Towards Trapped-Ion Thermometry Using Cavity-Based EIT

TL;DR

This work addresses the challenge of thermometry for trapped ions in cavity-QED systems by introducing a cavity-based electromagnetically induced transparency (EIT) scheme that leverages motional-state–dependent dephasing. By incorporating vibrational sidebands and coupling to a thermal reservoir, the authors show that the EIT transparency window linewidth directly encodes the mean phonon occupation $\bar{n}$ and, hence, the ion temperature, even in regimes approaching sub-Doppler cooling. They develop a full open-system model with a $\Lambda$-type ion coupled to a cavity, derive the rotating-frame Hamiltonian and Lindblad dynamics, and demonstrate, via numerical simulations, that the cavity-EIT linewidth broadens with increasing $n_{th}$ and can be used to map temperature in both single-ion strong-coupling and multi-ion collective regimes. The paper also outlines practical experimental pathways, including high-finesse near-concentric cavities for single ions and low-finesse cavities with many ions for collective coupling, providing a roadmap for real-time, minimally invasive thermometry and heating-monitoring in cavity-QED ion systems, with potential extensions to study measurement back-action and quantum network nodes.

Abstract

We present a technique for measuring ion temperature using cavity-based electromagnetically induced transparency (EIT) applicable for cavity-qed systems in the strong coupling regime. This method enables efficient extraction of the ion's phonon occupation number following sub-Doppler cooling close to the motional ground state. The proposed method relies on monitoring the cavity probe transmission while scanning the probe laser frequency once cavity EIT is established using the control beam, significantly simplifying the measurement procedure. We theoretically establish a model that demonstrates the influence of the thermal state of the trapped ion vis-a-vis the EIT linewidth measured. We show how the cavity EIT transmission may be used as a thermometry tool to deduce the ion temperature as well as the motional state for an ion in the sub-Doppler cooling regime, even for systems that are in the weak coupling regime. The current method can only be used for operation in the resolved-sideband regime, where individual motional states can be selectively addressed for all relevant transitions either by selecting appropriate energy levels for the three-level system or by employing strong confinement with high secular frequencies ($\sim 10 MHz$).

Figures (10)

• Figure 1: (a) Schematic of the cavity-based electromagnetically induced transparency (EIT) configuration, where a trapped ion interacts with a single cavity mode. (b) Energy-level diagram of the cavity-EIT scheme including vibrational states. The coupling laser drives the $|u\rangle \rightarrow |e\rangle$ transition with the addition of one vibrational quantum, while spontaneous decay to $|g\rangle$ preserves the vibrational level. The probe field and cavity mode couple the $|e\rangle$ and $|g\rangle$ states.
• Figure 2: Simulated cavity-EIT transmission spectrum for $\kappa=0.4\,\mathrm{MHz}$, $g=3\kappa$, $\gamma_{eg}=\gamma_{eu}=\kappa$, $\langle n \rangle = 1$, and $\gamma_b=0.25\kappa$. The observed linewidth ($0.55\,\mathrm{MHz}$) is narrower than the empty cavity linewidth $2\kappa=0.8\,\mathrm{MHz}$. The vacuum Rabi splitting peaks occur at $\pm\sqrt{g^2 + \Omega_c^2} = \pm1.56\,\mathrm{MHz}$.
• Figure 3: (a) Cavity-EIT linewidth from analytical expression as a function of control field Rabi frequency. (b) Comparison of linewidth without considering motional state vs with motional state for different temperatures of ion. The parameters for the plots chosen as follows, $\kappa=0.4 MHz$, $g=3\kappa$, $\gamma_{eg}=\gamma_{eu}=\kappa$, $\gamma_b=0.04\kappa$.
• Figure 4: (a) Cavity-EIT spectrum for different $n_{th}$ value (b) Ion temperature and average phonon occupancy ($\bar{n}$) as a function of cavity-EIT linewidth. Parameters used are: $\kappa=0.4 MHz$, $g=3\kappa$, $\gamma_{eg}=\gamma_{eu}=\kappa$, $n_{th}=\langle n\rangle=1$, $\gamma_b=0.25\kappa$.
• Figure 5: Cavity-EIT linewidth variation with the number of ions. The simulation parameters are shown in the inset.