Axion Detection via Atomic Excitations
Abstract
The possibility of axion detection by observing axion induced atomic excitations as recently suggested by Sikivie is discussed. The atom is cooled at low temperature and it is chosen to posses three levels. The first is the ground state, the second is completely empty chosen so that the energy difference between the two is close to the axion mass. Under the spin induced axionelectron interaction an electron is excited from the first to the second level. The presence of such an electron there can be confirmed by exciting it further via a proper tunable laser beam to a suitably chosen third level, which is also empty, and lies at a higher excitation energy. From the observation of its subsequent deexcitation one infers the presence of the axion. In addition the presence of the axion can be inferred from the deexcitation of the second level to the ground state. The system is in a magnetic field so that the energies involved can be suitably adjusted. Reasonable axion absorption rates have been obtained.
pacs:
93.35.+d 98.35.Gi 21.60.CsI Introduction
In the standard model there is a source of CP violation from the phase in the KobayashiMaskawa mixing matrix. This, however, is not large enough to explain the baryon asymmetry observed in nature.
Another source is the phase in the interaction between gluons (parameter), expected to be of order unity.
The non observation of elementary electron dipole moment limits its value to be . This has been known as the strong CP problem.
A solution to this problem has been the PQ (PecceiQuinn) mechanism.
In extensions of the SM, e.g. two Higgs doublets, the Lagrangian has a global PQ chiral symmetry , which is spontaneously broken, generating a Goldstone boson, the axion (a). In fact
the axion has been proposed a long time ago as a solution to the strong CP problem Peccei and Quinn (1977) resulting to a pseudo Goldstone Boson Weinberg (1978); Wilczek (1978). The two most widely cited models of invisible axions are the KSVZ (Kim, Shifman, Vainshtein and Zakharov) or hadronic axion models Kim (1979),Shifman et al. (1980)
and the DFSZ (Dine, Fischler, Srednicki and Zhitnitskij) or GUT axion model Dine et al. (1981),Zhitnisky (1980). This also led to the interesting scenario of the axion being a candidate for dark matter in the universe Abbott and Sikivie (1983); Dine and Fischler (1983); Preskill et al. (1983) and it can be searched for by real
experiments Asztalos et al. (2010); Duffy et al. (2005); Wagner et al. (2010); Irastorza and GarcÃa (2012). The relevant phenomenology has recently been reviewed Marsh (2016).
QCD effects violate the PQ symmetry and generate a potential for the axion field with axion mass with minimum at . Axions can be viable if the SSB (spontaneous symmetry breaking) scale is large GeV.
Thus the axion becomes a pseudoGoldstone boson
An initial displacement of the axion field causes an oscillation with frequency and energy density
The production mechanism varies depending on when SSB takes place, in particular whether it takes before or after inflation.
The axion field is homogeneous over a large de Broglie wavelength, oscillating in a coherent way, which makes it ideal cold dark matter candidate in the mass range eV eV. In fact it has been recognized long time ago as a prominent dark matter candidate by Sikivie Sikivie (1983), and others, see, e.g, Primack et al. (1988). The axions are extremely light. So it is impossible to detect axions as dark matter particles via scattering them off targets. They are detected by their conversion to photons in the presence of a magnetic field (Primakoff effect), see Fig. 1(a),(b). The produced photons are detected in a resonance cavity as suggested by Sikivie Sikivie (1983).
In fact various experiments
Since, however, the mass of the axion is not known, it is important to consider other processes, which may be accessible to a wider window of axion mass. In this paper we are going to discuss the possibility of axion detection by observing axion induced atomic excitations as recently suggested by Sikivie Sikivie (2014). The atom is cooled at low temperature and it is chosen to posses three levels. The first is the ground state. The second is completely empty, chosen such that the energy difference between the two is close to the axion mass. Under the spin induced axionelectron interaction an electron is excited from the first to the second level. The presence of such an electron there can be confirmed by exciting it further via a proper tunable laser beam to a suitably chosen third level, which is also empty, and lies at higher excitation energy. From the observation of its subsequent deexcitation one infers the presence of the axion (see Fig. 1c). In addition the presence of the axion can be inferred from the deexcitation of the second level to the ground state. The system is in a magnetic field so that the energies involved can be suitably adjusted.
Ii Our proposal
The axion is a pseudoscalar particle. We remind the reader that tts coupling to the electron takes the form of an axial current interaction:
where is a coupling constant and a scale parameter with the dimension of energy. For an axion with mass described by a a plane wave it easy to show:

the time component is given by:
which is negligible for .

The space component, , in the non relativistic limit is given by
where is the axion momentum.
This interaction has been proposed by Sikivie as a way of detecting the axion by causing atomic excitations
Thus the axion coupling to electrons in the non relativistic limit has the above form Sikivie (2014),
where is the axion decay constant, the axion momentum and the spin of the electron. is the relevant coupling constant to be determined by experiment.
