Plasma Effects on Lasing of Uniform Ultralight Axion Condensate

Lasing of ultralight axion condensate into photons can be sensitive to the presence of a background plasma owing to its coupling to electromagnetism. Such a scenario is particularly relevant for superradiant axion condensate around stellar mass black holes since the axion mass can be within a few orders of magnitude of the plasma frequency of the surrounding medium. In this paper I discuss the properties of the plasma around a black hole and analyze its effects on the lasing of a uniform axion condensate of mass of the order of the plasma frequency.


Introduction
Spinning black holes are known to exhibit superradiant instability for massive bosons with Compoton wavelength of the size of the horizon of the blackhole. The mechanism is based on the seminal work of [1] which showed that a wave scattering off of a spinning black hole gains in amplitude in the scattering event. Although this mechanism does not discriminate between fermionic and bosonic excitations, it is only for a massive bosons that a Bose-Einstein condensate with a large occupation number can form thereby causing the black hole to spin down [2][3][4]. This phenomenon has attracted significant interest of late in anticipation of upcoming gravitational wave observations [5][6][7][8][9][10][11][12][13] in particular involving stellar mass black holes. For black holes of mass M the typical mass of bosons that can condense are of the order of 1 GM ∼ 10 −10 eV M M where G is the Newton's constant. This allows for the attractive possibility of detection of scalars of very low mass through gravitational waves alone. An example of such a bosonic particle is the axion. There have been analysis of indirect signatures of the superradiant instability utilizing the spin distribution of black holes [9], direct detection of gravitational waves through axion level transitions [9], finite size effects [14] and axion annihilation to gravitons [9]. However, the prospects of electromagnetic signatures through lasing have not been explored in great detail except in [11,15]. The absence of such analysis for black holes of stellar mass or larger can partly be attributed to the extremely long wavelengths of photons under consideration which are ostensibly undetectable on earth based telescopes. However, it is important to note that the detectability of gravitational signatures from the axion condensate depends critically on the non-occurrence of lasing. A particularly gloomy prospect can involve complete depletion of the the axion condensate through lasing thereby eliminating any detectable gravitational wave signatures while the photons produced in the process are not observable either.
One of the key features of this problem is the contrast between the time scales associated with spontaneous and stimulated emissions. The spontaneous emission rate for axions, given by ∼ m 3 a f 2 a where m a and f a are the axion mass and the axion decay constant, is miniscule for a superradiant axion owing to its small mass. For QCD axions, this rate is further suppressed since m a and f a are related by m a f a ∼ λ 2 QCD . The associated decay time scale given by 10 64 M 10M 5 years, is much larger than the age of the universe. This may lead one to erroneously declare that the decay of superradiant axions to photons is irrelevant for any time scales of interest. The process of lasing however is largely dictated by the rate of stimulated emission as opposed to the rate of spontaneous emission. As I review in the main text of this paper, in the absence of any matter coupling this rate is ∼ α|φ| πfa m a where α is the fine-structure constant and φ is the axion condensate. The corresponding depletion time scale is found to be less than a second for a QCD axion condensate around a black holes of mass of M ≈ 10M =⇒ f a ≈ 10 18 GeV. Although this would doom direct detection of gravitational signatures from superradiant axion condensates, it naturally leads one to ponder the possibility of direct detection of the photons from the laser via telescopes. However, owing to their extremely long wavelengths these photons cannot propagate through the earth's ionosphere which renders them undetectable via radio telescopes. This would indeed paint a very grim picture for direct detection prospects of axion condensates via gravitational or electromagnetic signatures.
It is however important to note that the process of lasing can be sensitive to the medium around the black hole when coupling of electromagnetic fields to matter is taken into account. The frequency and wavelength dependent scattering of photons in matter can lead to the kinematic blocking of the two-photon decay mode [16] or detuning via damping thereby hindering the build up of the laser. Such obstruction to lasing which will prevent the rapid decay of the axion condensate can ensure its detectability through gravitational signatures. Therefore it is important to analyze the medium dependence of lasing of superradiant axion condensates in detail and this paper is a step in that direction.
