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| Fig. 1: Example of a two-encounter. The partner orbits differ noticeably only inside an encounter of two orbit stretches. The sketch greatly exaggerates the difference between the two partner orbits outside the encounter and greatly underemphasizes both the relative and absolute lengths of the orbit stretches outside the encounter. (Source: A. Rahman, after Müller. [6]) |
Most of the systems one encounters in the physical world - from the weather to black holes - are characterized by the presence of chaos. In systems well - described by classical mechanics, the presence of chaos can be diagnosed using any one of many equivalent definitions, the most famous being the butterfly effect: small changes to the system's initial conditions lead to exponentially different states at later times. Chaos is generic in physics, subsuming all but a small number of highly symmetric "integrable" systems characterized by the presence of an extensively large number of conserved quantities.
Quantum mechanics is currently our best paradigm for describing the physical world, but the generalization of the notion of chaos to the quantum regime encounters severe difficulties, not the least of which being that many of the definitions of chaos applicable in the setting of classical mechanics fail to even make sense for generic quantum systems. An example is the butterfly effect: the most straightforward analog is clearly inapplicable due to the linearity of the Schrodinger equation, while more nuanced analogs require the presence of additional structure such as large-N factorization.
A useful arena for learning how to diagnose quantum chaos is the semiclassical regime of quantum systems with chaotic classical limits. One hopes that a proper understanding of this situation (the object of study of the field of quantum chaology) will help us understand which properties of the quantum theory reflect the presence of chaos in the classical limit, and perhaps even understand which properties of a quantum mechanical system might diagnose the presence of chaos (nonintegrability) more generally.
A key issue one encounters right away is that many of the "standard" methods for studying the semiclassical regime, such as Bohr-Sommerfeld quantization, are not actually applicable to chaotic systems (essentially because the Bohr-Sommerfeld condition only applies to systems which can be decoupled into a product of systems with one degree-of- freedom each, which generally requires integrability). [1,2]
One method for studying the semiclassical regime which does work for nonintegrable systems is the periodic orbit quantization method pioneered by Gutzwiller in the 1970's. [2,3] The crux of this method is the determination, to leading order in 1/ℏ, of the spectrum of the quantum Hamiltonian in terms of the behavior of periodic solutions to the classical equations of motion ("periodic orbits"). The cornerstone of periodic orbit quantization is the Gutzwiller trace formula, which expresses the semiclassical approximation to the quantum Hamiltonian's resolvent in terms of a sum of one-loop amplitudes, one for each periodic orbit of the underlying classical system: [2]
Here Tγ is the period, Mγ the monodromy matrix, and mγ the Maslov index of the periodic orbit γ . The resolvent (specifically its discontinuity across the real E-axis) can then be be used to reconstruct the semiclassical density of states and hence the semiclassical approximation to the energy spectrum.
This method is particularly well adapted to the study of quantum chaos for two reasons. The first is that the behavior of periodic orbits is universal in classically chaotic systems. [4-6] The second is that the energy spectrum of a quantum system is precisely the quantity that one expects will diagnose the presence of quantum chaos, due to the BGS (Bohigas-Giannoni-Schmit) Conjecture of Random Matrix Universality, which posits that the excited energy levels of a chaotic quantum system should be distributed like those in a random matrix theory. [7] The particular universality class of random matrix theory describing the spectrum is expected to be determined only by the basic discrete symmetry properties of the system (such as e.g. the presence or absence of time-reversal symmetry). The BGS conjecture is supported by an overwhelming amount of numerical and experimental evidence, but its underlying theoretical foundations are to date poorly understood. [4]
A key diagnostic tool in the study of late-time quantum chaos is the spectral form factor, which is defined as follows: One begins by choosing some free parameter (e.g. a short time interval) over which to average observables in our system, and then calculates the two-point correlation function of the density of states with respect to this averaging procedure. This gives a quantity which probes the distribution of the system's energy levels and which can also easily be compared to the predictions of random matrix theory. The spectral form factor (SFF), K(τ), is defined to be the Fourier transform of the normalized connected density-density correlator
where τ = t / 2πℏρ(E).
