Dicke’s Superradiance in Astrophysics. I – The 21 cm Line Fereshteh Rajabi1 and Martin Houde1,2 1Department of Physics and Astronomy, The University of Western Ontario, London, ON, N6A 6 3K7, Canada 1 0 2Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, 2 CA 91125, USA n u J Abstract 1 ] A We have applied the concept of superradiance introduced by Dicke in 1954 to as- G trophysics by extending the corresponding analysis to the magnetic dipole interaction . h characterizing the atomic hydrogen 21 cm line. Although it is unlikely that superradi- p ance could take place in thermally relaxed regions and that the lack of observational - o evidence of masers forthis transition reduces the probability of detecting superradiance, r t in situations where the conditions necessary forsuperradiance are met (i.e., close atomic s a spacing, high velocity coherence, population inversion, and long dephasing time-scales [ compared to those related to coherent behavior), our results suggest that relatively low 2 levels of population inversion over short astronomical length-scales (e.g., as compared to v 7 those required formaser amplification) can lead tothe cooperative behavior required for 1 superradiance in the ISM. Given the results of our analysis, we expect the observational 7 1 properties of 21-cm superradiance to be characterized by the emission of high intensity, 0 spatially compact, burst-like features potentially takingplaceover short periodsranging . 1 from minutes to days. 0 6 1 Subject headings: atomic processes – ISM: atoms – radiation mechanisms: general : v i X r a 1. Introduction Itis generally assumedthat inmuch of theinterstellar medium (ISM)emissionemanating from atomic and molecular transitions within a radiating gas happen independently for each atom or molecule. From intensity measurements of such spectral lines, important parameters (e.g., density and temperature) can be determined and the physical conditions in a given environment thus characterized (Townes et al. 1955; Emerson 1996; Goldsmith et al. 1999; Irwin 2007). For example, in cases where the spectral lines are optically thin the intensity will be found to scale linearly with the number of atoms or molecules responsible for the detected radiation. The soundness of this approach rests mostly on the assumption that spontaneous emission from different atoms or molecules happens independently. – 2 – As was pointed out by R. H. Dicke in a seminal paper several decades ago (Dicke 1954), the assumption of independent spontaneous emission for the components of a gas does not apply in all conditions. As willbediscussed inthis paper, and following Dicke’s original analysis, closely packed atoms can interact with their common electromagnetic field and radiate coherently. That is, the spontaneous emission of atoms or molecules in such a gas will not be independent, but rather take place in a cooperative manner. In the ideal case, this phenomenon will lead to a much more intense and focussed radiation (proportional to the square of the number of atoms), which Dicke called superradiance. Since Dicke’s original proposal, the field of superradiance research has flourished and anabundant literature has developed within the physics community. The first experimental de- tection of superradiance in the laboratory was achieved by Skribanowitz et al. (1973), while several other independent verifications (Gross et al. 1976; Gibbs et al. 1977; Carlson et al. 1980; Moi et al. 1983; Greiner et al. 2000; Xia et al. 2012) have since been realized under a large domain of con- ditions and experimental setups (see Chap. 2 of Benedict et. al. 1996; MacGillivray & Feld 1976; Andreev et al. 1980; Gross & Haroche 1982 for reviews). While the reality of the superradiance phenomenon has long been clearly established in the laboratory, to the best of our knowledge, it has yet to be investigated within an astrophysical context. It appears to us important to do so since some of the requirements and conditions needed for the realization of a superradiant system are known to be satisfied in some regions of the ISM. More precisely, superradiance can arise in systems where there is a population inversion, and the effect will be much stronger and more likely to be realized when atoms or molecules are separated by approximately less than the wavelength of radiation (see below and Section 3.2). The population inversion condition is known to occur in the ISM and is partly responsible for the ubiquitous presence of masers (see Fish 2007; Watson 2009; Vlemmings 2012; Sarma 2012 for recent reviews). But it is also important to realize that, although it is a necessary condition, population inversion is not by itself sufficient to ensure superradiance. It is also required that there exists sufficient velocity coherence between the atoms partaking in the effect, and that any other dephasingtakes placeontime-scales longerthanthosecharacterizing superradiance. Whenallthese conditions are met a coherent behavior can be established between the atoms and superradiance can ensue. We note, however, that, as will be discussed later on, superradiance is unlikely to could take place in thermally relaxed regions of the ISM. This is because Doppler broadening resulting from, say, a Maxwellian velocity distribution would leave too few atoms with the required velocity coherence to allow superradiance to develop. Our analysis will therefore imply other types of environments where thermal equilibrium has not been reached. For example, any region in the ISM into which a significant amount of energy is being suddenly released (e.g., shocks or regions where