Cite this Article
Zhang Baolin, Huang Yao, Zhang Huaqing, Hao Yanmei, Zeng Mengyan, Guan Hua, Gao Kelin. Progress on the 40Ca+ ion optical clock. Chinese Physics B, 2020, 29(7): 074209  
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Progress on the 40Ca+ ion optical clock
Zhang Baolin1, 2, 3, Huang Yao1, 2, Zhang Huaqing1, 2, 3, Hao Yanmei1, 2, 3, Zeng Mengyan1, 2, 5, Guan Hua1, 2, †, Gao Kelin1, 2, 4, ‡
State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences (CAS), Wuhan 430071, China
Key Laboratory of Atomic Frequency Standards, Wuhan Institute of Physics and Mathematics, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430071, China
University of Chinese Academy of Sciences, Beijing 100049, China
Center for Cold Atom Physics, Chinese Academy of Sciences, Wuhan 430071, China
Huazhong University of Science and Technology, Wuhan 430074, China

 

† Corresponding author. E-mail: guanhua@wipm.ac.cn klgao@wipm.ac.cn

Project supported by the National Key Research and Development Program of China (Grant Nos. 2017YFA0304401, 2018YFA0307500, 2017YFA0304404, and 2017YFF0212003), the National Natural Science Foundation of China (Grant Nos. 11622434, 11774388, 11634013, 11934014, and 91736310), the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB21030100), the CAS Youth Innovation Promotion Association (Grant Nos. Y201963 and 2018364), and the Science Fund for Distinguished Young Scholars of Hubei Province, China (Grant No. 2017CFA040).

Abstract

Progress of the 40Ca+ ion optical clock based on the 42S1/2 – 3d 2D5/2 electric quadrupole transition is reported. By setting the drive frequency to the “magic” frequency Ω0, the frequency uncertainty caused by the scalar Stark shift and second-order Doppler shift induced by micromotion is reduced to the 10−19 level. By precisely measuring the differential static scalar polarizability Δα0, the uncertainty due to the blackbody radiation (BBR) shift (coefficient) is reduced to the 10−19 level. With the help of a second-order integrating servo algorithm, the uncertainty due to the servo error is reduced to the 10−18 level. The total fractional uncertainty of the 40Ca+ ion optical clock is then improved to 2.2× 10−17, whereas this value is mainly restricted by the uncertainty of the BBR shift due to temperature fluctuations. The state preparation is introduced together with improvements in the pulse sequence, and furthermore, a better signal to noise ratio (SNR) and less dead time are achieved. The clock stability of a single clock is improved to .

PACS: ;42.62.Fi;;37.10.Ty;;95.55.Sh;;06.20.-f;
Keyword:;40Ca+ ion optical clocks;;“magic” drive frequency;;frequency uncertainty;;frequency stability;
1. Introduction

Owing to their low uncertainty and instability, atomic optical frequency standards have found many irreplaceable and extremely important applications, such as in navigation and for tests of physical theories that need high precision. In the past two decades, the development of optical clocks has made great progress, surpassing the microwave Cs fountain clocks in terms of uncertainty and stability.

Optical frequency standards are mainly based on single ions trapped in ion traps such as Ba+, Sr+, Hg+, In+, Al+, Yb+, and Ca+, [1–5] or neutral atoms trapped in optical lattices such as Sr, Yb, and Hg.[6–10] Single-ion optical clocks always have simple ion-trap and optical systems as a single ion can be confined by a radio frequency (RF) potential for as long as several days or even a month. To date, the Al+ ion of NIST has achieved the smallest uncertainty of 9.4× 10−19.[11] Optical lattice clocks can provide better stability because of their large number of atoms. Benefiting from the development of the ultra-narrow-linewidth laser technology, the Sr lattice optical clock in JILA has achieved the best stability of with the use of an ultra-stable clock laser based on cryogenic silicon reference cavities.[12] Another good stability of has also been measured by NIST based on their Yb lattice optical clock.[8]

The optical clocks based on the 42S1/2–3d 2D5/2 electric quadrupole transition of a single trapped 40Ca+ ion have been studied by our group at the Wuhan Institute of Physics and Mathematics (WIPM), Chinese Academy of Sciences in China for many years. The reason why we chose the 40Ca+ ion is its relatively simple energy level structure and laser system in which all lasers mentioned in the energy-level scheme (Fig. 1) can be generated by diode lasers. However, compared with other ions, especially the Al+ ion, the 40Ca+ ion has a greater blackbody radiation (BBR) shift and uncertainty.

