The optical lattice clock has widely been recognized to be one of the most precise frequency standards in recent years.^[1–6] The two-valence-electron atoms, such as Sr, Yb, Hg, etc., can serve as stable and accurate frequency references because of their having narrow ¹S₀–³P₀ optical transitions. Based on the 5s^{2 1}S₀–5s5p ³P₀ transition of ultracold ⁸⁷Sr atoms confined in a one-dimensional optical lattice, the optical frequency standard pursued by Bloom et al.^[7] has been realized: the accuracy and stability are both at 10⁻¹⁸ level. It is widely known that atomic frequency reference is defined in a null-field and zero-temperature environment. However, in reality the optical clocks are mostly operated in an environment with the stray magnetic field at room temperature. Therefore the measured transition frequencies need correcting and the uncertainty of these corrections must be evaluated.

Previously, the blackbody-radiation (BBR) shift was estimated through the temperature measurement and theoretical calculation,^[8–12] and the temperature of the chamber was usually regarded as being isotropic. However, the experimental condition and the surrounding airflow can lead to the temperature inhomogeneity of the chamber, which can be obtained by more detailed measurements. Moreover, the temperature distribution of the outer surface may not reflect the real scenario inside the chamber. To deal with these problems, many efforts have been made to improve the BBR correction and its uncertainty. For the JILA ⁸⁷Sr optical lattice clock, there is a BBR enclosure surrounding the vacuum chamber to maintain the environmental temperature, and two temperature probes are placed inside the vacuum chamber to detect the thermal environment near the atoms.^[13] By using a different approach, the group of NIST developing the ¹⁷¹Yb optical lattice clocks placed a BBR shield into the vacuum chamber^[14] to reduce the fluctuation and improve the homogeneity of the environmental temperature around the cold atoms, which furnishes the atoms with a uniform, well-characterized radiative environment. Applying the technique of the moving lattice, the ⁸⁷Sr optical lattice clock located in RIKEN was examined in a cryogenic environment to reduce the BBR shift,^[15] which is also called the cryo-chamber.

In this paper, we accurately evaluate the blackbody-radiation shift and its uncertainty in the ¹⁷¹Yb optical lattice clock by numerically simulating the temperature distribution around the cold ytterbium atoms based on the measured temperatures on the surface of the vacuum chamber. At first, we review the theoretical calculation of the BBR shift and introduce the experimental setup to display the external radiation sources in our experiment. Secondly, we present the temperature measurement scheme with using seven sensors to determine the chamber temperature distribution. Then, we calculate the BBR shift according to the analyses of heat sources and the measured chamber temperatures. Finally, we estimate the temperature distribution inside the chamber by numerical simulation, so that the temperature gradient and fluctuation around the cold atoms can be inferred more accurately. Furthermore, the BBR shift in the ¹⁷¹Yb optical lattice clock is evaluated and discussed.

Theoretically, the blackbody radiation field is equal in any direction, making vector and tensor components zero. The temperature dependence of the clock scalar BBR shift is given by the formula:^[16,17]

Figure 1 shows the experimental setup of our ¹⁷¹Yb optical lattice clock. According to the path of the atoms, there are three main radiation sources obviously affecting the cold atoms: the heated atomic oven, the main vacuum chamber, and the room-temperature vacuum windows. Since a rather low density exists at room temperature, the ytterbium metal needs to be placed in the oven which is heated to a high temperature. Each part of the oven, including the body and nozzle, is uniformly heated and finally reaches a balanced temperature. To avoid the cold atoms in the optical lattice being kicked out by the hot atoms, a mechanical shutter is installed next to the oven nozzle, and it will be closed during the clock interrogation period. Since the hot Yb atoms are continuously accumulated on the metal plate when the shutter is closed, we install a recycled cooling water system surrounding the shutter to maintain its temperature nearly at the chamber temperature. After entering the main vacuum chamber, the Yb atoms will be trapped in the magneto–optical trap (MOT) with two-stage laser cooling, and finally they are loaded in the optical lattice for the clock interrogation.^[24] The vacuum chamber is made up of aviation aluminum with a high emissivity. The whole apparatus is exposed to the room temperature, while the room temperature BBR may enter into the chamber through quartz windows. To reduce the deposition of the atoms, the Zeeman window facing to the atomic beam needs to be heated to a certain temperature, which will make it another heat source. Fortunately, we do not need to heat the Zeeman window during the clock operation at this moment.