significant radiation flares occur) will be strongly out of equilibrium, and provide conditions that are potentially markedly different to those found in a thermal gas and may meet the requirements for superradiance. Also, although superradiance can also occur for large interatomic or molecular separations (i.e., greater than the wavelength of radiation; see Section 3.1.2), the aforementioned constraint of small interatomic or molecular separation, and its implication for the corresponding – 3 – densities, is likely to be met for only a limited number of spectral lines, but a few astrophysically important transitions are suitable candidates. One of these spectral lines is the 21 cm atomic hydrogen transition. Even though a 21 cm maser has yet to be discovered, which would also imply the realization of a population inversion for this spectral line, as will be seen through our analysis the length-scales required for superradiance at 21 cm are very small compared to those that would be needed for maser amplification in the ISM (Storer & Sciama 1968, and see below). It follows that although the lack of observational evidence of masers for this transition significantly affects the probability of detecting superradiance, it does not rule it out. Also, the existence of higher densities of atomic hydrogen in some parts of the ISM would increase the potential detectability of superradiance, if the other necessary conditions for its realization previously listed were also met. Furthermore, with the recent discoveries of radio bursts at frequencies close to 1400 MHz (Kida 2008; Thornton et al. 2013) the investigation of the properties of a transient phenomenon such as superradiance is timely. This is why in this first paper on the subject we chose to introduce the concept of superradiance to the ISM using this spectral line. Whether or not a population inversion can easily be realized for the energy levels leading to the 21 cm line , it has been considered in the existing literature (Shklovskii 1967; Storer & Sciama 1968; Dykstra & Loeb 2007) and we know of at least one region (the Orion Veil) where the kinetic temperatureislowerthanthe21-cmspintemperature, providingevidenceforapopulationinversion (Abel et al.2006). The main pumping process covered in the literature corresponds to the situation when a HI gas is close to a source of radiation that emits a field with an intensity Iv(ν) in the neighborhood of the Lyman α line. A hydrogen atom in the ground hyperfine state (n = 1, F = 0) can absorb a photon and become excited to the n = 2 level. Later on, the atom returns to the upper hyperfine state (n = 1, F = 1) emitting a slightly less energetic photon than the initial one absorbed by the atom. The same can happen for a hydrogen 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 level, emitting a slightly more energetic photon in the process. The absorption rate of the photons for both cases depends on the intensity of the radiation I (ν), but the return (emission) v process does not. Therefore, the F = 0 level will undergo more absorptions followed by a return to the (n = 1, F = 1) level whenever I (ν) harbors more blue than red photons, and will become v accordingly less populated than the F = 1 level (Wouthuysen 1952; Field 1958; Shklovskii 1967; Storer & Sciama 1968). Although Storer & Sciama (1968) concluded that it is unlikely to maintain apopulation inversion over an extendedregion needed forthemaser amplification withthis process, they also pointed out that an “appreciable” inversion can thus be realized over a region of thickness 6 10−5 pc. Giventheaboveinversionscenario,wewouldexpectthatenvironments locatedinthe ∼ × periphery or near boundaries of HII regions could provide conditions suitable for the development of superradiance, for example. The aforementioned evidence for a 21 cm population inversion in the Orion Veil brings support to this idea. Whatever the case, the 21 cm line will serve us as a starting point for the development of the superradiance formalism for the ISM (in the present – 4 – case for a magnetic dipolar transition), which will then be refined in the future and also applied to other (electric dipolar) spectral lines (e.g., the OH 1612-MHz, CH OH 6.7-GHz, and H O 22-GHz 3 2 maser transitions) where observational evidence for superradiance can be found in the literature (Rajabi & Houde 2016a,b). Itshould alsobe pointed out that superradiance is a fundamentally different phenomenon from the maser action, even though the two may seem similar at first glance. An astronomical maser is a collective but not coherent phenomenon. More precisely, for a maser a group of atoms, initially in their excited states, emit through the stimulated emission process but cannot be considered as a single quantum system. That is, it is possible to describe maser action through successive events where an excited atom is stimulated by the incident radiation and emits a photon, with the same stimulation/emission processes subsequently repeated for different atoms in the masing sample. On the other hand for superradiance, coherence emphasizes the fact that the group of atoms interacting with the radiation field behaves like a single quantum system (Nussenzveig 1973). That is, the superradiance emission process cannot be broken down into successive events as is the case of maser radiation. Finally, superradiance is a transient effect in which a strong directional pulse is radiated over a relatively short time-scale, while maser action operates more in a steady state regime as long as population inversion is maintained. The material covered in this paper goes as