The Astrophysical Journal, 826:216 (16pp), 2016 August 1 doi:10.3847/0004-637X/826/2/216 © 2016. The American Astronomical Society. All rights reserved.
DICKE’S SUPERRADIANCE IN ASTROPHYSICS. I. THE 21 cm LINE Fereshteh Rajabi1 and Martin Houde1,2 1 Department of Physics and Astronomy, The University of Western Ontario, London, ON, N6A 3K7, Canada 2 Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA 91125, USA Received 2015 April 21; revised 2016 May 29; accepted 2016 June 1; published 2016 August 1
ABSTRACT We have applied the concept of superradiance introduced by Dicke in 1954 to astrophysics by extending the corresponding analysis to the magnetic dipole interaction characterizing the atomic hydrogen 21 cm line. Although it is unlikely that superradiance could take place in thermally relaxed regions and that the lack of observational evidence of masers for this transition reduces the probability of detecting superradiance, in situations where the conditions necessary for superradiance are met (close atomic spacing, high velocity coherence, population inversion, and long dephasing timescales compared to those related to coherent behavior), our results suggest that relatively low levels of population inversion over short astronomical length-scales (e.g., as compared to those required for maser amplification) can lead to the cooperative behavior required for superradiance in the interstellar medium. Given the results of our analysis, we expect the observational properties of 21 cm superradiance to be characterized by the emission of high-intensity, spatially compact, burst-like features potentially taking place over short periods ranging from minutes to days. Key words: atomic processes – ISM: atoms – radiation mechanisms: general
1. INTRODUCTION While the reality of the superradiance phenomenon has long been clearly established in the laboratory, to the best of our It is generally assumed that in much of the interstellar knowledgeit has yet to be investigated within an astrophysical medium (ISM) emission emanating from atomic and molecular context. It appears to us important to do so since some of the transitions within a radiating gas happen independently for requirements and conditions needed for the realization of a each atom or molecule. From intensity measurements of such superradiant system are known to be satisfied in some regions spectral lines, important parameters (e.g., density and temper- of the ISM. More precisely, superradiance can arise in systems ature) can be determined and the physical conditions in a given where there is a population inversion, and the effect will be environment thus characterized (Townes & Schawlow 1955; ) much stronger and more likely to be realized when atoms or Emerson 1996; Goldsmith & Langer 1999; Irwin 2007 . For molecules are separated by approximately less than the example, in cases where the spectral lines are optically thin, the wavelength of radiation (see below and Section 3.2). intensity will be found to scale linearly with the number of The population inversion condition is known to occur in the atoms or molecules responsible for the detected radiation. The ISM and is partly responsible for the ubiquitous presence of soundness of this approach rests mostly on the assumption that masers (see Fish 2007; Watson 2009; Sarma 2012; Vlemmings spontaneous emission from different atoms or molecules 2012 for recent reviews). But it is also important to realize that, happens independently. although it is a necessary condition, population inversion is not As was pointed out by R. H. Dicke in a seminal paper by itself sufficient to ensure superradiance. It is also required ( ) several decades ago Dicke 1954 , the assumption of thatsufficient velocity coherence existsbetween the atoms independent spontaneous emission for the components of a partaking in the effect, and that any other dephasing takes place gas does not apply in all conditions. As will be discussed in this on timescales longer than those characterizing superradiance. ’ paper, and following Dicke s original analysis, closely packed When all these conditions are met, a coherent behavior can be atoms can interact with their common electromagnetic field and established between the atoms, and superradiance can ensue. radiate coherently. That is, the spontaneous emission of atoms We note, however, that, as will be discussed later on, or molecules in such a gas will not be independent, but rather superradiance is unlikely totake place in thermally relaxed take place in a cooperative manner. In the ideal case, this regions of the ISM. This is because Doppler broadening phenomenon will lead to a much more intense and focused resulting from, say, a Maxwellian velocity distribution would radiation (proportional to the square of the number of atoms), leave too few atoms with the required velocity coherence to which Dicke called superradiance. Since Dicke’s original allow superradiance to develop. Our analysis will therefore proposal, the field of superradiance research has flourished, and imply other types of environments where thermal equilibrium an abundant literature has developed within the physics has not been reached. For example, any region in the ISM into community. The first experimental detection of superradiance which a significant amount of energy is being suddenly in the laboratory was achieved by Skribanowitz et al. (1973), released (e.g., shocks or regions where significant radiation while several other independent verifications (Gross et al. 1976; flares occur) will be strongly out of equilibriumand provide Gibbs et al. 1977; Carlson et al. 1980; Moi et al. 1983; Greiner conditions that are potentially markedly different fromthose et al. 2000; Xia et al. 2012) have since been realized under a found in a thermal gas and may meet the requirements for large domain of conditions and experimental setups (see superradiance. Also, although superradiance can also occur for MacGillivray & Feld 1976; Chapter2 of Benedict et al. 1996; large interatomic or molecular separations (i.e., greater than the Andreev et al. 1980; Gross & Haroche 1982 for reviews). wavelength of radiation; see Section 3.1.2), the aforementioned
1 The Astrophysical Journal, 826:216 (16pp), 2016 August 1 Rajabi & Houde constraint of small interatomic or molecular separation, and its superradiance formalism for the ISM (in the present case for a implication for the corresponding densities, is likely to be met magnetic dipolar transition), which will then be refined in the for only a limited number of spectral lines, but a few future and also applied to other (electric dipolar) spectral lines astrophysically important transitions are suitable candidates. (e.g., the OH 1612 MHz, CH3OH 6.7 GHz, and H2O 22 GHz One of these spectral lines is the 21 cm atomic hydrogen maser transitions) where observational evidence for super- transition. radiance can be found in the literature (Rajabi & Houde 2016; Even though a 21 cm maser has yet to be discovered, which F. Rajabi & M. Houde 2016b, in preparation). would also imply the realization of a population inversion for It should also be pointed out that superradiance is a this spectral line, as will be seen through our analysis the fundamentally different phenomenon from the maser action, length-scales required for superradiance at 21 cm are very small even though the two may seem similar at first glance. An compared to those that would be needed for maser amplifica- astronomical maser is a collective but not coherent phenom- tion in the ISM (Storer & Sciama 1968, and see below).It enon. More precisely, for a maser, a group of atoms, initially in follows that, although the lack of observational evidence of their excited states, emit through the stimulated emission masers for this transition significantly affects the probability of process but cannot be considered as a single quantum system. detecting superradiance, it does not rule it out. Also, the That is, it is possible to describe maser action through existence of higher densities of atomic hydrogen in some parts successive events where an excited atom is stimulated by the of the ISM would increase the potential detectability of incident radiation and emits a photon, with the same / superradianceif the other necessary conditions for its realiza- stimulation emission processes subsequently repeated for tion previously listed were also met. Furthermore, with the different atoms in the masing sample. In contrast, for super- recent discoveries of radio bursts at frequencies close to radiance, coherence emphasizes the fact that the group of atoms fi 1400 MHz (Kida et al. 2008; Thornton et al. 2013), the interacting with the radiation eld behaves like a single ( ) investigation of the properties of a transient phenomenon such quantum system Nussenzveig 1973 . That is, the super- fi radiance emission process cannot be broken down into as superradiance is timely. This is why in this rst paper on the subject we chose to introduce the concept of superradiance to successive events, as is the case for maser radiation. Finally, the ISM using this spectral line. superradiance is a transient effect in which a strong directional Whether or not a population inversion can easily be realized pulse is radiated over a relatively short timescale, while maser