Higher-order adiabatic elimination in atom-cavity systems and its impact on spin-squeezing generation

Quantum Sci. Technol. 11, 035019 (2026) · DOI: 10.1088/2058-9565/ae799f · arXiv: 2506.22383

License: CC BY 4.0.

Abstract

Spin-squeezed states are metrologically useful quantum states where entanglement allows for enhanced sensing with respect to the standard quantum limit. Key challenges include the efficient preparation of spin-squeezed states and the scalability of estimation precision with the number N of probes. Recently, in the context of the generation of spin-squeezed states via coupling of three-level atoms to an optical cavity, it was shown that increasing the atom-cavity coupling can be detrimental to spin squeezing generation, an effect that is not captured by the standard second-order adiabatic cavity removal approximation. We describe adiabatic elimination techniques to derive an effective Lindblad master equation up to third order for the atomic degrees of freedom. Numerical simulations show that the spin squeezing scalability loss is correctly reproduced by the reduced open system dynamics, highlighting the role of higher-order contributions. Furthermore, we conjecture an extension beyond leading order of the adiabatic elimination technique to the case of conditional dynamics under quantum non-demolition continuous measurement and fast cavity loss, whose reliability is again confirmed by numerical simulation of the dynamics and the corresponding behavior of spin squeezing as a function of N.

Figures

16 panels with data across 4 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

• panel (a): Optimal spin-squeezing parameter $\xi^2_m$ against atom number $N$, on log-log axes, for the unconditional dynamics at $\epsilon = g/\kappa = 0.017$ and $d = 2\delta/\kappa = 0.80$: Full dynamics (circles), 2nd order (squares) and 3rd order (diamonds), with error bars. The dashed curve is the second-order asymptotic result and the horizontal dashed line the standard quantum limit (SQL). data.csv
• panel (b): Optimal spin-squeezing parameter $\xi^2_m$ against atom number $N$, on log-log axes, for the unconditional dynamics at $\epsilon = g/\kappa = 0.017$ and $d = 2\delta/\kappa = 1.00$: Full dynamics (circles), 2nd order (squares) and 3rd order (diamonds), with error bars. The dashed curve is the second-order asymptotic result and the horizontal dashed line the standard quantum limit (SQL). data.csv
• panel (c): Optimal spin-squeezing parameter $\xi^2_m$ against atom number $N$, on log-log axes, for the unconditional dynamics at $\epsilon = g/\kappa = 0.017$ and $d = 2\delta/\kappa = 1.67$: Full dynamics (circles), 2nd order (squares) and 3rd order (diamonds), with error bars. The dashed curve is the second-order asymptotic result and the horizontal dashed line the standard quantum limit (SQL). data.csv
• panel (d): Optimal spin-squeezing parameter $\xi^2_m$ against atom number $N$, on log-log axes, for the unconditional dynamics at $\epsilon = g/\kappa = 0.033$ and $d = 2\delta/\kappa = 0.80$: Full dynamics (circles), 2nd order (squares) and 3rd order (diamonds), with error bars. The dashed curve is the second-order asymptotic result and the horizontal dashed line the standard quantum limit (SQL). data.csv
• panel (e): Optimal spin-squeezing parameter $\xi^2_m$ against atom number $N$, on log-log axes, for the unconditional dynamics at $\epsilon = g/\kappa = 0.033$ and $d = 2\delta/\kappa = 1.00$: Full dynamics (circles), 2nd order (squares) and 3rd order (diamonds), with error bars. The dashed curve is the second-order asymptotic result and the horizontal dashed line the standard quantum limit (SQL). data.csv
• panel (f): Optimal spin-squeezing parameter $\xi^2_m$ against atom number $N$, on log-log axes, for the unconditional dynamics at $\epsilon = g/\kappa = 0.033$ and $d = 2\delta/\kappa = 1.67$: Full dynamics (circles), 2nd order (squares) and 3rd order (diamonds), with error bars. The dashed curve is the second-order asymptotic result and the horizontal dashed line the standard quantum limit (SQL). data.csv

