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Journal of Applied Physics

Comment on “Reduction of magnetic-field-induced shift in quantum frequency standards based on coherent population trapping” [J. Appl. Phys. 137, 164402 (2025)]

Eugene A. Tsygankov

Published Jun 23, 2026• 0 citations
Publisher

Abstract

In the work of Tsygankov et al. [Phys. Rev. A 109, 053703 (2024)], it was demonstrated that the phase modulation of the two-photon detuning leads to the multipeak structure of the coherent population trapping resonance. Recently, an article [Vishnyakov et al., J. Appl. Phys. 137, 164402 (2025)] was published where an interplay between side peaks from hyperfine transitions mFg=∓1⇆mFg=∓1 and the central one from the magneto-insensitive one in 87Rb atoms is used to reduce the frequency sensitivity of the resonance to magnetic field’s variations. In this Comment, I address an interpretation given in the work of Vishnyakov et al. [J. Appl. Phys. 137, 164402 (2025)] that the multipeak structure is due to additional sidebands of the optical field’s spectral components, which are provided by modulation. It is demonstrated that in this case, the peaks should have amplitudes that are not observed experimentally and contradict analytical results of Tsygankov et al. [Phys. Rev. A 109, 053703 (2024)].

DOI
10.1063/5.0307718

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Response to Comment on “Reduction of magnetic-field-induced shift in quantum frequency standards based on coherent population trapping” [J. Appl. Phys. 137, 164402 (2025)]

V. I. Vishnyakov, Denis V. Brazhnikov, Mikhail N Skvortsov

Dr. Tsygankov commented (see the preceding Comment) on our work Vishnyakov et al. [J. Appl. Phys. 137, 164402 (2025)], expressing disagreement with our physical interpretation of the multi-peak structure of the coherent population trapping (CPT) resonance, which can be observed under the Pound–Drever–Hall-like (PDH) regime of excitation. Below, we provide a brief response to the Comment, from which it follows that our physical interpretation remains unchanged.

Journal of Applied Physics•Jun 23, 2026•0 citations

Comment on “Light Shifts of Coherent Population Trapping Resonances in a Periodically Modulated Laser Field” (JETP Letters 122, 139 (2025))

Eugene A. Tsygankov

In this comment, I demonstrate that an addition to the light shift of the coherent population trapping resonance’s frequency, shown in [V. I. Yudin, M. Yu. Basalaev, A. V. Taichenachev, O. N. Prudnikov, M. D. Radchenko, and V. G. Palchikov, JETP Lett. 122, 144 (2025)], is not due to a new physical effect but is consequence of the asymmetry. To do this, the following model is considered. The phase of a monochromatic optical field is harmonically modulated at the frequency corresponding to the interval between ground-state levels of the Λ-scheme. The carrier is tuned to the low-frequency transition, while the first-order sideband is resonant to the high-frequency transition. I present curves demonstrating the opposite asymmetry of the resonance for the cases of the shift deviating from the light one in different directions.

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Coherent population trapping resonances on the D 1 line of rubidium

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In this communication we present our results related to examination of Coherent Population Trapping resonance profile on the D1 line of 87Rb when coupling two ground-state Zeeman sublevels belonging to different hyperfine ground-state levels to a common excited-state one. For this, two coherent laser fields with frequency difference of about 6.8 GHz are used. The resonance is observed as a narrow dip in the fluorescence in dependence on the modulation frequency (fm) of the laser light, centered at 2fm = &Dgr;vhf , where &Dgr;vhf is the frequency difference between the ground-state hyperfine levels of Rb. The resonance observation and its splitting in 7 components have been demonstrated under influence of nonshielded laboratory magnetic field (MF). It has been shown that different components are with different sensitivity to the MF gradients that is in agreement with the performed estimations.

