Evaluation of the blackbody radiation shift of an Yb optical lattice clock at KRISS

Stark shifts due to the blackbody radiation (BBR) in most optical clocks are at the 1 Hz level [1]. This shift is one of the significant effects contributing to the frequency uncertainty and also affects the long-term stability of optical clocks. The BBR shift of the clock transition (Δν_BBR) can be expressed as [2]:

$$\begin{align}\hfill {\Delta}{\nu }_{\text{BBR}}\left({T}_{\text{atom}}\right)& ={\Delta}{\nu }_{\text{stat}}{\left(\frac{{T}_{\text{atom}}}{{T}_{0}}\right)}^{4}\hfill \\ \hfill & \quad +{\Delta}{\nu }_{\text{dyn}}\left[{\left(\frac{{T}_{\text{atom}}}{{T}_{0}}\right)}^{6}+\mathcal{O}{\left(\frac{{T}_{\text{atom}}}{{T}_{0}}\right)}^{8}\right].\hfill \end{align} \tag{ 1 }$$

where T_atom is the temperature at the atom trap site, Δν_stat is the coefficient related to the differential static polarizability between the clock states, Δν_dyn is the coefficient associated with the small dynamic correction, and T₀ = 300 K. Since Δν_stat is well known [2–4], the uncertainty in the BBR shift is mainly determined by the uncertainties of T_atom and Δν_dyn. Much effort has been made for more accurate estimation of T_atom during the last decade. The BBR shift uncertainty of 10⁻¹⁷ to mid 10⁻¹⁸ level was attained by measuring the temperature distribution around the vacuum system and by estimating the BBR from the heat sources [5–10]. To further reduce the BBR shift uncertainty to a low 10⁻¹⁸ or 10⁻¹⁹ level, well-controlled thermal environments were required. For example, the clock operation in cryogenic environments could reduce the BBR shift and its uncertainty [11–13]. Another method was to install an in-vacuum thermal shield at room temperature for an isothermal environment, and the additional inhomogeneous heat sources were dealt with a finite-element (FE) radiation analysis using effective solid angles [2, 14, 15]. Also, there has been a method of measuring T_atom directly using a thermal probe [16] (and additional FE analysis in [17]) in a well-controlled thermal environment. In case of ion clocks [18, 19], the FE analysis with validation using a thermal imaging camera could reduce the uncertainty to 10⁻¹⁸ level.

This paper describes the BBR shift evaluation of KRISS-Yb2, a newly developed Yb optical lattice clock at KRISS (Korea Research Institute of Standards and Science). We installed an in-vacuum thermal shield for homogeneous temperature distribution around the atoms. This thermal shield was similar to that used in [2] but was modified to have one side exposed in the air for more convenient temperature control. The temperature at the position of atoms was measured using a black-sphere thermal probe (BSTP). Using this thermal probe, we also obtained the effective solid angles of dominant heat sources experimentally. This approach enabled us to determine directly T_atom without knowing exact values of geometric view factors and emissivities. The estimated uncertainty of T_atom was 13 mK.

To evaluate the BBR shift accurately, we installed an in-vacuum BBR shield chamber made of copper inside a dodecagon titanium main chamber, as in figure 1. The outer rim of the BBR shield served as a gasket of the CF fitting on the main vacuum chamber, as in figure 1(a). The BBR shield chamber was composed of the body part and the cap part, and they were assembled together with titanium bolts. Active temperature control of the BBR shield was carried out using a pair of heater films on the side wall of the body part (red lines in figure 1(a), which was exposed in the air as depicted in figure 1(a). Its temperature was controlled to be slightly higher than the room temperature. Two identical heater films with the same amount of currents in the opposite direction were overlapped to minimize the magnetic field induced by the currents. There were six platinum resistance thermometers (PRTs) installed in three deep holes and three shallow holes on the air-side of the BBR shield chamber to monitor the temperature. These PRTs and the precision resistance meter (milliK, Isotech) were calibrated with an uncertainty of 12.5 mK by the temperature standard group at KRISS. The PRT at the center acted as a reference probe for active temperature stabilization. The inner surface of the BBR shield was black-coated with Ultrablack^® of Acktar [20] (emissivity > 0.98 in the wavelength range 3 μm–10 μm) to reduce unwanted reflections. The QWP-mirror for retroreflection of cooling laser beams and the front window in figure 1(a) were indium-tin-oxide (ITO)-coated and will act as electrodes to generate an electric field for the evaluation of the DC Stark shift which may occur due to the unwanted patch charges. As seen in figure 1(b), four additional electrodes were installed on the cap part of the BBR shield for electric fields along the other two orthogonal axes.