for the energy levels leading to the 21 cm line, it has been action operates more in a steady-state regime as long as ( population inversion is maintained. considered in the existing literature Shklovskii 1967; Storer & Sciama 1968; Dykstra & Loeb 2007), and we know of at least The material covered in this paper goes as follows. We start one region (the Orion Veil) where the kinetic temperature is with a general discussion of the concept of superradiance for the so-called smalland largesamples, as originally discussed lower than the 21 cm spin temperature, providing evidence for by Dicke (1954, 1964), in Section 2. In Section 3, we examine a population inversion (Abel et al. 2006). The main pumping the possibility of building cooperative behavior in a H I sample process covered in the literature corresponds to the situation based on a comparative analysis of timescales for the 21 cm when a H I gas is close to a source of radiation that emits a field line in a H I gas, as well as present corresponding numerical with an intensity I n in the neighborhood of the Lyα line. A v () results. A discussion and short conclusion follow in Sections 4 fi (n = F = ) hydrogen atom in the ground hyper ne state 1, 0 can and 5, respectively, while the superradiance formalism and absorb a photon and become excited to the n=2 level. Later fi ( = = ) detailed derivations for the material discussed in the main on, the atom returns to the upper hyper ne state n 1, F 1 , sections of the paper will be found in appendices at the end. emitting a slightly less energetic photon than the initial one absorbed by the atom. The same can happen for a hydrogen 2. SUPERRADIANCE atom initially in the hyperfine state (n ==1, F 1) that returns to the ground (n = 1, F = 0) state after excitation to the n=2 2.1. Dicke’s Small-sample Model level, emitting a slightly more energetic photon in the process. Dicke originally proposed in 1954 a model where an The absorption rate of the photons for both cases depends on ensemble of N initially inverted two-level atoms interacting ( ) the intensity of the radiation Iv ()n , but the return emission with their common radiation field is considered as a single = process does not. Therefore, the F 0 level will undergo more quantum mechanical system (Dicke 1954). In his model, a two- absorptions followed by a return to the (n = 1, F = 1) level level atom is modeled as a spin-1/2 particle in a magnetic field whenever Iv ()n harbors more blue than red photons and will where the spin up configuration corresponds to the excited state become accordingly less populated than the F=1 level ∣eñ and the spin down to the ground state ∣gñ. Just as an (Wouthuysen 1952; Field 1958; Shklovskii 1967; Storer & ensemble of N spin-1/2 particles can be described using two ) ( ) Sciama 1968 . Although Storer & Sciama 1968 concluded quantum numbers s and ms, the eigenstates of the combined N that it is unlikely to maintain a population inversion over two-level atoms in Dicke’s model can also be labeled with two theextended region needed for the maser amplification with quantum numbers r and m such that 02rN and “ ” r this process, they also pointed out that an appreciable mrrrrr =-,1,,1, - + ¼ - , where inversion can thus be realized over a region of thickness ∼6× − 10 5 pc. Given the above inversion scenario, we would expect NNeg- mr = ,1() that environments located in the periphery or near boundaries 2 of H II regions could provide conditions suitable for the development of superradiance, for example. The aforemen- with Ne and Ng the number of particles in the excited and tioned evidence for a 21 cm population inversion in the Orion ground states, respectively. From the complete set of Veil brings support to this idea. Whatever the case, the 21 cm eigenstates characterizing this quantum mechanical system, line will serve us as a starting point for the development of the those symmetrical under the permutation of any pair of atoms
2 The Astrophysical Journal, 826:216 (16pp), 2016 August 1 Rajabi & Houde 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 aHI 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 smallsample of neutral atoms is confined to a volume l3 with the walls of the volume transparent to the Figure 1. Dicke states with r=N/2 for a system of N two-level (spin-1/2) radiation field. particles. Spontaneous radiation intensities are indicated on the right. 2. The N two-level atoms in the sample are separated by a distance much less than λ but distant enough not to worry are particularly important and are called Dicke states. The about any overlap between the wave functions of initial state ∣ee,,¼ñ , e of N fully inverted spin-1/2 particles neighboring atoms, which would require that the wave corresponding to N fully inverted two-level atoms is one such functions be symmetrized. 3. The ensemble of N initially inverted hydrogen atoms Dicke stateand is identified by r=N/2 and mr=N/2. When an atom in the ensemble decays to its ground state by emitting a possesses a permutation symmetry under the exchange of any pair of atoms in the sample. This is a restricting photon, the quantum number m is decreased by one while r r condition that could prove difficult to satisfy in general. remains unchanged, and the system moves to another 4. The transition between atomic levels takes place between symmetric state. Dicke showed that the radiation intensity nondegenerate levels, collisions between atoms do not from such an ensemble cascading from the initial affect their internal states, and collisional broadening is ( ) r ==NmN2, r 2 state down through an arbitrary state neglected as a result of the small size of the sample (r, mr) is (Dicke 1953). 5. Although it is mentioned in Dicke (1954) that the main IIrmrm=+()() -+12 () 0 rr results of his study are independent of the type of if the volume containing the ensemble of N two-level atoms is coupling between atoms and the field, the interaction of much smaller than l3, the cube of the wavelength of the the atoms with the radiation field in Dicke’s model is radiation interacting with the atoms. In Equation (2), I is the assumed to be electric dipolar. 0 fi radiation intensity due to spontaneous emission from a single 6. Finally, the radiation eld is assumed to be uniform through the smallsample, the electric dipoles associated two-level atom. This particular type of system and density withthe atoms are parallel, and propagation effects fi smallsample condition de nes a . This cascading process is neglected. depicted in Figure 1. Furthermore, Dicke pointed out that in the (r = N/2, mr = 0) Comparing a corresponding small sample of N neutral state, where the half of the atoms are in the ground state and the hydrogen atoms interacting with the 21 cm line in the ISM other half in the excited state, the radiation intensity of the with a Dicke sample, we can see that some of the assumptions system is maximum at made in the Dicke formalism hold and some do not. For example, the transitions between the hyperfine states of a ⎛NN⎞⎛ ⎞ ⎜⎟⎜ ⎟ hydrogen atom take place between nondegenerate levels since II=+0 13() ⎝ 22⎠⎝ ⎠ the external magnetic field in the ISM lifts the upper-level ( ) 2 degeneracy see Section 3 . Also, a small sample of H I atoms µNI0,4() found in many regions in the ISM would readily verify the 3 implying a significantly enhanced radiation beam, a phenom- criterion that N 1 in a volume < l and could thus be enon he named superradiance. This can be understood by the approximately assumed to experience the same 21 cm radiation fi fact that when the distance between neighboring atoms is much eld without consideration of propagation effects. On the other hand, unlike in Dicke’s sample, collisional and Doppler smaller than the wavelength of radiation, the photon emitted by broadening effects should, in the most general case, be one atom is seen to be in phase by neighboring atoms and can considered because, for example, collisions between hydrogen bring about the emission of a new photon of the same mode atoms affect the internal hyperfine states in their electronic and in the same direction as the initial photon. This process can ground state through spin de-excitation (Field 1958). Most continue through the whole ensemble, resulting in an intense importantly, it must also be noted that the type of coupling superradiant radiation pulse proportional to N2 (see between hydrogen atoms and the 21 cm line is magnetic dipolar Equation (4)). In contrast to thesuperradiance observed in a in nature.
