Stimulated Emission of Dark Matter Axion from Condensed Matter Excitations

We discuss a possible principle for detecting dark matter axions in galactic halos. If axions constitute a condensate in the Milky Way, stimulated emissions of the axions from a type of excitation in condensed matter can be detectable. We provide general mechanism for the dark matter emission, and, as a concrete example, an emission of dark matter axions from magnetic vortex strings in a type II superconductor are investigated along with possible experimental signatures.


Introduction
Recent discovery of the Higgs particle revealed that the standard model of particle physics correctly describes the fundamental constituents of our universe, up to the energy scale ∼ 1 TeV [1]. However, the current standard cosmology strongly indicates the existence of additional constituents, which are known as the dark matter [2,3]. In particular, observations of the rotation velocities of galaxies imply that there exists the dark matter halo around the Milky Way, whose mass density is given by Although there are various candidates of the dark matter, including the weakly interacting massive particles (WIMPs), the origin and properties of the dark matter remain a mystery, so far. (See [4,5] for the current status of the dark matter research.) Among them, the axion is one of the promising candidates for the dark matter in our universe. The axion is a hypothetical elementary particle, which gives the most plausible solution to the strong CP puzzle in quantum chromodynamics (QCD) [6,7]. Through various experiments and astrophysical observations for axion search, the properties of the axion, such as the mass and coupling strength, are severely constrained [1,8,9]. For example, the typical mass consistent with the experimental constraints can be summarized as 10 −6 eV/c 2 m a 10 −3 eV/c 2 , for the invisible QCD axion models [10,11,12,13]. This small mass distinguishes the axions from other WIMPs, whose masses are assumed to be m wimp 10 9 [eV/c 2 ]. In contrast to the standard WIMPs which are usually fermions, axions obey the Bose-Einstein statistics. In order to consistently explain the dark matter density in terms of the axion, a large number density of the bosonic axions is required in our galaxy, and the possibility of the Bose-Einstein condensation of dark matter axions due to such a large density has been discussed [14,15,16,17,18].
In this paper, we discuss a possible consequence of the Bose-Einstein condensation of the dark matter axions: stimulated emissions of the axions. Based on the coupling between the axion field and the electromagnetic field, we discuss the stimulated emissions of the axions from collective excitations in various condensed matter systems. In particular, as a concrete example, we investigate the stimulated emission from the magnetic vortex strings in type II superconductors, where a mobile vortex ensemble, such as vortex flow or vortex liquid, is realized near the critical temperature. The emission rate of the dark matter axions are estimated, and a possible experimental signature of the emission is discussed.

Properties of Dark Matter Axions
In this section, we briefly summarize the fundamental properties of the axion, in particular, the interaction between the axions and the electromagnetic fields. We also discuss the properties of the axion which make it to be a good candidate for the (cold) dark matter, with an emphasis on the difference from other dark matter candidates, such as WIMPs.

