Magnetic blackbody shift of hyperfine transitions for atomic clocks

Abstract

We derive an expression for the magnetic blackbody shift of hyperfine transitions such as the cesium primary reference transition which defines the second. The shift is found to be a complicated function of temperature, and has a T 2 dependence only in the high-temperature limit. We also calculate the shift of ground-state p1 /2 hyperfine transitions which have been proposed as new atomic clock transitions. In this case interaction with the p3 /2 fine-structure multiplet may be the dominant effect.PACS numbers: 06.20.fb,32.60.+i The frequency of the ground-state hyperfine transitions used in atomic clocks (such as the cesium primary standard) are known to be temperature dependent [1]. For this reason the SI second is defined at 0 K, and at any finite temperature the blackbody shift must be taken into account. Temperature fluctuation of the laboratory is a major portion of the clock error budget [2], therefore the NIST-F2 cesium fountain, currently under construction, will be cooled to 77 K to reduce the blackbody shift.Recently there was some disagreement in the literature over the size of the electric blackbody radiation shift in cesium. Early measurements and ab initio calculations support a value about 10% higher than later measurements and semiempirical calculations (see [3] for references). On the theory side, this seems to have been resolved [3,4,5] in favour of the larger values. As the temperature of the experiment is reduced in the future the magnetic blackbody shift (∼ T 2 ) will become more important relative to the electric shift (∼ T 4 ). Hence this reassessment of the magnetic blackbody shift.In this paper we present a derivation of the magnetic blackbody shift of ground-state hyperfine transitions that is valid at all temperatures (not just in the high-temperature limit). We calculate the effect for s1 /2 hyperfine transitions such as the 6s1 /2 (F = 3 → 4) 133 Cs transition which defines the second (there are many other such clocks, including 87 Rb, 171 Yb + , and 199 Hg + ). We find that the simple scaling law of the blackbody shift ∆ω hfs ∼ T 2 is only valid at high temperatures. Additionally we calculate the shift for p1 /2 hyperfine transitions which have been proposed as clock references [6]. We show that interaction with the p3 /2 fine-structure multiplet must be considered.The magnitude of the magnetic blackbody field is (atomic unitsh = e = m e = 1)An oscillating magnetic field B(ω) cos(ωt) affects an atomic energy level via the time dependent perturbationwhere µ is the magnetic dipole moment of the system. The energy is affected in the second order of perturbation theory (see, e.g. [7])For an atom with a single electron above closed shells µ = −µ B (L + g s S) with g s = 2 and µ B = α/2 in atomic units. A general expression for this case is presented in the Appendix. We first examine the case of a single s1 /2 orbital split by the hyperfine interaction with a nuclear spin I. In this case there are only two levels of interest, with F = I + 1 /2 and F = I − 1 /2; the next level will be ...