Super-exponential behaviors of out-of-time ordered correlators and Loschmidt echo in a non-Hermitian interacting system

Abstract

We investigate the out-of-time ordered correlators and Loschmidt echo in a non-Hermitian interacting system governed by a Gross-Pitaevskii map model, which incorporates a periodically modulated complex strength of the nonlinear interaction as delta kicks. We uncover that the time evolutions of the out-of-time ordered correlators follow that of the Loschmidt echo in certain situations. In particular, we find that both of them can exhibit a super-exponential growth with time, indicating the emergence of super-exponential scrambling and instability. Interestingly, after a proper scaling scheme, we find that all the super-exponential behaviors approximately collapse on a scaling-law curve that is independent on the non-Hermitian parameter as well as the effective Planck constant. The underlying mechanism is rooted in the super-exponentially fast diffusion of energy as well as the norm of quantum states. Our findings suggest a kind of fastest divergence of two nearby quantum states, which has implication in information scrambling.