Measurement induced faster symmetry restoration in quantum trajectories

TL;DR

We study measurement-induced global $U(1)$ symmetry restoration in a 1D spin-1/2 chain with $U(1)$-preserving Hamiltonian $H_S$ under continuous monitoring of the total number operator $\hat{N}$ (global) or local densities $\hat{n}_i$, using quantum jump (QJ) and quantum state diffusion (QSD) unravelings. For global monitoring, the symmetry-restoration timescale is governed by the smallest separation between number sectors in the initial superposition, with trajectories restoring faster when sectors are distant; the evolution of $|c_n^t|^2$ is driven by terms like $n^2-m^2$ (QJ) or $(n-m)^2$ (QSD), implying universality across protocols and extending to general POVMs in the weak-coupling limit. Under local monitoring, symmetry restoration depends on the overlap of local density profiles between sectors, enabling certain nearby-sector states to relax faster than under global monitoring, and in some configurations surpassing global rates. The ensemble-averaged GKSL dynamics reproduce the same overall relaxation pattern, with coherences decaying as $-(\gamma/2)(n-m)^2$ for global dephasing and with local dephasing showing analogous trends, highlighting a practical route to tailor measurement schemes to speed up relaxation through back-action.

Abstract

Continuous measurement of quantum systems provides a standard route to quantum trajectories through the successive acquisition of information which further results in measurement back-action. In this work, we harness this back-action as a resource for global $U(1)$ symmetry restoration where continuous measurement is combined with a $U(1)$-preserving unitary evolution. Starting from a $U(1)$ symmetry-broken initial state, we simulate quantum trajectories generated by continuous measurements of both global and local observables. We show that under global monitoring, states containing superpositions of distant charge sectors restore symmetry faster than those involving nearby sectors. We establish the universality of this behavior across different measurement protocols. Finally, we demonstrate that local monitoring can further accelerate symmetry restoration for certain states that relax slowly under global monitoring.

Figures (8)

• Figure 1: Schematic of the setup: one dimensional spin-1/2 lattice of length $L$, represented by XX Hamiltonian respecting $U(1)$ symmetry, is subjected to continuous monitoring of the (a) global operator $\hat{N}=\hat{S}_z+L/2$, which describes the total number of up spins in the lattice, and (b) local number operator/density $\hat{n}_i=\hat{S}_z^i+1/2$ at every lattice site. Starting from a broken $U(1)$ state, under such global or local continuous monitoring, $U(1)$ symmetry can be dynamically restored. We have schematically represented our findings in (c) & (d) for global and local monitoring, respectively. Under global monitoring, a symmetry broken state that contains distant number sectors restores symmetry faster than a state involving nearby number sectors. However in presence of local monitoring, the relaxation dynamics depends on the density profile differences across different number sectors. A state containing number sectors that have higher density difference relaxes faster than a state involving sectors with smaller density difference.
• Figure 2: Plot for the quantum jump (QJ) protocol: (a) The dynamics of the total number of up spins in the lattice i.e, $\langle \hat{N}\rangle=\langle\hat{S}_z\rangle+L/2$ is plotted for $100$ different trajectories for both initial states $|\Psi_0^{(1)}\rangle$ (green solid) and $|\Psi_0^{(2)}\rangle$ (brown solid), as mentioned in the main text. Here $\langle \hat{N}\rangle_{\rm ss}$ is the number of up spins in the lattice after the symmetry restoration which is either $n$ or $L-n$ in a trajectory. (b) The dynamics of fluctuation in $\hat{N}$ is plotted for the same initial states. The initial state $|\Psi_0^{(1)}\rangle$, which has larger initial fluctuation shows faster symmetry restoration than the state $|\Psi_0^{(2)}\rangle$ which has smaller number fluctuation. The inset in (b) represents the dynamics of local number operator $\langle \hat{n}_i\rangle= \hat{S}_z^i + 1/2$ at site $i=3$ which shows unitary dynamics with oscillations after the symmetry restoration. For the numerics, we have considered $L=5$, $\gamma=0.1$, $dt=0.01$. The unitary evolution is governed by the XX Hamiltonian with hopping strength $J=1$. The central results for $\langle \hat{N}\rangle$ and its fluctuations are however independent of the specific form of the lattice Hamiltonian.
• Figure 3: Plot for the quantum state diffusion (QSD) protocol: (a) The dynamics of $\langle \hat{N}\rangle=\langle\hat{S}_z\rangle+L/2$ is plotted for $100$ different trajectories for the same two initial states as Fig. \ref{['fig:QJ']}. (b) The fluctuation in $\hat{N}$ is plotted with time $t$ for those initial states. The initial state $|\Psi_0^{(1)}\rangle$, which has larger initial fluctuation shows faster symmetry restoration than the state $|\Psi_0^{(2)}\rangle$ which has smaller number fluctuation. The inset in (b) represents the behaviour of local magnetization $\langle \hat{n}_i\rangle=\hat{S}_z^i+1/2$ at site $i=3$ with time which shows unitary dynamics with oscillations after the symmetry restoration. The other parameters are the same as in Fig. \ref{['fig:QJ']}.
• Figure 4: Plot of total number fluctuations under local monitoring of operator $\hat{n}_i$ under QSD protocol [Eq. \ref{['eq:QSD_local_n']}]. The parameters used here: $L=5$, $\gamma=0.1$, $dt=0.01$, unitary part through XX Hamiltonian with $J=0.001$. We have plotted $100$ different trajectories. (a) Number fluctuation is plotted for two different initial states $|\Phi_0^{(1)}\rangle$ (main figure) and $|\Phi_0^{(2)}\rangle$ (inset). $|\Phi_0^{(1)}\rangle$ is restoring the symmetry faster than $|\Phi_0^{(2)}\rangle$. (b) We have compared the symmetry restoration in local (green) and global (brown) monitoring for initial states $|\Phi_0^{(1)}\rangle$ (top panel) and $|\Phi_0^{(2)}\rangle$ (bottom panel). The top panel shows that, for the initial state $|\Phi_0^{(1)}\rangle$, local monitoring (green) leads to a faster restoration of symmetry than global monitoring (brown). The bottom panel shows that, for the initial state $|\Phi_0^{(2)}\rangle$, both local (green) and global (brown) monitoring restores symmetry in equivalent timescales.
• Figure 5: Continuous monitoring of the total magnetization operator $\hat{S}_z$ under (a) quantum jump (QJ) process and (b) quantum state diffusion (QSD) process. The evolution is initialized from a non-$U(1)$ preserving initial state of the form $|\Psi_S\rangle=(|n-L/2\rangle+|-n+L/2\rangle)/\sqrt{2}$ where $|n-L/2\rangle$ corresponds to an eigenstate of the operator $\hat{S}_z$ with eigenvalue $(n-L/2)$ which contains $n$ number of up spins and $L-n$ number of down spins. Under continuous measurements combined with unitary evolution by $U(1)$ conserving $H_S$ (XX Hamiltonian with $J=1$ and $\Delta=0.1$), the quantum jump process is inefficient to restore the symmetry in (a). However, under QSD in (b), the $U(1)$ symmetry is restored and a seperation of time-scale is observed. The other parameters are $L=5$, $\gamma=0.1$, $dt=0.01$.