As is well known, the propagation of a quasi-monochromatic wave packet
$$E(t,z) = A(t,z)e^{{ – i\left( {\omega_{0} {\kern 1pt} t – k_{0} {\kern 1pt} z} \right)}} ,$$
(1)
(where \(\left| {{{dA} \mathord{\left/ {\vphantom {{dA} {dt}}} \right. \kern-\nulldelimiterspace} {dt}}} \right| < < \omega_{0} \left| A \right|\), \(\left| {{{dA} \mathord{\left/ {\vphantom {{dA} {dz}}} \right. \kern-\nulldelimiterspace} {dz}}} \right| < < k_{0} \left| A \right|\)) through a transparent medium of the length L, \(0 \le z \le L\), can be characterized by the group velocity, \(v_{g} \equiv \left. {\left( {{{d\omega } \mathord{\left/ {\vphantom {{d\omega } {dk^{\prime}}}} \right. \kern-\nulldelimiterspace} {dk^{\prime}}}} \right)} \right|_{{\omega = \omega_{0} }}\), subject to \(v_{g}\) does not depend on \(\omega\). Here \(\omega\) and \(k(\omega ) = k^{\prime}(\omega ) + ik^{\prime\prime}(\omega )\), where \(k^{\prime\prime} \ll k^{\prime}\), are the frequency and complex wavenumber of the wave packet Fourier constituent,
$$\overline{E}(\omega ,z) = \frac{1}{2\pi }\int\limits_{ – \infty }^{\infty } {E(t,z)e^{i\omega t} dt} ,$$
(2)
which can be written in the form \(\overline{E}(\omega ,z) = \overline{E}(\omega )\exp \left[ {ik(\omega )z} \right]\). In the case of a transparent medium, the group velocity of the wave packet is determined by the frequency dispersion of the medium, \(\omega \sqrt {\varepsilon^{\prime}(\omega )} \simeq ck^{\prime}(\omega )\) (where \(c\) is the speed of light in vacuum and \(\varepsilon = \varepsilon^{\prime} + i\varepsilon^{\prime\prime}\) (\(\varepsilon^{\prime\prime} \ll \varepsilon^{\prime}\)) is the medium dielectric permittivity at frequency \(\omega\)),
$$v_{g} = \frac{c}{{\sqrt {\varepsilon^{\prime}(\omega_{0} )} + \frac{{\omega_{0} }}{{2\sqrt {\varepsilon^{\prime}(\omega_{0} )} }} \cdot \frac{{d\varepsilon^{\prime}(\omega_{0} )}}{d\omega }}}.$$
(3)
As follows from Eq. (3), achieving a low group velocity implies a presence of a normal (\({{d\varepsilon ^{\prime}} \mathord{\left/ {\vphantom {{d\varepsilon ^{\prime}} {d\omega }}} \right. \kern-\nulldelimiterspace} {d\omega }}\) > 0) and steep (large derivative \({{d\varepsilon ^{\prime}} \mathord{\left/ {\vphantom {{d\varepsilon ^{\prime}} {d\omega }}} \right. \kern-\nulldelimiterspace} {d\omega }}\)) dispersion in a transparent medium. If a narrow dip is created within the absorption spectrum of a resonant quantum transition (spectral transparency window), then it is always accompanied by normal and steep dispersion, described by the Kramers–Kronig relation. For this reason, various techniques for achieving slow group velocity differ from each other primarily by the physical mechanism that provides the transparency window. For example, driving a resonant transition in a three-level medium by a sufficiently strong field can lead to transparency for the field resonant to an adjacent atomic transition due to interference of the induced atomic transitions (EIT) or, in the case of a significantly stronger field, due to the disappearance of the atom–field interaction as a result of Autler–Townes splitting. Both mechanisms cause light slowing2,17,18.