Fig. 1. Partial energy-level diagram of 40Ca+ showing the transitions used in cooling, repumping, and probing.

For optical clocks, the most important criteria are low uncertainty and instability. In 2015, we achieved an uncertainty of 6.5 × 10−16 and a stability of for a single 40Ca+ optical clock.[13] To improve these two criteria, many efforts have been adopted. On one hand, we introduce the “magic” RF drive frequency to reduce the micromotion-induced uncertainty,[3,14,15] a second-order integrating servo algorithm is applied to reduce the servo uncertainty,[14,16] and a 40Ca+ optical clock cryogenic system cooled by liquid nitrogen is setting up to reduce the uncertainty of the BBR field evaluation of temperature.[15] On the other hand, we apply a state-preparation technology to increase the SNR of the clock transition, and the high-resolution timing control system based on the field-programmable gate arrays (FPGAs) and direct digital synthesizers (DDSs) to reduce dead time.[6] The SNR and dead time are crucial for the stability of optical clocks. Finally, we have obtained a total frequency uncertainty of 2.2× 10−17 [15] and a stability of based on an asynchronous comparison of two independent clocks. In this paper, we will go through each of these improvements in detail.

2. Experimental setup

The 40Ca+ ion optical clock system mainly consists of two parts. One is the ion trap system, containing an ion trap, a vacuum chamber, magnetic shields, and fluorescence detection devices. The other is a laser system, for the 40Ca+ ion, which is composed of 397-nm, 866-nm, 854-nm, and 729-nm lasers and their frequency-stabilization devices.[7]

2.1. Ion trap system

In 2015, we used a miniature Paul ring trap to trap a 40Ca+ ion. To reduce the shift and its uncertainty due to the secular motion, an ion trap with a lower heating rate is required. A new trap originally designed by NIST[11] with small modifications is adopted. A schematic diagram of this trap is shown in Fig. 2, and the vacuum chamber of the 40Ca+ ion system is shown in Fig. 3. The outer vacuum chamber of the 40Ca+ ion system is made of stainless steel. There are 6 glass windows on the periphery for laser access, and a glass window on the upper part for collection of the ion fluorescence. The clock transition can be split into 10 Zeeman components with a weak (in μT level) magnetic field.[18] Owing to the spinless nucleus of the 40Ca+ ion, every component is very sensitive to the magnetic field. Small variations in the magnetic field could greatly broaden the spectrum of the clock transition; therefore, we installed two layers of magnetic shields outside the vacuum chamber to decrease this variation.

Fig. 2. Schematic diagram of the new linear Paul trap. V0 cos (Ω t) is the RF potential. V1 and V2 are the horizontal and vertical compensation voltages, which are used to compensate for the static electric stray field at the center of the ion trap.
Fig. 3. Vacuum chamber and fluorescence collection devices of a 40Ca+ ion optical clock.
2.2. Laser system

As shown in Fig. 1, the 40Ca+ ion has a simple energy level system. Only 4 lasers with wavelengths of 397 nm, 866 nm, 854 nm, and 729 nm are used. The strongly allowed 42S1/2 → 42P1/2 dipole transition is used for cooling, with a corresponding wavelength of 397 nm. For the cooling process, it is necessary to use an 866-nm laser to repump the ion from the metastable state 32D3/2 return to the S–P cooling branch, as there is a chance that the ion decays from the 42P1/2 state to the 32D3/2 state. After being excited to the 32D5/2 state by a probe pulse, the state of the ion will be read out by fluorescence detection. Another 854-nm laser is used to clear out the 32D5/2 metastable state after each interrogation. In the laser system, the 397-nm and 866-nm lasers are generated by commercial diode lasers and stabilized by an ultra-low expansion (ULE) cavity. The ultra-stable clock laser is stabilized by a 10-cm-long ULE reference cavity with a stability of ∼ 1 × 10−15 at 1 s–20 s. Because a high pumping efficiency can be obtained within a large frequency range for the 854-nm laser, there is no frequency-stabilization device for it. The frequencies and amplitudes of all lasers are manipulated by acousto-optic modulators (AOMs). We only need to change the RFs generated by the DDSs to change the frequencies and amplitudes of the laser beams or switch it on and off.

