A liquid nitrogen-cooled Ca + optical clock with systematic uncertainty of 3×10 -18

Here we present a liquid nitrogen-cooled Ca + optical clock with an overall systematic uncertainty of 3 × 10 -18 . In contrast with the room-temperature Ca + optical clock that we have reported previously, the temperature of the blackbody radiation (BBR) shield in vacuum has been reduced to 82(5) K using liquid nitrogen. An ion trap with a lower heating rate and improved cooling lasers were also introduced. This allows cooling the ion temperature to the Doppler cooling limit during the clock operation, and the systematic uncertainty due to the ion’s secular (thermal) motion is reduced to < 1 × 10 -18 . The uncertainty due to the probe laser light shift and the servo error are also reduced to < 1 × 10 -19 and 4 × 10 -19 with the hyper-Ramsey method and the higher-order servo algorithm, respectively. By comparing the output frequency of the cryogenic clock to that of a room-temperature clock, the differential BBR shift between the two was measured with a fractional statistical uncertainty of 7 × 10 -18 . The differential BBR shift was used to calculate the static differential polarizability, and it was found in excellent agreement with our previous measurement with a different method. This work suggests that the BBR shift of optical clocks can be well suppressed in a liquid nitrogen environment. This is advantageous because conventional liquid-helium cryogenic systems for optical clocks are more expensive and complicated. Moreover, the proposed system can be used to

as Al + that are not sensitive to BBR. In many state-of-the-art single-ion optical clocks and OLCs, the total systematic uncertainty is often dominated by BBR shift uncertainties [9,11,15]. Therefore, reducing the BBR shift is crucial in realizing high-precision atomic clocks.
For an atomic clock, the BBR shift would be [16] where Δα(0) is the differential scalar polarizability for the clock transition, h is the Plank's constant, 〈 2 〉 =[8.319430(15) V/cm] 2 (T/300 K) 4 is the mean-squared electric field in a BBR environment at the temperature T [17], η(T) gives a small dynamic correction to the BBR shift. The evaluation of ∆ needs both the knowledge of Δα(0), η(T) and the precise evaluation of the BBR environmental temperature T.
A possible approach to address these issues involves choosing optical standards that are not sensitive to BBR shifts. In particular, the BBR shift sensitivity for the clock transition in Al + is over hundreds of times smaller than that of other optical clocks like Sr, Yb, Sr + , Yb + , or Ca + [9,11,12,14,15,18], and it can be reduced to a low level even at room temperature. Previous studies also have been investigated using Hg [19], In + [20], and Lu + [21] clocks with low sensitivity to BBR.
Highly-charged-ion-based optical clocks with negligible sensitivity to BBR have also been studied [22][23][24]. However, for many of the systems described above, they often require using deepultraviolet light sources; for some of the clocks, they can only be achieved with quantum logic spectroscopy [12].
Another approach to realize accurate reference systems for optical clocks is to accurately measure the sensitivity parameters Δα(0) and η(T), in the meantime, precisely measuring the ambient temperature T and thus giving a precisely evaluated fractional BBR shift. However, evaluating the uncertainties in the fractional BBR shift in the 10 -18 level at room temperature requires very high temperature accuracy, even when Δα(0) and η(T) have already been known with a very high accuracy. Evaluating the uncertainties in the fractional BBR shift in the 10 -18 level at room temperature is challenging and requires either the dynamic temperature calibration [9] or BBR shields with accurate systematic error analysis [25]. In particular, the RF fields used to trap ions in single-ion-based optical clocks would heat the ion traps, which results in an uneven thermal environment, making the BBR shift estimation even more difficult. As stated in Eq. (1), the BBR shift is proportional to the fourth power of the temperature (T 4 ). Thus, decreasing the temperature of the BBR field can reduce the BBR shift and the corresponding temperature dependence ( ∝ 3 ). For example, the BBR shift for 40 Ca + at room temperature is ~345 mHz [26], an evaluated temperature uncertainty of ΔT≈0.1 K yields a fractional uncertainty at 1×10 -18 , while the BBR shift would be reduced to only 1.6 mHz at the liquid nitrogen temperature of ~77 K. The fractional uncertainty can be reduced to 1×10 -18 , as long as the temperature is within an uncertainty of 4 K. The cryogenic environment has been used in the Hg + optical clock, the Sr OLC, and the Cs fountain clock, which greatly suppresses the BBR shift and its uncertainty [10,13,27].