Similarly, piston-like vibration of a resonant two-level absorber along the direction of photon propagation can result in AIT37,46, which should also lead to a decrease in the group velocity of photons. In order to show this, let us calculate the susceptibility of the vibrating medium in the laboratory reference frame. For this purpose, following the references37,46, we find the resonant nuclear polarization \(P_{21}\) of quantum transition \(\left| 1 \right\rangle \to \left| 2 \right\rangle\) induced under the action of a monochromatic Fourier constituent Eq. (2) of the field Eq. (1) at the entrance of the absorber, \(z = 0\). The two-level absorber with quantum transition frequency ω21 sinusoidally vibrates as a whole (piston-like) along z-axis (Fig. 1),
$$z = z^{\prime} + R\sin (\Omega t + \vartheta ),$$
(4)
where z′ is the coordinate in the vibrating reference frame, Ω, R, and ϑ are the circular frequency, amplitude, and initial phase of vibration, respectively. Equation (4) describes the case when the thickness of the absorber, L, is much less than the wavelength of sound, \(L \ll 2\pi V_{sound} /\Omega\) (where Vsound is the speed of sound in the absorber) and its motion is non-relativistic,\(R\Omega \ll c\).
The induced resonant polarization in the nuclear absorber can be represented within the model of electric-dipole field-matter interaction (see26,30,31,37,46 and references therein): \(P_{21} = f_{a} Nd_{12} \rho_{21}\), where ρ21 is the induced coherence of the resonant nuclear transition, fa is the Lamb-Mössbauer factor accounting for the probability of recoilless absorption, N is the concentration of resonant nuclei, and \(d_{12} = d_{21}^{*}\) is the effective dipole moment of the resonant nuclear transition \(\left| 1 \right\rangle \leftrightarrow \left| 2 \right\rangle\). With an effective dipole moment, this model correctly describes the magnetic-dipole interaction of 14.4-keV photons with 57Fe nuclei25,26,30. Motion of the absorber leads to shifting its quantum transition frequency, ω21, relative to the spectral line of the motionless source due to the Doppler effect. The respective transition frequency of the moving absorber is \(\tilde{\omega }_{21} = \omega_{21} + k_{0} {\kern 1pt} v\), where v = dz/dt is the velocity of the absorber in the laboratory reference frame. As a result, the master equation for coherence ρ21 induced by the monochromatic Fourier constituent of the incident field Eq. (1), has the form
$$\frac{{\partial \rho_{21} }}{\partial t} + i\left[ {\omega_{21} + k_{0} R\Omega \cos (\Omega t + \vartheta )} \right]\rho_{21} + \gamma_{21} \rho_{21} = \frac{i}{\hbar }n_{12} d_{21} \overline{E}(\omega ,z)e^{ – i\omega t} ,$$
(5)
where n12 = ρ11 − ρ22 is the population difference between the states \(\left| 1 \right\rangle\) and \(\left| 2 \right\rangle\) and γ21 is the half-width at half-maximum (HWHM) of spectral line of the resonant transition, and the relation \(\Omega \ll \omega_{21} ,\omega\) is assumed. Since the considered x-ray field is too weak to change the populations of the states \(\left| 1 \right\rangle\) and \(\left| 2 \right\rangle\), we assume below that \(n_{12} = 1\). Solution of Eq. (5) can be searched in the form
