Demonstration of the Systematic Evaluation of an Optical Lattice Clock Using the Drift-Insensitive Self-Comparison Method

Abstract

The self-comparison method is a powerful tool in the uncertainty evaluation of optical lattice clocks, but any drifts will cause a frequency offset between the two compared clock loops and thus lead to incorrect measurement result. We propose a drift-insensitive self-comparison method to remove this frequency offset by adjusting the clock detection sequence. We also experimentally demonstrate the validity of this method in a one-dimensional ⁸⁷Sr optical lattice clock. As the clock laser frequency drift exists, the measured frequency difference between two identical clock loops is (240 ± 34) mHz using the traditional self-comparison method, while it is (−15 ± 16) mHz using the drift-insensitive self-comparison method, indicating that this frequency offset is cancelled within current measurement precision. We further use the drift-insensitive self-comparison technique to measure the collisional shift and the second-order Zeeman shift of our clock and the results show that the fractional collisional shift and the second-order Zeeman shift are 4.54(28) × 10⁻¹⁶ and 5.06(3) × 10⁻¹⁷, respectively.

1. Introduction

An optical lattice clock is not only a promising device to generate the second in the future due to its ultra-low uncertainty and instability [1,2,3,4], but also a powerful tool to observe physical phenomena such as verifying the general relativity [5,6,7], testing the Lorentz symmetry [8], detecting gravitational wave [9], and searching the dark matter [10,11,12]. The space optical lattice clock, which has been proposed by the European Space Agency (ESA) program [13,14,15], not only carries out geodesy and high-precision measurement of gravitational potential and gravitational redshift [16,17], but also improve the positioning accuracy of the global position system (GPS) and develops deep space navigation [18].

In terms of the uncertainty evaluation of an optical lattice clock, many systematic shifts are measured using the self-comparison method [19,20], such as the lattice AC Stark shift, the collisional shift, the clock laser AC Stark shift and the second-order Zeeman shift [21,22,23,24,25]. However, the drifts of the clock laser frequency and the stray electromagnetic field around the cold-atoms ensemble will lead to the self-comparison measurement error (SCME) [26]. The SCME, generally dominated by the clock laser frequency drift, leads to a frequency offset, of which the magnitude depends on the drift rate and the duration of the clock feedback cycle, and thus, causes incorrect measurement result of the self-comparison method. When the frequency drift rate changes regularly and slowly, this error can be reduced by adding a second-order integral loop to compensate the clock laser frequency using the acoustic optical modulator (AOM) [27], and the residual SCME can be well below 10⁻¹⁷ [24]. However, when the drift rate varies irregularly or fast, the residual SCME of this frequency compensation method could prevent the measurement accuracy of the self-comparison below 10⁻¹⁷. As the optical lattice clock operates outside the laboratory (the transportable optical lattice clocks and even the space optical lattice clocks), the complicated and changeable environment requires us to find more efficient techniques to eliminate the SCME.

In this paper, we propose a drift-insensitive self-comparison (DISC) method to cancel the SCME. By adjusting the interrogation sequence of the self-comparison method, the SCME is cancelled in every clock feedback cycle. The validity of this method is experimentally verified in ⁸⁷Sr optical lattice clock where the clock laser frequency drift dominates the drifts. Furthermore, with Rabi spectroscopy, the collisional shift and the second-order Zeeman shift in our system are carefully measured by using the DISC method.