The target is selected so that there exist two levels, say and which result from the splitting of the atomic levels by the magnetic field and they are characterized by the same and so that they can be connected by the spin operator. The lower one is occupied by electrons but the higher one is completely empty at sufficiently low temperature. It can be populated only by exciting an electron to it from the lower one by the axion field. The occurence of such an excitation is monitored by a tuned laser which excites such an electron from to a higher state , which cannot be reached in any other way, by observing its subsequent decay.
Since this is an one body transition the relevant matrix element takes the form:
(1) 
expressed in terms of the GlebschGordan coefficient and the nine j symbol.
Since the momentum transfer is small . So the matrix element becomes
(2) 
where
The cross section becomes
or
Then the event rate is given by:
(3)  
We will assume that the axion velocity distribution with respect to the galactic center is of the MaxwellBoltzmann type:
(4) 
In the local frame, ignoring the motion of the Earth, we have
(5) 
The integration over the velocity distribution takes the form:
(6) 
So the integration over the angles yields:

In the galactic frame
(7) which is symmetric.

in the local frame we get:
(8) that is
(9)
Th integration over the magnitude of the velocity is trivial due to the function appearing in Eq. (3). We thus get:
(10) 
Where is a normalized function of given by
(11) 
with
(12) 
The normalization factor , introduced to normalize , takes the values 0.33, for the galactic frame, while in the local frame 0.50 for and 0.36 for .
ii.1 The axion absorption widths
The above expression implies that the rate exhibits a resonance behavior in the variable , which depends on the energy difference. The latter depends on the magnetic quantum numbers qnd of the states involved. However there exists an additional dependence of on , which is mild. This behavior of the functions is exhibited in Fig.2. In fact we find that the characteristics of the resonance are:
This dependence of the widths on the magnetic quantum numbers of the states involved is a characteristic feature of the process and it can be exploited by experiments, if they reach adequate sensitivity in the measurement of the parameters of the width. Unfortunately the difference in the expected widths is small, about but the location of the resonances can differ by more than 10 (see Fig. 2). In the latter case one needs to correct for the dependence of on , since is not the same for both types of transitions.
We have seen that we have resonance behavior in the variable . Regarding the energy parameters we find
Both are very close to the axion mass, since
It is, of course, more practical to exhibit the function as a function of the the energy . This is exhibited in Fig. 3.
ii.2 Modulated widths due to the annual motion of the Earth
The width exhibits annual modulation due to the motion of Earth. This can be simply be included by making in the local frame we make the replacement :
We thus get a time variation of the width shown in Fig. 4. We see that the effect is small, the difference between the maximum and the minimum is less than , almost the same with that obtained in the axion to photon conversion Vergados and Semertzidis (2016). We note, however that, in addition to the seasonal dependence we have a dependence on the magnetic quantum numbers of the states involved The variation in the case of is almost twice as large compared to that with . This, in principle, can be exploited by the experiments.
It is amusing to know that the dispersion also exhibits a time dependence (see Fig. 5).
Iii Some estimates on the expected rates
Using the formulas of the previous section we can cast the event rate in the form:
(13) 
with is the number of atoms in the target. We separated out the resonance, which we have discussed above. It is affected by the assumed velocity distribution and the energy difference of he states involved.It depends mildly on the magnetic numbers of the states involve. The above expression can can be cast in the form:
(14) 
with to be determined (the limit has been set by CAST Barth et al. (2013) only involves its product with )
where is Avogadro’s number . The coefficients can easily be calculated and are given in tables 12 for some cases of practical interest. We should mention that does not contain the dependence on the substates and as well as the factor 0.1 coming from , see Fig. 3. The expected rate may thus be reduced, if one takes into account such a dependence and the selected states are not sstates.
The coupling is not known and, hopefully, it can be extracted from experiment. The coupling of axion to matter has been investigated di Cortona et al. (2016); Ringwald and Saikawa (2016), in particular in the context of the DFSZ axion models Dine and Fischler (1983); Zhitnisky (1980). This leads to:
(15) 
Where is the ratio of the vacuum expectation values of the two doublets of the mdel. This parameter is not known. It can be one, but it be quite large. Thus we find Adopting the higher value we find the scale of the expected rate rate to be:
(16) 
Iv Low temperature requirements
To achieve the condition that the second level must be essentially free of electrons the target material should be brought at low temperatures. The critical temperature depends on the axion mass to be explored. The average number of electrons expected to be at any given at the excited state is times the probability of occupancy which is given by the Boltzmann distribution probability . The number of electrons per year is expected to be . Let us take as expected number of real events those estimated above reduced by a factor of 100 due to angular momentum effect etc, i.e. . Thus
The condition on the temperature is given in Fig. 6
V Some experimental considerations
Thus in the search for for light axions one has to develop detectors involving materials, which at these low temperatures exhibit atomic structure. Ordinary atoms do not suffice. The ions of the crystal still exhibit atomic structure involving the bound electrons, as, e.g., the CUORE detector of Crystalline TeO at low temperatures .The electronic states probably won’t carry all the important quantum numbers as their corresponding neutral atoms. So one may prefer to consider targets which contain appropriate impurity atoms in a host crystal, e.g chromium in sapphire.
