DICKE'S SUPERRADIANCE IN ASTROPHYSICS. I. THE 21 cm LINE

Dicke originally proposed in 1954 a model where an ensemble of N initially inverted two-level atoms interacting with their common radiation field is considered as a single quantum mechanical system (Dicke 1954). In his model, a two-level atom is modeled as a spin-1/2 particle in a magnetic field where the spin up configuration corresponds to the excited state $| e\rangle$ and the spin down to the ground state $| g\rangle$. Just as an ensemble of N spin-1/2 particles can be described using two quantum numbers s and m_s, the eigenstates of the combined N two-level atoms in Dicke's model can also be labeled with two quantum numbers r and m_r such that $0\leqslant r\leqslant N/2$ and ${m}_{r}=-r,-r+1,\ldots ,r-1,r$, where

$$\begin{eqnarray}&&{m}_{r}=\displaystyle \frac{{N}_{e}-{N}_{g}}{2},\end{eqnarray} \tag{ 1 }$$

with N_e and N_g the number of particles in the excited and ground states, respectively. From the complete set of eigenstates characterizing this quantum mechanical system, those symmetrical under the permutation of any pair of atoms are particularly important and are called Dicke states. The initial state $| e,e,\ldots ,e\rangle$ of N fully inverted spin-1/2 particles corresponding to N fully inverted two-level atoms is one such Dicke state and is identified by r = N/2 and m_r = N/2. When an atom in the ensemble decays to its ground state by emitting a photon, the quantum number m_r is decreased by one while r remains unchanged, and the system moves to another symmetric state. Dicke showed that the radiation intensity from such an ensemble cascading from the initial ($r=N/2,{m}_{r}=N/2$) state down through an arbitrary state (r, m_r) is

$$\begin{eqnarray}&&I={I}_{0}(r+{m}_{r})(r-{m}_{r}+1)\end{eqnarray} \tag{ 2 }$$

if the volume containing the ensemble of N two-level atoms is much smaller than ${\lambda }^{3}$, the cube of the wavelength of the radiation interacting with the atoms. In Equation (2), I₀ is the radiation intensity due to spontaneous emission from a single two-level atom. This particular type of system and density condition defines a small sample. This cascading process is depicted in Figure 1.

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Furthermore, Dicke pointed out that in the (r = N/2, m_r = 0) state, where the half of the atoms are in the ground state and the other half in the excited state, the radiation intensity of the system is maximum at

$$\begin{eqnarray}&&I={I}_{0}\left(\displaystyle \frac{N}{2}\right)\left(\displaystyle \frac{N}{2}+1\right)\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\propto {N}^{2}{I}_{0},\end{eqnarray} \tag{ 4 }$$

implying a significantly enhanced radiation beam, a phenomenon he named superradiance. This can be understood by the fact that when the distance between neighboring atoms is much smaller than the wavelength of radiation, the photon emitted by one atom is seen to be in phase by neighboring atoms and can bring about the emission of a new photon of the same mode and in the same direction as the initial photon. This process can continue through the whole ensemble, resulting in an intense superradiant radiation pulse proportional to N² (see Equation (4)). In contrast to the superradiance observed in a perfectly coherent system, in a noncoherent system all atoms act independently with a radiation intensity scaling linearly with N. This possibility of coherent interactions is in contrast with the common assumption that, in the ISM, atoms, for example, mainly interact independently with the radiation field, such that the intensity of the radiation is a linear function of the atomic density.

In order to conduct a more careful investigation of the possibility of coherent interactions, especially superradiance in a H i gas, we will need to adapt Dicke's original theory to the corresponding astrophysical conditions. We therefore first need to carefully understand all the assumptions that lead to a symmetrical ensemble and superradiance in the original model of Dicke (1954). The main assumptions can be listed as follows:

• 1.  
  A small sample of neutral atoms is confined to a volume ${ \mathcal V }\ll {\lambda }^{3}$ with the walls of the volume transparent to the radiation field.
• 2.  
  The N two-level atoms in the sample are separated by a distance much less than λ but distant enough not to worry about any overlap between the wave functions of neighboring atoms, which would require that the wave functions be symmetrized.
• 3.  
  The ensemble of N initially inverted hydrogen atoms possesses a permutation symmetry under the exchange of any pair of atoms in the sample. This is a restricting condition that could prove difficult to satisfy in general.
• 4.  
  The transition between atomic levels takes place between nondegenerate levels, collisions between atoms do not affect their internal states, and collisional broadening is neglected as a result of the small size of the sample (Dicke 1953).
• 5.  
  Although it is mentioned in Dicke (1954) that the main results of his study are independent of the type of coupling between atoms and the field, the interaction of the atoms with the radiation field in Dicke's model is assumed to be electric dipolar.
• 6.  
  Finally, the radiation field is assumed to be uniform through the small sample, the electric dipoles associated with the atoms are parallel, and propagation effects neglected.

Comparing a corresponding small sample of N neutral hydrogen atoms interacting with the 21 cm line in the ISM with a Dicke sample, we can see that some of the assumptions made in the Dicke formalism hold and some do not. For example, the transitions between the hyperfine states of a hydrogen atom take place between nondegenerate levels since the external magnetic field in the ISM lifts the upper-level degeneracy (see Section 3). Also, a small sample of H i atoms found in many regions in the ISM would readily verify the criterion that $N\gg 1$ in a volume ${ \mathcal V }\lt {\lambda }^{3}$ and could thus be approximately assumed to experience the same 21 cm radiation field without consideration of propagation effects. On the other hand, unlike in Dicke's sample, collisional and Doppler broadening effects should, in the most general case, be considered because, for example, collisions between hydrogen atoms affect the internal hyperfine states in their electronic ground state through spin de-excitation (Field 1958). Most importantly, it must also be noted that the type of coupling between hydrogen atoms and the 21 cm line is magnetic dipolar in nature.

Above all, the permutation symmetry of atoms, which is a key assumption in the Dicke model, is difficult to preserve in an actual situation because of dipole–dipole interactions between the atoms. Dipole–dipole interactions have a ${r}^{\prime -3}$ dependency, and these short-range interactions become important in small samples where the distance between atoms ${r}^{\prime }$ is smaller than λ (see Section 3.1). In the Dicke model, the symmetry-breaking effect of dipole–dipole interactions is ignored. In later studies of superradiance (e.g., Gross & Haroche 1982), it has been shown that in general dipole–dipole interactions break the permutation symmetry, except in those configurations where all atoms have identical close-neighbor environments. This symmetry-breaking effect results in weakened correlations and a subsequent deviation from a perfectly symmetrical superradiance behavior (i.e., the I ∝ N² relation in Equation (4)). In a sample of N atoms, if s atoms ($s\lt N$) experience a similar close neighborhood, the correlation can build up among this group of atoms, and the intensity of radiation from the whole sample is expected to be larger than the intensity of a fully noncoherent system (${I}_{\mathrm{nc}}$) but smaller than that of a perfect superradiance system (I_SR).

In a small sample of N neutral hydrogen atoms in the ISM, it may thus appear possible to develop coherent behaviors if the permutation symmetry is conserved among a group of atoms in the sample. This is arguably a reasonable assumption on average for an ensemble of atoms within the small volumes discussed here. That is, the different atoms in the sample are likely to be subjected to the same conditions when averaged over time and space. Furthermore, we also note that in a H i sample the magnetic dipole–dipole interactions are definitely weaker than the electric dipole–dipole interactions discussed in the literature focusing on symmetry-breaking effects.

In his first paper on superradiance, Dicke also extended his formalism to a large sample, where the volume of the sample ${ \mathcal V }\gt {\lambda }^{3}$ and the interatomic distance ${r}^{\prime }$ between some atoms can be greater than λ. He showed that, in a large sample, coherent radiation can occur in a particular direction k in which the radiation from different atoms is in phase. When the phase-matching condition is satisfied in some direction k, the initial state of the system can be described by a correlated symmetric state of type $(r,{m}_{r})$, and the intensity of the radiation in a solid angle along k follows

$$\begin{eqnarray}&&I({\boldsymbol{k}})={I}_{0}({\boldsymbol{k}})[(r+{m}_{r})(r-{m}_{r}+1)],\end{eqnarray} \tag{ 5 }$$

similar to Equation (2) for a small sample. When a photon is emitted in the direction k, the system cascades to a lower state, obeying the selection rules ${\rm{\Delta }}r=0,{\rm{\Delta }}{m}_{r}=-1$, and similar to the case of a small sample, symmetrical states of the same r are coupled to each other through coherent transitions (see Section 3.1.2). On the other hand, when a radiated photon has a wave vector ${\boldsymbol{k}}^{\prime} \ne {\boldsymbol{k}}$, the states with different r (i.e., of different symmetry) can couple, and consequently the coherence is weakened in the system (Dicke 1954). It follows that in a large sample consisting of N inverted atoms, the radiation by one atom is only seen to be in phase by a group of atoms (contrary to a small sample, where the radiation field is assumed uniform over the whole sample), and correlation can only be developed among this group. This naturally results in a radiation intensity that is greater than that of the corresponding fully noncoherent system but smaller than the superradiance intensity of a perfectly coherent system consisting of N atoms.

Finally, in a large sample as a result of possibly large interatomic distances (i.e., $r^{\prime} \gt \lambda$), the symmetry-breaking effects of the dipole–dipole interactions are less important, whereas the propagation effects that are absent in a small sample cannot be neglected. The propagation of radiation over a large distance in a large sample results in the reabsorption and reemission of the photons and consequently leads to a nonuniform evolution of the atoms in the sample (see Section 3.2). Beyond these factors, Dicke's analysis of the large sample includes assumptions similar to those used for the small sample.

