The phase synchronization of an axion and a superconductor
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The phase synchronization of an axion and asuperconductor
Hideto Manjo , Koichiro Kobayashi , Kiyoshi Shiraishi Yamaguchi University, Yamaguchi-shi, Yamaguchi 753-8512, Japan Kyushu Institute of Technology, 680-4 Kawazu, Iizuka-shi, Fukuoka 820-8502, Japan ∗ E-mail: [email protected] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The effects of the axion field have been widely studied in theoretical physics, particularlyin particle physics. Considering the phase synchronization and the mean free path ofthe axion, the bulk of the phase coherent superconductor is regarded as the weak linkregion of the Josephson junction. It is expected that the axion mass influences theLondon penetration depth. There is a slight possibility of detecting this effect becausethe effect becomes more significant in superconductor with a low carrier density n s . Thedifferences due to the choice of axion model and the axion mass are discussed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Subject Index B04,B50,C15,I61 typeset using PTP
TEX.cls . Introduction
Many authors have proposed various theories of massive photons. In this paper, the topolog-ically massive model associated with the axion field is studied. The axion electrodynamics isan extension of Maxwell’s electromagnetic theory that includes the dynamical axion term.The presence of the dynamical axion term means that the photon becomes topologicallymassive in the axion electrodynamics.The axion term is also called the Chern-Simon (CS) term because of its origin. The chiralmagnetic effect (CME) [1, 2, 3] is a well-known topologically induced electromagnetic effectin the presence of the time-dependent CS term. The effect of the CS term has been reportedin the literature [4, 5, 6, 7]. In this connection, it is probable that the CS term affects theproperties of matter [8]. More detailed studies [9, 10, 11] have addressed the role of the CSterm or axion term in superconductors.It was reported recently that the axion mass can be estimated using resonant Josephsonjunctions, assuming a time-dependent axion field [12, 13]. These studies reported that theobserved Shapiro step anomalies in all four experiments consistently point toward an axionmass of (110 ± µ eV. As the author of [12, 13] pointed out, this result for the axion massalso needs to be examined from other viewpoints or experimentally.In present paper, it has been shown that the relation of ˙ θ and axion mass m a in the bulkof superconductor from the beginning of the phase synchronizing condition and the Londonequations. These findings suggest the axions penetrate deep inside of the superconductorand ˙ θ enhances in the superconductor, and besides, it is probable that the measurementof the following modified London penetration depth allows the checking of presence of thisphenomenon.Among the various possible effects of the axion field, we focus on the London penetra-tion depth. Superconductors have perfect diamagnetism, which is called the Meissner effect.Because of the Meissner effect, the magnetic field does not penetrate deep inside of thesuperconductor, and the depth is called the London penetration depth. It is inferred fromthe presence of enhanced ˙ θ in the superconductor that the London penetration depth isdue to the axion mass m a . This paper presents the simple classical results for the phasesynchronization of bulk of superconductor and the London penetration depth of a Type-Isuperconductor using the electromagnetic field theory, including the time-dependent axionfield.
2. Dynamics of photons and axions
The Lagrangian density for the axion electrodynamics (Maxwell–Chern–Simons equations[1, 3, 4]) and the axions dynamics is written as the sum of the Lagrangian densities forthe classical electromagnetism, the axion’s two-photon interaction and the axions dynamicalterm [14, 15]: L a = − F µν F µν + θ g γ e π F µν ˜ F µν + 12 f a ∂ µ θ∂ µ θ − f a m a θ − j µ A µ , (1)where g γ is a model-dependent coupling constant having a value of g γ = − .
97 for KSVZaxions [16, 17] or g γ = 0 .
36 for DFSZ axions [18, 19]. θ ( x ) = φ a ( x ) /f a is the misalignmentangle of the axion field φ a ( x ), and f a is the axion decay constant; − e is the charge of2n electron. The speed of light, the vacuum permittivity constant, the vacuum permeabil-ity constant, the reduced Planck constant are defined as c = 1, ε = 1, µ = 1 and ~ = 1,respectively. The gauge and Lorentz invariance cannot rule out a second term including θ ( x ).In other words, it is possible to allow a slight θ ( x ) dependence. Moreover, the behavior ofthis dynamical θ ( x ) is worth considering. It is a straightforward calculation to deduce theequations of motion from the Lagrangian (1): ∇ · E = ρ, (2) ∇ × B − ∂ t E = j − α ˙ θ B , (3)¨ θ + m a θ = − αf a E · B . (4)where assuming that θ depends only on the t coordinate, the differential term with respect tothe space coordinate, ∇ θ , can be eliminated, and ˙ θ ≡ ∂ t θ and α = ( g γ e ) / (4 π ) are defined.The other two expressions in Maxwell’s equation do not change ( ∇ · B = 0 , ∇ × E = − ∂ B ∂t ).The second term on the right-hand side in Eq. (3) represents the current, and this currentis called j CME (the chiral magnetic current) [3]. The purpose of this paper is to estimate theeffect of this term.