Coupling of Axion with Electromagnetic Field
Since the axion is a (pseudo) Nambu-Goldstone boson [19,20] of the spontaneous breaking of the Peccei-Quinn (PQ) symmetry [6,7], the axion field φ a couples to other particles through its derivative, originated from a universal coupling, where j µ PQ is the conserved current of the PQ-symmetry. Although the strength of this coupling is very small for the invisible axion models, various experiments to detect the axions using this type of coupling have been implemented so far [8,9].
The axion has an additional coupling in the leading order with the electromagnetic (EM) field, which is a parity-odd coupling originated from the quantum anomaly of the PQ symmetry: In this paper, we focus on this axion coupling (4) and discuss its consequence. *1 With this coupling, the action for the axion field and EM-field is given by where m a is the mass of the axion and the dots represent higher order terms with respect to the coupling g 0 . The coupling strength g 0 is constrained to be very small from various experiments and astronomical observations [1,8,9], and the current bound is given by where the first value is in the natural units, and the second one is in the SI units with C being Coulomb for the electric charge. (See the Appendix for the conversion between these *1 The experiments for the axion detection using this coupling have also been performed [8,9]. unit systems.) We optimistically take the value g 0 ≃ 10 −10 [GeV −1 ] in this paper. Due to this small value, the higher order terms with respect to g 0 can be completely neglected. Although the coupling (4) leads to the axion-photon scattering and the axion decay to two photons, the scattering and decay rates are very small and the lifetime of the axion is much longer than the current age of the universe. This indicates that the axions produced in the early universe with some mechanism, such as the misalignment of the vacuum angle and the cosmic string decay, can remain for a long time, and these relic axions can be promising candidates for the dark matter in our universe. Particularly, the production mechanisms are mainly non-thermal, and the axions, which are produced around the time of the QCD transition, behave as non-relativistic particles from the beginning. So, the axion is a good candidate for the cold dark matter [2,3].
If the axion accounts for the major part of the dark matter of our universe, the abundance of the axion in our galaxy should explain the mass density (1), which is determined by the observation of the rotation curve of various galaxies [4,5]. The velocity distribution of the dark matter in the galactic halo has also been discussed, and it is known that the violent relaxation by gravitational attraction [21] leads to the Maxwellian distribution, where the root-mean-squared velocity is given by v a ≃ 2.7 × 10 5 [m/sec] ∼ 10 −3 c .
As noted above, since v a /c ∼ 10 −3 ≪ 1, the axions in our galaxy behave as non-relativistic particles. Although the mass of the axion depends on the models beyond the standard model of the elementary particles, the plausible mass range is discussed from the various viewpoints including cosmology, astronomy and elementary particle physics [8]. In particular, in order to explain the dark matter abundance in our universe, the expected mass should be in the range (2). If we take m a ≃ 10 −6 [eV/c 2 ] as a typical value, we can calculate the number density n a of the dark matter axions in our galaxy,

Non-Relativistic Limit of Axion Field
Since the axions behave as non-relativistic particles (v a ∼ 10 −3 c), we can reasonably consider the non-relativistic limit of the axion field φ a (x, t). At first, we separate the rest mass contribution from the time dependence of the axion field as where φ † a = φ a is satisfied for the real scalar. Inserting this ansatz into the axion part of the Lagrangian (5), we obtain the following non-relativistic action where the second derivative term with respect to time is discarded. This is nothing but the Lagrangian of the Schrödinger field, which satisfies the non-relativistic Schrödinger equation.
The non-relativistic axion field ϕ a (x, t) becomes a complex scalar field, whose dimension is given by [L −3/2 ]. Here, we define the rest mass frequency as for the typical mass m a ≃ 10 −6 [eV/c 2 ].
In the non-relativistic limit, the axion coupling is also renormalized. Inserting the nonrelativistic ansatz of the axion field (9) into the original coupling (4), we obtain the axion coupling in the non-relativistic limit, Here, the non-relativistic coupling strength is defined as where the typical value m a ∼ 10 −6 [eV/c 2 ] is used. This is the basic formula describing the interaction between the dark matter axions and excitations in condensed matter systems.

