Axion dark matter detection by laser spectroscopy of ultracold molecular oxygen: a proposal

Generalizing Sikivie’s approach (Sikivie 2014 Phys. Rev. Lett.113 201301), according to which dark matter axions may induce transitions between Zeeman states in an atomic system, an experiment based on a molecular gas is proposed for a search in the axion mass range between 1.4 and 1.9 meV. In particular, a 16O2 sample is brought to the temperature of 280 mK via buffer-gas cooling and then subjected to an external magnetic field; axion-driven transitions are eventually detected by resonance-enhanced multi-photon ionization spectroscopy.


Introduction
An outstanding problem in contemporary physics is that of dark matter (DM) whose existence is generally accepted on the basis of strong astrophysical evidence, but the composition of which is still unclear [2,3]. The most authoritative candidates for DM are weakly interacting massive particles and axions [4]. The latter have been introduced to provide a solution to the strong CP problem [5][6][7][8] and, in view of their weak interaction with ordinary matter, have become an aspirant for the DM component of galactic halos. Several detection schemes have been proposed to detect DM axions [9], some of which are already being implemented. The conversion of axions into photons in a microwave cavity permeated by an external magnetic field, originally proposed by Sikivie [10,11], has been pursued by several groups [12][13][14]. Other proposed schemes include: the conversion of axions into magnons in a ferromagnet [15,16]; the study of the nuclear precession of a material which can be modified by its interaction with the axion [17,18]; the presence of energy shifts or nonzero nucleon electric dipole moments arising in atomic systems from the coupling of axions to gluons [19]; the excitation of a high n transition in a Rydberg atom [20]. Lately, it has also been suggested that DM axions may induce magnetic dipole (M1) transitions between atomic or molecular states that differ in energy by an amount equal to the axion mass [1].
Based on this idea, here we detail the proposal of an experiment which aims at detecting molecular transitions in a gas system where axions in the 10 −3 eV mass range are absorbed. For this purpose, first we have to identify a molecular system with two magnetic sublevels whose energy difference can be tuned via the Zeeman effect to the desired axion mass value. These two levels, let us say a and b, will respectively represent the lower and upper level of what we will hereafter call the axion transition. Then, by cooling the gas sample such that there are no molecules in b, as a result of the exposition to the DM axion field, a  b transitions will be induced with a unit-time rate, ab  , of few events per mole of molecules (provided that M1 selection rules are satisfied) [1]. Finally, an appropriate detection technique must be used to reveal with almost 100% efficiency the presence of the molecules promoted into level b. From an experimental point of view, this involves the preparation of a sub-Kelvin, mole-sized molecular gas sample through a buffer-gas-cooling (BGC) process in a 3 He environment. In addition, the molecular species should be sensitive enough to the application of an external magnetic field and well-suited to a single-particle spectroscopic interrogation technique as well. In this respect, the oxygen molecule, 16 O 2 , offers three major advantages. First, it is undoubtedly one of the best investigated molecules on Earth, in terms of spectroscopy as well [21]. In particular, it has already been used for spectroscopic tests of fundamental physics, such as the symmetrization postulate on spin-0 nuclei [22,23], and as well employed in a number of high-sensitivity laser-based detection schemes including magnetic rotation spectroscopy [24], polarization-dependent cavity ring-down spectroscopy [25], and resonance-enhanced multi-photon ionization (REMPI) spectroscopy [26]. Second, 16 O 2 is a prototypical paramagnetic molecule with a high magnetic moment (2μ B ). Third, it has already been cooled down to few Kelvin in BGC experiments [27].
The paper is organized as follows. In section 2, we focus on the choice of the 16 O 2 transition that is most eligible for the axion detection; crucial issues of the gas sample preparation in a BGC machine are also addressed from a theoretical viewpoint. Then, section 3 deals with the REMPI spectroscopic detection of the axioninduced absorption events. Following this discussion, section 4 gives the details of the experimental apparatus. Finally, in section 5, we conclude.

