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Characterization of a Two-Photon Rubidium-87 Optical Clock

Sophia Rojas

Published Jan 1, 2026• 0 citations
OpenAlex

Abstract

Atomic clocks are based on highly stable and well-characterized atomic transitions, mak-ing precise control and understanding of systematic frequency shifts essential for improved timekeeping performance. This thesis presents an experimental study of light-induced Stark shifts in rubidium and their impact on precision laser spectroscopy relevant to atomic frequency standards, as well as the effects of residual amplitude modulation on the laser lock point and the resulting sensitivity of both short- and long-term stability. A laser-based spectroscopy system was developed to probe rubidium transitions while enabling controlled variation of optical intensity. The optical beam was experimentally characterized through calculations of beam waist and measurements of power to accurately determine the intensity at the interaction region. Frequency shifts arising from the AC Stark effect were measured as a function of optical power, allowing extraction of the differential polarizability between atomic energy levels using a theoretical model relating energy shifts to electric-field intensity. Uncertainty analysis was performed through systematic propagation of experimental errors associated with beam geometry, optical power calibration, and frequency measurements. In addition, residual amplitude modulation generated by electro-optic phase modulation was investigated as a significant source of systematic error in frequency stabilization systems used in atomic clocks. The magnitude and behavior of RAM were experimentally characterized, and its impact on frequency stability was quantified by measuring the sensitivity of the laser lock point to controlled variations in RAM. Potential mitigation strategies were then explored through active pre-compensation of RAM using an amplitude electro-optic modulator in a closed-loop feedback system.

