Comment on "Gain-assisted superluminal propagation and rotary drag of photon and surface polaritons"

TL;DR

This paper critically examines a previous study on superluminal propagation and surface polaritons, highlighting that the parameter choices made in that work lead to physically irrelevant results due to the probe wavelength being in the decimeter range.

Contribution

The authors provide a critical commentary clarifying the physical relevance of parameter choices in the prior work on superluminal phenomena.

Findings

Previous parameter choices are physically irrelevant due to wavelength issues

The study's results do not apply to optical domain conditions

Highlights importance of realistic parameter selection in superluminal research

Abstract

In their study of superluminal propagation, rotary drag and surface polaritons [Phys. Rev. A 96, 013848 and 049906(E) (2017)], Khan et al. consider a four-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 and we point out that, in such conditions, all their results are irrelevant.

Equations

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

$${\left|{\wp }_{ac}\right|}^2=\left(\frac{3{\lambda }_p^3}{8{\pi }^2}\right)\hslash {\epsilon }_0{A}_{ac}$$

$$\chi =\left(\frac{3N{\lambda }_p^3}{32{\pi }^3}\right)\left(\frac{8\pi A_{ac}}{{\Omega }_p}\right){\rho }_{ac}.$$

$$u_{\chi }=\frac{3N{\lambda }_p^3}{32{\pi }^3}$$

Full text

In their study of superluminal propagation, rotary drag and surface polaritons art1 ; Art2 , Khan et al. consider a four-level atomic arrangement with transitions in the optical domain. See Fig. 1(a) in art1 . On the other hand, they specify in Art2 that all the (angular) frequencies are given in units of $\gamma=(2\pi)\times 1\,\mathrm{MHz}$ and that the probe frequency $\nu_{p}=1000\gamma$. The corresponding wavelength is thus $\lambda_{p}=30\,\mathrm{cm}$ (in the decimeter band!). As shown in the following this invalidates all the results given in art1 ; Art2 .

As correctly given in art1 , the electric susceptibility for the probe reads, in SI units,:

[TABLE]

where $N$ is the atomic number density, $a$ ($c$) 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. Expressing the susceptibility as a function of the probe wavelength as made to obtain Eq. (5) in art1 ; Art2 can be achieved by introducing the Einstein’s coefficient $A_{ac}$ associated with the transition $a\rightarrow c$. From its expression given in Art3 , we get:

[TABLE]

and finally

[TABLE]

The expression $\chi=\left(\frac{3N\lambda_{p}^{3}}{32\pi^{3}\Omega_{p}}\right)\rho_{ac}$ given by Eq. (5) in Art2 thus holds only if $\Omega_{p}$ is expressed in units of $8\pi A_{ac}$. According to the above choice of $\gamma$ as unit of (angular) frequency, this implies that $8\pi A_{ac}=\gamma$.

It is specified in Art2 that “the susceptibility and group index plotted versus probe detuning have units of $2N\left|\wp_{ac}\right|^{2}/(\varepsilon_{0}\hbar)$”. As shown in Eq.(1), this quantity has the dimension of an angular frequency and, for consistency, it should also be expressed in units of $\gamma$. It then reads $u_{\chi}=2N\left|\wp_{ac}\right|^{2}/(\varepsilon_{0}\hbar\gamma)$ and, taking into account the above relations,

[TABLE]

For wavelengths $\lambda_{p}$ in the visible domain and typical values of the atomic number density $N$, the susceptibility unit $u_{\chi}$ given by Eq. (4) is in the order of $3\times 10^{-3}$. On the other hand, for $\lambda_{p}=30\,\mathrm{cm}$ with $N=5\times 10^{12}\,\mathrm{cm^{-3}}$ as considered in Art2 , this unit rises to $u_{\chi}\approx 4\times 10^{14}$. Figure 2 in Art2 shows that the peak value of the relative susceptibility $\chi/u_{\chi}$ can exceed $5\times 10^{-3}$. The corresponding absolute susceptibility $\chi$ is then in the order of $10^{12}$. Such values are meaningless.

Although this point is less important, we note that, in SI units, the refractive index reads $n=\sqrt{1+\chi}$ and not $n=\sqrt{1+4\pi\chi}$ as used in art1 to determine the group index. Anyway the approximation $n\approx 1+2\pi\chi$ also made to obtain Eq. (6) in art1 fails when $\left|\chi\right|\gg 1$.

Without examining in detail the parts of art1 ; Art2 devoted to rotary drag and surface polaritons, we remark that these phenomena occur when the sample thickness $L$ is large compared to the probe wavelength $\lambda_{p}$ . According to Art2 , $L=10\,\mathrm{cm}$ and this condition is far from being fulfilled since this thickness is only one third of the probe wavelength. By the way, we also note the incompatibility of the figures 3(b) and 4 in Art2 which show rotary drags, respectively, in the order of $10^{-2}$ and $10^{-7}$ rad.

Khan et al. support their choice of the ratio $\nu_{p}/\gamma=1000$ by referring to a paper on the phase control of light velocity Art4 . The same ratio was actually considered in this paper but without specifying the absolute value of the frequencies. We, however, point out that, for a probe frequency in the visible domain, this ratio leads to lifetimes of the excited atomic levels which are fully unrealistic (in the subpicosecond domain).

Independently of the above criticisms, we remark that, quite generally, large negative group delays are not a sufficient condition to observe visible effects of superluminal propagation. A convincing demonstration of such effects would have required a comparison of the transmitted and incident pulses, which is not made in art1 ; Art2 .

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 (Project No. ANR-11-LABX-0007).