Recent Advances Concerning the ⁸⁷Sr Optical Lattice Clock at the National Time Service Center

Abstract

We review recent experimental progress concerning the ⁸⁷Sr optical lattice clock at the National Time Service Center in China. Hertz-level spectroscopy of the ⁸⁷Sr clock transition for the optical lattice clock was performed, and closed-loop operation of the optical lattice clock was realized. A fractional frequency instability of 2.8 × 10⁻¹⁷ was attained for an averaging time of 2000 s. The Allan deviation is found to be 1.6 × 10⁻¹⁵/τ^1/2 and is limited mainly by white-frequency-noise. The Landé g-factors of the (5s²)¹S₀ and (5s5p)³P₀ states in ⁸⁷Sr were measured experimentally; they are important for evaluating the clock’s Zeeman shifts. We also present recent work on the miniaturization of the strontium optical lattice clock for space applications.

1. Introduction

Because of their potential to provide high Q factors than those of microwave clocks, optical atomic clocks offer a tremendous advantage in both short-term and long-term stability in comparison with conventional microwave clocks [1,2]. In the last few decades, optical clocks employing neutral atoms [3,4,5,6] and single ions [7,8,9] have progressed remarkable regarding instability and achieving uncertainty levels of 10⁻¹⁸, surpassing the cesium primary frequency standard as the definition for the second of the International System of Units. The great progress in optical clocks over the past decade has immensely influenced developments in science and technology areas such as topological dark matter hunting [10], gravitational wave detection [11], fundamental physics measurement [12,13,14], quantum simulators and geodetic applications [15,16,17].

Alkaline earth(-like) atoms are the most promising candidates for optical lattice clock among possible atomic candidates because of their abundant and special energy levels. A large number of strontium optical lattice clocks [18] are currently under development world-wide with uncertainties of up to a few parts in 1 × 10⁻¹⁸ [3,5,6]. Recently, the team at the Joint Institute for Laboratory Astrophysics (JILA) obtained using an ⁸⁷Sr optical lattice clock [19] a measurement precision of 5 × 10⁻¹⁹ in 1 h of averaging time by taking advantage of the high, correlated density of a degenerate Fermi gas in a three-dimensional optical lattice to guard against on-site interaction shifts.

In this article, we report recent experimental progress concerning our ⁸⁷Sr optical lattice clock at the National Time Service Center. The organization of this paper is as follows. First, recent progress with the clock is described. Second, the experimental determination of the Landé 𝑔-factors for (5s²)¹S₀ and (5s5p)³P₀ states of ⁸⁷Sr is presented. Finally, we mention work on the miniaturization of strontium optical lattice clock for future space applications.

2. Optical Lattice Clock Based on ⁸⁷Sr

2.1. Level Structure and Cooling Transitions of ⁸⁷Sr

The relevant energy levels of ⁸⁷Sr with transitions and wavelengths used in an optical lattice clock are shown in Figure 1.

An oven, which is heated to a temperature of 500 °C, emits a strontium atomic beam at a velocity of 460 m/s. A Zeeman slower formed with ten coils and 461-nm laser is used to slow the velocity of the thermal atomic beam to about 50 m/s. The strontium atoms are further cooled and trapped in a two-stage magneto-optical trap (MOT). The strong dipole-allowed (5s²)¹S₀(F = 9/2)−(5s5p)¹P₁(F = 11/2) transition (461-nm, 32-MHz natural linewidth) is used for the first Doppler cooling stage; the corresponding cooling limit is roughly in the sub-millikelvin range. With the 679-nm and 707-nm lasers, optical repumping returns the (5s5p)³P₂ and (5s5p)³P₀ metastable states to the ground state (5s²)¹S₀ to cool and trap more atoms. The intercombination (5s²)¹S₀(F = 9/2)−(5s5p)³P₁(F = 11/2) transition (689-nm, 7.5-kHz natural linewidth) between the singlet and triplet states is used in the second Doppler cooling stage; the corresponding cooling limit is near the sub-microkelvin range. A stirring laser corresponding to the (5s²)¹S₀(F = 9/2)−(5s5p)³P₁(F = 9/2) transition provides a rapid randomization among the ten Zeeman sublevels of the (5s²)¹S₀ state. Additionally, it is also used as a pump laser to drive the entire population of the ten Zeeman sublevels to the m_F = +9/2-m_F = +9/2 and m_F = −9/2-m_F = −9/2 sublevels.