The theoretical model for the problem can be found in Appendix A, where the Hamiltonian for the two-level H i sample is developed and the main equations of superradiance are derived. To simplify our discussion, we have limited our analysis to the $| F=0,m=0\rangle \,\longleftrightarrow \,| F=1,m=+1\rangle$ transition through which a hydrogen atom emits a left circular polarization (LCP) photon, with its electric field vector rotating counterclockwise as seen by the observer facing the incoming wave. One of the main components of the Hamiltonian is the magnetic dipole energy term ${\hat{V}}_{\mathrm{MD}}$ that describes the interaction between the atoms composing the sample (see Equations (50) and (62)). We now focus on this interaction to get a sense of how the needed cooperative behavior for superradiance develops between atoms.

3.1.1. Hydrogen Atoms Separated by a Small Interatomic Distance $(r^{\prime} \lt \lambda )$

In order to have a better understanding of how cooperative behavior is built up in a sample of N atoms, it is helpful to first study the simpler case of two atoms. We specifically consider a system consisting of two hydrogen atoms, once again assuming each atom is a two-level system with the excited state $| e\rangle$ (F = 1) and the ground state $| g\rangle$ (F = 0). The two atoms are initially excited, and the state of the system is given by $| {e}_{1}\rangle \otimes | {e}_{2}\rangle =| {e}_{1}{e}_{2}\rangle$. Eventually one of the two atoms spontaneously decays to its ground state, emitting a photon with a wavelength λ and energy ${\hslash }\omega$. If the interatomic distance ${r}^{\prime }$ is much smaller than λ (i.e., ${{kr}}^{\prime }\ll 1$) and the two atoms are identical, then one cannot say which atom has emitted the photon nor which is in a given state. In the case of the two-level hydrogen atom discussed here, this decay rate must be related to that of the corresponding magnetic dipole transition given by (in MKS units)

$$\begin{eqnarray}&&{\rm{\Gamma }}=\displaystyle \frac{{\mu }_{0}{k}^{3}{| \langle e| \hat{{\boldsymbol{M}}}| g\rangle | }^{2}}{3{\hslash }\pi }.\end{eqnarray} \tag{ 6 }$$

We can furthermore express the state of the system by either a symmetric $| +\rangle$ or antisymmetric $| -\rangle$ combination of the $| {e}_{1}{g}_{2}\rangle$ and $| {g}_{1}{e}_{2}\rangle$ state vectors, such as

$$\begin{eqnarray}&&| +\rangle =\displaystyle \frac{1}{\sqrt{2}}(| {e}_{1}{g}_{2}\rangle +| {g}_{1}{e}_{2}\rangle )\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&| -\rangle =\displaystyle \frac{1}{\sqrt{2}}(| {e}_{1}{g}_{2}\rangle -| {g}_{1}{e}_{2}\rangle ),\end{eqnarray} \tag{ 8 }$$

which, at this stage of our analysis, have the same energy and are thus degenerate (see below).

We now refine this model by adding the magnetic dipole–dipole interaction term to the system's Hamiltonian. In this model, the magnetic dipole from one atom, say, ${\hat{{\boldsymbol{M}}}}_{1}$, interacts with the magnetic field ${\hat{{\boldsymbol{B}}}}_{2}$ due to the dipole of the other atom located at a position ${{\boldsymbol{r}}}^{\prime }={r}^{\prime }{{\boldsymbol{e}}}_{{r}^{\prime }}$ away in the near field, where ${{kr}}^{\prime }\ll 1$ (Jackson 1999):

$$\begin{eqnarray}{\hat{{\boldsymbol{B}}}}_{2}({{\boldsymbol{r}}}^{\prime }) & = & \displaystyle \frac{{\mu }_{0}}{4\pi }\left[\displaystyle \frac{\cos ({{kr}}^{\prime })}{{r}^{\prime 3}}+\displaystyle \frac{\sin ({{kr}}^{\prime })}{{r}^{\prime 2}}\right]\\ & & \times [3{{\boldsymbol{e}}}_{{r}^{\prime }}({{\boldsymbol{e}}}_{{r}^{\prime }}\cdot {\hat{{\boldsymbol{M}}}}_{2})-{\hat{{\boldsymbol{M}}}}_{2}].\end{eqnarray} \tag{ 9 }$$

It can be shown that when the two dipoles are aligned, the term of the interaction Hamiltonian that is relevant to the present discussion is

$$\begin{eqnarray}{\hat{H}}_{\mathrm{dd}} & = & -\displaystyle \frac{{\mu }_{0}{k}^{3}{\mu }_{{\rm{B}}}^{2}}{2\pi }(3{| \beta | }^{2}-1)\left[\displaystyle \frac{\cos ({{kr}}^{\prime })}{{({{kr}}^{\prime })}^{3}}+\displaystyle \frac{\sin ({{kr}}^{\prime })}{{({{kr}}^{\prime })}^{2}}\right]\\ & & \times ({\hat{R}}_{1}^{+}{\hat{R}}_{2}^{-}+{\hat{R}}_{1}^{-}{\hat{R}}_{2}^{+}),\end{eqnarray} \tag{ 10 }$$

with $\beta ={{\boldsymbol{e}}}_{L}\cdot {{\boldsymbol{e}}}_{{r}^{\prime }}$ . The raising/lowering operators ${\hat{R}}_{1}^{+}$, ${\hat{R}}_{1}^{-}$, and so on are defined in Equation (47) and the LCP unit vector state e_L in Equation (54), while μ_B is the Bohr magneton. It can further be shown, through a simple diagonalization exercise, that this interaction Hamiltonian lifts the degeneracy between the $| +\rangle$ and $| -\rangle$ states of Equations (7) and (8), with their corresponding energies becoming (Protsenko 2006)

$$\begin{eqnarray}&&{E}_{\pm }={E}_{0}\pm {\rm{\Delta }}E,\end{eqnarray} \tag{ 11 }$$

with E₀ the unperturbed energy of the states and

$$\begin{eqnarray}&&{\rm{\Delta }}E=\displaystyle \frac{{\mu }_{0}{k}^{3}{\mu }_{{\rm{B}}}^{2}}{2\pi }(3{| \beta | }^{2}-1)\left[\displaystyle \frac{\cos ({{kr}}^{\prime })}{{({{kr}}^{\prime })}^{3}}+\displaystyle \frac{\sin ({{kr}}^{\prime })}{{({{kr}}^{\prime })}^{2}}\right].\end{eqnarray} \tag{ 12 }$$

It is then also possible to verify through Equation (6), setting $\hat{{\boldsymbol{M}}}={\hat{{\boldsymbol{M}}}}_{1}+{\hat{{\boldsymbol{M}}}}_{2}$, that the transition rates Γ_± for the $| \pm \rangle$ states with the initial $| {e}_{1}{e}_{2}\rangle$ and final $| {g}_{1}{g}_{2}\rangle$ states are

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{+}=2{\rm{\Gamma }}\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{-}=0,\end{eqnarray} \tag{ 14 }$$

where Γ is the transition rate of a single atom acting independently.

The differentiation of energy levels brought about by the magnetic dipole–dipole interaction is, therefore, seen to be a function of β and the interatomic distance ${r}^{\prime }$. More important, however, is the doubling of the transition rate for the symmetric state $| +\rangle$ and the cancellation of that of the antisymmetric state $| -\rangle$. Such enhanced and reduced rates are respectively associated with superradiance and subradiance. This scenario for the two-atom system is depicted in Figure 3.

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This behavior can also be understood by considering the symmetry of the system's Hamiltonian and states. The fact that, as could easily be verified, the Hamiltonian of the system of two atoms (including the magnetic dipole term ${\hat{V}}_{\mathrm{MD}}$) is totally symmetric under the permutation of the two aligned atoms when ${{kr}}^{\prime }\ll 1$ implies that only states of like symmetry can be coupled. It follows that since the initial state $| {e}_{1}{e}_{2}\rangle$ of the fully inverted system is also symmetric, it can only couple to the $| +\rangle$ intermediate state, and from there to the symmetric ground state of the system $| {g}_{1}{g}_{2}\rangle$. Accordingly, it is interesting to note that under these conditions a system prepared in the intermediate antisymmetric state $| -\rangle$ will not decay to the ground state since Γ₋ = 0. This is evidently different from the case of a noncoherent system where both atoms eventually decay to their individual ground states $| g\rangle$ at the rate Γ. We therefore see that superradiance and subradiance are characteristics of a coherent system, where the intensity of radiation does not scale linearly with the number of atoms, as is the case for a noncoherent system.

When the effect discussed here is generalized to a sample composed of N atoms confined within a volume ${ \mathcal V }\ll {\lambda }^{3}$ (a small sample), we find that some of the conditions that prevailed for the two-atom case are not realized. Most importantly, Equations (11) and (12) indicate that this interaction leads to a distribution of energy levels in the system unless the atoms all have similar nearest neighborhoods (e.g., a ring-like periodic distribution of atoms; Gross & Haroche 1982). This spread in energy levels will tend to reduce the strength of the superradiance effect.