In this work, we report a liquid nitrogen-cooled Ca + optical clock with an overall systematic uncertainty of 3.0×10 -18 , which is approximately an order of magnitude lower than that for our previously reported Ca + optical clocks at room temperature [18,26]. The differential BBR shift between the two systems was determined by comparing the corresponding clock frequencies. The differential BBR shift was further used to derive the static differential polarizability, which was consistent with our previously reported result [18]. This provides a test for the measured Δα(0) with a different method.
Ca + is one of the candidates that have high-Q optical transitions, it's natural linewidth is ~0.14 Hz, suitable for building a high-performance optical clock. Its level scheme is relatively simple, only low-cost diode lasers would be needed. As a result, portable and robust clock can be made [26], these features would broaden the application for the optical clocks. In our previous work, both the laboratory Ca + optical clocks [18] and the portable Ca + optical clocks [26] have been made, with systematic uncertainty at the 10 -17 level. The excess micromotion induced 2 nd -order Doppler shift and Stark shift, the Stark shift due to the secular motion, and the 2 nd -order Doppler shift due to the secular motion (micromotion induced) have been canceled by choosing the magic RF trapping frequency [15,18]. The quadrupole shift, the 1 st -order Zeeman shift, and the tensor Stark shifts due to the ion motion and the lasers have been canceled by averaging the 3 pairs of Zeeman transitions [28][29][30]. For the previous made optical clocks, the total systematic uncertainty is limited by the BBR shift, the 2 nd -order Doppler shift due to the secular motion, and the ac Stark shift (light shift), etc.
In this article, we introduce a recently built, liquid nitrogen-cooled Ca + optical clock, for which the liquid nitrogen environment greatly reduced the BBR shift and its uncertainties. A new ion trap is introduced for lower heating rates. In addition, cooling lasers under an optimized working condition were adopted for cooling the ion trap close to the Doppler cooling limit. This reduces the 2 nd -order Doppler shift due to the ion's secular motion. The hyper-Ramsey interrogation scheme [14,31] was adopted to reduce the uncertainties in the frequency shift induced by the probe beam.
Optimized servos were adopted to reduce servo-introduced uncertainties. Moreover, the 1 st -order Doppler shift was eliminated by using two probe beams in opposite directions for the detection [12].
A BBR shield and a special designed vacuum system were constructed for creating a liquid nitrogen environment to reduce the BBR shift and its uncertainty ( collection. The windows were made of antireflection coated BK7 glass. The transmittance of the BK7 glass was ~0 for wavelengths above 3 μm [25], thereby eliminating the effect of window transmittance on the BBR at room temperature. Three platinum resistance temperature probes were installed in the shield chamber, one at the top, one at the bottom, and one near the ion trap for the temperature measurement. There are eight holes of diameter 5 mm on the shield for keeping the vacuum environment for the ion trap.
The vacuum chamber was made of stainless steel, and three pairs of coils were installed outside the chamber for adjusting the strength and direction of the magnetic field. To achieve a more stable magnetic field environment for Ca + , four layers of magnetic shielding were added. However, the magnetic field fluctuation would still broaden the observed clock transition, thus degrading the clock locking performance. Optimization has been made through fine adjustment of the magnetic field directions. Since the magnetic field fluctuation in the vertical direction is much stronger than that in the horizontal directions, the magnetic field direction is set to be horizontal, and the overall magnetic field amplitude is less sensitive to the vertical environmental magnetic field fluctuations.
The temperature inside the BBR-shielding varied between 77 K and 80 K during the consumption of liquid nitrogen. Considering the solid angle ratio of the small holes and the influence of the heat conduction of several thick wires of thickness ~1 mm, the BBR shift was estimated to be 3.0(1.1) mHz (see more details in Methods). Accordingly, the temperature-associated fractional uncertainty in the BBR shift was determined to be 2.7 ×10 -18 . A diamond wafer based linear ion trap was used in the clock, which is similar to the NIST ion trap [12], but with larger size and slightly different structure. The highly symmetry and high precision laser machining help reducing the heating rates of the ion trap. The relatively open structure was also well suited for fluorescence detection at large solid angles, three-dimensional detection, and compensation for the micromotion of ions. The ion-trap heating rate was experimentally determined to be less than 1 mK/s, and less than 5 quanta/s for the frequencies of the radial secular motion. This is two orders of magnitude smaller than that of our previously reported ion trap [18,26]. In the experiment, both the 397-nm cooling beam and the 866-nm repumping beam were stabilized to an ultra-low expansion reference cavity, and the laser cooling of the ion was optimized by adjusting the frequencies of the acousto-optic modulators (AOMs).