$$\rho_{21} = \sigma_{21} e^{{ – ik_{0} R\sin (\Omega t + \vartheta )}} e^{ – i\omega t} .$$
(6)
Then Eq. (5) for the amplitude of coherence, σ21, reads as
$$\frac{{\partial \sigma_{21} }}{\partial t} + i\left( {\omega_{21} – \omega } \right)\sigma_{21} + \gamma_{21} \sigma_{21} = \frac{i}{\hbar }n_{12} d_{21} \overline{E}\left( {\omega ,z} \right)e^{ip\sin (\Omega t + \vartheta )} ,$$
(7)
where \(p = k_{0} R\) is the modulation index of the absorber’s quantum transition frequency due to its vibration. Using the Jacobi-Anger expansion, \(e^{ \pm ip\sin \phi } = \sum\limits_{n = – \infty }^{\infty } {J_{n} (p)e^{ \pm in\phi } }\) (where Jn(p) is the n-th Bessel function of the first kind) one can find from Eq. (7) the amplitude of the induced resonant coherence σ21 as
$$\sigma_{21} = \frac{i}{{\hbar \gamma_{21} }}n_{12} d_{21} \overline{E}\left( {\omega ,z} \right)\sum\limits_{n = – \infty }^{\infty } {\eta_{n} J_{n} (p)e^{in(\Omega t + \vartheta )} } ,$$
(8)
where
$$\eta_{n} = \frac{1}{{1 + i{{\left( {\omega_{21} – \omega + n\Omega } \right)} \mathord{\left/ {\vphantom {{\left( {\omega_{21} – \omega + n\Omega } \right)} {\gamma_{21} }}} \right. \kern-\nulldelimiterspace} {\gamma_{21} }}}}.$$
(9)
So, according to Eqs. (6) and (8), the monochromatic incident field \(\overline{E}(\omega ,z)e^{ – i\omega t}\) induces nuclear coherence not only at the frequency of the field, \(\omega\), but also at the series of frequencies shifted by a multiple of the vibration frequency, \(\omega + q\Omega\), \(q \in {\mathbb{Z}},\;q \ne 0\):
$$\rho_{21} = \rho_{21}^{\left( 0 \right)} + \sum\limits_{\begin{subarray}{l} q = – \infty \\ \,\,q \ne 0 \end{subarray} }^{\infty } {\rho_{21}^{\left( q \right)} } ,$$
(10)
where
$$\rho_{21}^{\left( 0 \right)} = in_{12} e^{ – i\omega \;t} \frac{{d_{21} \overline{E}\left( {\omega ,z} \right)}}{{\hbar \gamma_{21} }}\sum\limits_{n = – \infty }^{\infty } {\eta_{n} J_{n}^{2} (p)}$$
(11)
is the nuclear coherence induced at the frequency of the monochromatic field, \(\omega\), and
$$\rho_{21}^{\left( q \right)} = in_{12} e^{{ – i\left( {\omega + q\Omega } \right)t – iq\vartheta }} \frac{{d_{21} \overline{E}\left( {\omega ,z} \right)}}{{\hbar \gamma_{21} }}\sum\limits_{n = – \infty }^{\infty } {\eta_{n} J_{n} (p)J_{n + q} (p)} ,\quad {\text{for}}\;q \ne 0,$$
(12)
is the nuclear coherence induced at the combination frequency, \(\omega_{q} = \omega + q\Omega\) due to the absorber vibration.
According to Eqs. (11), (12), the polarization with a comb-like spectrum transforms the incident monochromatic field into a multi-frequency field inside the absorber. The amplitudes of the central spectral component and sidebands of the nuclear response are determined by the modulation index \(p\). As can be seen from Eqs. (11), (12) and was shown in31,40,41,42,47, at some values of the modulation index (for example, at \(p = 1.84\)), the amplitudes of the central component and several sidebands of the nuclear response are comparable. This makes it possible to transform the incident monochromatic or quasi-monochromatic radiation Eq. (1) into a sequence of ultrashort pulses, which was theoretically studied in40,41,47 and experimentally implemented in31,42.
