Measurement-induced phase transition in interacting bosons from most likely quantum trajectory

Abstract

We propose a new theoretical method to describe the monitored dynamics of bosonic many-body systems based on the concept of the most likely trajectory. We show how such trajectory can be identified from the probability distribution of quantum trajectories, i.e. measurement readouts, and how it successfully captures the monitored dynamics beyond the average state. We prove the method to be exact in the case of Gaussian theories and then extend it to the interacting Sine-Gordon model. Although no longer exact in this framework, the method captures the dynamics through a self-consistent time-dependent harmonic approximation and reveals an entanglement phase transition in the steady state from an area-law to a logarithmic-law scaling.

Figures (7)

• Figure 1: Representation for the Free Bosons CFT for the specific case of $N=7$. The picture shows a chain of harmonic oscillators with periodic boundary conditions. Each oscillator, represented with a red sphere, has its momentum $\hat{p}_i$ measured independently at each time step from distinct measurement devices.
• Figure 2: Logarithmic Negativity for the Free Bosons CFT lattice model, the plot shows the coefficient for the fit $\log\mathcal{N}_{N/2}\sim c\log N$, which is always non-zero in this case. Inset: power-law decaying correlations in log-log scale.
• Figure 3: (a) Truncated representation for the Sine-Gordon model for the specific case of $N=7$. The picture shows a chain of harmonic oscillators with periodic boundary conditions. Each oscillator, represented with a red sphere, is subject independently to a cosine potential as shown for the zoomed window and has its momentum $\hat{p}_i$ measured at each time step from distinct measurement devices. (b) Truncated representation for the Sine-Gordon model in the SCTDHA for $N=7$. The picture shows a chain of 'dressed' harmonic oscillators in periodic boundary conditions. Each oscillator, represented with a red sphere, has its own time-dependent effective mass (the shade in the picture) and has its momentum $\hat{p}_i$ measured at each time step from distinct measurement devices. The zoomed picture shows the modified potential for a fixed time.
• Figure 4: Ratio between trajectory averaged results obtained from quantum state diffusion and $1000$ trajectories, and the results along the most-likely trajectory for (a) $\sigma_{xx}^{i,i}$, (b) $\sigma_{pp}^{i,i}$, (c) $\sigma_{xp}^{i,i}$. The parameters for the plot are $N=7$, $\omega/J=1/2$, $\alpha=2.1$. The legend is shared among the panels.
• Figure 5: Phase diagram of the model: the color code represents the fitting parameter $c$ such that $\log\mathcal{N} \sim c\,\log N$ for $\omega/J=1/2$. The plot shows the phase transition between the area-law massive region and the log-law massless region which reproduces the Free Bosons CFT results. The dashed line corresponds to the values of $\gamma/J$ and $\alpha/J$ used in Figs. \ref{['fig_SGbench']} and \ref{['fig_SG_mass_corr']}. The plot is obtained for $\omega/J=1/2$.