The out-of-time-order correlator (OTOC) is considered as a measure of quantum chaos. We formulate how to calculate the OTOC for quantum mechanics with a general Hamiltonian. We demonstrate explicit calculations of OTOCs for a harmonic oscillator, a particle in a one-dimensional box, a circle billiard and stadium billiards. For the first two cases, OTOCs are periodic in time because of their commensurable energy spectra. For the circle and stadium billiards, they are not recursive but saturate to constant values which are linear in temperature. Although the stadium billiard is a typical example of the classical chaos, an expected exponential growth of the OTOC is not found. We also discuss the classical limit of the OTOC. Analysis of a time evolution of a wavepacket in a box shows that the OTOC can deviate from its classical value at a time much earlier than the Ehrenfest time, which could be the reason of the difficulty for the numerical analyses to exhibit the exponential growth.The out-of-time-order correlator (OTOC) is considered as a measure of quantum chaos. We formulate how to calculate the OTOC for quantum mechanics with a general Hamiltonian. We demonstrate explicit calculations of OTOCs for a harmonic oscillator, a particle in a one-dimensional box, a circle billiard and stadium billiards. For the first two cases, OTOCs are periodic in time because of their commensurable energy spectra. For the circle and stadium billiards, they are not recursive but saturate to constant values which are linear in temperature. Although the stadium billiard is a typical example of the classical chaos, an expected exponential growth of the OTOC is not found. We also discuss the classical limit of the OTOC. Analysis of a time evolution of a wavepacket in a box shows that the OTOC can deviate from its classical value at a time much earlier than the Ehrenfest time, which could be the reason of the difficulty for the numerical analyses to exhibit the exponential growth.

arXiv:1703.09435Journal of High Energy PhysicsOct 1, 2017249 citations

From Classical to Quantum Chaos

George M. Zaslavsky

WORLD SCIENTIFIC eBooksSep 1, 199212 citations

超弦理論におけるカオスと乱流

Murata Keiju, Keiju Murata, 佳樹村田

(1)非時間順序相関関数に関する研究成果非時間順序相関関数(Out-of-time-order correlator, OTOC)は, 量子論におけるカオスの指標になると期待されている。我々は, 一般のハミルトニアンを持つ量子力学系においてOTOCを計算する手法を確立した。その手法を用いて, 調和振動子, 井戸型ポテンシャル, 円形ビリヤード, スタジアムビリヤードに対して, OTOCの計算を実際に行った。最初の三つは可積分系で, スタジアムビリヤードは非可積分系である。OTOCの初期挙動に対しては, 可積分性による定性的な違いは見られなかった。特に, スタジアムビリヤード系は典型的な古典カオス系であるが, OTOCに明確な指数増大は見られない。その理由を探るため, 我々は井戸型ポテンシャル中の波束に対してOTOCを計算し, 古典近似で得られる値と比較した。その結果OTOCは, エーレンフェスト時間よりも十分早いタイムスケールで, その古典的な値からずれる可能性があることが分かった。 (1)ゲージ・重力対応の物性系への応用に関する研究成果自由電子系に回転電場と印可すると, 価電バンドと伝導バンドが一点において接触する「ワイル半金属」が実現される。我々はAdS/CFT対応を用いて, その強結合領域を考えた。具体的には, N=2の超対称QCDに回転電場を掛けることで, そこに誘起される電流, ジュール熱, 温度などを調べた。また, その系にDC電場をprobeとして掛けることにより, ホール効果を見ることにも成功した。そのホール電流は磁場に寄るものと少し異なり, 振動成分も含まれる。また, 回転電場の振動数をうまく調節することでホール電流を最大化できることも分かった。この研究に動機付けられて, マックスプランク研究所の実験グループは, 回転電場によるワイル半金属の電気伝導度の測定を始めている。 (1)Out-of-time-order correlators in quantum mechanics The out-of-time-order correlator (OTOC) is considered as a measure of quantum chaos. We formulate how to calculate the OTOC for quantum mechanics with a general Hamiltonian. We demonstrate explicit calculations of OTOCs for a harmonic oscillator, a particle in a one-dimensional box, a circle billiard and stadium billiards. For the first two cases, OTOCs are periodic in time because of their commensurable energy spectra. For the circle and stadium billiards, they are not recursive but saturate to constant values which are linear in temperature. Although the stadium billiard is a typical example of the classical chaos, an expected exponential growth of the OTOC is not found. We also discuss the classical limit of the OTOC. Analysis of a time evolution of a wavepacket in a box shows that the OTOC can deviate from its classical value at a time much earlier than the Ehrenfest time. (2)Holographic Floquet states Using AdS/CFT correspondence, we evaluate nonlinear AC conductivities and optical Hall effect, for strongly coupled N=2 supersymmetric massless QCD at large Nc limit under a circularly-polarized external electric field. The nonlinear AC conductivity is found to be approximated by an Ohm's law for a high frequency region, while consistently reproduces the previously obtained DC conductivity. A weak AC probe analysis on the background strong circularly-polarized electric field shows a Hall AC current as a linear response, which is consistent with a weak coupling picture of Weyl semimetal.(1)非時間順序相関関数に関する研究成果非時間順序相関関数(Out-of-time-order correlator, OTOC)は, 量子論におけるカオスの指標になると期待されている。我々は, 一般のハミルトニアンを持つ量子力学系においてOTOCを計算する手法を確立した。その手法を用いて, 調和振動子, 井戸型ポテンシャル, 円形ビリヤード, スタジアムビリヤードに対して, OTOCの計算を実際に行った。最初の三つは可積分系で, スタジアムビリヤードは非可積分系である。OTOCの初期挙動に対しては, 可積分性による定性的な違いは見られなかった。特に, スタジアムビリヤード系は典型的な古典カオス系であるが, OTOCに明確な指数増大は見られない。その理由を探るため, 我々は井戸型ポテンシャル中の波束に対してOTOCを計算し, 古典近似で得られる値と比較した。その結果OTOCは, エーレンフェスト時間よりも十分早いタイムスケールで, その古典的な値からずれる可能性があることが分かった。 (1)ゲージ・重力対応の物性系への応用に関する研究成果自由電子系に回転電場と印可すると, 価電バンドと伝導バンドが一点において接触する「ワイル半金属」が実現される。我々はAdS/CFT対応を用いて, その強結合領域を考えた。具体的には, N=2の超対称QCDに回転電場を掛けることで, そこに誘起される電流, ジュール熱, 温度などを調べた。また, その系にDC電場をprobeとして掛けることにより, ホール効果を見ることにも成功した。そのホール電流は磁場に寄るものと少し異なり, 振動成分も含まれる。また, 回転電場の振動数をうまく調節することでホール電流を最大化できることも分かった。この研究に動機付けられて, マックスプランク研究所の実験グループは, 回転電場によるワイル半金属の電気伝導度の測定を始めている。 (1)Out-of-time-order correlators in quantum mechanics The out-of-time-order correlator (OTOC) is considered as a measure of quantum chaos. We formulate how to calculate the OTOC for quantum mechanics with a general Hamiltonian. We demonstrate explicit calculations of OTOCs for a harmonic oscillator, a particle in a one-dimensional box, a circle billiard and stadium billiards. For the first two cases, OTOCs are periodic in time because of their commensurable energy spectra. For the circle and stadium billiards, they are not recursive but saturate to constant values which are linear in temperature. Although the stadium billiard is a typical example of the classical chaos, an expected exponential growth of the OTOC is not found. We also discuss the classical limit of the OTOC. Analysis of a time evolution of a wavepacket in a box shows that the OTOC can deviate from its classical value at a time much earlier than the Ehrenfest time. (2)Holographic Floquet states Using AdS/CFT correspondence, we evaluate nonlinear AC conductivities and optical Hall effect, for strongly coupled N=2 supersymmetric massless QCD at large Nc limit under a circularly-polarized external electric field. The nonlinear AC conductivity is found to be approximated by an Ohm's law for a high frequency region, while consistently reproduces the previously obtained DC conductivity. A weak AC probe analysis on the background strong circularly-polarized electric field shows a Hall AC current as a linear response, which is consistent with a weak coupling picture of Weyl semimetal.

