Open Access
Add to BibSonomy
Abstract
The instability of the clock laser is one of the primary factors limiting the instability of the optical clocks. We present an ultra-stable clock laser based on a 30-cm-long transportable cavity with an instability of ∼3 × 10−16 at 1 s–100 s. The cavity is fixed by invar poles in three orthogonal directions to restrict the displacement, meeting the requirements of transportability and low vibration sensitivity. By applying the ultra-stable laser to a transportable 40Ca+ optical clock with a systematic uncertainty of 4.8 × 10−18 and using the real-time feedback algorithm to compensate the linear shift of the clock laser, the short-term stability of the transportable 40Ca+ optical clock has been greatly improved from $4.0 \times {10^{ - 15}}/\sqrt {\mathrm{\tau }/\textrm{s}} $ to $1.16 \times {10^{ - 15}}/\sqrt {\mathrm{\tau }/\textrm{s}} $, measured at ∼100 s–1000 s of averaging time, enriching its applications in metrology, optical frequency comparison, and time keeping.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical clocks can generate ultra-precise and ultra-stable optical frequencies [1], which are extensively utilized in time and frequency metrology [2], fundamental physics [3], relativistic geodesy [4], etc. Currently, the state-of-the-art single ion optical clocks based on 40Ca+, 27Al+, 171Yb+, 176Lu+ [5–8] and optical lattice clocks based on 171Yb, 87Sr [9–11] demonstrated fractional uncertainties of 10−18 or less, making them candidates for the redefinition of the SI second [2]. In terms of instability, the best 171Yb+ and 27Al+ optical clocks have demonstrated the instabilities of $1.0 \times {10^{ - 15}}/\sqrt {\mathrm{\tau }/\textrm{s}} {\; }$[7] and $1.2 \times {10^{ - 15}}/\sqrt {\mathrm{\tau }/\textrm{s }} $ [6], respectively, where $\mathrm{\tau }$ is averaging time. For optical lattice clocks, the 87Sr optical clocks have demonstrated the best instabilities of about $5 \times {10^{ - 17}}/\sqrt {\mathrm{\tau }/\textrm{s}} $ [12,13], and the 171Yb optical clock has demonstrated an instability of $6 \times {10^{ - 17}}/\sqrt {\mathrm{\tau }/\textrm{s}} $ [14]. In contrast to the high-performance optical clocks mentioned above, which are based on well controlled laboratories, the current transportable clocks for space applications and relativistic geodesy have achieved systematic uncertainties of mid 10−18 [15,16] but instabilities of $1.0 \times {10^{ - 15}}/\sqrt {\mathrm{\tau }/\textrm{s}} $ at best [17,18].
For transportable optical clocks used outside the laboratory, transportable ultra-stable lasers are critical to their instability performance and practical applications. To date, the best clock lasers have reached an instability of $3.5 \times {10^{ - 17}}$ at 124 K in the laboratory [19], while the instabilities of transportable clock lasers are typically one to two orders of magnitude worse because the reference cavities must be both robustly fixed and kept vibration insensitive. Among them, the frequency instability of RIKEN's optical cavity reaches $1 \times {10^{ - 15}}$ [17], which limits the instability of their transportable Sr optical clock. The instability of the transportable Ca+ optical clock is also limited by the reference cavity of the clock laser and has recently been reported to be $4.0 \times {10^{ - 15}}/\sqrt {\mathrm{\tau }/\textrm{s}} $ [15]. Since the length of transportable cavities is typically less than 10 cm [21–24], the thermal noise [25] of the cavity limits their reported frequency stability to a level of $> 3 \times {10^{ - 16}}$. Longer transportable cavities are then needed to further improve the instability of the optical clocks. The stability of the Physikalisch-Technische Bundesanstalt (PTB) transportable laser system has been improved to $1.6 \times {10^{ - 16}}\; $by locking the laser to a 20-cm-long transportable cavity [26]. A laser beating instability of $2 \times {10^{ - 16}}$ based on a 30-cm-long transportable cavity is reported later [27], which is close to the cavity’s thermal noise limit.
