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arXiv:2111.07351 · Physical review. D/Physical review. D.

Chaos and complexity for inverted harmonic oscillators

Le-Chen Qu, Jing Chen, Yu-Xiao Liu

Published Jun 16, 2022• 27 citations
arXivHTMLPDF

Abstract

We investigate the circuit complexity and Loschmidt echo for the (inverted) harmonic oscillators. Using a perturbative approach, we analytically derive the Lyapunov exponent and scrambling time of the inverted harmonic oscillators. We show that the circuit complexity and Loschmidt echo exhibit qualitatively similar behaviors, particularly the consistent Lyapunov exponent.

DOI
10.1103/physrevd.105.126015

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We investigate the circuit complexity and Loschmidt echo for the (inverted) harmonic oscillators. Using a perturbative approach, we analytically derive the Lyapunov exponent and scrambling time of the inverted harmonic oscillators. We show that the circuit complexity and Loschmidt echo exhibit qualitatively similar behaviors, particularly the consistent Lyapunov exponent.
The harmonic oscillator is the paragon of physical models; conceptually and computationally simple, yet rich enough to teach us about physics on scales that span classical mechanics to quantum field theory. This multifaceted nature extends also to its inverted counterpart, in which the oscillator frequency is analytically continued to pure imaginary values. In this article we probe the inverted harmonic oscillator (IHO) with recently developed quantum chaos diagnostics such as the out-of-time-order correlator (OTOC) and the circuit complexity. In particular, we study the OTOC for the displacement operator of the IHO with and without a non-Gaussian cubic perturbation to explore genuine and quasi scrambling respectively. In addition, we compute the full quantum Lyapunov spectrum for the inverted oscillator, finding a paired structure among the Lyapunov exponents. We also use the Heisenberg group to compute the complexity for the time evolved displacement operator, which displays chaotic behaviour. Finally, we extended our analysis to N-inverted harmonic oscillators to study the behaviour of complexity at the different timescales encoded in dissipation, scrambling and asymptotic regimes.
A bstract We examine the multifold complexity and Loschmidt echo for an inverted harmonic oscillator. We give analytic expressions for any number of precursors, implementing multiple backward and forward time evolutions of the quantum state, at the leading order in the perturbation. We prove that complexity is dominated by the longest permutation of the given time combination in an alternating “zig-zag” order, the exact same result obtained with holography. We conjecture that the general structure for multifold complexity should hold true universally for generic quantum systems, in the limit of a large number of precursors.
We study the connections between three quantities that can be used as diagnostics for quantum chaos, i.e., the out-of-time-order correlator (OTOC), Loschmidt echo (LE), and complexity. We generalize the connection between OTOC and LE for infinite dimensions and extend it for higher-order OTOCs and multi-fold LEs. Novel applications of this intrinsic relation are proposed. We also investigated the relationship between a specific circuit complexity and LE by using the inverted oscillator model and made a conjecture about their relationship. These relationships signal a deeper connection between these three probes of quantum chaos.
Abstract We study the connections between three quantities that can be used as diagnostics for quantum chaos, i.e., the out-of-time-order correlator (OTOC), Loschmidt echo (LE), and complexity. We generalize the connection between OTOC and LE for infinite dimensions and extend it for higher-order OTOCs and multi-fold LEs. Novel applications of this intrinsic relation are proposed. We also investigated the relationship between a specific circuit complexity and LE by using the inverted oscillator model and made a conjecture about their relationship. These relationships signal a deeper connection between these three probes of quantum chaos.
One of the fundamental manifestations of classical chaos is exponential sensitivity to initial conditions that is, two trajectories starting from nearly identical initial states diverge exponentially over time. This behavior is quantified by the Lyapunov exponents. Due to the unitary nature of quantum mechanics, such exponential divergence is elusive in quantum systems. As a result, several alternative quantities have been proposed and studied in recent years to capture analogous behavior. In this article, we present a pedagogical overview of three such quantities that have been the focus of intense research in recent years: the Loschmidt echo, out-of-time-order correlators (OTOCs), and Krylov complexity.
Recently, the out-of-time-ordered correlator(OTOC) and Krylov complexity have been studied actively as a measure of operator growth. OTOC is known to exhibit exponential growth in chaotic systems, which was confirmed in many previous works. However, in some non-chaotic systems, it was observed that OTOC shows chaotic behavior and cannot distinguish saddle-dominated scrambling from chaotic systems. For K-complexity, in the universal operator growth hypothesis, it was stated that Lanczos coefficients show linear growth in chaotic systems, which is the fastest. But recently, it appeared that Lanczos coefficients and K-complexity show chaotic behavior in the LMG model and cannot distinguish saddle-dominated scrambling from chaos. In this paper, we compute Lanczos coefficients and K-complexity in an inverted harmonic oscillator. We find that they exhibit chaotic behavior, which agrees with the case of the LMG model. We also analyze bounds on the quantum Lyapunov coefficient and the growth rate of Lanczos coefficients and find that there is a difference with the chaotic system. Microcanonical K-complexity is also analyzed and compared with the OTOC case.