It has nonetheless been observed through numerical calculations and experiments that coherent behaviors still apply to N-atom small-sample systems where radiation is of long enough wavelength (Gross et al. 1979), as is the case for the  21 cm line. For ${{kr}}^{\prime }\sim 1$ the ratio ${\rm{\Delta }}E/{E}_{0}\sim {\rm{\Gamma }}/\omega$ is exceedingly small for the 21 cm line, and the timescale associated with the energy shifts is on the order of ${\hslash }/{\rm{\Delta }}E\sim {({{kr}}^{\prime })}^{3}\,{{\rm{\Gamma }}}^{-1}$ (Benedict et al. 1996), which for the H i densities considered in this paper renders this type of dephasing negligible. As will be discussed later, dephasing due to collisions are more likely to set the timescale for homogeneous dephasing. The same is not necessarily true at short wavelengths, where it is very difficult to place a large number of atoms within a subwavelength dimension in a regular pattern, and in such a sample strong dipole–dipole interactions break the symmetry and terminate the coherent behavior by introducing large energy-level shifts. Thus most of the experimental observations of superradiance took place at longer wavelengths (i.e., in the infrared as opposed to optical; Benedict et al. 1996).

For an inverted N-atom small sample with initially uncorrelated dipoles, the first photon emitted by one of the atoms interacts with the dipole moments of the other atoms, resulting in the build-up of correlation between them. After some time, known as the delay time t_D, a very high degree of correlation is developed in the system, where, in the strongest superradiance regime, the N microscopic dipoles eventually act like one macroscopic dipole. The rate of emission is then enhanced to NΓ, while the radiation intensity is proportional to N² and becomes highly directional, being focused in a beam with a temporal half-width on the order of $1/(N{\rm{\Gamma }})$.

It should also be noted that the correlation between dipoles can be triggered by an external source, such as an input radiation field. This can happen if the input radiation field is stronger than the spontaneous fluctuations in the sample and the coupling of the dipoles to the external field leads to coherent behaviors. An enhancement of radiation through coupling to an external field is called triggered superradiance (Benedict et al. 1996).

It can be shown that the superradiance radiation intensity I_SR of an ideal H i small sample composed of N inverted atoms is given by (Dicke 1954; Gross & Haroche 1982; Benedict et al. 1996)

$$\begin{eqnarray}&&{I}_{\mathrm{SR}}={N}^{2}{\hslash }\omega {\rm{\Gamma }}{\cosh }^{-2}[N{\rm{\Gamma }}(t-{t}_{{\rm{D}}})],\end{eqnarray} \tag{ 15 }$$

where ${\hslash }\omega$ is the energy of the corresponding atomic transition and the aforementioned delay time ${t}_{{\rm{D}}}={(N{\rm{\Gamma }})}^{-1}\mathrm{ln}(N)$. In Figure 4 the radiation intensity of a H i small sample with N = 75 atoms confined within a cube of length 4 cm ($\simeq \lambda /5$ for the 21 cm line) is plotted as a function of time using Equation (15). The intensity is normalized to ${{NI}}_{\mathrm{nc}}$, where ${I}_{\mathrm{nc}}=N{\hslash }\omega {\rm{\Gamma }}$ is for the corresponding noncoherent small sample. It can be seen in Figure 4 that the energy stored in the small sample is radiated away in a single burst. After time t = t_D, the intensity reaches its maximum value, N times that of the noncoherent intensity, and the peak intensity of the normalized plot becomes equal to one. In this H i sample, Γ⁻¹ = 3.5 × 10¹⁴ s (Draine 2011), the delay time ${t}_{{\rm{D}}}=2.0\times {10}^{13}$ s, and the characteristic time of superradiance is ${T}_{{\rm{R}}}={(N{\rm{\Gamma }})}^{-1}=4.6\times {10}^{12}$ s. It should also be pointed out that in such a sample the correlation between dipoles is initiated by internal spontaneous fluctuations, and it is assumed that we are dealing with an ideal system, where the dipole–dipole symmetry-breaking effects are negligible and there are no other relaxation mechanisms (i.e., cooperative emission is the only decay mechanism).

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In a real system, there are some relaxation and dephasing effects that compete with the build-up of the correlation, and in order to subsequently have superradiance, its characteristic timescale T_R and delay time t_D must be shorter than (in some exceptional cases on the order of) the relaxation/dephasing timescales (Gross & Haroche 1982; Benedict et al. 1996). The nonideal case will be discussed in Section 3.2.1.

3.1.2. Two Hydrogen Atoms Separated by a Larger Interatomic Distance $(r^{\prime} \gt \lambda )$

Let us still assume that the atoms are prepared initially in their excited states, with the state of the two-atom system given by $| {e}_{1}{e}_{2}\rangle$. Similar to the subwavelength case, a first photon is radiated, leaving the system in an intermediate state, which, unlike for the subwavelength case, will be described with any combination of $| {e}_{1}{g}_{2}\rangle$ and $| {g}_{1}{e}_{2}\rangle$ states with each having equal probability contributions, that is, not only by the $| +\rangle$ and $| -\rangle$ states. More precisely, if we associate the general symmetric state

$$\begin{eqnarray}&&| S\rangle =\displaystyle \frac{1}{\sqrt{2}}(| {e}_{1}{g}_{2}\rangle +{e}^{i\phi }| {g}_{1}{e}_{2}\rangle )\end{eqnarray} \tag{ 16 }$$

with the intermediate state shown on the left side of Figure 5, then we should assign its orthogonal antisymmetric state

$$\begin{eqnarray}&&| A\rangle =\displaystyle \frac{1}{\sqrt{2}}(| {e}_{1}{g}_{2}\rangle -{e}^{i\phi }| {g}_{1}{e}_{2}\rangle )\end{eqnarray} \tag{ 17 }$$

to the intermediate state on the right side of the figure (Dicke 1964). In Equations (16) and (17), ϕ is a phase term discriminating between the multiple choices for the intermediate states. To get a better understanding of the transition probabilities for these states, it is useful to refer to Equation (62) for the magnetic dipole interaction term with a radiation field for two atoms separated by r' and for a given k. We then find that for coupling to, say, the $| {e}_{1}{e}_{2}\rangle$ state the following term comes into play:

$$\begin{eqnarray}&&{\hat{V}}_{\mathrm{MD}}\propto {\hat{R}}_{1}^{+}+{\hat{R}}_{2}^{+}{e}^{{{ikr}}^{\prime }\cos ({\theta }^{\prime })},\end{eqnarray} \tag{ 18 }$$

where θ' is the angle between k and r'. Given that the transition probability (and rates) are proportional to ${| \langle {e}_{1}{e}_{2}| {\hat{V}}_{\mathrm{MD}}| S\rangle | }^{2}$ and ${| \langle {e}_{1}{e}_{2}| {\hat{V}}_{\mathrm{MD}}| A\rangle | }^{2}$ (Grynberg et al. 2010), we calculate using Equations (16)–(18)

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{S}\propto {\cos }^{2}\left\{\displaystyle \frac{1}{2}[\phi -{{kr}}^{\prime }\cos ({\theta }^{\prime })]\right\}\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{A}\propto {\sin }^{2}\left\{\displaystyle \frac{1}{2}[\phi -{{kr}}^{\prime }\cos ({\theta }^{\prime })]\right\}.\end{eqnarray} \tag{ 20 }$$

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We therefore see that, although the first photon can be emitted in any direction θ', its direction of emission determines ϕ and the intermediate state of the system since the transition probabilities peak at $\phi -{{kr}}^{\prime }\cos ({\theta }^{\prime })=2m\pi$ for Γ_S and $\phi -{{kr}}^{\prime }\cos ({\theta }^{\prime })=n\pi$ for Γ_A (m and $n\ne 0$ are integers). Going through the same exercise for the $| {g}_{1}{g}_{2}\rangle$ state shows a similar dependency on θ' and ϕ as in Equations (19) and (20), which implies that these transition rates will also be likely to peak at the same value of ϕ. It follows that there is an angular correlation between two successive photons, where the direction of the second photon is correlated with the direction of the first. This angular correlation can take place even when the atoms are placed several wavelengths apart, as a result of their coupling to a common electromagnetic field, and favors intense radiation along elongated geometries (e.g., pencil-like or cylindrical structures; Dicke 1964).

Depending on the intermediate state of the system, two different classes of transitions are possible: coherent and noncoherent. If the emission of the first photon leaves the system in a symmetric intermediate state (e.g., $| S\rangle$), the symmetric coupling to the radiation field results in the coherent behavior, and consequently the system decays to the symmetric ground state $| {g}_{1}{g}_{2}\rangle$ with the corresponding transition rate shown in Figure 5. In contrast, if the system is in the antisymmetric intermediate state $| A\rangle$, the coupling to the radiation field will be antisymmetric under the exchange of the atoms as they interact with the nonuniform electromagnetic field, and the system decays noncoherently to the ground state $| {g}_{1}{g}_{2}\rangle$ with the decay rate Γ_A.

We can also explain this classically by considering two classical radiators separated by a distance r' > λ. Over large distances, the phase and the polarization of the radiation field emitted by each radiator varies from place to place. When the radiation from the two identical radiators interferes, the intensity of the total field is given by

$$\begin{eqnarray}&&{I}_{\mathrm{tot}}\propto \langle {({{\boldsymbol{B}}}_{1}+{{\boldsymbol{B}}}_{2})}^{2}\rangle \end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray}&&\propto \,\langle {B}_{1}^{2}\rangle +\langle {B}_{2}^{2}\rangle +2\langle {{\boldsymbol{B}}}_{1}\cdot {{\boldsymbol{B}}}_{2}\rangle ,\end{eqnarray} \tag{ 22 }$$

and can become as large as four times the intensity of a single radiator I₀ if

$$\begin{eqnarray}&&\langle {{\boldsymbol{B}}}_{1}\cdot {{\boldsymbol{B}}}_{2}\rangle \sim {I}_{0}.\end{eqnarray} \tag{ 23 }$$

If the phase of the radiation from different radiators does not match perfectly, the term containing the correlation $\langle {{\boldsymbol{B}}}_{1}\cdot {{\boldsymbol{B}}}_{2}\rangle$ in Equation (22) becomes smaller than I₀, and consequently the total intensity I_tot decreases until it reaches its minimum for completely out-of-phase radiators. Furthermore, the correlation term can vanish when radiators act independently. In this case, the total intensity becomes equal to the sum of the intensities of the two independent radiators (the so-called noncoherent system).