Dark Matter Axion as Condensate
Based on the discussion above, we are lead to a natural but somewhat surprising consequence: the dark matter axions can constitute a (Bose-Einstein (BE)) condensate. At first, the non-relativistic Lagrangian of the axion (10) essentially describes non-interacting particles obeying the Bose statistics. *2 Given the large number density (8) and applying the standard argument on the BE condensation to this axion system, the critical temperature is estimated by k B T c ≃ 2π 2 /m a n 2/3 a ∼ 10 7 [eV]. Furthermore, the thermal energy of dark matter axions, k B T a , is estimated from the average velocity (7) to be k B T a ∼ (1/2)m a v 2 a ≃ 10 −13 [eV]. Therefore, if the dark matter axions are in thermal equilibrium in our galaxy, the axion field is naturally expected to be in a BE condensate phase at the extreme low temperature *2 The coupling strength with themselves and other particles are so small that it can be essentially ignored.
T a (≪ T c ). The possibility of the thermalization of the dark matter axions by the small selfinteraction and the gravitational interaction has been recently discussed in the literatures [14,15,16,17,18], and some consequences from the resulting BE condensation have been explored. In this paper, we assume that the dark matter axions do thermalize and form the BE condensate in our galaxy. *3 Assuming that the dark matter axions are in a condensed phase, we can estimate the occupation number N k of the mode with the typical momentum p a ≃ m a v a ∼ 10 −9 [eV/c].
The corresponding thermal de Broglie wave length is given by λ a = 2π /p a ∼ 10 3 [m], which characterizes the quantum coherence scale of the BE condensation, as discussed in [18]. Using this wave length, the typical occupation number at the mode with k a = 2π/λ a becomes very large : As explained below, this large occupation number N k gives an enhancement factor for the transition probability, which is analogous to the case of the stimulated emission of photons, used in the LASER. Note that the occupation number N k is proportional to m −4 a .
In this condensed regime, the non-relativistic axion field ϕ a (x, t) behaves as a coherent classical field in the leading approximation, Using this condensed ansatz of the axion field, we can further rewrite the axion coupling in the following form: The factor √ n a also leads to an enhancement of the coupling strength g, and gives a timeperiodic coupling for collective excitations, as will be seen in the following.

Excitations Coupled with Axions in Condensed Matter
Based on the basic interaction (16), three types of excitations in condensed matter systems can possibly couple with the dark matter axions. *3 It has been argued the possibility that miniclusters composed of axions have been formed through the galaxy formation process [22,23]. (For the recent discussions, see [18,24,25].) In this case, we assume that a significant fraction of axions have been thrown off due to the collisions and mergings of such miniclusters, and some fraction of axions forms a smooth halo distribution.
(A) Excitations creating the electric field E under the external magnetic field B ex : For concreteness, we consider the homogeneous external magnetic field in the z-direction, B ex = (0, 0, B 0 ). In the external magnetic field, the excitations creating the z-component of electric field E z can couple with the axions. In this case, the basic coupling (16) approximately becomes Here, the electric field associated with the excitation is given by  (16) is given by where the magnetic field associated with the excitation is assumed to be B z = γa+γ * a † .
The candidates are magnons and domain walls in magnetic systems, etc.
(C) Excitations which can create both the electric and magnetic field: Recent studies on multiferroic and topological materials show that there exist the excitations creating both the electric field in response to an external magnetic field and the magnetic field in response to an external electric field. Through the magnetoelectric effect, these excitations have the coupling with the axions in the form of (17) or (18), depending on the external field. The interesting candidates for such excitations are skyrmions in chiral-lattice magnets [26] and condensed matter analogues of the axions in a topological insulator with time-reversal breaking (so-called topological magnetic insulator) [27,28]. *4 In the following, we focus on the case of the excitations of type A, since the discussions are almost parallel for those of type B and C. In particular, as a concrete example and an interesting possibility, the magnetic vortex strings in type II superconductors will be discussed thoroughly.
*4 A realization of axion electrodynamics is also discussed by using optical lattices [29].

Axion-Vortex Coupling in Type II Superconductors
In this section, we discuss the effects of the axion coupling (12) in the dynamics of type II superconductors and, in particular, magnetic vortex strings.