Multiple three-dimensional temperature measurements confirmed that the temperature of the Doppler-cooled ion was close to the Doppler cooling limit, and the temperature uncertainty was more than an order of magnitude lower than that in our previous room-temperature optical clock [18,26]. Considering the trap heating rates, the temperature was estimated to be 0.65-1.17 mK during the period for clock transition detection in the optical clock experiment, ie, = 0.91 (26) mK. Accordingly, the 2 nd -order Doppler shift due to the secular motion was evaluated as -3.1(9)×10 -18 .
Similar to the previous optical clock setup, we eliminated the electric quadrupole shift, the 1 storder Zeeman shift, and the ion motion-and laser-induced tensor Stark shift by averaging the frequencies of 3 pairs of Zeeman transitions with different 2 D5/2 state sublevels [28][29][30]. However, the slow drift in the electric quadrupole shift over time might still lead to a residual uncertainty. A specific magnetic field direction is chosen to achieve a magic angle between the magnetic field and electric field gradients for reducing the drift rate of the electric quadrupole shift. This further reduced the residual uncertainty, leading to a reduction in the electric quadrupole shift by ~2 orders of magnitude. However, the three pairs of Zeeman sublevels had different Rabi frequencies in this specific magnetic field direction for the same probe-beam intensity. The stability of the optical clock was optimized by varying the power of the probe beam for different Zeeman sublevels. Accordingly, the ac Stark shift caused by the probe beam was also different for different Zeeman sublevels. Thus, the electric quadrupole shift could not be eliminated by simply averaging the frequencies of the three Zeeman pairs. Furthermore, the collimation of the probe beam would change over time, which introduces an additional systematic uncertainty. Therefore, it is important to suppress the ac Stark shift caused by the probe beam, which was greatly suppressed in the present experiment by adopting the hyper-Ramsey interrogation scheme [14,31]. The hyper-Ramsey interrogation scheme basically eliminates the optical frequency shift, and solves the above problems fundamentally. The fractional uncertainty of the frequency shift was estimated to be < 10 -19 . For laser beams other than the probe beams, AOMs with mechanical shutters were used to eliminate the light shift [30].
When locking the frequency of the probe laser to the resonance of the clock transition, frequency drifts of the probe laser would cause a servo error, leading to a systematic frequency shift.
The systematic frequency shift can be evaluated by statistically analyzing the quantum jump imbalances as an indicator of the servo shift and uncertainty [30]. In our experiment, a higher-order servo loop was used to dynamically lock quantum jump imbalances to 0. Moreover, the drift rate of the probe beam frequency was measured to be less than 63 mHz/s, and accordingly, the upper limit of servo-induced frequency shift was simulated to be 0.16 mHz, or 4×10 -19 .
For our cryogenic clock, it is important for evaluating the 1 st -order Doppler shift since the ion trap still has a possible displacement during the clock running: the displacement might be very small and have a slow variation in speed, difficult to be observed. Besides, the photoelectric effect from laser beams may lead to changes in the stray electric field, which results in ion displacement in the ion trap. This in turn leads to a 1 st -order Doppler shift of up to order of 10 -17 when not suppressed [12]. In our experiment, two laser beams in opposite directions were used for independent and interleaved probing the ion, by averaging the two independent frequency measurements, the 1 storder Doppler shift can be eliminated. The clock transition frequencies observed in our experiment with two counter-propagated probe beams differed from each other within 1×10 -17 . The uncertainty in the 1 st -order Doppler shift was estimated to be 3×10 -19 with similar method to Ref [12].