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A nonlinear frequency shift caused by asymmetry of the coherent population trapping resonance: a generalization

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We investigate the coherent population trapping resonance induced by a polychromatic optical field with an asymmetric spectrum, i.e., whose sidebands equidistant from the carrier have unequal powers. A situation is considered where a modulation is used to provide the so-called in-phase and quadrature signals in the optical field's transmission, which are used for stabilization of the local oscillator's frequency in chip-scale atomic clocks. In a general case, the frequencies of the signals nonlinearly depend on the optical field's intensity due to the asymmetry of the resonance. In this work, we demonstrate a) that this effect stems from a multi-peak structure of the resonance; b) a linear dependence of the frequency on the optical field's intensity at high modulation values; c) that the regime of the resolved structure has more advantages than suppression of the frequency shift due to the spectrum asymmetry.

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Reduction of magnetic-field-induced shift in quantum frequency standards based on coherent population trapping

V. I. Vishnyakov, Denis V. Brazhnikov, Mikhail N Skvortsov

We investigate the magnetic-field-induced frequency shift (MFS) of the clock “0–0” transition in the microwave quantum frequency standard (atomic clock) based on coherent population trapping (CPT) in 87Rb vapor. To scan the CPT resonance and to form the error signal, a method analogous to the Pound–Drever–Hall (PDH) technique in the optical frequency range is employed, where the modulating frequency (fm) significantly exceeds the resonance linewidth (FWHM). The experiments demonstrate that this technique offers brilliant capabilities for controlling the sensitivity of the clock transition frequency to magnetic field variations in the vapor cell compared to the conventional method with low-frequency modulation (fm≪FWHM). Specifically, the PDH technique provides several optimal values of the bias magnetic field generated by the solenoid, at which the “0–0” transition frequency exhibits extremely low sensitivity to small variations in the external magnetic field. Furthermore, these magnetic field values can be easily adjusted by changing fm, which is relevant for the optimization of the atomic clock’s operating regime. The experimental results show that by using the PDH technique, the influence of MFS on the clock transition can be suppressed down to ≈3.2×10−13δB2mG−2. These findings can be leveraged both to relax stringent requirements for magnetic field shielding in state-of-the-art CPT-based miniature atomic clocks and to build a new generation of such clocks with long-term frequency stability better than 10−12.

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Reply to the Comment on “Light Shifts of Coherent Population Trapping Resonances in a Periodically Modulated Laser Field” (JETP Letters 122, 139 (2025))

V. I. Yudin, M. Yu. Basalaev, A. V. Taichenachev +3 more

This is the response to the comment by E.A. Tsygankov on our work [V. I. Yudin, M. Yu. Basalaev, A. V. Taichenachev, O. N. Prudnikov, M. D. Radchenko, and V. G. Palchikov, JETP Lett. 122, 144 (2025)]. It has been shown that the mathematical model used by the author is physically inadequate for describing coherent population trapping resonances under typical experimental conditions of an atomic gas with a temperature of $$T \gtrsim 300$$ K. The critical influence of the choice of the relaxation operator in the ground state on the description of the light shift and the shape of the coherent trapping resonance line has also been demonstrated.

