Initial atomic coherences and Ramsey frequency pulling in fountain clocks

In the uncertainty budget of primary atomic cesium fountain clocks, evaluations of frequency-pulling shifts of the hyperfine clock transition caused by unintentional excitation of its nearby transitions (Rabi and Ramsey pulling) have been based so far on an approach developed for cesium beam clocks. We re-evaluate this type of frequency pulling in fountain clocks and pay particular attention to the effect of initial coherent atomic states. We find significantly enhanced frequency shifts caused by Ramsey pulling due to sublevel population imbalance and corresponding coherences within the state-selected hyperfine component of the initial atom ground state. Such shifts are experimentally investigated in an atomic fountain clock and quantitative agreement with the predictions of the model is demonstrated.

Figure 1

Energy diagram of the relevant atomic states of the cesium atom. Transitions due to the microwave field component in the direction of the quantization axis ($\pi$ transitions) are shown by solid lines; those due to the microwave field component in the transverse direction ($\sigma$ transitions), by dotted lines. The first-order Zeeman effect is also shown. The g factor of the cesium ground state $F=4$ component is $g_F\approx 1/4$.

Figure 2

Time sequence of the fountain clock cycle (not to scale), indicating the time intervals for the state selection cavity traversal ${\tau }_S$, the free propagation between state selection and the Ramsey cavity $T_{SR}$, the Ramsey cavity traversal ${\tau }_R$, and the free propagation between the two Ramsey cavity traversals $T_R$. The $B_x^{\mathrm{RF}}$, $B_y^{\mathrm{RF}}$, and $B_z^{\mathrm{RF}}$ magnetic components of the microwave field are schematically represented by the shaded areas. The $B_y^{\mathrm{RF}}$ field component in the considered rectangular state selection cavities is 0 (see Appendix pp2).

Figure 3

Measured (gray curve) and calculated (red curve) clock transition probability as a function of the microwave frequency detuning for the fountain clock CSF2 in normal operation. The $\Delta m_F=0$ transitions are labeled “$\pi$”; the $\Delta m_F=\pm 1$ transitions are labeled “$\sigma$.” Due to the large frequency span, the Ramsey fringe structures appear as filled areas.

Figure 4

Diagram of the initial horizontal cloud positions (shaded area) and atomic states. The circle represents a Ramsey cavity aperture of radius $r_c$. Top labels: the absence (light gray) or presence (dark gray or gray gradient) of coherences and their phase dependence on the azimuthal angle (gray gradient). Bottom labels: maximum magnitudes of the Ramsey-pulling frequency shift for magnetic field $B_z=B_z^R$ values between 50 and 250 nT.

Figure 5

Ramsey-pulling shift as a function of the static magnetic field $B_z=B_z^R$ in and above the Ramsey cavity for a 1% population imbalance of the $|3,-1\rangle$ and $|3,+1\rangle$ states. The labels of the individual curves represent the initial conditions from Fig. 4: without coherences (cases A and B, left scale) and with coherences (case C, left scale; cases D–F, right scale).

Figure 6

Microwave amplitude dependence of the Ramsey-pulling shift for a 1% population imbalance of the $|3,-1\rangle$ and $|3,+1\rangle$ states. The labels of the individual curves represent the initial conditions from Fig. 4: without coherences (cases A and C, left scale) and with coherences (case B, left scale; cases D–F, right scale). The microwave amplitude is given with respect to normal operation with a microwave $\pi /2$ pulse area. Circles: odd multiples of a $\pi /2$ pulse area for case A.

Figure 7

Ramsey-pulling frequency shift of the fountain CSF1 as a function of the state selection microwave field detuning. Microwave pulse areas: $9\pi$ (state selection cavity) and $7\pi /2$ (Ramsey cavity). Data points: measurements. Dashed line: calculation based on the measured magnetic field value, $B_z^{SR}=97.2$ nT. The solid line and shaded area correspond to an equivalent magnetic field, $B_z^{SR}+B_z^{\mathrm{Stark}}\pm 0.5$ nT, accounting for the clearing pulse Stark shift. The shift is measured with respect to the fountain frequency without state selection field detuning.

Figure 8

Plot of the Ramsey-pulling frequency shift in the fountain CSF1 versus the static magnetic field $B_z^{SR}$ between the state selection and the Ramsey cavities. Microwave pulse areas: $9\pi$ (state selection cavity) and $7\pi /2$ (Ramsey cavity). Left scale: measurements, symbols. Right scale: calculations, lines. The state selection microwave frequency detuning is $+110\mathrm{Hz}$ [curves (a) and (c); red] or $-110\mathrm{Hz}$ [curves (b) and (d); blue]. Solid lines and shaded areas correspond to an equivalent magnetic field, $B_z^{SR}+B_z^{\mathrm{Stark}}\pm 0.5$ nT, accounting for the clearing pulse Stark shift. The shift is measured with respect to the fountain frequency without state selection field detuning.

Figure 9

Ramsey-pulling frequency shift as a function of the clearing pulse duration. Microwave pulse areas: $9\pi$ (state selection cavity) and $7\pi /2$ (Ramsey cavity). The corresponding lines are sinusoidal fits to guide the eye. Detuning: curve (a), (red) circles, +120 Hz; curve (b), (blue) squares, $-120$ Hz. The shift is measured with respect to the fountain frequency without state selection field detuning.