The case of AIT with preserving spectral-temporal characteristics of γ-ray photons was realized in37 at other values of the modulation index (see also46), namely at
$$p = p_{i} ,\;{\text{where}}\;p_{1} \approx 2.4,\,\,\,p_{2} \approx 5.5,\,\,\,\,p_{3} \approx 8.6, \ldots ,$$
(13)
corresponding to the amplitude of the absorber vibration
$$R = R_{i} ,\,\,{\text{where}}\,\,\,\,R_{1} \approx 0.38\lambda ,\,\,\,R_{2} \approx 0.88\lambda ,\,\,\,\,R_{3} \approx 1.37\lambda , \ldots$$
(14)
For these values, \(J_{0} (p_{i} ) = 0\) in Eqs. (11) and (12). If the absorber vibration frequency is large enough, \(\Omega \gg \gamma_{21}\), then in the case of the near-resonant monochromatic field, \(\left| {\omega_{21} – \omega } \right| \ll \Omega\), all other terms in sums of Eqs. (11) and (12) are negligible since \(\eta_{n} \approx \left( {\frac{{\gamma_{21} }}{n\Omega }} \right)^{2} – i\frac{{\gamma_{21} }}{n\Omega }\), \(n \ne 0\) (see also supplemental material in37). In other words, the nuclear response both at the frequency of the incident monochromatic field, \(\omega\), (Eq. (11)) and at the combination frequencies, \(\omega_{q}\), (Eq. (12)) is vanishing, i.e. the medium becomes transparent, preserving the spectral-temporal characteristics of the field.
Now consider the dependence of the nuclear response at the frequency of the incident monochromatic field, \(\omega\), Eq. (11), for an arbitrary absorber vibration frequency, \(\Omega\), and for a certain vibration amplitude, \(R = R_{1}\), corresponding to the modulation index \(p = p_{1}\) (the nuclear response at the combination frequencies, Eq. (12), will be considered afterwards). In this case, the nuclear susceptibility, \(\chi_{21} (\omega )\), follows from Eq. (11) according to the relation \(P_{21} = f_{a} Nd_{12} \rho_{21}^{(0)} = \chi_{21} (\omega )\overline{E}(\omega ,z)e^{ – i\omega t}\) and reads as
$$\chi_{21} \left( \omega \right) = \chi_{0} \sum\limits_{n = – \infty }^{\infty } {J_{n}^{2} \left( {p_{1} } \right)\frac{{{{\left( {\omega_{21} – \omega + n\Omega } \right)} \mathord{\left/ {\vphantom {{\left( {\omega_{21} – \omega + n\Omega } \right)} {\gamma_{21} }}} \right. \kern-\nulldelimiterspace} {\gamma_{21} }} + i}}{{{{\left( {\omega_{21} – \omega + n\Omega } \right)^{2} } \mathord{\left/ {\vphantom {{\left( {\omega_{21} – \omega + n\Omega } \right)^{2} } {\gamma_{21}^{2} }}} \right. \kern-\nulldelimiterspace} {\gamma_{21}^{2} }} + 1}}} ,\quad \chi_{0} = \frac{{f_{a} Nn_{12} \left| {d_{21} } \right|^{2} }}{{\hbar \gamma_{21} }}.$$
(15)
According to Eq. (15), the hilly-like response spectrum of the vibrating absorber at the frequency of the monochromatic field, \(\omega\), (Fig. 2) is the result of a weighted sum of Lorentzian contours separated by the vibration frequency, \(\Omega\). The imaginary part of the nuclear susceptibility, Eq. (15), has a dip centered at the nuclear transition frequency \(\omega_{21}\) between two absorption peaks shifted at the absorber vibration frequency, \(\pm \Omega\), forming the AIT spectral window (Fig. 2, red line). The depth and width of the AIT spectral window is determined by the ratio between the absorber vibration frequency, \(\Omega\), and the absorber transition halfwidth, \(\gamma_{21}\), similar to the transparency window induced due to Autler-Townes splitting by a strong driving field with frequency \(\Omega\)48. At large vibration frequency, \(\Omega \gg \gamma_{21}\), the width of the AIT spectral window tends to \(2\Omega\) and is much larger than the nuclear transition linewidth. In this case, the AIT is nearly perfect, \({\text{Im}} \left\{ {\chi_{21} \left( \omega \right)} \right\} \approx 2\chi_{0} \sum\limits_{n = 1}^{\infty } {J_{n}^{2} \left( {p_{1} } \right){{\gamma_{21}^{2} } \mathord{\left/ {\vphantom {{\gamma_{21}^{2} } {(n\Omega )^{2} }}} \right. \kern-\nulldelimiterspace} {(n\Omega )^{2} }}} \ll \chi_{0}\).
