Analysis tools for the accurate evaluation of a small frequency standard.

The short, optically pumped cesium beam tube developed at Laboratoire de l'Horloge Atomique has been carefully evaluated. For that purpose, we have developed a digital servo system that controls three parameters: the frequency of the ultra stable oscillator (USO), the microwave power of the signal experienced by the cesium atoms, and the static magnetic field applied to the atoms. The frequency standard shows a very satisfactory level of short- and medium-term frequency stabilities. A relative frequency offset, measured to be 4.10(-12 ), results mainly from the residual phase difference between the oscillatory fields in the two interaction regions, which is due to imperfection in cavity symmetry. We present two different means of analyzing the causes of this spurious frequency offset using theoretical and experimental considerations. First, a numerical simulation of the beam tube response is performed as a function of the microwave field amplitude for different values of the residual phase difference DeltaPhi. Results include the cavity-pulling effect. Compared with the measured frequency offset, the numerical simulation leads to a second-order Doppler shift of -3.3 mHz and a residual phase difference, DeltaPhi, between the fields interacting with the atoms in the second and first regions of the Ramsey cavity, amounting to +150 murad. Second, an experimental method of measurement of DeltaPhi without beam reversal is implemented. The latter yields DeltaPhi=155+/-17 murad. Finally, the clock accuracy is determined. It is equal to +/-14.10(-13).

Limitation of the frequency stability by local oscillator phase noise: new investigations and natural improvements.

In frequency standards in which the atoms have a continuous interaction with the probe signal, local oscillator phase noise may limit medium term frequency stability. This spurious effect cannot be suppressed whenever there Is any truncation in the spectrum of the resonator response. Nevertheless, a simultaneous processing of the probe signal, similar to that of the NIST, and of the resonator response (by means of an appropriate demodulation) makes it possible to reduce this limiting effect. Previously achieved with a square wave frequency modulation, this result is now extended to various frequency modulations. An uncontrolled distortion in the demodulation waveform may significantly degrade the performance. For the case of a square wave phase modulation, the limiting effect also exists, but it is smaller than for a frequency modulation. When the phase noise of the local oscillator is naturally "not flat", it is possible to easily reduce the spurious effect: using the quasi-static approximation, one can calculate various optimized demodulation waveforms and the corresponding improvements. For the simplest optimized demodulation (f (M), 3f(M)), theoretical predictions are experimentally confirmed for flicker phase noise and flicker frequency noise.

Controlling the microwave amplitude in optically pumped cesium beam frequency standards.

Assuming square wave frequency modulation, the response, versus the amplitude of the microwave field, of an optically pumped cesium beam tube is considered. The properties of the first maximum of this response are analyzed. The effect of the neighboring lines is taken into consideration, and a model of the profile of the microwave field in each interaction region is validated. A symmetry property of the response considered is pointed out. It enables us to implement a feedback control of the microwave amplitude with a large depth of the amplitude modulation. Residual frequency offsets that may occur as a consequence of a spurious amplitude modulation correlated with the frequency modulation are assessed. And, with a cavity designed such that sigma=pi between the two oscillatory fields, it also is possible to measure the microwave amplitude at the first maximum of the sole contribution of the reference atomic line.

Transient response following frequency or amplitude switching in a cesium beam tube.

The transient responses of an optically pumped cesium beam tube to square wave frequency and amplitude modulation is considered. The frequency transient is computed assuming a phase difference phi of either 0 or pi between the two oscillatory fields. We present theoretical and experimental data showing that, contrary to the frequency transient, the amplitude transient depends on the direction of switching. The knowledge of this property is useful for the design of the servo-loop controlling the amplitude of the microwave signal applied to the atomic resonator. A justification of this asymmetrical behavior is given. Experimental results confirm the theoretical predictions in the case phi=pi.

Frequency stability degradation of an oscillator slaved to a periodically interrogated atomic resonator.

Atomic frequency standards using trapped ions or cold atoms work intrinsically in a pulsed mode. Theoretically and experimentally, this mode of operation has been shown to lead to a degradation of the frequency stability due to the frequency noise of the interrogation oscillator. In this paper a physical analysis of this effect has been made by evaluating the response of a two-level atom to the interrogation oscillator phase noise in Ramsey and multi-Rabi interrogation schemes using a standard quantum mechanical approach. This response is then used to calculate the degradation of the frequency stability of a pulsed atomic frequency standard such as an atomic fountain or an ion trap standard. Comparison is made to an experimental evaluation of this effect in the LPTF Cs fountain frequency standard, showing excellent agreement.

Properties of an oscillator slaved to a periodically interrogated atomic resonator.

In advanced atomic resonators, such as those using a fountain of cold cesium atoms or an ensemble of stored ions, the atomic medium is interrogated periodically, and the control signal of the slaved oscillator is updated at equally spaced time intervals. We analyze the properties of the output frequency of these frequency standards. We establish the equations that describe the time behavior of this frequency. We give the stability condition and the transient response of the frequency feedback loop, the response to systematic frequency changes of the free running oscillator, the frequency stability for given free-running oscillator noise and given optical detection noise, and the limitation of the frequency stability by down-conversion of the intrinsic oscillator frequency noise (Dick effect). We point out that a second integration in the feedback loop may not improve significantly the rejection of slow perturbations, unless a condition relative to the timing of the atom-field interaction is verified.

Physical origin of the frequency shifts in cesium beam frequency standards-related environmental sensitivity.

When observed in a cesium beam frequency standard, the hyperfine transition frequency of the atoms differs slightly from the invariant transition frequency of the unperturbed atoms at rest. The various physical and technical origins of the frequency offsets are stated. They relate to fundamental physical effects, to the method of probing the atomic resonance and to the frequency control of the slaved oscillator. The variation of the frequency offsets under a change of the value of the internal operating characteristics is considered. The sensitivity to a change of the magnetic induction, the microwave power and the temperature is given. A comparison is made of the sensitivity of cesium beam frequency standards of the commercially available type, making use of magnetic state selection, and of devices under study in which the state preparation and detection is accomplished optically. The pathways between the external stimuli and the physical origin of the frequency offsets are specified.

New design for a high performance optically pumped cesium beam tube.

An optically pumped cesium beam resonator has been designed including three successive magnetic field regions. The optical interactions take place in the first and third regions, where the magnetic field has the required value of 3x10(-5) T. The microwave interaction occurs in the intermediate region, where the value of the C-field is typically set to 4x10(-6) T. It has been verified that the magnetic field profile along the cesium beam does not induce Majorana transitions. Using a single laser diode emitting at 852 nm with a linewidth of about 30 MHz, the resonator gives an excellent amplitude signal to noise ratio equal to 20000 in a 1-Hz bandwidth.