IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 50, NO. 2, APRIL 2001 503 Cavity Frequency Pulling in Cold Atom Fountains S. Bize, Y. Sortais, C. Mandache, André Clairon, and C. Salomon Abstract—We present an analytical model and a numerical cal- culation of the cavity frequency pulling effect in cold atom foun- tains. We report the first measurement of this effect in a Rb foun- tain. We find a quantitative agreement with theory. Index Terms—Atomic fountains, cavity frequency pulling. I. INTRODUCTION I N THE cold atom fountain geometry, microwave frequency standards now currently reach accuracies of the order of 10 [1], [2]. As shown in [3], when the microwave frequency is derived from a very low phase noise sapphire cryogenic os- cillator, short-term stabilities as low as ( at 1 day) can be obtained. Here, the stability is limited by the quantum projection noise which means that the fountain has to operate with a high number of detected atoms. This obvi- ously raises the question of atom number dependent systematic effects in atomic fountains. The two main atom number depen- dent effects in cold atom fountains are the collisional frequency shift due to interaction between atoms [4], [5] and the cavity fre- quency pulling. In this paper, we present a calculation and re- port the first measurement of the atom number dependent cavity pulling in a Rb cold atom fountain. In Rb, the cavity pulling is the dominant atom number effect, due to the smallness of the collisional frequency shift [6]. II. BASIC EQUATIONS In atomic fountains, the atomic transition is probed by a mi- crowave field sustained by a resonator with a typical quality factor of 10 000. As in hydrogen masers [7], the cavity fre- quency pulling effect is due to the interference inside the mi- crowave resonator between the field radiated by the input cou- pler and the field radiated by the atomic magnetic dipoles, when the atoms pass through the cavity. This interference induces a time dependent phase shift between the field inside the resonator and the signal delivered by the interrogation oscillator and thus a shift of the clock frequency. Equation (1) gives the general expression for the magnetic field amplitude inside the cavity mode of interest [7] Manuscript received May 14, 2000; revised November 10, 2000. S. Bize, Y. Sortais, and A. Clairon are with Laboratoire Primaire du Temps et des Fréquences, 75014 Paris, France (e-mail: sebastien.bize@obspm.fr). C. Mandache is with Institutul National de Fizica Laserilor, Plasmei si Radi- atiei, Bucuresti, Magurele, Romania. C. Salomon is with Laboratoire Kastler-Brossel, F-75231 Paris Cedex 05, France. Publisher Item Identifier S 0018-9456(01)02596-7. (1) In this formula, is the cavity eigenmode resonance fre- quency, represents the spatial dependence of the eigen- mode, normalized such that at the center of the cavity, the volume of the mode and the magnetic field being defined as and is the mode quality factor. The first term of the right-hand side in (1) represents the coupling from a microwave guide into the cavity. The second source term is due to the atomic cloud oscillating magnetic dipoles with a spatial distribution . Equation (1) assumes a stationary state of the microwave field. In atomic fountains, the damping time of the field s is much shorter than the time it takes for the atoms to cross the resonator ( 30 ms). The field thus responds quasi- instantaneously to the perturbation due to the atomic cloud and (1) can be rewritten as (2) where is the field amplitude in the absence of the atomic cloud. Here, variable refers to the slow evolution of and due to the cloud motion. The atomic internal state evolution is given by the Schrödinger equation with the time dependent Hamiltonian (3) where and are the ground and the excited clock states, respectively, and is the atomic transition energy. is the atomic magnetic dipole operator For the transition under interest, the mag- netic dipole is collinear with the vertical quantization axis im- posed by the bias C-field. In (3), is the oscillating 0018–9456/01$10.00 © 2001 IEEE