We can extend our discussion for the case of two distant atoms (i.e., ${r}^{\prime }\gt \lambda$) to a large sample consisting of N atoms distributed over a volume ${ \mathcal V }\gt {\lambda }^{3}$. As stated above, the build-up of correlations in an extended N-atom sample can be understood as a constructive interference of the radiation by different atoms. In a large sample as a result of propagation over large distances (i.e., larger than λ), the phase of the radiation varies throughout the sample (${{kr}}^{\prime }\gg 1$ and ${e}^{{ik}\cdot {r}^{\prime }}\nsim 1$). Consequently, the phase of the atomic magnetization differs with position. In an inverted large sample, the radiations from different atoms interfere with each other, and when the magnetizations of the radiators are perfectly in phase, an intense propagating wave is produced in one direction (the phase-matching condition cannot be satisfied in all directions).

In order to better understand the phase-matching process, it is useful to go back to the angular correlation effect described in the two-atom case. In a large sample of N inverted atoms, when the first photon is emitted, other atoms interact with its radiation field, and the direction of the next photon is affected by the first one. In a more general sense, when a photon is radiated in a particular direction k, it becomes more probable to observe the second photon in the same direction k than any other direction. Thus, as the atoms radiate, an angular correlation builds up in the sample that triggers the phase-matching process in a well-defined direction.

Ideal superradiance is the result of the symmetrical evolution of an atom–field system, and in a large sample, the propagation effects result in the nonuniform evolution of the atoms in the sample. In order to better understand propagation effects in a large sample, the atomic medium can be divided into small identical slices with dimensions larger than λ but much smaller than the length of the sample. A microscopic dipole is then associated with each slice with its magnitude being proportional to the number of excited atoms in the corresponding slice. At the beginning, the dipoles in different slices are independent and their radiation uncorrelated. After some time (or the so-called retarded time delay τ_D; see Equation (34)), as they interact with their common radiation field, the dipoles lock in to a common phase and act as a single macroscopic dipole radiating intensely with ${I}_{\mathrm{SR}}={{NfI}}_{\mathrm{nc}}$, where Nf is the enhancement factor of the superradiant intensity I_SR over the noncoherent intensity I_nc determined by the efficiency of the common phase-locking process (through the value of f ≤ 1). The enhancement factor Nf can become very large in samples with N ≫ 1, and it converges to N in an atomic system with dimensions of the order of λ, resulting in ${I}_{\mathrm{SR}}={{NI}}_{\mathrm{nc}}$ for the most efficient phase-locking process seen in a small sample. In other words, f < 1 implies a limited coherent behavior in a large sample, resulting in a smaller output intensity and weakened superradiance, whereas f = 1 indicates a fully coherent behavior leading to an intense radiation and perfect superradiance (MacGillivray & Feld 1976; Gross & Haroche 1982).

This approach has the shortcoming that it cannot explain the initiation of the radiation in the system by spontaneous fluctuations, and to overcome this problem phenomenological fluctuations of dipoles in the initial stages of the evolution can be added to the formalism. In contrast, triggered superradiance can be fully explained in this manner because the correlation process is initiated by an external field, which can be defined classically. It must be pointed out that the results of this method are valid only if the propagation time of the radiation ${\tau }_{{\rm{E}}}$ through a sample of length L (i.e., τ_E = L/c) is smaller than the superradiance characteristic time ${T}_{{\rm{R}}}$ given by

$$\begin{eqnarray}&&{T}_{{\rm{R}}}={\tau }_{\mathrm{sp}}\displaystyle \frac{16\pi }{3n{\lambda }^{2}L},\end{eqnarray} \tag{ 24 }$$

where τ_sp = 1/Γ is the spontaneous decay time of a single atom and n the density of inverted atoms (see MacGillivray & Feld 1976; Rosenberger & DeTemple 1981, and Appendix B). This condition (i.e., ${\tau }_{{\rm{E}}}\lt {T}_{{\rm{R}}}$) is known as the Arecchi–Courtens condition, and it ensures that the atomic magnetization in different parts of the sample can lock into a common phase and coherent behavior can develop through the sample.

In Appendix A, we derive the evolution equations for the radiation field and the atomic system using the Heisenberg representation, while in Appendix B we solve the corresponding Maxwell–Bloch system of equations, at resonance, within the framework of the slowly varying envelope approximation (SVEA). To do so, we adopted the following form for the radiation magnetic field and atomic magnetization:

$$\begin{eqnarray}&&{\hat{B}}_{L}^{\pm }({\boldsymbol{r}},t)={\hat{B}}_{0}^{\pm }({\boldsymbol{r}},t){e}^{\pm i({kz}-\omega t)}\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&{\hat{{ \mathcal M }}}^{\pm }({\boldsymbol{r}},t)={\hat{{ \mathcal M }}}_{0}^{\pm }({\boldsymbol{r}},t){e}^{\pm }i({kz}-\omega t),\end{eqnarray} \tag{ 26 }$$

with ${\hat{B}}_{0}^{\pm }$ and ${\hat{{ \mathcal M }}}_{0}^{\pm }$ corresponding to slowly varying envelope operators. The superradiance of a cylindrical large sample of length L under ideal conditions is then found to be determined by the following equations for, respectively, the magnetization, the population inversion, and the magnetic field:

$$\begin{eqnarray}&&{\hat{{ \mathcal M }}}_{0}^{+}=\displaystyle \frac{{\mu }_{{\rm{B}}}N}{2\sqrt{2}V}\sin (\theta )\end{eqnarray} \tag{ 27 }$$

$$\begin{eqnarray}&&\hat{{\mathbb{N}}}=\displaystyle \frac{N}{V}\cos (\theta )\end{eqnarray} \tag{ 28 }$$

$$\begin{eqnarray}&&{\hat{B}}_{0}^{+}=\displaystyle \frac{i{\mu }_{{\rm{B}}}}{2\sqrt{2}\gamma }\displaystyle \frac{\partial \theta }{\partial \tau },\end{eqnarray} \tag{ 29 }$$

where $\gamma ={\mu }_{{\rm{B}}}^{2}/2{\hslash }$. The solution for the Bloch angle θ as a function of the retarded time $\tau =t-L/c$ is obtained through the so-called Sine-Gordon equation:

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{2}\theta }{{{dq}}^{2}}+\displaystyle \frac{1}{q}\displaystyle \frac{d\theta }{{dq}}=\sin (\theta )\end{eqnarray} \tag{ 30 }$$

with

$$\begin{eqnarray}&&q=2\sqrt{\displaystyle \frac{z\tau }{{{LT}}_{{\rm{R}}}}}.\end{eqnarray} \tag{ 31 }$$

In Figure 6 we show the solution for the radiation intensity of such an ideal, cylindrical large sample of H i atoms of length $L=0.02\,{{cT}}_{{\rm{R}}}$, where c is the speed of the light, by numerically solving Equations (27)–(31). For these calculations we set a Fresnel number of unity to reduce the impact of diffraction losses, which are not taken into account in our model. This yields a cylinder of radius

$$\begin{eqnarray}&&w=\sqrt{\displaystyle \frac{\lambda L}{\pi }},\end{eqnarray} \tag{ 32 }$$

which for our ideal sample results in $w=0.036\,{({{cT}}_{{\rm{R}}})}^{1/2}$.

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In this sample, the Arecchi–Courtens condition is satisfied (i.e., ${\tau }_{{\rm{E}}}\ll {T}_{{\rm{R}}}$), allowing the use of the homogeneous condition θ₀ = 4.9 × 10⁻¹² rad for the initial value of the Bloch angle. More precisely, for the large sample used for the figure, we assumed that internal fluctuations dominate over triggered superradiance, and the initial Bloch angle was set with ${\theta }_{0}=2/\sqrt{N}$ (Gross & Haroche 1982).

In Figure 6, the retarded-time axis is scaled to T_R and the radiation intensity to NI_nc, that is, the number of inverted atoms times the corresponding noncoherent intensity that would otherwise be expected from such a sample. More precisely, for comparison purposes we consider the noncoherent intensity emanating through the sample's end-fire (i.e., the end facing the observer) of area A, within the superradiance radiation beam solid angle ϕ_D = λ²/A (in the direction k along which the phase-locking condition is satisfied) normalized to the solid angle associated with the total noncoherent radiation. We thus have

$$\begin{eqnarray}{I}_{\mathrm{nc}} & = & N{\hslash }\omega \left(\displaystyle \frac{1}{A{\tau }_{\mathrm{sp}}}\right)\left(\displaystyle \frac{{\phi }_{{\rm{D}}}}{4\pi }\right)\\ & = & \displaystyle \frac{4}{3}\displaystyle \frac{{\hslash }\omega }{{{AT}}_{{\rm{R}}}},\end{eqnarray} \tag{ 33 }$$

where Equation (24) was used for the last step, and $N{\hslash }\omega$ is the total energy initially stored in the sample. As shown in the figure, this energy is radiated away through multiple bursts, a phenomenon known as the ringing effect. This effect can be explained by the fact that atoms in different locations in the sample radiate at different times. In other words, an atom at location z = z₀, prepared in the excited state at τ = 0, radiates its energy away and decays to its ground state, then later on absorbs the energy radiated by another atom at a location z < z₀ and becomes excited, leading to another radiation event.

