Probability of transition gravitons into photons in a static magnetic field

WANG Ping，PAN Yu，ZHAO Fanghao

(1.College of Electronic and Information Engineering，Chongqing Three Gorges University，Chongqing 404000，P.R.China 2.College of Mathematics and Physics，Chongqing University of Posts and Telecommunications，Chongqing 400065，P.R.China)

1 Introduction

Relic gravitational waves(GWs)，predicted by the general theory of relativity，are very important sources for information on the early universe，whose physical behavior express the states and evolution of the early universe.However，because of its weakness interaction and technique restrictions，so far any direct detection of GWs has not been successful.Fortunately，because of the observation of gravitational radiation from the binary pulsar PSR 1913116 inspired greatly many physicists to do relevant researches，in spite of it providing only indirect evidence for the existence of GWs.Currently，many theory and experimental works in references［1－6］have entered into the mature period for detecting the gravitational waves，it is believed that along with development of new technology in future，the detection of GWs can become a possibility.Up to now，one of the best hopes to prove the presence of the graviton is to take the weak interaction of electromagnetic(EM)field and GWs field.In laboratory we can strength the signal flux by designing resonance and coherence of the electromagnetic waves and gravitational waves，it appears that GWs generated in the first cavity pass the second cavity，and generated the perturbative photon flux(PPF)，according to outgoing photon flux in a certain direction，we can get the information on the existing of graviton directly.In reference［6］，many properties of the EM field generated by the interaction between GWs and static magnetic field were considered by using purely classical method.The results bring very meaning reference value and it is worth attempting.

This paper presents the theoretical principles of conversion of gravitons into photons in a static magnetic field by using Feynman diagram techniques.By introducing some typical laboratorial parameters，an expected photon flux will be produced by using the transition expressions.The paper selects a metric ημνwith signature－2.Note that when upper index and lower index are sameness，it denotes the sum.Greek indices take value from 0 to 3，Latin indices take value from 1 to 3，e.g.，aμbμ=a0b0－a→·b→.

1 Production of gravition

The relic graviton energy density in quintessential inflationary models(QIM)，recent analyses in references［7－8］，depending upon the parameters of the model and upon the different inflationary dynamics，can be much more sizable than in standard inflationary models，for frequencies larger than 1 Hz，the maximal signal is localized in the GHz frequency band，where the energy density eight orders of magnitude larger than in ordinary inflationary models，therefore，smaller EM detection system can be used to detect the GHz frequency band GWs.

Reference［7］discussed the field operators of GWs by perturbing the Einstein Hibert action to second order in the amplitude of tensor fluctuations，whose Fourier expression can be written as

Therefore，the two－point correlation function of GWs operators in Fourier space is

According to the compute of Giovannini，one can get the relation between the logarithmic energy spectrum ΩGWand the spectrum density Snof the relic GW，

The relation curves of energy spectrum ΩGWand GWs frequency have showed in
Fig.1 from reference［5］，which will give the energy spectrum expected in the quintessential inflationary in this peak value region，one can estimation the spectrum density Sn.

Fig.1 Open superconducting cavity and coupling EM fields

In addition，the high frequency gravitational waves(HFGW)generation scheme of piezoelectriccrystal－resonators or ultra－high－intensity laser in references［9－10］have become promising in laboratory，whose action against targets emulates a double－star system and generates a GW flux at a focus midway between two such GW－generation lasers.The expression of the quadruple approximation to GW power can be written as

where r is the radius－of－gyration and Δf is the change in force at δm over the incremental time interval Δt(～33.9 ×10－15s)As derived in references［9－10］，the dimensionless amplitude is

Equation(6)is strictly valid for monochromatic or quasi－monochromatic GW.The frequency of high－frequency relic GWs in the QIM is～1010Hz，if the radius of gyration r=10 m and the change in force at δmover the incremental time interval Δt(～33.9×10－15s)，Δf=1.5 ×105N，the HFGW flux FGM～5.5 ×10－12which denotes the quantity of the outgoing gravitons～0.83 ×1012m－2·s－1.From Eq.(6)，one can estimate A～10－34，such a quantity magnitude of GWs amplitude can cover a wide range of amplitude.

2 Conversion of graviton into photon

Let us consider the gravitational waves，produced by ultra－high－intensity laser or piezoelectric－crystal－resonators，to travels through a same detection system presented in reference［6］，which consists of a superconducting wall and a fractal membrane as shown in Fig.1 .The producing perturbative photon flux with certain polarization directions and frequency will be many times reflected by the fractal membrane in an enough time intervals，and stored in the open cavity，at the same time photon flux with other polarization directions and frequencies will be transmitted by fractal membrane without reflection.Moreover，the strength of photon flux will keep invariant with the distance of one meter from the fractal－membrane surface.

Also，the interaction Lagrangian for the gravitational waves with the EM field in the linear approximation，has the following form references［2－3］

all other

here，the sinc function is defined by sin c(x)=For the electromagnetic wave，one choose the real photon polarization(ε0=0)，and for GWs，choose transverse traceless(TT)gauge condition of the GW，i.e.，

Considering the conversion of the graviton g with momentuminto the photon γ with momentum，one can get the transition matrix element by using Feynman rules in reference［11］

Using the usual procedure

here L is an arbitrarily length，and the following wellknown expression of summing up over the polarizations of photons and gravitons in references［3－4］

then applying it into the transition probability formula，we have

From Eq.(10)，it is seen that outgoing photon flux is constrained to travel the ±zdirections，namely

According to the transition probabilities，one can get outgoing photon flux，

Tab.1 Perturbative photon flux(PPF)and relevant parameters(A denotes the amplitude of GWs and FGW is HFGW flux)

Under the resonant condition，we have kg=ke=2π/λg=2π/λe=nπ/b.If we choose λg=3cm(νg=10 GHz)，this is typical frequency of the HFRGWs in QIM［5，7］，and n=70，then b=nλg/2=1.05 m，the photon in backward direction vanishes，so only the photon flux in forward direction is considered.In order to express detectable effect，we introduce some typical laboratorial parameters

(ii)νg=νe=10 GHz，the resonance frequency;

(iii)b=1.05 m，the distance between two parallel cavity walls;

(iv)ΔS=1 m2，the cross section of reflection plane，and the duration Δτ=1 s.

From the above parameters，and Eqs.(11)and(12)，one obtain some parameters as listed in Tab.1，where neglected the influence of other physical effect，such as noise problem.As is illustrated in Tab 1，for typical amplitude magnitude of the HFRGWs in the GHz band A～10－30，an strong outgoing photon flux TEMW(～1011)transmitted by fractal membrane would be generated in duration time Δτ

3 Conclusion

In this paper，the transition probability of one graviton is considered by using Feynman perturbative techniques in a static magnetic field.By introducing some typical laboratorial parameters，an expected photon flux will be produced.Although it is difficult to detect the photo flux presented in Table 1 by using current equipments，the work directs us to know the mechanism of detecting HFRGWs.Fortunately，coupling of the superconducting open cavity and a background Gaussian beam can further improve detecting sensitivity to the HFGW in the GHz band(A～10－30)［5，12－13］.This idea has been explored to certain extent，showing some advantages as a detection method for very high frequency GWs.It is also worth to attention that the quick development of super strength laser and the super strength magnetic techniques ［9－10，14］，and the study of advancing microwave photon flux detection system ［15－16］，provide the probable of detecting the graviton flux in experiment.

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