Ginzburg-Landau Description of Superconductor and Magnetic Vortex
We start with the Ginzburg-Landau (GL) equations, which give the effective description of type II superconductors [30,31].
Here, we denote the order parameter of superconductors as ψ, and e * and m * are the electric charge and mass of the order parameter, respectively. α and β correspond to the parameters which can be calculated from the microscopic theoryà la BCS.
From the second equation (20), we can read the supercurrent, where we insert the decomposition of the order parameter, ψ(x) = f (x) e iθ(x) , for the second equality.
As is well-known, the GL equations have the magnetic vortex solutions that are topologically stable [32,33]. For the later discussion, we briefly summarize the solitonic vortex solution. A straight single vortex, which is stretched in the z-direction, is described by the field configuration with the cylindrical symmetry: Here, (r, θ, z) are the cylindrical coordinates, and d z is the unit vector in the z-direction.
A(r) and f (r) are the functions of the radial coordinate r only, and the flux is normalized to be the minimal flux quantum, h/e * . With these ansatze, the GL equations are reduced to the following coupled ordinary differential equations with respect to r: Here, we define the coherence length for the order parameter: With the boundary conditions f (r) → 1 and A(r) → 1/r at r → ∞, *5 we can easily obtain the numerical solutions for these equations, which describe the straight vortex with minimal flux. Away from the vortex core (f (r) ≃ 1), we can obtain another characteristic length scale, which defines the penetration depth of the electromagnetic field in superconductors.

Vortex Dynamics with Axion Coupling
Now, we consider the interaction between the axion fields ϕ a and magnetic vortex strings based on the axion coupling (12). Since the vortex string is a typical example of the excitations of type A, we consider the external magnetic field B ex in the z-direction, B ex . Then, the axion coupling becomes Here, we derive the electric field created by the vortex motion in a simple set-up. For concreteness, we consider the limit of large penetration depth and small coherence length, compared to the size of a superconducting sample. In this situation, an external magnetic field becomes almost homogeneous in the sample and the core of magnetic vortex becomes infinitely thin. *6 Outside the core of vortex, the GL equation (20) and supercurrent (21) give A = ( /e * ) ∇θ. From the AC Josephson effect and gauge invariance, an electrostatic potential is also given by Φ = − ( /e * )θ. Since the motion of the vortex with (small) velocity v L implies a time-dependent phase θ = θ( x− v L t), the electric field originated from the vortex motion is obtained as where the multivalued property of the phase θ(x) is used, and d is a unit vector along the vortex axis, and the two-dimensional delta function has the support on the core of vortex.
As seen from the vortex electric field (28), the motion in the plane perpendicular to the vortex axis is important, and we focus on the perpendicular motion in the following.
*5 For non-singular solutions, the boundary condition f (r) → 0 at r → 0 is also imposed. *6 In the small ξ limit, we take the order parameter as ψ(x) = f e iθ(x) with a constant f .
Concretely, we regard the vortex string as a point particle on the perpendicular plane with a mass m vor , by integrating out the spatial coordinate dependence in the Lagrangian. Using the formula (28) and the homogeneous property (15) of ϕ a , the axion coupling can be written in the minimal coupling form, where ℓ vor is the length of the vortex string. Here, the effective gauge field A eff is introduced and its magnitude is given by The above analysis can be simply extended to the case of general type II superconductors, using the time-dependent Ginzburg-Landau theory [34,31].
This term and a standard kinetic term lead to the Lagrangian of a charged particle on the perpendicular plane, coupled with an AC electric field, Thus, our problem on the perpendicular motion of a rigid vortex with the axion coupling (27) is reduced to the problem of a charged particle coupled with an AC electric field, whose frequency is Ω ∼ 1 [GHz] and magnitude is given by where we have used the ansatz of the BE condensation of the axion field (15). If we put the parameters B 0 ∼ 10 [T], ℓ vor ∼ 10 −4 [m] as an example, the drift force by the effective "electric field" can be estimated as F drift ∼ 10 −27 [N], which is so small that it is hardly detectable.
Note that, for this coupling to work, we require the existence of the vortices or segments of the vortex which are non-parallel to the external magnetic field B ext , as is evident from (29). For this purpose, we consider the following types of configurations of the vortex strings: (a) Tilted vortex strings against a tuned external magnetic field, due to a boundary condition of a small superconducting sample.
(b) Wavy vortex strings with the non-paralell segments or the vortex strings with the kinklike segments, due to the small tension near the phase transition.
In the following, we discuss mainly the case of the tilted vortex strings of type (a). For the wavy or kink-like vortex strings, the calculations can be done in the same way, except for replacing the length and number of the vortex strings with those of the segments.

