Axion-Radiation Conversion by Super and Normal Conductors

We have proposed a method for the detection of dark matter axion. It uses superconductor under strong magnetic field. As is well known, the dark matter axion induces oscillating electric field under magnetic field. The electric field is proportional to the magnetic field and makes charged particles oscillate in conductors. Then, radiations of electromagnetic fields are produced. Radiation flux depends on how large the electric field is induced and how large the number of charged particles is present in the conductors. We show that the electric field in superconductor is essentially identical to the one induced in vacuum. It is proportional to the magnetic field. It is only present in the surface because of Meissner effect. On the other hand, although the magnetic field can penetrates the normal conductor, the oscillating electric field is only present in the surface of the conductor because of the skin effect. The strength of the electric field induced in the surface is equal to the one in vacuum. We obtain the electric field in the superconductor by solving equations of electromagnetic fields coupled with axion and Cooper pair described by Ginzburg-Landau model. The electric field in the normal conductor is obtained by solving equations of electromagnetic fields in the conductor coupled with axion. We compare radiation flux from the cylindrical superconductor with that from the normal conductor with same size. We find that the radiation flux from the superconductor is a hundred times larger than the flux from the normal conductor. We also show that when we use superconducting resonant cavity, we obtain radiation energy generated in the cavity two times of the order of the magnitude larger than that in normal conducting resonant cavity.

We have proposed a method for the detection of dark matter axion. It uses superconductor under strong magnetic field. As is well known, the dark matter axion induces oscillating electric field under magnetic field. The electric field is proportional to the magnetic field and makes charged particles oscillate in conductors. Then, radiations of electromagnetic fields are produced. Radiation flux depends on how large the electric field is induced and how large the number of charged particles is present in the conductors. We show that the electric field in superconductor is essentially identical to the one induced in vacuum. It is proportional to the magnetic field. It is only present in the surface because of Meissner effect. On the other hand, although the magnetic field can penetrates the normal conductor, the oscillating electric field is only present in the surface of the conductor because of the skin effect. The strength of the electric field induced in the surface is equal to the one in vacuum. We obtain the electric field in the superconductor by solving equations of electromagnetic fields coupled with axion and Cooper pair described by Ginzburg-Landau model. The electric field in the normal conductor is obtained by solving equations of electromagnetic fields in the conductor coupled with axion. We compare radiation flux from the cylindrical superconductor with that from the normal conductor with same size. We find that the radiation flux from the superconductor is a hundred times larger than the flux from the normal conductor. We also show that when we use superconducting resonant cavity, we obtain radiation energy generated in the cavity two times of the order of the magnitude larger than that in normal conducting resonant cavity.

I. INTRODUCTION
Axion is the Nambu-Goldstone boson [1] associated with Peccei-Quinn symmetry. The symmetry is a global U(1) chiral symmetry. It naturally solves the strong CP problem. Because the strong CP violation has not yet been observed, we need to explain why the CP violating term G µ,νG µ,ν is absent or extremely small in QCD Lagrangian where G µ,ν (G µ,ν ) denotes ( dual ) fields strength of gluons. The axion is a real scalar field which is the phase of a complex scalar field carrying the U(1) charge. It acquires the mass m a through chiral anomaly. That is, instantons in QCD give rise to the mass of the axion. The axion is called as QCD axion. In the paper we mainly consider the QCD axion. The axion is generated in early universe and becomes one of dominant components of dark matter in the present universe when the Peccei-Quinn symmetry is broken after inflation [2].
In the present day, the axion is one of the most promising candidates for the dark matter. There proceed many projects for the detection of the axion. There are three types of the projects; Haloscope, Helioscope and others. The dark matter axion in our galaxy is searched with Haloscope, while axion produced in the Sun is searched with Helioscope. Haloscope projects are ADMX [3], CARRACK [4], HAYSTAC [5], ABRACADABRA [6], ORGAN [7], etc. Helioscope projects are CAST [8], SUMICO [9], etc.. The others are LSW [10] ( Light Shining through a Wall ), VMB [11] ( Vacuum Magnetic Birefringence ) etc. Since axion mass is unknown, the mass range in the search is very wide, 10 −20 eV∼ 1keV, or more [12]. There are axion-like particles proposed whose masses are not limited, differently to the QCD axion mass. ( The axion-like particles are those which couple with electromagnetic fields just as the QCD axion does. ) But we assume the QCD axion in this paper and expect that the appropriate mass range is of the order of 10 −6 eV∼ 10 −4 eV, It comes from cosmological consideration [2] and simulation in latttice gauge theory [13].
