Optical atomic clocks stand at the forefront of precision measurement, pushing the boundaries of timekeeping and metrology while enabling transformative opportunities in fundamental physics and advanced technologies^[1–3]. By exploiting ultra-narrow atomic transitions at optical frequencies, these clocks have achieved unprecedented precision, enabling applications that span from geodesy and detection of gravitational waves to searches for dark matter and tests of variations in fundamental physical constants^[4–7]. These applications above are typically based on clock comparison experiments, where the stability and systematic uncertainty of the clocks play a pivotal role. Clock stability, in particular, determines the ability to resolve frequency shifts and directly impacts the efficiency of systematic uncertainty evaluation^[8]. Consequently, enhancing stability remains a central challenge in the development of optical atomic clocks.

The operation of optical clocks involves interrogating ultra-narrow optical atomic transitions with a clock laser stabilized to an optical reference. The fundamental limit of clock stability is quantum projection noise (QPN), which arises from the probabilistic nature of projective quantum state measurements^[9]. Additionally, the Dick effect, where the intermittent measurement process aliases the laser frequency noise into the measurement band during dead time between interrogations, imposes another practical limitation on stability^[10]. Optimizing the stability of an optical clock requires addressing two key aspects: the short-term and long-term stability. The short-term stability is primarily limited by the frequency noise of the clock laser, which contributes to the Dick effect and reduces the signal-to-noise ratio (SNR) of the clock spectrum. Besides, reducing the laser frequency noise can also extend the coherence time between the laser and the atomic ensemble, enabling higher spectral resolution^[8,11]. The long-term stability, on the other hand, is dominated by temporal fluctuations in systematic effects, such as blackbody radiation shifts, lattice light shifts, Zeeman shifts, and collision shifts, which can induce random walks in the atomic transition frequency and degrade clock stability over time^[11–14]. To achieve the clock stability at the 10−18 level or below demands precise identification, quantification, and suppression of these systematic effects^[15,16].

Among the leading optical atomic clocks, Yb171 optical lattice clocks have achieved fractional uncertainties and instabilities at the 10−18 level^[12,13,17]. The most advanced Yb171 clocks have demonstrated remarkable performance, with a short-term stability of 6×10−17/τ using a zero-dead-time scheme^[13] to suppress the Dick effect and a long-term stability of sub-10−18 at extended times^[12]. Despite these achievements, realizing such a performance across various research groups remains challenging due to the technical complexities and stringent experimental conditions involved. These challenges underscore the need for more general approaches to further improve clock stability.

In this work, we demonstrate the method to enhance the stability of a Yb171 optical lattice clock by decoupling the clock laser frequency noise and long-term systematic effects. First, we introduce our recently developed ultra-stable laser system (named FPC2). We optimized its performance by decoupling frequency noise and individually reducing each below the thermal noise limit, thereby improving the short-term stability. For the long-term stability, we present a method to decouple the systematic-effects-induced instability and demonstrate how this method guides our optimization of the Yb171 optical clock. Based on our two independent ytterbium optical clocks (Yb1 and Yb2), a long-term stability of 5×10−18 at 24000 s is measured by synchronous comparison initially. By decoupling the systematic effects, we deduce that the collision frequency shift is the limiting factor for the stability. With the compensation for the collision shift, the long-term stability of a single clock could be reduced to 3.6×10−18. Furthermore, by suppressing the fluctuations in the number of atoms and increasing the spectroscopy time, the synchronous comparison enables each clock to average at a rate of 3.2×10−16/τ and reach a stability of 2.4×10−18 at 8000 s.

The ultra-stable laser system FPC2 is based on a 30 cm notched Fabry-Perot (FP) cavity at 1156 nm. Its geometric structures are almost identical with our first system (FPC1), which has been reported in our previous work^[18,19]. The fused silica substrate of FPC2 has lower thermal noise than the ultra-low-expansion (ULE) glass substrate of FPC1^[20]. Besides, we optimized the vibration immunity of FPC2. When installing FPC2, we specifically optimized its symmetry and adopted a better vibration-isolation platform (AVI-200 LP with the outboard low-frequency suppression module LFS-3) to reduce the vibrations sensed by the FP cavity.

Generally, to obtain the noise characteristics of an individual ultra-stable laser, the three-cornered-hat method is required, in which at least three ultra-stable lasers with similar performance are required^[8]. In our condition, the noise characteristics of FPC2 are obtained by independently decoupling the contributions of each possible frequency noise source with the help of FPC1. The noise from the environment, such as vibration, and that from the Pound-Drever-Hall (PDH) servo loop, such as laser power fluctuation, residual amplitude modulation (RAM), and servo noise, are individually analyzed. The schematic diagram of the experimental setup for the noise decoupling is shown in Fig. 1(a).