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The in-vacuum temperature-stabilized BBR shield can ensure a homogeneous thermal environment. For the clock operation, we needed several optical and open accesses. The body part of the BBR shield has twelve 16 mm-diameter apertures. As seen in figure 1(c), two of them along the horizontal axis were open for the atomic beam and the Zeeman slowing laser, and they were extended by the 35 mm-long copper tubes with 8 mm inner diameter to reduce the BBR through the open apertures. Two apertures along the vertical axis were closed with copper plates with 4 mm-diameter holes at the center for the access of the lattice laser (the red dashed line in figure 1(c)). The blue dashed lines in figure 1(c) represent the magneto-optical trap (MOT) lasers, each of which transmits through a 4.5 mm-thick BK7 window and retroreflected on the QWP-mirror. Copper plates covered the rear sides of these two QWP-mirrors to block the external radiation. One aperture covered by a 4.5 mm-thick BK7 window was used for the viewport of CCD camera and another one covered by 10 mm-thick lens with the focal length of 35 mm for a fluorescence detection. The remaining two apertures were blocked by copper plates, one of which had an electrical feedthrough connected to the electrode on the QWP-mirror. The cap part of the BBR shield has one 16 mm-diameter aperture covered by a BK7 window. All the windows and a lens were made of BK7 glass to suppress the transmission of the external radiation above 3 μm, and their surfaces were coated with ITO material to drain out electrical charges from their surfaces and generate the electric field.

In summary, the BBR shield chamber has four apertures open for the entrance of atoms and the lattice laser beam, and five apertures (four on the side and one on the top) closed with BK7 optics. These apertures on the BBR shield can introduce additional thermal radiation different from the temperature of the BBR shield. Also the low thermal conductivity of BK7 windows induces the temperature inhomogeneity of the BBR shield chamber. Those effects were investigated using the BSTP described in the following.

Although the BBR shield drastically reduces the thermal environment's inhomogeneity, we cannot neglect the effects of external heat radiation sources and the residual inhomogeneity of the temperature and emissivity. To determine T_atom exactly, one usually needs to know exact values given for geometric view factors and emissivity of all the external heat sources, which is sometimes not feasible. Therefore, we decided to directly measure the temperature at the atom trap site using an in-vacuum black-coated spherical thermal probe (BSTP) and estimated the effective solid angles of dominant heat sources. We will discuss the design considerations of the BSTP, such as geometry and emissivity. Then we will describe its systematic errors.

The BSTP is a black-coated (Ultrablack^®, Acktar) 10 mm-diameter copper sphere with a PRT sensor (PT-111, Lakeshore) as in figure 2. The PRT sensor head (1.8 mm diameter and 5.0 mm length) was glued in the hole of the copper sphere using a thermally conductive epoxy. The resistance of the PRT was measured using four 200 mm-long 0.1 mm-diameter enameled Manganin^® leads connected to the electrical feedthroughs on the top of the main chamber using the four-point probe method. The BSTP was hung by gravity and was located at the center of the BBR shield. These Manganin wires are electrically conductive and thermally insulating with low thermal conductivity of 22 W m⁻¹ K⁻¹ at 300 K, which is about half the value of a widely used phosphor bronze wire.