In previous paper [14] we have proposed a method of the detection of radiations generated by dark matter axion. It is to use a superconductor of cylindrical shape. In general, the dark matter axion induces oscillating electric field under strong magnetic field. The oscillating electric field makes Cooper pair oscillate in the superconductor so that the radiations are emitted. Similarly, we expect that radiations are also emitted by electrons in normal metal under the magnetic field. In this paper we examine both cases in detail and show that the radiation flux from the metal is a hundred times smaller than that from the superconductor.
Our proposal is a type of Haloscope, in which the dark matter axion is converted to photon ( radio wave ) under magnetic field. How large amount of radiation is generated depends on materials we use [15]. Here, we consider superconductor and normal conductor ( sometimes we call it simply as metal ). In the superconductor there are Cooper pairs with large number density ∼ 10 22 cm −3 , while in the metal there are electrons with the number density almost identical to that of the Cooper pairs. These charged particles emit radiations when they are forced to oscillate by the oscillating electric fields. The strength of the electric fields are proportional to external magnetic field we impose. But, it is different depending on each material.
In this paper we show that the electric fields induced in the superconductor and the normal conductor are essentially identical to the one induced in the vacuum. It is proportional to the magnetic field in the conductors. However, the electric field in the superconductor is only present in the surface because the magnetic field is only present in the surface to a penetration depth owing to the Meissner effect. On the other hand, although the magnetic field penetrates the normal conductor, the oscillating electric field is only present in its surface because of the skin effect. Theses oscillating electric fields generate oscillating electric currents. The oscillating electric currents in both conductors only flow in their surfaces and emit electromagnetic radiations.
To find how strong electric field is generated in the superconductor, we analyze Ginzburg-Landau model for the superconductor. The superconductivity is represented as a condensed state of Cooper pair in the model. We solve equations of electromagnetic fields coupled with axion and Cooper pair described by the model. On the other hand, to find the electric field in the metal, we solve equations of electromagnetic fields coupled with axion in the normal conductor characterized by permeability µ and electric conductivity σ.
The presence of the oscillating electric fields leads to radiations from the conductors. We show that the radiation flux from the superconductor is two times the order of the magnitude larger than that from the normal conductor. The difference arises from the difference between the penetration depth and the skin depth. ( The penetration depth in the superconductor is shorter than the skin depth as long as the frequency of the radiation is approximately less than 100GHz. ) The radiation flux from the superconductor is so large that existing radio telescope such as one with parabolic dish antenna of radius ∼ 10m can easily observe the radiation, even when the superconductor is put at a hundred meter away from the telescope. Indeed, the radiation flux from the cylindrical superconductor with radius ∼ 1cm and length ∼ 10cm under magnetic field ∼ 5T is of the order of ∼ 10 −18 W.
We also show that radiation energy generated in superconducting cavity is two times of the order of the magnitude larger than that in normal conducting cavity, e.g. copper. The difference arises from the difference in the depth from the surface in which the oscillating electric current flows.
In the next section, we introduce axion photon coupling and find electric field in vacuum induced by axion under magnetic field. In the section (III), we introduce Ginzburg-Landau model for superconductor, in which Cooper pair is described by the field Φ. The model represents the coupling with electromagnetic field and the Cooper pair, and describes Meissner effect characterizing the superconductivity. We derive electric field in the superconductor induced by axion under magnetic field. In the section(IV) we solve the equations of electromagnetic fields in normal conductor coupled with axion. We find that electric field in normal conductor induced by axion is only present in the surface of the conductor. In the section(V) we numerically show the radiation fluxes from the superconductor and normal conductor of cylindrical shape. In the section(VI) we estimate radiation energy generated in superconducting resonant cavity. In the final section(VII) summary and discussion follow.

II. ELECTRIC FIELD IN VACUUM
First we show that the coherent axion induces an electric field in vacuum under a magnetic field. It is well known that the axion a( x, t) couples with both electric E and magnetic fields B in the following, with the decay constant f a of the axion and the fine structure constant α ≃ 1/137, where the numerical constant k a depends on axion models; typically it is of the order of one. The standard notation g aγγ is such that g aγγ = k a α/f a π ≃ 0.14(m a /GeV 2 ) for DFSZ model [16] and g aγγ ≃ −0.39(m a /GeV 2 ) for KSVZ model [17]. In other words, k a ≃ 0.37 for DFSZ and k a ≃ −0.96 for KSVZ. The axion decay constant f a is related with the axion mass m a in the QCD axion; m a f a ≃ 6 × 10 −6 eV × 10 12 GeV. We show that the coupling parameter k a αa/f a π in the Lagrangian eq(1) is extremely small for the dark matter axion a(t). Furthermore, the dark matter axion a(t) can be treated as a classical field because the axions are coherent.