It is, also, possible that one may be able to employ at low temperatures some exotic materials used in quantum technologies (for a recent review see Doherty et al. (2013)) like nitrogenvacancy (NV), i.e. materials characterized by spin , which in a magnetic field allow transitions between and . These states are spin symmetric. Antisymmetry requires the space part to be antisymmetric, i.e. a wave function of the form
Of special interest are the cases:
Then the spin matrix element takes the form:
The reduced matrix elements are given in table 3. the full matrix element for the most important case is also shown in 3
Footnotes
 Permanent address, University of Ioannina, Ioannina, Gr 451 10, Greece
 Heavier axions with larger mass in the 1eV region produced thermally ( such as via the mechanism), e.g. in the sun, are also interesting and are searched by CERN Axion Solar Telescope (CAST) Aune et al. (2011). Other axion like particles (ALPs), with broken symmetries not connected to QCD, and dark photons form dark matter candidates called WISPs (Weakly Interacting Slim Particles) , see, e.g.,Arvanitaki et al. (2010), are also being searched.
References
 R. Peccei and H. Quinn, Phys. Rev. Lett 38, 1440 (1977).
 S. Weinberg, Phys. Rev. Lett. 40, 223 (1978).
 F. Wilczek, Phys. Rev. Lett. 40, 279 (1978).
 J. E. Kim, Phys. Rev. Lett. 43, 137 (1979).
 M. A. Shifman, A. Vainshtein, and V. I. Zakharov, Nuc. Phys. B 166, 493 (1980).
 M. Dine, W. Fischler, and M. Srednicki, Phys. Lett. B 104, 199 (1981).
 A. Zhitnisky, Sov. J. Nuc. Phys. 31, 260 (1980), in Russian.
 L. F. Abbott and P. Sikivie, Phys. Lett. B120, 133 (1983).
 M. Dine and W. Fischler, Phys. Lett. B120, 137 (1983).
 J. Preskill, M. B. Wise, and F. Wilczek, Phys. Lett. B120, 127 (1983).
 S. J. Asztalos et al., Phys. Rev. Lett. 104, 041301 (2010), the ADMX Collaboration, arXiv:0910.5914 (astroph.CO).
 L. Duffy et al., Phys. Rev. Lett. 95, 09134 (2005), for the ADMX Collaboration.
 A. Wagner et al., Phys. Rev. Lett. 105, 171801 (2010), for the ADMX collaboration; arXiv:1007.3766 (astroph.CO).
 I. G. Irastorza and J. A. GarcÃa, JCAP 1210, 022 (2012), arXiv:1007.3766 (astroph.IM).
 D. J. E. Marsh, Phys. Rep. D 643, 1 (2016).
 P. Sikivie, Phys. Rev. Lett. 51, 1415 (1983).
 J. Primack, D. Seckel, and B. Sadoulet, Ann. Rev. Nuc. Par. Sc. 38, 751 (1988).
 S. Aune et al., Phys. Rev. Let. 107, 261302 (2011), arXiv:1106.3919; CAST Collaboration.
 A. Arvanitaki, S. Dimopoulos, S. Dubovsky, N. Kaloper, and J. MarchRussell, Phys. Rev. D 81, 123530 (2010), arXiv:0905.4720.
 I. P. Stern, ArXiv 1403.5332 (2014) physics.ins–det, on behalf of ADMX and ADMXHF collaborations, Axion Dark Matter Searches.
 G. Rybka, The Axion Dark Matter Experiment, IBS MultiDark Joint Focus Program WIMPs and Axions, Daejeon, S. Korea October 2014.
 Center for Axion and Precision Physics research (CAPP), Daejeon 305701, Republic of Korea. More information is available at http:.
 S. J. Asztalos et al., Nucl. Instr. Meth. in Phys. Res. A656, 39 (2011), arXiv:1105.4203 (physics.insdet).
 G. Raffelt, Astrophysical Axion Bounds , IBS MultiDark Joint Focus Program WIMPs and Axions, Daejeon, S. Korea October 2014.
 P. Sikivie, Phys. Rev. Lett. 113, 201301 (2014).
 J. D. Vergados and Y. Semertzidis, Nuc. Pgys. B 897, 821 (2016), arXiv:1601.04765 (hepph).
 K. Barth et al., JCAP 1305, 010 (2013), arXiv:1302.6283.
 G. G. di Cortona, E. Hardy, J. P. Vega, and G. Villadoro, JHEP 01, 034 (2016), arXiv:1511.02867 , (hepph), (hepex), (heplat).
 A. Ringwald and K. Saikawa, Phys. Rev. D 93, 085031 (2016), arXiv:1512.06436 [hepph].
 M. W. Doherty, N. B. Manson, P. Delaney, F. Jelezko, J. Wrachtrup, and L. C. L. Hollenberg, Phys. Rep. 528, 1 (2013), arXiv:1302.3288.