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Higher-Order Adiabatic Elimination in Atom-Cavity Systems and Its Impact on Spin-Squeezing Generation

Stefano Giaccari, Giulia Dellea, Marco G. Genoni, Gianluca Bertaina

TL;DR

This work shows that spin squeezing generated in atom–cavity systems cannot be reliably captured by leading-order adiabatic elimination when the atom–cavity coupling is not deeply in the bad-cavity regime. By deriving a third-order effective master equation for the atomic subsystem and proposing a corresponding stochastic master equation for conditional dynamics, the authors demonstrate via numerical simulations that beyond-leading-order terms crucially alter the scaling of spin squeezing with the atom number $N$, often erasing the $N^{-2/5}$ trend predicted by second-order theory. The third-order corrections introduce a $S_z^3$ term in the Hamiltonian and a refined jump operator, yielding improved agreement with full atom–cavity dynamics and revealing a possible optimal $N$ for squeezing under realistic dissipation. These results emphasize the limits of standard reductions for metrology-oriented protocols and guide future efforts to design scalable spin-squeezing schemes under finite cavity losses and measurement backaction.

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 $N$.

Higher-Order Adiabatic Elimination in Atom-Cavity Systems and Its Impact on Spin-Squeezing Generation

TL;DR

This work shows that spin squeezing generated in atom–cavity systems cannot be reliably captured by leading-order adiabatic elimination when the atom–cavity coupling is not deeply in the bad-cavity regime. By deriving a third-order effective master equation for the atomic subsystem and proposing a corresponding stochastic master equation for conditional dynamics, the authors demonstrate via numerical simulations that beyond-leading-order terms crucially alter the scaling of spin squeezing with the atom number , often erasing the trend predicted by second-order theory. The third-order corrections introduce a term in the Hamiltonian and a refined jump operator, yielding improved agreement with full atom–cavity dynamics and revealing a possible optimal for squeezing under realistic dissipation. These results emphasize the limits of standard reductions for metrology-oriented protocols and guide future efforts to design scalable spin-squeezing schemes under finite cavity losses and measurement backaction.

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 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 .

Paper Structure

This paper contains 14 sections, 83 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Physical model and overview of our main results
  3. Adiabatic elimination in the bad-cavity limit
  4. Conditional dynamics under continuous measurement
  5. Main result: third-order effective master equations
  6. Numerical results for the spin-squeezing parameter
  7. Unconditional dynamics dominated by the OAT Hamiltonian
  8. QND conditional dynamics at zero detuning
  9. QND conditional dynamics at finite detuning
  10. Adiabatic elimination for bipartite open quantum systems
  11. The atom-cavity system
  12. Conclusions
  13. Solution to the invariance condition up to third order
  14. Third-order expansion of reduced Lindblad superoperator $\mathcal{L}_s$ for the atom-cavity system

Figures (3)

  • Figure 1: Values of the optimal spin-squeezing parameter with increasing number of atoms $N$, for coupling parameters $\epsilon=g/\kappa = 0.017$ (upper row) and $0.033$ (lower row), and driving laser detunings $d=2\delta/\kappa=0.8$ (first column), $1.0$ (second column), and $1.67$ (third column). Circles: solution of the full unconditional master equation \ref{['eq:MEoriginal']}. Squares: solution of the second order master equation from Eq. \ref{['eq:MEsecondordert']}. Diamonds: solution of the third order master equation \ref{['eq:MEtotalPositive']}. Dashed line: second-order asymptotic result, Eq. \ref{['eq:scalingbarberena']}. Horizontal dashed line: standard quantum limit.
  • Figure 2: Values of the optimal (average) spin-squeezing parameter with increasing number of atoms $N$, for the conditional dynamics described by Eq. \ref{['eq:StochasticRedMasterEq']} with coupling parameters $\epsilon = 0.017$ (panel a), and $\epsilon = 0.033$ (panel b), for driving laser detuning $d=0$, homodyne phase $\phi=0$, and efficiency $\eta=1$. Circles: solution of the full Eq. \ref{['eq:StochasticFullMasterEq']}. Diamonds: solution of the third order Eq. \ref{['eq:StochasticRedMasterEq']}. Squares: solution of Eq. \ref{['eq:StochasticRedMasterEq']} truncated to second order. Dashed line: second-order asymptotic result in the case $d=0$, Eq. \ref{['eq:scalingcaprotti']}. Horizontal dashed line: standard quantum limit.
  • Figure 3: Values of the optimal (average) spin-squeezing parameter with increasing number of atoms $N$, for the conditional dynamics with coupling parameters and driving laser detunings arrayed as in Fig. \ref{['fig:compare_unconditional']}, the optimal homodyne phase of Eq. \ref{['eq:optimalphase']}, and efficiency $\eta=1$. Circles: solution of the full Eq. \ref{['eq:StochasticFullMasterEq']}. Diamonds: solution of the third order Eq. \ref{['eq:StochasticRedMasterEq']}. Squares: solution of Eq. \ref{['eq:StochasticRedMasterEq']} truncated to second order. Dashed line: second-order asymptotic result in the case $d=0$, Eq. \ref{['eq:scalingcaprotti']}. Horizontal dashed line: standard quantum limit.