[Paper Review] Stark effect of the cesium ground state: electric tensor polarizability and shift of the clock transition frequency

Simone Ulzega, A. Hofer+2 more|ArXiv.org|Apr 28, 2006

Advanced Frequency and Time Standards3 citations

TL;DR

This paper presents a refined third-order perturbation theory calculation of the Stark effect in cesium's ground state, including previously neglected matrix elements involving hyperfine and Stark interactions. It resolves a 40-year discrepancy in tensor polarizability and provides a clock transition frequency shift in excellent agreement with recent experiments, challenging the long-standing value used for black-body radiation corrections in atomic clocks.

ABSTRACT

We present a theoretical analysis of the Stark effect in the hyperfine structure of the cesium ground-state. We have used third order perturbation theory, including diagonal and off-diagonal hyperfine interactions, and have identified terms which were not considered in earlier treatments. A numerical evaluation using perturbing levels up to n=18 yields new values for the tensor polarizability $α_2(6S_{1/2})$ and for the Stark shift of the clock transition frequency in cesium. The polarizabilities are in good agreement with experimental values, thereby removing a 40-year-old discrepancy. The clock shift value is in excellent agreement with a recent measurement, but in contradiction with the commonly accepted value used to correct the black-body shift of primary frequency standards.

Motivation & Objective

• To resolve a longstanding theoretical-experimental discrepancy in the tensor polarizability of cesium's ground state.
• To provide a precise theoretical description of the F- and M-dependent Stark shifts in the cesium clock transition.
• To improve the accuracy of the Stark shift for the F=4, M=0 ↔ F=3, M=0 hyperfine transition, critical for primary frequency standards.
• To re-evaluate the role of third-order perturbation theory including diagonal and off-diagonal hyperfine matrix elements previously omitted.
• To assess the impact of these corrections on the black-body radiation shift in cesium atomic clocks.

Proposed method

• Third-order perturbation theory is applied to the energy shift of the 6S_{1/2} ground state, including both hyperfine and Stark interactions.
• The perturbation Hamiltonian W = H_hf + H_St is expanded in the energy shift formula, with selection rules ΔL = ±1 for Stark and ΔL = 0, ±2 for hyperfine interactions.
• Diagrams involving two Stark matrix elements and one hyperfine matrix element are evaluated, with contributions from nS, nP, and nD states up to n=18.
• Diagonal hyperfine matrix elements are derived from measured hyperfine splittings; off-diagonal elements are computed using Schrödinger wave functions with core polarization and spin-orbit corrections.
• The tensor polarizability α₂ and clock transition shift are extracted from the energy shift expressions, with uncertainties estimated from wave function accuracy (4–8%).
• Results are compared with experimental data and prior theoretical values, including rescaling of older approximations for consistency.

Experimental results

Research questions

• RQ1 Why has there been a persistent 40-year discrepancy between theory and experiment for the tensor polarizability of cesium's ground state?
• RQ2 What contributions to the Stark shift were omitted in prior third-order perturbation treatments of the hyperfine structure?
• RQ3 How do diagonal and off-diagonal hyperfine matrix elements affect the F- and M-dependent Stark shifts in cesium?
• RQ4 What is the precise value of the clock transition frequency shift due to static electric fields, and how does it compare with recent measurements?
• RQ5 How do these improved calculations affect the black-body radiation shift correction used in primary cesium frequency standards?

Key findings

• The calculated tensor polarizability α₂(F=I±J) = ∓3.72(25)×10⁻² Hz/(kV/cm)² is in excellent agreement with experimental values from Carrico et al. (1968) and Gould et al. (1969).
• The Stark shift of the clock transition frequency is Δν₀₀/ℰ² = -2.06(1) Hz/(kV/cm)², in excellent agreement with the recent measurement by Godone et al. (2005).
• The theoretical uncertainty in the clock shift is significantly smaller than in the tensor polarizability due to reliance on experimental input for key matrix elements.
• The inclusion of previously neglected matrix elements in third-order perturbation theory resolves the long-standing discrepancy in tensor polarizability.
• The result contradicts the commonly used value for black-body radiation shift correction, which is based on older experimental data with a 21-fold discrepancy relative to the theoretical uncertainty.
• The calculation shows that 90–95% of the clock shift contribution comes from n=6–8 states, with dominant contributions from diagrams A and 1 involving experimentally known matrix elements.