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We found that the resistance value of the PRT sensor changed after gluing it inside the copper sphere. Thus, the PRT sensor installed in the copper sphere was altogether calibrated by the temperature standard group at KRISS with an uncertainty of 12.5 mK. Repeatability and self-heating of the PRT in vacuum were measured to be 2 mK and 5.5 mK, respectively.

Due to the conduction loss through the Manganin wires, the temperature, T_BSTP, measured by the BSTP can be different from T_atom. Thus, we investigated the dependence of the temperature, especially the conduction loss, on the size and the emissivity of the BSTP by the numerical simulation under simplified thermal environments, as seen in figure 3(a). The outer and inner concentric shells corresponded to the main vacuum chamber and the BBR shield, respectively, and their radii were 100 mm and 50 mm, respectively. The temperature T_VC on the outer spherical shell was fixed at 20 °C, and the temperature T_BBRC on the inner shell is at 30 °C. A Manganin wire connected the copper sphere with the outer shell without touching the inner shell through the 4 mm-diameter hole. There are two 10 mm-diameter apertures along the y-axis of the inner shell as in the BBR shield.

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Figure 3(b) shows the calculated temperature T_wire(z) along the wire. When we neglected the radiation from the wire surface by setting the emissivity zero, the temperature dropped linearly from the BSTP to the end of the wire at the outer sphere. But with the finite emissivity, the part inside the inner shield absorbed the heat radiation, and the temperature changed more slowly. This radiative absorption significantly reduced the conduction loss from the BSTP, which can be clearly seen in figures 4(a) and (b). Figures 4(a) and (b) shows the dependence of T_BSTP against T_BBRC on the emissivity and the radius of the BSTP, respectively, with different values of ɛ_wire. T_BSTP without the wire corresponds to T_atom (represented by black-filled squares in figures 4(a) and (b)). As shown in the insets, T_atom differed from T_BBRC due to the external radiation coming from the outer shell even without the conduction loss.

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For a given value of ɛ_wire, T_BSTP deviated more from T_atom for smaller values of the emissivity ɛ_BSTP or radius r_BSTP of BSTP. As shown in figure 4(a), with r_BSTP = 5 mm and ɛ_wire = 0.9, the difference of T_BSTP from T_atom was smaller than 5 mK for ɛ_BSTP > 0.8. Also, for given ɛ_BSTP = 0.9 and ɛ_wire = 0.9, the difference of T_BSTP from T_atom is smaller than 5 mK for r_BSTP > 4 mm, as shown in inset of figure 4(b).

Although the larger values of r_BSTP, ɛ_BSTP and ɛ_wire are of help to make T_BSTP be closer to T_atom, it should be noted that too big r_BSTP will not well represent atoms at the trap site. Considering these aspects, we chose the BSTP with ɛ_BSTP > 0.9 and r_BSTP = 5 mm in our experiment. In this condition, the offset of T_BSTP from T_atom which can be thought as the black line, due to the conduction loss by wire was expected to be much smaller than the calibration accuracy of the BSTP, 12.5 mK.

Next, we experimentally verified the effect of the conduction loss of the BSTP. We changed the temperature T_end at the end of the wire by varying the temperature of the electrical feedthroughs connected with the four-point probe wires of the BSTP using a heater with a PRT from 22 °C to 34 °C as in figure 5. The temperature of the BBR shield was kept at 24 °C, and that of the outer vacuum chamber was around 21.3 °C. The measured sensitivity of T_BSTP on the temperature of electrical feedthroughs at the end of four-point probe wires, was −0.00014(36) K/K as a result of a linear curve fit shown in figure 5. Temperature offset ΔT_conduction of T_BSTP from T_atom due to conduction loss is

$$\begin{equation}{\Delta}{T}_{\text{conduction}}=-0.000\,14\left(36\right)\times {\Delta}{T}_{\text{wire}},\end{equation} \tag{ 2 }$$

where ΔT_wire = T_BSTP − T_end ≈ T_BBR − T_end.