3. London penetration depth
The London equations describe the Meissner effect phenomenologically [20]. From theLondon equations, the magnetic field is written as a rotation of the current, B = − m e n s e ( ∇ × j ) , (5)and Eq. (5) represents perfect diamagnetism. In the axion electrodynamics, the additionalmagnetic field on the right-hand side of Eq. (3) is in the same direction as the current.Namely, it is necessary to calculate the London penetration depth under slightly unusualconditions. This paper shows a method of deriving the modified London penetration depthunder simple assumptions.Let us now apply a rotation to both sides of Eq. (3) and substitute the perfect magnetism(5) in the equation. The magnetic field equation is written as ∇ B = β B + α ˙ θ ∇ × B , (6)where β = ( n s e ) / ( m e ), and ∂ t E = 0 are defined. Note that the second term of Eq. (6)depends on ˙ θ .Next, the following two conditions apply. First, the superconductor is placed in the region x >
0. Second, the magnetic field and current depend only on the x direction. Then, themagnetic field B and current j are expressed as B = B y ( x ) e y + B z ( x ) e z and j = j y ( x ) e y + j z ( x ) e z , respectively, in Cartesian coordinates. From Eq. (5), the y and z components of Eq.(6) are rewritten as ∂ B y ( x ) ∂x = βB y ( x ) − α ˙ θ ∂∂x B z ( x ) , (7) ∂ B z ( x ) ∂x = βB z ( x ) + α ˙ θ ∂∂x B y ( x ) . (8)3ere, we introduce the ratio of B y and B z : B z ( x ) B y ( x ) = tan η. (9)This equation states that the magnetic field decreases, whereas the magnetic field ratio (9)inside the superconductor is maintained. Substituting Eq. (9) into Eqs. (7) and (8), we obtain βB y ( x ) = ∂ ∂x B y ( x ) − η ) α ˙ θ ∂∂x B y ( x ) , (10)where the function 2Γ( η ) = (1 − tan η ) / (1 + tan η ) is defined as that composed of the ratiovariable η . It is important to note that the magnetic field becomes zero deep inside ofthe superconductor to estimate the London penetration depth. This boundary condition islim x →∞ B = 0. Assuming lim x → B y ( x ) = B y , Eq. (10) yields a simple magnetic solution: B y ( x ) = B y e − xλa , (11)1 λ a = q β + Γ( η ) α ˙ θ − Γ( η ) α ˙ θ. (12)When ˙ θ = 0, this equation corresponds to the original London penetration depth λ L .
4. Phase synchronization of the superconductor
In the literature [12], the author states that the axions into weak link region of the Josephsonjunction immediately decays. In present paper, we consider a bulk of phase coherent super-conductor that lies in θ space, instead of considering the Josephson junction. Assuming thesuperconductor synchronize with the axion field, it is found that the axion phase ˙ θ enhancesin the superconductor. Our result is base on a new point of view, i.e., the London equations.Let us now consider the phase synchronization of the superconductor and axion decay.The superconductor is under the status of Bose-Einstein condensates (BEC), that impliesthe the phase ϕ of the wave function Ψ sc = | Ψ sc | e iϕ is synchronized on the bulk of thesuperconductor. Also it means that the superconductor is the microscopic quantum object.Now, we define that the wave function of the exterior of the superconductor Ψ ext = | Ψ ext | e iθ and the interior of the superconductor Ψ int = | Ψ int | e iϕ where the phase difference of boththe wave functions is δ = θ − ϕ . If the incoming axions enter the superconductor, then theregion around axions is the different phase to the external θ vacuum space. Therefore, thephase difference δ emerges, and produces the weak link like region of the Josephson junctionin the superconductor.Grant that this axion generated region is considered as weak link region of the Josephsonjunction, the following equations holds, d δ d t = 2 eV. (13)where V is the difference of a voltage of the inside of superconductor and the outside of thesuperconductor. From this relation, it is found that the time derivatives of the phase satisfythe relation ˙ δ = ˙ θ − ˙ ϕ = 2 eV . Assume that the initial condition of the phase of the super-conductor ϕ = ϕ , we get the phase synchronization condition ˙ δ = ˙ θ = 2 eV , that equivalentsto the Beck’s phase synchronization condition [12] about the weak link region.4he main key for describing the enhancement of the phase ˙ θ is the relation of the elec-tromagnetic field and the current, i.e., the London equations. The electric