Quantum-classical correspondence in a nonlinear Gross–Pitaevski system

Abstract

We investigate the quantum-classical correspondence of the Bose–Einstein condensate (BEC) which is described by the Gross–Pitaevski equation with the periodic, both temporally and spatially, modulation of nonlinear interaction. It is found that this system is equivalent to a generalized kicked rotor (GKR) model. Interestingly, the classical dynamics of the GKR system exhibits the regular-to-chaotic transition as time evolves, and the corresponding energy diffusion is exponentially fast. For the BEC system, the energy diffusion can be predicted by the classical mapping equation of the GKR model. However, the ratio of the energy between the BEC system and the classical GKR model is not unity, even if the effective Planck constant is very small, which is contradictory to our conventional understanding of quantum-classical correspondence. We find that the BEC system is not exactly a quantum counterpart of the classical GKR system. We construct a better quantum counterpart and derive an analytical expression for the exponentially-fast diffusion of energy, which is in perfect agreement with numerical results.

Super-exponential scrambling of Out-of-time-ordered correlators

“…The norm of this state has the expression 0) , which is just the mean energy at the time t + . Our pervious investigation has shown that the mean energy exponentially increases [29,30], i.e.,…”

“…We have proved that the system in Eq. ( 1) is mathematically equivalent to a generalized kicked rotor (GKR) model [29][30][31], whose Hamiltonian takes the form…”

“…A significant feature of this system is the exponential localization of quantum states, i.e., |ψ(p)| 2 ∼ exp(−|p|/ξ) with ξ being the localization length. By simple calculation one can obtain that the quantum correlation has also the exponential decay, i.e., K n ∝ exp(−|p|/ξ) [29,30]. Then, a rough estimation of the kicking strength in the GKR model (see Eq.…”

“…Then, a rough estimation of the kicking strength in the GKR model (see Eq. ( 8)) is K(t) = n nK n (t) ∝ n ne −|n| eff /ξ ∝ ξ. Perviously, we have predicted the exponentially-fast increase of the localization length ξ ∝ e γt by means of the hybrid quantum-classical theory [29,30]. As a consequence, the kick strength exponentially grows with time, i.e., K ∝ e γt .…”

Super-exponential diffusion in nonlinear non-Hermitian systems

“…Nonlinear effects appear in a broad range of systems, for instance the Bose-Einstein condensates [25] and the nonlinear optics [26]. Up to now, a wide spectrum of diffusion processes, from power-law diffusion [27][28][29][30][31][32][33][34] to exponential diffusion [35][36][37][38] have been found in nonlinear systems (see the NL-H zone in Fig. 1).…”