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When ΔT_wire ≈ 10 K, ΔT_conduction was around 1.4 mK. This is in good agreement with the numerical calculation in figure 4, where ΔT_conduction ≈ 2 mK for ɛ_wire = 0.9, ɛ_BSTP = 0.9, r_BSTP = 5 mm, and ΔT_wire ≈ 10 K. In the clock operation condition, ΔT_wire ≈ 1 K, we expect ΔT_conduction ≈ 0.14(36) mK. The high emissivity of the enamel coating on the wire, ɛ_wire ≈ 0.9, plays a significant role in reducing the BSTP's conductive loss, and we could neglect the conduction loss of the BSTP for the estimation of T_atom from T_BSTP.

The whole experimental setup is shown in figure 6 with the BSTP installed. The BSTP will be removed in the actual clock operation. Atoms can see external BBRs from various parts. In general, the temperature at atoms, T_atom, can be derived with

$$\begin{equation}{T}_{\text{atom}}^{4}=\sum\limits _{i}{{\Omega}}_{\text{eff}}^{i}{T}_{i}^{4}.\end{equation} \tag{ 3 }$$

${{\Omega}}_{\text{eff}}^{i}$ is a normalized effective solid angle of a heat source i, satisfying ${\sum }_{i}{{\Omega}}_{\text{eff}}^{i}=1$ and T_i is its temperature. ${{\Omega}}_{\text{eff}}^{i}$ includes the effects of the geometric view factor and the emissivities of all the relevant parts. Dominant heat radiation (and therefore the largest ${{\Omega}}_{\text{eff}}^{i}$) were from the temperature-controlled BBR shield chamber whose temperature was set slightly higher than the main chamber by 1 K–3 K. In our experimental condition, temperature of each part of the main chamber was similar within 0.5 K and could be represented by the temperature of the main chamber, T_main, which is the average value of temperatures on the top, at the side, and at the front of the main chamber. The radiation from the main chamber has effects on BSTP in several ways. One is the direct radiations through open apertures of the BBR shield and transmitted radiation through BK7 optics on the BBR shield, which is significantly suppressed by BK7 glass. The other indirect effect is absorption and emission by BK7 optics on the BBR shield, which causes temperatures of BK7 optics lower than that of the BBR shield due to poor thermal conductivity of BK7 material and imperfect thermal contact with the BBR shield. All of these contribute to the effective solid angle of T_main. In addition, we considered the temperature T_nipple of the nipple at the right of the main chamber separately because its radiation can be transferred directly through the open aperture of the BBR shield. It is noted that the conduction and radiation from the heated atom oven (T = 673 K) were suppressed with the bellow and the shutter [21] which is closed during the clock spectroscopy.

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Because T_atom can be estimated from the temperature of the BSTP using equation (3), we have

$$\begin{equation}{T}_{\text{atom}}^{4}={\left({T}_{\text{BSTP}}-{\Delta}{T}_{\text{self}}-{\Delta}{T}_{\text{conduction}}\right)}^{4}=\sum\limits _{i}{{\Omega}}_{\text{eff}}^{i}{T}_{i}^{4}.\end{equation} \tag{ 4 }$$

ΔT_self is the self-heating offset, 5.5(2.0) mK, as described previously. To obtain ${{\Omega}}_{\text{eff}}^{i}$ we varied experimental conditions (stabilized temperature of the BBR shield chamber, anti-Helmholtz coil current, ZS duty cycle, cooling water temperature, etc) as described in table 1, and measured T_i and T_BSTP simultaneously and obtained 29 samples of distributed temperature (open circles in figure 7). From the multi-variable non-linear fit of these samples using equation (4), we obtained ${{\Omega}}_{\text{eff}}^{i}$ as in table 1. Normalized solid angles are given in table 1 and summed up close to 1, confirming that we did not omit significant heat sources.