After this two-stage cooling and trapping, roughly 3.5 × 10⁶ atoms are trapped in the red MOT with a temperature of 3.9 μK. The one-dimensional (1-D) horizontally oriented optical lattice is operated at 813 nm and traps the cooled atoms in the Lamb-Dicke regime [20]. The (5s²)¹S₀−(5s5p)³P₀ transition at 698 nm is used as the clock line because of its narrow natural linewidth of about 1 mHz.

2.2. Fiber Phase Noise Cancellation

The 698-nm grating-stabilized diode laser is stabilized to an ultra-low-expansion (ULE) cavity, having a finesse of about 4 × 10⁵ with the Pound-Drever-Hall (PDH) technique. The length and orientation of the ULE cavity at 698 nm placed in vacuum are 10 cm and horizontal, respectively. The linewidth of the clock laser is about 1 Hz and the power is 5 mW. The clock laser beam is sent to the cold atoms for probing using a 5-m-long polarization-maintaining optical fiber with a transmission efficiency of about 80%. The fiber introduces a lot of phase noise, which translates into frequency noise and eventually a loss of spectral purity. To maintain the optical coherence of the clock laser source with the strontium cold atoms, we applied an efficient fiber-noise-cancellation (FNC) system. We introduce noise compensation directly at the input of the fiber with an acousto-optic modulator (AOM), where a part of the light comes from the reflected light that is inputted and reflected back from the output end of the fiber; the rest part of the light is used as a reference. It is possible to obtain a beat signal introduced by the fiber phase noise from the beat signal of the two parts but has a high frequency [21,22,23].

The error signal at low frequency can be obtained directly by mixing the beat signal with a local reference oscillator. The error signal is then used to modulate a voltage-controlled-oscillator (VCO) that drives the AOM around 80 MHz.

Figure 2a shows the frequency fluctuation of the beat signal with and without the FNC system. With the FNC active, the frequency fluctuation caused by the residual phase noise is reduced below 40 MHz and is sufficiently low for transferring a 1-Hz linewidth laser beam. Figure 2b shows the optical field spectral density of the beat signal with and without FNC, measured with a spectrum analyzer. Figure 2c shows the frequency instability caused by fiber noise when the FNC system is active, obtained by measuring directly the Allan deviation for our phase-locked system. Affected by white phase noise, the Allan deviation of the phase noise cancellation system follows a τ⁻¹ behavior.

2.3. Lifetime of Atoms in the Lattice

A laser operating at wavelength 813 nm is used to configure a 1-D horizontally oriented optical lattice. The 813-nm ECDL is stabilized to an ULE optical reference cavity, having a finesse of about 3 × 10⁴, using the PDH technique. The 813-nm lattice laser beam was focused to a waist size of 120 μm; the depth of the trap lattice is about 56 Er, where Er is the lattice photon recoil energy. About 10⁴ atoms are then loaded into the 1-D optical lattice at a magic wavelength of 813.4 nm.

An image of the 1-D optical lattice with a falling time of 30 ms is shown in Figure 3a. The line above is an in-situ image of the atoms trapped in the lattice; the cloud below is the atomic distribution after time-of-flight imaging. From fitting an exponential decay function (Figure 3b), the 1/e lifetime of the 1-D lattice is about 1.6 s. A long lattice lifetime is useful in resolving the narrow spectroscopy of the clock transition.

