A Dyson Brownian motion model for weak measurements in chaotic quantum systems

TL;DR

This paper models monitored dynamics in chaotic quantum systems using a Dyson Brownian motion approach, deriving exact solutions for eigenvalue distributions and analyzing entanglement evolution under different measurement regimes.

Contribution

It introduces a novel Dyson Brownian motion model for weak measurements in chaotic quantum systems, providing exact solutions for eigenvalue distributions and entanglement dynamics.

Findings

Eigenvalue distribution follows inverse Marchenko Pastur law under dephasing.

Exact joint probability distribution for eigenvalues during finite-time evolution.

Characterization of entanglement entropy in different measurement regimes.

Abstract

We consider a toy model for the study of monitored dynamics in a many-body quantum systems. We study the stochastic Schrodinger equation resulting from the continuous monitoring with a rate $\Gamma$ of a random hermitian operator chosen at every time from the gaussian unitary ensemble (GUE). Due to invariance by unitary transformations, the dynamics of the eigenvalues $\{\lambda_\alpha\}_{\alpha=1}^n$ of the density matrix can be decoupled from that of the eigenvectors. Thus, stochastic equations are derived that exactly describe the dynamics of $\lambda$'s. We consider two regimes: in the presence of an extra dephasing term, which can be generated by imperfect quantum measurements, the density matrix has a stationary distribution, and we show that in the limit of large sizes the distribution of $\lambda$'s is described by an inverse Marchenko Pastur distribution. In the case of perfect measurements instead, purification eventually occurs and we focus on finite-time dynamics. In this case, remarkably, we find an exact solution for the joint probability distribution of $\lambda$'s at each time $t$ and for each size $n$. Two relevant regimes emerge: at small times $t\Gamma= O(1)$, the spectrum is in a Coulomb gas regime, with a well-defined continuous spectral distribution in the limit of $n\to\infty$. In that case, all moments of the density matrix become self-averaging and it is possible to characterize the entanglement spectrum exactly. In the limit of large times $t \Gamma = O(n)$ one enters instead a regime in which the eigenvalues are exponentially separated $\log(\lambda_\alpha/\lambda_\beta) = O(\Gamma t/n)$, but fluctuations $\sim O(\sqrt{\Gamma t/n})$ play an essential role. We are still able to characterize the asymptotic behaviors of entanglement entropy in this regime.