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We investigate, both analytically and numerically, the dynamics of quantumchaos and quantum scrambling under many-body interaction effects via anon-Hermitian Gross-Pitaevski map model, incorporating a periodicallymodulated, complex strength of nonlinear interaction as delta kicks. Weestablish a theoretical equivalence between a particular out-of-time orderedcorrelator and the fidelity of a quantum state, which is proportional to theLoschmidt echo. Both exhibit a double exponential growth in relation to time,indicating the emergence of superexponential instability andsuperexponentially-fast scrambling. The underlying mechanism is rooted in thesuperexponentially-fast diffusion of energy, arising from the interplay betweenthe positive feedback of nonlinear interaction and the growth of the amplitudeof quantum state due to non-Hermiticity. Our findings suggest a kind of fastestdivergence of two nearby quantum states, which has implication in informationscrambling and brachistochrone evolution of quantum states.

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