2.4. Clock Transition Spectra of ⁸⁷Sr

Using the normalized shelving method [5,6], resolved sideband spectroscopy of the ⁸⁷Sr clock transition was performed (Figure 4a). The clock transition spectroscopy indicates that the atomic longitudinal temperature is about 4.2 μK. With decreasing power and frequency range of the clock laser, a high-resolution spectrum of the clock transition (Figure 4b) was obtained when the atoms–laser interaction time was 150 ms and shows a linewidth of 6.7 Hz.

With a small bias magnetic field of 0.3 G applied along the polarization axis of the lattice light, spectra of the π transitions were observed and ten Zeeman peaks were separated (Figure 5a). The intensity of individual π-polarized transitions depends on values of the Clebsch-Gordan coefficients. To avoid wasting the atomic population, atoms in all the m_F states were prepared in the (5s²)¹S₀(m_F = +9/2) or (5s²)¹S₀(m_F = −9/2) stretched states via the 689-nm σ⁺ or σ⁻ polarized laser resonance excitation. The pumping laser beam propagates along the polarization axis of clock and lattice lasers and its polarization is switched between σ⁺ and σ⁻ polarization using a liquid crystal variable wave plate. As a result, about 90% of the population were driven into the two sublevels; the population of the excited clock states is about seven times higher than that without the pumping laser beam. Finally, the linewidth of the two magnetic sublevels transition are 6.8 Hz and 6.2 Hz respectively (Figure 5b). The narrowest linewidth of the magnetic sublevel transitions that we can obtain at present is 4.9 Hz with a probing time of 300 ms (Figure 5c).

2.5. Closed-Loop Operation of ⁸⁷Sr Optical Lattice Clock

In the experimental setup for our ⁸⁷Sr optical lattice clock (Figure 6), a clock laser operating at 698-nm, locked to an ULE cavity using the PDH technique, delivers a beam to a cold atom sample. The clock transition is detected using the electron-shelving technique for the transition (5s²)¹S₀-(5s5p)¹P₁. A normalization scheme is used to eliminate the shot-to-shot noise [24]. To stabilize the clock laser to resonate at the atomic transition of (5s²)¹S₀-(5s5p)³P₀, we applied the modulation method. We stabilized the clock laser to the average frequency of the two m_F = +9/2-m_F = +9/2 and m_F = −9/2-m_F = −9/2 Zeeman transitions. The error signal is calculated digitally from four clock cycle. After filtering using a PID (proportional, integral, derivative) controller, feedback is provided by a frequency correction signal to modulate the AOM2 [25].

The error signal was obtained after the clock laser had been locked to the atomic transition (Figure 7a) along with the in-loop error signal stability of the ⁸⁷Sr optical lattice clock (Figure 7b). Short-term stability is limited by the clock laser and is affected by the length fluctuation of the ULE cavity; long-term stability is also limited by the clock transition of ⁸⁷Sr. The fractional frequency instability obtained was 2.8 × 10⁻¹⁷ for an averaging time of 2000 s. The Allan deviation is fitted with 1.6 × 10⁻¹⁵/τ^1/2, which is limited mainly by white-frequency-noise.

3. Experimental Determination of the Landé g-Factors for (5s²)¹S₀ and (5s5p)³P₀ States of ⁸⁷Sr

Because the clock transition frequency is sensitive to both first- and second-order Zeeman shifts, the value of the Landé g-factors of the clock states need to be determined accurately [26]. We measured experimentally the g-factors of the ground state (5s²)¹S₀ and the differential g-factor δg between the (5s5p)³P₀ and (5s²)¹S₀ states of ⁸⁷Sr with π-polarized and σ^±-polarized interrogation beams, respectively, at different magnetic field strengths [27]. A typical spectrum of the π-transition is shown in Figure 5a. When the polarization direction of the clock laser is perpendicular to the direction of the magnetic field, the σ^±-transition spectrum is obtained (Figure 8a). These peaks correspond to the transitions between the ground and excited sublevels. From the σ^±-transition spectrum, δg can be expressed as

$$\delta g=\frac{g\left(5s^2{}^{1}S{}_0\right)}{\frac{f_{d,m_F}}{2f_{{\sigma }^{\pm },m_F}}-1}=\frac{\left(1-{\sigma }_d\right)g_I}{\frac{f_{d,m_F}}{2f_{{\sigma }^{\pm },m_F}}-1},$$