Stimulated Emission of Axions from Vortices
In this section, we estimate the emission probability of axions from a vortex motion, using the first order perturbation theory, i.e. the Fermi's golden rule. In order to realize the stimulated emission process by using the coupling (29), which is originated from the axion coupling (4), we require that there exist the excited states of the vortex dynamics having the energy level ∆E ∼ Ω, which enables the emission of the non-relativistic axions with the energy, E a ≃ m a c 2 + ma 2 v 2 a . Furthermore, as in the case of the LASER, the non-equilibrium situation between the vortex system and the axions is also required, and the enough number of states of the vortices should be excited from the ground state. This implies that the vortex system should have a higher temperature than both the axion temperature T a and T gap = ∆E/k B . Therefore, we make the following assumptions on the dynamics of vortices in type II superconductors: (I) The energy spectrum of the vortex dynamics has the excitations of the order of Ω ∼ 10 −6 eV, which corresponds to the energy of the condensed axions.
(II) The vortices in the superconductor are in thermal equilibrium of the temperature T SC , higher than T a and T gap , without considering the effect of the dark matter axions.
To realize the assumption (I), we consider the magnetic vortices in mobile vortex systems where the excitaions with such a small energy level can be realized and observed as a resonance in experiments with microwave around GHz. Such mobile vortices can be realized in vortex creep, vortex flow, and vortex liquid in a type of superconductors [35]. For the assumption (II), we take the typical temperature of a mobile vortex ensemble in superconductors as T SC ∼ 4 K, *7 where the thermal energy of the vortex is typically k B T SC ≃ 10 −4 eV. It should be noted that the energy spectrum of the vortex dynamics in mobile vortex systems is expected to be broad around ∆E, and the emission can take place within a certain range of the axion mass.
In the second quantized language, the non-relativistic axion fields ϕ a has the Fourier expansion, Here, V a is the volume of a box where the axion field is defined, and we finally take the limit coupling becomes where n A = ( B ex × d )/ |B ex | is a unit vector in the direction of the effective gauge field.
Here, we used the low-energy approximation, e i k· x ∼ 1 and Ω ≫ ω, and the axion distribution, which is highly concentrated around k ≃ k a based on the assumption of the axion condensation.
Since this coupling is the same form as the standard minimal coupling to the EM-field, the interaction Hamiltonian, which contributes the perturbative calculation, simply becomes From this interaction Hamiltonian, we can estimate the emission probability within the firstorder perturbation, by using the Fermi's golden formula, where the momentum integration is performed over the final states of the emitted axion.
Since we assume that the axions form the BE condensate, the problem is rather simplified.
In the condensed phase, the state can be expressed by a coherent state, which is an eigenstate of the annihilation operator: From this property, the matrix elements for the axion sector is evaluated as where N k is the occupation number of the axions with a condensed momentum k a .
Inserting the explicit form of A eff (t) into the formula (37) and taking the emission part proportional to δ(E i − E f − Ω), the probability becomes where the matrix element for the vortex sector is denoted as p L ≡ i, vor| p L |f, vor . Since we assume that the temperature of the vortex system T SC ∼ 4 K is much higher than the temperature of the condensed axions T a ∼ 10 −9 K, the dominant process is the axion emission to the condensate. Note that the (apparent) dependence of the volume V a is cancelled in the resulting probability formula.
Further assuming the matrix element p L is independent of the momentum of the emitted axion in the low-energy region, we can perform the momentum integration, Gathering the above formulas, we have the expression for the emission probability of the axion from a single vortex, where we defined the vortex mass density ρ vor = m vor /ℓ vor [kg/m].
Finally, in order to evaluate the matrix element | p L | 2 , we consider a simple model of the vortex dynamics, for concreteness. The model is described by the Hamiltonian with an effective harmonic potential [35]: *8 Our result implies that, in an appropriate circumstances, the emission probability of the dark matter axions can be sizable, even if the axion coupling strength g 0 is quite small.
For this type of process, the Bose statistics and small mass of the dark matter axions are crucial, which distinguishes from other dark matter candidates, such as WIMPs. We comment here the difference from the conventional conversion process of axions into photons with a background magnetic field through the axion coupling. In this process, a mode mixing between the coherent axion field and the EM-field is essential, where the both fields are treated as classical waves, and the aforementioned stimulated emission does not play any role [36,37]. In our axion emission process, the vortex is a heavy localized object compared to the axion, and thus the spontaneous emission of axions can be discussed in analogy with the case of spontaneous radiation from an atom.