If the absorber vibration frequency is on the order of the nuclear transition linewidth, \(\Omega \sim \gamma_{21}\), the AIT window becomes narrower than \(2\Omega\) (Fig. 2, red line), whereas the resonant nuclear absorption (the imaginary part of the absorber susceptibility at \(\omega = \omega_{21}\) in Eq. (15)) increases and strongly depends on the absorber vibration frequency. For this reason, it is convenient to introduce an effective resonant optical depth (effective Mössbauer thickness) of the vibrating absorber,
$$T_{a}^{{\left( {eff} \right)}} \equiv T_{a}^{{\left( {eff} \right)}} (\omega_{21} ) = {{4\pi \omega_{21} {\text{Im}} \left[ {\chi_{21} \left( {\omega_{21} } \right)} \right]L} \mathord{\left/ {\vphantom {{4\pi \omega_{21} {\text{Im}} \left[ {\chi_{21} \left( {\omega_{21} } \right)} \right]L} c}} \right. \kern-\nulldelimiterspace} c},$$
(16)
generalizing the usual resonant optical depth (Mössbauer thickness) of the motionless absorber,
$$T_{a} = \frac{{4\pi \omega_{21} L}}{c}\chi_{0} .$$
(17)
According to Eqs. (15), (16), the effective resonant optical depth can be expressed as
$$T_{a}^{{\left( {eff} \right)}} = T_{a} \sum\limits_{n = – \infty }^{\infty } {\frac{{J_{n}^{2} \left( {p_{1} } \right)}}{{\left( {{{n\Omega } \mathord{\left/ {\vphantom {{n\Omega } {\gamma_{21} }}} \right. \kern-\nulldelimiterspace} {\gamma_{21} }}} \right)^{2} + 1}}} = \left\{ \begin{gathered} T_{a} ,\quad \Omega = 0; \hfill \\ \frac{{T_{a} \gamma_{21}^{2} }}{{\Omega^{2} }}\sum\limits_{\begin{subarray}{l} n = – \infty \\ \,\,n \ne 0 \end{subarray} }^{\infty } {\frac{{J_{n}^{2} \left( {p_{1} } \right)}}{{n^{2} }}} \ll T_{a} , \, \Omega \gg \gamma_{21} . \hfill \\ \end{gathered} \right.$$
(18)
As shown in Fig. 2 by blue line, the dispersion of the nuclear resonant transition (the real part of susceptibility in Eq. (15)) is normal and almost linearly depends on the frequency with a steep slope over most of the AIT spectral window. In the case of a sufficiently small absorption of the field inside the AIT window, \(T_{a}^{{\left( {eff} \right)}} \ll 1\), the slope of the dispersion curve determines the group velocity of a narrowband wave packet, the spectral width of which is much less than the width of the AIT window. The corresponding condition is \(\Delta \ll \Omega\), where \(\Delta\) is the HWHM of the field spectrum. In the case of a large absorber vibration frequency, \(\Omega \gg \gamma_{21}\), the narrowband wave packet can be essentially broader than the absorber linewidth, \(\gamma_{21} \ll \Delta \ll \Omega\). In the case \(\Omega \sim \gamma_{21}\), the narrowband wave packet should meet the condition \(\Delta \ll \gamma_{21}\).
