Out-of-time-ordered correlators in Bose-Hubbard systems

Out-of-time-ordered correlators (OTOCs) are quantum objects defined as the square modulus of the commutator of a time-evolved and an unevolved operator. They can be used as a probe for chaos in quantum system. We study them in the context of 1D Bose-Hubbard systems, which consist of interacting bosons in a 1D lattice. We will first look at OTOCs from a quantum perspective, then focus on their classical analogues. The goal is to be able to distinguish which features of the dynamics of the system is quantum, and which can be explained from the underlying classical mechanics. To this end, we derived a classical OTOC following two approaches: a direct classical limit using Wigner-Moyal formalism, valid only in the short-time regime, and a more sophisticated way using the semiclassical van Vleck-Gutzwiller propagator plus the diagonal approximation [1]. The first one allows to recover the exponential growth, but fails to saturate. The second one allows in addition to obtain a finite value in the long time limit, thus predicting a saturation from a non-quantum origin.
[1] 1.Rammensee, J., Urbina, J.-D. & Richter, K. Many-Body Quantum Interference and the Saturation of Out-of-Time-Order Correlators. Phys. Rev. Lett. 121, 124101 (2018)