Analysis of spin-squeezing generation in cavity-coupled atomic ensembles with continuous measurements

Quantum Sci. Technol. 9, 035032 (2024) · DOI: 10.1088/2058-9565/ad4584

License: CC BY 4.0.

Abstract

We analyze the generation of spin-squeezed states via coupling of three-level atoms to an optical cavity and continuous quantum measurement of the transmitted cavity field in order to monitor the evolution of the atomic ensemble. Using analytical treatment and microscopic simulations of the dynamics, we show that one can achieve significant spin squeezing, favorably scaling with the number of atoms N. However, contrary to some previous literature, we clarify that it is not possible to obtain Heisenberg scaling without the continuous feedback that is proposed in optimal approaches. In fact, in the adiabatic cavity removal approximation and large N limit, we find the scaling behavior N − 2 / 3 for spin squeezing and N − 1 / 3 for the corresponding protocol duration. These results can be obtained only by considering the curvature of the Bloch sphere, since linearizing the collective spin operators tangentially to its equator yields inaccurate predictions. With full simulations, we characterize how spin-squeezing generation depends on the system parameters and departs from the bad cavity regime, by gradually mixing with cavity-filling dynamics until metrological advantage is lost. Finally, we discuss the relevance of this spin-squeezing protocol to state-of-the-art optical clocks.

Figures

13 panels with data across 7 figures. Each panel page shows the plot, its columns and its files; each data.csv begins with a header naming the paper, the panel, the source, the license and the provenance route.

Fig. 1

Illustrative figure, no extractable data. Shown in the paper PDF.

Fig. 2

Illustrative figure, no extractable data. Shown in the paper PDF.

Fig. 3

• panel (1): Scaled moments against $\tilde{\kappa} t$ on log-log axes, cavity-removal simulation with $N = 160$ and $\eta = 1$: for each of 100 trajectories $2\Delta^2 J_x/J$, $2\Delta^2 J_y/J$, $2\Delta^2 J_z/J$, the three scaled covariances and the contrast $\mathcal{C}$, with the average E$[\xi^2]$ (dark solid) and the analytic curves (dashed). data.csv

Fig. 4

• panel (1): $\xi^2(t=t_m)$ on a log axis against $|\langle \hat{J}_z(t=t_m)\rangle_c / J|$ for 100 cavity-removal trajectories each at $N = 400$, 200, 100, 50 and 20, each with its analytic curve (gray dashed). Dotted vertical lines mark the initial standard deviation of $\hat{J}_z$, dotted horizontal lines the optimal average $\xi^2_m$. data.csv

Fig. 5

• panel (a): Photon number $\langle n\rangle_c/n_0$ against $t\,\Delta$ in the bad-cavity regime, $\kappa = 0.4\Delta$: 400 trajectories (thin gray) and their average E$[n]/n_0$ (red dashed) with its band. The green dash-dotted line is the stationary value $n_0$. data.csv
• panel (b): Spin $\langle\hat{J}_z\rangle_c/J$ against $t\,\Delta$ in the bad-cavity regime, $\kappa = 0.4\Delta$: 400 trajectories (thin gray) and their average E$[J_z]/J$ (red dashed) with its band. Green dash-dotted lines mark the standard deviation of $\hat{J}_z$ in the initial coherent spin state. data.csv
• panel (c): Squeezing $\xi^2$ against $t\,\Delta$ in the bad-cavity regime, $\kappa = 0.4\Delta$: 400 trajectories (thin gray), their average E$[\xi^2(t)]$ (red dashed) with its statistical-uncertainty band, the SQL (black dashed), the cavity-removal average (purple dotted), the analytic curve (green solid) and $\xi^2_F$ (blue dash-dotted), the last three shifted by the cavity-filling time. data.csv
• panel (d): Photon number $\langle n\rangle_c/n_0$ against $t\,\Delta$ out of the bad-cavity regime, $\kappa = 0.04\Delta$: 400 trajectories (thin gray) and their average E$[n]/n_0$ (red dashed) with its band. The green dash-dotted line is the stationary value $n_0$. data.csv
• panel (e): Spin $\langle\hat{J}_z\rangle_c/J$ against $t\,\Delta$ out of the bad-cavity regime, $\kappa = 0.04\Delta$: 400 trajectories (thin gray) and their average E$[J_z]/J$ (red dashed) with its band. Green dash-dotted lines mark the standard deviation of $\hat{J}_z$ in the initial coherent spin state. data.csv
• panel (f): Squeezing $\xi^2$ against $t\,\Delta$ out of the bad-cavity regime, $\kappa = 0.04\Delta$: 400 trajectories (thin gray), their average E$[\xi^2(t)]$ (red dashed) with its statistical-uncertainty band, the SQL (black dashed), the cavity-removal average (purple dotted), the analytic curve (green solid) and $\xi^2_F$ (blue dash-dotted), the last three shifted by the cavity-filling time. data.csv

Fig. 6

• panel (1): Optimal squeezing $\xi^2_m$ against $g^2/\kappa\Delta$ on log-log axes at $N = 45$ for five $(\kappa, \varepsilon)$ combinations, with error bars; empty symbols are outside the bad-cavity regime. The vertical dashed line is the bad-cavity boundary, the dotted line the cavity-removal result and the horizontal dashed line the SQL. data.csv

Fig. 7

• panel (1): Optimal squeezing $\xi_m^2$ against atom number $N$ on log-log axes: cavity-removal simulations (red diamonds) and full simulations for four $(\kappa, \varepsilon)$ combinations, empty symbols outside the bad-cavity regime, with error bars. The dash-dotted line is $(3/2)/N^{2/3}$, the dotted line $e/N$ and the dashed line the SQL. data.csv

Fig. 8

• panel (1): Scaled optimal time $\tilde{\kappa}\,t_m$ against atom number $N$ on log-log axes: cavity-removal simulations (red diamonds) and the full simulations for four $(\kappa, \varepsilon)$ combinations, empty symbols outside the bad-cavity regime, with error bars, and the $1/N^{1/3}$ scaling (dash-dotted). The main axes of the printed figure. data.csv
• panel (1-inset): $\tilde{\kappa}(t_m - 2c/\kappa)$ against atom number $N$ on log-log axes: cavity-removal simulations (red diamonds) and the full simulations for four $(\kappa, \varepsilon)$ combinations, empty symbols outside the bad-cavity regime, with error bars, and the $1/N^{1/3}$ scaling (dash-dotted). The inset of the printed figure, hosted as its own panel. data.csv

Fig. 9

• panel (1): Scaled optimal time $\tilde{\kappa}\,t_m\,N^{\beta}$ against $\tilde{\kappa}/\kappa$ on log-log axes for the full dynamics, $N = 45$ (diamonds) and $N = 20$ (crosses) at several $(\kappa, \varepsilon)$ combinations, empty symbols outside the bad-cavity regime, with error bars. The dotted line is $t_m = b/(\tilde{\kappa}N^{\beta}) + 2c/\kappa$. data.csv

Cite

A Caprotti, M Barbiero, M G Tarallo, M G Genoni, G Bertaina. Analysis of spin-squeezing generation in cavity-coupled atomic ensembles with continuous measurements. Quantum Sci. Technol. 9, 035032 (2024). https://doi.org/10.1088/2058-9565/ad4584

When you use hosted data, cite the original paper and give the panel's URL so a reader can find the exact values you used.

All papers · About · Statistics · Submit data · API