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Atomic clocks are based on highly stable and well-characterized atomic transitions, mak-ing precise control and understanding of systematic frequency shifts essential for improved timekeeping performance. This thesis presents an experimental study of light-induced Stark shifts in rubidium and their impact on precision laser spectroscopy relevant to atomic frequency standards, as well as the effects of residual amplitude modulation on the laser lock point and the resulting sensitivity of both short- and long-term stability. A laser-based spectroscopy system was developed to probe rubidium transitions while enabling controlled variation of optical intensity. The optical beam was experimentally characterized through calculations of beam waist and measurements of power to accurately determine the intensity at the interaction region. Frequency shifts arising from the AC Stark effect were measured as a function of optical power, allowing extraction of the differential polarizability between atomic energy levels using a theoretical model relating energy shifts to electric-field intensity. Uncertainty analysis was performed through systematic propagation of experimental errors associated with beam geometry, optical power calibration, and frequency measurements. In addition, residual amplitude modulation generated by electro-optic phase modulation was investigated as a significant source of systematic error in frequency stabilization systems used in atomic clocks. The magnitude and behavior of RAM were experimentally characterized, and its impact on frequency stability was quantified by measuring the sensitivity of the laser lock point to controlled variations in RAM. Potential mitigation strategies were then explored through active pre-compensation of RAM using an amplitude electro-optic modulator in a closed-loop feedback system.
We implement a feedback protocol to suppress the ac Stark shift in a two-photon rubidium optical frequency reference, reducing its sensitivity to optical power variations by a factor of 1000. This method alleviates the tradeoff between short-term and long-term stability imposed by the ac Stark shift, enabling us to simultaneously achieve instabilities of 3 × 10 − 14 at 1 s and 2 × 10 − 14 at 10 4 s. We also quantitatively describe, and experimentally explore, a stability limit imposed on clocks using this method by frequency noise on the local oscillator.
Two-photon vapor cell-based optical clocks are strong candidates for next-generation portable atomic standards, offering simplicity, compactness, and high performance. Their narrow clock transitions with counter-propagating beams enable first-order Doppler-free operation. However, systematic perturbations such as the AC Stark shift, temperature-induced shift, and drifts resulting from the laser system pose challenges cause instabilities to medium- to long-term performance. This paper provides a comprehensive overview of Rb two-photon vapor cell optical standards, focusing on the long-term performance-limiting effects and potential mitigation strategies, aiming for clock stabilities better than 1 × 10−15 over the averaging time of a day and beyond.
The two-photon transition in rubidium is an attractive candidate for use in the next generation of field-deployable atomic clocks and frequency references. However, this transition is known to suffer from a large AC Stark shift, through which fluctuations in the intensity of the interrogation laser cause fluctuations of the clock frequency, often limiting long-term stability. We implement an interrogation protocol known as the auto-compensated shift (ACS) in a two-photon rubidium optical frequency reference, reducing its sensitivity to optical intensity fluctuations by a factor of 1000. ACS alleviates the tradeoff between short-term and long-term stability imposed by the light shift and enables us to simultaneously achieve a fractional instability of 3 x 10^-14 at an integration time of 1 s and 2 x 10^-14 at an integration time of 10,000 s. We also examine a stability limit imposed on clocks using ACS by frequency noise on the laser.
The invention of the atomic clock has enabled applications like the global navigation satellite system, are important for telecommunication and are used for the definition of the SI second. Nowadays, optical clocks allow for measurement of height differences on the centimeter scale and can be employed for tests of the standard model of particle physics. Among the various implementations of optical clocks, this thesis focuses on an Al⁺ clock. For this work an optical resonator is examined, which enables the simultaneous frequency stabilization of two laser on the same mirror pair through the use of a dual-wavelength coating. Various contributions on the frequency stability are characterized and correlations are investigated. This cavity enables high fidelities in coherent manipulation of trapped ions, as required to operate the Al⁺ clock. This thesis partly builds upon the findings presented in Ref. Kramer, 2023. The setup and the systematic shifts are revisited and a detailed evaluation of the shifts is conducted, resulting in a total fractional frequency uncertainty of 1.6×10⁻¹⁸. The biggest contribution is from an ac-Stark shift induced by cooling during the clock interrogation, resulting in a shift of −9.3(1.1) × 10⁻¹⁸. The cooling reduces the second order Doppler shift to −1.6(0.2)×10⁻¹⁸. Additionally, a position dependent micromotion of the ion was investigated, which contributes an uncertainty of 0.38 × 10⁻¹⁸. Moreover, the first-order Doppler shift, servo error, ac-Stark shift of the clock laser and the phase chirp are determined for the first time, showing uncertainties below 10⁻¹⁸. The first frequency comparison between the Al⁺ clock and the ¹¹⁵In⁺, ¹⁷¹Yb⁺ and ⁸⁷Sr clocks at PTB are demonstrated. Measurements against Sr reveal a stability of the Al+ clock of 5.9 × 10⁻¹⁶ at 1s , which is one of the best single ion clock stabilities to date. All measured frequency ratios fall within the 10⁻¹⁸ uncertainty regime. The ratio between Sr/Al⁺ exhibits a significant deviation of more than 8.6𝜎 from published measurement data, underscoring the importance of inter-institutional frequency ratio measurements to verify the uncertainty budgets and reproducibility of optical clocks.
Abstract We analyze an optical atomic clock using two-photon 5 S 1 / 2 → 4 D J transitions in rubidium. Four one- and two-color excitation schemes to probe the 4 D 3 / 2 and 4 D 5 / 2 fine-structure states are considered in detail. We compare key characteristics of Rb 4 D J and 5 D 5 / 2 two-photon clocks. The 4 D J clock features a high signal-to-noise ratio due to two-photon decay at favorable wavelengths, low dc electric and magnetic susceptibilities, and minimal black-body shifts. Ac Stark shifts from the clock interrogation lasers are compensated by two-color Rabi-frequency matching. We identify a ‘magic’ wavelength near 1060 nm, which allows for in-trap, Doppler-free clock-transition interrogation with lattice-trapped cold atoms. From our analysis of clock statistics and systematics, we project a quantum-noise-limited relative clock stability at the 10 − 13 / τ ( s ) -level, with integration time τ in seconds, and a relative accuracy of ∼ 10 − 13 . We describe a potential architecture for implementing the proposed clock using a single telecom clock laser at 1550 nm, which is conducive to optical communication and long-distance clock comparisons. Our work could be of interest in efforts to realize small and portable Rb clocks and in high-precision measurements of atomic properties of Rb 4 D J -states.