Finite element analysis of blackbody radiation environment for an ytterbium lattice clock operated at room temperature

Optical clocks based on lattice-trapped neutral atoms or single trapped ions have reached frequency uncertainty levels in the 10⁻¹⁸ range [1–6], surpassing the cesium primary frequency standard by orders of magnitude. The superior performance of optical clocks has motivated serious considerations of a future redefinition of the SI second in terms of an optical transition. In fact, the International Committee for Weights and Measures has defined five milestones toward such a redefinition [7]. Absolute frequency measurements of the ⁸⁷Sr and ¹⁷¹Yb clock transitions have achieved an uncertainty level at low 10⁻¹⁶, limited only by the performance of Cs fountain clocks. Combined with optical frequency ratios measured via direct optical comparison, the frequency measurements of ⁸⁷Sr, ¹⁷¹Yb and Cs clocks have realized a loop closure [8], consistent at the uncertainty limit of the current realization of SI second. In parallel, much efforts have been devoted to incorporate optical frequency standards into existing time scales [9–15]. Besides time and frequency metrology, ultra-precise optical clocks have found wide applications for relativistic geodesy [16, 17], measurement of possible variations of fundamental physical constants [18–20], dark matter detection [21–24] and gravitational wave astronomy [25].

For optical lattice clocks based on Sr or Yb atoms, blackbody radiation (BBR) can induce a Hz-level Stark shift at room temperature. The uncertainty from this frequency shift usually dominates the uncertainty budget table of a clock. Well controlled and characterized BBR environment is important for improving both the frequency uncertainty and long-term stability. Since it scales quarticly with the absolute temperature T of the environment, the BBR Stark shift of the clock transition will be reduced if the atoms are interrogated in a cold environment. Equipped with a cryogenic chamber, Sr clocks at RIKEN that transport ultracold atoms into the cryogenic environment for spectroscopic interrogation reached an uncertainty level of 9 × 10⁻¹⁹ for BBR Stark shift [4]. Other strategies have been used to improve BBR shift uncertainty for clocks operating at room temperature. The technique of in situ temperature monitoring has enabled accurate determination of the BBR temperature felt by the atoms in a Sr clock at JILA [2]. An alternative strategy has been developed for the Yb clocks at NIST. An in-vacuum BBR shield enclosing the atoms ensures a uniform, well characterized BBR environment, yielding a BBR shift uncertainty of the order of 10⁻¹⁹ [26].

We have built an ¹⁷¹Yb optical lattice clock, and its closed-loop operation was reported elsewhere [27]. Since we did not place a BBR shield inside the vacuum, the Yb atoms are exposed to thermal radiation of the vacuum chamber while the whole apparatus sits in the air of the laboratory. For an lattice clock operated at room temperature environment like ours, the BBR temperature can be estimated with uncertainties at low kelvin level [28–30] or sub-kelvin level [31, 32], depending on the degree of temperature homogeneity. We have performed a finite element (FE) analysis to the BBR environment of the ultracold Yb atoms in our setup. The main heat source and the air convection around the main vacuum chamber have also been taken into consideration. We show that the effective temperature felt by the atoms can be determined to an accuracy level better than 200 mK, despite of the temperature inhomogeneity on the vacuum chamber. Our Yb clock can thus achieve a BBR shift uncertainty below 10⁻¹⁷.

The paper is organized as follows. In section 2, the experimental setup is described. In section 3, we introduce the FE analysis for the BBR radiation of our vacuum chamber, and give the effective solid angles of surrounding vacuum parts. We then describe the FE calculation of the effective temperature in section 4. In section 5, we discuss various factors that contribute to the uncertainty budget of the effective temperature. In section 6, we present a simple method to estimate the effective temperature with a good accuracy. Finally, we conclude and discuss our results.