Fig. 2

• panel (a): Optimal average spin-squeezing parameter $\xi^2_m$ against atom number $N$, on log-log axes, for the conditional dynamics at $\epsilon = 0.017$, $d = 0$, $\phi = 0$ and $\eta = 1$: Full dynamics (circles), 2nd order (squares) and 3rd order (diamonds), with error bars. The dashed curve is the second-order asymptotic result and the horizontal dashed line the standard quantum limit (SQL). data.csv
• panel (b): Optimal average spin-squeezing parameter $\xi^2_m$ against atom number $N$, on log-log axes, for the conditional dynamics at $\epsilon = 0.033$, $d = 0$, $\phi = 0$ and $\eta = 1$: Full dynamics (circles), 2nd order (squares) and 3rd order (diamonds), with error bars. The dashed curve is the second-order asymptotic result and the horizontal dashed line the standard quantum limit (SQL). data.csv

Fig. 3

• panel (a): Optimal average spin-squeezing parameter $\xi^2_m$ against atom number $N$, on log-log axes, for the conditional dynamics at $\epsilon = 0.017$ and $d = 0.80$, optimal homodyne phase, $\eta = 1$: Full dynamics (circles), 2nd order (squares) and 3rd order (diamonds), with error bars. Dashed curve: second-order asymptotic result for $d = 0$; horizontal dashed line: SQL. data.csv
• panel (b): Optimal average spin-squeezing parameter $\xi^2_m$ against atom number $N$, on log-log axes, for the conditional dynamics at $\epsilon = 0.017$ and $d = 1.00$, optimal homodyne phase, $\eta = 1$: Full dynamics (circles), 2nd order (squares) and 3rd order (diamonds), with error bars. Dashed curve: second-order asymptotic result for $d = 0$; horizontal dashed line: SQL. data.csv
• panel (c): Optimal average spin-squeezing parameter $\xi^2_m$ against atom number $N$, on log-log axes, for the conditional dynamics at $\epsilon = 0.017$ and $d = 1.67$, optimal homodyne phase, $\eta = 1$: Full dynamics (circles), 2nd order (squares) and 3rd order (diamonds), with error bars. Dashed curve: second-order asymptotic result for $d = 0$; horizontal dashed line: SQL. data.csv
• panel (d): Optimal average spin-squeezing parameter $\xi^2_m$ against atom number $N$, on log-log axes, for the conditional dynamics at $\epsilon = 0.033$ and $d = 0.80$, optimal homodyne phase, $\eta = 1$: Full dynamics (circles), 2nd order (squares) and 3rd order (diamonds), with error bars. Dashed curve: second-order asymptotic result for $d = 0$; horizontal dashed line: SQL. data.csv
• panel (e): Optimal average spin-squeezing parameter $\xi^2_m$ against atom number $N$, on log-log axes, for the conditional dynamics at $\epsilon = 0.033$ and $d = 1.00$, optimal homodyne phase, $\eta = 1$: Full dynamics (circles), 2nd order (squares) and 3rd order (diamonds), with error bars. Dashed curve: second-order asymptotic result for $d = 0$; horizontal dashed line: SQL. data.csv
• panel (f): Optimal average spin-squeezing parameter $\xi^2_m$ against atom number $N$, on log-log axes, for the conditional dynamics at $\epsilon = 0.033$ and $d = 1.67$, optimal homodyne phase, $\eta = 1$: Full dynamics (circles), 2nd order (squares) and 3rd order (diamonds), with error bars. Dashed curve: second-order asymptotic result for $d = 0$; horizontal dashed line: SQL. data.csv

Fig. 4

• panel (a): Spin-squeezing parameter $\xi^2$ against time $t\kappa$, on a logarithmic $\xi^2$ axis, for the unconditional dynamics with $N = 110$, $\epsilon = g/\kappa = 0.017$ and $d = 2\delta/\kappa = 1.0$: Full dynamics (solid), 2nd order (dotted) and 3rd order (dot-dashed). The horizontal dashed line is the standard quantum limit (SQL). data.csv
• panel (b): Trajectory-averaged spin-squeezing parameter $\xi^2$ against time $t\kappa$, on a logarithmic $\xi^2$ axis, for the conditional dynamics with $N = 72$, $\epsilon = 0.033$, $d = 1.67$ and $\phi = \theta_s$: Full dynamics (solid), 2nd order (dotted) and 3rd order (dot-dashed); horizontal dashed line: SQL. Error bars are the standard deviation of the mean, narrower than the lines, as in the print. data.csv

Cite

Stefano Giaccari, Giulia Dellea, Marco G Genoni, Gianluca Bertaina. Higher-order adiabatic elimination in atom-cavity systems and its impact on spin-squeezing generation. Quantum Sci. Technol. 11, 035019 (2026). https://doi.org/10.1088/2058-9565/ae799f

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