The black hole environment typically consists of an accretion disc and resembles the interstellar medium outside of the disc. The transport of electrons in the accretion disc is dominated by collisions (electron-ion collisions), whereas electron transport in interstellar medium (ISM) is mostly collision-less. Note that, in a collision-less plasma, axions cannot decay to two photons if the plasma mass of the photons is much larger than the axion mass. Similarly in the collision dominated regime, propagating electromagnetic modes are damped at a rate set by the d.c. conductivity (conductivity at zero frequency) σ dc . In this regime, photon detuning due to the dc conductivity obstructs the growth of the laser provided κ Γ ≥ 1 where Γ is the exponential growth rate of the laser in the absence of any matter coupling and κ is the rate of photon detuning. I find that in the accretion disc the d.c conductivity is indeed much larger than any other scale of interest. Hence one concludes that there is no lasing inside of the accretion disc. The region outside of the accretion disc which resembles the interstellar medium is more interesting. There exists regions of the parameter space for which the plasma mass of the ISM is much greater than the axion mass which ensures the absence of any lasing. However, for stellar mass black holes of low mass, the plasma frequency of the interstellar medium can be comparable to the axion mass. Similarly, if the medium around a relatively massive black hole is exceptionally dilute, the plasma frequency of the black hole's environment can be comparable to the axion mass in question. In this region of the parameter space, the details of lasing including the laser growth rate should depend on both the plasma frequency as well as the strength of the axion condensate. The medium response discussed in this paper pays particularly close attention to this regime of parameter space.
Typically lasing of axions to photons is dealt with by solving Maxwell's equations in the presence of a source term driven by the axions. These equations supplemented with an appropriate term for the conductivity can then be used to describe medium response. In this paper I estimate the conductivity of the medium around black hole and solve the corresponding Maxwell's equations in the presence of an axion condensate. It is evident that the detailed electromagnetic response of interest to this paper is a very involved problem in its full generality which requires numerical work. However, such a treatment is beyond the scope of this paper. Here I work with a simple toy model that can clearly explain the physics in question. I begin with a short review of the superradiant instability and lasing of axions in the absence of any matter. This is followed by estimates of the conductivity around the black hole and solution to the lasing problem in the presence of a finite conductivity. I assume a spatially uniform condensate and ignore back-reaction of the electromagnetic field on the axion condensate. I also ignore self interactions of the axions if any.

Maxwell's equations for the axion cloud
Before delving into the relevant equations of motions for lasing of axions, let us first review the basics of the superradiant instability. In the process we will also estimate the order of magnitude of some of the parameters involved in the problem. As we will see, the most generic solution to the equations of motion are not very enlightening and one needs to identify small parameters in the problem so as to understand better the conditions under which lasing of axions to photons can take place. As stated earlier, an axion condensate forms around a spinning black hole of mass M when axion mass m a is of the order of m 2 P M where m P is the Planck mass. The condensate is well described by hydrogen atom wave functions with a coupling constant α M ∼ maM m 2 P . The corresponding spectrum is where n, l, m are the hydrogen quantum numbers. The axion cloud extracts angular momentum from a maximally rotating black hole as long as the superradiance condition is satisfied For QCD axions, φ fa ranges from 1 to 100 for black hole masses between 100M to M . As we will see below the axion coupling to photons given by ∼ αφ fa , α being the fine-structure constant, is one of the possible small parameters which we will eventually expand in to make sense of the results. With this brief review let us now write down the axion-photon Lagrangian [5,17] Here C is some order one numerical constant [5]. Ignoring the back-reaction of the electromagnetic field on the evolution of axion field and assuming a spatially uniform axion condensate the equations of motion can be expressed as In order to describe the physics of lasing one has to express the electric, magnetic and axion fields in their second quantized form in terms of creation and annihilation operators and then solve for their expectations values in coherent states according to Eq. 2.2. It is the expectation values of the gauge fields in coherent states that are expected to exhibit