In a large sample, just as in a small sample, internal field fluctuations or an external field trigger superradiance, and after the delay time ${\tau }_{{\rm{D}}}$ the atoms radiate coherently. But contrary to a small sample, the large-sample delay time depends on the initial conditions and is given by (Benedict et al. 1996)

$$\begin{eqnarray}&&{\tau }_{{\rm{D}}}\simeq \displaystyle \frac{{T}_{{\rm{R}}}}{4}{\left|\mathrm{ln}\left(\displaystyle \frac{{\theta }_{0}}{2\pi }\right)\right|}^{2}.\end{eqnarray} \tag{ 34 }$$

In Figure 6 the first burst of radiation occurs after ${\tau }_{{\rm{D}}}\simeq 160\,{T}_{{\rm{R}}}$, which is consistent with the value one finds using Equation (34). As can also be seen, this first intensity burst only carries out a fraction of the total energy stored in the sample, while the remaining radiation happens through subsequent bursts. The number of burst events depends on the length of the sample, and as the length is increased radiation emanates through a larger number of bursts, while the peak intensity of consecutive burst events gradually drops. This is a consequence of energy conservation and the fact that, in a larger (i.e., longer) sample, radiations from different groups of atoms along the sample arrive at the end-fire at different times, and the process of absorbing the radiation, developing correlations between the dipoles, and eventually reemitting the radiation repeats multiple times over a very long period of time. On the other hand, when the length of the sample is decreased, the ringing effect becomes weaker until, for a small sample of dimension of order of λ, it totally washes out and we only observe a single burst of radiation carrying away all the energy stored in the system (as in Figure 4). Finally, we note that although the maximum radiation intensity seen in Figure 6 seems to imply that $f\sim 0.001\ll 1$, the large number of atoms present in the sample ensures that ${I}_{\mathrm{SR}}\gg {I}_{\mathrm{nc}}$ (see Section 4). We should also note, however, that the Sine-Gordon equation is very sensitive to initial conditions. It therefore follows that the exact shape of the intensity curve, for example, the number of bursts in Figure 6, is also strongly dependent on θ₀.

3.2.1. Nonideal Case: Dephasing Effects

As was mentioned earlier, the characteristic timescale of superradiance T_R and the delay time τ_D (for a large sample) must be shorter than the relaxation/dephasing timescales to allow the build-up of correlations in a nonideal sample. These effects include natural broadening due to the spontaneous decay timescale τ_sp of a single atom and collisional broadening related to the mean time between collisions τ_coll for an atom in the sample. Although, as was stated in Section 1, our analysis is aimed at regions of the ISM where thermal equilibrium has not been reached and where consequently the assignation of a temperature to determine, for example, collision rates is perhaps ill-defined, we will nonetheless adopt such a procedure for the rest of our discussion to get a sense of the timescales involved. Accordingly, in a H i gas different types of collisions can take place depending on the temperature and density. For environments of temperatures ranging from approximately 10 to 300 K, which are the focus of our analysis, collisions between two neutral hydrogen atoms (H–H collisions) dominate and fall into two categories: elastic and inelastic. During an elastic H–H collision, the spacing between the atomic energy levels is slightly affected, but no transition between them is induced. The change in energy spacing occurs as a result of short-range interaction forces between the two colliding particles and induces a phase shift in the wave function of the scattered atoms. After a number of elastic collisions, an atom can lose coherence with the interacting radiation field as a result of the randomness in the perturbations. In contrast, in an inelastic H–H collision, the internal energy of the hydrogen atoms will be changed. This occurs when the two hydrogen atoms with oppositely directed electron spins approach each other at distances less than approximately 10⁻⁸ cm. This process is known as electron exchange or the spin de-excitation effect. As a result of such a collision, the induced phase shift can lead to a change in the internal spin states (Wittke & Dicke 1956), and it is found that spin de-excitation is the dominating relaxation process in a high-density collision-dominated H i gas. We therefore find that H–H collisions not only can affect the strength of a potential coherent 21 cm radiation by reducing the population of the excited hyperfine states, they also contribute to the line breadth and can change the shape of the spectral line by affecting the spacing between internal energy levels. For example, the timescale of H–H collisions is estimated to be on the order of 10⁸ s in the case of elastic scattering and 10⁹ s for spin de-excitations, using the mean effective collisional cross sections given in Irwin (2007) for a H i gas at $T=100\,{\rm{K}}$ and n = 10 cm⁻³. The mean time between collision ${\tau }_{\mathrm{coll}}$ is thus set to the shortest of these timescales and must at least be larger than T_R and ${\tau }_{{\rm{D}}}$ to allow coherent behavior (see Section 4).

In addition, other broadening effects, such as Doppler broadening, are further dephasing mechanisms that can destroy cooperative behavior if their timescales (importantly the so-called Doppler dephasing time, i.e., the reciprocal of the Doppler width) are smaller than T_R and τ_D (Bonifacio & Lugiato 1975; Meziane et al. 2002). In a thermally relaxed gas, thermal motions are probably the most important dephasing effects and result in line broadenings that correspond to very short dephasing timescales (e.g., T_therm ∼ 10⁻³ s at T = 100 K). In the presence of such strong dephasing effects, correlations cannot develop between the dipoles, and any coherent interaction will be terminated right from the start. Hence our earlier comment that we do not expect to find superradiance under conditions of thermal equilibrium, but potentially only in (out-of-equilibrium) regions where strong velocity coherence can be maintained along the line of sight. Furthermore, this condition may be only met among a group of atoms in such regions, therefore reducing the number of inverted atoms in the sample that could participate in coherent interactions. However, we know from maser observations that a high level of velocity coherence can be achieved in some regions of the ISM, and we expect that superradiance could happen under similar conditions. As was mentioned in Section 1, the main inversion pumping mechanism likely involved for the 21 cm transition points to the surroundings of H ii regions as potential sites for superradiance in this spectral line. It follows that we should also anticipate an analogous (very small) volume filling factor for the emitting regions of superradiant sources as for masers.

In Figure 7 the intensity of the H i large sample discussed in Section 3.2 (and presented in Figure 6 for the ideal case) is plotted as a function of the retarded time τ for the special case where dephasing/relaxation effects are included and characterized by a single timescale set to ${T}^{\prime }=541\,{T}_{{\rm{R}}}$ (see Section 4 and Appendix B). These results were obtained by once again numerically solving Equation (30), while the magnetization and population inversion are given by

$$\begin{eqnarray}&&{\hat{{ \mathcal M }}}_{0}^{+}=\displaystyle \frac{{\mu }_{{\rm{B}}}N}{2\sqrt{2}V}\sin (\theta ){e}^{-\tau /{T}^{\prime }}\end{eqnarray} \tag{ 35 }$$

$$\begin{eqnarray}&&\hat{{\mathbb{N}}}=\displaystyle \frac{N}{V}\cos (\theta ){e}^{-\tau /{T}^{\prime }},\end{eqnarray} \tag{ 36 }$$

and the dimensionless parameter

$$\begin{eqnarray}&&q=2\sqrt{\displaystyle \frac{z{\tau }^{\prime }}{{{LT}}_{{\rm{R}}}}}\end{eqnarray} \tag{ 37 }$$

with

$$\begin{eqnarray}&&{\tau }^{\prime }={T}^{\prime }(1-{e}^{-\tau /{T}^{\prime }}).\end{eqnarray} \tag{ 38 }$$

The magnetic field is once again given by Equation (29). The intensity and time axes are scaled like those in Figure 6 for the ideal H i sample. We can see from Figure 7 that the ringing effect seen in the ideal sample is also present here but is weakened by the dephasing and basically terminated after $\tau \sim 1000\,{T}_{{\rm{R}}}$ (i.e., approximately the dephasing timescale). The dephasing effects also affect the maximum energy radiated away through each burst event and result in slightly weaker intensities.

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We follow Dicke (1954) and approximate the Hamiltonian for a sample of N hydrogen atoms with each atom acting as a two-level system, while taking into account the magnetic nature of the dipole-radiation interaction applicable to this case:

$$\begin{eqnarray}&&\hat{H}={\hat{H}}_{0}+{\hat{H}}_{\mathrm{rad}}+{\hslash }\displaystyle \sum _{j=1}^{N}\,{\omega }_{j}\,\left({\hat{R}}_{j3}+\displaystyle \frac{\hat{1}}{2}\right)-\displaystyle \sum _{j=1}^{N}\,{\hat{{\boldsymbol{M}}}}_{j}\cdot \hat{{\boldsymbol{B}}}({{\boldsymbol{r}}}_{j}).\end{eqnarray} \tag{ 42 }$$

In this Hamiltonian equation, ${\hat{H}}_{0}$ contains the translational and interatomic interaction energies of the atoms, ${\hat{H}}_{\mathrm{rad}}$ is the radiation field Hamiltonian term, ${\hslash }{\omega }_{j}({\hat{R}}_{j3}+\hat{1}/2)$ is the internal energy of the $j\mathrm{th}$ two-level atom ($\hat{1}$ is the unit operator), which has the eigenvalues 0 and ${\hslash }{\omega }_{j}$, and the last term stands for the interaction between the electromagnetic field and the magnetic dipole of the jth atom ${\hat{{\boldsymbol{M}}}}_{j}$. Since this Hamiltonian is written under the magnetic dipole approximation, it implies that the magnetic field $\hat{{\boldsymbol{B}}}$ does not change considerably over the size of the atom and is determined by its value at the position of the center of mass of the atom, r_j. Finally, the effects of the hyperfine interaction between the proton and electron spins within a single hydrogen atom and the Zeeman interaction due to an external magnetic field would be included in the frequency ${\omega }_{j}$ of the atomic transition.

