1. Introduction

The electric field experienced by an atom, in a sufficiently dense medium, is not simply the macroscopic field appearing in Maxwell’s wave-equation. It is rather the local electric field, which includes, inter alia, the contribution from the surrounding atoms volume polarization. [1 – 4] These contributions are refered to in the litterature as Local Field Corrections (LFC). At the relatively high atomic densities required to have appreciable LFC, and in the absence of quantum interference effects which may reduce absorption, [5, 6] the absorption length for an electromagnetic field resonant with the dipole-allowed transition is of the order of 1/10 of the transition wavelength. In this optically dense medium, the usual Slowly Varying Envelope Approximation in Space (SVEAS) is not valid as the field magnitude varies appreciably over a wavelength. As a result, Maxwell’s wave-equation can not be reduced to a first order differential equation in space and time (we are neglecting the transverse variation of the dynamical variables in this paper). Physically, a short (relative to the wavelength) absorption length implies that both the forward and backward waves are important in the propagation analysis. The integro-differential form of the Maxwell-Bloch equations proved an adequate tool for developing a numerical solution to the propagation problem in those cases where both the forward and backward waves are of comparable magnitudes. [7 – 10]

We have previously shown that the effects of the backward-wave can substantially alter the results for the reflection coefficient at a dielectric-dense atom interface. In particular, we showed that a delayed reflectivity can result in the case that the transverse relaxation time is equal or shorter than the inhomogeneous dephasing time. [10] In this paper, we propose to investigate the Dynamical Lorentz Shift, [11 – 14] in an optically dense superradiant amplifier [15]. In particular, we study the spectral distributions of the reflected and transmitted signals.

In a superradiant amplifier, depending on the longitudinal length of the amplifier’s active medium, there are extended ranges for the medium length where the atomic polarization is an eigenfunction of the operator of spatial inversion around the midpoint of the sample with positive or negative eigenvalues. Separating these opposite parity symmetry sectors, there are transition domains with asymmetrical spatial distributions. We show here that as a result of the asymmetrical distribution in the population difference in these transition domains, the spectral distributions for the transmitted and reflected signals possess different Dynamical Lorentz Shifts.

2. Maxwell-Bloch equations

The dynamics of the interaction of the electromagnetic field with the two-level atoms system is described by Maxwell-Bloch equations. We shall neglect the counter-rotating term in the Hamiltonian, i.e. replacing the electromagnetic real field (Ee ^-ω_ct + E ^* e ^+iω_ct by Ee ^-iω_ct where ω_c is the carrier frequency. The Bloch’s equations, including the local field correction and the quantum mechanical collision correction, are given by:

where the fast time phasor was factored out of the x-y components of the Bloch vector, p = p_x + ip_y , n is the population difference between the ground and excited states, ω_o is the atomic bare resonance frequency, ω_L is the Lorentz shift (including both the classical or static term and the quantum mechanical correction), ω_D is the Doppler shift due to the atomic thermal motion, γ ₁ and γ ₂ are the longitudinal and transverse decay rates and d is the atomic transition dipole moment. In the binary collision regime, the collisional width (i.e. the transverse decay rate) is linear in the atomic number density. If we assume that the atomic resonance corresponds to a j = 0 → j = 1 transition, then ${\gamma }_2=1.15\left(\frac{2}{3}\pi \right)\beta$ and ${\omega }_L=\left(\frac{4}{3}-0.22\right)\beta$, where β = πρd ²/2h. [2] The (1.15) factor in γ ₂ is the correction due to a proper inclusion of the time-ordering in the expression for the collision amplitude, [16] and the (4/3) term in ω_L is the static contribution to the Lorentz shift, while the (0.22) term is that due to the quantum mechanical collision contribution. [2, 3, 17]

If we normalize the above equations to the time-scale of the inhomogeneous width (which, for a specific gas, depends exclusively on the gas temperature) and to an appropriate dimensionless field (corresponding to the associated Rabi frequency in units of the gas inhomogeneous width), Bloch’s equations can be written as:

where the normalized dynamical variables and coordinates are defined as:

