Parameter estimation with one- and two-time measurements on the emission field of the boundary time crystal

Abstract

Many-body quantum systems can exhibit collective effects that enhance the sensitivity of parameter estimation protocols. An example is provided by resonantly driven two-level atoms subject to collective dissipation, which can display a transition between a stationary phase and a time-crystal one. Previous work has shown that the light emitted in the time-crystal phase can be harnessed for parameter estimation using continuous monitoring protocols, such as photon counting or homodyne detection, which under ideal conditions yield a quadratic enhancement of sensitivity with the number of particles. In this work, we explore what is the minimal information about the emission field that needs to be accessed in order to resolve collective effects and exploit them for parameter estimation. We show that, for short probing times, a single-time measurement of the emission field already captures the collective behavior emerging at the nonequilibrium transition. In contrast, within the time-crystal phase, exploiting collective effects requires at least two-time measurements. To this end, we introduce a family of correlated intensity measurements that extract the relevant information and can be implemented using an interferometric setup. While the ultimate sensitivity bound remains size independent, as recently established within the framework of noisy quantum metrology, our analysis shows that these protocols utilize collective effects to yield a transient increase in sensitivity with particle number.

Figures (12)

• Figure 1: Sketch of the system and measurement protocol. We consider an ensemble of $N$ two-level atoms, emitting collectively and driven at resonance with Rabi frequency $\omega$. We are interested in the estimation of small variations of $\omega$ by analyzing the light that is emitted collectively. The two-time measurement protocol could be implemented through a Mach-Zehnder interferometer. The collective emission of the system is input into arm '0', while vacuum is input into arm '1'. Photodetectors, are placed at the output arms '4' and '5'. The difference in path length $l_1-l_2$ is chosen such that we can probe the field emitted at two times of interest.
• Figure 2: Sketch of the discrete time representation of the input-output field. The input-output field is discretized in time bins of length $\Delta t$ such that $t_n=n\Delta t$ with $n=1,2,\dots$ Each bin is represented by an independent bosonic mode (time-bin modes), that interacts with the system for a time window $\Delta t$ and with strength $\sqrt{\Gamma/\Delta t}$. In the sketch the system has already interacted with the first $n$ time-bin modes. Hence, the output field is formed by the time-bin modes $[1,n]$, which are generally in a nonseparable state which is also correlated with the system itself (here pictorially represented as a blue shadow). The input field is given by bins $[n+1,\infty)$, which are in a product vacuum state. The reduced state of the time-bin mode $n_2$, which has already interacted with the system, is denoted by $\hat{\mu}_{[n_2]}$. The reduced joint state of time-bin modes $n_1$ and $n_2$ ($n_2>n_1$), that have already interacted with the system, is denoted by $\hat{\mu}_{[n_1,n_2]}$. In this work we focus on parameter estimation based on the information contained in these reduced states of the emitted light field.
• Figure 3: QFI in the small-$\Delta t$ regime. (a) $\mathcal{F}_{[n_1]}(\omega,\Delta t)/\Delta t$ for the one time-bin reduced state (in the stationary state, $n_1\gg1$). (b) $\mathcal{F}_{[n_1,n_2]}(\omega,\Delta t)/\Delta t$ for the two time-bin reduced state optimized over the time difference between time bins. In both panels $\omega=2\omega_\mathrm{c}$. The dashed horizontal lines are a guide to the eye. The results of this figure indicate that for sufficiently small $\Delta t$ the QFI scales linearly with the interaction time $\Delta t$.
• Figure 4: QFI per unit of time for one time-bin mode. (a) $\mathcal{F}_{[n_1]}(\omega)$ varying $\omega/\omega_\mathrm{c}$ and $N$ in the long-time limit, $n_1\gg 1$. (b) Scaling of $\mathcal{F}_{[n_1]}(\omega)$ with $N$ for $\omega/\omega_\mathrm{c}=0.5$ (blue circles), $\omega/\omega_\mathrm{c}=1$ (orange squares) and $\omega/\omega_\mathrm{c}=2$ (green triangles). The dashed lines correspond to a fit $N^\alpha$ of the largest $N$ points, with exponents $\alpha=(0.01,0.93,-0.12)$ for $\omega/\omega_\mathrm{c}=(0.5,1,2)$, respectively. Black triangles correspond to $\mathcal{F}_{\mathrm{SE}}(\omega)$ for $\omega/\omega_\mathrm{c}=1$ obtained in Ref. Cabot2024. In this figure we have fixed $\Gamma\Delta t=10^{-5}$.
• Figure 5: QFI per unit of time for the two time-bin modes state. (a) Green solid line: $\mathcal{F}_{[n_1,n_2]}(\omega)/2$ for $\omega=\omega_\mathrm{c}$, $N=50$, and varying the time between the modes $\tau=(n_2-n_1-1)\Delta t$. Black dashed line: $\mathcal{F}_{[n_1]}(\omega)$ for the same case. (b) $\mathcal{F}_{[n_1,n_2]}(\omega)/2$ for $\omega=\omega_\mathrm{c}$ and varying $N$ for two different cases: $n_2=n_1+1$ (orange triangles), and for the optimal $\tau=\tau^*$ (blue squares) when the QFI is maximal. Black triangles correspond to $\mathcal{F}_\mathrm{SE}(\omega)$ for the same parameter values Cabot2024. Dashed lines correspond to a fit $\propto N^\alpha$ to the largest system sizes. In panel (a) and (b) we have fixed $\Gamma\Delta t=10^{-5}$. (c) and (d) Same quantities but for the time-crystal phase $\omega=2\omega_\mathrm{c}$. In this case, in panel (c) various $N$ are chosen. In panel (c) and (d) we have fixed $\Gamma\Delta t=2.5\cdot 10^{-5}$.