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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2401.00822 |
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| _version_ | 1866909233518215168 |
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| author | Gerbino, Federico Doussal, Pierre Le Giachetti, Guido De Luca, Andrea |
| author_facet | Gerbino, Federico Doussal, Pierre Le Giachetti, Guido De Luca, Andrea |
| contents | 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 $\{λ_α\}_{α=1}^n$ of the density matrix can be decoupled from that of the eigenvectors. Thus, stochastic equations are derived that exactly describe the dynamics of $λ$'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 $n\to\infty$. 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 $t Γ= O(n)$ one enters instead a regime in which the eigenvalues are exponentially separated $\log(λ_α/λ_β) = O(Γt/n)$, but fluctuations $\sim O(\sqrt{Γt/n})$ play an essential role. We are still able to characterize the asymptotic behaviors of entanglement entropy in this regime. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_00822 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Dyson Brownian motion model for weak measurements in chaotic quantum systems Gerbino, Federico Doussal, Pierre Le Giachetti, Guido De Luca, Andrea Statistical Mechanics 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 $\{λ_α\}_{α=1}^n$ of the density matrix can be decoupled from that of the eigenvectors. Thus, stochastic equations are derived that exactly describe the dynamics of $λ$'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 $n\to\infty$. 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 $t Γ= O(n)$ one enters instead a regime in which the eigenvalues are exponentially separated $\log(λ_α/λ_β) = O(Γt/n)$, but fluctuations $\sim O(\sqrt{Γt/n})$ play an essential role. We are still able to characterize the asymptotic behaviors of entanglement entropy in this regime. |
| title | A Dyson Brownian motion model for weak measurements in chaotic quantum systems |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2401.00822 |