Compared to other transportable optical clocks [17,18,28–31], the 40Ca+ optical clock has a relatively simple laser system. All the lasers used in the optical clock are accessible by diode lasers, and the laser power requirements are only in the hundreds of microwatts, which is a significant advantage compared to optical lattice clocks. This facilitates the design and imple mentation of an integrated and transportable system. Recently, a Ca+ optical clock towards a transportable clock with a systematic uncertainty of 4.8 × 10−18 has been developed. However, its instability only reaches to $4 \times {10^{ - 15}}/\sqrt{\mathrm{\tau }}$. The main factor limiting instability is the performance of the transportable clock laser. For the 40Ca+ optical clock, the 729 nm ultra-stable laser is used to interrogate the 42S1/2-32D5/2 clock transitions of the 40Ca+ ion. Its instability was approximately $1 \times {10^{ - 15}}$ at 1 s-100 s [20] by locking to a 10-cm-long cavity. Obviously, to promote the application of high accuracy transportable 40Ca+ optical clock in the field of precision measurement in the future, improving the performance of the transportable laser is of prime concern. In this paper, we focus on improving the instability of the clock laser to make the ion optical clocks comparable to other transportable neutral atomic lattice clocks. Here, we developed a 30-cm-long transportable ultra-low expansion (ULE) cavity with fused silica (FS) substrates and a stable supporting structure. By beating it with the other 30-cm-long cavity in the laboratory, we achieved clock laser frequency instability of approximately $3 \times {10^{ - 16}}$ at 1 s -100 s of averaging time. This is the first application in transportable optical clock for this 30-cm-long design of transportable cavity. As far as we know, this cavity has the best stability published for transportable ion optical clocks [30–33].
Based on the new clock laser and the implementation of the real-time feedback algorithm for the clock laser’s linear shift, the instability of the transportable 40Ca+ optical clock has been improved to $1.16 \times {10^{ - 15}}/\sqrt {\mathrm{\tau }/\textrm{s}} $ at an averaging time of 100 s - 1000 s. This is the best instability result among transportable ion clocks. Recently, the 40Ca+ optical clock and the cavity have been transported 1200 km to the National Institute of Metrology, China (NIM) and successfully locked. The next step of the work is already in progress, and it is expected to play an important role in achieving higher-accuracy clock comparisons with optical clocks from other laboratories.
2. Design and properties of the cavity
2.1 Design of the cavity
The cavity’s structure is depicted in Fig. 1. To reduce vibration noise on the cavity length, a reasonable rigidly-mounted structure was designed using finite element analysis (FEA), and this structure restricts all degrees of freedom to meet the transportable requirements. As shown in Fig. 1(a), the convex and concave parts of the cavity are used for fixing. The thickness of the central convex part is 10 mm, and the distance between the concave parts on both sides of the cavity and the polished surface of the cavity mirror bonding is 60.3 mm. In Ref. [27], the supporting structure consist of invar blocks that are glued to the spacer. The invar material has a large thermal expansion rate, which could lead to an increased thermal sensitivity. Furthermore, the adhesive used in the bonding process may undergo changes in its properties as a result of the baking process of vacuum preparation and temperature cycles from room temperature to zero thermal expansion temperature (ZTET). These changes could alter the bonding surface and consequently impact the stability of the supporting structure. In this paper, we optimize the design of the cavity. Supporting surfaces are all cut from a ULE material to ensure greater temperature stability and a more reliable supporting structure. Eight long Invar poles of 168.5 mm and sixteen short Invar poles of 36.5 mm are used for fixing the spacer in three orthogonal directions, restricting the displacement and rotation. Fluorine rubber balls are used as the buffer between the posts and the spacer. The rods are fixed to a stainless-steel frame, as depicted in Fig. 1(b). Based on the results of the simulated supporting positions, there are several threaded holes on the stainless-steel frame at the corresponding positions to fix the Invar rods. Symmetry in both the geometry and support ensures that the cavity is insensitive to inertial forces. By optimizing the parameters, the displacement along the optical axis of the cavity mirror is calculated to be lower than $2 \times {10^{ - 11}}$ under the 1 g acceleration. The optical resonator consists a 30-cm-long ULE glass spacer and two cavity mirrors made of fused silica (FS) substrates. Two ULE rings are attached to the outside surface of the substrates to compensate the deformation caused by the difference in thermal expansion coefficients. The vacuum chamber, thermal shields and temperature control are the similar to Ref. [27]. The vacuum chamber size is $0.47 \times 0.25 \times 0.195\; {m^3}$.
Fig. 1. Schematic of the ultra-stable cavity. (a) Spacer and supporting structure of the cavity. (b) Supporting frame of the cavity. (c) Section view of the vacuum chamber and thermal shields of the cavity. The gray and yellow color indicate the vacuum chamber and the thermal shields, respectively.
