Non-Markovian continuous-time quantum walks on lattices with dynamical noise (1510.08652v2)

Abstract: We address the dynamics of continuous-time quantum walks on one-dimensional disordered lattices inducing dynamical noise in the system. Noise is described as time-dependent fluctuations of the tunneling amplitudes between adjacent sites, and attention is focused on non-Gaussian telegraph noise, going beyond the usual assumption of fast Gaussian noise. We observe the emergence of two different dynamical behaviors for the walker, corresponding to two opposite noise regimes: slow noise (i.e. strong coupling with the environment) confines the walker into few lattice nodes, while fast noise (weak coupling) induces a transition between quantum and classical diffusion over the lattice. A phase transition between the two dynamical regimes may be observed by tuning the ratio between the autocorrelation time of the noise and the coupling between the walker and the external environment generating the noise. We also address the non-Markovianity of the quantum map by assessing its memory effects, as well as evaluating the information backflow to the system. Our results suggest that the non-Markovian character of the evolution is linked to the dynamical behavior in the slow noise regime, and that fast noise induces a Markovian dynamics for the walker.

• The paper explores continuous-time quantum walks (CTQWs) on disordered lattices with dynamical noise, identifying how different noise regimes (slow/fast) affect walker dynamics and diffusion.
• The study confirms that non-Markovian memory effects are prominent in the slow noise regime, using trace distance and dynamical map analysis.
• Fast noise leads to a transition from quantum ballistic to classical diffusive behavior, while slow noise causes Anderson-like localization on the lattice.

Non-Markovian Continuous-Time Quantum Walks on Lattices with Dynamical Noise

The paper of quantum walks (QWs), as the quantum equivalent of classical random walks, provides significant insight into quantum algorithms and their potential applications across various fields, such as quantum computing and biological process modeling. This paper explores the dynamics of continuous-time quantum walks (CTQWs) influenced by non-Markovian noise on one-dimensional disordered lattices. Specifically, it addresses the interplay between quantum walks and environmental noise described by time-dependent fluctuations in tunneling amplitudes, with a focus on the impact of non-Gaussian telegraph noise.

Key Findings

1. Noise Regimes and Walker Dynamics:
   • The paper identifies two primary noise regimes: slow noise, which reflects strong coupling with the environment, confines the walker to a few lattice nodes. Conversely, fast noise indicative of weak coupling facilitates a transition between quantum and classical diffusion.
   • The transition between these regimes hinges on the ratio of the noise's autocorrelation time and the walker's environmental coupling.
2. Non-Markovianity and Memory Effects:
   • The analysis confirms that the non-Markovian character is predominantly associated with the slow noise regime, exhibiting significant memory effects and information backflow to the QW system.
   • The research leverages the concepts of trace distance and the violation of dynamical map compositions to establish the non-Markovian nature of the dynamics.
3. Quantum to Classical Transition:
   • Under fast noise conditions, the quantum ballistic propagation of the wave packet transitions to classical diffusive behavior. This is evidenced by the particle's probability distribution gravitating towards a Gaussian shape over time.
   • Slow noise results in Anderson-like localization, preventing the walker from exhibiting extensive diffusion over the lattice.

Implications and Future Directions

The findings have theoretical and practical implications for quantum information processing, particularly in reservoir engineering and noise management. By providing a more nuanced understanding of noise-induced transitions in QWs, the research suggests pathways for controlling quantum systems and enhancing computational tasks. The outcomes also underscore the role of non-Markovian effects in maintaining quantum coherence over longer scales, which could be pivotal for realizing scalable quantum technologies.

This paper's methodology, involving stochastic modeling and numerical analyses, establishes a foundation for further inquiries into more complex lattice configurations and noise models, which can simulate realistic conditions in quantum information systems. Future work could explore multi-dimensional lattices and the interplay between different noise types to further refine quantum walk models and optimize their application in larger, more complex quantum networks.