exponential growth as a signature of lasing. Let us now write down the second quantized gauge field as well as a spatially uniform second quantized axion field Here, the time dependence ofα k (t) is slow compared to the photon frequency of ω k . This time dependence is intended to eventually capture the physics of exponentially growing laser. On the other hand, ignoring the back-reaction of the gauge field on the axion condensate leads toφ being independent of time. Let us now concentrate on a particular mode of the gauge field given bŷ Substituting Eq. 2.4 and 2.5 in Eq. 2.2, I find Eq 2.6 which excludes medium response will produce lasing solutions with the following ansatz where φ R 0 is a real constant and f (t) is a complex valued spatially uniform function of time given by with f 0 , β and λ being real numbers. It is easy to see that the phase shift between the two orthogonal components of the gauge field can be adjusted to absorb any U (1) phase of the axion field operator expectation in the coherent state which is why I choose φ to be real without any loss of generality. Eq 2.6 can be solved for any wavelength k which will relate the growth rate λ to the wave number k, the axion mass and the axion condensate field φ R 0 . The growth rate is found to be maximum at k = ma 2 for Cα|φ| πfa 1. Solving Eq. 2.6 for |k| = ω = m a /2, the growth rate can be written as λ ≈ m a Cα 2πfa |φ R 0 | in the limit of Cα|φ| πfa 1. For θ = 0, π, in the limit of Cα|φ| πfa 1, one finds |β| ≈ 1. Similarly for θ = ± π 2 , the growing modes correspond to |β| ∼ 4πfa Cα|φ| and |β| ∼ Cα|φ| 4πfa .

Estimates of the conductivity
The solution in the previous section of course ignores medium effects completely. Incorporating these effects in the equations of motion in principle involves augmenting the equations with appropriate constitutive relations for the current density in linear response to an external electromagnetic field. Such a linear response current is well approximated by j medium = σE where σ is the conductivity of the medium. The response to electromagnetic waves typically carries frequency dependence unless one is in the collision dominated regime. To be specific, the dependence of the conductivity on the frequency of interest can be approximately expressed as where n e is the density of electrons in the medium, m e is the mass of an electron and τ coll is the inverse collision frequency. In the limit ωτ coll 1, one is in the collision dominated regime where conductivity is independent of the frequency and is given by 4πnee 2 τ coll me . In the collision-less limit with ωτ coll 1, the conductivity is frequency dependent and is given by i4πnee 2 ωme . In principle one can solve Maxwell's equations as a function of an arbitrary conductivity. However, this is cumbersome and unnecessary in the present context. Instead it is much more useful to begin with an estimate of the conductivity in regions around the black hole which simplifies the equations significantly. With this in mind, I now proceed to estimate the conductivity of the black hole environment. In the presence of a thin accretion disc, the region around a black hole outside of the disc is expected to resemble the interstellar medium to first approximation. Interstellar medium typically consists of hot ionized hydrogen plasma with a density of about 1/cc − 0.001/cc and temperature of about 10 4 K − 10 6 K. In order to determine the conductivity of this region I have to first estimate the collision frequency of electrons in it. The collision frequency is related to the mean free path of electrons which is given by λ mfp = T 2 n e πe 4 ln(Λ) (3.2) for coulomb collisions in a neutral plasma. Here T is the temperature, n e is the density of electrons and the Coulomb logarithm ln(Λ) ∼ 10. Assuming the density to be one particle per cubic centimeter n e ∼ 8 × 10 −15 eV 3 and a temperature of 10 4 K ∼ 1eV, the mean free path is given by λ ∼ 4.72 × 10 14 1/cc ne T 2 (1eV 2 ) eV −1 . The collision frequency is related to the mean free path as ν coll ∼ v e (λ) −1 where v e is the speed of an electron v e ∼ T /m e . For the temperature and densities under consideration v e ∼ 10 −3 eV and the collision frequency is ∼ 10 −18 1eV T 3/2 ne 1/cc eV. Since the frequency of the lasing photons is set by the axion mass ∼ 10 −11 eV 10M M , it is clear that one is in the collision-less limit outside of the accreting disc. The conductivity is then given by i 4πnee 2 ωme upto corrections of the order m a τ coll . Maxwell's equations in the collisionless limit with φ = 0 has propagating modes only for frequencies larger than the plasma frequency given by ω 2 P = ωIm[σ] = 4πnee 2 me . The corresponding dispersion relation is given by Similarly an estimate of the collision frequency can be obtained if there is accreting matter around a black hole. In the Eddington limit the quantity of matter around a black hole is related to its mass M , the event horizon r g and accretion time τ accr as δM = M The accreting matter can be assumed to be ionized hydrogen to a good approximation.