(1)

where the diamagnetic correction σ_d = 0.00345 [26], f_σ^±_,mF is the frequency gap between neighboring lines for the σ^± transitions, f_d,mF is the frequency difference between σ⁺ and σ⁻ transition from the same sublevel, g_I = μ_I/Iμ₀ with μ₀ the vacuum permeability, and the nuclear magnetic moment μ_I = −1.0924(7) μ_N [28] with μ_N the nuclear magneton. By accurately measuring the frequency interval between the two neighboring peaks, the Landé g-factor g((5s²)¹S₀) (Figure 8b) and the differential g-factor δg (Figure 8c) may be obtained. The values of g((5s²)¹S₀) and δg may be determined without the calibration of the external magnetic field from the σ^±-transition spectra. A final value of δg = −7.72(43) × 10⁻⁵ is obtained that agrees very well with a previous report [26].

4. Miniaturization of Strontium Optical Lattice Clock for a Space Optical Clock

Because space is a microgravity environment, an optical clock should achieve higher stability and accuracy than on Earth [29,30]. An optical clock in space may be used in tests of fundamental physics, such as Einstein’s theory of relativity, positioning, time and frequency transfers, accurate geoid determination, and monitoring and deep space navigation.

Making optical clocks compact and robust for transport is the first step towards applications of optical clocks in space [31,32,33]. We have produced a compact, robust laser source system operating at 461 nm along with a vacuum setup system (Figure 9). With these two systems, the first stage cooling and trapping of the ⁸⁷Sr atoms have been accomplished [34]. The ECDL 461-nm laser is locked to the (5s²)¹S₀−(5s5p)¹P₁ transition line of ⁸⁸Sr. After lock, the residual power of the 461-nm laser is 35 mW. More power is obtained from two other injection-locked slave lasers. The vacuum system includes an atomic oven, a two-dimensional collimating cavity, a Zeeman slower and a MOT cavity. We designed and constructed the Zeeman slower with permanent magnets [34]. With a MOT coil current of 2 A, an axial magnetic field gradient of 43 Gs/cm was obtained in the center of the MOT cavity and satisfies the requirements for the first-stage Doppler cooling very well.

The internal diameter is 1.5 mm and the final temperature of the blue MOT is approximately 10.6 mK, which was determined by observing the expansion of the atoms in this MOT. The number of cold atoms finally trapped in the blue MOT is 1.6 × 10⁶ for ⁸⁸Sr and 1.5 × 10⁵ for ⁸⁷Sr. With two repumping lasers operating at 679 nm and 707 nm, the typical number of loaded atoms increased typically five-fold compared with before.

5. Conclusions

We achieved the clock laser stabilization to the atomic transition in an optical atomic clock over a continuous operation of over 8 h. A Hertz-level spectrum of the ⁸⁷Sr clock transition was obtained. A fractional frequency instability of 2.8 × 10⁻¹⁷ was attained with an averaging time of 2000 s. We plan to evaluate the systematic uncertainty of ⁸⁷Sr optical lattice clock and measure the absolute frequency of the clock transition using an optical comb that locks to a H-maser. The Landé g-factors of the (5s²)¹S₀ and (5s5p)³P₀ states and the differential g-factor δg between the two states of the ⁸⁷Sr atom were measured accurately in experiments and are in good agreement with those of a previous report. We have achieved a compact and robust laser source system operating at 461 nm and a vacuum system. To date, the first-stage cooling and trapping of ⁸⁷Sr atoms have been accomplished.