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Journal of Experimental and Theoretical Physics Letters

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, O. N. Prudnikov, M. D. Radchenko, V. G. Pal'chikov

Published Apr 1, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1134/s0021364026600436

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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

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.

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An effective model for describing coherent population trapping resonances, which correctly takes into account the off-resonant frequency components in periodically modulated laser field

Valeriy I. Yudin, M. Yu. Basalaev, A. V. Taichenachev +1 more

We have developed an effective mathematical model to calculate the coherent population trapping (CPT) resonance in periodically modulated light, when the modulation frequency $f$ varies near the fractional part of hyperfine splitting in the ground state $Δ_{\rm hfs}/N$ (where $N=1,2,...$). In such polychromatic field, only two frequency components that are most resonant with atomic optical transitions are taken into account accurately, while all other off-resonant components are taken into account using the second-order perturbation theory in the field. Within the presented concept, equation for atomic density matrix is obtained, in which the contribution of all off-resonant components is reduced to the appearance of two new operators (non-diagonal, in general case): the shift operator and relaxation operators. In the case of three-level $Λ$-system, the adequacy of presented effective model was verified by numerical calculations of various dependencies, in which we did not find visual differences from the exact calculations. In addition to a significant mathematical simplification, our model provides a clear physical picture of various features of CPT spectroscopy in a periodically modulated laser field, including effects that have not been discussed in the scientific literature before. In particular, we show that the widespread viewpoint that the CPT resonance shift is determined by usual 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 $Δ_{\rm hfs}$ between different off-resonant frequency components can be comparable (or even dominate) with the standard ac Stark shift. Therefore, even if we have detailed information on the modulated field spectrum (e.g. using a spectrum analyzer), this is, in general, absolutely insufficient to determine the light shift of CPT resonance.

arXiv:2505.01924•May 3, 2025•0 citations

Efficient bichromatic model for describing coherent population trapping resonances in a periodically modulated laser field

Valeriy I. Yudin, M. Yu. Basalaev, А. В. Тайченачев +1 more

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 $f$ is close to the fractional part of the hyperfine splitting frequency (HFS) in the ground state ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{HFS}}/N$ (where $N=1,2,3,\dots{}$). 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.

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Transient processes under dynamic excitation of a coherent population trapping resonance

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Addition to the dynamic Stark shift of the coherent population trapping resonance

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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 $Λ$ 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.

quant-pharXiv:2601.10319•Physical Review A•Jan 16, 2026•0 citations
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.
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 developed an effective mathematical model to calculate the coherent population trapping (CPT) resonance in periodically modulated light, when the modulation frequency fff varies near the fractional part of hyperfine splitting in the ground state Δhfs/NΔ_{\rm hfs}/NΔhfs​/N (where N=1,2,...N=1,2,...N=1,2,...). In such polychromatic field, only two frequency components that are most resonant with atomic optical transitions are taken into account accurately, while all other off-resonant components are taken into account using the second-order perturbation theory in the field. Within the presented concept, equation for atomic density matrix is obtained, in which the contribution of all off-resonant components is reduced to the appearance of two new operators (non-diagonal, in general case): the shift operator and relaxation operators. In the case of three-level ΛΛΛ-system, the adequacy of presented effective model was verified by numerical calculations of various dependencies, in which we did not find visual differences from the exact calculations. In addition to a significant mathematical simplification, our model provides a clear physical picture of various features of CPT spectroscopy in a periodically modulated laser field, including effects that have not been discussed in the scientific literature before. In particular, we show that the widespread viewpoint that the CPT resonance shift is determined by usual 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 ΔhfsΔ_{\rm hfs}Δhfs​ between different off-resonant frequency components can be comparable (or even dominate) with the standard ac Stark shift. Therefore, even if we have detailed information on the modulated field spectrum (e.g. using a spectrum analyzer), this is, in general, absolutely insufficient to determine the light shift of CPT resonance.
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.
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.