The details of our experimental setup have been described elsewhere [33, 34]. Our vacuum chamber enclosing the ultracold atoms are assembled from commercial stainless-steel vacuum parts and Conflat silica viewports (see figure 1). Thirteen platinum RTDs distributed throughout the external surface provide an real-time measure of the chamber's absolute temperature. One additional RTD is place near the vacuum system to monitor the room temperature. All these RTDs (Omega; PT100) have been calibrated at National Institute of Metrology of China. For a sense current of 1 mA through an RTD, the self-heating reaches a steady-state value of ∼6 mK with a time constant of about 2 s. In practice, the sense current is applied to each RTD only for a short time of 1 s when temperature measurement is required. Therefore, the self-heating effect is actually less than 3 mK. The chamber is connected to the Zeeman slower through a stainless-steel tube. Local heat sources which affect the chamber's temperature include a pair of coils for MOT, Zeeman slower coils and an atomic oven. The MOT coils are water cooled, and its mount is very close to the main chamber, as shown in figure 1. Despite the water cooling, the coil mount surfaces close to the main chamber are still noticeably hotter than the chamber body. So the MOT coils dominate the heating effect although there is no mechanical contact between them. The readout of the RTDs shows a non-uniform temperature distribution, and the maximum temperature difference is as large as ∼2 K.

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As the atoms are illuminated by the thermal radiation of the chamber, they feel a temperature which depends upon the local field energy density u in the form of cu/(4σ), where c is the speed of light and σ is the Stephan–Boltzmann constant. Due to temperature inhomogeneity, the thermal radiation of the chamber deviates from an ideal BBR environment. To account for departures from an isothermal environment, each radiating surface can be described by its own temperature and effective solid angle viewed by the atoms. The effective radiation temperature at the position of the atoms, T_eff, can thus be expressed in terms of effective solid angles of the surrounding surfaces [26]:

$$\begin{equation}{T}_{\text{eff}}^{4}=\sum\limits _{i}\left(\frac{{{\Omega}}_{i}^{\text{eff}}}{4\pi }\right){T}_{i}^{4},\end{equation} \tag{ 1 }$$

where ${{\Omega}}_{i}^{\text{eff}}$ and T_i are the effective solid angle and the temperature of surface i, respectively, and i runs over all enclosure surfaces. Similar to geometric solid angles, effective solid angles satisfy the normalization condition ${\sum }_{i}{{\Omega}}_{i}^{\text{eff}}=4\pi$. Apparently, ${{\Omega}}_{i}^{\text{eff}}$ represents the weight of the corresponding surface i in the calculation of T_eff. Different from geometric solid angles, an effective solid angle depends on the geometry and emissivity of all other surfaces. Only in the limit of a completely black environment, ${{\Omega}}_{i}^{\text{eff}}$ reduces to the geometric solid angle Ω_i .

As shown in the work mentioned above [26], both the effective temperature and effective solid angles can be obtained from a steady-state thermal analysis based on FE method. Such an analysis can take into account all sorts of heat transfers, including thermal radiation, heat conductivity, as well as convection of the air. We reconstruct in SolidWorks the mechanical structure of the vacuum chamber, and then load it to ANSYS steady-state thermal [35] where the simulation is carried out. All the internal surfaces of the chamber and optical windows are treated as opaque, diffuse graybody surfaces with temperature-independent emissivities. On the walls of the chamber, there are two apertures that provide access to the Zeeman slower and the ion pump, respectively. We treat each aperture as a disk-shaped blackbody surface with the same size, which is reasonable because each aperture opens up to a large volume of the closed vacuum system. In our model, ytterbium atoms are replaced by a small blackbody sphere with a diameter of 2 mm, which acts as a probe. When the entire system has reached thermal equilibrium, the temperature of the probe represents the effective temperature felt by the atoms.