2.2 Properties of the cavity
To evaluate the performance of the transportable cavity, the other 30-cm-long cavity with a similar configuration has been set up. The two cavities have the same supporting structure, and the only difference between them is whether the two thermal shields are screwed or separated by polytetrafluoroethylene balls. The screwed cavity is transportable (cavity 1), while the other cavity separated by polytetrafluoroethylene balls can only be used in the laboratory (cavity 2).
Setting the ULE temperature to its ZTET can significantly reduce the sensitivity of cavity length to temperature. The temperature fluctuation of the cavity is precisely controlled to 0.005°C by using a commercial temperature control circuit (Wavelength LFI-3751). The temperatures of cavity 1 and cavity 2 are maintained at 15°C and 7°C, respectively. Since both of cavities are under temperatures lower than the room temperature, heat accumulation during the temperature control process will lead to ineffective temperature control. To reduce the heat accumulation, an additional set of thermoelectric coolers is installed outside the vacuum chamber. Furthermore, to ensure a stable external environment, two cavities are placed inside two aluminum insulation boxes with a thickness of 10 mm. The insulation boxes are also coated with thermal insulation material on the outer surface. The insulation box size is $0.7 \times 0.7 \times 0.7\; {m^3}$. Liquid-cooling devices are installed on the inner surface to control the temperature inside the insulation boxes at 20 °C, ensuring that the change in interior temperature is less than 0.5 °C. By employing multistage thermal control, a long-term stability at the level of 10−16 for the reference cavity can be achieved. To minimize vibration noise, insulation boxes with soundproofing material and an active vibration isolation (AVI) platform are used. The properties of the two cavities have been evaluated and are presented in Table 1. Compared to the best transportable cavities [22,26], whose vibration sensitivities do not exceed $3 \times {10^{ - 11}}/g$, the vibration sensitivities of the two cavities in this paper were evaluated to be lower than $4 \times {10^{ - 10}}/g$ in the three Cartesian directions. The fixed accuracy of the invar poles limits vibration sensitivity, including the deformation of the fluorine rubber balls and manual installation errors.
Table 1. The properties of the two cavities, including vibration sensitivity, linewidth, vacuum and fraction of laser power coupled into the cavity
3. Configuration and performance of the clock laser
3.1 Configuration and evaluation system of the clock laser
The schematic of the experimental system setup is shown in Fig. 2, which includes the frequency locking of the laser to two cavities and the measurement of the frequency-difference between the two lasers. A commercial CW continuously tunable Ti:Sapphire laser (M Squared) is used as the laser source. The linewidth of the laser is sub-50 kHz and the output power is approximately 1.3 W near 729 nm.
Fig. 2. Schematic of the experimental setup. The abbreviations in the figure mean follows: $\mathrm{\lambda }/2$: half-wave plate; PBS: polarizing beam splitter; $\mathrm{\lambda }/4$: quarter-wave plate; PD: photo detector; GL: Glan prism; EOM: electro-optic modulator; AOM: acoustic optical modulator; SG: signal generator; PID: proportional-integral-derivative circuit.
The light emitted by the laser is split into two paths through the PBS and directed into two cavities. The first path passes through AOM1, and is then transmitted to cavity 2 via a single-mode polarization maintaining fiber. AOM2 is used to compensate for laser frequency drift. During laser transmission in the fiber, pressure, temperature, and vibration generate phase noise, which compromises the stability of the laser frequency [34]. In our experiment, the laser is incident from the angled physical contact (APC) end of the fiber and emitted from the physical contact (PC) end. Part of the light is reflected back from the PC end, and the fiber noise is obtained by beating with the laser that has not passed through the fiber. Frequency modulation is added to AOM3 for fiber noise compensation. To confirm the elimination effect, we beat the laser before and after fiber transmission. The frequency instability introduced by the fiber noise is lower than $4 \times {10^{ - 18}}$ from 1s to 100 s after locking the fiber noise cancellation (FNC) circuit. The laser is then locked to cavity 2 using the Pound-Drever-Hall (PDH) method [35]. The fast frequency loop is fed back to both AOM1 and AOM4 simultaneously. The slow frequency loop is fed back to the Ti:Sapphire laser. In this way, the second path is pre-stabilized by cavity 2. Similarly, the pre-stabilized laser is incident to cavity 1 after eliminating the fiber noise. In this path, AOM5 is used to eliminate the phase noise during fiber transmission and stabilize the incident light power. The pre-stabilized laser is locked to cavity 1 using the PDH method and the error signal is fed back to AOM6. By recording the frequency on AOM6, the frequency difference of the laser frequency locked to two cavities can be determined.