As we will be working in a regime where the pulse width and relaxation times are much larger than the inverse resonant frequency, we can assume that the Slowly Varying Envelope Approximation in Time (SVEAT) for the electric field envelope and polarizability is valid. On the other hand, the SVEAS is not valid since the small signal cw absorption length is approximately equal to l/10 the wavelength which implies large changes of the field amplitude over a wavelength. Therefore, we write Maxwell's equation as:

where z̅ = k_cz, B = (πρd ² T ₂/ h($T_2^{*}$/ T ₂) = 1.66 ($T_2^{*}$/ T ₂), ρ is the atomic number density, ($T_2^{*}$)^-1 is the inhomogeneous width, and 〈χ〉_Δ represents the normalized polarizability averaged over the Doppler distribution given by:

Eq.(3) can also be written, for the case that L / c is much smaller than the pulse-width or the atomic relaxation times, as an integral equation,

where L̅ is the normalized sample length.

The boundary conditions for the problem are:

where ${\mathrm{\varphi }}_{\mathit{\text{in}}}^0$ = ∑ $T_2^{*}$ / (√πτ), and ∑ is the incoming pulse area, assumed to be of the same order as that of the vacuum field fluctuation (≈ 10^-8)

for all z̅ > 0, and where U = 0^- refers to the time prior to the incident signal being switched-on. For a Gaussian pulse centered at the origin of time, we take this time to correspond to at least five times the pulse width.

If we denote the ratio of $\frac{\partial \varphi }{\partial \bar{z}}\left(0,U\right)$ to ϕ(0,U) by iη, then the reflection coefficient at the dielectric-resonant gas interface is:

where ϕ_refl(0, U) = r(U) ϕ_in (0, U) and n_r is the index of refraction of the container dielectric. We shall illustrate here the results for the case that the dielectric index of refraction is one. In this case, the reflected field is given by:

3. Results

In fig.1, we plot the spatio-temporal dependence of the field-atoms variables for the superradiant amplified bursts for L = (p ± ¼)λ. We note that the polarizability has a definite parity with respect to the z = L / 2 axis, and so does the field distribution. They possess alternatively negative or positive parity for the consecutive cases of the L = (p ± ¼)λ sequence. The number of the local extrema in these distributions, along the U = U _max line, where (U _max is the coordinate of the intensity of the outgoing burst at its maximum value), increases by one as we move to the next term of the sequence, as a standing-wave picture would imply. The figures also illustrate the result that along the U = U _max line, the positions of the maxima (minima) of the field intensity correspond to those of the minima (maxima) of the population difference distributions.

The above symmetry results can be summarized by the expressions:

Similar transformation properties hold, as well, for the polarization.

So far in fig.1, we have considered only the case that T ₂ < $T_2^{*}$, i.e. the classically defined homogeneously broadened regime. If we compare the distributions of the polarizability when T ₂ is larger than $T_2^{*}$ , i.e. the intermediate and the inhomogeneously broadened regimes, we note, as shown in fig.2, that as long as the system is in the superradiant transition regime, and that the transmitted and reflected bursts are equal, the number of extrema in the respective polarizability distributions do not change, even though their values may have changed by six orders of magnitude.

 

Fig. 1. The spatio-temporal dependence of the field and atomic variables are plotted for a typical value of L = (p + ¼)λ, T ₁ = 30 ns , $T_2^{*}$ = 300 ps , T ₂ = 100 ps , τ = 30 ps.

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Fig. 2. The normalized field amplitude is plotted as a function of z / L , at the value of U corresponding to the location of the maximum of the transmitted field for different values of the transverse relaxation time. T ₁ = 30 ns , $T_2^{*}$ = 300 ps , τ = 30 ps, L = 5.25 λ

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In fig.3, we illustrate the extent of the symmetry sectors by plotting the magnitude of the field for two values of the sample length located near the edges of a particular symmetry sector. We observe that, while the magnitude of the amplitude of the normalized field distributions changes, because the value of αL corresponding to these different lengths vary, the curves spatial symmetry and the number of extrema do not.

