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

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 $Γ$ 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 \λ_α\α=1ⁿ of the density matrix can be decoupled from that of the eigenvectors. Thus, stochastic equations are derived that exactly describe the dynamics of $λ$'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 $λ$'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 $λ$'s at each time t and for each size n. Two relevant regimes emerge: at small times $tΓ= O(1)$, the spectrum is in a Coulomb gas regime, with a well-defined continuous spectral distribution in the limit of n→∞. 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 $t Γ= O(n)$ one enters instead a regime in which the eigenvalues are exponentially separated log(λ_α/λ_β) = O(Γt/n), but fluctuations ~ O(√Γt/n) play an essential role. We are still able to characterize the asymptotic behaviors of entanglement entropy in this regime.