EOM is used in the PDH method to generate phase modulation. However, it will cause additional residual amplitude modulation (RAM), which will reduce the frequency stability of the laser [36]. We made significant efforts to reduce the RAM: the EOM’s surface has a transmission-enhancing coating at 729 nm, and the output surface of the EOM crystal is wedged at a 4° angle. Furthermore, two optical isolators are placed after the EOM. These efforts help prevent laser reflection between various surfaces; To ensure that the polarization direction of the incident laser aligns with the main axis of the EOM crystal, a Glan prism is added to optimize the angle and control the EOM temperature. By adjusting the angle of the Glan prism, the position of the EOM, and the temperature of the EOM, we can minimize the RAM. Here we evaluated the RAM by measuring the signal output from the mixer after the PD when the laser frequency is far detuned from the cavity resonance. In these ways, the frequency instability introduced by RAM is reduced to below $4 \times {10^{ - 17}}$.
Due to the relatively high thermal expansion of the FS mirror, it is necessary to stabilize the laser intensity inside the cavity to prevent fluctuations in laser power from affecting frequency stability. In our case the incident laser power is stabilized to be ∼30 ${\mathrm{\mu} \mathrm{W}}$. According to the measurement, the stability of the laser will be reduced to $5 \times {10^{ - 16}}$ at 1 s if laser power stabilization is not implemented. The laser transmitted through the cavity is used for power stabilization. Amplitude modulation is applied to AOM3 in order to stabilize the laser power. Assuming that the change in laser frequency due to the laser power is proportional to the change in power, the conversion coefficient for converting power jitter to laser frequency jitter is about 79 Hz/V. The fractional voltage jitter of the PD that detects the laser power is about $4\sim 6 \times {10^{ - 5}}$ after the power stabilization. Subsequently, the frequency instability caused by changes in laser power in the cavity is evaluated to lower than $8 \times {10^{ - 18}}$.
To enhance the stability of the optical path, we placed the entire optical path on the AVI platform, and enclosed the optical components in an acrylic box to minimize the effects of air disturbance on the optical path.
3.2 Beat results
As shown in Fig. 2, the frequency of AOM6 represents the frequency difference between the two lasers locked to the cavities. The beat signal is down-converted to approximately 900 Hz using a continuously varying signal to compensate linear drift. The signal is then sent to the Dynamic Signal Analyzer SR785 to acquire the linewidth of the beat signal, with a resolution band width (RBW) of 0.125 Hz and a measurement time of 8 s. A total of 1329 measurements are taken. Each measurement was fitted with a Lorentz function as shown in Fig. 3(a). The probability distribution of the beat signal linewidth is depicted in Fig. 3(b). By using a Gaussian function to fit the data in Fig. 3(b), we can obtain the most probable linewidth of the beat signal is $0.33 \pm 0.05$ Hz.
Fig. 3. The linewidth of the beat signal. (a) The red line is the Lorentz fitting of the beat signal with the 0.34${\pm} $0.01 Hz linewidth. (b) The probability distribution of the beat signal, and the red line is the Gaussian fitting of the distribution.
The frequency difference was measured using a frequency counter (Keysight 53210A) referenced to a hydrogen maser, with a gate time of 0.2 s. Assuming the performance of the two cavities is similar since they have the similar design of configuration, each frequency instability can be obtained by dividing the beat signal instability with $\sqrt 2 $. As shown in Fig. 4(a), the instability of the clock laser reaches about $3 \times {10^{ - 16}}$ at 1 s -100 s. At $< \; $0.5 s and 5 s - 100 s, the instability is below $3 \times {10^{ - 16}}$ and reaches $2.3 \times {10^{ - 16}}$ minimum. The deterioration of instability after 100 s may be due to temperature change. The bulge near 1 s may be due to the air disturbance. The noises of fiber, laser power, and RAM are suppressed below $5 \times {10^{ - 17}}$, a level that will not compromise the stability of the laser. The vibration noise is still higher than the calculated thermal noise limit of the cavity, which mainly limits the stability of the laser frequency, as shown in Fig. 4(b). More precise control of the environmental temperature and vibration around the cavity and reduction of the effects of air disturbances could further improve the stability.