To determine the mean free path of electrons in the accreting matter, one needs to know the density of electrons in it as well as its temperature. It is known that in the Eddington limit, assuming black body radiation, the temperature of the accreting matter around a black hole at a radial distance r is given by T = 3M 2 G N MeV −1 (3.5) and the corresponding collision frequency is given by Hence we see that in the accreting region, ω/ω accr coll 1 for ω ∼ m a ∼ 10 −10 eV and the conductivity is In the absence of an axion condensate the collision dominated limit of Maxwell's equations is accompanied by damping of electromagnetic waves. In this regime for a conductivity of σ, the characteristic length over which electromagnetic waves are damped is given by σ 2 for σ ω and √ σω/ √ 2 for ω σ.

Axion lasing in matter background
Having obtained an estimate of the conductivity we can now solve Maxwell's equations coupled to an axion condensate in the presence of a finite conductivity The ansatz of Eq. 2.7 and 2.8 solve Eq. 4.2 as well. As stated earlier the solutions are not particularly enlightening as a function of an arbitrary conductivity. Relatively simple expressions are obtained in the strictly collision-less and the strictly collision dominated limit. From the estimates of the conductivity above, it is clear that most of the black hole environment for a thin accretion disc is extremely well described by the collision-less limit. Similarly, the accretion disc is very well described by the collision dominated limit. Let us first consider Eq. 4.2 in the collision dominated regime relevant for the accretion disc. It is important to note that in the accretion disc, the conductivity as obtained in Eq. 3.7 is much larger than the axion mass scale. Plugging the ansatz of Eq. 2.7 and 2.8 in Eq. 4.2 one can see that |σE| is parametrically larger (by a factor of σaccr ma ) compared to the rest of the terms. As a result no lasing can take place in the accreting region.
Let us now concentrate on the region outside of the accretion disc for a thin disc. As we will see at the end of this section, the possibility of lasing in the collision-less limit depends on the relative magnitude of the plasma frequency to the axion mass and that indeed they can be comparable in a region of the parameter space relevant to superradiant condensates. Analyzing Eq. 4.1 and Eq. 4.2 in the collision-less limit again leads to exponentially growing solutions. Just like the solution of Eq.2.6, solving Eq. 4.2 will lead to a relation between the wavelength of the photons, the axion mass, the axion condensate and the lasing rate. Defining Cα πfa ≡ ξ, I find exponential growth for momentum ) are taken to be less than 1. For the small ξ|φ| limit ( √ m 2 a −4ω 2 P ma > ξ|φ|), the maximum growth rate is found to be at |k| ≡ k max,1 ≈ m 2 a 4 − ω 2 P . The corresponding growth rate is given by The expansion in |φ|ξ for the maximum growth rate λ k max,1 breaks down when ξ|φ| ∼ √ This gives the maximum possible growth rate in the small |φ|ξ limit to be These growing modes are given by ξ|φ|, maximum growth rate is achieved at |k| ≡ k max,2 ≈ m a ξ|φ| 1+2ξ 2 |φ| 2 2+8ξ 2 |φ| 2 and the maximum growth rate is given by λ k max,2 ≈ m a ξ 2 |φ| 2 ( 2 + 4ξ 2 |φ| 2 − 1 + 4ξ 2 |φ| 2 ) (1 + 4ξ 2 |φ| 2 )(3 + 8ξ 2 |φ| 2 − 2 √ 2 1 + 6ξ 2 |φ| 2 + 8ξ 4 |φ| 4 ) . One can see that the growth rate given by Eq. 4.6 which is obtained in the limit ξ|φ|, is independent of the plasma frequency whereas the growth rate in the limit of small ξ|φ| given by Eq. 4.4 is not. Fig. 1  . For fixed ξ|φ|, lasing becomes more and more disfavored as the black hole mass increases which drives the axion mass to smaller values thus forcing m 2 a (1 + ξ 2 |φ| 2 ) < 4ω 2 P eventually. Plugging in the density for the ISM I find that the plasma frequency is given by ω 2 p ∼ 20 × 10 −21 ne 1/cc eV 2 . Comparing the plasma frequency with frequency of the photon ∼ 10 −11 10M M eV one can conclude that lasing of a spatially uniform condensate of superradiant of axions around low mass stellar black holes (10M ) can take place for background densities smaller than ≤ 10 −3 /cc. Note that the growth rate of the electromagnetic field is about 10 −12 eV for a QCD axion mass of 10 −10 