3 The Astrophysical Journal, 826:216 (16pp), 2016 August 1 Rajabi & Houde Above all, the permutation symmetry of atoms, which is a key assumption in the Dicke model, is difficult to preservein an actual situation because of dipole–dipole interactions between the atoms. Dipole–dipole interactions have a r¢-3 dependency, and these short-range interactions become impor- tant in smallsamples where the distance between atoms r¢ is smaller than λ (see Section 3.1). In the Dicke model, the symmetry-breaking effect of dipole–dipole interactions is ( Figure 2. Energy level diagram for the H I 21 cm line in the presence of a ignored. In later studies of superradiance e.g., Gross & Zeeman-splitting external magnetic field. 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 radiation intensity that is greater than that of the corresponding environments. This symmetry-breaking effect results in fully noncoherent system but smaller than the superradiance weakened correlations and a subsequent deviation from a intensity of a perfectly coherent system consisting of N atoms. perfectly symmetrical superradiance behavior (i.e., the I ∝ N2 Finally, in a largesample as a result of possibly large relation in Equation (4)). In a sample of N atoms, if s atoms interatomic distances (i.e., r¢>l), the symmetry-breaking (s < N) experience a similar closeneighborhood, the correla- effects of the dipole–dipole interactions are less important, tion can build up among this group of atoms, and the intensity whereasthe propagation effects that are absent in a small of radiation from the whole sample is expected to be larger than sample cannot be neglected. The propagation of radiation over the intensity of a fully noncoherent system (Inc) but smaller a large distance in a largesample results in the reabsorption than that of a perfect superradiance system (ISR). and reemission of the photons and consequently leads to a In a smallsample of N neutral hydrogen atoms in the ISM, it nonuniform evolution of the atoms in the sample (see may thus appear possible to develop coherent behaviors if the Section 3.2). Beyond these factors, Dicke’s analysis of the permutation symmetry is conserved among a group of atoms in largesample includesassumptions similar to those used for the the sample. This is arguably a reasonable assumption on small sample. average for an ensemble of atoms within the small volumes discussed here. That is, the different atoms in the sample are 3. THE TWO-LEVEL H ISAMPLE likely to be subjected to the same conditions when averaged over time and space. Furthermore, we also note that in a Let us consider an ensemble of neutral hydrogen atoms in H Isample the magnetic dipole–dipole interactions are defi- the electronic ground state in some region of the ISM, where it nitely weaker than the electric dipole–dipole interactions can emit or absorb photons at the λ=21 cm wavelength. The discussed in the literature focusing on symmetry-breaking hydrogen 21 cm line is perhaps the most important source of effects. information in radio astronomy and arises from the transition between two levels of the hydrogen atom in the 1s ground state. 2.2. Dicke’s Large-sample Model The interaction between the electron spin and the proton spin in the nucleus of the atom splits the otherwise degenerate 1s In his first paper on superradiance, Dicke also extended his = = energy level into the two F 0 and F 1 sublevels. The formalism to a large sample, where the volume of the sample F =«10transition in the absence of an external magnetic 3 ¢ > l and the interatomic distance r between some atoms can field produces the 21 cm line, corresponding to a frequency be greater than λ. He showed that, in a largesample, coherent ν= k 1420.406 MHz. radiation can occur in a particular direction in which the Considering a more realistic case, the magnetic field in a radiation from different atoms isin phase. When the phase- μ fi k cold neutral gas is generally on the order of 10 G matching condition is satis ed in some direction , the initial (Crutcher 2012), and the energy level corresponding to state of the system can be described by a correlated symmetric F=1 splits into three sublevels identified by mF=−1, 0, state of type ()rm, r , and the intensity of the radiation in a solid = = k and 1. The interaction between the F 0 and F 1 levels angle along follows becomes more complicated as this splitting provides three fi I(kk )=+-+ I0 ( )[( rmrmrr )(1, )] ( 5 ) possible hyper ne transitions, as shown in Figure 2. These hyperfine transitions link states of like parity and obey the similar to Equation (2) for a small sample. When a photon is general magnetic-dipole selection rules ΔF=0,±1 and emitted in the direction k, the system cascades to a lower state, Δm=0,±1. Based on these rules, all of the three transitions obeying the selection rules Drm=D=-0, r 1, and similar to shown in Figure 2 are allowed;however, depending on the the case of a small sample, symmetrical states of the same r are relative orientation (or the polarization) of the magnetic coupled to each other through coherent transitions (see component of the radiation field to the quantization axis of Section 3.1.2). On the other hand, when a radiated photon the atom, some transitions may be favored. In the more general has a wave vector kk¢¹ , the states with different r (i.e., of case, there is a mixture of all three transitions, with each transition exhibiting particular polarization properties. In order different symmetry) can couple, and consequently the coher- ( ) to better understand the coherent and cooperative evolution of a ence is weakened in the system Dicke 1954 . It follows that in fi sample of N hydrogen atoms coupled to its radiation eld, it a large sample consisting of N inverted atoms, the radiation by will be simpler to focus our analysis on only one of these one atom is only seen to be in phase by a group of atoms transitions and consider the atomic system as an ensemble of (contrary to a small sample, where the radiation field is two-level atoms. Although this model represents a significant assumed uniform over the whole sample), and correlation can simplification, the two-level atom approximation is extensively only be developed among this group. This naturally results in a used for, and its results are wellverified in, laboratory
4 The Astrophysical Journal, 826:216 (16pp), 2016 August 1 Rajabi & Houde (inMKS units) m ke32∣∣áñMˆ ∣∣ g G= 0 .6() 3p We can furthermore express the state of the system by either a symmetric ∣+ñ or antisymmetric ∣-ñ combination of the ∣eg1 2ñ and ∣ge1 2ñ state vectors, such as 1 ∣(∣∣)()+ñ =eg1 21 ñ + ge2 ñ 7 2 1 ∣(∣∣)()-ñ =eg1 21 ñ - ge2 ñ ,8 2 which, at this stage of our analysis, have the same energy and are thus degenerate (see below). kr¢ Figure 3. Two-hydrogen-atom system. When 1, the upper and lower We now refine this model by adding the magnetic dipole– symmetric states ∣ee12ñ and ∣gg12ñ, respectively, couple to the intermediate symmetric state ∣+ñ at the enhanced transition rate 2Γ, where Γ is the transition dipole interaction term to the system’s Hamiltonian. In this rate of a single atom acting independently. In contrast, the antisymmetric state model, the magnetic dipole from one atom, say, Mˆ1, interacts ∣-ñ cannot couple to the upper and lower states because of the cooperative with the magnetic field Bˆ2 due to the dipole of the other atom behavior between the two atoms. The energy level shifts DE for the ∣ñ fi states are also shown. located at a position r¢ = r¢er¢ away in the near eld, where kr¢ 1 (Jackson 1999): experiments involving more complicated atomic or molecular ⎡ ¢ ¢ ⎤ ˆ m0 cos()kr sin ()kr ( ) Br2 ()¢ = ⎢ + ⎥ systems with more complex energy levels Mandel 2010 . 4p ⎣ r¢32r¢ ⎦ ˆ ˆ ´-[(·3.9eerr¢¢ M22) M]() – 3.1. Magnetic Dipole Dipole Interaction It can be shown that when the two dipoles are aligned, the between Hydrogen Atoms term of the interaction Hamiltonian that is relevant to the The theoretical model for the problem canbe found present discussion is in Appendix A, where the Hamiltonian for the two-level mmk 3 2 ⎡ ¢ ¢ ⎤ 0 B 2 cos()kr sin ()kr H I sample is developed and the main equations of super- Hˆdd =-(∣∣31b - )⎢ + ⎥ radiance arederived. To simplify our discussion, we have 2p ⎣ ()kr¢ 32()kr¢ ⎦ limited our analysis to the ∣Fm==ñ0, 0 ⟷ +- -+ ´+(RRˆˆ RR ˆˆ)(),10 ∣Fm==+ñ1, 1 transition through which a hydrogen atom 12 12 ( ) + - emits a left circular polarization LCP photon, with its electric with b = ee· ¢ . The raising/lowering operators Rˆ , Rˆ , and fi L r 1 1 eld vector rotating counterclockwise as seen by the observer so onare defined in Equation (47) and the LCP unit vector state facing the incoming wave. One of the main components of the eL in Equation (54), while μB is the Bohr magneton. It can Hamiltonian is the magnetic dipole energy term VˆMD that further be shown, through a simple diagonalization exercise, describes the interaction between the atoms composing the that this interaction Hamiltonian lifts the degeneracy between sample (see Equations (50) and (62)). We now focus on this the ∣+ñ and ∣-ñ states of Equations (7) and (8), with their interaction to get a sense of how the needed cooperative ( ) behavior for superradiance develops between atoms. corresponding energies becoming Protsenko 2006 EE =D0 E,11() with E the unperturbed energy of the states and 3.1.1. Hydrogen Atoms Separated by a Small 0 2 ⎡ ⎤ Interatomic Distance ()r¢ 5 The Astrophysical Journal, 826:216 (16pp), 2016 August 1 Rajabi & Houde ∣-ñ. Such enhanced and reduced rates are respectively associated with superradiance and subradiance. This scenario for the two-atom system is depicted in Figure 3. 