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arXiv:2011.04707 · Physical Review Letters

Kolmogorov-Arnold-Moser Stability for Conserved Quantities in Finite-Dimensional Quantum Systems

Daniel Klaus Burgarth, Paolo Facchi A, Hiromichi Nakazato, Saverio Pascazio, Kazuya Yuasa

Published Apr 12, 2021• 10 citations
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Abstract

We show that for any finite-dimensional quantum systems the conserved quantities can be characterized by their robustness to small perturbations: for fragile symmetries, small perturbations can lead to large deviations over long times, while for robust symmetries, their expectation values remain close to their initial values for all times. This is in analogy with the celebrated Kolmogorov-Arnold-Moser theorem in classical mechanics. To prove this result, we introduce a resummation of a perturbation series, which generalizes the Hamiltonian of the quantum Zeno dynamics.

DOI
10.1103/physrevlett.126.150401

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We show that for any finite-dimensional quantum systems the conserved quantities can be characterized by their robustness to small perturbations: for fragile symmetries, small perturbations can lead to large deviations over long times, while for robust symmetries, their expectation values remain close to their initial values for all times. This is in analogy with the celebrated Kolmogorov-Arnold-Moser theorem in classical mechanics. To prove this result, we introduce a resummation of a perturbation series, which generalizes the Hamiltonian of the quantum Zeno dynamics.
The classical Kolmogorov-Arnold-Moser (KAM) theorem provides the underlying mechanism for the stability of the solar system under some small chaotic perturbations. Despite many previous efforts, any quantum version of the KAM theorem remains elusive In this work, we provide a quantum KAM theorem in the context of the anisotropic Dicke model which is the most important quantum optics model. It describes a single mode of photons coupled to $ N $ qubits with both a rotating wave (RW) term and a counter-RW (CRW) term. As the ratio of the CRW over the RW term increases from zero to one, the systems evolves from quantum integrable to quantum chaotic. We establish a quantum KAM theorem to characterize such a evolution quantitatively by both large $ N $ expansion and Random Matrix Theory and find agreement from the two complementary approaches. Connections and differences between the Dicke models and Sachdev-Ye-Kitaev (SYK) or hybrid SYK models are examined. Possible Quantum KAM theorem in terms of other quantum chaos criterion such as quantum Lyapunov exponent is also discussed.
We investigate possible quantum analog of Kolmogorov-Arnold-Moser (KAM) theorem in two types of hybrid SYK models which contain both $ q=4 $ SYK with interaction $ J $ and $ q=2 $ SYK with an interaction $ K $ in type-I or (q=2)2(q=2)^2(q=2)2 SYK with an interaction $ \sqrt{K} $ in type-II. These models include hybrid Majorana fermion, complex fermion, and bosonic SYK. For the Majorana fermion case, we discuss both $ N $ even and $ N $ odd case. We introduce a new universal ratio which is the ratio of the next nearest neighbour (NNN) energy level spacing to characterize the KAM theorem. We make exact symmetry analysis on the possible symmetry class of both types of hybrid SYK in the 10-fold way by random matrix theory (RMT) and also work out the degeneracy of each energy level. We perform exact diagonalization to evaluate both the known NN ratio and the new NNN ratio, then use both ratios to study Chaotic to Integrable transitions (CIT) in both types of hybrid SYK models. After defining the quantum analog of the KAM theorem in the context of RMT, we show that the KAM theorem holds when $ J/K \sim \sqrt{N} e^{-N} $ and shrinks exponentially fast in the thermodynamic limit. In the GOE or GSE case in type-I, we find a lower bound of the dual form of the KAM theorem which states the stability of quantum chaos. While the stability of quantum chaos in all the other cases are much more robust than the KAM theorem in the integrable side. We explore some intrinsic connections between the two complementary approaches to quantum chaos: the RMT and the Lyapunov exponent by the $ 1/N $ expansion in the large $ N $ limit at a suitable temperature range. We stress the crucial difference between the quantum phase transition (QPT) characterized by renormalization groups at $ N=\infty $, $ 1/N $ expansions at a finite $ N $, and the CIT characterized by the RMT at a finite $ N $.
We consider a system of linear oscillators, or quantum states, described by random matrix theory and analyze how its time evolution is affected by a nonlinear perturbation. Our numerical results show that above a certain chaos border a weak or moderate nonlinearity leads to a dynamical thermalization of a finite number of degrees of freedom with energy equipartition over linear eigenmodes as expected from the laws of classical statistical mechanics. The system temperature is shown to change in a broad range from positive to negative values, and the dependence of system characteristics on the initial injected energy is determined. Below the chaos border the dynamics is described by the Kolmogorov-Arnold-Moser integrability. Owing to universal features of random matrix theory we argue that the obtained results describe the generic properties of its nonlinear perturbation.
Abstract The results of numerical computations are presented for the Bohmian trajectories of the family of different one‐ and two‐dimensional anharmonic oscillators, which exhibit regular or chaotic motion in both classical and quantum domains, depending on the values of the parameters appearing in the respective Hamiltonians. Quantum signatures of the Kolmogorov–Arnold–Moser (KAM) transition from the regular to chaotic classical dynamics of these oscillators are studied using a quantum theory of motion (QTM) as developed by de Broglie and Bohm. A phase space distance function between two initially close Bohmian trajectories, the associated Kolmogorov–Sinai–Lyapunov (KSL) entropy, the phase space volume, the autocorrelation function, the associated power spectrum, and the nearest‐neighbor spacing distribution, clearly differentiate the quantum analogues of the corresponding regular and chaotic motions in the classical domain. These quantum anharmonic oscillators are known to be useful in several diverse branches of science. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004