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Physical Review A

Quantum analog of the Kolmogorov-Arnold-Moser theorem in the anisotropic Dicke model and its possible implications in the hybrid Sachdev-Ye-Kitaev models

Yu Yi-Xiang, Jinwu Ye, Wu-Ming Liu, Cunlin Zhang

Published Aug 19, 2022• 4 citations
Publisher

Abstract

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 an 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. A possible quantum KAM theorem in terms of other quantum chaos criteria such as the quantum Lyapunov exponent is also discussed.

DOI
10.1103/physreva.106.022213

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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 NNN 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 an evolution quantitatively by both large-NNN 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. A possible quantum KAM theorem in terms of other quantum chaos criteria such as the quantum Lyapunov exponent is also discussed.
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 $.
Here 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 $ SYK with an interaction $ \sqrt{K} $ in Type II . These models include hybrid Majorana fermion, complex fermion and bosonic SYK. We first introduce a new universal ratio which is the ratio of the next nearest neighbour (NNN) energy level spacing to characterize the random matrix behaviours. We make exact symmetry analysis on the possible symmetry class of both types of hybrid SYK in the 10 fold way and also work out the degeneracy of each energy levels. We perform exact diagonalization to evaluate both the known NN ratio and the new NNN ratio. In Type I, as $ K/J $ changes, there is always a chaotic to non-chaotic transition (CNCT) from the GUE to Poisson in all the hybrid fermionic SYK models, but not the hybrid bosonic SYK model. In Type II, there are always CNCT from the corresponding GOE, GUE or GSE dictated by the symmetry of the $ q=4 $ SYK to the Poisson dictated by $ ( q=2 )^2 $ SYK. When the double degeneracy at the $ q=4 $ ( or $ (q=2)^2 $ ) side is broken by the $ q=2 $ ( or $ q=4 $ ) perturbation in Type I ( or Type II), the new NNN ratio can be most effectively to quantify the stability of quantum chaos ( or the KAM ). We compare the stability of quantum chaos and KAM theorem near the integrability in all these hybrid SYK models. We also discuss some possible connections between CNCT characterized by the random matrix theory and the quantum phase transitions (QPT) characterized by renormalization groups. Quantum chaos in both types of hybrid SYK models are also contrasted with that in the $ U(1)/Z_2 $ Dicke model in quantum optics.
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 $.
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