Effective solid angle is only related to thermal radiation properties of the surrounding surfaces. Therefore, in the calculation of ${{\Omega}}_{i}^{\text{eff}}$, we do not need to consider the air convection and heat conductivity for the chamber body. Noting that the walls of our vacuum chamber and optical windows have absolute temperatures very close to the room temperature, we can use effective solid angles in the case of uniform temperature distribution to characterize the realistic system. The expression of ${{\Omega}}_{i}^{\text{eff}}$ can be derived from equation (1), and written as

$$\begin{equation*}{{\Omega}}_{i}^{\text{eff}}/(4\pi )={\left(\frac{{T}_{\text{eff}}}{{T}_{i}}\right)}^{3}\frac{\partial {T}_{\text{eff}}}{\partial {T}_{i}}.\end{equation*}$$

In fact, ${{\Omega}}_{i}^{\text{eff}}$ can be simplified to ∂T_eff/∂T_i because of T_eff/T_i ≈ 1. In our model, the vacuum chamber is set to a steady state in which the whole chamber holds an identical temperature except the vacuum component i of interest. We add a small gap of 0.1 mm between component i and the chamber body in order to avoid mechanical contact. Doing so, we can set a temperature T_i for component i that may be different from the chamber temperature. The value of T_i is chosen from a narrow range around the chamber temperature. Then T_eff is calculated for each given T_i . Finally, we deduce the value of ${{\Omega}}_{i}^{\text{eff}}$ from the slope of the T_eff(T_i ) curve.

In table 1 we have listed the effective solid angles for all the vacuum parts. In the calculation typical emissivity values have been adopted for different surface materials. For comparison, geometric solid angles have also been calculated using unit-emissivity (ɛ = 1) for all surfaces. As predicted, the effective solid angle of a vacuum part is not equal to its geometric solid angle. Particularly for optical windows, effective solid angles are much larger than the corresponding geometric solid angles due to the low emissivity of stainless steel walls. The total effective solid angle of the thirteen optical windows, ${{\Omega}}_{\text{win}}^{\text{eff}}$, is about 17.7% of the full solid angle.

Table 1. Effective and geometric solid angles of the enclosure surfaces of our vacuum chamber. In the calculation, the emissivities of fused-silica windows and stainless steel are set to be 0.94 and 0.4, respectively.

Vacuum parts	${{\Omega}}_{i}^{\text{eff}}/4\pi$	${{\Omega}}_{i}^{\text{geo}}/4\pi$
Fused-silica windows
1.33'' CF ( × 8)	0.012 26	0.010 63
2.75'' CF ( × 3)	0.041 82	0.023 68
4.5'' CF (top)	0.118 68	0.041 94
4.5'' CF (bottom)	0.004 26	0.002 79
Subtotal	0.177 02	0.079 04
Stainless steel walls	0.822 99	0.920 96
Total	1.000 01	1.000 00

In our model, all radiation surfaces have been assumed to be opaque to the room-temperature BBR which has a broad infrared spectrum peaked at nearly 10 μm. In fact, a fused silica window is not completely opaque, and a small fraction of thermal radiations can pass through it. According to the reported transmittance of fused silica over the wavelength region above 2 μm [36], the transmittance of our 3 mm thick fused silica windows is estimated to be t = 1.4%. The small residual transmission has no significant effect on the effective temperature T_eff for the following reasons: First, the optical windows cover a rather small effective solid angle compared to the stainless steel walls. Secondly, since the chamber's temperature is pretty close to the room temperature, the incoming and outgoing thermal radiations are nearly balanced. Explicitly, as viewed by the atoms, the residual thermal radiation imbalance should cause a small change of T_eff which can be written as

$$\begin{equation*}\delta {T}_{\text{eff}}^{\text{win}}=({T}_{\text{room}}-{T}_{\text{eff}})\frac{{{\Omega}}_{\text{win}}^{\text{eff}}}{4\pi }t.\end{equation*}$$

For a temperature difference as large as 1 K between T_room and T_eff, $\delta {T}_{\text{eff}}^{\text{win}}$ is only 2.5 mK. It is clear that the opaque gray body assumption is reasonable in the calculation of T_eff, although a small correction must be considered.

