Multilevel Electromagnetically Induced Transparency Cooling

Abstract

Electromagnetically Induced Transparency (EIT) cooling is a well-established method for preparing trapped ion systems in their motional ground state. However, isolating a three-level system, as required for EIT cooling, is often challenging or impractical. Nonetheless, multilevel systems can inherently host dark states. In this work, we extend the EIT cooling framework to such multilevel systems. We develop a formalism to accurately determine the cooling rate in the weak sideband coupling regime and provide an approximate estimate for cooling rates beyond this regime, without the need for explicit simulation of the motional degree of freedom. We clarify the connection between the cooling rate and the absorption spectrum, offering a pathway for efficient near-ground-state cooling of ions with complex electronic structures.

Figures (15)

• Figure 1: Level scheme of single electromagnetically induced transparency cooling: Solid arrows connecting the $|g_{j}\rangle \rightarrow |e\rangle$ represent the two lasers with Rabi frequencies $\Omega_j$, detuned by $\Delta_j$. Dashed arrows represent decay from $|e\rangle \rightarrow |g_j\rangle$ with decay rate $\Gamma_j$.
• Figure 2: (a) Absorption profile of the rest ion and (b) corresponding cooling dynamics for (1) $\Omega_2/2\pi$ = 1 MHz, (2) $\Omega_2/2\pi$ = 2 MHz, and (3) $\Omega_2/2\pi$ = 3 MHz. Dashed lines indicate the positions of the carrier (black), motion removing (red), and motion adding (blue) sideband transitions. Parameters: $\eta = 0.01$, $\omega_0/2\pi = 1.1$ MHz, $\Omega_1/2\pi = 17$ MHz, $\Delta_d/2\pi = 70$ MHz, and $\Gamma/2\pi = 20$ MHz with $\Gamma_1/\Gamma_2 = 1$.
• Figure 3: Single EIT cooling: A comparison of the performance of the cooling rate estimation methods for varying Rabi frequency ratios $\Omega_2/\Omega_1$. $W_{abs}$ is the rest ion rate, Eq.\ref{['wabs']}, $W$ denotes the result from the mean phonon rate equation, Eq.\ref{['cooling_gio']}, and $W_{\text{exp}}$ corresponds to the fitted, Eq.\ref{['exp_decay']}, decay rate to numerical cooling dynamics, which is used to benchmark the methods. The shaded region indicates where $W_{\text{abs}}$ takes negative values. Note that $W_{\text{abs}}$ has been multiplied by a factor of 2 to match the other methods factor_2. Parameters: $\eta = 0.01$, $\omega_0/2\pi = 2$ MHz, $\Omega_1/2\pi = 17$ MHz, $\Delta_d/2\pi = 70$ MHz, and $\Gamma/2\pi = 20$ MHz with $\Gamma_1/\Gamma_2 = 1$.
• Figure 4: Absorption profiles for single EIT cooling, based on (a) the steady-state population of the rest ion's excited state, (b) the real part of the fluctuation spectrum of $V_1$, (c) the absorption rate of fictitious lasers. Dashed lines indicate the positions of the carrier (black), motion removing (red), and motion adding (blue) sideband transitions. (d) The corresponding cooling dynamics for three scenarios: (1) $\Omega_2/\Omega_1 = 0.5$, (2) $\Omega_2/\Omega_1 = 0.75$, and (3) $\Omega_2/\Omega_1 = 1$. (e) Cooling rates across all cases, where $W_{\text{abs}}$, $W$, $W_{\text{fic}}$ are obtained by spectra (a),(b), (c), respectively, as described in the main text. $W_{\text{exp}}$ is obtained by fiitng Eq.\ref{['exp_decay']} to the cooling dynamics. The shaded region highlights negative values of $W_{\text{abs}}$. Note that $W_{\text{abs}}$ has been multiplied by a factor 2. Parameters: $\eta = 0.01$, $\omega_0/2\pi = 2$ MHz, $\Omega_1/2\pi = 17$ MHz, $\Delta_d/2\pi = 70$ MHz, nd $\Gamma/2\pi = 20$ MHz with $\Gamma_1/\Gamma_2 = 1$.
• Figure 5: Single EIT cooling: (a) Cooling rates for different decay rates at $\eta = 0.01$. Second row: Cooling rates as functions of the Lamb-Dicke parameter $\eta$ for (b) $\Gamma/2\pi = 1$ MHz, (c) $\Gamma/2\pi = 20$ MHz, and (d) $\Gamma/2\pi = 50$ MHz. $W$ represents the rate equation result, Eq.\ref{['cooling_gio']}, $W_{\text{fic}}$ corresponds to the fictitious laser absorption rate, Eq.\ref{['fic_rate']}, and $W_{\text{exp}}$ is the numerically calculated rate. Parameters: $\omega_0/2\pi = 2$ MHz, $\Omega_1/2\pi = \Omega_2/2\pi = 17$ MHz, $\Delta_d/2\pi = 70$ MHz, and $\Gamma_1/\Gamma_2 = 1$.