We note that the energy density of the dark matter axion is given by where a(t) = a 0 cos(t m 2 a + p 2 a ) ≃ a 0 cos(m a t), because the velocity p a /m a of the axion is about 10 −3 in our galaxy. The local energy density of dark matter in our galaxy is supposed such as ρ a ≃ 0.3GeV cm −3 ≃ 2.4 × 10 −42 GeV 4 . Assuming that the density is equal to that of the dark matter axion, we find extremely small parameter a/f a ≃ √ ρ a /(m a f a ) ∼ 10 −19 . The energy density also gives the large number density of the axions ρ a /m a ∼ 10 15 cm −3 (10 −6 eV/m a ), which causes their coherence. This allows us to treat the axions as the classical axion field a(t). Anyway, we find that the parameter k a αa(t)/f a π = g aγγ a(t) is extremely small.
The interaction term in eq(1) between axion and electromagnetic field slightly modifies Maxwell equations in vacuum, From the equations, we approximately obtain the electric field E generated by the axion a under the static background magnetic field B( x), assuming the parameter k a αa/f a π extremely small. This is the electric field in vacuum induced by the dark matter axion a(t, x) = a 0 cos(ω a t − p a · x) ≃ a 0 cos(m a t) with ω a = m 2 a + p 2 a ≃ m a . We note out that the magnetic field configuration is arbitrary.

III. ELECTRIC FIELD IN SUPERCONDUCTOR
Now we proceed to examine the electric field induced in superconductor. Especially, we suppose that the superconductor is present at x > 0 and uniform in y and z directions. The magnetic field is parallel to the surface of the superconductor. It is also uniform in y and z directions, and points to z direction. We suppose that the superconductor is described by Ginzburg-Landau model with Cooper pair Φ, with electric charge q = 2e ( electron charge e ) and coupling constant h, where A 0 and A denote gauge fields; When we include the effect of the Cooper pair, the modified Maxwell equations are led to the following, In order to see magnetic field configuration in the superconductor, we solve the second equation (7) with external magnetic field B = (0, 0, B z (x)) with B z (x) = B 0 for x < 0, neglecting the axion field, with Φ = v 0 and the penetration depth λ = ( √ 2qv 0 ) −1 . We have taken the value Φ = v 0 of Cooper pair in the supercondoctor. The equations (6) and (8) trivially hold when A 0 = 0 and Φ 0 = v 0 .
We find the magnetic field configuration in the superconductor; it penetrates the surface to the depth λ. That is, it represents the Meissner effect. Then, we expect that the electric field induced by the axion is also present only in the surface.
We will derive the electric field E a induced by the axion under the magnetic field B(x) = (0, 0, B z (x)). Supposing the parameter k a αa/f a π extremely small and small momentum of axion ∂a = 0, we derive the equations for δΦ and δ A from Ginzburg-Landau Lagrangian in eq(5) and modified Maxwell eq (7) with the gauge condition ∂ · δ A = 0, where δΦ and δ A is of the order of k a αa/f a π.
In addition to eq(10) and eq(11), we have the equation (6) for the gauge field A 0 = δA 0 , All of the variables δ A, δA 0 , and δΦ may oscillate according to the oscillation a(t) ∝ cos(m a t). Because we assume that the fields are uniform in y and z directions, we simplify the equation(11) by taking δ A = (0, 0, δA z ) as well as It is easy to obtain a solution of eq(10), eq(12) and eq (13), with arbitrary amplitude A ′ 0 and frequency ω, where we have λ ′ ≃ λ(1 + ω 2 λ 2 /2), assuming ω 2 ≪ λ −2 = 2q 2 v 2 0 . Namely, the electric field is The first term in E a,z represent a radiation with frequency ω entering from outside the superconductor, while the second term represents the electric field E a = −g aγγ a(t) B(x) induced by the dark matter axion. It is just equal to the one in eq(4) in the vacuum. The radiation entering the superconductor from outside only penetrates the surface of the superconductor to the penetration depth λ ′ ≃ λ owing to the Meissner effect It is instructive to see that we have small suppression factor (m a λ) 2 in the electric field E a when the second term (13) is absent. Namely, E a = −(λm a ) 2 g aγγ a(t) B. ( Typically , λ ∼ 10 −6 cm and m −1 a ∼ 10cm. ) The second term represents the effect of the Cooper pair and cancels the derivative ∂ 2 x δ A in eq(13) because B z ∝ exp(−x/λ). Thus, we find that the suppression factor is absent in the superconductor; E a (x, t) = −g aγγ a(t) B(x). The electric field E a is only present in the surface of the superconductor because the magnetic field B is only present in the surface. These results were used in the previous paper [14].