Fig. 1.

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	Fig. 1. Experimental setup of the 171Yb optical lattice clock. The 171Yb atoms from the heated oven pass through the two-dimensional (2D) collimation, Zeeman slower system, and finally enter into the magneto–optical trap (MOT) area. The atomic oven is heated to 673.15 K (400 °C) during the operation, and both the Zeeman slower and MOT chamber are exposed to the room temperature.

As mentioned above, the temperature of the radiation source will directly affect the BBR shift. It is necessary to define the temperature of the heat source. The atomic oven is covered by a sleeve which is heated by electrodes. Two thermocouples are inserted into the sleeve for monitoring the oven temperature in time. The oven temperature can be regarded as being the same as the sleeve temperature because they are tightly attached to each other. The beam shutter is located between the oven nozzle and the Zeeman slower, and its temperature is very close to the oven temperature. To determine the actual temperature of the vacuum chamber, seven platinum resistance thermometers (PRTs) are tightly attached to the outer surface at different spots of the metal chamber to obtain both temperature values and their fluctuations. The sensors are made by Heraeus Sensor Technology, and the accuracy of the PRTs is class 1/10B (AAA) which leads to the temperature measurement error being ± 0.04 K less than the standard value. The three performance coefficients of each PRT are calibrated by the Shanghai Institute of Measurement and Testing Technology (SIMT). The discrepancies among seven PRTs are shown in Fig. 2(a), where the largest difference is 0.026 K. Figure 2(b) shows the distribution diagram of seven PRTs on the surface of the vacuum chamber. Most of the measuring points are located at the upper and lower parts. There are three PRTs on the lower part because of the space limitation while there are four PRTs on the upper part. The temperature values are calculated through the conversion formula of PT100 resistance which is measured through the 4-wire circuit by using the digital multimeter (KEITHLEY model 2000).

Fig. 2.

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Fig. 2. (a) Discrepancies among seven PRTs tested by SIMT (Shanghai Institute of Measurement and Testing Technology). The largest resistance difference is 0.01Ω which will lead the temperature difference of ∼ 0.026 K. (b) Distribution diagram of the PRTs on the outside surface of the vacuum chamber. There are only three points on the lower part because of the space limitation while there are four on the upper part.

We monitor the temperature of the vacuum chamber in the whole clock operation process. Despite the fact that we use the recycled water of 293.15 K (20 ° C) to cool the apparatus, the temperature of the chamber is still about 6 K higher than the cooling water temperature when the MOT coils run with a current of ∼ 8 A.

Figure 3 shows the chamber temperatures measured by seven PRTs (Nos. 1–7) and the environmental temperature surrounding the chamber, measured by the extra PRT (No. 8) during the clock-operation process. The results also show the temperature inhomogeneity of the vacuum chamber, including the discreteness of the PT100s. The platinum resistance thermometers 1–7 are fixed on the outer surface of the vacuum chamber while the uncalibrated PRT 8 hangs in the air near the chamber. The absolute temperature of PRT 8 may be unreal, because it has not been calibrated in advance, while the temperature fluctuation of 0.5 K is authentic.

Fig. 3.

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	Fig. 3. Variations of chamber temperature with time, measured by seven PRTs (Nos. 1–7) during the clock-operation period. The extra PRT (No. 8) is utilized to measure the environmental temperature surrounding the chamber.

The measured values are listed in Table 1, from which we can obtain the temperature fluctuation in different parts of the chamber. The mean temperature of PRT 1 is the highest as it is located near the Zeeman slower and the upper MOT coils. The fluctuation ranges of PRTs 6 and 7 are larger than others’ and deviate from the average range obviously. These big observed temperature fluctuations are real because the temperature fluctuations do not appear when all these sensors are attached at the point 2 to measure the temperature by turns. We find that during the clock operation period the mean temperature of the vacuum chamber is 299.43 ± 0.17 K.