Since the ambient temperature for our optical clock changes very slowly, the heat transfer of the vacuum chamber is actually in a steady state at a given time. In our FE analysis program, the thirteen RTDs attached to the chamber are modeled as small stainless steel disks with a diameter of 2 mm and a thickness of 1 mm. Temperatures of all the fourteen RDTs, including the one used to monitor the room temperature, are fixed to their measured values in the simulation. Since convection of the air is also considered in our program, the room temperature and convection heat transfer coefficients around the external surfaces are required as input parameters. As the dominating heat source, the MOT coils affect the temperature distribution of the main chamber through its thermal radiations. This effect cannot be totally reflected by the nearby temperature sensors. Therefore, we include also the MOT coils in the FE analysis for the completeness of our model. For the normal power dissipation of MOT coils, the coil mount surfaces close to the chamber reach a temperature of 309 K, about 11 K higher than that of the chamber. The coil mount is made of aluminum alloy, and its surfaces are polished. Its emissivity is set to be 0.1 in the calculation.

Initially the vacuum chamber and the probe are assigned a uniform temperature value that is close to the room temperature. As the simulation proceeds, the temperature distribution of the vacuum chamber will approach a steady state, while the effective temperature T_eff converges to a fixed value. The steady-state temperature distribution, as well as the corresponding T_eff, are thus obtained after sufficient steps of iterations.

A source of trouble is the lack of knowledge about coefficients of heat convection. Part of the external surfaces of our chamber (the blue area in figure 1) is covered by the closely located coil mount, and the air convection in this area is weakened to some extent. In contrast, other part of external surfaces are exposed to a large volume of the air, which facilitates the air convection. Accordingly, we assign a larger coefficient of heat convection, h₁, to the open surfaces, and a smaller one, h₂, to the covered surfaces. The following procedure is used to determine the optimum values of h₁ and h₂. Among the thirteen RTDs on the chamber, we use eleven of them to calculate the steady-state temperature distribution, i.e., the other two RTDs are spare ones and their measured temperature values are ignored by the FE program. By adjusting h₁ and h₂, we let the calculated temperature values on positions of the two spare RTDs match well with the measured values, and a pair of optimum values of h₁ and h₂ is then obtained. Choosing different RTDs as the two spare ones, we repeat the calculation, and obtain a new pair of optimum values. The averaging for four times repeated calculation yields h₁ = 3.3 W (m²K)⁻¹ and h₂ = 0.4 W (m²K)⁻¹. We have adopted this pair of values in the calculation of T_eff. Note that h₁ here is pretty close to the recommended value for natural convection in air (4 W (m²K)⁻¹) [37].

Figure  2 displays a typical steady-state temperature distribution of the vacuum chamber (MOT coils are not shown for clarity). The CF flanges connected to the Zeeman slower are the hottest vacuum parts. Since the main chamber is a bit hotter than the ambient air, the calculated effective temperature is higher than the measured room temperature. A default mesh size of 4 mm has been used in the FE analysis. In order to know the influence of mesh size on T_eff, we repeat the calculation with different mesh sizes while keeping all other conditions unchanged (see figure 3). For decreasing mesh size, the effective temperature does not converge to a fixed value, possibly due to the larger error accumulation at smaller mesh sizes. Despite this, the discrepancy in T_eff is less than 60 mK. If the T_eff value at the mesh size of 4 mm is regarded as the recommended value, then an uncertainty of 47 mK can cover the whole discrepancy range. For comparison, we also calculated T_eff in absence of air convection, which is about 150 mK higher than the case with air convection.