Fig. 4. Characterization of the clock laser. (a) Allan deviation of system frequency instability. The blue line represents the frequency instability of the clock laser by dividing the beat signal instability with $\sqrt 2 $. The green and red lines indicate the RAM and laser power instability of the cavity 1 respectively. The orange line indicates the instability of the fiber noise cancellation. (b) Frequency noise spectrum of the clock laser. The blue line represents the thermal noise limit of the cavity. The orange line represents the frequency noise spectrum caused by the vibration noise. The black line represents the laser frequency noise.
3.3 Linear shift compensation
The instability is an important indicator for an optical clock, and is often quantified as fractional frequency fluctuations. For single-ion optical clocks, the main limitation is quantum projection noise (QPN), which can be expressed as [37]
where ${\nu _0}$ is the clock transition frequency, ${T_p}$ is the probe time, ${T_c}$ is the clock cycle time, which includes the probe time and the dead time in a clock cycle, $\chi $ is a numerical factor near unity that is determined by the line shape of the clock transition spectroscopy, and $\tau $ is the averaging time (in seconds). Increasing the probe time can improve the instability of optical clocks due to a lower QPN limit.For achieving longer probe time, low linear drift rates are also necessary for the clock laser to ensure that the frequency shift of the clock laser has a negligible effect on the spectrum during the locking process. To achieve this, we implemented a real-time feedback algorithm to compensate the linear shift of the clock laser.
The linear drift rate is obtained by performing a linear fit on the most recent 10 feedback frequencies. The drift rate is then applied to an AOM to compensate for the linear shift of the clock laser. Specifically, when the current linear rate of the modulated RF frequency is denoted as “k”, and the latest compensation frequencies are represented as${\; }[{{f_1}, \ldots ,{f_{10}}} ]$, which indicate the frequency deviation between the clock laser with line drift compensation and the undisturbed clock transition. The new compensation drift can be obtained as follows:
where ${k_m}$ is the measured linear drift, derived from a linear fitting between the frequencies $[{{f_1}, \ldots ,{f_{10}}} ]$ and the corresponding timestamps.Figure 5 shows the impact of the real-time feedback algorithm. By introducing this algorithm, the linear drift rate is reduced by more than an order of magnitude from 26 mHz/s to 1.2 mHz/s. For the current locking parameters of the optical clock, the linear drift rate is already sufficiently small. This is because the feedback period of the optical clock is only about 10 s, resulting in a frequency shift drift of the clock laser of only 12 mHz during this period. This drift is small enough when compared to the locking linewidth at the Hertz level, and the server error is also less than $1 \times {10^{ - 19}}$ [5]. A second-order integrating servo algorithm is expected to further reduce the linear drift rate of clock lasers.
4. Stability improvement of 40Ca+ optical clock
The ultra-stable clock laser was utilized to improve the stability of a room-temperature transportable 40Ca+ optical clock. Based on spectroscopic analysis of the clock transition, the frequency of the AOM7 in front of the ion is adjusted. For the 40Ca+ optical clock, the partial energy level diagram is shown in Fig. 6(b). The 397 nm and 866 nm lasers are used for both laser cooling and fluorescence detection. Their frequencies are referenced to a multiple wavelength ULE cavity, with sub-kilohertz frequency jittering. The 42S1/2-32D5/2 electric quadrupole transition is the clock transition. After clock interrogation, another 854 nm laser is used to excite the ion from 32D5/2 to 42P3/2.
Fig. 6. Schematic of the 40Ca+ optical clock system and the partial energy level of the 40Ca+ optical clock.
The ions are loaded using the laser ablation technique [38], and then a single ion can be obtained by reducing the amplitude of the RF signal. Furthermore, in order to increase the fidelity of quantum states, the ion was prepared to 2S1/2(m = 1/2) or 2S1/2(m = -1/2) state with ∼95% probability by the method of state preparation before each interrogation [39]. Algorithms for avoiding fringe slips in the Ramsey interrogation and automatic searching for resonance using the Rabi interrogation are used during the locking process [40]. The frequency of the clock laser is referenced to the central fringe of the Ramsey spectra, as shown in Fig. 7. At each frequency point, the laser interacts with the ion 30 times to obtain the transition probability. The real-time feedback algorithm compensates for linear shift of the clock laser, resulting in a more stable locking process.
Fig. 7. The Ramsey spectra of the 40Ca+ clock transition. (a) and (b) represent the spectroscopy based on the old and new cavity for the transportable Ca+ optical clock. The blue points and the orange solid lines depict the measured transition probabilities and the sine fitting results, respectively. (a) and (b) are obtained with a probe time of 40 ms and 320 ms, corresponding to a linewidth of 13.2 Hz and 2.0 Hz, respectively.