follows, we start with a general discussion of the concept of superradiance for the so-called small- and large-samples, as originally discussed by Dicke (1954, 1964), in Section 2. In Section 3, we examine the possibility of building cooperative behavior in a HI sample based on a comparative analysis of time-scales for the 21 cm line in a HI gas, as well as present corresponding numerical results. A discussion and short conclusion follow in Sections 4 and 5, respectively, while the superradiance formalism and detailed derivations for the material discussed in the main sections of the paper will be found in appendices at the end. 2. Superradiance 2.1. Dicke’s Small-sample Model 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 and the spin down to | i the ground state g . Just as an ensemble of N spin-1/2 particles can be described using two | i quantum numbers s and m , the eigenstates of the combined N two-level atoms in Dicke’s model s can also be labelled with two quantum numbers r and m such that 0 r N/2 and m = r r ≤ ≤ r, r+1,...,r 1,r, where − − − N N e g m = − , (1) r 2 – 5 – r = N m = N 2 r m = N −21 NI0 r 2 2(N−1)I0 3(N−2)I0 . . . m =−N +1 r m =2 −N NI0 r 2 Fig. 1.— Dicke states with r = N/2 for a system of N two-level (spin-1/2) particles. Spontaneous radiation intensities are indicated on the right. with N and N the number of particles in the excited and ground states, respectively. From the e g completesetofeigenstatescharacterizingthisquantummechanicalsystem,thosesymmetricalunder the permutation of any pair of atoms are particularly important and are called Dicke states. The initial state e,e,...,e of N fully inverted spin-1/2 particles corresponding to N fully inverted two- | i level atoms is one such Dicke state, and is identified by r = N/2 and m = N/2. When an atom in r the ensemble decays to its ground state by emitting a photon, the quantum number m is decreased r 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 = N/2) r state down through an arbitrary state (r, m ) is r I = I (r+m )(r m +1) (2) 0 r r − if the volume containing the ensemble of N two-level atoms is much smaller than λ3, the cube of the wavelength of the radiation interacting with the atoms. In Equation (2), I is the radiation 0 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. Furthermore, Dicke pointed out that in the (r = N/2, m = 0) state, where the half of the r atoms are in the ground state and the other half in the excited state, the radiation intensity of the system is maximum at N N I = I +1 (3) 0 2 2 (cid:18) (cid:19)(cid:18) (cid:19) N2I , (4) 0 ∝ implying asignificantly enhanced radiation beam, aphenomenon henamed superradiance. This can – 6 – 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 atomsandcanbringabouttheemissionofanewphotonofthesamemodeandinthesamedirection 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 superradiance observed in a perfectly coherent system, in a non-coherent 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. Inordertoconduct amorecarefulinvestigation ofthepossibility ofcoherent interactions, espe- cially superradiance in a HI 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: A small-sample of neutral atoms is confined to a volume λ3 with the walls of the volume • V ≪ transparent to the radiation field. The N two-level atoms in the sampleare separated by adistance 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. TheensembleofN initiallyinvertedhydrogenatomspossessesapermutation symmetryunder • the exchange of any pair of atoms in the sample. This is a restricting condition that could prove difficult to satisfy in general. The transition between atomic levels takes place between non-degenerate 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). 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. Finally, the radiation field is assumed to be uniform through the small-sample, the electric • dipoles associated to the atoms are parallel, and propagation effects neglected. Comparingacorrespondingsmall-sampleofN neutralhydrogenatomsinteractingwith21cmlinein the ISM with aDicke sample, we can seethat some of the assumptions made in the Dicke formalism holdandsomedonot. Forexample, thetransitions betweenthehyperfinestatesofahydrogenatom take place between non-degenerate levels sincethe external magnetic field inthe ISM lifts the upper – 7 – leveldegeneracy (seeSection 3). Also,asmall-sampleof HIatoms foundinmany regions intheISM would readily verify the criterion that N 1 in a volume < λ3, and could thus be approximately ≫ V 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 theinternal hyperfinestates intheirelectronic ground statethroughspinde-excitation (Field1958). 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 be preserved in 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 important in small-samples where the distance between atoms r′ is smaller than λ (see Section 3.1 below). In the Dicke model, the symmetry breaking effect of dipole-dipole interactions is ignored. Inlater studies of superradiance (e.g., Gross & Haroche 1982), ithas beenshown 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 < 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 non-coherent system (I ) but smaller than that of a perfect superradiance system (I ). nc 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 HI-sample the magnetic dipole-dipole interactions are definitely weaker than the electric dipole-dipole interactions discussed in the literature focusing on symmetry breaking effects. 