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 VˆMD) is totally symmetric under the permutation of the two aligned atoms when kr¢ 1 implies that only states of like symmetry can be coupled. It follows that since the initial state ∣ee12ñ of the fully inverted system is also symmetric, it can only couple to the ∣+ñ intermediate state, and from there to the symmetric ground state of the system ∣gg12ñ. Accordingly, it is interesting to note that under these conditions a system prepared in the intermediate antisymmetric state ∣-ñ will not decay to the ground state since Γ−=0. This is evidently different from the case of a noncoherent system whereboth atoms eventually decay to their individual ground states ∣gñ at the rate Γ. We therefore see that superradiance and subradiance are characteristics of a Figure 4. The ideal H I small-sample superradiant system. The radiation intensity is plotted as a function of time t, for N=75 atoms confined within a coherent system, where the intensity of radiation does not scale 13 cube of L=4 cm. After the delay time tD=2.0×10 s the system radiates linearly with the number of atoms, as is the case for a coherently in a single burst of radiation. noncoherent system. When the effect discussed here is generalized to a sample It should also be noted that the correlation between dipoles composed of N atoms confined within a volume l3 (a can be triggered by an external source, such as an input small sample),wefind that some of the conditions that radiation field. This can happen if the input radiation field is prevailed for the two-atom case are not realized. Most stronger than the spontaneous fluctuations in the sampleand importantly, Equations (11) and (12) indicate that this the coupling of the dipoles to the external field leads to interaction leads to a distribution of energy levels in the coherent behaviors. An enhancement of radiation through system unless the atoms all have similar nearest neighborhoods coupling to an external field is called triggered superradiance (e.g., a ring-like periodic distribution of atoms; Gross & ( ) ) Benedict et al. 1996 . Haroche 1982 . This spread in energy levels will tend to reduce It can be shown that the superradiance radiation intensity I the strength of the superradiance effect. SR of an ideal H I small sample composed of N inverted atoms is It has nonetheless been observed through numerical given by (Dicke 1954; Gross & Haroche 1982; Benedict calculations and experiments that coherent behaviors still apply et al. 1996) to N-atom small-sample systems where radiation is of long 22- enough wavelength (Gross et al. 1979), as is the case for the INSR =Gw cosh[( Ntt G-D )] , () 15 21 cm line. For kr¢ ~ 1 the ratio DEE~G w is exceedingly 0 where w is the energy of the corresponding atomic transition small for the 21 cm line, and the timescale associated with the -1 energy shifts is on the order of D~Ekr()¢ 3 G-1 (Benedict and the aforementioned delay time tND =G()ln () N.In et al. 1996), which for the H I densities considered in this paper Figure 4 the radiation intensity of a H I small sample with renders this type of dephasing negligible. As will be discussed N=75 atoms confined within a cube of length 4 cm (l 5 for later, dephasing due to collisions are more likely to set the the 21 cm line) is plotted as a function of time using timescale for homogeneous dephasing. The same is not Equation (15). The intensity is normalized to NInc, where necessarily true at short wavelengths, where it is very difficult Inc =GNw is for the corresponding noncoherent small to place a large number of atoms within a subwavelength sample. It can be seen in Figure 4 that the energy stored in dimension in a regular pattern, and in such a sample strong the small sample is radiated away in a single burst. After time dipole–dipole interactions break the symmetry and terminate t=tD, the intensity reaches its maximum value, N times that of the coherent behavior by introducing large energy-level shifts. the noncoherent intensity, and the peak intensity of the Thus most of the experimental observations of superradiance ( normalized plot becomes equal to one. In this H I sample, took place at longer wavelengths i.e., in the infrared as Γ−1= × 14 ( ) ) 3.5 10 s Draine 2011 , the delay time opposed to optical; Benedict et al. 1996 . 13 tD =´2.0 10 s, and the characteristic time of superradiance For an inverted N-atom small sample with initially -112 uncorrelated dipoles, the first photon emitted by one of the is TNR =G=() 4.6 ´ 10 s. It should also be pointed out atoms interacts with the dipole moments of the other atoms, that in such a sample the correlation between dipoles is initiated resulting in the build-up of correlation between them. After by internal spontaneous fluctuations, and it is assumed that we – some time, known as the delay time tD, a very high degree of are dealing with an ideal system, where the dipole dipole correlation is developed in the system, where, in the strongest symmetry-breaking effects are negligible and there are no other superradiance regime, the N microscopic dipoles eventually act relaxation mechanisms (i.e., cooperative emission is the only like one macroscopic dipole. The rate of emission is then decay mechanism). enhanced to NΓ, while the radiation intensity is proportional to In a real system, there are some relaxation and dephasing N2 and becomes highly directional, being focused in a beam effects that compete with the build-up of the correlation, and in with a temporal half-width on the order of 1 ()NG . order to subsequently have superradiance, its 6 The Astrophysical Journal, 826:216 (16pp), 2016 August 1 Rajabi & Houde Equations (16)–(18) 2 1 GµS cos [()]()fq -kr¢¢cos 19 {}2 2 1 GµA sin [()]()fq -kr¢¢cos . 20 {}2 We therefore see that, although the first photon can be emitted in any direction θ′, its direction of emission determines f and the intermediate state of the system since the transition probabilities peak at f - kr¢¢cos()qp= 2 m for ΓS and f - kr¢¢cos ()qp= n for ΓA (m and n ¹ 0 are integers). Going through the same exercise for the ∣gg12ñ state shows a similar dependency on θ′ and f as in Equations (19) and (20), ′>λ Figure 5. Two-hydrogen-atom system with r . The upper and lower which implies that these transitionrates will also be likely to symmetric states, ∣eeñ and ggñ, respectively, couple to the intermediate 12 ∣ 12 f symmetric ∣Sñ and antisymmetric ∣Añ states with the corresponding transition peak at the same value of . It follows that there is an angular rates ΓS and ΓA. The direction of emission of the first photon determines the correlation between two successive photons, where the intermediate state of the system, and the direction of the second photon is direction of the second photon is correlated with the direction correlated with that of the first. 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 characteristictimescale TR and delay time tD must be shorter than (in some exceptional cases on the order of) the relaxation/ intense radiation along elongated geometries (e.g., pencil-like dephasing timescales (Gross & Haroche 1982; Benedict or cylindrical structures; Dicke 1964). et al. 1996). The nonideal case will be discussed in Depending on the intermediate state of the system, two Section 3.2.1. different classes of transitions are possible:coherent and noncoherent. If the emission of the first photon leaves the 3.1.2. Two Hydrogen Atoms Separated by a system in a symmetric intermediate state (e.g., ∣Sñ), the fi Larger Interatomic Distance ()r¢>l symmetric coupling to the radiation eld results in the coherent behavior, and consequently the system decays to the symmetric Let us still assume that the atoms are prepared initially in ground state ∣gg12ñ with the corresponding transition rate shown their excited states, with the state of the two-atom system given in Figure 5. In contrast, if the system is in the antisymmetric fi by ∣ee12ñ. Similar to the subwavelength case, a rst photon is intermediate state ∣Añ, the coupling to the radiation field will be radiated, leaving the system in an intermediate state, which, antisymmetric under the exchange of the atoms as they interact unlike for the subwavelength case, will be described with any with the nonuniform electromagnetic field, and the system combination of ∣eg1 2ñ and ∣ge1 2ñ states with each having equal decays noncoherently to the ground state ∣ggñ with the decay probability contributions, that is, not only by the ∣+ñ and ∣-ñ 12 rate ΓA. states. More precisely, if we associate the general symmetric We can also explain this classically by considering two state classical radiators separated by a distance r′>λ. Over large distances, the phase and the polarization of the radiation field 1 if ∣(∣∣)()Segegeñ=1 21 ñ+2 ñ 16 emitted by each radiator varies from place to place. When the 2 radiation from the two identical radiators interferes, the with the intermediate state shown onthe left side ofFigure 5, intensity of the total field is given by then we should assign its orthogonal antisymmetric state 2 Itotµá()BB 1 + 2 ñ ()21 1 if ∣(∣∣)()Aegegeñ=1 21 ñ-2 ñ 17 2 2 2 µáBB1 ñ+á2 ñ+2, áBB12·() ñ 22 to the intermediate state on the right side of the figure (Dicke 1964). In Equations (16) and (17), f is a phase term and can become as large as four times the intensity of a single discriminating between the multiple choices for the intermedi- radiator I0 if ate states. To get a better understanding of the transition áñ~BB·()I .23 probabilities for these states, it is useful to refer to Equation (62) 12 0 fi for the magnetic dipole interaction term with a radiation eld If the phase of the radiation from different radiators does not for two atoms separated by r′ and for a given k. We then find match perfectly, the term containing the correlation áBB12· ñ in that for coupling to, say, the ∣eeñ state the following term 12 Equation (22) becomes smaller than I0, and consequently the comes into play: total intensity Itot decreases until it reaches its minimum for ++ikr¢¢cos ()q completely out-of-phase radiators. Furthermore, the correlation VRReˆˆˆMD µ+ ,18() 12 term can vanish when radiators act independently. In this case, where θ′ is the angle between k and r′. Given that the transition the total intensity becomes equal to the sum of the intensities of ( ) ˆ 2 probability and rates are proportional to ∣áñee12∣ V MD∣∣ S and the two independent radiators (the so-called noncoherent 2 ∣áñee12∣ Vˆ MD∣∣ A (Grynberg et al. 2010), we calculate using system). 