We lock two independent lasers onto the cavities separately, corresponding to the switches in Fig. 1 set to position ①. For the FPC1 system, the external cavity diode laser (ECDL) generates the 1156 nm laser with 200 mW power, utilizing periodically poled lithium niobate (PPLN) to generate the 578 nm laser for atomic clock operation. For the FPC2 system, the 1156 nm laser is supplied by a distributed feedback fiber laser (FL) with 10 mW power. The 1156 nm laser beam passes the acoustic-optic modulators (AOMs) to match the cavity resonance frequency. The lasers are locked onto FP cavities with the standard PDH technique^[21] where we use a wedged electro-optic modulator (EOM) to achieve phase modulation of the laser to obtain the PDH error signal. We suppress the RAM through various means^[22–25]. The PDH signal from FPC1 is sent to a fast analog controller and a two-way-control lock is realized by regulating the current and piezoelectric ceramic (PZT) of the ECDL in series. A similar two-way-control lock of FPC2 is actuated by regulating the built-in AOM and PZT of the FL. The FL-based system shows better robustness and is less susceptible to external vibrations and sound interference, which can ensure long-term continuous locking for over 15 days.

The two systems are installed in different rooms that are tens of meters apart. We transmit the ultra-stable laser between two rooms through a 60 m optical fiber. The fiber is routed through the ceiling space and protected by conduits. A part of the FL laser beam transmits through the AOM3 and couples into a 60 m fiber to reach the Yb171 clock room. The fiber noise is suppressed with the fiber noise cancellation (FNC) technique^[26], in which the beam reflected by the ultra-physical contact (UPC) fiber end beats with the reference beam via PD3 to form a phase-sensitive signal. The AOM3 compensates for the phase noise induced by the fiber. Beat frequency measurements show that the linewidth induced by the fiber phase noise is <20mHz. At the fiber output, the laser beams from FPC1 and FPC2 heterodyne beat via PD4 for frequency noise measurements.

The main idea of decoupling laser frequency noise is to identify the factors that cause laser frequency noise and analyze their contributions. Besides the thermal noise^[27,28], the noise sources of the ultra-stable laser system can be classified into two categories: 1) the external environment source, such as the vibration, temperature, and vacuum where the FP cavities are loaded; 2) the non-ideal characteristics in the PDH technique, such as the RAM, input laser power fluctuation, and servo error. Regardless of which type of noise is evaluated, the goal is to obtain the sensitivity coefficients of the laser frequency with respect to that noise.

Taking vibration as an example, the AVI vibration isolation platform allows an excitation signal to be applied that can be used for a shaker in each direction (H1, H2, and V). We applied vibrations of ∼10−4g to FPC2. A g-meter (Wilcoxon 731 A) records the vibration-modulated acceleration on the AVI platform while a frequency counter (K+K FXE) records the beat frequency fluctuation. The frequency-vibration sensitivity is determined by their ratio as 1.8×10−10/g for H1, 6.2×10−11/g for H2, and 2.3×10−10/g for V. For the noise arising from the FPC2’s PDH servo loop, the slope of the error signal is used as the coefficient for their evaluation. By pre-stabling the ECDL on FPC1 and heterodyne locking the FL, the linewidth of FL reaches the several-hertz magnitude, yielding a high-resolution PDH error signal of the FPC2 system. Subsequently, measuring the power spectral density (PSD) of the error signal in the lock-in state, we inferred the noise produced within the servo link. In the unlocked state, the PSD of the baseline fluctuations of the error signal at the far-off resonance was measured to acquire the combined noise resulting from RAM and electrical noise. Figure 1(b) summarizes all the frequency noise components of FPC2 from their corresponding sources. By carefully suppressing each noise to a level below the thermal noise of FPC2, the sum of all noise contributions (red line) approaches the thermal noise limit (black line).

In consideration of the improved performance of FPC2, our two Yb171 lattice clocks share the clock laser from the FPC2 system when determining the stability of the optical clocks. The clock laser scheme is that the FL locks onto the FPC2 and heterodyne locks the ECDL, corresponding to the switches selected to ② in Fig. 1(a).

