Quantum walk coherences on a dynamical percolation graph

Abstract

Coherent evolution governs the behaviour of all quantum systems, but in nature it is often subjected to influence of a classical environment. For analysing quantum transport phenomena quantum walks emerge as suitable model systems. In particular, quantum walks on percolation structures constitute an attractive platform for studying open system dynamics of random media. Here, we present an implementation of quantum walks differing from the previous experiments by achieving dynamical control of the underlying graph structure. We demonstrate the evolution of an optical time-multiplexed quantum walk over six double steps, revealing the intricate interplay between the internal and external degrees of freedom. The observation of clear non-Markovian signatures in the coin space testifies the high coherence of the implementation and the extraordinary degree of control of all system parameters. Our work is the proof-of-principle experiment of a quantum walk on a dynamical percolation graph, paving the way towards complex simulation of quantum transport in random media.

Figure 1

Implementation Scheme: (a) One pattern of a dynamically changing graph with 3 sites used in the experiment (shown explicitly for 3 steps). The full set of patterns used in the experiment is denoted by for n steps. The input state (blue arrow) is evolved (red arrow) and measured tomographically at every step. (b) Implementation scheme of the example, the and operators are represented by filled and hollow diamonds, respectively. (c) Setup scheme of the time-multiplexed PQW. According to the implementation scheme the walker always alternates between paths A and B. The colour coding is used to mark corresponding entities in both panels. We average over all patterns to obtain the open system’s dynamics.

Figure 2

Spatial Distributions: Measured spatial distributions versus the step number for 3 example graph patterns (shown on the upper left panel) from . Note that the height of a bar is unchanged from one step to the next if the site is disconnected. The high similarity (on average 95.6%) between the empirically observed probabilities and those from the ideal process makes a graphical comparison unnecessary.

Figure 3

Coherence Measurements: (a) Values of the Stokes parameters of the reduced state of the coin: observed (solid red lines), ideal model (dashed lines), and realistic model (blue, solid lines). (b) Hilbert–Schmidt distance between and , the maximally mixed, asymptotic state: observed data (red), ideal model (turquoise), and realistic model (blue). The insets show the experimental density matrix for the three chosen steps. Statistical errors are smaller than the symbol size. The depicted error bars are calculated using a numerical simulation of all relevant systematic errors and are discussed in detail in the Methods section.