7 The Astrophysical Journal, 826:216 (16pp), 2016 August 1 Rajabi & Houde radiating intensely with ISR = NfInc, where Nf is the enhance- ment factor of the superradiant intensity ISR over the noncoherent intensity Inc 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 ISR= NI 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 largesample, 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 & Har- oche 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 Figure 6. The ideal cylindrical H I largesample. The radiation intensity, scaled fl uctuations of dipoles in the initial stages of the evolution can to NInc, is plotted vs. the retarded time t =-tLcnormalized to the be added to the formalism. In contrast, triggered superradiance superradiance characteristic timescale TR. The length and radius of the cylinder 1 2 can be fully explained in this manner becausethe correlation are, respectively, L = 0.02 cTR and wcT= 0.036()R . process is initiated by an external field, which can be defined classically. It must be pointed out that the results of this method 3.2. The N-atom Large Sample () > l3 are valid only if the propagation time of the radiation tE through a sample of length L (i.e., τE = L/c) is smaller than the We can extend our discussion for the case of two distant superradiance characteristic time TR given by atoms (i.e., r¢ > l) to a largesample consisting of N atoms 3 16p distributed over a volume > l . As stated above, the build- TRsp= t ,24() up of correlations in an extended N-atom sample can be 3nLl2 understood as a constructive interference of the radiation by where τ =1/Γ is the spontaneous decay time of a single sp different atoms. In a large sample as a result of propagation n ( ( λ) atom and the density of inverted atoms see MacGillivray & over large distances i.e., larger than , the phase of the Feld 1976; Rosenberger & DeTemple 1981, and Appendix B). ( ¢ ik· r¢ ) radiation varies throughout the sample kr 1 and e 1 . This condition (i.e., t < T ) is known as theArecchi– Consequently, the phase of the atomic magnetization differs ER Courtens condition, and it ensures that the atomic magnetiza- with position. In an inverted largesample, the radiations from different atoms interfere with each other, and when the tion in different parts of the sample can lock into a common magnetizations of the radiators are perfectly in phase, an phase and coherent behavior can develop through the sample. intense propagating wave is produced in one direction (the In Appendix A, we derive the evolution equations for the fi phase-matching condition cannot be satisfied in all directions). radiation eld and the atomic system using the Heisenberg In order to better understand the phase-matching process, it representation, while in Appendix B we solve the corresp- – is useful to go back to the angular correlation effect described onding Maxwell Bloch system of equations, at resonance, N within the framework of the slowly varying envelope in the two-atom case. In a large sample of inverted atoms, ( ) when the first photon is emitted, other atoms interact with its approximation SVEA . To do so, we adopted the following fi radiation field, and the direction of the next photon is affected form for the radiation magnetic eld and atomic magnetization: fi by the rst one. In a more general sense, when a photon is BtBteˆ ()rr,,= ˆ ()-ikz()w t () 25 radiated in a particular direction k, it becomes more probable to L 0 observe the second photon in the same direction k than any ˆ ˆ ()rr,,tteikzt=-0 () (w ) ,26 () other direction. Thus, as the atoms radiate, an angular ˆ ˆ correlation builds up in the sample that triggers the phase- with B0 and 0 corresponding to slowly varying envelope matching process in a well-defined direction. operators. The superradiance of a cylindrical largesample of Ideal superradiance is the result of the symmetrical evolution length L under ideal conditions is then found to be determined –fi of an atom eld system, and in a large sample, the propagation by the following equations for, respectively, the magnetization, effects result in the nonuniform evolution of the atoms in the the population inversion, and the magnetic field: sample. In order to better understand propagation effects in a largesample, the atomic medium can be divided into small ˆ + mBN 0 = sin()q ( 27 ) identical slices with dimensions larger than λ but much smaller 22V than the length of the sample. A microscopic dipole is then N associated witheach slice with its magnitude being propor- ˆ = cos()q ( 28 ) tional to the number of excited atoms in the corresponding V slice. At the beginning, the dipoles in different slices are ˆ + imB ¶q B0 = ,29() independent and their radiation uncorrelated. After some time 22g ¶t (or the so-called retarded time delay τ ; see Equation (34)),as D 2 θ they interact with their common radiation field, the dipoles lock where g = mB 2 . The solution for the Bloch angle as a in to a common phaseand act as a single macroscopic dipole function of the retarded time t =-tLcis obtained through 8 The Astrophysical Journal, 826:216 (16pp), 2016 August 1 Rajabi & Houde the so-called Sine-Gordon equation: initial conditions and is given by (Benedict et al. 1996) 2 ⎛ ⎞ 2 d qq1 d TR0⎜⎟q +=sin()q ( 30 ) tD ln .34() dq2 q dq 4 ⎝2p ⎠ fi with In Figure 6 the rst burst of radiation occurs after tDR 160 T , which is consistent with the value one finds using zt Equation (34). As can also be seen, this first intensity burst q = 2.() 31 LTR only carries out a fraction of the total energy stored in the sample, while the remaining radiation happens through In Figure 6 we show the solution for the radiation intensity subsequent bursts. The number of burst events depends on of such an ideal, cylindrical largesample of H I atoms of length the length of the sample, and as the length is increased radiation L = 0.02 cTR, where c is the speed of the light, by numerically emanates through a larger number of bursts, while the peak ( )–( ) solving Equations 27 31 . For these calculations we set a intensity of consecutive burst events gradually drops. This is a Fresnel number of unity to reduce the impact of diffraction consequence of energy conservation and the fact that, in a losses, which are not taken into account in our model. This larger (i.e., longer) sample, radiations from different groups of yields a cylinder of radius atoms along the sample arrive at the end-fire at different times, lL and the process of absorbing the radiation, developing w = ,32() correlations between the dipoles, and eventually reemitting p the radiation repeats multiple times over a very long period of 12 time. On the other hand, when the length of the sample is which for our ideal sample results in w = 0.036 ()cTR . In this sample, the Arecchi–Courtens condition is satisfied decreased, the ringing effect becomes weaker until, for a small λ (i.e., tER T ), allowing the use of the homogeneous condition sample of dimension of order of , it totally washes out and we −12 θ0=4.9×10 rad for the initial value of the Bloch angle. only observe a single burst of radiation carrying away all the More precisely, for the largesample used for the figure, we energy stored in the system (as in Figure 4). Finally, we note assumed that internal fluctuations dominate over triggered that although the maximum radiation intensity seen in Figure 6 superradiance, and the initial Bloch angle was set with seems to imply that f ~ 0.001 1, the large number of atoms q = 2 N (Gross & Haroche 1982). 