Addition to the dynamic Stark shift of the coherent population trapping resonance

TL;DR

Problem: The light-induced CPT resonance shift includes the conventional dynamic Stark shift $\delta_{AC}$ and an additional distortion-driven shift $\delta_D$ from off-resonant transitions. Approach: A four-level Λ-type model with a bichromatic field is analyzed under the adiabatic and uniform-decay approximations to derive an analytic expression for $\delta_D$ in the weak-coupling limit and to explore nonlinearities in the strong-coupling regime. Contributions: In the weak-coupling regime, the total shift is $\delta_0 = \delta_{AC} + \delta_D$ with $\delta_D = p_1 p_2 G \gamma_D \frac{\Omega_2^2 - \Omega_1^2}{\Omega_1^2 + \Omega_2^2}$ and $G = \frac{\Gamma \omega_{34}}{\Gamma^2 + \omega_{34}^2}$, and observable signals peak at this position. In the strong-coupling regime, $\delta_0$ can deviate from linear intensity dependence, with parameter regions where the shift is suppressed (or enhanced) by choosing the relative Rabi frequencies; the transition toward a three-level Λ with no light shift can occur for large couplings. Significance: The results reveal a substantial, controllable distortion-driven contribution to CPT shifts, with potential impact on CPT-based quantum frequency standards and possible extensions to Ramsey interrogation methods.

Abstract

This paper presents a theoretical study of the light-induced shift of the coherent population trapping resonance. An analytical model is proposed that describes the interaction of two radiation components with an atomic system using a $Λ$ scheme and takes into account an additional level of excited state. Both weak and strong coupling regimes with off-resonant transitions are considered. It is shown that, in addition to the conventional dynamic Stark shift, an extra shift arises due to the distortion of the resonance line shape when bichromatic laser radiation interacts with off-resonant atomic transitions. An analytical expression for this additional shift is derived in the weak-coupling limit, and its significant impact on the resonance shape and sensitivity to the intensities of the laser field components is demonstrated. It is found that under strong coupling conditions, the additional shift can deviate substantially from a linear dependence on light intensity, suggesting new opportunities for controlling light shifts in precision atomic devices such as quantum frequency standards.

Figures (7)

• Figure 1: Four-level $\Lambda$-type atomic configuration interacting with two-frequency radiation. Solid arrows indicate near-resonant driven transitions. Dashed arrows indicate off-resonant driven transitions. See text for notation.
• Figure 2: Dependence of the real and imaginary parts of $\tilde{\rho}_{12}$ and $\tilde{\rho}_{12}^{(0)}$ in relative units on the two-photon detuning $\delta$ for $\omega_{34} = 10\Gamma$, $\Omega_1 = 3\Omega_2$, $|\Omega_1|^2 + |\Omega_2|^2 = 10^{-4}\Gamma^2$, $\Gamma_{12} = 0$, $p_1 = 1$, $p_2 = -1$.
• Figure 3: Dependence of the total shift $\delta_0$ on the parameter $p_2$ in units of $\gamma_D$ for different $p_1$. The remaining parameters are the same as in the caption to Fig. \ref{['rho12_delta']}
• Figure 4: Dependence of the CPT resonance shapes observed in the signal ${{\rho }_{\text{exc}}}$ on the excited state splitting ${{\omega }_{34}}$ for cases (a) ${{p}_{1}}={{p}_{2}}=1$ and (b) ${{p}_{1}}=-1,{{p}_{2}}=1$ at $\Omega _{1}^{2}=10\Omega _{2}^{2}$, $\Omega _{1}^{2}+\Omega _{2}^{2}=0.1{{\Gamma }^{2}}$.
• Figure 5: Dependence of the shift $\delta_0$ on $p_2$ in units of $\gamma_D$ for different $p_1$ in the strong coupling regime with the fourth level ($\omega_{34}=0.5\Gamma$) for case $\Omega _{1}=\Omega _{2}$, $\Omega _{1}^{2}+\Omega _{2}^{2}=10^{-4}{{\Gamma }^{2}}$.