The group velocity of the narrowband wave packet resonant to the nuclear transition, \(\omega_{0} = \omega_{21}\), follows from Eqs. (3) and (15) with accounting for \(\varepsilon \left( \omega \right) = 1 + 4\pi \chi_{21} \left( \omega \right)\),
$$v_{g} = \frac{c}{{1 + c\frac{{f_{a} Nn_{12} }}{{2\gamma_{21} }}\sigma_{0} \sum\limits_{n = – \infty }^{\infty } {J_{n}^{2} \left( {p_{1} } \right)\frac{{\left( {{{n\Omega } \mathord{\left/ {\vphantom {{n\Omega } {\gamma_{21} }}} \right. \kern-\nulldelimiterspace} {\gamma_{21} }}} \right)^{2} – 1}}{{\left\{ {\left( {{{n\Omega } \mathord{\left/ {\vphantom {{n\Omega } {\gamma_{21} }}} \right. \kern-\nulldelimiterspace} {\gamma_{21} }}} \right)^{2} + 1} \right\}^{2} }}} }},$$
(19)
where \({{\sigma_{0} = 4\pi \omega_{21} d_{21}^{2} } \mathord{\left/ {\vphantom {{\sigma_{0} = 4\pi \omega_{21} d_{21}^{2} } {(c\hbar \gamma_{21} )}}} \right. \kern-\nulldelimiterspace} {(c\hbar \gamma_{21} )}} = 2.56 \times 10^{ – 18} {\text{cm}}^{{2}}\) is the cross-section of the resonant 14.4-keV transition of 57Fe nucleus, assumed to be naturally broadened, \({{\gamma_{21} } \mathord{\left/ {\vphantom {{\gamma_{21} } {(2\pi )}}} \right. \kern-\nulldelimiterspace} {(2\pi )}} = 0.56{\text{ MHz}}\). Below we consider the absorber in the form of a stainless-steel foil at room temperature. In this case, the typical value of the Lamb-Mössbauer factor is \(f_{a} = 0.75\).
As follows from Eq. (19), the group velocity of the narrowband wave packet does not depend on either the optical depth or physical thickness (see also Fig. 4d below), but only on the nuclear parameters and the concentration of nuclei. The minimum group velocity is achieved at maximum concentration of 57Fe nuclei, \(N\). Therefore, we consider hereinafter a stainless-steel foil Fe70Cr19Ni11, with 95% of 57Fe in iron fraction30,37,49,50 corresponding to \(N \approx 5.5 \times 10^{22} {\text{cm}}^{{ – 3}}\). In this case, the term before the sum in Eq. (19) equals 4.5 × 108. According to Eq. (18), the transparency condition \(T_{a}^{{\left( {eff} \right)}} \ll 1\) limits the absorber vibration frequency in an optically deep absorber, \(T_{a} > 1\), to approximately \(\Omega \ge 3\gamma_{21}\). Thus, for \(\Omega = 3\gamma_{21}\), the group velocity of a narrowband wave packet, \(\Delta \ll \Omega ,\gamma_{21}\) is \(v_{g} \approx 12\,{{\text{.4 m}} \mathord{\left/ {\vphantom {{\text{.4 m}} {\text{s}}}} \right. \kern-\nulldelimiterspace} {\text{s}}}\). Increase in the frequency of the absorber vibration leads to increase in the group velocity. However, the group velocity remains significantly less than the speed of light, approaching the latter only at \({\Omega \mathord{\left/ {\vphantom {\Omega {\gamma_{21} }}} \right. \kern-\nulldelimiterspace} {\gamma_{21} }} > 10^{4}\).
