Fourier transform of the spectrum of a microwave resonator shaped as a quarter Sinai billiard. Each peak corresponds to a periodic orbit of the billiard. For the bouncing ball orbit, labelled by "1", three peaks associated with repeated orbits are clearly visible. The smooth part of the density of states is responsible for the increase of \(|\hat{\rho}(k)|^2\) for small lengths.

Semiclassical quantum mechanics relates the spectrum to the classical periodic orbits of the system, i. e. to individual system properties. In view of this fact one may wonder where the universal features discussed in Section 3 come in. Let as have again a look on Eq. (4) to answer this question. For short orbits \(\hat{\rho}(l)\) exhibits a well-resolved length spectrum. This is the individual regime. But with increasing length the peaks become denser and denser, until they eventually cannot be resolved any longer. This is the universal regime. It needed more then 25 years of research to prove the equivalence between random matrix theory and semiclassical quantum mechanics in the universal regime explicitly.


Occasionally people doubt whether chaotic systems really need an extra quantum-mechanical treatment. The Schrödinger equation after all gives exact results both for regular and chaotic systems. Why should one resort to old-fashioned techniques which had been abandoned already 80 years ago, after the development of "correct" quantum mechanics had been completed?

The answer is simple: the numerical solution of the Schrödinger equation means a black-box calculation, and the human brain is not adopted to perform Fourier transforms. This is why spectra as the one shown in Figure 3 seemingly do not contain any relevant information. But the brain is extremely good in identifying paths and trajectories, and therefore the representation of the spectra in terms of classical trajectories, as shown in Figure 5 allows an immediate suggestive interpretation.

Figure 8: Snapshot of the pulse propagation in a dielectric quadrupole cavity made of teflon (length of the long axis l=113 mm). The upper figure shows the pulse intensity inside the teflon at the moment of strongest emission in a colour plot. In addition the Poynting vector is shown in the region outside of the teflon. The lower figure shows the Husimi distribution of the pulse in a Poincarè plot. In addition the unstable manifold of the rectangular orbit is shown. See Media:movie.gif for the complete sequence (Schäfer et al. 2006, New J. of Physics 8, 46).

All this is not just l'art pour l'art, as shall be demonstrated by one example.

The relation between wave propagation and classical trajectories had become of practical importance for the optimization of the emission behaviour of microlasers. Again this shall be demonstrated by a microwave study. The upper part of Figure 8 shows the snapshot of the pulse propagation in a dielectric quadrupole resonator made of teflon. The pulse starts as an outgoing circular wave from an antenna close to the boundary in the lower part of the cavity, but already after a short time only two pulses survive circulating clock- and counter-clockwise close to the border. The figure shows a moment where there is a particularly strong emission to the outside. Contrary to intuition the strongest emission does not occur a the point of largest curvature.

In the lower part of the figure the same situation is shown in a Poincarè plot, with the polar angle as the abscissa, and the sine of the incidence angle as the ordinate (in a Poincarè plot each trajectory is mapped onto a sequence of points representing the reflections at the boundary). The tongue-like structure is the instable manifold of the rectangular orbit. It had been obtained by pursuing a trajectory starting with a minute deviation from the ideal orbit. In addition the Husimi representation of the pulse is shown (a Husimi representation is a convenient tool to embed wave functions into the classical phase space).

Now the observed emission behaviour can be explained. Teflon has an index of refraction of n=1.44 meaning a \(\sin\chi_{\rm crit}=0.69\) for the critical angle of total reflection. Thus the circulating pulses are trapped by total reflection. But whenever the critical line of total reflection is surpassed, there is a strong escape. This happens exactly in the region of the most pronounced tongues of the instable manifold of the rectangular orbit. Meanwhile "phase-space engineering" has become a standard tool in shape optimization of microcavities (Kwon et al. 2010).

The number of activities on the transport of different types of waves, light, seismic waves, water waves, sound waves etc. through disordered media is steadily increasing, many of them based on theories and techniques originally developed in wave and quantum chaos. After several decades of basic research the time for applications has come.

