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Journal of Experimental and Theoretical Physics Letters

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

Eugene A. Tsygankov

Published Apr 1, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1134/s0021364025609169

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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))

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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

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

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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.

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We present the research of dynamic continuous-wave spectroscopy of the coherent population trapping resonance at the phase-jump modulation. Λ system of levels supplemented by a nonabsorbing state and bichromatic optical field, whose spectral components have different intensities, are considered. We demonstrate that the asymmetry leads to an additional nonlinear shift of the error-signal frequency under unisotropic relaxation of the ground-state density-matrix elements. We also investigate the conventional approach where the frequency difference of the optical field components is harmonically modulated to obtain the error signal. Comparison demonstrates that in the high-frequency modulation regime the corresponding frequency shift is more linear than at the phase-jump modulation for nonshort integration times.

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The appearance of the population difference in the ground state of the $$\Lambda $$ -system due to the spontaneous decay of the upper level into lower ones is considered. The resulting asymmetry of the coherent population trapping resonance is shown to lead to a frequency shift of the quadrature signal, which nonlinearly depends on the optical field intensity. In this case, this contribution is not suppressed due to the use of high modulation frequencies compared to the relaxation rate of ground state polarization (analogue of the Pound–Drever–Hall technique). A significant smallness of this shift when using a binary mixture of noble buffer gases is discussed.

Journal of Experimental and Theoretical Physics Letters•Jul 1, 2024•1 citation
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.
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 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.
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 present the research of dynamic continuous-wave spectroscopy of the coherent population trapping resonance at the phase-jump modulation. Λ system of levels supplemented by a nonabsorbing state and bichromatic optical field, whose spectral components have different intensities, are considered. We demonstrate that the asymmetry leads to an additional nonlinear shift of the error-signal frequency under unisotropic relaxation of the ground-state density-matrix elements. We also investigate the conventional approach where the frequency difference of the optical field components is harmonically modulated to obtain the error signal. Comparison demonstrates that in the high-frequency modulation regime the corresponding frequency shift is more linear than at the phase-jump modulation for nonshort integration times.
The appearance of the population difference in the ground state of the Λ\Lambda Λ -system due to the spontaneous decay of the upper level into lower ones is considered. The resulting asymmetry of the coherent population trapping resonance is shown to lead to a frequency shift of the quadrature signal, which nonlinearly depends on the optical field intensity. In this case, this contribution is not suppressed due to the use of high modulation frequencies compared to the relaxation rate of ground state polarization (analogue of the Pound–Drever–Hall technique). A significant smallness of this shift when using a binary mixture of noble buffer gases is discussed.