Here we make a brief comment that similar electric field E a = −g aγγ a(t) B is induced around magnetic vortices in type 2 superconductor. As we know, the magnetic field penetrates the type 2 superconductor and forms vortices. They are described as classical solutions in the Ginzburg-Landau model. A solution of the magnetic vortex located at ρ = 0 is characterized in the following, with integer n, where the cylindrical coordinate (ρ, θ, z) is used. The important feature of the vortex is the flux quantization; dρdθB v z = 2nπ/q. The type 2 superconductor has a property that the penetration depth λ is larger than the coherent length ξ of the Cooper pair; λ > ξ. Thus, we may approximate these fields such as |Φ v (ρ)| = v 0 and B v z ∝ exp(−ρ/λ) for ρ > λ. In the region ρ > λ, we have a similar equation to eq(13) for δ A ≡ A − A 0 = (0, 0, δA z ) in the cylindrical coordinate by putting δΦ = δA 0 = 0 and Then, we find that the solution of the electric field E a = −∂ t δ A is given such as E a (ρ) = −g aγγ a B v (ρ). The solution holds only in the region ( ρ > ξ ) outside the vortex core. This is the electric field expected naively when the magnetic field B v (ρ) is present. Obviously, it is attached to the vortex.
We would like to mention that the current density J c induced by the axion is given such as The current J c ∝ sin(m a t) has no dissipation of its energy; 2π/ma 0 dtJ c (t)E a (t) = 0. Namely, the current is a superconducting current even in the presence of the electric field. The fact is valid both for the surface current in the superconductor and the current flowing along the magnetic vortex.
We note that the penetration depth is rewritten such as λ = 1/2q 2 v 2 0 = m c /q 2 n c in terms of the number density n c , where we use the relation n c = 2m c v 2 0 mentioned above. Then, the current density is also rewritten such We will explain later that the formula can be obtained by using the Drude model. The model describes the classical motion of Cooper pair under the electric field E a .
In this section, we find that the electric field induced in the superconductor is essentially identical to the one induced in the vacuum, although the electric field is confined to the surface of the superconductor. It oscillates with the frequency identical to the one of the axion field a(t) ∝ cos(ω a t). Similarly, the current density J c also oscillates with the same frequency.

IV. ELECTRIC FIELD IN NORMAL CONDUCTOR
Next, we derive the electric field in the normal conductor ( metal ) with permeability µ and electric conductivity σ. The configuration of the metal is the same as the case of the superconductor. That is, the metal is present at x > 0 and uniform in y and z directions. We impose magnetic field B = (0, 0, B z ) parallel to the metal. The magnetic field can penetrate the metal. We denote the magnetic field inside as B in . Because the component of the field H in = B in /µ parallel to the surface ( at x = 0 ) of the metal is continuous at the surface, the magnetic field B in is obtained such that B in = Bµ/µ 0 where we denote the vacuum permeability µ 0 ( µ 0 = 1 in natural unit ). For instance, µ ≃ µ 0 for copper or µ ∼ 5000 × µ 0 for iron. We should note that in general, the permeability µ depends on the magnetic field B and that the strength B in = Bµ(B)/µ 0 saturates as the external field B increases. The maximal magnetic field B in is at most 1T∼ 2T. Thus, we cannot make B in increase unlimitedly. Hereafter, we assume that µ is independent of B and we use natural unit µ 0 = 1.
The electromagnetic fields in the metal are described by the modified Maxwell equations including non trivial permeability of the metal, where H in = µ −1 B in and D = ǫ E with permittivity ǫ. The permittivity ǫ is nearly equal to 1 in the metal for radiations with frequency ∼ 1GHz. In eq (18) we have included the current J e = σ E induced by electric field E, but have neglected external current generating the background magnetic field B.
When we neglect axion contribution, we obtain magnetic field B 0 in = µB uniform inside the metal where B is external magnetic field imposed. Obviously, there is no electric field inside the metal. When we take into account the axion contribution, the oscillating electric field is induced. Naively we expect that the electric field is given such that E in a = −g aγγ a(t)µ B/ǫ. But, this is not correct as we show below. Assuming the parameter g aγγ a( x, t) small and noting that the electric field is the order of g aγγ a( x, t), we derive the equations, with δ E = E − E in a and δ B ≡ B in − µ B = B in − B 0 in , where we have used the relation ∂ × E in a = 0 because ∂ × B = 0 inside the metal. Here, we should note that δ B is the order of g aγγ a( x, t). The electric field E in a is the naive one expected in the metal.