According to the BGS conjecture, we expect a chaotic system with Hamiltonian H to fall into one of three classes: [6] unitary (no time reversal symmetry), orthogonal (time-reversal symmetry squaring to the identity), or symplectic (time-reversal symmetry squaring to minus the identity), with random matrix theory predictions [8]
for 0 < τ < 1 . For chaotic systems, it is expected that the spectral form factor will begin to resemble the random matrix theory prediction for τ larger than a nonuniversal time-scale known as the "ramp" or "Thouless" time. [9]
It's a remarkable fact that, for chaotic systems with ergodic and hyperbolic classical dynamics, one can actually derive the random matrix theory prediction for the spectral form factor directly from the Gutzwiller trace formula. [5,6] One does this by using the Gutzwiller trace formula to yield the semiclassical expansion of the density of states: [5]
and then taking the semiclassical limit ℏ → 0, TH; → ∞, Tγ /TH constant. In this limit, the leading contributions to the semiclassical spectral form factor come from families of pairs of orbits (γ,γ') with small action difference |Sγ - Sγ'| of order ℏ. Specifically, we only need to consider the contribution to the spectral form factor from pairs (γ,γ') of orbits with γ and γ' differing from one another only within a close l-encounter, where a close l-encounter is a short stretch of configuration space along which two distinct segments of a given orbit run alongside one another (before they begin to diverge exponentially due to chaotic Lyapunov behavior). [5,6] From these families of pairs of orbits alone, one can completely reproduce the random matrix theory prediction for the spectral form factor, after averaging over a small time interval.
As an example, note that, for systems described by the unitary or orthogonal classes, we can recover the first term of the random matrix theory prediction from the terms with Sγ = Sγ'. For systems without time reversal symmetry, this is simply given by taking γ' = γ. Classical ergodicity allows us to apply the Hannay-Ozorio de Almeida (HOdA) sum rule and so we find that [10]
as predicted by random matrix theory. This is the famous "diagonal" approximation of Berry. [11] For systems with time reversal symmetry (since we are considering systems with conventional classical limits, time reversal symmetry will square to the identity and we are dealing with the orthogonal class), there is an overall factor of 2 which comes from an additional, equal contribution to the semiclassical spectral form factor from pairs with γ' the time reverse of γ. This matches the prediction of random matrix theory for the orthogonal class.
© A. Rahman. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] A. Einstein, "Zum Quantensatz von Sommerfeld und Epstein," Verhandl. Deutsh. Phys. Ges. 19, 82 (1917).
[2] M. C. Gutzwiller, "Periodic Orbits and Classical Quantization Conditions," J. Math. Phys. 12, 342 (1971).
[3] M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics, (Springer, 1991).
[4] F. Haake, S. Gnutzmann, and M. Kuś, Quantum Signatures of Chaos, 4th Ed. (Springer, 2019).
[5] S. Müller et al., "Semiclassical Foundation of Universality in Quantum Chaos," Phys. Rev. Lett. 93, 014103 (2004).
[6] S. Müller et al., "Periodic-Orbit Theory of Universality in Quantum Chaos," Phys. Rev. E 72, 046207 (2005).
[7] O. Bohigas, M. J. Giannoni, and C. Schmit, "Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws," Phys. Rev. Lett. 52, 1 (1984).
[8] M. L. Mehta, Random Matrices, 3rd Ed. (Academic Press, 2004).
[9] H. Gharibyan et al., "Onset of Random Matrix Behavior in Scrambling Systems," J. High Energy Phys. 2018, 124 (2018).
[10] J. H. Hannay and A. M. Ozorio de Almeida, "Periodic Orbits and a Correlation Function for the Semiclassical Density of States," J. Phys. A: Math. Gen. 17, 3429 (1984).
[11] M. V. Berry, "Semiclassical Theory of Spectral Rigidity," Proc. R. Soc. Lond. A 400, 229 (1985).