Following Dicke, we define the operators ${\hat{R}}_{x},{\hat{R}}_{y},{\hat{R}}_{3}$, and ${\hat{R}}^{2}$ such that

$$\begin{eqnarray}&&{\hat{R}}_{K}({\boldsymbol{r}})=\displaystyle \sum _{j=1}^{N}\,{\hat{R}}_{{jK}}\delta ({\boldsymbol{r}}-{{\boldsymbol{r}}}_{j}),\quad K=x,y,3\end{eqnarray} \tag{ 43 }$$

$$\begin{eqnarray}&&{\hat{R}}^{2}({\boldsymbol{r}})={\hat{R}}_{x}^{2}+{\hat{R}}_{y}^{2}+{\hat{R}}_{3}^{2}\end{eqnarray} \tag{ 44 }$$

$$\begin{eqnarray}&&[{\hat{R}}^{2}({\boldsymbol{r}}),{\hat{R}}_{K}({{\boldsymbol{r}}}^{\prime })]=0\end{eqnarray} \tag{ 45 }$$

$$\begin{eqnarray}&&[{\hat{R}}_{a}({\boldsymbol{r}}),{\hat{R}}_{b}({{\boldsymbol{r}}}^{\prime })]=i{\varepsilon }_{{abc}}{\hat{R}}_{c}({\boldsymbol{r}})\delta ({\boldsymbol{r}}-{{\boldsymbol{r}}}^{\prime }),\quad a,b,c=x,y,3,\end{eqnarray} \tag{ 46 }$$

which are similar to the relations found in the spin or general angular momentum formalisms. We can also define the raising and lowering operators

$$\begin{eqnarray}&&{\hat{R}}^{\pm }({\boldsymbol{r}})={\hat{R}}_{x}({\boldsymbol{r}})\pm i{\hat{R}}_{y}({\boldsymbol{r}}),\end{eqnarray} \tag{ 47 }$$

which further verify the following commutation relations:

$$\begin{eqnarray}&&[{\hat{R}}^{\pm }({\boldsymbol{r}}),{\hat{R}}_{3}({\boldsymbol{r}}^{\prime} )]=\mp {\hat{R}}^{\pm }({\boldsymbol{r}})\delta ({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} )\end{eqnarray} \tag{ 48 }$$

$$\begin{eqnarray}&&[{\hat{R}}^{+}({\boldsymbol{r}}),{\hat{R}}^{-}({\boldsymbol{r}}^{\prime} )]=2{\hat{R}}_{3}({\boldsymbol{r}})\delta ({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} ).\end{eqnarray} \tag{ 49 }$$

It is clear from the form of the Hamiltonian and the commutation relations between the operators that ${\hat{R}}^{2}$ and ${\hat{R}}_{3}$ commute with $\hat{H}$ and, therefore, share the eigenfunctions $| r,{m}_{r}\rangle$ introduced in Section 2.1 to describe the state of the system.

The atomic hydrogen transitions at 21 cm are magnetic dipolar in nature and bring into consideration the next leading term in our analysis, the magnetic dipole interaction found on the right-hand side of Equation (42):

$$\begin{eqnarray}&&{\hat{V}}_{\mathrm{MD}}=-\displaystyle \sum _{j=1}^{N}\,{\hat{{\boldsymbol{M}}}}_{j}\cdot \hat{{\boldsymbol{B}}}({{\boldsymbol{r}}}_{j}),\end{eqnarray} \tag{ 50 }$$

which is at the center of our analysis. In general, the magnetic dipole operator ${\hat{{\boldsymbol{M}}}}_{j}$ of the $j\mathrm{th}$ atom can be written as (Condon & Shortley 1935)

$$\begin{eqnarray}&&{\hat{{\boldsymbol{M}}}}_{j}={\mu }_{F}{\hat{{\boldsymbol{F}}}}_{j}\end{eqnarray} \tag{ 51 }$$

$$\begin{eqnarray}&&{\mu }_{F}\simeq {g}_{J}\left[\displaystyle \frac{F(F+1)+J(J+1)-I(I+1)}{2F(F+1)}\right],\end{eqnarray} \tag{ 52 }$$

where $\hat{{\boldsymbol{J}}}$ is the sum of the electronic orbital $\hat{{\boldsymbol{L}}}$ and spin $\hat{{\boldsymbol{S}}}$ angular momenta (i.e., $\hat{{\boldsymbol{J}}}=\hat{{\boldsymbol{L}}}+\hat{{\boldsymbol{S}}}$), and $\hat{{\boldsymbol{F}}}$ is the sum of $\hat{{\boldsymbol{J}}}$ and the nuclear spin $\hat{{\boldsymbol{I}}}$ (i.e., $\hat{{\boldsymbol{F}}}=\hat{{\boldsymbol{J}}}+\hat{{\boldsymbol{I}}}$). For the hyperfine levels of the ground state of the hydrogen atom, we have $F=0\ \mathrm{and}\ 1$, J = S = 1/2, I = 1/2, and ${\mu }_{F}\simeq {g}_{J}/2$, whereas ${g}_{J}={g}_{e}{\mu }_{{\rm{B}}}/{\hslash }$. In this equation, ${g}_{e}\simeq 2$ and μ_B is the Bohr magneton. The operator ${\hat{{\boldsymbol{F}}}}_{j}$ can also be written in terms of pseudospin operator ${\hat{{\boldsymbol{R}}}}_{j}$ as ${\hat{{\boldsymbol{F}}}}_{j}={\hslash }\,({\hat{{\boldsymbol{R}}}}_{j}+\hat{1}/2)$, allowing us to write

$$\begin{eqnarray}&&{\hat{{\boldsymbol{M}}}}_{j}={\mu }_{{\rm{B}}}\left({\hat{{\boldsymbol{R}}}}_{j}+\displaystyle \frac{\hat{1}}{2}\right).\end{eqnarray} \tag{ 53 }$$

For the $| F=0,m=0\rangle \longleftrightarrow | F=1,m=+1\rangle$ LCP transition we consider, the circular polarization state of radiation can be defined using the corresponding unit vectors (Grynberg et al. 2010):

$$\begin{eqnarray}&&{{\boldsymbol{e}}}_{L}=-\displaystyle \frac{1}{\sqrt{2}}({{\boldsymbol{e}}}_{x}+i{{\boldsymbol{e}}}_{y}),\end{eqnarray} \tag{ 54 }$$

$$\begin{eqnarray}&&{{\boldsymbol{e}}}_{R}=\displaystyle \frac{1}{\sqrt{2}}({{\boldsymbol{e}}}_{x}-i{{\boldsymbol{e}}}_{y}),\end{eqnarray} \tag{ 55 }$$

which with e₃ can be used to write the pseudospin operator as

$$\begin{eqnarray}&&{\hat{{\boldsymbol{R}}}}_{j}=\displaystyle \frac{1}{\sqrt{2}}(-{\hat{R}}_{j}^{-}{{\boldsymbol{e}}}_{L}+{\hat{R}}_{j}^{+}{{\boldsymbol{e}}}_{R})+{\hat{R}}_{j3}{{\boldsymbol{e}}}_{3}.\end{eqnarray} \tag{ 56 }$$

In a H i gas, the LCP magnetic component of the radiation propagating along the k direction interacts with the magnetic dipole of a hydrogen atom, resulting in a transition between the two hyperfine levels. The corresponding magnetic field operator can be expressed as

$$\begin{eqnarray}&&{\hat{{\boldsymbol{B}}}}_{L}({\boldsymbol{r}},t)=\displaystyle \sum _{{\boldsymbol{k}}}\,[{\hat{B}}_{{Lk}}^{+}({\boldsymbol{r}}){e}^{-i{\omega }_{k}t}{{\boldsymbol{e}}}_{L}+{\hat{B}}_{{Lk}}^{-}({\boldsymbol{r}}){e}^{i{\omega }_{k}t}{{\boldsymbol{e}}}_{L}^{\ast }],\end{eqnarray} \tag{ 57 }$$

where

$$\begin{eqnarray}&&{\hat{B}}_{{Lk}}^{+}({\boldsymbol{r}})=\displaystyle \frac{1}{c}\sqrt{\displaystyle \frac{{\hslash }{\omega }_{k}}{2{\epsilon }_{0}V}}{\hat{a}}_{{Lk}}{e}^{{ik}\cdot {\boldsymbol{r}}}\end{eqnarray} \tag{ 58 }$$

$$\begin{eqnarray}&&={({\hat{B}}_{{Lk}}^{-})}^{\dagger },\end{eqnarray} \tag{ 59 }$$

and V is the arbitrary volume of quantization. In Equations (58) and (59), ${a}_{{Lk}}$ and ${a}_{{Lk}}^{\dagger }$ are, respectively, the LCP second quantization field annihilation and creation operators, and they obey the following commutation relation:

$$\begin{eqnarray}&&[{\hat{a}}_{{Lk}},{\hat{a}}_{{Lk}^{\prime} }^{\dagger }]=\hat{1}{\delta }_{{{kk}}^{\prime }}.\end{eqnarray} \tag{ 60 }$$