There are a few other contributions for the systematic uncertainty, while they contribute < 1×10 -19 to the total uncertainty. Since the determination of the blackbody environment is very important to the uncertainty budget of an optical clock, also for the verification of the reliability of our previous results [18], we further compared the frequencies of the cryogenic and the room-temperature optical clocks. Based on the frequency difference, the BBR shift could be determined directly, which could be used to further derive the sensitivity parameters Δα(0). The sensitivity parameters in our study were compared with those previously result for cross-verification. It was difficult to further improve the accuracy in our previous studies by using the magic trapping frequency measurements [18].
However, such a comparison is necessary, as it provides an independent method to verify the evaluation results.
The ion trap used in the room-temperature optical clock is similar to that in the NIST optical clock [12]. The diamond material and silver-plated oxygen-free copper support employed has high thermal conductivity, thereby facilitating heat conduction after RF heating of the ion trap.
Furthermore, the ion-trap electrodes are coated with a thicker layer of gold to reduce the heating effect due to the RF field. In addition, the magic trapping frequency (~2π×24.8 MHz) allows the optical clock to work at low RF power (< 2 W). This further reduces the RF heating to the ion trap.
However, fluctuations in room temperature cause the vacuum chamber temperature fluctuating by ±1.5 K, leading to a total systematic uncertainty of 1.7×10 -17 . The total systematic uncertainty is dominated by room temperature variations.
In the clock comparison experiment, a probe laser was used to synchronously probe both clocks [30]. However, both clocks were mutually independent in terms of the cooling, repumping, and quenching laser beams. Liquid nitrogen was completely consumed approximately 36 h after the liquid nitrogen container was filled, thus requiring the container to be filled once every day. This caused an interruption to the experiment for more than an hour every day. In addition to the systematic and statistical uncertainties, it is necessary to consider the gravitational redshift in the clock comparison experiment [5]. The two clocks were separated by about 8 m in the same laboratory with an ion-trap height difference of 15(1) cm as measured by a spirit level, equivalent to a frequency shift of 7.0(5) mHz or 1.7(1) ×10 -17 . Fig. 2 shows the frequency comparison results, each data point shows the average of a measurement last for ~18 000 s, the error bar shows the calculated stability for the measurement. The systematic shifts other than the BBR shift, including the gravitational shift have been corrected in the figure. Taken our previous measured Δα(0) and calculated η(T) [18], for our Ca + clocks with cryogenic temperature and room temperature, the BBR shift difference would be calculated as -342 (8)   In summary, we have developed a liquid nitrogen-cooled Ca + clock with an uncertainty of the order of 10 -18 . The differential BBR shift between the cryogenic clock (~82 K) and the roomtemperature clock (~293 K) was directly determined through a comparative experiment, and the result was used to further derive the static differential polarizability Δα(0) of the Ca + clock transition. In the future, further simulation and analysis will be performed on the liquid-nitrogen cryogenic shield to further reduce the temperature evaluation uncertainty [32]. Once the total uncertainty is dominated by uncertainties in the 2 nd -order Doppler shift, it is necessary to further decrease the ion-trap temperature below the Doppler cooling limit with 3-dimentional sideband cooling [33] or electromagnetically induced transparency cooling [34]. The total uncertainty can be further reduced to within 1×10 -18 . By introducing a probe laser with stability in the order of 1×10 -16 at 1-200 s, the probe time can be prolonged to ~ 1 s, it is possible to further improve the stability of Ca + optical clock, close to the single ion quantum projection noise limit [35]. The stability would be at the 10 -18 level with an averaging time of a day. It is also possible to take multiple ions as a reference to further improve the clock stability. Moreover, we are planning to develop a new cryogenic system to achieve a longer continuous operation period. Developing a cryogenic system that can continuously work without interruption is also in consideration. Afterwards, clock comparison with other optical clocks would be made for testing the uncertainty evaluation and the measurement of the clock frequency ratios [36].

