One-Dimensional Nonlinear Quantum Walks

We explore a continuous-time quantum walk starting at a single vertex on the discrete path and cycle with a cubic nonlinearity. Such nonlinearities arise in Bose-Einstein condensates described by the Gross-Pitaevskii equation or by nonlinear optical waveguide arrays. We analytically prove that the nonlinear quantum walk can be trapped to arbitrary fidelity depending on the coefficient of the nonlinear term. This contrasts with linear quantum walks, which are known for spreading quickly in one dimension. We propose that this trapping can be used for timing in quantum state transfer, where a qubit is held at a node until it is ready to be transferred, and it can also be held again at the receiving node. This scheme can also be interpreted as a form of quantum memory, with the trap and transfer corresponding to the storage and release of quantum information.