3. Time sequence control system

In this work, the improvements on the pulse sequence mainly focus on a reduction in dead time and the introduction of state-preparation technology. The new control system based on the FPGAs and DDSs can alter the RFs applied to the AOMs within a few microseconds. Then, we can change the frequency and power of the cooling laser beam many times within one cycle to realize optimal cooling and fluorescence-detection efficiency simultaneously. The dead time caused by modifying all RFs is negligible. Moreover, with the implementation of the state preparation nearly double the previous excitation probability on the resonance of the clock transition can be achieved, which means a lower instability.

3.1. Cooling and detection pulses

Figure 4 shows the spectrum of the transition of 42S1/2 → 42P1/2 used for cooling and fluorescence detection in 40Ca+ ion optical clocks. Three detuning frequencies (precooling, Doppler-cooling, and PMT shown in Fig. 4) are used for cooling and detection. First, the cooling laser is far-red detuned (precooling point), then near-red detuned (Doppler-cooling point) by approximately half the linewidth from the cooling transition line center to further cool the ion. In the cooling stage, the 866-nm laser beam is switched on to repump the ion to the 42P1/2 state, due to the branching fraction of the decay of the 42P1/2 state to the 42S1/2 state. One reason for dividing the cooling process into two stages is that the 854-nm-laser pulse is inevitably required to clear out the 2D5/2 state after each interrogation, so the first cooling process can be merged into it. The other is that if the ion is very hot, Doppler cooling will be less efficient than precooling. The measured temperature of the cooled ion is 1.0(5) mK, which is close to the Doppler cooling limit.[18]

Fig. 4. Spectrum of the cooling transition of 40Ca+ ion. There are three frequency detuning points, which are used in precooling, Doppler cooling, and PMT detection. The goal is to achieve optimal efficiency for each process.

As shown in Fig. 4, the strongest fluorescence signal can be obtained at the PMT point.[3] The FPGAs and DDSs control systems can implement the cooling and detection pulses at different frequencies without introducing additional dead time. The final state of the 40Ca+ ion after interrogation can be distinguished by a state detection pulse. If the 40Ca+ ion is in metastable state 2D5/2, only the background fluorescence signal can be obtained. By optimizing the imaging mirror and the related parameters of the 397-nm and 866-nm lasers, we obtained an average fluorescence count of 30 in a 0.5-ms-long detection, with a background count of less than 6.

3.2. State preparation

For the 40Ca+ ion, the ground state 2S1/2 has two magnetic sublevels with equal probability, so the greatest excited probability for each Zeeman component is only ∼ 0.5, as there is only a 50% chance that the ion is in the 2S1/2(m = +1/2) or 2S1/2(m = –1/2) states initially. Initial state preparation can increase the transition probability twice and improve the stability by times.[16,19] Many methods can be used to implement state preparation, depending on the energy-level structure of the ion and the requirements of the experiment.[19] In our lab, we chose the frequency-resolved optical pumping on the clock transition. Compared with the optical pumping of circularly polarized light on the cooling transition, it takes more time but only requires a simple setup.[20] Specifically, as shown in Fig. 5, the probe laser’s frequency is set at the resonance of one of the two transitions of 2S1/2(m = +1/2) → 2D5/2(m = +1/2) or 2S1/2(m = –1/2) → 2D5/2(m = –1/2) state to pump the ion from the 2S1/2 to the 2D5/2 state. An 854-nm-laser pulse will pump the ion from the 2D5/2 state to the 2P3/2 state and then it will decay to the two 2S1/2(m = ± 1/2) states with equal probability. By repeating this cycle for approximately 5 times, we obtained a > 95% probability of preparing the 40Ca+ ion in one of the two ground states. Figure 6 shows the Zeeman-resolved spectrum with and without state preparation.