Table 1.

Table 1.

Table 1. Temperature differences among seven PRTs during the clock-operation period. “Range” refers to the difference between maximum and minimum temperatures of each PRT. “Average” means the mean temperature of each PRT. . PT100 No. Mean/K Standard deviation Range/K 1 299.69336 0.01797 0.09512 2 299.49976 0.02060 0.09982 3 299.43957 0.02186 0.11325 4 299.49424 0.02038 0.10631 5 299.47941 0.01734 0.08591 6 299.17457 0.02878 0.17116 7 299.24417 0.03222 0.18088 Average 299.43215 ± 0.17351 0.18088

Table 1. Temperature differences among seven PRTs during the clock-operation period. “Range” refers to the difference between maximum and minimum temperatures of each PRT. “Average” means the mean temperature of each PRT. .

Table 2 shows the uncertainty of the sensor, including the calibration process and the temperature measurement. The uncertainty of the temperature turns out to be 0.42 K. We apply the confidence interval of 99% with the coverage factor of k = 2.58 and obtain the expanded uncertainty of ∼ 1 K, which is adopted by most groups.

Table 2.

Table 2.

Table 2. Uncertainties of the chamber temperature for calibrated PRT. The uncertainty values in the calibration are offered by SIMT. The temperature fluctuation uncertainty of the temperature measurement comes from Table 1, which is the largest in seven sensors’. . Effect Uncertainty/K Calibration Discreteness 0.03 Sensor self-heating 0.036 Resistance measurement 0.002 Temperature experiment Statistics 0.17 Repeatability of measurement 0.03 Temperature fluctuation 0.09 Temperature inhomogeneity 0.37 Total 0.42

Table 2. Uncertainties of the chamber temperature for calibrated PRT. The uncertainty values in the calibration are offered by SIMT. The temperature fluctuation uncertainty of the temperature measurement comes from Table 1, which is the largest in seven sensors’. .

According to the structure of our experimental apparatus, the BBR shift in the ¹⁷¹Yb optical lattice clock mainly results from three radiation sources: the room temperature effect through the windows of the vacuum chamber, the radiation reflection of the cavity inner surface, and the radiation emitted from heated atomic oven. We simply model the vacuum chamber as a spherical cavity with a radius of 9.5 cm. There are two kinds of windows on the chamber: (i) seven 1.5-inch (1 inch = 2.54 cm) windows for the cooling lasers at 399 nm and 556 nm; (ii) twelve 0.5-inch windows for the optical-lattice lasers at 759 nm, the clock laser at 578 nm, the repumping lasers at 649 nm and 770 nm, and the detection laser at 399 nm.

In some cases, the windows of the vacuum chamber are assumed to be not transparent for the room temperature BBR,^[2] but the effect exists because of the transmissivity of the window glass. The corresponding frequency shift can be calculated according to the sold angle of the window:

Most of the radiation entering into the chamber is absorbed by the inner surfaces of both the chamber and the windows, while a small part of the radiation may affect the atoms due to its reflection. Here, we apply a model including the inner surface absorption coefficients of both the cavity (a) and windows (b) under three assumptions: firstly, the total energy entering into the cavity is the same as that leaking out of the cavity during the clock-operation period; secondly, we assume that the diffuse reflection from the inner wall and the energy density ρ_cav are spatially homogeneous and isotropic; the last assumption is that the temperature of the cavity is finally balanced at T_cavity after being heated by the MOT coils nearly for 1.5 h. Under these assumptions, we can obtain the following energy rate equations:^[25]

Fig. 4.

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	Fig. 4. (a) Dependence of the BBR shift on the absorption coefficient a of the metal chamber while the coefficient b = 0.8. (b) Dependence of the BBR shift on the absorption coefficient b of the vacuum windows while the coefficient a = 0.9.