As a result, one can express the magnetic dipole interaction term in Equation (50) for transitions involving only LCP photons as

$$\begin{eqnarray}&&{\hat{V}}_{\mathrm{MD}}=-{\mu }_{{\rm{B}}}\sum _{j=1}^{N}\,{\hat{{\boldsymbol{R}}}}_{j}\cdot {\hat{{\boldsymbol{B}}}}_{L}({{\boldsymbol{r}}}_{j})\end{eqnarray} \tag{ 61 }$$

$$\begin{eqnarray}&&=\,\displaystyle \frac{{\mu }_{{\rm{B}}}}{\sqrt{2}c}\displaystyle \sum _{j=1}^{N}\,\displaystyle \sum _{k}\,\sqrt{\displaystyle \frac{{\hslash }{\omega }_{k}}{2{\epsilon }_{0}V}}({\hat{R}}_{j}^{+}{\hat{a}}_{{Lk}}{e}^{{ik}\cdot {{\boldsymbol{r}}}_{j}}+{\hat{R}}_{j}^{-}{\hat{a}}_{{Lk}}^{\dagger }{e}^{-{ik}\cdot {{\boldsymbol{r}}}_{j}}),\end{eqnarray} \tag{ 62 }$$

using $\hat{{\boldsymbol{R}}}\cdot {{\boldsymbol{e}}}_{L}=-{\hat{R}}^{+}/\sqrt{2}$ and $\hat{{\boldsymbol{R}}}\cdot {{\boldsymbol{e}}}_{L}^{* }=-{\hat{R}}^{-}/\sqrt{2}$. It will also prove useful to write Equation (62) in the following form:

$$\begin{eqnarray}&&{\hat{V}}_{\mathrm{MD}}=-{\int }_{V}\hat{{\boldsymbol{ \mathcal M }}}({\boldsymbol{r}})\cdot {\hat{{\boldsymbol{B}}}}_{L}({\boldsymbol{r}}){d}^{3}r,\end{eqnarray} \tag{ 63 }$$

which allows a definition of the transverse macroscopic magnetization operator $\hat{{\boldsymbol{ \mathcal M }}}({\boldsymbol{r}})$ in terms of the raising and lowering density operators ${\hat{R}}^{+}$ and ${\hat{R}}^{-}$ as follows:

$$\begin{eqnarray}&&\hat{{\boldsymbol{ \mathcal M }}}({\boldsymbol{r}})=-\displaystyle \frac{{\mu }_{B}}{\sqrt{2}}[{\hat{R}}^{-}({\boldsymbol{r}}){{\boldsymbol{e}}}_{L}+{\hat{R}}^{+}({\boldsymbol{r}}){{\boldsymbol{e}}}_{L}^{* }]\end{eqnarray} \tag{ 64 }$$

$$\begin{eqnarray}&&\equiv \,{\hat{{\boldsymbol{ \mathcal M }}}}^{+}({\boldsymbol{r}})+{\hat{{\boldsymbol{ \mathcal M }}}}^{-}({\boldsymbol{r}}).\end{eqnarray} \tag{ 65 }$$

Neglecting any inhomogeneous broadening effect (i.e., we omit ${\hat{H}}_{0}$ and set ${\omega }_{j}={\omega }_{0}$) and inserting Equation (62) in Equation (42), the Hamiltonian of the H i-sample system interacting with the 21 cm line via the ($F,{m}_{F}\,:0,0\,\longleftrightarrow \,1,+1$) transition becomes

$$\begin{eqnarray}\hat{H} & = & {\hslash }{\omega }_{0}\displaystyle \sum _{j=1}^{N}\,\left({\hat{R}}_{j3}+\displaystyle \frac{\hat{1}}{2}\right)+{\hat{H}}_{\mathrm{rad}}+\displaystyle \frac{{\mu }_{B}}{\sqrt{2}c}\displaystyle \sum _{k}\,\sqrt{\displaystyle \frac{{\hslash }{\omega }_{k}}{2{\epsilon }_{0}V}}\\ & & \times \displaystyle \sum _{j=1}^{N}\,({\hat{R}}_{j}^{+}{\hat{a}}_{{Lk}}{e}^{{ik}\cdot {{\boldsymbol{r}}}_{j}}+{\hat{R}}_{j}^{-}{\hat{a}}_{{Lk}}^{\dagger }{e}^{-{ik}\cdot {{\boldsymbol{r}}}_{j}}).\end{eqnarray} \tag{ 66 }$$

The radiation Hamiltonian term ${\hat{H}}_{\mathrm{rad}}$ can be expressed in terms of the second quantized operators ${\hat{a}}_{{Lk}}$ and ${\hat{a}}_{{Lk}}^{\dagger }$, and ${\hat{a}}_{R{\boldsymbol{k}}}$ and ${\hat{a}}_{{Rk}}^{\dagger }$ associated with left  and right circular-polarized radiation states, respectively, with

$$\begin{eqnarray}&&{\hat{H}}_{\mathrm{rad}}={\hslash }\omega \displaystyle \sum _{k}\,\left[\left({\hat{a}}_{{Lk}}^{\dagger }{\hat{a}}_{{Lk}}+\displaystyle \frac{\hat{1}}{2}\right)+\left({\hat{a}}_{{Rk}}^{\dagger }{\hat{a}}_{{Rk}}+\displaystyle \frac{\hat{1}}{2}\right)\right].\end{eqnarray} \tag{ 67 }$$

As discussed in the literature (Gross & Haroche 1982; Benedict et al. 1996), the evolution of the atomic system can be calculated using the Heisenberg equation of motion for the operator $\hat{X}$ in a system described by the Hamiltonian $\hat{H}$ with

$$\begin{eqnarray}&&\displaystyle \frac{d\hat{X}}{{dt}}=\displaystyle \frac{1}{i{\hslash }}[\hat{X},\hat{H}].\end{eqnarray} \tag{ 68 }$$

One can then readily find the following equations of motions for ${\hat{R}}^{+}$, ${\hat{R}}^{-}$, and ${\hat{R}}_{3}$:

$$\begin{eqnarray}&&\displaystyle \frac{d{\hat{R}}^{+}}{{dt}}=i{\omega }_{0}{\hat{R}}^{+}-\displaystyle \frac{i\sqrt{2}{\mu }_{B}}{{\hslash }}\,{\hat{R}}_{3}{\hat{B}}_{L}^{-}\end{eqnarray} \tag{ 69 }$$

$$\begin{eqnarray}&&\displaystyle \frac{d{\hat{R}}^{-}}{{dt}}=-i{\omega }_{0}{\hat{R}}^{-}+\displaystyle \frac{i\sqrt{2}{\mu }_{B}}{{\hslash }}\,{\hat{R}}_{3}{\hat{B}}_{L}^{+}\end{eqnarray} \tag{ 70 }$$

$$\begin{eqnarray}&&\displaystyle \frac{d{\hat{R}}_{3}}{{dt}}=-\displaystyle \frac{i{\mu }_{B}}{\sqrt{2}{\hslash }}\,({\hat{R}}^{+}{\hat{B}}_{L}^{+}-{\hat{R}}^{-}{\hat{B}}_{L}^{-}),\end{eqnarray} \tag{ 71 }$$

where, for simplicity, we now set ${\hat{B}}_{L}^{\pm }$ for the value of the LCP component of the magnetic field averaged over the positions of the atoms. In a similar way, we can write the following equations of motion for the raising and lowering magnetization operators ${\hat{{\boldsymbol{ \mathcal M }}}}^{+}({\boldsymbol{r}})$ and ${\hat{{\boldsymbol{ \mathcal M }}}}^{-}({\boldsymbol{r}})$ using Equation (64) as

$$\begin{eqnarray}&&\displaystyle \frac{d{\hat{{\boldsymbol{ \mathcal M }}}}^{+}}{{dt}}=-i{\omega }_{0}{\hat{{\boldsymbol{ \mathcal M }}}}^{+}-\displaystyle \frac{i{\mu }_{{\rm{B}}}^{2}}{{\hslash }}\,({{\boldsymbol{e}}}_{L}\cdot {\hat{{\boldsymbol{B}}}}_{L}^{+}){\hat{R}}_{3}{{\boldsymbol{e}}}_{L}\end{eqnarray} \tag{ 72 }$$

$$\begin{eqnarray}&&\displaystyle \frac{d{\hat{{\boldsymbol{ \mathcal M }}}}^{-}}{{dt}}=i{\omega }_{0}{\hat{{\boldsymbol{ \mathcal M }}}}^{-}+\displaystyle \frac{i{\mu }_{{\rm{B}}}^{2}}{{\hslash }}\,({{\boldsymbol{e}}}_{L}^{* }\cdot {\hat{{\boldsymbol{B}}}}_{L}^{-}){\hat{R}}_{3}{{\boldsymbol{e}}}_{L}^{* }.\end{eqnarray} \tag{ 73 }$$

