Many-Body Dynamics in Monitored Atomic Gases without Postselection Barrier

We study the properties of a monitored ensemble of atoms driven by a laser field and in the presence of collective decay. The properties of the quantum trajectories describing the atomic cloud drastically depend on the monitoring protocol and are distinct from those of the average density matrix. By varying the strength of the external drive, a measurement-induced phase transition occurs separating two phases with entanglement entropy scaling subextensively with the system size. Incidentally, the critical point coincides with the superradiance transition of the trajectory-averaged dynamics. Our setup is implementable in current light-matter interaction devices, and most notably, the monitored dynamics is free from the postselection measurement problem, even in the case of imperfect monitoring.

Figure 1

Results of QJ simulations. (a) Long-time average of the half-chain entanglement entropy $\overline{S_{N/2}}$ as a function of ${\omega }_0/\kappa$, for system sizes $N=20$, 40, 80, 160, 320, 640. Inset: the numerical derivative of the long-time averaged entanglement entropy highlights the critical point; the peak height grows logarithmically with $N$. (b) Scaling of $\overline{S_{N/2}}$ as a function of $N$ at fixed values of ${\omega }_0/\kappa$. At the critical point, the long-time averaged entanglement entropy grows logarithmically with $N$ (gray triangles). In the phase ${\omega }_0/\kappa <1$ the behavior is area law (black circles). The scaling in the region ${\omega }_0/\kappa >1$ is sublogarithmic (orange squares). Numerically, one can fit the curves as $\overline{S_{N/2}}\sim {\ln }^{\beta }N$ with a nonuniversal exponent $\beta <1$ ($\beta \sim 0.2$ at ${\omega }_0/\kappa =2$). The exponent decreases when moving away from the critical point. (c) Long-time dynamics of the entanglement entropy for several system sizes and several values of $\kappa /{\omega }_0$. The entanglement entropy grows as $\ln t$ before saturating. (d) Dependence of the saturation time on the system size $N$ (in all cases, $\tau \sim \ln N$).

Figure 2

Results of QSD simulations. (a) Long-time averaged entanglement entropy $\overline{S_{N/2}}$ as a function of ${\omega }_0/\kappa$ for various system sizes $N$. The entanglement develops a peak at the critical point ${\omega }_0=\kappa$. (b) Scaling of $\overline{f_Q}=\overline{F_Q}/N$ as a function of $N$ at the fixed values of ${\omega }_0/\kappa$. At the critical point, the long-time averaged Fisher density grows $\propto \sqrt{N}$ (gray line). In the phase ${\omega }_0<\kappa$, $\overline{f_Q}$ saturates to the constant 1 (black line). For ${\omega }_0>\kappa$, the Fisher density scales logarithmically with the system size. Therefore, the system is multipartite entangled at the critical point and in the nontrivial phase. As discussed in the text, we compare the Fisher density for (c) efficient ($\eta =0$) and (d) inefficient detectors ($\eta =0.7$). The blue-dashed line is the constant value $\overline{f_Q}=1$. While quantitative changes are present, the qualitative features of the phase diagram are preserved.

Figure 3

(a) Rescaled trajectory variance of $⟨{\widehat{J}}_z⟩$ for various system sizes and along the phase diagram for the quantum jump evolution. The trajectory histogram of $⟨{\widehat{J}}_z⟩$ is experimentally observable; cf. Ref. [78]. (b) Rescaled trajectory variance of $⟨{\widehat{J}}_x⟩$ for the quantum state diffusion. This quantity is directly obtainable from the homodyne current [2].