Open quantum systems and random matrix theory

Abstract

A simple model for open quantum systems is analyzed with Random Matrix Theory. The system is coupled to the continuum in a minimal way. In this paper the effect of opening the system on the level statistics is seen. In particular the ∆ 3 (L) statistic, the width distribution and the level spacing are examined as a function of the strength of this coupling. The emergence of a superradiant transition is observed. The level spacing and ∆ 3 (L) statistic exhibit the signatures of missed levels or intruder levels as the super-radiant state is formed. * Electronic address: mulhalld2@scranton.edu

Spectral properties of the temporal evolution of brain network structure

“…In nuclear physics systems, the energy levels always fluctuate around the mean behaviors. 32,50,51 In the brain network, this spectral fluctuation also exists and can be influenced by brain activities, such as visual stimulation. 25 The RMT was proposed initially to measure the spectral fluctuation and has found widespread applications in different real systems.…”

“…25 The RMT was proposed initially to measure the spectral fluctuation and has found widespread applications in different real systems. [21][22][23][24][25][26][27][28][29][30][31][32][33] Therefore, we used the RMT method to further explore the dynamic brain network. NNSD is the distribution of spacing between eigenvalues and only accounts for the relationship between neighboring eigenvalues.…”

“…32,50,51 According to principal component analysis (PCA), 52 eigenvalues represent the contribution of brain regions to the brain network in the brain system. Our results indicate that the shifting of eigenvalue distribution strongly reveals a changed eigenvalue structure, which demonstrates the changed contribution for brain regions.…”

“…21,25,30,32,45 To obtain more universal properties of spectral fluctuation, all eigenvalues must be placed on the same footing and work at a constant spectral density on average, which is performed by removing the spurious effects caused by variations in spectral density. 25 Thus, it is customary in RMT to unfold eigenvalues k through a transformation 33,45 …”

“…20 Recent studies have indicated that RMT is also useful for characterizing the spectral properties of an adjacency matrix corresponding to a complex network and provides important information about network structure and dynamics; for example, RMT has been widely applied in different real systems, such as price fluctuations in the stock market, [21][22][23] atmosphere, 24 human EEG data, 25 biological networks, [26][27][28] and various other model networks. [29][30][31][32][33] Compared to the complex network method, RMT has a great advantage in its ability to separate the complex system-dependent data from the random universal data, which results from the complexity of the system. Depending on the property of the spectrum, RMT is effective for studying the intrinsic properties of complex networks, such as the topological structure of the network.…”