The group velocity of propagation of the narrowband wave packet through the absorber of thickness \(L\) can also be characterized by the group delay, \(\tau_{g}\), at the exit from the absorber relative to the transmission time, \({L \mathord{\left/ {\vphantom {L c}} \right. \kern-\nulldelimiterspace} c}\), in free space,
$$v_{g} = \frac{L}{{\tau_{g} + {L \mathord{\left/ {\vphantom {L c}} \right. \kern-\nulldelimiterspace} c}}}\mathop \simeq \limits^{{\tau_{g} \gg {L \mathord{\left/ {\vphantom {L c}} \right. \kern-\nulldelimiterspace} c}}} {L \mathord{\left/ {\vphantom {L {\tau_{g} }}} \right. \kern-\nulldelimiterspace} {\tau_{g} }}.$$
(20)
Following Eqs. (18)–(20), the group delay can be estimated as
$$\tau_{g} = \frac{{T_{a} }}{{2\gamma_{21} }}\sum\limits_{n = – \infty }^{\infty } {J_{n}^{2} \left( {p_{1} } \right)\frac{{\left( {{{n\Omega } \mathord{\left/ {\vphantom {{n\Omega } {\gamma_{21} }}} \right. \kern-\nulldelimiterspace} {\gamma_{21} }}} \right)^{2} – 1}}{{\left[ {\left( {{{n\Omega } \mathord{\left/ {\vphantom {{n\Omega } {\gamma_{21} }}} \right. \kern-\nulldelimiterspace} {\gamma_{21} }}} \right)^{2} + 1} \right]^{2} }}} < \frac{{T_{a}^{{\left( {eff} \right)}} }}{{2\gamma_{21} }} \, \to ^{{\Omega \gg \gamma_{21} }} \, \frac{{T_{a}^{{\left( {eff} \right)}} }}{{2\gamma_{21} }}.$$
(21)
As follows from Eq. (21), due to the AIT condition \(T_{a}^{{\left( {eff} \right)}} \ll 1\), the group delay of the narrowband wave packet cannot exceed the decay time of the resonant nuclear transition, namely, \(\tau_{g} \ll \left( {2\gamma_{21} } \right)^{ – 1}\).
Now let us consider the nuclear response to the incident monochromatic field at the combination frequencies, \(\omega_{q} = \omega + q\Omega\), \(q \in {\mathbb{Z}},\;q \ne 0\), Eq. (12). The nuclear susceptibility \(\chi_{21}^{(q)} \left( \omega \right)\) at the combination frequency \(\omega_{q}\) follows from Eq. (12) according to the relation \(P_{21}^{(q)} = f_{a} Nd_{12} \rho_{21}^{(q)} = \chi_{21}^{q} (\omega )\overline{E}(\omega ,z)e^{{ – i\omega_{q} t}}\) and reads as
$$\chi_{21}^{\left( q \right)} \left( \omega \right) = \chi_{0} e^{ – iq\vartheta } \sum\limits_{n = – \infty }^{\infty } {J_{n} (p_{1} )J_{n + q} (p_{1} )\frac{{{{\left( {\omega_{21} – \omega + n\Omega } \right)} \mathord{\left/ {\vphantom {{\left( {\omega_{21} – \omega + n\Omega } \right)} {\gamma_{21} }}} \right. \kern-\nulldelimiterspace} {\gamma_{21} }} + i}}{{{{\left( {\omega_{21} – \omega + n\Omega } \right)^{2} } \mathord{\left/ {\vphantom {{\left( {\omega_{21} – \omega + n\Omega } \right)^{2} } {\gamma_{21}^{2} }}} \right. \kern-\nulldelimiterspace} {\gamma_{21}^{2} }} + 1}}} .$$
(22)
Comparison of Eq. (22) with Eq. (15) shows that the nuclear response appearing at the combination frequencies is comparable to the nuclear response at the frequency of the incident field and hence should be taken into account in the case of relatively small vibration frequency \(\Omega \sim \gamma_{21}\). The field appearing at the combination frequencies can distort the shape of the transmitted narrowband wave packet in such a way that the resulting propagation velocity and propagation delay may differ from the group velocity and group delay.
It should be noted that existing sources of 14.4-keV radiation considered below have the linewidths comparable to or larger than the linewidth of the 57Fe absorber (as shown in Fig. 2 by black dashed line). This means that in the most interesting case of a relatively low absorber vibration frequency promising the slowest group velocity, the group velocity and group delay model itself can be invalid for such relatively broadband wave packets. The real propagation velocity and propagation delay of the broadband wave packet in the vibrating absorber can differ from the group velocity and group delay of the narrowband wave packet. Nevertheless, as shown above, the steep resonant dispersion within the AIT window remains the basic physical mechanism of slowing down and delaying the incident wave packet once the major part of its spectrum is inside the AIT window.