Most experimental examples presented in this article have been obtained in the author's group at the university of Marburg. I want to thank all my coworkers, in particular my senior coworker U. Kuhl. The experiments had been funded by the Deutsche Forschungsgemeinschaft by numerous grants, amongst others via the research group 760 Scattering systems with complex dynamics.


  • Beenakker, C. W. J. (1997). Rev. Mod. Phys. 69: 731.
  • Bohigas, O.; Giannoni, M. J. and Schmit, C. (1984). Phys. Rev. Lett. 52: 1.
  • Chladni, E. F. F. (1802). Die Akustik. Breitkopf und Härtel, Leipzig.
  • Gutzwiller, M. C. (1990). Chaos in Classical and Quantum Mechanics, Interdisciplinary Applied Mathematics, Vol. 1. Springer, New York.
  • Haake, F. (2001). Quantum Signatures of Chaos, 2nd edition. Springer, Berlin.
  • Mehta, M. L. (1991). Random Matrices. 2nd edition. Academic Press, San Diego.
  • Mitchell, G. E.; Richter, A. and Weidenmüller, H. A. (2010). Random Matrices and Chaos in Nuclear Physics: Nuclear reactions. Rev. Mod. Phys. 82: 2845.
  • Kwon, O.; An, K. and Lee, B. (Eds.) (2010). Trends in Nano- and Micro-Cavities. Sharjah, U.A.E.: Bentham Science Pub.
  • Stöckmann, H.-J. (1999). Quantum Chaos - An Introduction. University Press, Cambridge.

Further reading

  • Heller, E. J. (1984). Bound-state eigenfunctions of classically chaotic Hamiltonian systems: Scars of periodic orbits. Phys. Rev. Lett. 53, 1515. In this paper the term "scar" had been introduced, and the phenomenon had been described for the first time.
  • Kuhl, U.; Stöckmann, H.-J. and Weaver, R. (2005). Classical wave experiments on chaotic scattering. J. Phys. A 38: 10433. A discussion of the scattering aspects of wave chaotic systems.
  • Nöckel, J. U. and Stone, A. D. (1997). Ray and wave chaos in asymmetric resonant cavities. Nature 385, 45. Here the relevance of the classical phase-space properties for the emission behaviour of microcavities had been pointed out for the first time.
  • Richter, A. (1999). Playing billiards with microwaves - quantum manifestations of classical chaos. In: Hejhal et al.: Emerging Applications of Number Theory. The IMA Volumes im Mathematics and its Applications, Vol 109. Springer, New York. A report on the microwave experiments of the Darmstadt group.
  • Sirko, L.; Koch, P.M. and Blümel, R. (1997). Experimental identification of non-Newtonian orbits produced by ray splitting in a dielectric-loaded microwave cavity. Phys. Rev. Lett. 78: 2940. Microwave results on a generalization of Gutzwiller’s trace formula to ray splitting occurring in systems with sharp interfaces.
  • Stöckmann, H.-J. (2007). Chladni meets Napoleon. Eur. Phys. J. Special Topics 145: 17. A report on Chladni's Paris visit 1809.

In addition to the groups in Marburg and Darmstadt represented by experimental results in the article, there are a number of other microwave laboratories listed here with representative publications. Only groups where there are at least two publications are mentioned:

  • Kudrolli, A.; Kidambi, V. and Sridhar, S. (1995). Experimental studies of chaos and localization in quantum wave functions. Phys. Rev. Lett. 75: 822. The Boston group.
  • So, P.; Anlage, S. M.; Ott, E. and Oerter, R. N. (1995). Wave chaos experiments with and without time reversal symmetry: GUE and GOE statistics. Phys. Rev. Lett. 74: 2662. The Maryland group
  • Hul, O.; Tymoshchuk, O.; Bauch, S.; Koch, P. and Sirko, L. (2005). Experimental investigation of Wigner's reaction matrix for irregular graphs with absorption. J. Phys. A 38: 10489. The Warsaw group.
  • Barthèlemy, J.; Legrand, O. and Mortessagne, F. (2005). Complete S-matrix in a microwave cavity at room temperature. Europhys. Lett. 70:162. The Nice group.

Internal references

External links

Website of the Marburg quantum chaos group

See also internal links

Bohigas-Giannoni-Schmit conjecture, Gutzwiller trace formula, Random matrix theory

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