0 present in the sample ensures that ISR I nc (see Section 4).We In Figure 6, the retarded-time axis is scaled to TR and the should also note, however, that the Sine-Gordon equation is radiation intensity to NI , that is, the number of inverted atoms nc very sensitive to initial conditions. It therefore follows that the times the corresponding noncoherent intensity that would exact shape of the intensity curve, for example, the number of otherwise be expected from such a sample. More precisely, for θ comparison purposes we consider the noncoherent intensity bursts in Figure 6, is also strongly dependent on 0. emanating through the sample’send-fire (i.e., the end facing the observer) of area A, within the superradiance radiation 3.2.1. Nonideal Case: Dephasing Effects f =λ2/ ( k beam solid angle D A in the direction along which As was mentioned earlier, the characteristictimescale of fi ) the phase-locking condition is satis ed normalized to the solid superradiance TR and the delay time τD (for a largesample) angle associated withthe total noncoherent radiation. We thus must be shorter than the relaxation/dephasing timescales to have allow the build-up of correlations in a nonideal sample. These effects include natural broadening due to the spontaneous ⎛ ⎞⎛ ⎞ τ 1 fD decay timescale sp of a single atom and collisional broadening INnc = w ⎜ ⎟⎜ ⎟ τ ⎝ At ⎠⎝ 4p ⎠ related to the mean time between collisions coll for an atom in sp the sample. Although, as was stated in Section 1, our analysis is 4 w = ,33() aimed at regions of the ISM where thermal equilibrium has not 3 ATR been reached and where consequently the assignation of a temperature to determine, for example, collision rates is where Equation (24) was used for the last step, and Nw is the perhaps ill-defined, we will nonetheless adopt such a procedure total energy initially stored in the sample. As shown in the for the rest of our discussion to get a sense of the timescales figure, this energy is radiated away through multiple bursts, a involved. Accordingly, in a H I gas different types of collisions can take place depending on the temperature and density. For phenomenon known as theringing effect. This effect can be environments of temperatures ranging from approximately 10 explained by the fact that atoms in different locations in the to 300 K, which are the focus of our analysis, collisions sample radiate at different times. In other words, an atom at between two neutral hydrogen atoms (H–H collisions) = τ= location z z0, prepared in the excited state at 0, radiates dominate and fall into two categories: elastic and inelastic. its energy away and decays to its ground state, then later on During an elastic H–H collision, the spacing between the absorbs the energy radiated by another atom at a location atomic energy levels isslightly affected, but no transition z<z0 and becomes excited, leading to another radiation event. between them is induced. The change in energy spacing occurs In a largesample, just as in a small sample, internal field as a result of short-range interaction forces between the two fluctuations or an external field trigger superradiance, and after colliding particles and induces a phase shift in the wave the delay time tD the atoms radiate coherently. But contrary to function of the scattered atoms. After a number of elastic a small sample, the large-sample delay time depends on the collisions, an atom can lose coherence with the interacting 9 The Astrophysical Journal, 826:216 (16pp), 2016 August 1 Rajabi & Houde radiation field as a result of the randomness in the perturba- tions. 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 isestimated to be on the order of 8 9 Figure 7. The nonideal H I cylindrical largesample. All parameters are as in 10 s in the case of elastic scattering and 10 s for spin de- / excitations, using the mean effective collisional cross sections Figure 6, except that dephasing relaxation effects are included for the special case where they are characterized by a single timescale TT¢ = 541 R. The given in Irwin (2007) for a H I gas at T = 100 K and n=10 ringing effect is weakened as a result of the dephasing. −3 cm . The mean time between collision tcoll is thusset to the shortest of these timescales and must at least be larger than TR ( ) N -t T ¢ and tD to allow coherent behavior see Section 4 . ˆ = cos()q e , ( 36 ) In addition, other broadening effects, such as Doppler V broadening, are further dephasing mechanisms that can destroy and the dimensionless parameter cooperative behavior if their timescales (importantly the so- called Doppler dephasing time, i.e., the reciprocal of the zt¢ ) τ ( q = 237() Doppler width are smaller than TR and D Bonifacio & LT Lugiato 1975; Meziane et al. 2002). In a thermally relaxed gas, R thermal motions are probably the most important dephasing with effects and result in line broadenings that correspond to very ¢ −3 t¢ = Te¢()1.- -t T () 38 short dephasing timescales (e.g., Ttherm ∼ 10 satT = 100 K). In the presence of such strong dephasing effects, correlations The magnetic field is once again given by Equation (29). The cannot develop between the dipoles, and any coherent intensity and time axes are scaled likethose inFigure 6 for the interaction will be terminated right from the start. Hence our ideal H I sample. We can see from Figure 7 that the ringing earlier comment that we do not expect to find superradiance under conditions of thermal equilibrium, but potentially only in effect seen in the ideal sample is also present here but is (out-of-equilibrium) regions where strong velocity coherence weakened by the dephasing and basically terminated after ( ) can be maintained along the line of sight. Furthermore, this t ~ 1000 TR i.e., approximately the dephasing timescale . The condition may be only met among a group of atoms in such dephasing effects also affect the maximum energy radiated regions, therefore reducing the number of inverted atoms in the away through each burst eventand result in slightly weaker sample that could participate in coherent interactions. However, intensities. we know from maser observations that a high level of velocity coherence can be achieved in some regions of the ISM, and we 4. DISCUSSION: COOPERATIVE BEHAVIOR IN A H I expect that superradiance could happen under similar condi- GAS IN THE ISM tions. As was mentioned in Section 1, the main inversion As is evident from our previous discussions, the character- pumping mechanism likely involved for the 21 cm transition istic timescale of superradiance TR is a fundamental criterion to points to the surroundings of H II regions as potentialsites for consider in the investigation of this cooperative behavior. For superradiance in this spectral line. It follows that we should 3 an ideal small sample of volume <~l3410 cm , for total also anticipate ananalogous (very small) volume filling factor -3 −3 hydrogen densities 1 cm < 10 The Astrophysical Journal, 826:216 (16pp), 2016 August 1 Rajabi & Houde is collisional de-excitation, the rate for which is given by intensity of a smallor a large sample isonly affected slightly as long as TPD< t . The main effect of the finite pumping time is 1 then an increase in the delay time of the superradiant process in tcoll = .41() ( ) nH10k comparison to what one expects from Equation 34 : the actual τ delay time t¢D will be longer than the theoretical delay time D Values for the collisional de-excitation rate coefficient κ10 (MacGillivray & Feld 1981). for the hyperfine state F=1 over a range of temperatures can However, if pumping occurs at an approximately constant be found in Zygelman (2005), and for 10 K< 11 The Astrophysical Journal, 826:216 (16pp), 2016 August 1 Rajabi & Houde η=0.01, implying an inverted population of 940 atoms in a publications to tackle these questions and investigate super- volume λ3. Our results indicate that radiation bursts over radiance in other important astronomical spectral lines (e.g., the timescales on the order of days (i.e., from Figure 7 a few OH 1612 MHz, CH3OH 6.7 GHz, and H2O 22 GHz maser 3 4 hundred times TR = 10 s, while átDñ=5.2 ´ 10 s) can be transitions) where observational evidence for superradiance can associated withcylindrical H I samples of length and radius be found in the literature (Rajabi & Houde 2016; F. Rajabi & L≈1011 cm and w≈9×105 cm, respectively, while bursts M. Houde 2016b, in preparation). It would further be beneficial over timescales on the order of minutes (i.e., TR = 1 s and to broaden the scope of our analysis to include a wider range of 14 átDñ=66 s) can be associated with samples withL≈10 cm conditions for such effects as pumping, diffraction losses, and (approximately equal to 6×10−5 pc; see Storer & Sciama different sample geometries associated withdifferent Fresnel 1968) and w≈3×107 cm. In all cases, we have numbers. T¢ »>áñttcoll D over a wide range of conditions, ensuring that dephasing effects will not destroy atomic coherent We thank M. Harwit for bringing this research topic to our behaviors, and we found f≈10−4 (from Figure 7) with an attentionand J. Zmuidzinas for helpful discussions. We are efficiency factor Nf ranging from approximately 1012 to 1024, also grateful to T. Troland for alerting us to the evidence for a from the shortest to the longest sample length L. These results 21 cm population inversion in the Orion Veil. MHʼs research is imply a corresponding amplification factor of 1010–1022 over funded through the NSERC Discovery Grant and the Western the corresponding noncoherent intensity of such samples Strategic Support. (taking into account the noninverted population). Although the samples considered above would probably not APPENDIX A yield strong detections (e.g., for the sample of length THEORETICAL MODEL L≈1014 cm and w ≈ 9×107 cm, we calculate an integrated − − − A.1. The Hamiltonian and the Maxwell–Bloch Equations flux ∼10 22 erg s 1 cm 2 at a distance of 400 pc), given the small radii considered here it is unlikely that only a single We follow Dicke (1954) and approximate the Hamiltonian superradiant system would be realized in a region harboring an for a sample of N hydrogen atoms with each atom acting as a inverted population. That is, if we assume a reasonable maser two-level system, while taking into account the magnetic nature