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As discussed in the previous section, we roughly divided the external surfaces of the vacuum chamber into two regions according to the air convection ability. The coefficients of heat convection were inferred from the self-consistency of temperature distributions. So far we do not know to what extent this simple model has accurately characterized the air convection. In order to estimate the error of T_eff due to the lack of knowledge of air convection, we have checked the dependence of T_eff on the coefficients of heat convection. To avoid underestimating the error, h₁ and h₂ are allowed to vary in a wide range. Specifically, both h₁ and h₂ are varied from 0, the lowest possible value, to a maximum which is three times the optimum value obtained in the previous section. As shown in figure 4, the effective temperature changes with the two convection coefficients. The recommended value of T_eff corresponds to the pair of recommended convection coefficients. In the considered range of h₁ and h₂, the deviation of T_eff from its recommended value can be up to 143 mK. We can safely set an uncertainty of 143 mK for T_eff to account for the limited knowledge of air convection. This item dominates the uncertainty budget of T_eff as shown in table 2.

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The effective temperature is also affected by the two emissivities, ɛ_SST and ɛ_win, which correspond to the stainless steel surfaces and fused-silica windows, respectively. Note that ɛ_SST depends strongly upon the surface quality of the stainless steel. In reference [38], ɛ_SST for stainless steel conductors is estimated to be 0.25 by IR camera measurement, while ɛ_SST for polished stainless steel is measured to be 0.09 in reference [39]. Our vacuum chamber is unpolished, and it has been baked in air before assembling. The surfaces of the chamber show a gray color. Considering the roughness of internal surfaces, we adopted a value of 0.4 for ɛ_SST in the calculations. In the meantime, considering the strong absorption of the optical windows to the BBR radiation, we adopted a value of 0.94 for ɛ_win. We find that T_eff increases monotonically with ɛ_SST (see figure 5). In contrast, T_eff is much less sensitive to ɛ_win, and it decreases monotonically with ɛ_win. We conservatively set an uncertainty range of 0.15 to 0.6 for ɛ_SST, and then the deviation of T_eff from its typical value at ɛ_SST = 0.4 is less than 49 mK. Similarly, given an uncertainty range of 0.7 to 1.0 for ɛ_win, the maximum deviation of T_eff is only 1.4 mK.

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The temperature uncertainty of the MOT coil mount should also be considered. Our simulation shows that T_eff increases linearly with the mount temperature. An uncertainty of 1 K in the mount temperature leads to an uncertainty of 3.8 mK in T_eff. To further understand the thermal radiation effect of the MOT coils mount, we have also calculated T_eff when the thermal radiation of the mount is ignored. In this case, T_eff is slightly shifted down by 29 mK. This difference is much smaller than the uncertainty caused by the air convection.

The aperture opening up to the Zeeman slower has an effective solid angle of ${{\Omega}}_{\text{slower}}^{\text{eff}}/4\pi \simeq 0.001\enspace 92$. In the FE simulation of temperature distribution, the Zeeman slower's flange connected to the vacuum chamber is modeled as a blank flange, which means that the aperture has been replaced by stainless steel disk of the same size. However, the Zeeman slower is hotter than the disk, and its thermal radiation should shift T_eff up. This shift, ${\Delta}{T}_{\text{eff}}^{\text{slower}}$, can be expressed in the following form:

$$\begin{equation*}{\Delta}{T}_{\text{eff}}^{\text{slower}}\simeq \frac{{{\Omega}}_{\text{slower}}^{\text{eff}}}{4\pi }({T}_{\text{slower}}-{T}_{\text{eff}}),\end{equation*}$$

where T_slower is the temperature of the Zeeman slower. In fact, the Zeeman slower has an inhomogeneous temperature distribution. If its average temperature is used as the typical temperature, i.e., T_slower = 306 K, we then have to assign a large uncertainty of 8 K to T_slower. The corresponding shift in T_eff is ${\Delta}{T}_{\text{eff}}^{\text{slower}}\simeq 15\enspace \mathrm{m}\mathrm{K}$ with an uncertainty of 15 mK.