Atomic oven is heated up to 673.15 K during the clock operation and becomes a radiation source near the main chamber. Fortunately, there is a beam shutter in front of the oven preventing the hot atoms from shocking the cold atoms and blocking most of the BBR photons from being scattered along the Zeeman slower pipe towards the optical lattice. Also, the beam shutter and the Zeeman slower are both installed with the recycled cooling water to reduce the influence of the heated oven. The radiation is limited by the aperture of the link pipe, which connects the Zeeman slower and main chamber. The residual radiation from the heated oven can be regarded as a surface emitting a different radiation spectrum at the temperature of the link pipe T_link. We define the representative temperature T_link as being the same as the temperature of sensor PRT (NO. 1) (299.69 K), whose position is quite close to the link pipe. The frequency shift induced by the oven is then determined by

Combing the effects of the three radiation sources, the total BBR induced frequency shift turns out to be −1.282(17) Hz with an uncertainty of 3.28 × 10⁻¹⁷. Here, we use the standard deviation to characterize the total uncertainty. The uncertainty comes from three factors: Δα(0), T, and η, and can be expressed as

Figure 5 shows the δ(Δν) ∼ ΔT dependence. The temperature uncertainty ΔT will directly determine the δ(Δν). Obviously, the minimum value is 0.62 mHz when ΔT = 0 K, and the corresponding uncertainty is 1.2 × 10⁻¹⁸.

Fig. 5.

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	Fig. 5. Dependences of δ(Δν) ∼ ΔT from 0 K∼1 K while the room temperature is 299.43 K. The diagram on the top left corner shows δ(Δν) ∼ ΔT dependence from 0 K∼0.1 K where the shift is 0.62 mHz when ΔT = 0 K.

Certainly, the discussion above is based on the simplified hollow sphere model which is far from the real case. To address this issue, we build the chamber model by Solid Works according to the trim size with twenty two sections which is shown in Fig. 6(a). The main chamber is originally made up of a one-piece of the aluminum block, while the window channels come from the drilling of it. We simplify the Zeeman channel into a big window to establish a closed chamber. Twenty two windows are assembled into the hollow metal chamber, with ignoring the flanges, the structures of the Zeeman slower channel and the Zeeman window. All the windows are assembled into the hollow metal chamber in the face-face coincident manner. The model is incorporated into ANSYS APDL for steady-state temperature analysis. We consider the case where no radiation source exits near the vacuum chamber except the room temperature. All the chamber and the atmosphere inside it will reach an equilibrium temperature similar to the ambient temperature. Therefore, we set the initial temperatures of all the structures (including vacuum chamber and the windows) to be room temperature which is 293.15 K. The Zeeman slower and atomic oven are considered separately and none of them is included in our model. For the outer surfaces, we adjust the convective heat transfer coefficient to make the simulated temperatures at the measurement spots close to the experimental values, which can also judge the facticity of the simulation. Since the temperatures are sensitive to the MOT coils during the clock operation, we can regard the two MOT coils as heating sources, specifically, the upper and the lower surfaces. To determine the heating temperature of the two surfaces, we extract the temperature of point 1 and obtain a near-linear dependence while the heating temperature is changing. From the fitting curve, we find that the heating temperature is 299.71 K (26.56 °C) while the temperature of point 1 is 299.39 K (26.54 °C) which is the measured temperature value as shown in Table 1. The temperature differences at other points between the measured values and the simulation results at the same position is less than 0.1 K, which is limited by the meshing process and the position picking. The simulated temperature values are mostly determined by the heat conduction and convection as shown in Fig. 6(b). The blue part means lower temperature due to the higher convective heat transfer coefficient of 200 W/(m²·K), while the red part refers to higher temperature due to the lower convective heat transfer coefficient of 13 W/(m²·K). Both two coefficients are determined by attempt until the temperatures at the measurement spots are close to the experimental values.

Fig. 6.