Sample preparation
Let us start by considering the X 0 )state up to a magnetic field intensity of B=20 T by [25]. At this point in the discussion, we choose the rotational level (N=1, J=0, M J =0) ≡ a and (N=1, J=1, M J =−1) ≡ b as the lower and upper level of the axion transition, respectively. This determines the maximum axion mass range that can be explored according to the following reasoning. In order to realize a mole-sized population of 16 with k B being the Boltzmann constant, N A the Avogadro number, and T the gas temperature [1]. Now, as explained later, the minimum T value which provides a helium-3 vapour pressure that is capable to effectively cool N A oxygen molecules is T He ; 277 mK; as a result, B min =12 T is found, corresponding to W ba (B min )=11 cm −1 (1.4 meV) with Q (B min , T He )=0.09. Concerning the upper end of the axion mass interval, it is worth remarking that, besides technical difficulties, the Zeeman splitting W ba (B) tends to saturate for magnetic fields above B max =18 T; this amounts to W ba (B max )=15.5 cm −1 (1.9 meV) with Q (B max , T He )=1.5×10 −10 .
The above argument presupposes, however, complete rotational thermalization of the molecular sample within the BGC source under reasonable experimental conditions. To ascertain this point, we have to get a little more into the details of the BGC process. Here, both translational and internal degrees of freedom of the desired molecular species, at initial temperature T 0 , are cooled in a cryogenic cell via collisions with a thermal bath of helium-3 (buffer gas) at temperature T He and density n He . After a characteristic number of collisions N coll , the molecular temperature becomes arbitrarily close to equilibrium with the buffer gas, ) ( · )(m and m He denote the molecular and the helium-3 mass, respectively). With this in mind, let us now calculate the minimum 16 O 2in- 3 He collisional path-length (random walk) which is needed to thermalize N A molecules. Since the order of magnitude for He-molecule elastic-collision (rotational-relaxation) cross sections is 10 −18 m −2 (10 −20 m −2 ) [28][29][30], around 100 collisions are required to relax a rotational state. Denoting with P(E i ) the probability that the ith oxygen molecule has not reached thermalization after traveling the distance L in the buffer gas (E i is the corresponding event), the probability N A  that even one molecule (in N A molecules) evades the thermalization process is given by ) According to Boole's inequality, this probability is smaller than the sum of the probabilities of the separated events, P E P.
This latter quantity is given by where M is a number high enough and the P(E i )ʼs are weighted by the Maxwell-Boltzmann distribution of molecular velocities: Equation (1) is then computed by a direct Monte Carlo simulation for T He =280 mK and the corresponding vapor density, n He =3×10 16 cm −3 [31]; for L=1 cm, a value close to 10 −6 is already found for P s . Then, if P 1 s  , a fortiori this will be true for . Finally, the integral of the Maxwell-Boltzmann distribution is computed from v Hẽ to ¥ and multiplied by N A to provide the number of He-3 atoms exceeding this velocity. The result, which represents an overestimation of the number of spurious collision excitation events, is well below the unity. Furthermore, blackbody radiation effects are quantitatively addressed. For this purpose, the power emitted by the total internal surface area of the experimental setup, , is first calculated by the Stefan-Boltzmann law as P T .  s −1 is found; since only a very small fraction of N bb is absorbed by the molecular oxygen due to the low strength of the axion transition, this implies a negligible blackbody-radiation induced transition rate. In conclusion, it is worth pointing out that, from an experimental point of view, an effective way to discriminate against any spurious effect is to exploit the directional character of the interaction between axions and ordinary matter by counter-checking the signal at a second angle (at 90 • with respect to the first one) with a rotatable apparatus.

REMPI spectroscopic detection
The next step is to detect the molecules promoted by the DM axion field into level b. Since very few events are expected, an extremely sensitive spectroscopic technique must be implemented, which is particularly able to detect an axion absorption occurrence before collisions with surrounding particles may cause decay from level b. As already mentioned, molecular oxygen is well-suited to the (2+1)REMPI interrogation scheme. In general, this method involves a resonant single or multiple photon absorption to an electronically excited intermediate state followed by another photon which ionizes the molecule; the ionization product is then detected with almost 100% efficiency usually by a micro-channel plate (MCP). Specifically, the (2+1) symbol means that two photons must be absorbed to excite the molecule in the intermediate state and one photon to ionize the molecule. With reference to figure 2, in the 16 [32,33]. Anomalously, the rotational levels belonging to this component are characterized by far long lifetimes (in other words, they are much less predissociative than usual), thus favorably exhibiting enough narrow line widths (< 6 cm −1 ) [34]. Therefore, calling w laser the laser linewidth and w rot int the width of the narrowest rotational level in

