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Journal of Applied Physics

A multi-factor model for the coherent population trapping resonance

Sergey Petropavlovsky, VP Yakovlev, Dmitry S. Chuchelov, Maria I. Vaskovskaya, Vitali V. Vassiliev, Sergei A. Zibrov, Vladimir L Velichansky

Published Mar 2, 2026• 0 citations
Publisher

Abstract

A systematic semi-analytical study of the coherent population trapping resonance is presented. The emphasis is put on quantifying the importance of different experimental factors potentially affecting the shape of the resonance, and especially the magnitude and structure of the light shift. In particular, we consider the impact of FM modulation, spectrum asymmetry, light detuning, and an optically thick atomic medium. The proposed model allows for a quick and reliable check whether a zero light shift can be reached in a realistic experimental setting. The results of modeling are compared with the experiment carried out with a diode laser (vertical-cavity surface emitting laser) and 87Rb atoms.

DOI
10.1063/5.0307357

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A systematic semi-analytical study of the coherent population trapping resonance is presented. The emphasis is put on quantifying the importance of different experimental factors potentially affecting the shape of the resonance, and especially the magnitude and structure of the light shift. In particular, we consider the impact of FM modulation, spectrum asymmetry, light detuning, and an optically thick atomic medium. The proposed model allows for a quick and reliable check whether a zero light shift can be reached in a realistic experimental setting. The results of modeling are compared with the experiment carried out with a diode laser (vertical-cavity surface emitting laser) and 87Rb atoms.
Abstract Motivated by recent developments in atomic frequency standards employing the effect of coherent population trapping (CPT), we propose a theoretical framework for describing the CPT resonances. Our extended model includes several realistic experimental factors not accounted for in the standard approaches, namely, the FM-based technique for generating the error signal, skewed laser spectrum, etc. We provide a simple yet non-trivial analytical solution revealing the magnitude and structure of the light shift. The performance of the model is checked against numerical simulations, the agreement is good to perfect.
Abstract We develop a theory of modulation response of the coherent population trapping (CPT) resonance. We consider the simplest three-level atom, but take into account the polychromatic spectrum of the pumping and probing light produced by pure FM modulation of single frequency field. The analysis based on the density matrix equations rigorously includes the most important for applications range of modulation parameters where modulation frequency and deviation are comparable with or greater than the CPT linewidth, so the response of the atomic medium is not adiabatic. Some theoretical results, in particular the possibility of using a quadrature response to modulation to suppress light shift, are compared with the experiment carried out with a diode laser (VCSEL) and 87 Rb atoms.
A method of detecting Coherent Population Trapping resonance by phase modulation (PM) was studied to expand the frequency locking range (LR). We calculated error signals by density matrix analysis using an eigenvector algorithm and verified the calculated result using an 87Rb vapor cell and a Rb-D1 vertical cavity surface-emitting laser. By comparing the error signal for PM with that for conventional frequency modulation (FM), it was found that the LR with PM is one order of magnitude wider than that with FM without degrading the frequency stability.
This paper presents a theoretical study of the light-induced shift of the coherent population trapping resonance. An analytical model is proposed that describes the interaction of two radiation components with an atomic system using a ΛΛΛ scheme and takes into account an additional level of excited state. Both weak and strong coupling regimes with off-resonant transitions are considered. It is shown that, in addition to the conventional dynamic Stark shift, an extra shift arises due to the distortion of the resonance line shape when bichromatic laser radiation interacts with off-resonant atomic transitions. An analytical expression for this additional shift is derived in the weak-coupling limit, and its significant impact on the resonance shape and sensitivity to the intensities of the laser field components is demonstrated. It is found that under strong coupling conditions, the additional shift can deviate substantially from a linear dependence on light intensity, suggesting new opportunities for controlling light shifts in precision atomic devices such as quantum frequency standards.
Motivated by recent developments in atomic frequency standards employing the effect of coherent population trapping (CPT), we propose a theoretical framework for the frequency modulation spectroscopy of the CPT resonances. Under realistic assumptions we provide simple yet non-trivial analytical formulae for the major spectroscopic signals such as the CPT resonance line and the in-phase/quadrature responses. We discuss the influence of the light shift and, in particular, derive a simple expression for the displacement of the resonance as a function of modulation index. The performance of the model is checked against numerical simulations, the agreement is good to perfect. The obtained results can be used in more general models accounting for light absorption in the thick optical medium.
We have developed and tested an efficient mathematical model for calculating a coherent population trapping (CPT) resonance in a periodically modulated laser field when the modulation frequency fff is close to the fractional part of the hyperfine splitting frequency (HFS) in the ground state ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{HFS}}/N$ (where N=1,2,3,…N=1,2,3,\dots{}N=1,2,3,…). In such a polychromatic light, only two frequency components that are most resonant with the working optical transitions in the atom are taken into account exactly, while all other off-resonant frequency components are taken into account by using the second-order perturbation theory in the field magnitude. Within the framework of our concept, an equation for the atomic density matrix is obtained, in which all off-resonant frequency components are described by two operators (which are nondiagonal in the general case): a shift operator and a relaxation operator. This model has been verified by numerical calculations for various dependencies, in which we have not found visual differences from the results of exact calculations. Thus, in our bichromatic model, the infinite variety of periodically modulated fields in CPT spectroscopy can be uniformly described by only five parameters. In addition to a significant mathematical simplification, our model provides a clear physical picture of the various features of CPT spectroscopy in a periodically modulated laser field, including some effects that have not yet been discussed in the scientific literature. In particular, we show a widespread viewpoint that the CPT resonance shift is determined by standard ac Stark shifts of the lower levels is, in general, fundamentally incorrect, since the contribution to the light shift due to beats at the frequency ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{HFS}}$ between different off-resonant frequency components can be comparable to the value of standard ac Stark shift (or even dominate it). Therefore, even if we have detailed information on the spectral composition of the modulated field (e.g., using a spectrum analyzer), this is, in general, absolutely insufficient for determining the light shift of the CPT resonance.