2.2. Dicke’s Large-sample Model In his first paper on superradiance, Dicke also extended his formalism to a large-sample, where thevolumeofthesample > λ3 andtheinteratomic distance r′ betweensomeatoms canbegreater V 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 are 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 ), and the intensity of the radiation in a solid angle along k follows r – 8 – I(k) = I (k)[(r+m )(r m +1)], (5) 0 r r − 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 ∆r = 0, ∆m = 1, and similar to r − 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 k′ = k, the states with different r (i.e., of different symmetry) can couple and consequently 6 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 non-coherent 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′ > λ) 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 re-absorption and re-emission of the photons and consequently leads to a non-uniform evolution of the atoms in the sample (see Section 3.2). Beyond these factors, Dicke’s analysis of the large-sample includes similar assumptions as those used for the small-sample. 3. The Two-level HI-sample Let us consider an ensemble of neutral hydrogen atoms in the electronic ground state in some region of the ISM,where it can emit or absorb photons at the λ = 21 cm wavelength. The hydrogen 21 cm line is perhaps the most important source of information in radio astronomy and arises from the transition between two levels of the hydrogen atom in the 1s ground state. The interaction between the electron spin and the proton spin in the nucleus of the atom splits the otherwise degenerate 1s energy level into the two F = 0 and F = 1 sub-levels. The F = 1 0 transition ↔ in the absence of an external magnetic field produces the 21 cm line corresponding to a frequency ν = 1420.406 MHz. Considering a more realistic case, the magnetic field in a cold neutral gas is generally on the order of 10 µG (Crutcher 2012), and the energy level corresponding to F = 1 splits into three sub-levels identified by m = 1,0, and1. The interaction between the F = 0 and F = 1 levels F − becomesmorecomplicatedasthissplittingprovidesthreepossiblehyperfinetransitions,asshownin Figure 2. These hyperfine transitions link states of like parity and obey the general magnetic-dipole selectionrules ∆F = 0, 1 and∆m = 0, 1. Basedontheserules, allofthethree transitions shown ± ± in Figure 2 are allowed, however, depending on the relative orientation (or the polarization) of the – 9 – m = 1 F F = 1 m = 0 F m = −1 F 1s F = 0 m = 0 F Fig. 2.— Energy level diagram for the HI 21 cm line in the presence of a Zeeman-splitting external magnetic field. magnetic component of the radiation field to the quantization axis of the atom, some transitions may be favored. In the more general case, there is a mixture of all three transitions with each transition exhibiting particular polarization properties. In order to better understand the coherent and cooperative evolution of a sample of N hydrogen atoms coupled to its radiation field, it will be simpler for us to focus our analysis on only one of these transitions and consider the atomic system as an ensemble of two-level atoms. Although this model represents a significant simplification, the two-level atom approximation is extensively used for, and its results well-verified in, laboratory experiments involving more complicated atomic or molecular systems with more complex energy levels (Mandel 2010). 3.1. Magnetic Dipole-dipole Interaction Between Hydrogen Atoms The theoretical model for the problem will be found in Appendix A, where the Hamiltonian for the two-level HI-sample is developed and the main equations of superradiance derived. To simplify our discussion we have limited our analysis to the F = 0, m = 0 F = 1, m = +1 transition | i ←→| i throughwhichahydrogenatomemitsaleftcircularpolarization(LCP)photon, i.e.,withitselectric field vector rotating counter-clockwise as seen by the observer facing the incoming wave. One of the main components of the Hamiltonian is the magnetic dipole energy term Vˆ that describes MD the interaction betweens the atoms composing the sample (see Equations [A9] and [A21]). 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′ < λ) 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 – 10 – |e1e2! 2Γ |+! ∆E ∆E |−! 2Γ |g1g2! Fig. 3.— The two-hydrogen-atom system. When kr′ 1 the upper and lower symmetric states ≪ e e and g g , respectively, couple to the intermediate symmetric state + at the enhanced 1 2 1 2 | i | i | i transition rate 2Γ, where Γ is the transition rate of a single atom acting independently. In contrast, the antisymmetric state cannot couple to the upper and lower states because of the cooperative |−i behavior between the two atoms. The energy level shifts ∆E for the states are also shown. ± |±i with the excited state e (F = 1) and the ground state g (F = 0). The two atoms are initially | i | i excited and the state of the system is given by e e = e e . Eventually one of the two 1 2 1 2 | i ⊗ | i | i atoms spontaneously decays to its ground state emitting a photon with a wavelength λ and energy ~ω. If the interatomic distance r′ is much smaller than λ (i.e., kr′ 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 (MKS units) 2 µ k3 e Mˆ g 0 h | | i Γ = . (6) (cid:12)3~π (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) We can furthermore express the state of the system by either a symmetric + or antisymmetric | i |−i combination of the e g and g e state vectors such as 1 2 1 2 | i | i 1 + = ( e g + g e ) (7) 1 2 1 2 | i √2 | i | i 1 = ( e g g e ), (8) 1 2 1 2 |−i √2 | i−| i which, at this stage of our analysis, have the same energy and are thus degenerate (see below).