2. The Principle of the Drift-Insensitive Self-Comparison Method and the Experimental Setup

2.1. The Principle of the Drift-Insensitive Self-Comparison Method

The self-comparison method is that two clock loops (R₁ and R₂) alternately operate in the time domain. As shown in Figure 1a, a clock feedback cycle of the traditional self-comparison (TSC) method contains four clock detection cycles (the duration of the clock detection cycle is 1 s in this experiment). In the first and the second clock detection cycles, where the clock operates in the systematic parameter of para.1, the initial clock laser frequency is set to f₀₁ − δ/2 and f₀₁ + δ/2, respectively, where the δ is the full width at half maximum (FWHM) of the spectral peak, and f₀₁ corresponds to the center frequency of the spectral peak of the R₁. After clock excitations, the excitation fractions of P₁ and P₂ are obtained, and thus the frequency correction can be calculated by Δf₁ = (P₂ − P₁)δ/(2P_max), where P_max is the maximum excitation fraction. The corrected frequency of f_01N = f₀₁ + Δf₁ is closer to the resonance of the R₁. In the same way, after the third and fourth clock detection cycles, where the clock operates in the systematic parameter of para.2, f_02N = f₀₂ + Δf₂ can be obtained. The frequency difference between the R₁ and R₂ can be expressed by Δυ = f_02N − f_01N which is determined every four clock detection cycles. By closed-loop operation, the uncertainty of the Δυ can be reduced, and thus the influence of a certain parameter on the clock transition frequency can be measured with high precision.

Obviously, in the process of TSC method, because the R₁ and R₂ successively operate in the time domain and the amount of the frequency drift accumulates over time, the frequency correction, caused by the frequency drift, of R₂ is twice that of R₁. To take the measurement of the collisional shift as an example, in the first two clock detection cycles, the atomic density is higher (H), while in the second two clock detection cycles, the atomic density is lower (L). If the drift rate is ζ Hz/s and the measurement sequence is HHLL, the drift-induced frequency offset for R₁ is about Δ_s1 = (ζ + 2ζ)/2 = 3ζ/2, and in terms of R₂, the drift-induced frequency offset is about Δ_s2 = (4ζ + 3ζ)/2 = 7ζ/2. Therefore, the SCME is about Δ_s = Δ_s2 − Δ_s1 = 2ζ as the density shift is measured by the TSC method.

According to the previous analysis, a simple but high-efficiency way (DISC method), which is realized by changing the detection sequence as HLLH (or LHHL) (as shown in Figure 1b), can be used to cancel the SCME and achieve drift-insensitive self-comparison measurement. When the detection sequence is HLLH, the drift-induced frequency offset is ${\Delta }_{\mathrm{s}1}^{\prime }=(\zeta +4\zeta )/2=5\zeta /2$ for R₁ and ${\Delta }_{\mathrm{s}2}^{\prime }=(2\zeta +3\zeta )/2=5\zeta /2$ for R₂. Thus, the SCME is ${\Delta }_{\mathrm{s}1}^{\prime }={\Delta }_{\mathrm{s}1}^{\prime }-{\Delta }_{\mathrm{s}2}^{\prime }=0$, indicating that the SCME is cancelled. The DISC method will not complicate the existing devices or consume extra time. Meanwhile, in terms of the cancellation of the SCME, the DISC method, which updates the frequency drift rate every four clock detection cycles, is expected to work better than the traditional way described in reference [27], which typically needs forty clock detection cycles to update the frequency drift rate. It’s worth noting that the DISC method cannot be used when the drift rate is very high (for example, the total frequency drift is larger than the FWHM of the spectrum in one clock feedback cycle) due to bad locking.

2.2. Description of the Experimental Setup of ⁸⁷Sr Optical Lattice Optical Clock

We experimentally verify the validity of the DISC method based on the one-dimensional ⁸⁷Sr optical lattice clock, the details of which is described in reference [26,28]. After two-stage laser cooling, about 8 × 10⁴ atoms are loaded into a horizontal one-dimensional optical lattice. Using the Pound–Drever–Hall (PDH) technique, the lattice laser wavelength is stabilized to an ultra-low expansion (ULE) cavity at 813.42 nm where the atomic polarizability of the clock ground state and the excited state is same, the so called “magic wavelength” [29]. Following that, the atoms are spin-polarized to the Zeeman sublevels of $|{}^1{S}_0,m_{\mathrm{F}}=+9/2\rangle$ with a spin-polarized purity of more than 99%. The clock laser is locked to an ULE cavity with a finesse of 400,000 by PDH stabilization at 698.44 nm corresponding to the 5s^{2 1}S₀→5s5p ³P₀ transition. The line-width of the clock laser is about 1 Hz obtained by beating with another similar clock laser system. The polarization of the clock laser and the lattice laser is linear and the direction is along the one of the magnetic field quantization axis which is parallel to the gravity. According to the resolved sideband spectroscopy, the longitudinal and radial temperatures of atoms trapped in the lattice are 2.9 and 2.7 μK, respectively. Additionally, the misalignment angle between the clock laser beam and the lattice light is about 6 mrad extracted from the Rabi oscillation of the carrier transition [30,31].

