Equation 123 · Part 3 · The Clock That Comes Back Wrong by Exactly Its Mass

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k_{\rm eff}\approx9.00\times10^6\,\mathrm{m^{-1}}

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What this part means

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

Its job in the formula

A raised mark can be a power or an index. Its position and the surrounding notation determine which.

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The passage around this formula

The scale of what needs to be canceled is worth stating plainly, using the same illustrative parameters as the previous section. The laser/center-of-mass term for a single-photon transition near 698\, $\mathrm{nm}$ ( $k_{\rm eff}$ $\approx$ 9.00 $\times$ 10^6\, $\mathrm{m^{-1}}$ ) scales as $k_{\rm eff}$ g $T^2$ $\approx$ 8.8 $\times$ 10^{7}\, $\mathrm{rad}$ for T=1\, $\mathrm s$ . The recoil term for a strontium-mass atom ( m $\approx$ 1.443 $\times$ 10^{-25}\, $\mathrm{kg}$ ) scales as $\hbar$ $k_{\rm eff}^2$ T/m $\approx$ 5.9 $\times$ 10^{4}\, $\mathrm{rad}$ . The clock term computed above is $\approx$ 0.29\, $\mathrm{rad}$ . Both confound terms are, respectively, roughly eight and five orders of magnitude larger than the signal, which is exactly why…

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An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

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