The probe time is limited by the coherence time of the laser and ion interaction. As shown in Fig. 7, after greatly suppressing the magnetic noise [41], the probe time of the transportable 40Ca+ optical clock has been prolonged from 40 ms to 320 ms by introducing the new cavity. Figure 8 illustrates the improvement of the Allan deviation of the 40Ca+ optical clock instability with an extended probe time, measured using the interleaved-comparison method.
Fig. 8. Allan deviation measured by the interleaved-comparison method with a probe time 320 ms. The purple line represents the QPN limitation on the experimental conditions. The blue and green dots represent the stability of a single optical clock obtained by dividing the frequency comparison results by $ \sqrt 2 $ of the old and new cavity. The red line is the 1/$\sqrt{\mathrm{\tau }}$ fitting for the green dots, showing an instability of $1.16 \times {10^{ - 15}}/\sqrt {\mathrm{\tau }/\textrm{s }} $ at an averaging time of 100 s – 1000 s.
The data indicates a single 40Ca+ optical clock instability of $1.16 \times {10^{ - 15}}/\sqrt {\mathrm{\tau }/\textrm{s}} $ at an averaging time of 100 s – 1000 s, which is slightly greater than the calculated quantum projection noise (QPN) limit of $8.6 \times {10^{ - 16}}/\sqrt {\mathrm{\tau }/\textrm{s}} $. The gap between them is mainly caused by magnetic noise, which reduces the contrast of spectral lines. Benefiting from an extremely narrow linewidth, this result represents the lowest instability for 40Ca+ optical clocks. This represents more than threefold improvement compared to previous results.
The improvement of the clock laser prolongs the probe time and narrows the transition linewidth, thereby improving the short-term frequency stability of the 40Ca+ optical clock. However, because the first order Zeeman shift is very sensitive to the magnetic field, the residual clock laser and magnetic noises are now the main factors that limit the instability of the 40Ca+ optical clock.
Compared to ion optical clocks, neutral atomic lattice clocks have higher stability because they have a large number of atoms. However, the stabilities of transportable atomic clocks used for long-distance optical clocks comparisons are still in the 10−15$/\sqrt {\mathrm{\tau }/\textrm{s}} $ level [17,18], limited by experimental conditions. The achievement of the excellent transportable ultra-stable cavity has enabled the 40Ca+ optical clock comparable with the neutral atomic lattice clocks in both stability and uncertainty.
5. Conclusion
In summary, we have developed an ultra-stable clock laser based on a 30-cm-long transportable ultra-stable cavity with an instability of ${\sim} 3 \times {10^{ - 16}}$ at 1 s–100 s and improved the instability of the transportable 40Ca+ optical clock to $1.16 \times {10^{ - 15}}/\sqrt {\mathrm{\tau }/\textrm{s}} $ at an averaging time of 100 s–1000 s by applying this ultra-stable clock laser after real-time compensating the linear shift. Transportable ultra-stable cavities are essential for improving the performance of transportable optical clocks which play a key role in higher-accuracy optical frequency comparisons with other types of optical clocks in different laboratories, and then promote the time definition from microwave to optical frequency. Further improvement of the transportable clock laser can be achieved through precise control of temperature, airflow, and vibration environment. Thus, with the laser improvement and the suppression of magnetic noise, it is envisaged that the probe time can be prolonged to 500 ms or even longer, and the instability of the 40Ca+ optical clock is expected to be below $1.0 \times {10^{ - 15}}/\sqrt {\mathrm{\tau }/\textrm{s}} $.
Funding
National Key Research and Development Program of China (2018YFA0307500, 2022YFB3904001, 2022YFB3904004); National Natural Science Foundation of China (11934014, 12022414, 12121004); Natural Science Foundation of Hubei Province (2022CFA013); CAS Youth Innovation Promotion Association (Y201963, Y2022099); CAS Project for Young Scientists in Basic Research (YSBR-055); Interdisciplinary Cultivation Project of the Innovation Academy for Precision Measurement of Science and Technology (S21S2201).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this may be obtained from the authors upon reasonable request.