Experimental apparatus and results
A schematic representation of the experimental apparatus is shown in figure 3. Machined from Cryo-G10, the BGC cell is thermally anchored to the mixing chamber of a dilution refrigerator by high-purity copper braids [35]. Thermal conductivity along the cell length is provided by copper wires running vertically along the outer Cryo-G10 walls. A hole with a diameter of d=15 cm is made in one of the walls to let the 16 O 2 molecules escape from the cell into a second chamber whose internal surface is covered by a layer of activated charcoal. At cryogenic temperatures, this acts as a pump with a speed of a few thousands dm 3 s −1 , creating a high-vacuum region. The (2+1)REMPI interrogation then takes place in the bore of 57 cm diameter superconducting magnet (coaxial to the BGC cell) which creates a homogeneous magnetic field of up to 18 T along a traveling distance h=12 cm. Two facing high-reflectivity (R m =99.995%) mirrors are used to accomplish multiple reflections of the laser beam so as to draw a pattern of a large number of spots, N refl , through the trapezoidal section of the (truncated-cone-shaped) molecular beam. In this way, the laser radiation interacts with a fraction N w h d h tan ) ( )of the molecular flux, where w denotes the laser beam waist and θ=60°is the molecular beam divergence. Then, for w=1 mm and N refl =13 500, 1   is found. At the end of the magnet, a micro-channel plate is then employed to collect the (2+1)REMPI ionization products. Capillary filling is used to inject both the molecular oxygen and the He-3 buffer gas, coming from room-temperature bottles, into the 280 mK BGC cell. For an efficient cooling process, the oxygen molecules should enter the cell with a temperature of 60 K, which ensures an appreciable vapor density (8 10 17 cm −3 ). The same type of consideration applies to the helium-3 buffer gas; here, as already mentioned, an inlet temperature of 280 mK gives rise to a vapor pressure of 3 10 16 cm −3 . These injection temperatures can be obtained by choosing, for each of the two gas lines, the most appropriate thermal-exchange configuration (both in terms of geometry and materials) through all the stages of the refrigerator as well as on the BGC cell walls. The whole is housed in a cylindrical vacuum chamber equipped with a number of optical windows for laser spectroscopy. As demonstrated in [26], for the case illustrated in figure 2,   two identical lasers (instead of one) can be used to draw the whole pattern of multiple reflections. In the above configuration, the number of oxygen molecules that have been exposed during a second to the axion field is given by ) ( )and n n 1 30 10 max He 15 =  ( ) · cm −3 is the maximum molecular density that can be effectively cooled down to T He [28]. Therefore, the number of axion-induced absorption events is calculated as ab A  = Then, for an acquisition time of n days =10, N 1 event  is found. However, as W ba (B) increases, higher temperatures can be used for the He-3 bath. In particular, for W ba (B max )=15.5 cm −1 (B max =18 T), the condition Q(B, T)<0.1 is met up to a temperature of 390 mK which corresponds to a He-3 vapour density of 3 10 17 - cm −3 ; this oneorder-of-magnitude enhancement in the n max value (and hence in N molec by virtue of equation (4)) can be exploited to reduce the acquisition time in equation (5) down to 1 day (for the same number of events).

Conclusion
A novel experimental approach has been described to expose DM in the mass range from 1.4 to 1.9 meV through a laser-spectroscopy detection of the absorption events induced by cosmic axions between Zeeman states in a molecular gas system. Although, in principle, the outlined measurement scheme applies to nearly all species, the oxygen molecule seems to magically combine several crucial features: first, its favourable Zeeman energy-level diagram in the X 0 g 3 u S = -( )ground state has already been calculated and measured with great accuracy; second, its very low boiling point in conjunction with a large rotational collision cross-section allows the cooling of a mole-sized gas sample down to 280 mK via BGC; last but not least, highly sensitive, rotationally-resolved REMPI spectroscopy has already been carried out starting from the aforementioned axion transition's levels. Due to the expected far low absorption rate, related to the inherent weakness of matter-DM interaction, very stringent requirements must be matched in order to yield a few counts for reasonable acquisition times. Accordingly, the presented setup involves more than one state-of-the-art technology, the main elements being an effective He-3 refrigerator, a high-performance ultraviolet spectrometer, and a cutting-edge superconducting magnet. It is worth stressing that, while constituting a challenging mix of skills, the different components are essentially already available within minor improvements; also, scalable technological advances can make the proposed experiment even more attractive.