Classical limit of measurement-induced transition in many-body chaos in integrable and non-integrable oscillator chains

Abstract: We discuss the dynamics of integrable and non-integrable chains of coupled oscillators under continuous weak position measurements in the semiclassical limit. We show that, in this limit, the dynamics is described by a standard stochastic Langevin equation, and a measurement-induced transition appears as a noise- and dissipation-induced chaotic-to-non-chaotic transition akin to stochastic synchronization. In the non-integrable chain of anharmonically coupled oscillators, we show that the temporal growth and the ballistic light-cone spread of a classical out-of-time correlator characterized by the Lyapunov exponent and the butterfly velocity, are halted above a noise or below an interaction strength. The Lyapunov exponent and the butterfly velocity both act like order parameter, vanishing in the non-chaotic phase. In addition, the butterfly velocity exhibits a critical finite size scaling. For the integrable model we consider the classical Toda chain, and show that the Lyapunov exponent changes non-monotonically with the noise strength, vanishing at the zero noise limit and above a critical noise, with a maximum at an intermediate noise strength. The butterfly velocity in the Toda chain shows a singular behaviour approaching the integrable limit of zero noise strength.