[Paper Review] Correspondence between non-Hermitian topology and directional amplification in the presence of disorder

Wanjura, Clara C., Brunelli, Matteo+1 more|arXiv (Cornell University)|Oct 27, 2020

Quantum Mechanics and Non-Hermitian Physics68 citations

TL;DR

This paper establishes that non-Hermitian (NH) topological directional amplification remains robust in disordered driven-dissipative cavity arrays, even under arbitrarily strong on-site disorder. By deriving analytic bounds on scattering matrix elements, it proves that exponential forward gain and exponential reverse gain suppression—hallmarks of NH topology—persist with system size, confirming the topological correspondence is preserved despite disorder.

ABSTRACT

In order for non-Hermitian (NH) topological effects to be relevant for practical applications, it is necessary to study disordered systems. In the absence of disorder, certain driven-dissipative cavity arrays with engineered non-local dissipation display directional amplification when associated with a non-trivial winding number of the NH dynamic matrix. In this work, we show analytically that the correspondence between NH topology and directional amplification holds even in the presence of disorder. When a system with non-trivial topology is tuned close to the exceptional point, perfect non-reciprocity (quantified by a vanishing reverse gain) is preserved for arbitrarily strong on-site disorder. For bounded disorder, we derive simple bounds for the probability distribution of the scattering matrix elements. These bounds show that the essential features associated with non-trivial NH topology, namely that the end-to-end forward (reverse) gain grows (is suppressed) exponentially with system size, are preserved in disordered systems. NH topology in cavity arrays is robust and can thus be exploited for practical applications.

Motivation & Objective

• . The research aims to determine whether non-Hermitian (NH) topological amplification remains stable in the presence of disorder.
• . The problem is that while NH topology is promising for applications, real systems inevitably contain disorder, and its impact on topological transport is poorly understood.
• . The objective is to extend the correspondence between NH topology and directional amplification to disordered systems.
• . The study focuses on how disorder affects the scattering matrix, particularly end-to-end forward and reverse gains.
• . It aims to prove that the exponential scaling of gain with system size—key to topological robustness—survives in disordered settings.

Proposed method

• . The authors model a 1D array of driven-dissipative cavities with engineered non-local dissipation and on-site disorder.
• . They derive the equations of motion for cavity amplitudes, incorporating disorder via complex i.i.d. random variables ξj for decay rates and cavity frequencies.
• . The system's transport is analyzed through the scattering matrix S(ω) = 1 + γM⁻¹(ω), where M(ω) is the dynamic matrix.
• . They derive analytic bounds on the probability distribution of scattering matrix elements, particularly the end-to-end gain, under the assumption of compactly supported disorder.
• . The bounds rely on decomposing the susceptibility matrix χ into disorder-free and disorder-perturbed parts, using series expansions and matrix inversion techniques.
• . They distinguish between rate disorder (real part of ξj) and frequency disorder (imaginary part), showing their different impacts on gain distribution and phase.

Experimental results

Research questions

• RQ1 . Does the correspondence between non-trivial NH topology and directional amplification survive in the presence of strong on-site disorder?
• RQ2 . Can the exponential scaling of forward gain and suppression of reverse gain be preserved when disorder is introduced?
• RQ3 . How do rate disorder and frequency disorder differentially affect the distribution of scattering matrix elements?
• RQ4 . Are the topological features of directional amplification robust in the thermodynamic limit (N → ∞)?
• RQ5 . What are the analytic bounds on the probability distribution of the scattering matrix elements under bounded disorder?

Key findings

• . The correspondence between non-Hermitian topology and directional amplification is preserved even under arbitrarily strong on-site disorder.
• . Perfect non-reciprocity, quantified by vanishing reverse gain, is maintained for any disorder type and strength when the system is tuned near the exceptional point.
• . The bounds on the scattering matrix elements are analytically derived and depend only on the compact support of the disorder.
• . The exponential scaling of forward gain with system size is preserved, as the lower bound on gain grows exponentially with N.
• . The reverse gain is exponentially suppressed with system size, maintaining directional amplification in disordered systems.
• . The difference between rate and frequency disorder is geometrically interpreted: rate disorder affects gain magnitude with minimal phase shift, while frequency disorder affects phase more strongly but keeps gain magnitude nearly unchanged.