Table 1. Experimental conditions and determined normalized solid angles. Numbers not parenthesized in the 'Temperature range' column were ranges for data set to obtain solid angles, and those parenthesized were for verification data set.

Heat sources	Temperature range/°C	Normalized effective solid angle ${{\Omega}}_{\text{eff}}^{i}$
BBR shield (TBBR) a	22.0–24.0 (22.0–24.0)	0.97814(15)
Right nipple (Tnipple) b	20.9–21.8 (21.3–21.9)	0.00229(85)
Main chamber (Tmain) c	21.3–21.9 (21.2–22.7)	0.01911(78)
Zeeman slower (TZS) d	21.2–33.9 (21.5–34.2)	0.000116(26)
Zeeman slower laser input window (TZSLW)	20–200 (20–200)	0.000041(1)
Atom oven	20–400 (room temperature)	Suppressed
Total		0.9997(12)
TBSTP	21.9–23.9 (22.0–24.0)

^aVaried by the control of the stabilized temperature of the BBR shield. ^bVaried by anti-Helmholtz coil current, cooling water temperature, and room temperature. ^cVaried by anti-Helmholtz coil current, cooling water temperature, and room temperature. ^dVaried by ZS coil current and room temperature.

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Within the range given in table 1, the temperature at atoms can be estimated using the following equation,

$$\begin{equation}{T}_{\text{atom}}=\sqrt[4]{{{\Omega}}_{\text{eff}}^{\text{BBR}}{T}_{\text{BBR}}^{4}+{{\Omega}}_{\text{eff}}^{\text{nipple}}{T}_{\text{nipple}}^{4}+{{\Omega}}_{\text{eff}}^{\text{main}}{T}_{\text{main}}^{4}+{{\Omega}}_{\text{eff}}^{\text{ZS}}{T}_{\text{ZS}}^{4}+{{\Omega}}_{\text{eff}}^{\text{ZSLW}}{T}_{\text{ZSLW}}^{4}}\end{equation} \tag{ 5 }$$

The difference between T_atom measured from T_BSTP and the predicted values using equation (5) was (0.0 ± 2.0) mK as shown in figure 7 within the experimental condition in table 1.

The total uncertainty of the atomic temperature under normal operating conditions for the Yb optical lattice clock is given in table 2.

To evaluate the BBR shift of the Yb optical lattice clock (KRISS-Yb2), we adopted an in-vacuum BBR shield and radiation thermometry using a BSTP. We could obtain the effective solid angles of heat sources experimentally, which did not require exact knowledge of emissivities and geometric view factors of heat sources. To do this, it was important to estimate T_atom from the thermal probe measurement. We carried out the numerical simulation of the conduction effect, which implied that high-emissivity surface of the lead wires mitigated the conduction effect significantly, and this was in good agreement with the experimental observation. At the normal operating condition for the Yb optical lattice clock, we obtained the uncertainty of 13 mK, which corresponds to an uncertainty of 0.22 mHz in clock frequency, or 4.2 × 10⁻¹⁹ in terms of a fractional frequency. When we combine this with the contribution by the dynamic correction from the atomic response Δν_dyn in equation (1) [2], the total BBR shift uncertainty is 9.5 × 10⁻¹⁹. This will pave the way for out Yb optical clock to reach the uncertainty of low 10⁻¹⁸ level.

The authors thank Young Hee Lee, Inseok Yang, and Wukchul Joung for the calibration of the temperature sensors, and Ki-Lyong Jeong for the transmission measurement of the vacuum view-ports. We thank Kwang Min Yu for the calibration of the multimeter of PRTs. This work was supported by Research on Measurement Standards for Redefinition of SI Units funded by Korea Research Institute of Standards and Science (KRISS-2022-GP2022-0001).