Fig. 5. Schematic diagram of the state preparation. Firstly the 40Ca+ ion is excited to the metastable 2D5/2(m = +1/2) state form the 2S1/2(m = +1/2) state by the 729-nm laser, then to the 2P3/2 state by the 854-nm laser, and spontaneously decay to the two grounds with equal probability from the 2P3/2 state. After several times, the 40Ca+ ion would be pumped to the 2S1/2(m = –1/2) state.
Fig. 6. Zeeman-resolved spectra of the 2S1/2 → 2D5/2 transition of a single 40Ca+ ion obtained at the same probe laser power. The ground state sublevel mJ = ± 1/2 and the upper state sublevel mJ′ (2D5/2) are shown below and above each arrow respectively. (a) Spectrum shows 10 Zeeman components completely and is obtained without state preparation. In the (b) spectrum, the 40Ca+ ion has been pumped to the 2S1/2(m = +1/2) state; thus, only 5 Zeeman components are obtained, but with a higher transition probability.
3.3. Probing pulse

In the interrogation, other lasers, except for the probe laser, have been switched off by mechanical shutters instead of the AOMs to cancel other lasers’ AC Stark shift.[14,21] It adds a 5-ms-long dead time twice, for opening or closing the mechanical shutter before or after the integration, respectively. In this experiment, the two clocks share the same clock laser and timing control system, i.e., for the two clocks, the interrogation is asynchronous. To improve the stability compared to the previous result,[13] we choose the Ramsey scheme with which a single π-pulse is separated equally by a free evolution time Tfree. For a sufficiently short pulse duration, the linewidth of the central fringe is independent of the Rabi pulsation and is equal to 1/2Tfree.[6] This is much narrower than the Rabi result on the condition of the same time spent in the interrogation.

3.4. Clock transition spectroscopy

For a single ion system, to obtain the transition probability at each frequency point, the ion needs to be interrogated multiple times with the same parameters, and the successful excitations will be counted. For the 2S1/2 → 2D5/2 transition, if the ion is found in the 2D5/2 state by detection after interrogation, it has been excited successfully.[22] Thirty integrations at each frequency point were taken, and a Ramsey spectrum with a free evolution time of 80 ms is obtained, as shown in Fig. 7. More integrations can reduce the uncertainty of the transition probability, but take longer time.

Fig. 7. The pulse sequence for one interrogation cycle. In this experiment, the free evolution time is Tfree = 80 ms and the time for one cycle is Tc = 99.6 ms. The fluorescence is collected at D1 for state detection and at D2 to verify that the ion has returned to the S state after an 854-nm pulse.
4. Evaluation of systematic frequency shifts

A variety of potential sources might be associated with the 2S1/2 → 2D5/2 electric-quadrupole transition of a laser-cooled trapped 40Ca+ ion, causing a frequency shift and uncertainty. They have been evaluated in our previous work.[14,21] In this paper, we report the recent progress on the evaluation of the scalar Stark shift and the second-order Doppler shift induced by excess micromotion, the BBR shift, and the servo shift.

4.1. Excess micromotion uncertainty induced by scalar Stark shift and second-order Doppler shift

Figure 2 shows a schematic diagram of the latest ion trap, modified by a linear Paul trap. The potential near the axis of the trap can be approximated by[23]

where V0 cos (Ω t) and U0 are the potential added to the electrodes and endcaps, respectively, R is the perpendicular distance from the trap axis to the trap electrodes, κ (<1) is a geometrical factor, and Z0 is the distance between the two cap electrodes. After solving the Mathieu equation, we obtain the position of the ion in a typical case with |qi|≪ 1 and |ai|≪ 1 [23]

where ui is the position of the ion (i = x,y,z), φSi is the initial phase, and is the secular motion angular frequency. u1i is the amplitude of the secular motion, reflecting the ion temperature, which can be reduced by Doppler or sideband cooling.