Jan 1, 20170 citations

More on OTOCs and Chaos in Quantum Mechanics -- Magnetic Fields

Cameron Beetar, Jeff Murugan

We revisit thermal out-of-time-order correlators (OTOCs) in single-particle quantum systems, focusing on magnetic billiards. Using the stadium billiard as a testbed, we compute the thermal OTOC $C_T(t) = -\langle [x(t), p]^2 \rangle_β$ and extract Lyapunov-like exponents $λ_L$ that quantify early-time growth. We map out $λ_L(T, B)$ , revealing a crossover from quantum chaos to magnetic rigidity. In parallel, we compute an alternative OTOC built from guiding-center operators, which exhibits qualitatively distinct dynamics and no exponential growth. Our results offer a controlled framework for probing scrambling, temperature dependence, and the interplay of geometry and magnetic fields in quantum systems.We revisit thermal out-of-time-order correlators (OTOCs) in single-particle quantum systems, focusing on magnetic billiards. Using the stadium billiard as a testbed, we compute the thermal OTOC $C_T(t) = -\langle [x(t), p]^2 \rangle_β$ and extract Lyapunov-like exponents $λ_L$ that quantify early-time growth. We map out $λ_L(T, B)$, revealing a crossover from quantum chaos to magnetic rigidity. In parallel, we compute an alternative OTOC built from guiding-center operators, which exhibits qualitatively distinct dynamics and no exponential growth. Our results offer a controlled framework for probing scrambling, temperature dependence, and the interplay of geometry and magnetic fields in quantum systems.

hep-thCross-listedarXiv:2602.05322Feb 6, 20260 citations

Aspects of Quantum Chaos

Sandro Wimberger

Graduate texts in physicsJan 1, 20220 citations

Aspects of Quantum Chaos

Sandro Wimberger

Graduate texts in physicsJan 1, 20140 citations

We revisit thermal out-of-time-order correlators (OTOCs) in single-particle quantum systems, focusing on magnetic billiards. Using the stadium billiard as a testbed, we compute the thermal OTOC

C_T(t) = -\langle [x(t), p]^2 \rangle_β

and extract Lyapunov-like exponents

λ_L

that quantify early-time growth. We map out

λ_L(T, B)

, revealing a crossover from quantum chaos to magnetic rigidity. In parallel, we compute an alternative OTOC built from guiding-center operators, which exhibits qualitatively distinct dynamics and no exponential growth. Our results offer a controlled framework for probing scrambling, temperature dependence, and the interplay of geometry and magnetic fields in quantum systems.