Scattering and perturbation theory for discrete-time dynamics

TL;DR

This paper develops a formal framework for analyzing scattering processes in discrete-time quantum systems, introducing perturbative techniques and comparing continuous and discrete models, with applications to quantum simulators like quantum walks.

Contribution

It introduces a systematic approach to scattering in discrete-time quantum systems, including perturbative methods and a comparison framework for continuous and discrete models.

Findings

Conservation of quasi-energy modulo 2π established.

Two perturbative techniques for scattering operator expansion developed.

Comparison between continuous and discretized models validated for bounded Hamiltonians.

Abstract

We present a systematic treatment of scattering processes for quantum systems whose time evolution is discrete. We define and show some general properties of the scattering operator, in particular the conservation of quasi-energy which is defined only modulo $2 \pi$. Then we develop two perturbative techniques for the power series expansion of the scattering operator, the first one analogous to the iterative solution of the Lippmann-Schwinger equation, the second one to the Dyson series of perturbative Quantum Field Theory. We use this formalism to compare the scattering amplitudes of a continuous-time model and of the corresponding discretized one. We give a rigorous assessment of the comparison for the case of bounded free Hamiltonian, as in a lattice theory with a bounded number of particles. Our framework can be applied to a wide class of quantum simulators, like quantum walks and quantum cellular automata. As a case study, we analyse the scattering properties of a one-dimensional cellular automaton with locally interacting fermions.