Using the Ohm's law J e = σ E in eqs (19), we derive the equation for E, where we note that E in a ∝ cos(m a t). Then, we find the solution, with arbitrary field strength E 0 , and frequency ω. The skin depth δ is given by δ = 2/σµω. In the derivation, we have neglected the term µǫ∂ 2 t E in the left hand side of eq (20), which is much smaller than the term σµ∂ t E in the right hand side. Namely, we have used the inequality µσ ≫ ω (∼ m a ) because the electric conductivity σ ∼ 10 4 eV in copper or iron is much larger than the axion mass m a = 10 −6 eV ∼ 10 −4 eV under consideration.
The first term in the solution E represents oscillating electric field with the skin depth δ, while the second term represents the oscillating electric field E in inside the metal; E in ≡ ǫ σ ∂ t E in a . The first term is only present in the surface of the metal and represents a radiation with frequency ω entering from outside the metal. The strength E 0 is determined by the boundary condition at the surface x = 0. In our case it is determined by electric field induced outside the metal, i.e. electric field in the vacuum. It is just E a = −g aγγ a(t) B. Because of the continuity of the electric field parallel to the surface, the electric field in the surface is given by E suf = −g aγγ a 0 B exp(− x δ ) cos(m a t− x δ ). ( Similar consideration in the superconductor leads to the electric field E a derived in the previous section. ) On the other hand, the electric field E in present inside the metal is the one induced by the dark matter axion. It is suppressed by the factor m a /σ compared with the naive one E in a . This oscillating electric field induces the oscillating current J e = σ E in = ǫ∂ t E in .
In general, electric field is absent inside conductor with finite size because the field is screened by the electric field produced by surface charge. It is induced on the surface of the boundary between the conductor and vacuum. ( Even if the electric field is present inside the metal, free electrons move to make the field cancelled by the surface charge. ) In our case there is no such surface charge because we consider the conductor extended infinitely in the direction z and y. Thus we have non vanishing electric field E in inside the metal.
When the conductor has finite size, the electric field E in is absent. But we show that the oscillating electric current J e = ǫ∂ t E in a is present. We suppose that the shape of the metal is cylindrical and the metal has finite size; the metal has upper and down surfaces. The external magnetic field B imposed parallel to the cylinder is perpendicular to the upper and down surfaces. The perpendicular magnetic field is continuous at the surfaces. Then, the electric field E just outside the upper or down surfaces induced by the axion is given by is the magnetic field inside the metal. It is also the magnetic field just outside the metal. Differently to the case of the magnetic field, the electric field is absent inside the metal. Thus the electric field perpendicular to the surfaces is discontinuous at the surfaces. Then, there are surface charges density σ f on the upper and down surfaces; σ f = ±E in a . Because the field E in a oscillates, the charge density also oscillates. It means that an oscillating electric current is produced [18] such as J e = ∂ t σ f = ∂ t E in a . This current flows in the surface to the skin depth. It is the physical reason why the oscillating current J e is generated in the metal. ( We have set ǫ = 1 in the argument because ǫ ≃ 1 for the electric field with the frequency m a /2π ∼ 1GHz. ) We make a comment that the electric field E in = ǫ σ ∂ t E in a vanishes in the limit of infinite conductivity, σ → ∞. Namely, there is no electric field in perfect conductor. It means that the above formulation cannot be applied to the superconductor, although the superconductor is perfect conductor. We have the electric field E a in the surface of the superconductor, as we have shown. We need a model like Ginzburg-Landau model for the superconductor to appropriately describe electromagnetic fields in the superconductor.
The current density J e = ∂ t E in a = m a µg aγγ a 0 B sin(m a t) flows in the surface of the metal, while there is an additional current J suf e flowing in the surface. It is given such that J suf e = σ E suf = −σg aγγ a 0 B exp(− x δ ) cos(m a t − x δ ), because the electric field E suf is present in the surface. Obviously, the current density J suf e is much larger than J e . Differently to the superconducting current J c , the energy of the current J suf e is dissipated; Because the current oscillates, it generates dipole radiation from the metal.
In this section we find that electric field E suf = −g aγγ a 0 B exp(− x δ ) cos(m a t − x δ ) is induced in the surface of the metal, which produces the surface current J suf e = σ E suf . The strength of the electric field E suf a is almost identical to that of the electric field E a in the superconductor; E suf a (x = 0) = E a .

V. RADIATION FLUX FROM CYLINDRICAL CONDUCTORS
We proceed to show how large amount of radiation is emitted from the superconductor as well as the normal conductor. The radiation is generated by the oscillating current J c ( J suf e ) in the superconductor ( normal conductor ) induced by the oscillating electric fields E a ( E suf a ). Because the oscillation is harmonic, the radiation is dipole radiation. The current is carried by Cooper pair ( electrons ) in the superconductor ( normal conductor ).