References
1. A. D. Ludlow, M. M. Boyd, J. Ye, et al., “Optical atomic clocks,” Rev. Mod. Phys. 87(2), 637–701 (2015). [CrossRef]
2. F. Riehle, P. Gill, F. Arias, et al., “The CIPM list of recommended frequency standard values: guidelines and procedures,” Metrologia 55(2), 188–200 (2018). [CrossRef]
3. R. Lange, N. Huntemann, J. M. Rahm, et al., “Improved limits for violations of local position Invariance from atomic clock comparisons,” Phys. Rev. Lett. 126(1), 011102 (2021). [CrossRef]
4. T. E. Mehlstäubler, G. Grosche, C. Lisdat, et al., “Atomic clocks for geodesy,” Rep. Prog. Phys. 81(6), 064401 (2018). [CrossRef]
5. Y. Huang, B. Zhang, M. Zeng, et al., “Liquid-nitrogen-cooled Ca+ optical clock with systematic uncertainty of 3 × 10−18,” Phys. Rev. Appl. 17(3), 034041 (2022). [CrossRef]
6. S. M. Brewer, J.-S. Chen, A. M. Hankin, et al., “27Al+ Quantum-Logic Clock with a Systematic Uncertainty below 10− 18,” Phys. Rev. Lett. 123(3), 033201 (2019). [CrossRef]
7. S. Dörscher, N. Huntemann, R. Schwarz, et al., “Optical frequency ratio of a 171Yb+ single-ion clock and a 87Sr lattice clock,” Metrologia 58(1), 015005 (2021). [CrossRef]
8. Z. Q. Zhang, K. J. Arnold, R. Kaewuam, et al., “176Lu+ clock comparison at the 10−18 level via correlation spectroscopy,” Sci. Adv. 9(18), 1971 (2023). [CrossRef]
9. W. F. McGrew, X. Zhang, R. J. Fasano, et al., “Atomic clock performance enabling geodesy below the centimetre level,” Nature 564(7734), 87–90 (2018). [CrossRef]
10. T. Bothwell, D. Kedar, E. Oelker, et al., “JILA SrI optical lattice clock with uncertainty of 2 × 10−18,” Metrologia 56(6), 065004 (2019). [CrossRef]
11. I. Ushijima, M. Takamoto, M. Takamoto, et al., “Cryogenic optical lattice clocks,” Nat. Photonics 9(3), 185–189 (2015). [CrossRef]
12. R. Schwarz, S. Dörscher, A. Al-Masoudi, et al., “Long term measurement of the 87Sr clock frequency at the limit of primary Cs clocks,” Phys. Rev. Res. 2(3), 033242 (2020). [CrossRef]
13. E. Oelker, R. B. Hutson, C. J. Kennedy, et al., “Demonstration of 4.8 × 10−17 stability at 1 s for two independent optical clocks,” Nat. Photonics 13(10), 714–719 (2019). [CrossRef]
14. M. Schioppo, R. C. Brown, W. F. McGrew, et al., “Ultrastable optical clock with two cold-atom ensembles,” Nat. Photonics 11(1), 48–52 (2017). [CrossRef]
15. M. Zeng, Y. Huang, B. Zhang, et al., “Toward a Transportable Ca+ Optical Clock with a Systematic Uncertainty of 4.8×10−18,” Phys. Rev. Appl. 19(6), 064004 (2023). [CrossRef]
16. N. Ohmae, M. Takamoto, Y. Takahashi, et al., “Transportable strontium optical lattice clocks operated outside laboratory at the level of 10−18 uncertainty,” Adv. Quantum. Technol. 4(8), 2100015 (2021). [CrossRef]
17. M. Takamoto, I. Ushijima, N. Ohmae, et al., “Test of general relativity by a pair of transportable optical lattice clocks,” Nat. Photonics 14(7), 411–415 (2020). [CrossRef]
18. J. Grotti, I. Nosske, S. B. Koller, et al., “Long-distance chronometric leveling with a transportable optical clock,” arXiv, arXiv:2309.14953v2 (2023). [CrossRef]
19. D. Kedar, J. Yu, E. Oelker, et al., “Frequency stability of cryogenic silicon cavities with semiconductor crystalline coatings,” Optica 10(4), 464–470 (2023). [CrossRef]
20. W. Bian, Y. Huang, H. Guan, et al., “1 Hz linewidth Ti:sapphire laser as local oscillator for 40Ca+ optical clocks,” Rev. Sci. Instrum. 87(6), 063121 (2016). [CrossRef]
21. D. R. Leibrandt, J. C. Bergquist, and T. Rosenband, “Cavity-stabilized laser with acceleration sensitivity below 10−12g-1,” Phys. Rev. A 87(2), 023829 (2013). [CrossRef]
