Reply

The relative frequency uncertainty due to the distributed cavity phase (DCP) shift, which is equal to 3 × 10−16, is the biggest contribution to the current error budget of the Rb fountain frequency standard. The experimental data presented in our paper [1] enable us to conclude that this is a safe uncertainty for the DCP frequency shift. In response to the comment above, we will further demonstrate this by providing limits for the DCP shift in terms of different m-components of a Fourier expansion of the phase distribution of the microwave field inside the cavity, as was suggested in [2]. m = 0 phase variation: The theory [2, 3] predicts a very strong dependence of the corresponding frequency shift on the amplitude of the microwave field. Given that the measured dependence of the DCP shift shows no relative shift between the 3π /2 and 5π /2 excitation pulses within the combined uncertainty 2.1 × 10−15, the theoretical dependence (dashed line in figure 1 of the comment) predicts the m = 0 DCP shift from a π /2 pulse to be about 1.0 × 10−17. Taking into account that the corresponding dependences look similar for three different Cs cavities (of SYRTE, PTB and NPL), the theory must give the right order of magnitude of the shift for our cavity. m = 1 phase variation: This shift was measured by asymmetric excitation of the Ramsey cavity in the presence of a 1 mrad tilt of the fountain along the feeds. (The angle was determined from the tilt-sensitive microwave leakage shift of our fountain.) At fully asymmetric excitation of the cavity, the corresponding frequency shift is about 1.5×10−15. Therefore, under normal conditions, when the fountain is excited symmetrically with amplitudes of the opposite waves balanced within 1%, the corresponding shift should be ∼1.5 × 10−17. The loaded quality factor of our Ramsey cavity, Qc = 28 500, is very close to the theoretically calculated unloaded quality factor of an ideal cylindrical cavity of the same size, Qu = 32 100, and has remained constant since the cavity was built. Therefore, it is unlikely that this cavity can have any significant variations of the surface resistance, caused by copper oxidation or deposition of Rb atoms, which would lead to the appearance of additional running waves inside the resonator. The weak coupling of the cavity also provides more accurate balancing of the feeds by standard methods, which do not demand tilting of the fountain. m = 2 phase variation: This shift was never measured experimentally because of its smallness <1.0 × 10−16 and its value was based solely on calculations [2, 3]. There are no reasons why it should be larger in our fountain, which has a large radius of the initial atomic cloud (3 mm) and large waist radius of the detection laser beams (25 mm at 1/e2 intensity level). Therefore, the total DCP frequency shift in our fountain is more likely to be ∼1.0 × 10−16 and the given uncertainty of 3 × 10−16 is a rather conservative value. A more complete characterization of this shift, as is done in [3], should reduce its uncertainty down to ∼1.0 × 10−16, which will reduce the total frequency uncertainty of the Rb frequency standard down to 2.3 × 10−16. We have to admit that in our paper [1] the uncertainty of the DCP frequency shift was estimated based on the phenomenological DCP model [4], which, as it was confirmed by recent measurements [3], is not accurate enough. For our fountain, the frequency shift due to the microwave lensing effect of the cavity field, which is calculated according to formula (5) of the work [5], is equal to 8.9×10−17, which is within the ‘other’ frequency shifts in the current error budget. In addition, this shift can be completely eliminated by periodic changing of the initial hyperfine state of the launched atoms. Such a switching will lead to a change of sign of the corresponding frequency shift, which will lead to averaging it away. We do admit that the increased temperature of the launched atoms will not decrease the DCP frequency shift. On the other