If we consider a uniform static electric field Edc, equation (4) becomes[23]

where

The field Edc displaces the averaging position u0i (i = x,y,z) from the center of the trap, and induces a micromotion with an amplitude of and an angular frequency Ω. This excess micromotion is caused by a uniform static electric field Edc and the AC electric fields of the trap. From the expression of its amplitude, one can ascertain that it cannot be reduced by cooling, but it can be evaluated by monitoring the change of the ion’s place with a high-resolution camera, while the trap depth is alternated between strong and weak. Moreover, by varying the trim electrode voltages, we can find a set of ideal parameters where the position of the ion does not change with the trap depth. This was the most effective way to minimize excess micromotion in our previous work.[14,21]

Ion movement can create a second-order Doppler (time-dilation) shift and a scalar AC Stark shift due to the AC field the ion experiences. If we consider only the shifts caused by the excess micromotion, they are approximately[3,15,23]

where Δα0 = α0 (D5/2) – α0(S1/2) is the differential static scalar polarizability of the clock transition.

From Eqs. (1) and (5), we can obtain the AC electric field and the velocity of the ion. By placing the position obtained in Eq. (5) to the AC electric field, equation (7) can be simplified to[3,15,23]

where Ri is the micromotion sideband-to-carrier intensity ratio of one Zeeman component. For the 40Ca+ ion, Δα0 is a negative value, so we can choose a drive frequency of to cancel the scalar Stark shift and second-order Doppler shift induced by excess micromotion. The frequency Ω0 is called the “magic” trap drive frequency.

The experiment of measuring the frequency Ω0 is based on the comparison of two clocks (clock 1 and clock 2). Clock 1 is used to provide a constant standard frequency reference, and clock 2 is a test trap to study the change in micromotion shifts with trap drive frequency Ω. To begin with, all indicators of frequency shift and uncertainty of both clocks were optimized to the best with a fractional frequency uncertainty of ∼ 3× 10−17. We can guarantee that the transition frequency of clock 1 is a constant value within the uncertainty level, and that any frequency change exceeding the uncertainty level of clock 2 is entirely caused by micromotion. The DC offset voltage of clock 2 is trimmed to induce a relatively large micromotion level in the ion for an obvious comparison. Then, the frequency difference of the two clocks is due to the micromotion shift of clock 2. The drive frequency of clock 2 is tuned from 22 MHz to 27 MHz and the frequency difference at each point was measured, as shown in Fig. 9.[15]

Fig. 8. The high-resolution spectrum with a free evolution time of 80 ms. The black points are the probabilities of the clock transition and the red solid line is the fitting result of a Ramsey line shape, corresponding to a linewidth of approximately 6.5 Hz.
Fig. 9. Frequency difference between the two 40Ca+ ion clocks as a function of the RF drive frequency of clock 2. The red curve is a fit to the data and gives Ω0 = 2π × 24.81(1) MHz. The shaded area illustrates the uncertainty in the theoretical result.[15]

To obtain the value of Δα0 with a lower uncertainty by Ω0, we have to consider the micromotion at the trap drive harmonics nΩ. The total scalar Stark shift and the time-dilation shift can be expressed as[3,15]

By solving Eq. (9), Δα0 is calculated to be –7.2677(21)× 10−40 J⋅m2⋅V−2.[15] From Eq. (8) and the previous evaluation method, we evaluated an excess micromotion fractional uncertainty of 4× 10−19.[15]

4.2. Blackbody radiation (BBR) shift of coefficient ±Δα0

To date, the BBR shift uncertainty is the main source of the total uncertainty for the 40Ca+ ion.[24] The BBR shift of the S–D transition can be expressed as[25–27]

where T is the BBR field temperature, h is Planck’s constant, and η is a temperature-dependent dynamic correction to Δα0. From Eq. (10), the BBR shift uncertainty is caused by two parts: the uncertainty of coefficient Δα0 and temperature T. By using the value of –7.2677(21)× 10−40 J⋅m2⋅V−2, the uncertainty caused by the BBR shift (coefficient Δα0) at room temperature has been reduced to 3× 10−19,[15] benefiting from the calculated coefficient Δα0 having a lower uncertainty than the theoretical value.

