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Joint Conference of the European Frequency and Time Forum and IEEE International Frequency Control Symposium (EFTF/IFCS)

Transparency waveforms in square wave frequency modulated vapor cell passive frequency standards, and their influence on frequency noise aliasing

Andrea De Marchi, Massimo Ortolano, N. Beverini

Published Jan 22, 2003• 0 citations
Publisher

Abstract

The need for a dynamical analysis of the atomic system response in square wave frequency modulated vapor cell passive frequency standards was recently recognized in connection with the problem of characterizing the mechanism by which frequency noise of the local oscillator (LO) is partially transferred onto the output frequency of the standard. In this paper, waveforms calculated with a dynamic analysis based on the density matrix formalism are reported and successfully compared with experimental ones obtained at NIST in a Rb cell standard. A theoretical description of how the dynamical response affects the noise transfer mechanism is briefly sketched and preliminary conclusions are drawn.

DOI
10.1109/freq.1999.840718

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The need for a dynamical analysis of the atomic system response in square wave frequency modulated vapor cell passive frequency standards was recently recognized in connection with the problem of characterizing the mechanism by which frequency noise of the local oscillator (LO) is partially transferred onto the output frequency of the standard. In this paper, waveforms calculated with a dynamic analysis based on the density matrix formalism are reported and successfully compared with experimental ones obtained at NIST in a Rb cell standard. A theoretical description of how the dynamical response affects the noise transfer mechanism is briefly sketched and preliminary conclusions are drawn.
Frequency noise of the local oscillator at even harmonics of the modulation frequency is known to contribute to the short-term instability of passive frequency standards. The mechanism by which this noise transfer takes place in Rb-cell standards is described here in a comprehensive approach that includes both signal theory methods and a time-dependent quantum mechanical analysis. The resulting calculated transfer coefficients for the various even harmonics are discussed in relationship with measured coefficients in an actual experiment.
The error signal which is used to control the frequency of the quartz crystal oscillator of a passive rubidium cell frequency standard is considered. The value of the slope of this signal, for an interrogation frequency close to the atomic transition frequency is calculated and measured for various phase (or frequency) modulation waveforms, and for several values of the modulation frequency. A theoretical analysis is made using a model which applies to a system in which the optical pumping rate, the relaxation rates and the RF field are homogeneous. Results are given for sine-wave phase modulation, square-wave frequency modulation and square-wave phase modulation. The influence of the modulation frequency on the slope of the error signal is specified. It is shown that the modulation frequency can be chosen as large as twice the non-saturated full-width at half-maximum without a drastic loss of the sensitivity to an offset of the interrogation frequency from center line, provided that the power saturation factor and the amplitude of modulation are properly adjusted.
This paper gives a theory of the signal strength in rubidium vapor frequency standards for various situations encountered in practice. The density matrix formalism is used and the signal strength is given as a function of light intensity, rubidium density, absorption cell length, and applied microwave power. The results obtained are found to be in good agreement with published data.A theory is also developed giving the strength and the phase of the signal as a function of the modulation frequency and of the frequency excursion amplitude. This theory uses a model of a simple harmonic oscillator. The results are similar to those obtained by previous authors using a more complex model.
We present a multilevel theoretical approach to describe the behavior of a pulsed optically pumped (POP) 87Rb^{87}\mathrm{Rb}87Rb frequency standard based on a microwave cavity-vapor cell arrangement. The full Zeeman manifold of the ground-state hyperfine levels is then considered, and the dynamics induced among them by relaxation processes (buffer gas, spin exchange, and cell-walls collisions) is taken into account. The model includes as well the absorption of the pumping laser along the cell and the cavity feedback on the atoms; the effects related to a nonuniform excitation of the atomic sample are also discussed. Theoretical predictions are proven with a laboratory prototype of POP passive maser in which the clock transition is excited by means of the Ramsey technique. The agreement between theory and experiment is very satisfactory from both a quantitative and qualitative point of view, in terms of the shape of the Ramsey fringes, microwave power delivered by the atoms, and short-term frequency stability. In particular, a frequency stability of $1.2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}12}$ at one second has been measured, in very good agreement with the expected value. On this basis, the model has been extended to predict the short-term frequency stability of a POP frequency standard operating with optical detection. A shot-noise limited frequency stability (Allan deviation) of ${\ensuremath{\sigma}}_{y}(\ensuremath{\tau})\ensuremath{\approx}3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}14}{\ensuremath{\tau}}^{\ensuremath{-}1∕2}$ is foreseen. This value can be degraded by microwave phase noise and laser fluctuations so that the overall predicted clock stability is ${\ensuremath{\sigma}}_{y}(\ensuremath{\tau})\ensuremath{\approx}1.5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}{\ensuremath{\tau}}^{\ensuremath{-}1∕2}$.
In this paper we calculate the effect of spin-exchange collisions in alkali-metal vapors. In the framework of the high-energy approximation, we evaluate the spin-exchange cross sections related to the line broadening and to the frequency shift of the ground state hyperfine transition. We do the calculation for the four isotopes, 23Na^{23}\mathrm{Na}23Na, 39K^{39}\mathrm{K}39K, 87Rb^{87}\mathrm{Rb}87Rb, and 133Cs^{133}\mathrm{Cs}133Cs. The results are used in particular to evaluate the spin-exchange frequency shift in Rb vapor cell frequency standards used in many applications. It turns out that, due to possible fluctuations in the atomic density, spin exchange may affect significantly the medium and long term frequency stability of the frequency standard.
One of the important subjects in coherent-population- trapping-based (CPT-based) vapor cell frequency standards is the light shift (ac Stark shift). We calculate the light shift using a numerical method and perturbation approximation. Experimentally, we measure light shift using a pair of phase-locked diode lasers as well as a frequency-modulated diode laser. There is good agreement between theory and experiment. A method of controlling light shift in CPT-based frequency standards is proposed and implemented in Agilent Laboratories. The short-term stability of our CPT-based rubidium vapor cell frequency standard is measured as 1.3 x 10(-12) tau (-1/2), which is limited by the phase noise of the reference local oscillator used.