According to the Drude model, we give a simple argument for the form of the current density J c = q 2 E a n c /m a m c We make a comment that the current J c = qn c v c = q t dt ′ E a (t ′ )/m c ∝ sin(m a t) has a phase different by π/2 with the electric field E a ∝ cos(m a t). It leads to the dissipation less current in the superconductor, as we have shown in previous section. On the other hand, the current J e = en e v e = e 2 n e E suf a τ /m e has the identical phase to that of the electric field E suf a so that the current in the normal conductor is dissipative. The current oscillates with the frequency m a /2π and flows in the direction of the magnetic field. The spectrum of the axions has the peak frequency m a /2π with small bandwidth ∆ω ≃ 10 −6 × m a , because of the velocity v a ∼ 10 −3 of the dark matter axion in our galaxy. Thus, the oscillating current has the same spectrum as that of the axion.
We have proposed a method for the conversion of the dark matter axions to electromagnetic waves. We use a superconductor of cylindrical shape, on which the magnetic field B parallel to the direction along the length of the superconductor is imposed. We take the direction as z direction. The magnetic field is expelled from the superconductor. But the field penetrates into the superconductor to the depth λ = m c /q 2 n c ( London penetration depth ). Thus, the oscillating current is present only in the surface to the depth λ. In general, the oscillating current in conductors is only present in the surface with the skin depth δ = 2/µωσ. But, the skin depth is larger than the penetration depth, only to which the magnetic field is present. Typically, the penetration depth is λ ∼ 10 −5 cm, while the skin depth is δ ∼ 10 −4 cm for copper with the frequency ω = 1GHz. Thus, the oscillating current J c ∝ B in the superconductor is only present to the penetration depth. Now, we estimate the radiation flux emitted by the cylindrical superconductor under the magnetic field B. We suppose that the superconductor has radius R = 1cm and length l = 10cm. Then, the flux of the dipole radiation is given by with E a ≡ E 0 cos(m a t) ( E 0 = −k a αa 0 B/f a π = −g aγγ a 0 B ), where we have used the formulae of the penetration depth λ = m c /q 2 n c . When we use the superconductor Nb 3 Sn, the penetration depth is about λ = 8 × 10 −6 cm. Then, we numerically estimate the flux S, with k a ≃ 0.37 for DFSZ model and k a ≃ −0.96 for KSVZ model, where there is no dependence on the axion mass.
The spectrum of the radiation shows a sharp peak at the frequency m a /2π with the bandwidth ∆ω ∼ 10 −6 m a /2π. It is remarkable that the radiation flux in eq(23) is four times of the order of magnitude larger than that obtained in resonant cavity [3,20]. The flux is obtained by integrating a Poynting vector over the sphere with radius r ≫ 2π/m a around the superconductor; S c = S c (θ, r)r 2 dΩ = S c (θ, r)r 2 sin θdθdφ, where where we have taken the polar coordinate. The dipole radiation is emitted mainly toward the direction ( θ = π/2 ) perpendicular to the electric current flowing in z direction. Thus, when we measure the radiation emitted in the direction, we receive relatively strong flux density. For example, when we observe the radiation using the radio telescope of parabolic dish antenna with the diameter 32m by putting the cylindrical superconductor 100m away from the telescope ( e.g. Yamaguchi 32-m radio telescope of National Astrophysical Observatory of Japan ), the observed flux per frequency P c is given by with δ t ≃ 16m/100m = 0.16 and ∆ω = 10 3 Hz m a /(6 × 10 −6 eV) , where the center of the parabolic antenna is set in the direction θ = π/2. Thus, we find that the antenna temperature is approximately T a ≡ P η ≃ 1.5K with the unit k B = 1, assuming the antenna efficiency η ≃ 0.6. Therefore, the radiation can be observed with the radio telescope with diameter such as 32m.
We estimate the detection sensitivity. When we observe the radiation over time t with bandwidth δω, the ratio of signal to noise is given by S/N =S c Tsys t/δω whereS c = 3δ 2 t S c /8 ∼ 10 −2 S c denotes the radiation flux received by the telescope 100m away from the superconductor and T sys is the system noise temperature. For instance, T sys = 40K for Yamaguchi 32-m radio telescope. Therefore, we find that with T sys = 40K, where we have taken the physical parameters B = 7T, R = 2cm and l = 20cm to have better detection sensitivity. Here, we put g 15 ≡ g aγγ /(10 −15 GeV −1 ) and m 6 ≡ m a /(10 −6 eV). Thus, even for DFSZ axion ( ( g15 m6 ) 2 ≃ 0.1 ), we reach the sensitivity S/N ∼ 4 when we observe the radiation over 1 second with the bandwidth δω = 1MHz. In the formula we do not use the relation m a f a ≃ 6 × 10 −6 eV × 10 12 GeV specific to the QCD axion. The formula holds even for axion-like particle.