4.3. Servo

In contrast to optical lattice atomic clocks, a single interrogation of a single ion can only provide one binary unit of information. In one frequency feedback cycle, the ion should be interrogated multiple times on the two sides of each Zeeman component to obtain the transition probabilities or error signal. The probe laser’s frequency is f and the ion is interrogated n times at the frequencies of f+ = f + δm/2 and f− = f – δm/2 respectively, where δm is the linewidth. After counting the successful excitations n+ at f+ and n– at f–, the error signal and the frequency modification of the probe laser can be expressed as follows:[16,22]

where g is a numerical coefficient, called “gain”. A frequency correction g ⋅ e is added to the laser frequency by an AOM to compensate for any frequency change of the clock laser stabilized by an ULE reference cavity. The servo algorithm is used to lock the frequency of the probe laser to the resonance of the clock transition. In this section, the clock laser refers to the laser beam stabilized by the reference cavity, and the probe laser represents the beam that passes through the position of the ion after modulated by an AOM.

Generally, the frequency of the clock laser is subject to drift with a nonlinear rate, similar to the probe laser between each frequency modification. However, for an ion optical clock, each feedback takes about a few seconds or more. The probe laser’s frequency may change significantly and randomly owing to the noise of the reference cavity. This will cause a significant servo shift and error. A second-order integrating servo algorithm is introduced to obtain the current drift rate by the analysis of the latest frequencies, which reduces the servo error[14,16,28]

where edr refers to the current drift rate, [f1,…, fN] are the latest N frequencies, and N is determined by the drift characteristics of the clock laser. Then, the modified frequency of the probe laser in the next servo cycle can be expressed as[16]

where g1 is another numerical coefficient.

Benefiting from the implementation of a second-order integrating servo algorithm and a longer probe time, the servo-error uncertainty is reduced to 7.6× 10−18.[15]

4.4. 40Ca+ uncertainty budget

Table 1 summarizes the latest results of the systematic shifts and their uncertainties for the 40Ca+ ion in our group. The total fractional uncertainty obtained by adding the contributions in quadrature is 2.2× 10−17. The BBR shift (temperature) is the largest contribution to the total uncertainty, but it can be reduced by lowering the temperature of the ion surroundings or a more accurate evaluation of the temperature.

Table 1.

System shifts and uncertainties for the evaluation of the 40Ca+ ion.[5]

.
5. Stability of 40Ca+ ion optical clock

Stability is an important criterion for optical clocks besides uncertainty. To measure the frequency stability of a 40Ca+ ion optical clock, we compared the frequency difference between two independent clocks continuously for approximately 38 h. In this experiment, both clocks have independent physical and optical systems, except for the probe laser, which is stabilized by a 10-cm-long ULE cavity.[7] Both 40Ca+ ions are interrogated asynchronously. To cancel the electric quadrupole shift and quadratic Stark shift, we lock the probe laser to two pairs of the Zeeman components.[14,29] One pair is 2S1/2(m = +1/2) → 2D5/2(m = +1/2) and 2S1/2(m = –1/2) → 2D5/2(m = –1/2), and the other is 2S1/2(m = –1/2) → 2D5/2(m = –3/2) and 2S1/2(m = +1/2) → 2D5/2(m = +3/2). The pulse sequence is shown in Fig. 7. Eight frequency points on the left and right sides of the 4 Zeeman components are interrogated. To obtain the excited probability on each point or the total error signal, the same interrogation at each point is repeated 10 times.

