Comment on ''Inverse Doppler shift and control field as coherence generators for the stability in superluminal light''

TL;DR

The paper critiques a previous study on inverse Doppler shift and superluminal light, arguing that the parameters used do not produce observable superluminal effects and that the Doppler shifts are negligible.

Contribution

It clarifies that the parameter choices in the criticized study do not demonstrate superluminal light or significant Doppler shifts.

Findings

Doppler shifts are negligible at the given parameters.

The probe wavelength is in the decimeter band, not supporting superluminal effects.

Simulations do not evidence superluminal light phenomena.

Abstract

In their study of inverse Doppler shift and superluminal light [Phys. Rev. A 91, 053807 (2015)], Ghafoor et al. consider a three-level atomic arrangement with transitions in the optical domain. In fact, the values they give to the parameters lead to a probe wavelength lying in the decimeter band. We point out that the Doppler shifts are then negligible and remark that the simulations performed by Ghafoor et al. do not evidence any superluminal effect.

Equations

$$\chi =\frac{2N{\left|{\wp }_{ac}\right|}^2{\rho }_{ac}}{{\epsilon }_0\hslash {\Omega }_p}$$

$$\Gamma =\frac{{\left|{\wp }_{ac}\right|}^2{\omega }_{ac}^3}{{\epsilon }_0\hslash c^3}=O\left(\frac{N{\left|{\wp }_{ac}\right|}^2}{{\epsilon }_0\hslash }\right)$$

$$N=O\left(\frac{8{\pi }^3}{{\lambda }^3}\right)$$

Full text

In their study of inverse Doppler shift and superluminal light art1 , Ghafoor et al. consider a three-level atomic arrangement with transitions in the optical domain, referring in particular to the sodium $D_{2}$ line at $586.9\,\mathrm{nm}$. On the other hand, they specify in the caption of their Fig.2 that all the (angular) frequencies are given in units of $\Gamma=2\pi\times 1\,\mathrm{MHz}$ and that the frequency of the probe transition $\omega_{ac}=1000\Gamma$. The corresponding wavelength is thus $\lambda=30\,\mathrm{cm}$ (in the decimeter band).

The first consequence of the large value of the probe wavelength is that the Doppler broadening $V_{D}$ is very small, in the order of $2\pi\times 1\,\mathrm{kHz}$ . The consideration of $V_{D}$ going from $2$ to $12\,\mathrm{MHz}$ as made in art1 is meaningless and the so-called inverse Doppler shift, claimed as the novelty of the article, is in fact negligible.

A second point is that the calculations developed in art1 lead to fully unrealistic values of the atomic number density $N$. As correctly given in the article, the electric susceptibility for the probe reads in SI units:

[TABLE]

where $a$ ($c$) is the upper (lower) level of the probe transition, $\wp_{ac}$ ($\rho_{ac}$) is the corresponding matrix element of the dipole moment (of the density operator) and $\Omega_{p}$ is the Rabi (angular) frequency of the probe. From the involved discussion following this equation, it results that

[TABLE]

and that

[TABLE]

For $\lambda=30\,\mathrm{cm}$, we get an atomic number density in the order of $10^{-2}\mathrm{cm^{-3}}$, which is 12 orders of magnitude lower than those attainable in the best vacuum devices.

As a third point, we remark that, contrary to the claim made in the article title, the simulations made in art1 do not evidence any superluminal effect, namely an advance of the intensity profile of the transmitted pulse on that of the incident one (see Fig.5). The calculation itself raises some questions. The transmitted field is actually the inverse Fourier transform of $S_{in}(\omega)H(\omega)$, where $S_{in}(\omega)$ and $H(\omega)$ are, respectively, the Fourier transform of the incident field and the transfer function of the medium. Insofar as $S_{in}(\omega)$ is Gaussian and $H(\omega)$ is the exponential of a polynomial of degree 3, the result cannot be that given by Eq.(15) in art1 but necessarily involves an Airy function. We also note that the transfer function $H(\omega)$ considered by Ghafoor et al. neglects the frequency dependence of the medium transmission that can considerably affect the profile of the transmitted pulse art2 .

For completeness, we mention that some equations in art1 seem to be dimensionally inhomogeneous, that the Einstein’s coefficient given below Eq.(4) is erroneous (see art3 for its exact value in SI units) and that Eq.(1) and Eq.(8) mix results that hold, respectively, in SI and in electrostatic units (the corresponding susceptibilities differ by a factor of $4\pi$).

We finally point out that the atomic number density given by Eq.(3), anomalously weak in the conditions considered in art1 , raises on the contrary to values $N=O(10^{15}\mathrm{cm^{-3}})$ which are too large when the probe wavelength $\lambda$ is that of the sodium $D_{2}$ line. On another hand, the fixed ratio $\omega_{ac}/\Gamma=1000$ leads then to lifetimes of the excited atomic states which are fully unrealistic (in the subpicosecond range).

We thank Shubhrangshu Dasgupta from the Indian Institute of Technology Ropar for helpful information. This work has been partially supported by the Ministère de l’Enseignement Supérieur, de la Recherche et de l’Innovation , the Conseil Régional des Hauts de France and the European Regional Development Fund (ERDF) through the Contrat de Projets État-Région (CPER) 2015–2020, as well as by the Agence Nationale de la Recherche through the LABEX CEMPI project (ANR-11-LABX-0007).