spot size for the population-inverted region (e.g., wspot ∼ 1au), of the dipole-radiation interaction applicable to this case: then it becomes possible that a very large number of ⎛ ⎞ N 1ˆ N superradiant systems could simultaneously erupt HHˆˆ=+ H ˆ + w ⎜ R ˆ +⎟ -MBrˆ · ˆ (). (w w ~ 106) and render a strong detection more like- 0rad ååjj⎝ 3 ⎠ jj spot j==1 2 j 1 lywhen the conditions for superradiance are met (Rajabi & ()42 Houde 2016). This leads us to suggest that, despite the simplicity of and the approximations used in our model, In this Hamiltonian equation, Hˆ0 contains the translational and significant intensity variability due to superradiance could be interatomic interaction energies of the atoms, Hˆrad is the detectable for the 21 cm line in some regions of the ISM. ˆ radiation field Hamiltonian term, wjj(Rˆ 3 + 12) is the internal energy of the jth two-level atom (1ˆ is the unit operator), which 5. CONCLUSIONS has the eigenvalues 0 and wj, and the last term stands for the We have applied the concept of superradiance introduced by interaction between the electromagnetic field and the magnetic Dicke (1954) to astrophysics by extending the corresponding dipole of the jth atom Mˆ j. Since this Hamiltonian is written analysis to the magnetic dipole interaction characterizing the under the magnetic dipole approximation, it implies that the atomic hydrogen 21 cm line. Although it is unlikely that magnetic field Bˆ does not change considerably over the size of superradiance could take place in thermally relaxed regions and that the lack of observational evidence of masers for this the atom and is determined by its value at the position of the r transition reduces the probability of detecting superradiance, center of mass of the atom, j. Finally, the effects of the in situations where the necessary conditions are met (close hyperfine interaction between the proton and electron spins atomic spacing, high velocity coherence, population inversion, within a single hydrogen atom and the Zeeman interaction due and long dephasing timescales compared tothose related to to an external magnetic field would be included in the coherent behavior), our results suggest that relatively low levels frequency wj of the atomic transition. of population inversion over short astronomical length-scales fi Rˆˆˆ,,RR Rˆ2 ( fi ) Following Dicke, we de ne the operators xy3, and e.g., as compared to those required for maser ampli cation such that can lead to the cooperative behavior required for superradiance N in the ISM. Given the results of our analysis, we expect the ˆ ˆ observational properties of 21 cm superradiance to be char- RRK ()rrr=-=å jKd () j ,,,343 Kxy () acterized by the emission of high-intensity, spatially compact, j=1 222 burst-like features potentially taking place over periods ranging RRRRˆ2 ()r =++ˆˆˆ ()44 from minutes to days. xy3 This first paper on this topic has, in part, served as an 2 [RRˆ ()rr,0ˆK ()]¢ = ( 45 ) introduction to superradiance in astrophysics, butmuch remains to be done. For example, we have not attempted to [Rˆa()rr,,,,,,3, Rˆ b()]¢ =- ied abc Rˆ c () rrr (¢ ) abc= xy characterize the shapes of superradiant spectral lines or their ()46 polarization properties, which for the 21 cm line would necessitate the consideration of all hyperfine F =«10 which are similar to the relations found in the spin or general transitions. We thus intend to extend our analysis in subsequent angular momentum formalisms. We can also define the raising 12 The Astrophysical Journal, 826:216 (16pp), 2016 August 1 Rajabi & Houde and lowering operators operator can be expressed as + - ˆ ˆ ˆ ˆ ˆ -itwwkkˆ it* RRiR()rr=xy() () r,47 ( ) BrL (),,57tBeBe=+å [Lk() r eL Lk() r eL ] ( ) k which further verify the following commutation relations: where [RRˆ ()rr,48ˆ3 ()]¢= Rˆ () rrrd ( -¢ ) ( ) ˆ+ 1 wk ik·r ˆ+-ˆ ˆ BLk()r = aeˆLk ()58 [RR()rr,2()]¢= R3 () rrrd ( -¢ ) . ( 49 ) cV20 It is clear from the form of the Hamiltonian and the ˆ- † 2 =(BLk)(),59 commutation relations between the operators that Rˆ and Rˆ3 commute with Hˆ and, therefore, share the eigenfunctions and V is the arbitrary volume of quantization. In Equations (58) ( ) † ∣rm, rñ introduced in Section 2.1 to describe the state of the and 59 , aLk and aLk are, respectively, the LCP second system. quantization field annihilation and creation operators, and The atomic hydrogen transitions at 21 cm are magnetic theyobey the following commutation relation: dipolar in nature and bring into consideration the next leading † ˆ term in our analysis, the magnetic dipole interaction found on [aaˆˆLk,1.Lk¢] = d kk¢ () 60 ( ) the right-hand side of Equation 42 : As a result, one can express the magnetic dipole interaction N term in Equation (50) for transitions involving only LCP ˆˆˆ VMD =-å MBrjj· (),50 ( ) photons as j=1 N ˆˆˆ which is at the center of our analysis. In general, the magnetic VMD =-mB å RBrjLj· () (61 ) j=1 dipole operator Mˆ j of the jth atom can be written as (Condon & ) N Shortley 1935 m wk +- B ˆ ik·rrjjˆ † -ik· =+åå (Raj ˆLk e Raj ˆLk e )(),62 ˆˆ 2 c 20V MFj = mF j ()51 j=1 k ⎡ ⎤ ˆ ˆ+ ˆ ˆ- FF()()()++111 JJ +- II + using R · eL =-R 2 and R · eL* =-R 2 . It will also m g ⎢ ⎥,52() ( ) FJ⎣ 21FF()+ ⎦ prove useful to write Equation 62 in the following form: ˆ ˆ ˆ 3 where Jˆ is the sum of the electronic orbital Lˆ and spin Sˆ VdrMD =- ()·rBrL (),63 ( ) òV angular momenta (i.e., JLSˆˆ=+ˆ), and Fˆ isthe sum of Jˆ and fi the nuclear spin Iˆ (i.e., Fˆˆˆ=+JI). For the hyperfine levels of which allows a de nition of the transverse macroscopic ˆ the ground state of the hydrogen atom, we have F = 0and1, magnetization operator ()r in terms of the raising and = = / = / lowering density operators Rˆ+ and Rˆ- as follows: J S 1 2, I 1 2, and mF gJ 2, whereas ggJe= mB . In this equation, g 2 and μ is the Bohr magneton. The m e B ˆ B ˆ-+ˆ * ˆ ()rrere=- [RR()L + ()L ] (64 ) operator Fj can also be written in terms of pseudospin operator 2 Rˆ Fˆ =+ Rˆ ˆ +- j as jj( 12), allowing us to write º+ˆ ()rrˆ ().65 ( ) ⎛ ˆ ⎞ ˆˆ⎜ 1 ⎟ Neglecting any inhomogeneous broadening effect (i.e., we MRjj=+mB .53() ⎝ 2 ⎠ omit Hˆ0 and set wj = w0) and inserting Equation (62) in Equation (42), the Hamiltonian of the H I-sample system For the ∣Fm==ñ==+ñ0, 0⟷∣ Fm 1, 1 LCP trans- interacting with the 21 cm line via the (Fm,:0,01,1F ⟷ + ) ition we consider, the circular polarization state of radiation can transition becomes be defined using the corresponding unit vectors (Grynberg ⎛ ⎞ et al. 2010): N 1ˆ m w HRHˆˆ=+++w ⎜ ⎟ ˆ B k 0 åå⎝ j3rad⎠ 1 j=1 2 2 c k 20V eeeLxy=-() +i ,54 () N 2 +- ˆ ik·rrjjˆ † -ik· ´+å (Raj ˆLk e Raj ˆLk e )().66 j=1 1 The radiation Hamiltonian term Hˆrad can be expressed in eeeRxy=-()i ,55 () † 2 terms of the second quantized operators aˆLk and aˆLk, and aˆRk † e and aˆRk associated with left and right circular-polarized radia- which with 3 can be used to write the pseudospin operator as tion states, respectively, with ˆ 1 ˆˆ-+ˆ ⎡⎛ ⎞ ⎛ ⎞⎤ Reeej =-+( RRj L j Rj) + R33.56() 1ˆ 1ˆ 2 Haaaaˆ =+++w ⎢⎜ ˆˆ††⎟ ⎜ ˆˆ ⎟⎥.67() rad å ⎢⎝ Lk Lk ⎠ ⎝ Rk Rk ⎠⎥ k ⎣ 2 2 ⎦ In a H I gas, the LCP magnetic component of the radiation propagating along the k direction interacts with the magnetic As discussed in the literature (Gross & Haroche 1982; dipole of a hydrogen atom, resulting in a transition between the Benedict et al. 1996), the evolution of the atomic system can be two hyperfine levels. The corresponding magnetic field calculated using the Heisenberg equation of motion for the 13 The Astrophysical Journal, 826:216 (16pp), 2016 August 1 Rajabi & Houde operator Xˆ in a system described by the Hamiltonian Hˆ with ˆ ˆ using the SVEA, where B0 and 0 are slow-varying envelope dXˆ 1 operators multiplied by fast-oscillating exponential terms = [XHˆˆ,.]() 68 propagating in the positive z direction. Within the context of dt i the SVEA, we assume that the Bˆ and ˆ significantly One can then readily find the following equations of motions 0 0 change over timescales much longer than 1 w and length- ˆ+ ˆ- ˆ for R , R , and R3: scales much larger than 1/k (Gross & Haroche 1982). Upon ˆ+ applying the SVEA, Equation (79) is simplified to dR ˆˆˆ+ i 2 mB - =-iRw03RBL ()69 dt ⎛ ¶ 1 ¶ ⎞ imw ⎜ + ⎟Bˆ 0 ˆ .82() ˆ- ⎝ ¶zct¶ ⎠ 0 2c 0 dR ˆˆˆ- i 2 mB + =-iRw03 + RBL ()70 dt In the derivation of Equation (82), we neglected any transverse ˆ fi ( dR3 imB + + - - effects on the radiation eld and magnetization i.e., =-(RBˆˆ - RB ˆˆ)(),71 LL ˆˆ) dt 2 ¶Bx00¶»¶ By ¶»0 and ¶00¶»¶xy ¶»0 , ˆ which should be included in numerical calculations for a true where, for simplicity, we now set BL for the value of the fi three-dimensional sample. LCP component of the magnetic eld averaged over the Using Equations (80) and (81), we can rewrite positions of the atoms. In a similar way, we can write the Equations (76), (77), and (78) at resonance, that is, when following equations of motion for the raising and lowering w = w0, in the reduced form of ˆ + ˆ - magnetization operators ()r and ()r using ( ) dˆ Equation 64 as 0 = iBg ˆ ˆ ()83 dt 0 ˆ + im2 d ˆ + B ˆ + ˆ =-iw0 - (·eBL L )R3 eL ()72 dˆ 2i - + + - dt =-(ˆ BBˆ ˆ ˆ )().84 dt 00 00 ˆ - im2 d ˆ - B ˆ - ˆ ( )–( ) – =+iw0 (·eBLL**)R3 e L.73() Equations 82 84 are known as the Maxwell Bloch dt equationsand can be solved simultaneously to determine the fi It is also useful to define the operator