 

Fig. 3. The normalized field amplitude is plotted as a function of z / L, at the value of U corresponding to the location of the maximum of the transmitted field for values of the sample length within the same symmetry sector.

T ₁ = 30 ns , $T_2^{*}$ = 300 ps , T ₂ = 100 ps, τ = 30 ps.

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Having determined the parity eigenvalues at the midpoints of two adjoining symmetry sectors, now we examine the dynamical development of the field distribution, for a sample’s length, at the center of an in-between asymmetrical transition domain. We plot in fig.4, the meshes of the system’s dynamical variables for L = 4.5λ. The system’s dynamical variables are not eigenfunctions of the previously defined spatial parity operator.

In fig.5, we illustrate, the evolution of the system’s dynamical variables within the same asymmetrical transition domain of fig.4. We plot the time integrated energy flux of the field $\left(\underset{-U_w}{\overset{U_w}{\int }}{\mid \varphi \left(z,U\right)\mid }^2\mathit{dU}\right)$, normalized to the initially stored energy density in the atomic system, as function of z (where the U integration is over the size of the computational temporal window, which is much smaller than the atomic longitudinal relaxation time). The pattern of variation in the asymmetry structure show the details of the flip from a parity eigenvalue to another, and the mechanism by which the energy flux change from having a local maximum at the sample midpoint into having a minimum at that point. Furthermore, we also illustrate how this mechanism accounts for the unequal transmitted and reflected bursts, when the sample length places it in an asymmetrical domain. The extent of the asymmetrical domains and the symmetry sectors can be obtained by examining figs. 3 & 5. The asymmetrical transition domains are much narrower than λ / 2, their individual width are approximately equal to 0.2 λ, while the symmetric sectors individual width are approximately equal to 0.3λ.

 

Fig. 4. The spatio-temporal dependence of the field and atomic variables are plotted for sample length within an asymmetrical transition region.

T ₁ = 30 ns , $T_2^{*}$ = 300 ps , T ₂ = 100 ps, τ = 30 ps.

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Fig. 5. The time-integrated field energy flux inside the resonant medium, normalized to the initially stored energy density in the atomic system, is plotted as a function of z / L , for different values of the sample length within the asymmetrical transition region centered at L = 4.5 λ.

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Next, we investigate the effects of the LFC. As can be seen by an examination of Eq.(1), the Local Field Corrections lead to a shift of the atomic resonance by an amount proportional to n. Consequently, as we analyze the spectral distribution of the superradiant transmitted signal, we should look for the Dynamical Lorentz Shift resulting from the specific spatio-temporal distribution of the population difference. In fig.6, we plot the difference in the first moment of the spectral distributions of the transmitted and reflected signal as function of the length of the sample for the two cases of including and excluding the LFC in the Maxwell-Bloch system. We note that this difference, in normalized frequency units, is bounded by the quantity ${\Omega }_c={\omega }_c{T}_2^{*}=\frac{{\omega }_c}{{\gamma }_2}\frac{T_2^{*}}{T_2}$. For the parameters of fig.6, the absolute value of this quantity is equal to 1.3825. This value is obtained, as shown in the figure, in the limit of an ultrathin sample. The observed differences in the frequency shifts between the spectral distributions of the transmitted and reflected signals occur in the transition domains where an asymmetry in the population difference distributions leads to different Dynamical Lorentz Shifts.

 

Fig. 6. The difference in the values of the first moment of the transmitted and reflected fields spectral distributions, between the models including and excluding the Local Field Corrections, is plotted as a function of L / λ

T ₁ = 30 ns , $T_2^{*}$ = 300 ps , T ₂ = 100 ps, τ = 30 ps.

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