field in thesuperconductor obeys the London equation for the electric field: E = 1 β d J d t + ∇ ρ. (14)This equation implies that the time derivative of the density of the current produces theelectric field E , and the static current does not produce the electric field in the superconduc-tor. From the phase synchronization condition ˙ δ = ˙ θ , we get the relation of θ and J , that are˙ θ = (2 ed ˙ J ) / ( β ), ¨ θ = (2 ed ¨ J ) / ( β ), and θ = (2 edJ ) / ( β ) + c where ∇ ρ = 0, V = Ed . If J = 0and δ = 0 satisfies, then δ = c − ϕ = 0, this means c = ϕ . Moreover, the magnetic fieldis written as B = 12 ed ˙ J βα (cid:18) β ¨ J + J (cid:19) , (15)from Eq.(3) and Eq.(14) where the electric field E is in the same direction as the the magneticfield B . Substitute these ˙ θ , ¨ θ , Eq.(14) and Eq.(15) to Eq.(4), the equation of motion of axionis rewritten as the current equation: " f a (cid:18) ed (cid:19) ¨ J + " m a + βf a (cid:18) ed (cid:19) J = 0 , (16)where ϕ ≪ d isunknown. Solving Eq.(16), we now get the oscillation solution J = J sin ( ωt + δ ) , and ω is written as ω = h m a + β e d f a ih e d f a i , (17)about the current J where J and δ are constant. This equation means that the current isoscillation in the superconductor. Here, if this current J is regard as the Josephson junctioncurrent, we obtain δ = ωt + δ and ˙ δ = ∂ t ( ωt + δ ) = ω = 2 eV . This result implies that, inthe region of surface to length d , the current obey J = β ed ( ωt + δ ) , (18)since the relation of θ and J , and θ = δ . Using this relation, the entering axion generatedmagnetic field are written as B = 12 ed βα t = 2 π g γ n s em e d t, (19)in the weak link like region from Eq.(14) and Eq.(15) for J | t =0 = δ = 0.Next, the length of the axions decay is estimated. We consider the situation that an axionis placed the surface of the superconductor at t = 0 and has the velocity v a . From Eq.(19),the time average of magnetic field in the weak link like region becomes¯ B = π ~ g γ n s em e d T = π ~ g γ n s em e v a , (20)in SI units where d = v a T is used. Putting typical values for the superconducting electrondensity n s = 10 m − , the axion velocity v a = 2 . × m / s and the axion mass m a c =5110 ± µ eV. As the numerical example, the magnetic field ¯ B = − . × T for theKSVZ axion is found. The Primakoff effect is estimated by using this result. The probabilityof axion decay is given by following equation [14]: P a → γ = 14 v a (cid:0) g ¯ BL (cid:1) sin qL qL ! , (21)where q is axion-photon momentum transfer, L is an axion flying distance, and g =( g γ e ) / (4 π f a ) = α/f a . In P a → γ = 1, an expression for the mean free path in the weak linklike region is L = 64 µ c ~ m e f a e n s v a , (22)for qL ≪ ~ in SI units. As the numerical example, we obtain the long distance L = 10 m.Note that the distance is independent on the axion model since the mean free path L doesnot include g γ .Here, we consider that an entering axion decay at L (See Fig.1). In the region more deeper Fig. 1
The phase synchronization of the axion and the superconductor. The axion decaysat L .than distance L , it can be assumed that the phase θ that is propagated an axion does notexist, which implies that the weak link like region depth d is equals to the mean free path L for the superconductor that has a thickness D > L , namely d = L . In that case, the angularvelocity is rewritten as ω = h m a c ~ + c ~ Lf a ) n s m e ih c ~ µ Lf a e ) i , (23)in SI units. As the order estimation, the first term of above is 10 s − , the second term ofabove is 10 − s − , and the second term of below is 10 − . Therefore, the second term ofabove and the second term of below are clearly negligible. Note that if the superconductorhas a thickness D < L , then the edges effect act on the angular velocity ω from (17). Howeverin this case, these edge effects is very tiny, still the second term of above and the second termof below have no more than 10 − s − and 10 − for D = 1 m. Therefore, it is reasonable tosuppose that ω ∼ ( m a c ) / ( ~ ) in SI units is consistent.6s the summary in this section, it is found that the bulk of superconductor for D < L isregarded as the bulk of the weak link like region, and the axion mass m a is related to theangular velocity of the Josephson junction like current, that is to say ω = m a c ~ = ˙ δ = ˙ θ = 2 eV ~ , (24)in SI units.