ˆ time evolution of the radiation eld, magnetization, and excitation state for an ideal sample. ˆ = 274Rˆ3 () N A.2. Dephasing Effects and Pumping ˆ =-2,å R3jjd ()rr () 75The previous derivations for the ideal case must be j=1 augmented appropriately when dealing with more realistic which can be interpreted as a population inversion density conditions for the ISM, where dephasing and relaxation effects operator considering that Rˆ3j has eigenvalues of 1 2 and the cannot be neglected and continuous pumping of the atomic system can take place. One can phenomenologically add the eigenvalue of Rˆˆ= åR is equal to half of the population 33j corresponding terms to the atomic equations as follows difference between the excited level Fm==1,F 1 and the (Mandel 2010): ground level F=0, mF=0 at time t. Using (75) one can show ( )–( ) ˆ + that Equations 71 73 can be rewritten as d0 + 1 ˆ + =-iBg ˆ ˆ - +LM ()85 dt 0 T 0 dˆ 2i + + - - 2 =-ˆ BBˆ ˆ ˆ ( · LL· )()76 ˆ dt d 21i ˆ - ˆ + ˆ + ˆ - ˆ =---(00BB 00)(eq)(),86 ˆ + dt T d ˆ + ˆ + ˆ 1 =-iiwg0 - (·eBL L ) eL ()77 dt where T1 and T2 are the characteristic timescales for, ˆ - respectively, population decay and demagnetization, eq is d ˆ - ˆ - ˆ =+iiwg0 (·eBLL**) e L,78() “ ” ˆ dt the equilibrium value for obtained in the absence of interaction with the coherent field Bˆ , andL represents any where g = m2 2 . L M B source term of magnetization. Furthermore, in the Heisenberg representation one can derive The one-dimensional magnetic field Equation (82) can also theequation be adapted to the more realistic conditions by adding a 2 1 ¶ Bˆ correction term to account for the loss of radiation due to -2Bˆ + L =-m 2ˆ ()79 L 2 2 0 transverse effects and diffraction, which depend on the shape ct¶ and symmetry of the sample. These are characterized by the for the evolution of the magnetic component of the radiation Fresnel number fi fi eld when de ning A Fn = ,87() ˆˆ -ikz()w t lL BtBteL ()rr,,= 0 () () 80 where A and L, respectively, stand for the crosssection and ˆ ()rr,,,tte= ˆ ()-ikz()w t () 81 0 length of the sample. For samples of cylindrical symmetry with 14 The Astrophysical Journal, 826:216 (16pp), 2016 August 1 Rajabi & Houde aFresnel number smaller than one, transverse effects of the in the retarded time frame. Taking the spatial derivative of field are negligible, whereas the diffraction of radiation along Equation (96) we can write the propagation axis can play an important role. Gross & + ( ) + ¶Bˆ im ¶2q Haroche 1982 have shown that a damping term B0 Ldiff can 0B= ,98() be included in the field equation to take into account diffraction ¶z 22g ¶¶z t effects in samples with F 1. For such a sample, n ( ) Equation (82) can be approximately augmented to which when compared to Equation 97 yields the following nonlinear equation ⎛ ¶ 1 ¶ ⎞ + 1 imw + ⎜ + ⎟ Bzˆ (), tt+ Bz+ (), 0 ˆ ,88() ⎝ ⎠ 0 0 0 ¶2q mmw2 N ¶zct¶ Ldiff 2c = 0 B sin()q ( 99 ) ¶¶z t 4cV where Ldiff FL n 0.35 (Gross & Haroche 1982). fi ( ) ( ) ( ) The atom- eld Equations 85 , 86 , and 88 also form a upon using g = m2 2 . This equation is further transformed – B Maxwell Bloch system of equations and provide a more with the introduction of a new dimensionless variable (Gross & complete and realistic picture for the evolution of the system. Haroche 1982) This set of equations can be numerically solved for a given set of parameters TT,, ,and Λ . zt 12 eq M q = 2 ,() 100 LT APPENDIX B R THE SINE-GORDON SOLUTION to The set of Equations (85), (86), and (88) can only be solved d2qq1 d analytically for a few special cases (Mandel 2010).Wefirst +=sin()q , ( 101 ) 2 consider the ideal condition, where the dephasing/relaxation, dq q dq diffraction, and pumping terms are neglected (i.e., TT==¥L =¥ L=) with TR the characteristic time for superradiance given by 12 , diff , and M 0 . Effecting a change of ( ) ( ) variable from t to the retarded time t =-tzcyields Equation 24 . Equation 101 is the so-called Sine-Gordon equation (Gross & Haroche 1982). This equation can be ⎛ ¶ 1 ¶ ⎞ ¶ θ ⎜ + ⎟ = ()89 numerically solved and the corresponding solution for ⎝ ¶zct¶ ⎠ ¶ z substituted back into Equation (96) to determine the field + ¶ ¶ amplitude Bˆ emerging from the sample (at z = L) as a = ,90() 0 τ ˆ + ¶t ¶t function of the retarded time . Knowing B0 ()L, t , the output which can be used to simplify the set of Maxwell–Bloch radiation intensity I is given by Equations (82)–(84) to + c ˆ 2 I = ∣B0 ∣(). 102 2m ¶ˆ 0 0 = iBg ˆ ˆ ()91 ¶t 0 A more realistic case where dephasing/relaxation is included ( ˆ with a single timescale i.e., TTT¢ ==¹¥12 , Ldiff =¥, ¶ 2i ˆ - ˆ + ˆ + ˆ - =-(00BB 00)()92 and L=Meq =0) can be dealt with in a similar manner. We ¶t then have the corresponding definitions for Equations (94) and ( ) ¶Bzˆ (), t imw 95 : 00 ˆ .93() ¶z 2c 0 ˆ + mBN -t T ¢ 0 = sin()q e ( 103 ) The form of Equations (91) and (92) implies that 22V ˆ +-22ˆ 2 ˆ 2 ∣∣∣++∣(mB 4 )∣∣ is a conserved quantity and N -t T ¢ allows us to redefine ˆ and as ˆ = cos()q e , ( 104 ) 0 V ˆ + mBN + 0 = sin()q ( 94 ) which also lead to Equation (96) for Bˆ . Performing a spatial 22V 0 derivative on Equation (96) yields N ˆ = cos()q , ( 95 ) V ¶2q mmw2 N = 0 B sin()q e-t T ¢ . ( 105 ) where N is the number of inverted atoms in the sample at t = 0 ¶¶z t 4cV θ and is the so-called Bloch angle. A comparison of Equation(105) with (99) shows that the Taking these solutions into account, at resonance ( ) ( ) presence of dephasing implies a source term containing a Equations 91 and 93 are transformed to decaying exponential. This exponential factor can be removed from this equation through the following change of variable: ˆ + imB ¶q B0 = ()96 22g ¶t ¢ tt⟶(¢ = Te¢ 1- -t T ) , () 106 + ¶Bˆ iNmwm 00B= sin()q ( 97 ) which allows us to transform Equation (105) to the Sine- ¶z 42cV Gordon equation (i.e., Equation (101)) by redefining the 15 The Astrophysical Journal, 826:216 (16pp), 2016 August 1 Rajabi & Houde dimensionless parameter q with the following: Gross, M., Fabre, C., Pillet, P., & Haroche, S. 1976, PhRvL, 36, 1035 Gross, M., & Haroche, S. 1982, PhR, 95, 301 zt¢ Grynberg, G., Aspect, A., & Fabre, C. 2010, Introduction to Quantum Optics: q = 2 .() 107 from the Semi-classical Approach to Quantized Light (Cambridge: LTR Cambridge Univ. Press) Irwin, J. A. 2007, Decoding the Cosmos (Chichester: Wiley) Jackson, J. D. 1999, Classical Electrodynamics (New York: Wiley) REFERENCES Kida, S. U. M. I. K. O., Niinuma, K., Suzuki, S., et al. 2008, NewA, 13, 519 MacGillivray, J. C., & Feld, M. S. 1981, PhRvA, 23, 1334 Abel, N. P., Ferland, G. J., O’Dell, C. R., Shaw, G., & Troland, T. H. 2006, MacGillivray, J. C., & Feld, M. S. 1976, PhRvA, 14, 1169 ApJ, 644, 344 Mandel, P. 2010, Nonlinear Optics (Weinheim: Wiley-VCH) Andreev, A. V., Emelianov, V. I., & Ilinskii, I. A. 1980, UsFiN, 131, 653 Meziane, J., Oullemine, S., Amezian, K., & Boursey, E. 2002, CPL, Benedict, M. G., Ermolaev, A. M., Malyshev, V. A., Sokolov, I. V., & 363, 573 Trifonov, E. D. 1996, Super-radiance Multiatomic Coherent Emission Moi, L., Goy, P., Gross, M., et al. 1983, PhRvA, 27, 2043 (Bristol: IOP Publishing Ltd) Nussenzveig, H. M. 1973, Introduction to Quantum Optics (New York: Bonifacio, R., & Lugiato, L. A. 1975, PhRvA, 11, 1507 Gordon and Breach) Carlson, N. W., Jackson, D. J., Schawlow, A. L., Gross, M., & Haroche, S. Protsenko, I. E. 2006, J. of Russian Laser Research, 27, 414 1980, OptCo, 32, 350 Rajabi, F., & Houde, M. 2016, ApJ, in press (arXiv:1601.01718) Condon, E. U., & Shortley, G. H. 1935, The Theory of Atomic Spectra Rosenberger, A. T., & DeTemple, T. A. 1981, PhRvA, 24, 868 (Cambridge: Cambridge Univ. Press) Sarma, A. P. 2012, in IAU Symp. 287, Cosmic Masers: from OH to H0, ed. Crutcher, R. M. 2012, ARA&A, 50, 29 R. S. Booth, E. M. L. Humphreys, & W. H. T. Vlemmings (Cambridge: Dicke, R. H. 1953, PhRv, 89, 472 Cambridge Univ. Press),41 Dicke, R. H. 1954, PhRv, 93, 99 Shklovskii, I. S. 1967, SvA, 11, 240 Dicke, R. H. 1964, QuEle, 1, 35 Skribanowitz, N., Herman, I. P., MacGillivray, J. C., & Feld, M. S. 1973, Draine, B. T. 2011, Physics of the Interstellar and Intergalactic Medium PhRvL, 30, 309 (Princeton, NJ: Princeton Univ. Press) Storer, S. H., & Sciama, D. W. 1968, Natur, 217, 1237 Dykstra, M., & Loeb, A. 2007, NewA, 13, 395 Thornton, D., Stappers, B., Bailes, M., et al. 2013, Sci, 341, 53 Emerson, D. 1996, Interpreting Astronomical Spectra (New York: Wiley) Townes, C. H., & Schawlow, A. L. 1955, Microwave Spectroscopy (New Field, G. B. 1958, PIRE, 46, 240 York: McGraw Hill) Fish, V. L. 2007, in IAU Symp. 242, Astrophysical Masers and their Vlemmings, W. H. T. 2012, in IAU Symp. 287, Cosmic Masers: from OH to Environments, ed. J. M. Chapman & W. A. Baan (Cambridge: Cambridge H0, ed. R. S. Booth, E. M. L. Humphreys, & W. H. T. Vlemmings Univ. Press), 71 (Cambridge: Cambridge Univ. Press),54 Gibbs, H. M., Vrehen, Q. H. F., & Hikspoors, H. M. 1977, PhRvL, 39, 547 Watson, W. D. 2009, RMxAC, 36, 113 Goldsmith, P. F., & Langer, W. D. 1999, ApJ, 517, 209 Wittke, J. P., & Dicke, R. H. 1956, PhRv, 103, 620 Greiner, C., Boggs, B., & Mossberg, T. W. 2000, PhRvL, 85, 3793 Wouthuysen, S. A. 1952, AJ, 57, 31 Gross, M., Goy, P., Fabre, C., Haroche, S., & Raimond, J. M. 1979, PhRvL, Xia, H., Svidzinsky, A. A., Yuan, L., et al. 2012, PhRvL, 109, 093604 43, 343 Zygelman, B. 2005, ApJ, 622, 1356 16