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

Understand

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.