Journal of Experimental and Theoretical Physics Letters•Apr 1, 2026•0 citations
In the work of Tsygankov et al. [Phys. Rev. A 109, 053703 (2024)], it was demonstrated that the phase modulation of the two-photon detuning leads to the multipeak structure of the coherent population trapping resonance. Recently, an article [Vishnyakov et al., J. Appl. Phys. 137, 164402 (2025)] was published where an interplay between side peaks from hyperfine transitions mFg=∓1⇆mFg=∓1 and the central one from the magneto-insensitive one in 87Rb atoms is used to reduce the frequency sensitivity of the resonance to magnetic field’s variations. In this Comment, I address an interpretation given in the work of Vishnyakov et al. [J. Appl. Phys. 137, 164402 (2025)] that the multipeak structure is due to additional sidebands of the optical field’s spectral components, which are provided by modulation. It is demonstrated that in this case, the peaks should have amplitudes that are not observed experimentally and contradict analytical results of Tsygankov et al. [Phys. Rev. A 109, 053703 (2024)].
In this comment, I demonstrate that an addition to the light shift of the coherent population trapping resonance’s frequency, shown in [V. I. Yudin, M. Yu. Basalaev, A. V. Taichenachev, O. N. Prudnikov, M. D. Radchenko, and V. G. Palchikov, JETP Lett. 122, 144 (2025)], is not due to a new physical effect but is consequence of the asymmetry. To do this, the following model is considered. The phase of a monochromatic optical field is harmonically modulated at the frequency corresponding to the interval between ground-state levels of the Λ-scheme. The carrier is tuned to the low-frequency transition, while the first-order sideband is resonant to the high-frequency transition. I present curves demonstrating the opposite asymmetry of the resonance for the cases of the shift deviating from the light one in different directions.
In this communication we present our results related to examination of Coherent Population Trapping resonance profile on the D1 line of 87Rb when coupling two ground-state Zeeman sublevels belonging to different hyperfine ground-state levels to a common excited-state one. For this, two coherent laser fields with frequency difference of about 6.8 GHz are used. The resonance is observed as a narrow dip in the fluorescence in dependence on the modulation frequency (fm) of the laser light, centered at 2fm = &Dgr;vhf , where &Dgr;vhf is the frequency difference between the ground-state hyperfine levels of Rb. The resonance observation and its splitting in 7 components have been demonstrated under influence of nonshielded laboratory magnetic field (MF). It has been shown that different components are with different sensitivity to the MF gradients that is in agreement with the performed estimations.
We investigate the coherent population trapping resonance induced by a polychromatic optical field with an asymmetric spectrum, i.e., whose sidebands equidistant from the carrier have unequal powers. A situation is considered where a modulation is used to provide the so-called in-phase and quadrature signals in the optical field's transmission, which are used for stabilization of the local oscillator's frequency in chip-scale atomic clocks. In a general case, the frequencies of the signals nonlinearly depend on the optical field's intensity due to the asymmetry of the resonance. In this work, we demonstrate a) that this effect stems from a multi-peak structure of the resonance; b) a linear dependence of the frequency on the optical field's intensity at high modulation values; c) that the regime of the resolved structure has more advantages than suppression of the frequency shift due to the spectrum asymmetry.
We investigate the magnetic-field-induced frequency shift (MFS) of the clock “0–0” transition in the microwave quantum frequency standard (atomic clock) based on coherent population trapping (CPT) in 87Rb vapor. To scan the CPT resonance and to form the error signal, a method analogous to the Pound–Drever–Hall (PDH) technique in the optical frequency range is employed, where the modulating frequency (fm) significantly exceeds the resonance linewidth (FWHM). The experiments demonstrate that this technique offers brilliant capabilities for controlling the sensitivity of the clock transition frequency to magnetic field variations in the vapor cell compared to the conventional method with low-frequency modulation (fm≪FWHM). Specifically, the PDH technique provides several optimal values of the bias magnetic field generated by the solenoid, at which the “0–0” transition frequency exhibits extremely low sensitivity to small variations in the external magnetic field. Furthermore, these magnetic field values can be easily adjusted by changing fm, which is relevant for the optimization of the atomic clock’s operating regime. The experimental results show that by using the PDH technique, the influence of MFS on the clock transition can be suppressed down to ≈3.2×10−13δB2mG−2. These findings can be leveraged both to relax stringent requirements for magnetic field shielding in state-of-the-art CPT-based miniature atomic clocks and to build a new generation of such clocks with long-term frequency stability better than 10−12.
This is the response to the comment by E.A. Tsygankov on our work [V. I. Yudin, M. Yu. Basalaev, A. V. Taichenachev, O. N. Prudnikov, M. D. Radchenko, and V. G. Palchikov, JETP Lett. 122, 144 (2025)]. It has been shown that the mathematical model used by the author is physically inadequate for describing coherent population trapping resonances under typical experimental conditions of an atomic gas with a temperature of T≳300T \gtrsim 300T≳300 K. The critical influence of the choice of the relaxation operator in the ground state on the description of the light shift and the shape of the coherent trapping resonance line has also been demonstrated.