Collective Dissipation and Parameter Sensitivity in Trapped Ions Coupled to a Common Thermal Reservoir

TL;DR

This work analyzes two trapped ions coupled to a common thermal reservoir, deriving a cross-damped Langevin framework that reveals collective decay channels and a decoherence-free subspace when cross-damping matches local damping. By computing the classical Fisher information for phonon-number measurements, it identifies regimes where reservoir correlations boost parameter estimability and sustain information about initial states, bridging metrology with dissipative state engineering. The study also shows that reservoir-mediated correlations can generate and stabilize Gaussian entanglement, with the strongest effects near the decoherence-free condition, providing a unified picture of how dissipation can serve as a tunable resource in trapped-ion platforms.

Abstract

We investigate the dynamics of two trapped ions interacting with a common thermal reservoir, focusing on how cross-correlated dissipation influences heating, steady-state behavior, and parameter sensitivity. Starting from a microscopic system--reservoir model, we derive the corresponding Heisenberg--Langevin equations and show that reservoir-induced correlations generate collective decay channels and, when the cross-damping rate matches the local damping, a decoherence-free normal mode that preserves memory of the initial excitations. Using the Fisher information associated with motional population measurements, we identify the parameter regimes in which cross-damping enhances the estimability of both system and reservoir properties. For nonclassical initial states, we also show that reservoir-mediated correlations can generate or maintain entanglement, with the strongest effects occurring near the decoherence-free condition.

Figures (9)

• Figure 1: (Color online) Time evolution of the mean phonon number of ion 1 for several values of the normalized cross-damping rate $\gamma_{12}/\gamma$. The curves, from top to bottom, correspond to $\gamma_{12}/\gamma = 0.00, 0.50, 0.80, 1.00$. The ions are initially prepared in thermal state with $\bar{n}_1=0.35$ and $\bar{n}_2=2.3$. The black dashed line marks the thermal occupation $\bar{N}$ of the reservoir for $\hbar \omega_{0}/k_{B}T = 0.10$, and the black dotted line marks the steady-state population in the fully correlated limit $\gamma_{12}=\gamma$. Panels (a) and (b) correspond to $\gamma/\Omega = 0.05$ and $\gamma/\Omega = 0.10$, respectively. For $\gamma_{12}<\gamma$, both ions relax exponentially to the thermal value $\bar{N}$. When the critical condition $\gamma_{12}=\gamma$ is reached, a decoherence-free normal mode emerges and part of the initial excitation remains trapped, leading to the modified steady-state value given in Eq. \ref{['eq:steadyState']}.
• Figure 2: (Color online) Fisher information element $F_{11}$, quantifying the sensitivity of the phonon-number distribution to the initial occupation $n_1(0)$, for several values of the normalized cross-damping rate $\gamma_{12}/\gamma$ with the initial conditions $\bar{n}_{1}=0.35,\ \bar{n}_{2} = 2.3,\ \hbar\omega_{0}/k_{B}T = 1.0,\ \gamma/\Omega = 0.02$ and $\Omega = 3.1\pi k Hz$. The curves, from bottom to top, correspond to $\gamma_{12}/\gamma = 0.00, 0.50, 0.80, 1.00$. For weakly correlated dissipation, $F_{11}$ displays damped Rabi oscillations whose envelope decays at rate $\gamma$. As $\gamma_{12}$ increases, the decay slows down and the short-time peak broadens. When the fully correlated condition $\gamma_{12}=\gamma$ is reached, a decoherence-free subspace forms and $F_{11}$ saturates to a nonzero long-time plateau, indicating persistent information about the initial state. The inset shows the long-time behavior toward the respective steady-state values
• Figure 3: (Color online) Fisher information element $F_{12}$, describing correlations between the estimators of the initial occupations $n_1(0)$ and $n_2(0)$ for several values of the normalized cross-damping rate $\gamma_{12}/\gamma$ with the initial conditions $\bar{n}_{1}=0.35,\ \bar{n}_{2} = 2.3,\ \hbar\omega_{0}/k_{B}T = 1.0,\ \gamma/\Omega = 0.02$ and $\Omega = 3.1\pi k Hz$. The curves, from bottom to top, correspond to $\gamma_{12}/\gamma = 0.00, 0.50, 0.80, 1.00$. For $\gamma_{12}=0$, $F_{12}$ oscillates around zero as excitations are periodically exchanged between ions through the coherent coupling $\Omega$. Increasing $\gamma_{12}$ leads to an asymmetric oscillation pattern and an overall enhancement of short-time sensitivity. In the fully correlated case $\gamma_{12}=\gamma$, a nonzero long-time plateau emerges due to the decoherence-free subspace, indicating that correlations between the initial occupations remain encoded in the asymptotic state.
• Figure 4: (Color online) Fisher information element $F_{66}$ associated with the estimability of the reservoir thermal occupation $\bar{N}$ for several values of the normalized cross-damping rate $\gamma_{12}/\gamma$ with the initial conditions $\bar{n}_{1}=0.35,\ \bar{n}_{2} = 2.3,\ \hbar\omega_{0}/k_{B}T = 1.0,\ \gamma/\Omega = 0.02$ and $\Omega = 3.1\pi k Hz$. The curves, from top to bottom, correspond to $\gamma_{12}/\gamma = 0.00, 0.50, 0.80, 1.00$. For all values of the correlation parameter $\gamma_{12}/\gamma$, $F_{66}$ monotonically increases at short times and saturates to a finite constant. This reflects that the steady-state phonon number always carries information about the reservoir temperature, whereas the influence of correlated dissipation affects only the transient dynamics. Unlike the FI elements associated with initial occupations, $F_{66}$ does not vanish at long times even when the bath is uncorrelated.
• Figure 5: (Color online) Fisher information element $F_{33}$ associated with estimating the coherent coupling $\Omega$ for several values of the normalized cross-damping rate $\gamma_{12}/\gamma$ with the initial conditions $\bar{n}_{1}=0.35,\ \bar{n}_{2} = 2.3,\ \hbar\omega_{0}/k_{B}T = 1.0,\ \gamma/\Omega = 0.02$ and $\Omega = 3.1\pi k Hz$. The curves, from top to bottom, correspond to $\gamma_{12}/\gamma = 0.00, 0.50, 0.80, 1.00$. The FI displays oscillations at short and intermediate times, reflecting the sensitivity of $n_{1}(t)$ to the coherent exchange of excitations. Increasing the cross-damping $\gamma_{12}$ slows the decay of the envelope and enhances the time window over which $\Omega$ can be accurately inferred. For $\gamma_{12}<\gamma$, $F_{33}$ vanishes at long times since the steady state does not retain information about the coherent coupling.