Possible Signature of Dark Matter Axions
An immediate consequence from the stimulated emission of dark matter axions is the axiondriven non-equilibrium states of the vortex ensemble in type II superconductors [30,31]. Another possibility is to detect the transport phenomena of the magnetic vortices in type II superconductors, such as the vortex Hall effect from the vortex motions, in a nonequilibrium situation [30,31,35].

Stimulated Emission from Other Excitations in Condensed Matter
So far, we have intensively discussed the stimulated emission from the magnetic vortices in superconductors, and estimated the emission rates, as a concrete example. As discussed in Section 3, there are other types of excitations which can couple with the dark matter axions via the coupling (16). If such excitations have the energy level E = Ω ∼ 10 −6 [eV], the same arguments as the vortex case can be applied straightforwardly.
At first, we consider an excitation in type A with the external magnetic field B ex = (0, 0, B 0 ). In the linear approximation, the electric field created by such an excitation is represented by E z = γ a+ γ * a † , with the annihilation and creation operators of the excitation mode. *9 In this case, the stimulated emission rate is generally given by the following formula Here, the number of the excitations in the sample is given by N exc , and the excitation level is assumed to be Ω ≃ m a c 2 . For the excitations of type B, which create a magnetic field under the background electric field E ext = (0, 0, E 0 ), similar calculations based on the coupling (18) can be applied, and we can obtain the emission rate, where N exc is the total number of the excitations in the sample.
If the excitations of type C in a multiferroic (or topological) material also have the energy spectrum of the order 10 −6 [eV], the stimulated emission of axions can be realized with either an external electric field or a magnetic field via the coupling (17) or (18). The emission rate is given by the formulas (47) or (48) in the same way.

Summary and Discussion
We have discussed the coupling between dark matter axions and excitations in various condensed matter systems, and physical consequences from the coupling. In particular, we investigated the stimulated emission of dark matter axions, which are expected to take place BE condensates in our galaxy, from the various excitations in condensed matter systems through the axion coupling. As a concrete example, the emission from magnetic vortex strings of mobile vortex systems in superconductors was investigated, and a possible experimental There is another interesting problem: The excitation of type C, such as an analogue of the axion in a topological magnetic insulator, has the same effective coupling as the axion coupling. Thus the mixing between the dark matter axions and condensed matter axions can occur in principle. Physical consequences from such a mixing will be discussed in a future work.
From the perspective of thermodynamics, the axion condensate plays the role of heat bath of an extremely low temperature for the condensed matter excitations discussed in this paper.
Therefore, the mechanism investigated in this paper can lead to a novel energy conversion process from the condensed matter excitations.
As is well-known, the dimensions of the EM-field are determined from the equations of motion for the charged particles, From these relations, we obtain the dimension of the vector potential, [ A] = M LT −1 Q −1 .
From the above dimensions, we have the dimension of the following term (operator), where c = 3.00 × 10 8 m/sec is the light velocity.
Thus, the axion coupling strength in the natural units g nat is given by The upper bounds of this coupling strength originate from various astronomical observations of sun, neutron stars, and supernovae, which lead to This means quite tiny coupling strength.