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arXiv:2309.06761 · Physical Review A

Derivation of coherent-population-trapping resonance signals from density-matrix equations with all relevant sublevels of Cs atoms and confirmation of experimental results

Kenta Matsumoto, Sota Kagami, Takahiro Fujisaku, Akihiro Kirihara, Shinya Yanagimachi, Takeshi Ikegami, Atsuo Morinaga

Published Aug 27, 2024• 1 citation
arXivHTMLPDF

Abstract

Coherent-population-trapping resonance in vapor cell is a quantum interference effect that appears in the two-photon transitions between the ground-state hyperfine levels of alkali-metal atoms and is often utilized in miniature and centimeter-scale clock devices. To quantitatively understand and predict the performance of this phenomenon, it is necessary to consider the transitions and relaxations between all hyperfine Zeeman sublevels involved in the different excitation processes of the atom. In this paper, we constructed a computational multilevel atomic model of the Liouville density-matrix equation for 32 Zeeman sublevels involved in the ${D}_{1}$ line of $^{133}\mathrm{Cs}$ irradiated by two frequencies with circularly polarized components and then simulated the amplitude and shape of the resonance spectrum of the transmitted light through centimeter-scale Cs-vapor cells. We show that the numerical solutions of the equation and analytical investigations adequately explain a variety of the characteristics observed in the experiment.