Obviously, the larger radiation flux can be obtained when we put the superconductor nearer the radio telescope than 100m. Furthermore, even when we use a radio telescope with smaller radius than 32m, large S/N ratio can be achieved if we put the apparatus near the telescope. The radiation flux is determined by the solid angle of the parabolic dish antenna viewed from the superconductor. The larger solid angle leads to larger radiation flux. The merit of our proposal is that we can simultaneously search wide bandwidth of the radio frequency without tuning the shape of the superconductor. In this way, we can observe the radiation from the dark matter axion.
We make a comment on the actual setup for the observation. The strong magnetic field B parallel to the direction along the length of the cylindrical superconductor is produced by coils surrounding it. The coils should have open space for the dipole radiations to escape outside the coils and reach the telescope. In particular, the open space should be present in the θ = π/2 ± δ directions ( e.g. δ ≃ 16cm/100cm = 0.16 ) perpendicular to the cylindrical superconductor. That is, the coils are composed of two parts; one covers the upper side of the superconductor and the other one covers the lower side. Then, there is an open space in the coils through which the radiations can escape from the coils. Furthermore, the whole of the apparatus must be cooled. We need to use a glass container for liquid helium so as for the radiation to pass the container and reach radio telescope.
We make an additional comment on radiation from magnetic vortex. We use type 2 superconductor for the apparatus in order for the strong magnetic field not to break the superconductivity. Then, the magnetic vortices are formed inside the superconductor and the electric field E a is induced around the vortices. But, the radiations from magnetic vortices do not arise. This is owing to the fact that each vortex is surrounded by the superconducting state Φ = 0. The electromagnetic waves do not pass the state.
For comparison, we estimate the radiation flux S e from the normal conductor with the shape identical to the one in the superconductor.
with E 0 = g aγγ a 0 µB and δ = 2/m a σ, where we put µ = 1 for simplicity. Differently to the case of the superconductor, the radiation flux depends on the axion mass. This is because the skin depth δ depends on the frequency of the radiation. The difference between the flux from the superconductor and that from the normal conductor comes from the difference in the depth in which the currents flow. This difference causes the big difference in the flux. That is, the radiation flux S e emitted from the normal conductor is about a hundred times smaller than S c from the superconductor because typically δ ∼ 10 −4 cm and λ ∼ 10 −5 cm.

VI. SUPERCONDUCTING RESONANT CAVITY
We would like to briefly show radiation energy generated in superconducting resonant cavity. The cylindrical resonant cavity ( tube ) has been considered for the detection of the dark matter axion for more than 30 years ago [19]. The cavity is formed of normal conductor, e.g. cupper. In this section we consider superconducting resonant cavity instead of the normal conductor. We should note the difference between the superconductor and the normal conductor. The difference is the currents flowing the conductors. That is, the electric currents J c = E a /(m a λ 2 ) flows in the surface in the superconductor and J e = 2E suf a /(m a δ 2 ) does in the surface of the metal. We remind that J c = q 2 n c E a /(m a m c ) ( J e = σE suf a ) with λ −2 = q 2 n c /m c ( δ −2 = µm a σ/2 ). The difference in the depth of the current flow leads to the difference in their fluxes as shown in above. ( We remember E a ≃ E suf a . ) In the derivation [20] of the radiation energy inside the cavity, we solve equations of electromagnetic fields coupled with the axion by taking account of the effect of the normal conducting cavity. The effect is taken by introducing the current J e = σE a ≃ 2E a /(m a δ 2 ) in the equations.
