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Journal of Applied Physics

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

Published Jun 23, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1063/5.0328065

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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

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)].

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

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

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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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.

Journal of Experimental and Theoretical Physics Letters•Apr 1, 2026•0 citations

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We have experimentally demonstrated that magnetic-field-insensitive coherent-population-trapping (CPT) resonances are generated between the ground hyperfine levels on the ${D}_{1}$ line of $^{133}\mathrm{Cs}$ 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 ${D}_{1}$ line of $^{133}\mathrm{Cs}$ excited by a simple lin||lin scheme will be one of the best candidates for frequency reference of miniature atomic clocks.

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Strong Mitigation of the Magnetic-Field-Induced Frequency Shift in Coherent-Population-Trapping Atomic Clocks

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

We study the magnetic-field-induced frequency shift (MFS) of the clock (``0--0'') transition in coherent-population-trapping (CPT) microwave atomic clock. It is shown that the use of the Pound-Drever-Hall-like (PDH) technique for frequency locking provides brilliant opportunities for mitigating the MFS. Using a $0.125$ cm$^3$ rubidium vapor cell with a buffer gas, we have measured a residual sensitivity of the clock transition frequency to be $\approx\,72$ $μ$Hz/mG over $\approx\,$$6$ mG interval. It means that a fractional frequency shift is extremely small ($\approx\,$$1$$\,\times\,$$10^{-14}$ mG$^{-1}$). The results contribute to the development of a new-generation CPT-based miniature atomic clock (MAC) with improved long-term frequency stability. The proposed method is quite general and can be used for other excitation schemes in atomic clocks, including Ramsey-like techniques.

arXiv:2506.05865•Jun 6, 2025•0 citations
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.
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)].
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.
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.
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.
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.
We study the magnetic-field-induced frequency shift (MFS) of the clock (``0--0'') transition in coherent-population-trapping (CPT) microwave atomic clock. It is shown that the use of the Pound-Drever-Hall-like (PDH) technique for frequency locking provides brilliant opportunities for mitigating the MFS. Using a 0.1250.1250.125 cm3^33 rubidium vapor cell with a buffer gas, we have measured a residual sensitivity of the clock transition frequency to be ≈ 72\approx\,72≈72 μμμHz/mG over ≈ \approx\,≈666 mG interval. It means that a fractional frequency shift is extremely small (≈ \approx\,≈111 × \,\times\,×10−1410^{-14}10−14 mG−1^{-1}−1). The results contribute to the development of a new-generation CPT-based miniature atomic clock (MAC) with improved long-term frequency stability. The proposed method is quite general and can be used for other excitation schemes in atomic clocks, including Ramsey-like techniques.