eV for n 10 −3 /cc or ω P m a . Hence I find that the characteristic time scale over which a QCD axion condensate gets completely depleted in the regime of ω p m a is less than a millisecond. For a fixed αφ πfa , although this rate decreases with m a , it remains substantially fast for M > 100M . This implies that in the absence of a plasma a condensate of axions around black holes would decay to photons escaping detection through gravitational signatures as well as electromagnetic signatures. However, as my estimates above suggests, the plasma frequency of the interstellar medium is larger than the axion mass for most of the parameter range. It is only for stellar mass black holes (M < 100M ) that there is a possibility of lasing and the rate of lasing is given by λ M AX . Hence, it can be concluded that lasing is unlikely to disrupt axion condensates around black holes of mass M > 100M unless the density around is exceptionally low. The possibility of lasing does exist for 2M < M < 100M . In these cases if ω p < 0.1m a , lasing can deplete the condensate in less than a second releasing a significant fraction of the energy of the condensate ( of the order of the mass of the black hole) in photons of long wavelength. This would indeed imply absence of gravitational signatures from deformation of axion cloud in binary mergers of low mass black holes or monochromatic gravitational wave emissions from axion anihilation etc. However, if lasing occurs during a low mass BH-BH merger event, it can leave its imprints in the gravitational wave form which will indicate a sudden loss of mass of one or both of the binary constituents.

conclusion
In this paper I analyzed the possibility of lasing of superradiant axion condensate around black holes in the presence of matter coupling to Maxwell's equations. In the simple toy model assumed in this paper, I include medium effects by considering Maxwell's equations with a finite conductivity. The accretion region is dominated by collisions and described by the collision-dominated limit of the conductivity. Owing to its large conductivity, the accretion disc is found to hinder lasing. For a thin accretion disc, outside of the accreting region, densities are sufficiently small so as to allow for the possibility of lasing provided the plasma frequency of this region is smaller than the axion mass. This region is mostly collision-less and is expected to approximately resemble the interstellar medium. Estimating the plasma frequency of the interstellar medium I find that lasing of axion condensates is unlikely to occur around black holes of mass greater than 100M . For smaller black holes 2M < M < 10M lasing is possible provided the density around the black hole is smaller than 0.001/cc. I calculate the range of allowed wavelengths for growing modes as well as their growth rates as a function of the plasma frequency for ω P ≈ O(m a ).
The calculations in this paper can be improved using models that capture the features of the black hole environment as well as the axion condensate in greater detail. The simple toy model in this paper assumes a spatially uniform axion condensate whose time dependence is solely dictated by the mass of the axion. I also ignore the effect of the curvature of the metric on lasing and assume the magnitude of the condensate to be constant in time. In principle axion lasing can take place while the superradiant instability itself is arising and an accurate analysis of this will have to involve solving coupled equations describing simultaneous growth or decay of both superradiance and the lasing processes. Ideally, to decide the fate of superradiant axions including the possibility of lasing requires solving the equations of motion of axion Lagrangian coupled to electromagnetism in a Kerr metric. This should be addressed in future work. Similarly, the spatial dependence of the axion condensate can be taken into account by using in the axion wave function instead of a spatially uniform axion condensate in the Maxwell's equations.

acknowledgement
I would like to thank David B. Kaplan, Sanjay Reddy and Tom Quinn for insightful discussions. This work was supported by U.S. Department of Energy under grant Contract Number DE-FG02-00ER41132.