Schwinger-Keldysh path integral formalism for a Quenched Quantum Inverted Oscillator

TL;DR

This paper develops a Schwinger-Keldysh path integral approach to analyze out-of-equilibrium quantum correlations and chaos in a quenched inverted oscillator, deriving general expressions for correlators and Lyapunov exponents.

Contribution

It introduces a novel formalism for studying time-dependent quantum correlations in inverted oscillators with quenched parameters, including chaos diagnostics.

Findings

Derived general expressions for generating functions and OTOCs.

Analyzed early, intermediate, and late time dynamics of OTOCs.

Computed quantum Lyapunov exponent indicating chaotic behavior.

Abstract

In this work, we study the time-dependent behaviour of quantum correlations of a system of an inverted oscillator governed by out-of-equilibrium dynamics using the well-known Schwinger-Keldysh formalism in presence of quantum mechanical quench. Considering a generalized structure of a time-dependent Hamiltonian for an inverted oscillator system, we use the invariant operator method to obtain its eigenstates and continuous energy eigenvalues. Using the expression for the eigenstates, we further derive the most general expression for the generating function as well as the out-of-time-ordered correlators (OTOC) for the given system using this formalism. Further, considering the time-dependent coupling and frequency of the quantum inverted oscillator characterized by quench parameters, we comment on the dynamical behaviour, specifically the early, intermediate and late time-dependent features of the OTOC for the quenched quantum inverted oscillator. Next, we study a specific case, where the system of inverted oscillator exhibits chaotic behaviour by computing the quantum Lyapunov exponent from the time-dependent behaviour of OTOC in presence of the given quench profile.