Cancellation of the Collisional Frequency Shift in Caesium Fountain Clocks

We have observed that the collisional frequency shift in primary caesium fountain clocks varies with the clock state population composition and, in particular, is zero for a given fraction of the $|F=4,m_F=0⟩$ atoms, depending on the initial cloud parameters. We present a theoretical model explaining our observations. The possibility of the collisional shift cancellation implies an improvement in the performance of caesium fountain standards and a simplification in their operation.

Figure 1

The theoretical collision rate coefficients as a function of collision energy $E$. The collision energy is given in terms of the Boltzmann constant $k_B$.

Figure 2

Evolution of the local energy during the ballistic flight (Monte Carlo simulations). The atoms pass through the Ramsey cavity at $t_{\mathrm{up}}=0.165\mathrm{s}$ and $t_{\mathrm{down}}=0.665\mathrm{s}$ in the NPL-CsF1 ($t_{\mathrm{up}}=0.130\mathrm{s}$, $t_{\mathrm{down}}=0.695\mathrm{s}$ in the PTB-CSF1). The black lines correspond to the initial cloud size of ${\sigma }_0=1\mathrm{mm}$ and represent various temperatures at launch (solid line $1\mu \mathrm{K}$; dashed line $2\mu \mathrm{K}$; dotted line $5\mu \mathrm{K}$). The gray solid line corresponds to ${\sigma }_0=5\mathrm{mm}$ and initial temperature of $1\mu \mathrm{K}$.

Figure 3

Calculated values of the cancellation point for various initial cloud sizes (${\sigma }_0$, spherical cloud shapes assumed) and two initial temperatures ($T_0=1\mu \mathrm{K}$: circles; $T_0=2\mu \mathrm{K}$: triangles). The points calculated for the same $T_0$ are linked to guide the eye. Note that for ${\sigma }_0>1\mathrm{mm}$ ($T_0\ge 1\mu \mathrm{K}$) the cancellation does not occur. Inset: Calculated collisional frequency shift as a function of the fraction of the $|4,0⟩$ state population. The position of the zero-shift (cancellation) point ${\rho }_{40}^{(C)}$ depends on the initial parameters ${\sigma }_0$ and $T_0$ and the timing $t_{\mathrm{up}}$ and $t_{\mathrm{down}}$. The slope of the linear function is proportional to the number of atoms in the fountain, with other parameters unchanged.

Figure 4

Measurement of the collisional frequency shift as a function of the population composition during the Ramsey time. The experimental data are fitted by a linear function. (a) Data from the NPL-CsF1, and (b) data from the PTB-CSF1 (in gray are plotted data points taken for the fountain operating at $1<x<2$). A typical atomic density in the center of the cloud, averaged over the Ramsey time, was ${10}^7{\mathrm{cm}}^{-3}$ for both setups.