Theoretical Derivation of $1/f$ Noise in Quantum Chaos

It was recently conjectured that $1/f$ noise is a fundamental characteristic of spectral fluctuations in chaotic quantum systems. This conjecture is based on the power spectrum behavior of the excitation energy fluctuations, which is different for chaotic and integrable systems. Using random matrix theory, we derive theoretical expressions that explain without free parameters the universal behavior of the excitation energy fluctuations power spectrum. The theory gives excellent agreement with numerical calculations and reproduces to a good approximation the $1/f$ ($1/f^2$) power law characteristic of chaotic (integrable) systems. Moreover, the theoretical results are valid for semiclassical systems as well.

Figure 1

Theoretical power spectrum of the ${\delta }_q$ function for GUE and integrable systems (solid lines), compared to numerical averages calculated using 500 GUE matrices of dimension $N=1000$ (circles) and 500 Poisson level sequences of length $N=1000$ (triangles).

Figure 2

Numerical average power spectrum of the ${\delta }_q$ function for ${}^{34}\mathrm{N}\mathrm{a}$, calculated using 25 sets of 256 consecutive levels from the high level density region, compared to the parameter free theoretical values (solid line) for GOE.

Figure 3

Numerical average power spectrum of ${\delta }_q$ for a rectangular billiard, calculated using 25 sets of 256 consecutive levels, compared to the parameter free theoretical values (solid line) for integrable systems.