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Fig. 6. (a) Chamber model built according to the trim size. The flanges for pressing the windows are ignored because of the complex structures. (b) Temperature distribution of the metal chamber caused by heat conduction and convection while the environmental temperature is 293.15 K. The chamber is placed on the platform with one side exposing near the air outlet. That is the reason why the left side (blue) in the graph is cooler than the right side (red), while the upper side (green) is cooler than the lower side (red). (c) The temperature distribution around the cold atoms while the environmental temperature is 293.15 K.

The vacuum part is empty while the atoms are replaced by a blackbody sphere. The setting values of the material model in ANSYS are limited to an effective range. The radiation loading is added to the inner surface of the chamber so that the temperature gradient around the cold atoms can be inferred. The convective heat transfer coefficient is not uniform because of the unilateral bigger airflow. The convergent nonlinear analysis gives the temperature distribution near the cold atoms, which is shown in Fig. 6(c). The temperature is not uniform around the atoms because of the inhomogeneous temperature of the chamber. The vacuum part prevents the atoms from exchanging energy directly with chamber. Most heat of the chamber is taken away by the environment while small fraction of the heat transfers through the vacuum and the temperature near the atoms rises. We take the average temperature to represent T_atoms which is 299.25 ± 0.14 K and the BBR shift is −1.264(7) Hz. The average temperature at the cold atom cloud is lower than that of the chamber because of the isolation of the vacuum. For representing the temperature of the vacuum part, we fill the chamber with an entity with the characteristics of the vacuum and a blackbody sphere representing the atoms. We suppose an ideal condition and set the density and the convective heat transfer coefficient of the vacuum part to be very small (∼ 0) while the specific heat capacity is set to be very big (∼ ∞). The software identifies the extreme values automatically. We define the temperature gradient along the lattice laser as coordinate X, while other two vertical directions are defined as Y and Z which are indicated in Fig. 7(a). With the path generation block, we obtain the temperature gradients near the cold atoms along the local coordinates which are shown in Fig. 7(b). The solid lines represent the temperature variations in the uniform case with the convective heat transfer coefficient of 100 W/(m²·K), while the dash lines represent the inhomogeneous case with the convective heat transfer coefficients of 13 W/(m²·K) and 200 W/(m²·K). The temperatures at the center position of the atomic cloud are added with the same offsets so that the differences between the gradients are seen more obviously. The biggest difference is at coordinate Z (blue lines) whose direction is perpendicular to the heat convection surface.

Fig. 7.

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Fig. 7. (a) Defined local coordinate system where X is along the lattice laser. (b) Differences of the temperature gradients near the cold atoms where the convection is uniform or not. The solid lines represent that the heat convection is uniform with the convective heat transfer coefficient of 100 W/(m2·K). The dash lines represent that the heat convections are inhomogeneous with the convective heat transfer coefficients of 13 W/(m2·K) and 200 W/(m2·K), respectively.

We simulate the temperature distribution of the blackbody sphere when the environmental temperature changes from 292.65 K to 294.65 K. Finally we take the largest temperature difference value of 0.082 K all around as the temperature variation at the cold atoms. Adding the effect of the room temperature through the vacuum windows and the heated oven, the total BBR shift obtained from the simulation is −1.289(7) Hz, and the corresponding uncertainty of the BBR shift is 1.25 × 10⁻¹⁷. For comparison, if directly using the measured temperatures of the outer surface without simulation, the total BBR shift is −1.282(17) Hz with an uncertainty of 3.28 × 10⁻¹⁷. Obviously, they are different, and the former is more accurate than the latter, because the presented method is more realistic to describe the temperature distribution around the cold atoms.