It is also useful to define the operator $\hat{{\mathbb{N}}}$

$$\begin{eqnarray}&&\hat{{\mathbb{N}}}=2{\hat{R}}_{3}\end{eqnarray} \tag{ 74 }$$

$$\begin{eqnarray}&&=2\displaystyle \sum _{j=1}^{N}\,{\hat{R}}_{3j}\delta ({\boldsymbol{r}}-{{\boldsymbol{r}}}_{j}),\end{eqnarray} \tag{ 75 }$$

which can be interpreted as a population inversion density operator considering that ${\hat{R}}_{3j}$ has eigenvalues of $\pm 1/2$ and the eigenvalue of ${\hat{R}}_{3}=\sum {\hat{R}}_{3j}$ is equal to half of the population difference between the excited level $F=1,{m}_{F}=1$ and the ground level F = 0, m_F = 0 at time t. Using (75) one can show that Equations (71)–(73) can be rewritten as

$$\begin{eqnarray}&&\displaystyle \frac{d\hat{{\mathbb{N}}}}{{dt}}=\displaystyle \frac{2i}{{\hslash }}({\hat{{\boldsymbol{ \mathcal M }}}}^{+}\cdot {\hat{{\boldsymbol{B}}}}_{L}^{+}-{\hat{{\boldsymbol{ \mathcal M }}}}^{-}\cdot {\hat{{\boldsymbol{B}}}}_{L}^{-})\end{eqnarray} \tag{ 76 }$$

$$\begin{eqnarray}&&\displaystyle \frac{d{\hat{{\boldsymbol{ \mathcal M }}}}^{+}}{{dt}}=-i{\omega }_{0}{\hat{{\boldsymbol{ \mathcal M }}}}^{+}-i\gamma ({{\boldsymbol{e}}}_{L}\cdot {\hat{{\boldsymbol{B}}}}_{L}^{+})\hat{{\mathbb{N}}}{{\boldsymbol{e}}}_{L}\end{eqnarray} \tag{ 77 }$$

$$\begin{eqnarray}&&\displaystyle \frac{d{\hat{{\boldsymbol{ \mathcal M }}}}^{-}}{{dt}}=i{\omega }_{0}{\hat{{\boldsymbol{ \mathcal M }}}}^{-}+i\gamma ({{\boldsymbol{e}}}_{L}^{* }\cdot {\hat{{\boldsymbol{B}}}}_{L}^{-})\hat{{\mathbb{N}}}{{\boldsymbol{e}}}_{L}^{* },\end{eqnarray} \tag{ 78 }$$

where $\gamma ={\mu }_{{\rm{B}}}^{2}/2{\hslash }$.

Furthermore, in the Heisenberg representation one can derive the equation

$$\begin{eqnarray}&&-{{\rm{\nabla }}}^{2}{\hat{{\boldsymbol{B}}}}_{L}^{\pm }+\displaystyle \frac{1}{{c}^{2}}\displaystyle \frac{{\partial }^{2}{\hat{{\boldsymbol{B}}}}_{L}^{\pm }}{\partial {t}^{2}}=-{\mu }_{0}{{\rm{\nabla }}}^{2}{\hat{{\boldsymbol{ \mathcal M }}}}^{\pm }\end{eqnarray} \tag{ 79 }$$

for the evolution of the magnetic component of the radiation field when defining

$$\begin{eqnarray}&&{\hat{B}}_{L}^{\pm }({\boldsymbol{r}},t)={\hat{B}}_{0}^{\pm }({\boldsymbol{r}},t){e}^{\pm i({kz}-\omega t)}\end{eqnarray} \tag{ 80 }$$

$$\begin{eqnarray}&&{\hat{{ \mathcal M }}}^{\pm }({\boldsymbol{r}},t)={\hat{{ \mathcal M }}}_{0}^{\pm }({\boldsymbol{r}},t){e}^{\pm i({kz}-\omega t)},\end{eqnarray} \tag{ 81 }$$

using the SVEA, where ${\hat{B}}_{0}^{\pm }$ and ${\hat{{ \mathcal M }}}_{0}^{\pm }$ are slow-varying envelope operators multiplied by fast-oscillating exponential terms propagating in the positive z direction. Within the context of the SVEA, we assume that the ${\hat{B}}_{0}^{\pm }$ and ${\hat{{ \mathcal M }}}_{0}^{\pm }$ significantly change over timescales much longer than $1/\omega$ and length-scales much larger than 1/k (Gross & Haroche 1982). Upon applying the SVEA, Equation (79) is simplified to

$$\begin{eqnarray}&&\left(\displaystyle \frac{\partial }{\partial z}+\displaystyle \frac{1}{c}\displaystyle \frac{\partial }{\partial t}\right){\hat{B}}_{0}^{\pm }\simeq \pm \displaystyle \frac{i{\mu }_{0}\omega }{2c}{\hat{{ \mathcal M }}}_{0}^{\pm }.\end{eqnarray} \tag{ 82 }$$

In the derivation of Equation (82), we neglected any transverse effects on the radiation field and magnetization (i.e., $\partial {B}_{0}^{\pm }/\partial x\approx \partial {B}_{0}^{\pm }/\partial y\approx 0$ and $\partial {\hat{{ \mathcal M }}}_{0}^{\pm }/\partial x\approx \partial {\hat{{ \mathcal M }}}_{0}^{\pm }/\partial y\approx 0$), which should be included in numerical calculations for a true three-dimensional sample.

Using Equations (80) and (81), we can rewrite Equations (76), (77), and (78) at resonance, that is, when $\omega ={\omega }_{0}$, in the reduced form of

$$\begin{eqnarray}&&\displaystyle \frac{d{\hat{{ \mathcal M }}}_{0}^{\pm }}{{dt}}=\mp i\gamma {\hat{B}}_{0}^{\pm }\hat{{\mathbb{N}}}\end{eqnarray} \tag{ 83 }$$

$$\begin{eqnarray}&&\displaystyle \frac{d\hat{{\mathbb{N}}}}{{dt}}=\displaystyle \frac{2i}{{\hslash }}({\hat{{ \mathcal M }}}_{0}^{-}{\hat{B}}_{0}^{+}-{\hat{{ \mathcal M }}}_{0}^{+}{\hat{B}}_{0}^{-}).\end{eqnarray} \tag{ 84 }$$

Equations (82)–(84) are known as the Maxwell–Bloch equations and can be solved simultaneously to determine the time evolution of the radiation field, magnetization, and excitation state for an ideal sample.

The previous derivations for the ideal case must be augmented appropriately when dealing with more realistic conditions for the ISM, where dephasing and relaxation effects cannot be neglected and continuous pumping of the atomic system can take place. One can phenomenologically add the corresponding terms to the atomic equations as follows (Mandel 2010):

$$\begin{eqnarray}&&\displaystyle \frac{d{\hat{{ \mathcal M }}}_{0}^{+}}{{dt}}=-i\gamma {\hat{B}}_{0}^{+}\hat{{\mathbb{N}}}-\displaystyle \frac{1}{{T}_{2}}{\hat{{ \mathcal M }}}_{0}^{+}+{{\rm{\Lambda }}}_{{\rm{M}}}\end{eqnarray} \tag{ 85 }$$

$$\begin{eqnarray}&&\displaystyle \frac{d\hat{{\mathbb{N}}}}{{dt}}=\displaystyle \frac{2i}{{\hslash }}({\hat{{ \mathcal M }}}_{0}^{-}{\hat{B}}_{0}^{+}-{\hat{{ \mathcal M }}}_{0}^{+}{\hat{B}}_{0}^{-})-\displaystyle \frac{1}{{T}_{1}}(\hat{{\mathbb{N}}}-{{\mathbb{N}}}_{\mathrm{eq}}),\end{eqnarray} \tag{ 86 }$$

where T₁ and T₂ are the characteristic timescales for, respectively, population decay and demagnetization, ${{\mathbb{N}}}_{\mathrm{eq}}$ is the "equilibrium" value for $\hat{{\mathbb{N}}}$ obtained in the absence of interaction with the coherent field ${\hat{B}}_{L}$, and ${{\rm{\Lambda }}}_{{\rm{M}}}$ represents any source term of magnetization.

The one-dimensional magnetic field Equation (82) can also be adapted to the more realistic conditions by adding a correction term to account for the loss of radiation due to transverse effects and diffraction, which depend on the shape and symmetry of the sample. These are characterized by the Fresnel number

$$\begin{eqnarray}&&{F}_{{\rm{n}}}=\displaystyle \frac{A}{\lambda L},\end{eqnarray} \tag{ 87 }$$

where A and L, respectively, stand for the cross section and length of the sample. For samples of cylindrical symmetry with a Fresnel number smaller than one, transverse effects of the field are negligible, whereas the diffraction of radiation along the propagation axis can play an important role. Gross & Haroche (1982) have shown that a damping term ${B}_{0}^{+}/{L}_{\mathrm{diff}}$ can be included in the field equation to take into account diffraction effects in samples with ${F}_{{\rm{n}}}\ll 1$. For such a sample, Equation (82) can be approximately augmented to

$$\begin{eqnarray}&&\left(\displaystyle \frac{\partial }{\partial z}+\displaystyle \frac{1}{c}\displaystyle \frac{\partial }{\partial t}\right)\,{\hat{B}}_{0}^{+}(z,\tau )+\displaystyle \frac{1}{{L}_{\mathrm{diff}}}\,{B}_{0}^{+}(z,\tau )\simeq \displaystyle \frac{i{\mu }_{0}\omega }{2c}{\hat{{ \mathcal M }}}_{0}^{+},\end{eqnarray} \tag{ 88 }$$

where ${L}_{\mathrm{diff}}\simeq {F}_{{\rm{n}}}L/0.35$ (Gross & Haroche 1982).

The atom-field Equations (85), (86), and (88) also form a Maxwell–Bloch system of equations and provide a more complete and realistic picture for the evolution of the system. This set of equations can be numerically solved for a given set of parameters ${T}_{1},{T}_{2},\ {{\mathbb{N}}}_{\mathrm{eq}}$, and Λ_M.