Method
More details about the systematic uncertainty evaluation for the liquid nitrogen clock Ion trap. The ion trap we used in the experiment is shown in Fig. 3. The ion trap is similar to the one used in NIST Al + clocks [12], while with a modified design, the trap size is a bit larger. The ion trap is made of laser machined diamond wafer, while part of the diamond surface is gold plated to be used as trap electrodes, with a coating thickness of ~5 μm. The thickness of the diamond wafer is ~ 400 μm, while the distance between the RF electrodes and the trapped ion is ~ 400 μm, the two pairs of RF electrodes are designed fully symmetrical, for achieving smaller differential-RF -phase-   Table II gives the evaluated solid angle percentage view by the ion, the evaluated temperature, and the contributions to the BBR shift evaluation uncertainty. Excess micromotion induced shift. Excess micromotion exists when the ion's position is pushed away from the trap center by the stary field [37], the RF phase difference between the RF electrodes would also give rise to the micromotion [38]. The micromotion would cause both the 2 nd -order  [15,18], thus frequency shift uncertainty due to the excess micromotion was kept at a very low level, ~2×10 −19 .
Secular motion induced shift. During the clock run, the trapped ion is firstly laser cooled to a relatively low temperature (lower amplitude of secular motion), then the cooling lasers are blocked for avoiding the light shift, after that the state preparation, the clock interrogation, and the state detection are carried out. The ion' temperature can be evaluated by measuring the amplitude ratio between the clock transition carrier and its 1 st -order red secular sidebands [39]. When the cooling lasers are blocked, the ion trap would heat the ion, making the ion's temperature higher with longer state preparation and interrogation time. Both the ion's temperature and the ion trap's heating rate are measured every day. Fig. 4 shows the measured ion's temperature for the cryogenic clock during the clock comparison experiments. The ion's temperature is between 0.65 and 1.17 mK, close to the Doppler cooling limit, we take 0.91(26) mK for the evaluation of the secular motion induced shifts.
The secular motion would cause three kinds of shifts: the 2 nd -order Doppler shift, the Stark shift, and another different 2 nd -order Doppler shift due to the micromotion (secular motion induced).
Under the magic RF frequency, the Stark shift, and another different 2 nd -order Doppler shift due to the micromotion (secular motion induced) would cancel each other [15,18], taking the evaluated ion temperature, the total secular motion induced shift would be -3.1(9)×10 −18 .  (2 −1) where dEz/dz is the electric field gradient，β is the angle between the magnetic field and the electric field gradient, Θ(D,J) is the quadrupole moment. Since the electric field gradient is difficult to be precisely controlled or measured, it is difficult to precisely measure the quadrupole moment. Thus, in our experiment, the quadrupole shift is not evaluated by using Eq. (2), while it is eliminated by averaging mJ sub levels [28][29][30] 1 st -order Doppler shift. The photoelectric effect from laser beams may lead to changes in the stray electric field, which results in ion displacement in the ion trap. In the experiment, two laser beams in opposite directions were used for interleaved probing to eliminate such frequency shifts. For every probe pulse, only one beam is frequency tuned to the transition resonance, while the other is tuned to be 1 MHz red detuned. The two beams are well overlapped, and the angle between two opposite propagated beams is evaluated to be < 1 degree. The clock transition frequencies observed in the present experiment using two probe beams differed from each other within 1×10 -17 . Compared to Ref [12], the reason why no obvious 1 st -order Doppler shift is observed in our experiment is remain unclear, however, one possible explanation is the ultraviolet laser beams may cause a more obvious frequency shift. With our measured ion displacement, the uncertainty in the 1 st -order Doppler shift was estimated to be 3×10 -19 with similar method to Ref [12].
Other systematic shifts Other systematic shifts are also evaluated including: light shift due to lasers other than the probe laser, uncertainty due to the knowledge of the differential scalar polarizability and the dynamic correction [18] we have built another ion trap, identical to the room temperature clock, but with 3 in-vacuum temperature sensors, for directly measuring the temperature of the ion trap: one is installed at the diamond wafer, the other two are installed at the post for holding the trap. For the sensors on the post, one is ~ 1 cm away from the trap, the other is ~ 10 cm, close to the vacuum chamber. An RF field with the same frequency and amplitude is applied by checking the ion's secular frequencies, a temperature rise of 0.1(1) K is measured. In conclusion, we take 0.2 K as the upper limit of the temperature difference between the ion trap and the vacuum chamber.
Systematic uncertainty budget for the room temperature clock. For the room temperature clock, the systematic shift evaluation methods are mostly identical to the cryogenic clock, except for the BBR shift evaluation. The systematic uncertainty budget for the room temperature clock is shown in Table III.