To evaluate the accuracy of the presented method, we simulate the temperature gradient and compare the results with the measured values at the Sr clock in JILA.^[7] Firstly, we build a simplified vacuum chamber according to the shape described in Ref. [13], in which the flanges and small windows in the horizontal direction are ignored. Then we apply the loadings according to the measurement conditions. As mentioned in Ref. [7], the temperature of the bottom window was varied from 301 K to 302 K, and the temperature gradient near the atoms was measured by two sensors. Thus the temperature loadings in our model are in two areas on the flanges of the bottom window with a temperature of 302 K. The initial convective heat transfer coefficient is set uniformly to be 13 W/(m²·K) because there is a BBR shield around the whole chamber. The vacuum part is also filled with an entity with the characteristics of the vacuum. The radiation loadings are added on the inner surface of the metal chamber with an absorption coefficient of 0.8 initially. The temperature distribution of the metal chamber is shown in Fig. 8(a) and obviously the temperature decreases from the bottom to the top. The temperature distribution of the vacuum entity is shown in Fig. 8(b) while the centers of the bottom and the top each have a lower temperature because of the relatively lower thermal conductivity of the quartz window. Because the values of the radiation coefficient and the convective heat transfer coefficient may influence the gradients, we follow the control variable method to simulate the temperature distribution and discuss the results. As described in Figs. 8(c) and 8(d), the gradients decrease while the radiation coefficient and the convective heat transfer coefficient increase. The relative temperature difference between the position 0 cm and 2.5 cm is ∼ 0.06 K while the radiation coefficient is 0.8 and the convective heat transfer coefficient is 13 W/(m²·K), which is very close to the measured value of 0.04 K. The reproduction above also proves the reliability of the presented simulation method.

Fig. 8.

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Fig. 8. (a) Temperature distribution of the metal chamber. (b) Temperature distribution of the vacuum entity. (c) The relationship between the radiation coefficient and temperature gradients with the convective heat transfer coefficient being 13 W/(m2·K). (d) The relationship between the convective heat transfer coefficient and temperature gradients with the absorption coefficient being 0.8. HF means the convective heat transfer coefficient and it is in units of W/(m2·K).

Through the simulation and analysis above, we find that the main drawback is the temperature inhomogeneity of the vacuum chamber. Therefore, in the next research we need to improve the homogeneity of the vacuum chamber, say, by rearranging the positions of the air conditioners. Furthermore, we plan to increase the flow rate of the cooling water so that the heating effect of the MOT coils could be reduced and the vacuum chamber temperature can be lowered. To improve the present status further, a radiation shield could be built inside the vacuum chamber to avoid the room temperature influence, and thus the temperature uncertain value will only be limited by the inhomogeneity of the shield which may be reduced down to 0.1 K. In this case, the uncertainty of the BBR shift could reach 1.14 × 10⁻¹⁷. On the other hand, we will improve the simulation. For instance, we will optimize the chamber model and surface parameters to take more radiation factors into account. Ultimately, we will upgrade our apparatus, such as installing the temperature probers inside the vacuum chamber or making the cyro-chamber, etc, which have been done successfully by other groups.^[13–15]

The BBR shift results from various sources which come from inside and outside of the experimental apparatus, and at present it is one of major obstacles for improving the accuracy of the optical lattice clocks. Many schemes of dealing with the BBR shift come up, each with its own advantages. For the JILA Sr lattice clock, the inner sensors can accurately measure the temperature near the atoms during the experiment so that the BBR correction can be accurately calculated.^[13] The BBR shield installed inside the vacuum chamber of the NIST Yb clock provides a uniform radiative environment for the atoms at room temperature, so that the temperature fluctuation around the cold atoms is reduced.^[14] The cryo-chamber realized by the group of RIKEN Sr clock is an effective method to reduce the BBR shift while the environmental temperature can be maintained at 95 K by the compressor with a negligible temperature fluctuation.^[15] In this paper, we present a more precise evolution of the BBR shift by using the numerical simulation combined with the temperature measurement on the outer surface of the chamber. With the assistance of the numerical simulation, we obtain the temperature distribution inside the vacuum chamber and calculate the BBR correction more accurately, which is −1.289(7) Hz with an uncertainty of 1.25 × 10⁻¹⁷ governed by a temperature inhomogeneity of ΔT = 0.3791 K. Definitely the simulation could help us know the radiation environment around the cold atoms more unmistakably. Finally, we believe that this method will be quite suitable for precisely evaluating the BBR shift of the optical lattice clock where there are no sensors inside the vacuum chamber.