3. Experimental Results

3.1. Self-Comparison Measurement Error Cancellation using the Drift-Insensitive Self-Comparison Method

The SCME is evaluated by measuring the frequency difference between the two identical clock loops (the systematic parameters of R₁ and R₂ are the same) with the TSC and DISC methods, respectively. If there are no drifts at all, the expected frequency difference between R₁ and R₂ is zero. Figure 2a shows the frequency difference between R₁ and R₂ by TSC and DISC methods, respectively, without changing any experimental parameters but the detection order. The SCME of the TSC method is (240 ± 34) mHz, while the SCME measured using the DISC method is (−15 ± 16) mHz, where the measurement uncertainty is given by the last point of their respective total Allan deviation of the self-comparison instabilities. With the DISC method, the SCME is consistent with zero, indicating that the DISC method almost completely removes the frequency offset caused by the drifts even if the clock laser frequency drift is nonlinear and dramatically change as shown in the inset of Figure 2a. Furthermore, the comparison data shown in Figure 2a also demonstrate that the magnitude of the frequency difference fluctuation with the DISC method is significantly smaller than the TSC method. Therefore, the self-comparison instability of the DISC method is lower than the one using the TSC method. As shown in Figure 2b, the self-comparison instability of the TSC method is 5.6 × 10⁻¹⁵ τ^−0.5 (τ is the averaging time), while the self-comparison instability of the DISC method is 3.1 × 10⁻¹⁵ τ^−0.5, indicating that the DISC method cannot only cancel the SCME, but also improve the measurement accuracy when the averaging time exceeds 100 s. In order to eliminate the frequency offset caused by the drifts, we also try to use the AOM to compensate the frequency drift of the clock laser in the TSC method, where the frequency sweeping rate is calculated every 40 s. However, the frequency compensation method cannot completely remove the frequency offset and the residual frequency offset is more than 30 mHz which is caused by the rapidly changed drift rate in our system.

3.2. The Collisional Shift Evaluation

With Rabi spectrum, we further measure the collisional shift using the DISC method in our ⁸⁷Sr clock. The clock transition benefits from the detection of thousands of trapped cold-atoms simultaneous optical transitions, resulting in optical lattice clocks with ultra-low quantum projection noise limit. On the other hand, for a one-dimensional optical lattice clock, large number of atoms in the same lattice site will cause the collisional shift. Even for Fermions, the collisional shift can be more than 1 × 10⁻¹⁶ [32,33], indicating that the collisional shift should be carefully evaluated. As the trap potential and atomic temperature keep unchanged, the collisional shift depends on the atomic density and the excitation fraction (P_e). The way of measuring this shift is shown in Figure 1b, where we set Para.1 as high density (I₁) and Para.2 as low density (I₂) and the atomic density is changed by varying the current of the Zeeman slower, which changes the loaded atoms of the first-stage cooling and eventually changes the total atomic number of the lattice. I₁ and I₂, which are in direct proportion to the atomic density, represent the fluorescence intensity collected by the photomultiplier (PMT). In order to avoid the influence of the variation of the total atomic number, for the ith clock feedback cycle, the comparison result is divided by ΔI_i = I_1i − I_2i, to obtain the collisional shift of unit fluorescence intensity (marked by Δ_uf) [20,24], where the I_1i and I_2i correspond to the fluorescence intensity under the conditions of Para.1 and Para.2, respectively. Additionally, in terms of the regular operation of the clock, the collisional shift is calculated by multiplying the Δ_uf by the intensity I_nor that corresponds to the fluorescence intensity as the clock regularly operates. The gain of the PMT and the detection laser intensity remain constant during the whole measurement process.