DOI
10.1103/physreva.110.023113

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arXiv:2110.02466•Physical Review A•Feb 22, 2022•6 citations
Coherent-population-trapping resonance in vapor cell is a quantum interference effect that appears in the two-photon transitions between the ground-state hyperfine levels of alkali-metal atoms and is often utilized in miniature and centimeter-scale clock devices. To quantitatively understand and predict the performance of this phenomenon, it is necessary to consider the transitions and relaxations between all hyperfine Zeeman sublevels involved in the different excitation processes of the atom. In this paper, we constructed a computational multilevel atomic model of the Liouville density-matrix equation for 32 Zeeman sublevels involved in the D1{D}_{1}D1​ line of 133Cs^{133}\mathrm{Cs}133Cs irradiated by two frequencies with circularly polarized components and then simulated the amplitude and shape of the resonance spectrum of the transmitted light through centimeter-scale Cs-vapor cells. We show that the numerical solutions of the equation and analytical investigations adequately explain a variety of the characteristics observed in the experiment.
We study the peculiarities of the coherent population trapping (CPT) dark resonance in atomic vapor confined in a finite-size cell. Taking into account the scattering of atoms on anti-relaxation coating of the walls, we analyze the line shape of the CPT resonance in 133 C s depending on various parameters, such as the width of laser radiation spectrum, the type of pumping, and the magnitude of static magnetic field for both lin || lin and l i n ⊥ l i n polarization schemes. We show that for different parameters, the dependence of the line shape on the size of a cell is different, i.e., different factors affect the CPT resonance nonadditively. Thus, in the case when CPT resonance is excited by a narrowband laser, the shape of the resonance weakly depends on the size of a cell due to the laser-induced narrowing effect. We also show that the distortion of the line shape of the CPT resonance caused by nonresonant hyperfine sublevels of atomic excited states is modified with the size of a cell. Our calculation reveals the important role of accounting for the hyperfine and Zeeman structure of atomic levels. We have shown that the contrast of the CPT resonance differs about 5 times from the prediction of the simple three-level Λ model. The obtained results can be used for the optimization of the quantum frequency standards based on the CPT effect. In the final part of the paper, on the basis of the shot noise approximation we estimate the short-term stability of such standards.
SummaryCurrent compact atomic clock technologies rely on coherent population trapping (CPT), which employs a microwave resonance to stabilize a crystal oscillator. Standard CPT, while a simple and easy-to-implement optical preparation scheme in alkali metal atoms, suffers from low signal-to-noise ratios (SNR). The primary contributor to the low signal amplitude is the collection of atoms at the extreme ends of the Zeeman sublevels in the hyperfine levels of the alkali ground state, so-called “trap states.” Solutions to the trap-state problem have been reported previously, though the redistribution of atomic population into the CPT signal states with these methods has not yet been carefully examined. This work aims to provide an experimental basis to evaluate trap-state recovery schemes.
The evolution of the populations of the magnetic sublevels of the cesium atom (133Cs isotope) in resonant laser fields with linear polarization is analyzed using the equations for the density matrix. Analytic expressions are derived for stationary populations resulting from laser-induced optical transitions on the hyperfine structure components F g = 3 ↔ F e = 2, 3 and F f = 4 ↔ F e = 3, 4 of lines D1 (62S1/2 → 62P1/2) and D2 (62S1/2 → 62P3/2) depending on the initial values of the populations. The numerical solution of the evolution equations gives the characteristic times of stabilization of the steady regime as functions of laser field intensities and detuning from optical resonance. We determine the sequences of optical transitions increasing (by more than an order of magnitude) the population of the lower sublevel 62S1/2F g = 3 M = 0 of the “clock” microwave transition F g = 3 M = 0 ↔ F f = 4 M = 0 in the cesium frequency standard, which increases the signal intensity in the recording system by the same proportion.
State-of-the-art miniature atomic clocks (MACs) are based on the phenomenon of coherent population trapping (CPT) in alkali-metal atomic vapors (Rb\mathrm{Rb}Rb or Cs\mathrm{Cs}Cs). Improving the stability of the clock frequency is a crucial issue; resolving it would result in significant progress in numerous applications. Here, we examine the configuration of the light field, which is composed of two bichromatic beams with opposing handedness of circular polarization. The beams are in resonance with optical transitions in the Cs\mathrm{Cs}Cs D1{D}_{1}D1​ line ($\ensuremath{\lambda}\ensuremath{\approx}895$ nm). This configuration has previously been shown to produce observed CPT resonances with higher contrast than a standard single-beam scheme; however, our scheme implements pump and probe beams that operate independently, with the transmission of the probe beam being monitored separately. This approach differs from previous studies. The experiments are carried out using a cubic $5\ifmmode\times\else\texttimes\fi{}5\ifmmode\times\else\texttimes\fi{}5\phantom{\rule{0.2em}{0ex}}{\mathrm{mm}}^{3}$ vapor cell filled with buffer gas. It is shown that the line shape of the resonance acquires an asymmetry that can be efficiently controlled by a microwave (Raman) phase between the beams. As a proof of concept, we study the way in which this asymmetry can help to mitigate the influence of the ac Stark (light) shift on the long-term frequency stability of CPT clocks based on vapor microcells ($V\ensuremath{\ll}1\phantom{\rule{0.2em}{0ex}}{\mathrm{cm}}^{3}$). Experimental verification is performed with both a distributed-Bragg-reflector laser and a vertical-cavity surface-emitting laser. The latter is of particular importance for the development of MACs. The experimental results are in qualitative agreement with the analytical theory based on a double-$\mathrm{\ensuremath{\Lambda}}$ scheme of atomic energy levels.
In this paper we present our spectroscopic studies on Coherent Population Trapping (CPT) in micro-fabricated Caesium cells and our evaluation of its application in miniature atomic frequency standards (atomic clocks). We observe the CPT signal on the Cs D1-line by coupling two hyperfine ground-state Zeeman sub-levels to a common excited state using two coherent electromagnetic fields created with a modulated DFB laser. Contrarily to double resonance, CPT does not require any microwave cavity, which should facilitate the miniaturization of a future atomic clock device. We study and report here on the light shift phenomena at different cell temperatures and laser wavelength. We also present resonance shifts due to cell temperature variations and clock frequency stability measurements. To the best of our knowledge, this article is the first report on light shift with Cs D1 line in a CPT vapour-cell atomic clock.
We have experimentally demonstrated that magnetic-field-insensitive coherent-population-trapping (CPT) resonances are generated between the ground hyperfine levels on the D1{D}_{1}D1​ line of 133Cs^{133}\mathrm{Cs}133Cs using a two-photon $\mathrm{\ensuremath{\Lambda}}$ scheme excited by lin||lin polarizations. The frequency shift of the CPT resonance is 0.04 Hz for the deviation of $1\phantom{\rule{0.16em}{0ex}}\ensuremath{\mu}\mathrm{T}$ at a ``magic'' magnetic field of $139\phantom{\rule{0.16em}{0ex}}\ensuremath{\mu}\mathrm{T}$ and is 50 times smaller than that of the conventional clock transition at a typical bias magnetic field for the clock operation. The amplitude of the CPT spectrum excited by lin||lin polarizations is enhanced as the excitation intensity increases and is well explained by the three-level model without trap levels. Thus, the CPT resonance on the D1{D}_{1}D1​ line of 133Cs^{133}\mathrm{Cs}133Cs excited by a simple lin||lin scheme will be one of the best candidates for frequency reference of miniature atomic clocks.