The schematic diagram of the two optical clock setups is shown in Fig. 2(a). The time sequence is summarized in Fig. 2(b). Here, we supply the details of atomic systems of two Yb171 optical clocks, which are important for measuring optical clock stability. Both optical clocks share the consistent parameters and are placed on the same optical platform. The levelness of this optical platform was strictly calibrated during installation to rule out the gravitational redshift caused by the height difference. The Yb171 atoms from a 400°C oven are Zeeman slowed and cooled to a sub-millikelvin temperature in 300 ms by a three-dimensional 399 nm magneto-optical trap (MOT). Subsequently, a two-stage 556 nm MOT is employed. Notably, the first stage utilizes higher power with a relatively large detuning from resonance, while the second stage features lower power and a smaller detuning. Each stage lasts 50 ms and further cools the atoms to a few microkelvin. The atomic sample is loaded into the optical lattice at the “magic wavelength”. The one-dimensional lattice is formed by reflecting the laser back onto itself using a concave mirror and oriented in the ⟨1,1,1⟩ direction relative to gravity. The lattice waist with the 1/e2 power radius is 50 µm. The lattice beam ensures linear polarization through a Glan-Thompson prism with a polarization ratio >104. The optical lattice lasers are provided by two independent Ti: sapphire lasers (M squared, SolsTiS). The lattice laser of Yb1 is locked onto the optical comb. Meanwhile, the lattice laser of Yb2 is heterodyne-locked to Yb1’s lattice laser. To eliminate potential frequency shifts caused by the residual background spectrum of the lattice laser, we use Bragg gratings (ONDAX, bandwidth <0.15nm) to purify the spectrum of each lattice laser. Both lattices operate at a trap depth of 225 Er (here Er is the lattice photon recoil energy) to compromise between the atom number and the Raman scattering of lattice photons as well as lattice-induced shifts. The atoms are spin-polarized with the 556 nm S10→3P1 (F=1/2−F′=1/2) transition. The atoms are then cooled to a longitudinal temperature of 0.8 µK by 100 ms sideband cooling.

The clock laser is referenced to FPC2 and delivered to the Yb1 and Yb2 optical clocks. Resonance with the atomic transition is detected by observing fluorescence collected onto a photo-multiplier tube (PMT). The fluorescence signals are digitized and processed by computers, which give a correction frequency to their AOMs. The atomic number is also calculated from the PMT signal. We optimized the optical path for clock interrogation, as shown in Fig. 2(a). The waist of the 578 nm clock laser beam is prepared to be four times the lattice waist. Its polarization is regulated parallel to the magnetic field B→ and the lattice polarization with a Glan-Thompson prism. The clock beam is counter-propagating with the lattice beam. The zero-degree reflector is the reference for phase noise cancellation of fiber and the free-space optical path. An optical attenuator is applied to reduce the power to several nanowatts to operate Rabi spectroscopy. The 578 nm reflector, attenuator, and 759 nm concave mirror are mounted on the same bracket to limit the potential Doppler shift from lattice motion to the clock laser^[12,29].

The 578 nm π-pulse of a laser is applied to excite the clock transition. By stepping the radio frequencies (RFs) of AOMYb1 and AOMYb2, the Rabi spectrum can be obtained. The typical Rabi spectroscopy for 200 and 400 ms is displayed in Fig. 3, and the full width at half-maximum (FWHM) of the observed spectral lines is Fourier-limited. With a longer interrogation time, the linewidth is above the Fourier limit. Figure 3(a) shows the 200 ms Rabi spectroscopy with 20 times repeated single-sweeps of the spectrum overlapped, and Fig. 3(b) is the 10 times repeated single-sweeps of 400 ms Rabi spectroscopy. The position of the central peak is aligned by post-processing of the data. The overlapped spectrum demonstrates high reproducibility of the spectral lines. Figures 3(c) and 3(d) give the single-sweep spectroscopy. Benefiting from the improvement of FPC2, the SNR of the Rabi spectral line has been significantly enhanced.

To assess the frequency stability of the two optical clocks, we carried out the synchronous comparison between Yb1 and Yb2. The time sequence in Fig. 2(c) shows the comparison running process. Each cycle of the clock operation requires four rounds of atom preparation and detection. For the first two preparations, the atomic spins are polarized to the ground state mF=+1/2, and the last two for mF=−1/2. The frequency deviation is inferred by detecting the excitation rate difference at the FWHM positions and averaging the frequency of two nuclear spin states to feed back to the AOM. This synchronization mechanism helps reduce a portion of the common-mode noise originating from the shared clock laser and yields better sensitivity to the clock transition shift caused by the systematic effects.

We conducted a continuous comparison for approximately 15 h with a clock-interrogation time of 200 ms. The frequency difference of the optical clocks is shown in Fig. 4(a). The invalid data caused by issues such as laser loss locking were excluded. Assuming that both clocks have equal frequency stability, and all noise processes are uncorrelated between the two systems, the single clock’s stability is just given by dividing the combined stability by 2. A single clock’s stability is evaluated as 5×10−16/τ as shown in Fig. 4(b) (black data). The individual stability of the single Yb171 lattice clock can be averaged down to 5×10−18 with the long-term comparison.

The long-term measurement requires both optical clock systems to be unaffected by systematic-effects-induced frequency shifts. To further improve the stability of the optical clock, it is of significance to clarify what factors limit the long-term stability. The frequency of the Yb optical clocks may be affected by various effects, among which collision shift, blackbody radiation shift, Zeeman shift, and lattice light shift are the most noteworthy. Correspondingly, in the comparison experiment, we recorded the number of atoms, Zeeman splitting, temperature of the science chamber, and lattice frequency difference of the two optical clocks. In this way, we can decouple the contribution of each factor to the comparison stability.