Similarly, we may solve the equations taking into account the superconducting effect just as we have performed in the section(III). In the case of the cavity, we use cylindrical coordinate (ρ, θ, z) in which the the cavity is located at ρ = R c . Then, the equation (13) holds in the superconducting region ρ > R c with B z = B 0 exp(−(ρ − R c )/λ). We solve the equations of electromagnetic fields coupled with axion inside the tube ρ < R c and determine the fields using the boundary conditions which relate the fields in ρ < R c and those in ρ > R c . Then, we derive the electromagnetic energy in the tube. But, there is a simple way of the derivation. It is simply replacing the current J e ( = 2E a /(m a δ 2 ) ) in the normal conductor with J c ( = E a /(m a λ 2 ) ) in the superconductor ( that is, replacing δ with √ 2λ ). The energy U e = dV ( E 2 + B 2 ) /2 of a radiation, i.e. TM mode in the metal resonant cavity has already been derived such that when the resonant condition J 0 (R c ω a ) = 0 is satisfied, with the volume V of the cavity, where we assume 2π/(m a δ) < ω a /∆ω ∼ 3 × 10 6 with ω a = m a /2π. J 0 (x) denotes a Bessel function of the first kind. Here, the time average is taken; ωa 2π 2π/ωa 0 dt Q = Q Then, when we use the superconducting resonant cavity, we have with typical values δ ∼ 10 −4 cm and λ ∼ 10 −5 cm. Therefore, a hundred times larger amount of the radiation energy is generated when we use the superconducting resonant cavity. Contrary to the radiation energy inside the cavity, the flux absorbed in the superconducting cavity vanishes because the time average of the Poynting vector vanishes at the surface ρ = R c of the cavity; 2π/ωa 0 dt δ E × δ B = 0. This is owing to the fact that the superconducting current is dissipationless. To find the radiation flux absorbed in the cavity we need to take into account the effect of electrons remaining in the superconductor without forming Cooper pairs at nonzero temperature. The electrons absorb the radiation.

VII. SUMMARY AND DISCUSSION
We have shown that electric field induced by axion in superconductor is essentially identical to the one in vacuum. The electric field is proportional to magnetic field in the superconductor. The electric field is only present in the surface of the superconductor because of the Meissner effect; magnetic field is expelled from the superconductor. The result is obtained by analyzing equations of electromagnetic fields coupled with the axion and Cooper pairs. The Cooper pairs are described by a Ginzburg Landau model.
On the other hand, although the magnetic field is present inside normal conductor, the electric field induced by axion is absent in normal conductor. It is only present in the surface of the conductor because of skin effect of the oscillating electric field. The strength of the electric field is almost equal to the one in vacuum. The result is obtained by analyzing equations of electromagnetic fields in the metal coupled with the axion.
These electric fields oscillate with the frequency given by the axion mass so that Cooper pairs ( electrons ) in the superconductor ( normal conductor ) are forced to oscillate and emit radiations with the frequency. We have estimated the radiation fluxes from the cylindrical conductors. In particular, the flux from the superconductor is sufficiently large to be observed by existing radio telescopes. For instance, even when the superconductor with radius 1cm and length 10cm is put 100m away from the radio telescope of the parabolic dish antenna with the diameter 32m, the flux received by the telescope is four times of the order of the magnitude larger than that in the resonant cavity recently used [3] in ADMX.
The flux from the superconductor or metal is inversely proportional to the square of the penetration depth λ in the superconductor or the skin depth δ in the normal conductor, respectively. In general, the penetration depth ∼ 10 −5 cm is shorter than the skin depth ∼ 10 −4 cm for frequency ∼ 1GHz. This difference results in the difference of each flux.
In this paper we have mainly considered the QCD axion whose mass is expected in the range from 10 −6 eV ∼ 10 −4 eV. This expectation comes from the previous our paper [21], in which we have predicted the axion mass ∼ 7 × 10 −6 eV. The prediction comes from the analysis of the spectrum of fast radio bursts ( FRBs ) [22,23]. The FRBs are radio bursts with typical frequency 1GHz, flux ∼ 10 40 erg/s and duration ∼ 1ms. It is still a mysterious phenomena in astrophysics. Our model [24] for the FRBs is that the fast radio bursts arise from the collision between axion star [25] and magnetized dense electron gases such as neutron star or geometrically thin accretion disk around black hole with larger mass than ∼ 10 3 M ⊙ . The axion star is gravitationally bound state of axions, which is more dense than the dark matter axion ρ a under the consideration. The emission mechanism of radiations from the astrophysical objects is identical to the one discussed in the present paper. That is, the strongly magnetized electron gas emit radiations when they collide the dense axions. For this reason, our main interest in the axion mass is in the range mentioned above.
Obviously, our method for the detection of the dark matter axion can be applicable for much wide mass range beyond the range 10 −6 eV ∼ 10 −4 eV. We need sensitive receiver for the capture of the radiations from the cylindrical superconductor. The receiver should have surface area with the large solid angle as possible viewed from the superconductor. We also need to fabricate appropriate magnet, and glass container for liquid helium, in order for the radiations to pass through the magnet and the container and to reach the receiver. Then, we can search the wide range of the axion mass with high sensitivity.
The author expresses thanks to Alexander John Miller, Izumi Tsutsui and Osamu Morimatsu for useful comments and discussions. Especially, he expresses great thank to Yasuhiro Kishimoto for useful comments. This work was supported in part by Grant-in-Aid for Scientific Research ( KAKENHI ), No.19K03832.