[Paper Review] High-Accuracy Microwave Atomic Clock via Magic Optical Lattice

Xiaoji Zhou, Xuzong Chen+1 more|arXiv (Cornell University)|Dec 25, 2005

Advanced Frequency and Time Standards3 citations

TL;DR

This paper proposes a high-accuracy microwave atomic clock using Rb and Cs atoms trapped in an optical lattice at magic wavelengths, where the ac Stark shift of the clock transition is canceled. By suppressing key systematic shifts—such as cavity pulling, collisions, Doppler effects, and second-order Zeeman shifts—the scheme achieves a projected fractional frequency uncertainty of better than $2 \times 10^{-17}$, matching the precision of state-of-the-art optical clocks.

ABSTRACT

A microwave atomic clock scheme based on Rb and Cs atoms trapped in optical lattice with magic wavelength for clock transition is proposed. The ac Stark shift of clock transition due to trapping laser can be canceled at some specific laser wavelengths. Comparing with in fountain clock, the cavity related shifts, the collision shift, and the Doppler effect are eliminated or suppressed dramatically in atomic clock when the magic optical lattice is exploited. By carefully analyzing various sources of clock uncertainty, we conclude that a microwave atomic clock with an accuracy of better than $2 imes10^{-17}$ is feasible, which is of the same accuracy as the expected best optical atomic clock.

Motivation & Objective

• To overcome the fundamental limitations of atomic fountain clocks that restrict microwave clock accuracy to $\sim 10^{-16}$.
• To identify magic wavelengths for Cs and Rb clock transitions where the ac Stark shift vanishes due to matched dynamic polarizabilities of the clock states.
• To demonstrate that a microwave atomic clock in an optical lattice can achieve accuracy comparable to the best optical clocks, specifically $2 \times 10^{-17}$.
• To reduce the size and complexity of primary microwave clocks while improving stability and systematic uncertainty control.
• To enable a practical, compact, and highly accurate microwave standard suitable for space-based applications and future SI second redefinition.

Proposed method

• Use of a three-dimensional optical lattice formed by a red-detuned laser beam at a specific 'magic' wavelength to trap Cs and Rb atoms without inducing a net ac Stark shift on the clock transition.
• Calculation of differential dynamic polarizabilities ($\delta\alpha$, $\delta C$, $\delta B$, $\delta\gamma$) between the hyperfine ground states using quantum defect theory and transition matrix elements from Kurucz and Bell data.
• Employment of the ac Stark shift formula $\delta\nu = -\frac{1}{2}(\delta\alpha/h)E_z^2 - \frac{1}{4}(\delta C/h)E_{zz}^2 - \cdots$ to determine the wavelength at which $\delta\nu = 0$ for the clock transition.
• Numerical evaluation of scalar light shifts from $6P_{1/2}$, $6P_{3/2}$, $7P$, and higher excited states to identify cancellation points in the spectrum.
• Comparison of systematic uncertainties between fountain and lattice configurations using a detailed error budget, including blackbody radiation, gravitational redshift, microwave cavity pulling, and second-order Zeeman effects.
• Estimation of trap depth ($167$ kHz at $604.8$ nm, $10$ kW/cm²) and vibrational frequency ($\sim 34$ kHz) to assess stability and coherence time.

Experimental results

Research questions

• RQ1 Can the ac Stark shift of the microwave clock transition in Cs and Rb be canceled using a magic-wavelength optical lattice?
• RQ2 What specific wavelengths enable cancellation of the differential light shift for Cs and Rb clock transitions?
• RQ3 How does the uncertainty budget of a lattice-based microwave clock compare to that of an atomic fountain clock?
• RQ4 To what extent can systematic shifts such as second-order Zeeman effect, collisions, and Doppler shifts be suppressed in the lattice configuration?
• RQ5 Is it feasible to achieve a microwave clock accuracy of $2 \times 10^{-17}$ using trapped atoms in an optical lattice?

Key findings

• The paper identifies multiple magic wavelengths for Cs: 402.2 nm, 427.0 nm, 493.0 nm, and 604.8 nm, and for Rb: 370.4 nm, 395.6 nm, 452.3 nm, and 549.7 nm, where the ac Stark shift of the clock transition vanishes.
• The tuning rate of the clock frequency with respect to laser detuning is as low as $6 \times 10^{-14}$, indicating high stability against laser wavelength drift.
• The trap depth reaches 167 kHz (8 $\mu$K) at 604.8 nm with 10 kW/cm² intensity, and the vibrational frequency is approximately 34 kHz, enabling long coherence times.
• Systematic uncertainties are significantly reduced in the lattice configuration: second-order Zeeman shift is suppressed due to tight spatial confinement, reducing magnetic field sensitivity.
• The residual first-order Doppler shift is negligible due to atomic motion averaging over lattice sites, and the second-order Doppler shift is reduced to below $10^{-21}$, far below the fountain clock limit.
• The total estimated fractional uncertainty of the lattice-based microwave clock is $2.0 \times 10^{-17}$, matching the performance of the best optical clocks and surpassing current fountain clocks by an order of magnitude.