The atomic oven (not shown in figure 1) is further away from the chamber, and it subtends a very small geometric solid angle of 4π × 3.5 × 10⁻⁶. The effective solid angle can be approximated by the geometric solid angle because its thermal radiation is almost totally absorbed by the optical window located in the opposite end. In operation state, the oven works at a high temperature of 673 K with an uncertainty of 20 K. Similar to the Zeeman slower, the atomic oven should shift T_eff up by an amount of 7 mK with an uncertainty of 1 mK.

The uncertainty budget of the effective temperature has been listed in table 2. The air convection is the dominant contribution, while the second and third largest terms arises from the inaccuracy of stainless steel emissivity and the FE analysis error, respectively. Considering the fact that an effective temperature uncertainty of 1 K sets a fractional frequency uncertainty of 3.3 × 10⁻¹⁷ for an Yb clock [40], the uncertainty level of 160 mK for our vacuum chamber represents a BBR shift uncertainty of 5.3 × 10⁻¹⁸.

During each day the room temperature of our laboratory fluctuates slowly with a peak-to-peak value of ∼1 K. We have calculated the T_eff values within 24 hours for discrete times with a half-hour interval. As shown in figure 6, the effective temperature roughly follows the slow fluctuation of room temperature, but with a smaller peak-to-peak variation of about 0.4 K.

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Now that the effective temperature changes with time, the optical clock requires real-time corrections to the BBR Stark shift according to the varying T_eff. However, frequent FE calculations are time-consuming and inconvenient. We need a faster and simpler method to calculate T_eff with sufficient accuracy. Note that the thermal radiation power of a surface element scales as T⁴, the effective temperature can be expressed as a function of the temperatures at the monitored points:

$$\begin{equation}{T}_{\text{eff}}=\sqrt[4]{\sum\limits _{i=1}^{14}{\alpha }_{i}{T}_{i}^{4}},\end{equation} \tag{ 2 }$$

where T_i 's refer to the temperatures of the fourteen RTDs at a given time, and α_i 's are fitting parameters. The data points of T_eff obtained by the FE program, here denoted by ${T}_{\text{eff}}^{\text{FE}}$ for clarity, have been used to determine these fitting parameters. Once the α_i 's are set, formula (2) enable us to know the value of T_eff without FE analysis any more. We find that the effective temperatures calculated by formula (2), denoted by ${T}_{\text{eff}}^{\text{fit}}$, reproduce the corresponding values of ${T}_{\text{eff}}^{\text{FE}}$ with an accuracy level better than 0.8 mK (see figure 6).

For our ¹⁷¹Yb optical clock, the characterization of BBR environment is complicated by the inhomogeneous temperature distribution of the vacuum chamber enclosing the atoms. By using steady-state thermal analysis based on FE method, we have calculated the effective solid angles of the vacuum parts, the steady-state temperature distribution, and the effective temperature T_eff. The limited knowledge of air convection is found to be the main source of error for T_eff. The total uncertainty of T_eff allow us to know the BBR Stark shift with a fractional frequency uncertainty of 5.3 × 10⁻¹⁸. In addition, we have also found that the relation between T_eff and temperatures of the RTDs can be well described by a simple formula, which enables real-time thermometry of the BBR environment, and hence real-time corrections to the BBR Stark shift.

Theoretically, adding more RTDs through the external surface of the chamber would allow more accurate simulation of the temperature distribution, and make T_eff less sensitive to the heat convection coefficients. This approach, however, will increase the complexity of the apparatus. We are planning to build a new ytterbium optical clock with improved BBR environment. The atoms will be enclosed by an in-vacuum BBR shield, where the temperature homogeneity is expected to be orders of magnitude better.

This work is supported by the National Key Research and Development Program of China under Grant No. 2017YFA0304402, by the Natural Science Foundation of China under Grant No. U20A2075, by the Strategic Priority Research Program of the Chinese Academy of Sciences under Grant No. XDB21030100, and in part by the National Natural Science Foundation of China (Grants No. 91636215, No. 11574352, and No. 11803072). We thank Yi-Ge Lin and Zhan-Jun Fang for helpful discussions.