The center frequency of the 2S1/2 → 2D5/2 transition for the 40Ca + ion can be computed by the frequencies of the four Zeeman components[14,30]

where f1 and f2 are the frequencies of the transitions of 2S1/2(m = ± 1/2) → 2D5/2(m = ± 1/2), and f3 and f4 are the frequencies of the transitions of 2S1/2(m = ± 1/2) → 2D5/2(m = ± 3/2). The frequency difference of the two clocks is obtained from the RF frequencies applied to the AOMs as they share the same clock laser. Figure 11 shows the results of the frequency difference of each comparison. The stability of each Ca+ optical clock is , which is deduced from the Allan deviation of the comparison between the two clocks, as shown in Fig. 12.

Fig. 10. Measurement of Ω0 obtained by scanning the RF drive frequency of clock 2. To reduce the measurement statistical uncertainty of Ω0, we repeated this scanning randomly from 24.7 MHz to 24.95 MHz 15 times. Fifteen scans were recorded (data shown as black squares). The corresponding weighted mean of Ω0 is shown by the red circle. The inset shows the data taken in one of the scans and a linear fit through the data as shown in Fig. 8. and obtained the result of Ω0 = 2π × 24.801(2) MHz.[15]
Fig. 11. Frequency difference between two 40Ca+ ion optical clocks.
Fig. 12. Normal Allan deviation is measured for a single clock. The stability of a single clock is achieved by dividing the Allan deviation of the comparison of two clocks. The result is .
6. Summary and prospects

To improve the performance of the 40Ca+ optical clock, several efforts have been made in recent years. For the uncertainty, we precisely measured the “magic” trap drive frequency Ω0 at which the shift of the scalar Stark and second-order Doppler induced by excess micromotion can cancel each other, corresponding to an uncertainty of below 1× 10−18. Then, the differential static scalar polarizability Δα0 is obtained by using the measured “magic” trap drive frequency Ω0, with an uncertainty about 40 times smaller than the value of the theoretical calculation. Benefiting from the more accurate value of Δα0, the uncertainty due to the BBR shift (coefficient) has also been evaluated to be 3× 10−19. Additionally, the servo error uncertainty is also reduced to the 10−18 level by introducing a second-order integrating servo algorithm. Finally, the total uncertainty of the 40Ca+ ion optical clock is evaluated to be 2.2× 10−17. For stability, we increased the pulse length to 80 ms and obtained a high-resolution Ramsey spectrum with a linewidth of approximately 6.5 Hz. The time spent on cooling and detection has been reduced significantly by changing the frequency of the cooling laser on the precooling, Doppler, and detection pulses. Furthermore, by preparing 40Ca+ ion in one of the two ground states, the transition probability is double than before, which greatly reduces the QPN. With these improvements, the Allan deviation is measured to be by a Ramsey-excited comparison of two independent 40Ca+ ion optical clocks.

For the BBR shift due to temperature fluctuations, according to Eq. (10), the frequency shift is related to T4. Then, lowering the temperature around the ion can significantly reduce the BBR shift (temperature) as well as its uncertainty. In our laboratory environment, the temperature is measured as 21.0(1.5) °C (294.15(1.5) K), corresponding to a 7.8-mHz frequency uncertainty.[8] It is difficult to evaluate the room temperature with a lower uncertainty under existing experimental conditions. Therefore, we are setting up a liquid nitrogen cryogenic system as shown in Fig. 13, which puts the ion at the temperature of liquid nitrogen of approximately –196 °C (77.15 K). The shift at liquid nitrogen temperature can be reduced by two orders of magnitude smaller to the uncertainty under the same temperature uncertainty assessment. The new system is expected to be able to evaluate the uncertainty of the BBR shift (temperature) to the 10−18 level.

Fig. 13. The liquid nitrogen cryogenic 40Ca+ ion optical clock system.

The stability of the 40Ca+ ion optical clocks is ultimately limited by its metastable state lifetime of 1.16 s.[31] For Rabi interrogation, this limit is for a probe time of ≈ 1.5 s. For Ramsey interrogation, this limit is for a free evolution time of ≈ 1 s.[16,19]

Lower instability for single-ion optical clocks can be obtained by a longer probe time, which will be adopted in the future, with the help of a lower frequency noise by the clock laser and a shorter dead time.

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