With Rabi spectrum, the collisional shifts under different excitation fraction are measured as shown in Figure 3a. Typically, the atoms are prepared in ¹S₀ state (the ground state), but those points that the excitation fractions are larger than 0.4 are realized by preparing the atoms in ³P₀ state (the excited state). By linear fitting [34,35], the relationship of Δ_uf and P_e is determined as Δ_uf = 18.5(18)P_e − 11.6(7). Thus, for the regular clock operation of which the expected collisional shift is about 45Δ_uf, 1% change in the excitation fraction will lead to a change in fractional collisional shift of 2 × 10⁻¹⁷.

As the clock regularly operates, the average excitation fraction P_e-nor is about 0.4 which corresponds to the point of half the maximum excitation. Thus, we carefully evaluate the collisional shift when the excitation fraction is P_e-nor as shown in Figure 3b. The weighted mean of the ten measurements is 4.31(27) mHz per unit fluorescence intensity, where the measurement uncertainty has been multiplied by the square root of the reduced-chi-square ${\chi }_{\mathrm{red}}^2$ = 2.06. Thus, as the clock regularly runs, the corresponding fractional collisional shift is 4.54 × 10⁻¹⁶ with an uncertainty of 2.8 × 10⁻¹⁷. The residual SCME, caused by fast changed drift, will lead random frequency offset to each measurement, which eventually increases the uncertainty of the measurements. However, the experimental result shows that with the DISC method, the collisional shift is obtained with a precision of 2.8 × 10⁻¹⁷, though we have not compensated the frequency drift.

3.3. The Second-Order Zeeman Shift Evaluation

The DISC method can also be applied in the closed-loop operation of optical lattice clocks. For removing the first-order Zeeman shift, the lattice light vector shift and line pulling shift, the clock laser frequency of the ⁸⁷Sr optical lattice clock is usually stabilized to the average frequency of the transitions of m_F = +9/2→m_F = +9/2 and m_F = −9/2→m_F = −9/2. The magnitude of the bias magnetic field, which defines the magnetic field quantization axis, can be accurately extracted from the frequency gap between the two transitions. Benefiting from the DISC method, this frequency gap can be precisely extracted without the frequency offset caused by the clock laser frequency drift. Figure 4 shows the frequency gap of the m_F = +9/2 and m_F = −9/2 during the closed-loop operation of our clock with the DISC operation. Herein, Para.1 and Para.2 correspond to the m_F = +9/2→m_F = +9/2 and m_F = −9/2→m_F = −9/2 transitions, respectively, and the clock detection cycle is 0.6 s (the duration of the clock laser is 0.15 s). The average frequency gap is 297.8(8) Hz, where the uncertainty indicates the 95% confidence interval of the mean, and the corresponding magnetic field intensity is 305.2(8) mG [36]. As the second-order Zeeman coefficient is −23.37(3) MHz/T² [37], the fractional second-order Zeeman shift is 5.06(3) × 10⁻¹⁷.

4. Conclusions

In summary, we propose and experimentally demonstrate a drift-insensitive self-comparison method to eliminate the self-comparison measurement error and evaluate the systematic shifts. By measuring the frequency difference between two identical clock loops, the self-comparison measurement error is (240 ± 34) mHz using the traditional method, while it is (−15 ± 16) mHz using the drift-insensitive method. Based on the drift-insensitive self-comparison technique, we use Rabi spectrum to evaluate the collisional shift of our clock. With the DISC method, the collisional shift and the second-order Zeeman shift are evaluated as 4.54(28) × 10⁻¹⁶ and 5.06(3) × 10⁻¹⁷, respectively. The lattice light and the clock laser AC Stark shifts can also be measured by this method. The DISC method could be widely used in the evaluation of the space clocks that have less possibility to operate the clock under a condition of no drift or a linear drift due to complicated space environment. Combined with the drift compensation process, the SCME can be further suppressed even in a harsh experimental environment.