A general theory is developed for the evolution of the cell order (CO) distribution in planar granular systems. Dynamic equations are constructed and solved in closed form for several examples, including systems under compression, dilation of very dense systems, and the general approach to steady state. We find that all steady states are stable and satisfy a detailed balance-like condition when the CO ≤ 6. Illustrative numerical solutions of the evolution are presented. Our theoretical results are validated against extensive simulations of a sheared system. The formalism can be readily extended to other structural characteristics, paving the way for a general theory of the structural organization of granular systems.
Topological framework for directional amplification in driven-dissipative cavity arrays
Clara C. Wanjura,
Matteo Brunelli,
Andreas Nunnenkamp
Nature Communications
11
3149
(2020)
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Directional amplification, in which signals are selectively amplified depending on their propagation direction, has attracted much attention as a key resource for applications, including quantum information processing. Recently, several physically very different directional amplifiers have been proposed and realized in the lab. In this work, we present a unifying framework based on topology to understand non-reciprocity and directional amplification in driven-dissipative cavity arrays. Specifically, we unveil a one-to-one correspondence between a non-zero topological invariant defined on the spectrum of the dynamic matrix and regimes of directional amplification, in which the end-to-end gain grows exponentially with the number of cavities. We compute analytically the scattering matrix, the gain, and the reverse gain, showing their explicit dependence on the value of the topological invariant. Parameter regimes achieving directional amplification can be elegantly obtained from a topological ‘phase diagram,’ which provides a guiding principle for the design of both phase-preserving and phase-sensitive multimode directional amplifiers.
Kontakt
ForschungsgruppeClara Wanjura
Max-Planck-Institut für die Physik des Lichts Staudtstr. 2 91058 Erlangen