In our condition, the collisions between lattice-trapped atoms contribute the majority to clock instability and limit it at 5×10−18. The sufficient sideband cooling and spin polarization promise a good suppression of s-wave collisional shifts; we applied the linear model of p-wave cold collisions^[30]. The fluctuation of the atom number over the comparison is recorded in Fig. 4(c). We obtained the coefficient k=−3.7(0.1)×10−5Hz per atom by actively changing the number of atoms at the 225 Er lattice, as shown in Fig. 4(d). Each data point represents the average value over 1200 s, and the error bars are the Allan deviation. Based on the number of atoms and the collision frequency shift coefficient, we can conclude that the collision frequency shift’s contribution affects the optical clocks’ long-term stability on a time scale of more than 5000 s, as shown in Fig. 4(e) (green data). If we compensated for the collision frequency shift based on the number of atoms, the stability of a single clock could be enhanced to 3.6×10−18, as shown in Fig. 4(b) (red data).

Besides the collision shift, we also analyze the instability arising from the black body radiation (BBR) shift, lattice light shift, and Zeeman effect. The evaluated instability components are summarized in Fig. 4(e). The purple square data show the BBR shift resulting in an instability exceeding the QPN limit when the measurement is over thousands of seconds. Our current science chamber does not adopt a thermal shielding system^[31,32] but applies a simulation model with seven platinum temperature detectors on the surface of the science chamber. We infer the temperature around atom samples with finite element simulation with chamber surface temperature and further determine the BBR stark shift. The simulation model has been carefully calibrated in our previous work^[33]. In future measurements with higher precision, we will try to make corrections to this term. With the lattice shift effects we have characterized before^[34], we infer that the instability from the lattice laser shift is small as long as the laser is well-locked at the “magic wavelength”^[35], as shown in Fig. 4(e) (blue inverted triangle data). In the clock comparison, the first-order Zeeman shift is canceled by iteratively interrogating the two nuclear spin states and taking the average. The fluctuations of the environmental magnetic field are about 2 mG, so the second-order Zeeman-effect-induced instability is also small as shown in Fig. 4(e) (brown triangle data).

Through the method above, we achieved the decoupling of the instability induced by systematic effects. This helps to evaluate the contribution of each term separately, guiding further improvement of the stability in a targeted manner. Accordingly, we focused on addressing the issue of atom number fluctuations in the optical lattice. The main improvement was to enhance the stability of the power and frequency stabilization for the 556 nm MOT and 759 nm lattice laser. We employ FPC1 to lock the 1112 nm seed laser for the 556 nm cooling laser, thereby minimizing its drift to a mere 5 kHz per day. The pump laser of 759 nm Ti: sapphire lasers is replaced with a more stable fiber laser. These improvements have been proven to effectively reduce the fluctuations of the number of atoms in lattices.

The results of the improved synchronous comparison with a 400 ms clock-laser pulse are shown in Fig. 5. The frequency difference between the two optical clocks is shown in Fig. 5(a). The fluctuations in the number of atoms during the comparison are presented in Fig. 5(b), the fluctuations are suppressed to a certain extent. As the green data in Fig. 5(c) show, the instability caused by collisions is slightly higher than the quantum projection noise limit, but it no longer deteriorates the comparison results. Finally, the stability of the single optical clock is 3.2×10−16/τ, with an average time of 8000 s; the long-term stability reaches 2.4×10−18.

In conclusion, we have demonstrated a method to enhance optical clock stability by decoupling both the laser frequency noise of the ultra-stable laser system and systematic frequency shift effects. By developing a noise model, we systematically suppressed individual noise contributors below the cavity’s thermal noise limit, thereby improving the SNR of the clock transition spectrum and enhancing the short-term stability. Our approach, which employs two sets of cavities, provides a practical alternative to the three-cornered-hat measurements, particularly in scenarios where three ultra-stable lasers are unavailable. To some extent, this allows the laser system used as the reference to have slightly inferior performance, and it does not even need to be as good as our FPC1. Furthermore, we demonstrated a method to decouple the systematic frequency shift effects affecting the long-term stability of Yb171 lattice clocks. We identified the collision frequency shift as the primary limiting factor in our clocks. Additionally, through optimized atom number fluctuations in the optical lattices and extended clock pulse durations, we achieved the synchronous clock comparisons with an averaging stability of 3.2×10−16/τ and a long-term stability of 2.4×10−18 at 8000 s. Our work establishes a universally applicable decoupling framework that enhances both the short-term and long-term stability of optical clocks, unlocking the full potential of optical clocks for next-generation timekeeping, gravitational wave detection, and tests of fundamental physics.