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Journal of High Energy Physics

Out-of-time-ordered-correlators for the pure inverted quartic oscillator: classical chaos meets quantum stability

Paul Romatschke

Published Oct 17, 2024• 6 citations
Publisher

Abstract

A bstract Out-of-time-ordered-correlators (OTOCs) have been suggested as a means to diagnose chaotic behavior in quantum mechanical systems. Recently, it was found that OTOCs display exponential growth for the inverted quantum harmonic oscillator, mirroring the fact that this system is classically and quantum mechanically unstable. In this work, I study OTOCs for the inverted anharmonic (pure quartic) oscillator in quantum mechanics, finding only oscillatory behavior despite the classically unstable nature of the system. For higher temperature, OTOCs seem to exhibit saturation consistent with a value of –2⟨ x 2 ⟩ T ⟨ p 2 ⟩ T at late times. I provide analytic evidence from the spectral zeta-function and the WKB method as well as direct numerical solutions of the Schrödinger equation that the inverted quartic oscillator possesses a real and positive energy eigenspectrum, and normalizable wave-functions.

DOI
10.1007/jhep10(2024)126

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A bstract Out-of-time-ordered-correlators (OTOCs) have been suggested as a means to diagnose chaotic behavior in quantum mechanical systems. Recently, it was found that OTOCs display exponential growth for the inverted quantum harmonic oscillator, mirroring the fact that this system is classically and quantum mechanically unstable. In this work, I study OTOCs for the inverted anharmonic (pure quartic) oscillator in quantum mechanics, finding only oscillatory behavior despite the classically unstable nature of the system. For higher temperature, OTOCs seem to exhibit saturation consistent with a value of –2⟨ x 2 ⟩ T ⟨ p 2 ⟩ T at late times. I provide analytic evidence from the spectral zeta-function and the WKB method as well as direct numerical solutions of the Schrödinger equation that the inverted quartic oscillator possesses a real and positive energy eigenspectrum, and normalizable wave-functions.
Out-of-time-ordered-correlators (OTOCs) have been suggested as a means to diagnose chaotic behavior in quantum mechanical systems. Recently, it was found that OTOCs display exponential growth for the inverted quantum harmonic oscillator, mirroring the fact that this system is classically and quantum mechanically unstable. In this work, I study OTOCs for the inverted anharmonic (pure quartic) oscillator in quantum mechanics, finding only oscillatory behavior despite the classically unstable nature of the system. For higher temperature, OTOCs seem to exhibit saturation consistent with a value of −2⟨x2⟩T⟨p2⟩T-2 \langle x^2 \rangle_T \langle p^2 \rangle_T−2⟨x2⟩T​⟨p2⟩T​ at late times. I provide analytic evidence from the spectral zeta-function and the WKB method as well as direct numerical solutions of the Schrödinger equation that the inverted quartic oscillator possesses a real and positive energy eigenspectrum, and normalizable wave-functions.
Out-of-time-ordered correlators (OTOCs) have been suggested as a means to study quantum chaotic behavior in various systems. In this work, I calculate OTOCs for the quantum mechanical anharmonic oscillator with quartic potential, which is classically integrable and has a Poisson-like energy-level distribution. For low temperature, OTOCs are periodic in time, similar to results for the harmonic oscillator and the particle in a box. For high temperature, OTOCs exhibit a rapid (but power-like) rise at early times, followed by saturation consistent with 2 < x2 >T < p2 > T at late times. At high temperature, the spectral form factor decreases at early times, bounces back and then reaches a plateau with strong fluctuations.
Abstract Out-of-time-ordered correlators (OTOCs) have been suggested as a means to study quantum chaotic behavior in various systems. In this work, I calculate OTOCs for the quantum mechanical anharmonic oscillator with quartic potential, which is classically integrable and has a Poisson-like energy-level distribution. For low temperature, OTOCs are periodic in time, similar to results for the harmonic oscillator and the particle in a box. For high temperature, OTOCs exhibit a rapid (but power-like) rise at early times, followed by saturation consistent with 2〈x2〉T〈p2〉T at late times. At high temperature, the spectral form factor decreases at early times, bounces back and then reaches a plateau with strong fluctuations.
We calculate the third-order out-of-time-order correlator (OTOC) of a simple harmonic oscillator with an additional quartic interaction using the second quantization method. We obtain analytic relations for the spectrum, Fock space states, and matrix elements of the coordinate, which are then used to numerically evaluate the OTOC. We observe that after the scrambling, the OTOC becomes a fluctuation around a saturation point at later times, which is associated with quantum chaotic behavior in systems that exhibit chaos. We analyze the early-time properties of the OTOC and find that in systems with sufficiently strong quartic interactions, an exponential growth curve fitting over a long time window clearly emerges in the third-order perturbation
The out-of-time-order correlator (OTOC) of simple harmonic oscillator with extra anharmonic (quartic) interaction are calculated by the second quantization method. We obtain the analytic formulas of spectrum, Fock space states and matrix elements of coordinate to the second order of anharmonic interaction. These relations clearly reveal the property that the correction of the interaction is proportional to the quantum number of the energy level, and shows the enhancement for some physical quantities. The analytic results explicitly show that the OTOC is raising in the quadratic power law at early times. Then, we use the formulas to do numerical summation to calculate the OTOC, which shows that, at late times, while the first-order perturbative OTOC is oscillating as that in a simple harmonic oscillator, the second-order perturbative OTOC saturates to a constant value. We compare it with 2⟨x2⟩T⟨p2⟩T2\langle x^2\rangle_T\langle p^2\rangle_T2⟨x2⟩T​⟨p2⟩T​, which is associated with quantum chaotic behavior in systems that exhibit chaos, and discuss the validity of the second-order perturbation.
In this paper, the out-of-time-order correlators (OTOC) in quantum harmonic oscillators are calculated analytically by second quantization method in perturbative approximation. We consider the coupled harmonic oscillators and anharmonic (quartic) oscillators. The analytic formulas of microcanonical OTOC in the leading order of interaction are obtained. From these results we clearly see that the interactions can enhance the correlation to very large values over time and is proportional to the energy level. The growth of OTOC signatures the quantum chaos therein. The microcanonical OTOC which is an increasing oscillation becomes a simple increasing function in thermal OTOC after sum over all energy level, as different level oscillates with different frequency. The property that, at late time stage OTOC saturates to a constant value, however, does not show in the first-order perturbation studied in this paper.