22. S. Webster and P. Gill, “Force-insensitive optical cavity,” Opt. Lett. 36(18), 3572–3574 (2011). [CrossRef]
23. S. Wang, J. Cao, J. Yuan, et al., “Integrated multiple wavelength stabilization on a multi-channel cavity for a transportable optical clock,” Opt. Express 28(8), 11852–11860 (2020). [CrossRef]
24. X. Chen, Y. Jiang, B. Li, et al., “Laser frequency instability of 6 × 10−16 using 10-cm-long cavities on a cubic spacer,” Chin. Opt. Lett. 18(3), 030201 (2020). [CrossRef]
25. K. Numata, A. Kemery, and J. Camp, “Thermal-Noise Limit in the Frequency Stabilization of Lasers with Rigid Cavities,” Phys. Rev. Lett. 93(25), 250602 (2004). [CrossRef]
26. S. Herbers, S. Häfner, S. Dörscher, et al., “Transportable clock laser system with an instability of 1.6 ×10−16,” Opt. Lett. 47(20), 5441–5444 (2022). [CrossRef]
27. R. Xiao, Y. Xu, Y. Wang, et al., “Transportable 30 cm optical cavity based ultrastable lasers with beating instability of 2 × 10−16,” Appl. Phys. B 128(12), 220 (2022). [CrossRef]
28. S. B. Koller, J. Grotti, S. Vogt, et al., “Transportable Optical Lattice Clock with 7 × 10−17 Uncertainty,” Phys. Rev. Lett. 118(7), 073601 (2017). [CrossRef]
29. F. Guo, W. Tan, C. Zhou, et al., “A proof-of-concept model of compact and high-performance 87Sr optical lattice clock for space,” AIP Adv. 11(12), 125116 (2021). [CrossRef]
30. S. Hannig, L. Pelzer, N. Scharnhorst, et al., “Towards a transportable aluminium ion quantum logic optical clock,” Rev. Sci. Instrum. 90(5), 053204 (2019). [CrossRef]
31. J. Stuhler, M. A. Hafiz, B. Arar, et al., “Opticlock: Transportable and easy-to-operate optical single-ion clock,” Measurement:Sensors 18, 100264 (2021). [CrossRef]
32. Y. Huang, H. Zhang, B. Zhang, et al., “Geopotential measurement with a robust, transportable Ca+ optical clock,” Phys. Rev. A 102(5), 050802 (2020). [CrossRef]
33. J. Cao, J. Yuan, S. Wang, et al., “A compact, transportable optical clock with 1 × 10−17 uncertainty and its absolute frequency measurement,” Appl. Phys. Lett. 120(5), 054003 (2022). [CrossRef]
34. L. Ma, P. Jungner, J. Ye, et al., “Delivering the same optical frequency at two places: accurate cancellation of phase noise introduced by an optical fiber or other time-varying path,” Opt. Lett. 19(21), 1777–1779 (1994). [CrossRef]
35. R. W. P. Drever, J. L. Hall, F. V. Kowalski, et al., “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31(2), 97–105 (1983). [CrossRef]
36. N. C. Wong and J. L. Hall, “Servo control of amplitude modulation in frequency-modulation spectroscopy: demonstration of shot-noise-limited detection,” J. Opt. Soc. Am. B 2(9), 1527–1533 (1985). [CrossRef]
37. T. L. Nicholson, M. J. Martin, J. R. Williams, et al., “Comparison of two independent Sr optical clocks with 1×10−17 stability at 103 s,” Phys. Rev. Lett. 109(23), 230801 (2012). [CrossRef]
38. H. Shao, M. Wang, M. Zeng, et al., “Laser ablation and two-step photo-ionization for the generation of 40Ca+,” J. Phys. Commun. 2(9), 095019 (2018). [CrossRef]
39. M. Zeng, Y. Huang, H. Shao, et al., “Improvement of stability of 40Ca+ optical clock with state preparation,” Chin. Phys. Lett. 35(7), 074202 (2018). [CrossRef]
40. B. Zhang, Y. Huang, Y. Hao, et al., “Improvement in the stability of a 40Ca+ ion optical clock using the Ramsey method,” J. Appl. Phys. 128(14), 143105 (2020). [CrossRef]
41. M. Zeng, Z. Ma, R. Hu, et al., “A combined magnetic field stabilization system for improving the stability of 40Ca+ optical clock,” Chin. Phys. B 32(11), 110704 (2023). [CrossRef]