5. Shapiro step of the axion
We consider the junction that is made of this synchronized superconductors and normalmetal. In this section, we use SI units. If an axion enter the superconductor on the otherhand either, and the phase of axion θ synchronize with the superconductor (See Fig.2).Then the phase difference ˙ δ = ω = ( m a c ) / ~ as many as an axion emerges on the weak link Fig. 2
S/N/S junction with an incoming axion.region of normal metal, which make the super current of Josephson junction and the axiongenerated voltage V a = ( m a c ) / (2 e ) = ( ω a ~ ) / (2 e ). So, the junction voltage bias is writtenas V = V ± ( ω a ~ ) / (2 e ), which implies δ = δ + (2 eV / ~ ± ω a ) t . Inserting this δ into J = J sin δ , this means a dc component only when 2 eV ± ω a ~ = 0, i.e., when the dc voltage hasthe Shapiro step values V = ( ∓ ~ ω a ) / (2 e ). Therefore, it is a possible that our scenario alsodescribe the observed Shapiro step [21] and the unknown differential conductance peek [22]in the S/N/S junction experiment.
6. Discussion
The effect of the axion field about the London penetration depth is estimated from theresults. Assume that the magnetic field ratio variable η →
0, namely, 2Γ →
1, the externalmagnetic field has only a y component. In SI units, the modified London penetration depthare rewritten as 1 λ a = vuut µ n s e m + 14 µ ~ g γ e ˙ θ π ! − µ ~ g γ e ˙ θ π . (25)Next, the relationship between the axion mass m a and ˙ θ is considered. The frequency, ω = m a c / ~ = ˙ δ = ˙ θ , is given by the axion mass. From the literature [13], the axion mass is7 a c = (110 ± µ eV, which implies ˙ θ ∼ . × s − . Assume this value and the super-conducting electron density of niobium n s (Nb) ∼ . × m − , from the value of λ L in theliterature [23]. Then, the difference between the modified London penetration depth λ a andthe original London penetration depth λ L is ∆ λ = λ L − λ a , where ∆ λ KSVZ ∼ . × − m,and ∆ λ DFSZ ∼ − . × − m. It seems that the effect of the light axion on typical super-conductors is insignificant. However, the effect of the axion for m a c > n s except for ˙ θ as a physical variable. Theaxion field becomes significant if n s has a very low value. Therefore, there is some possibilityof detecting the effect of the light axion on superconductors having a low carrier density n s . If the axion mass is m a c = (110 ± µ eV, it is expected that the effect of the axionmass in both the KSVZ and DFSZ models becomes prominent in the region n s < m − .The sign of ∆ λ is positive for KSVZ axions and negative for DFSZ axions, which show thephotons in the superconductor become heavier than typical photons in the KSVZ modelbut lighter than typical photons in the DFSZ model. That is, this effect provides a method Μ eV m a c = T yp i ca l s up e r c ondu c t o r s Nb, Al, Pd, Cd, Sn, etc ...
KSVZ axionD FSZ axion Μ eV1 Μ eV1 neV 1 Μ eV - - n s @ (cid:144) m D D Λ (cid:144) Λ L Fig. 3 ∆ λ/λ L for KSVZ axions and DFSZ axions, and the relationship between n s and∆ λ/λ L . The dash-dotted lines show various axion masses m a c ; the thick line shows m a c =110 µ eV.of cross-checking the axion mass and selecting the axion model in principle. However, thislow carrier density, n s < m − , is not realistic, and we would like to emphasize that thedifficulty lies in detecting this effect. 8 . Conclusion Considering the phase synchronization and the London equations in the superconductor, it isfound that the time derivative of the phase ˙ θ enhances in the superconductor, and this valueis related to the axion mass m a . The London penetration depth of a Type-I superconductorwas calculated from the axion electrodynamics in the presence of a time-dependent axionfield. There is a slight possibility of detecting this effect because the effect becomes moresignificant in superconductor with a low carrier density n s . The London penetration depthbecomes shorter in the KSVZ model but longer in the DFSZ model than the typical Londonpenetration depth. Acknowledgments
We thank M. Kuniyasu, Professor T. Asahi and K. I. Nagao for valuable discussions, andalso thank T. Takahashi who teach us the useful view for our study.
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