Probing axion mediated fermion--fermion interaction by means of entanglement

We propose a new approach in the investigation and detection of axion and axion-like particles based on the study of the entanglement for two interacting fermions. We study a system of two identical fermions with spin-1/2, and we show that the fermion-fermion interaction mediated by axions leads to a non-zero entanglement between the fermions. Even if also the magnetic dipole-dipole interaction can contribute to the entanglement, by setting opportunely the duration of the experiment its effect can be suppressed. Therefore, an entanglement measurement can reveal such an interaction and the existence of the axions. We introduce a two-body correlation function which can be directly observed in an experiment and plays the role of an entanglement witness. The results of our analysis can be in principle tested with today technologies.

Introduction -The Standard Model of particles (SM) provides a satisfactory description of fundamental particles and interactions, able of explaining several observed phenomena involving the electromagnetic and nuclear forces. In spite of its success, this is not the ultimate theory of elementary particles, since a wide class of phenomena, from particle mixing [1][2][3][4][5] to the quantum features of gravitation [6], cannot be explained within the SM. Among the shortcomings of the theory is the so-called strong CP problem, i.e., the absence of CP violation in Quantum chromodynamics (QCD) [7]. Axions were first introduced by Peccei and Quinn in the '70s to tackle this problem [8]. Known as the QCD axions, these particles can be described as the pseudo Nambu-Goldstone bosons of the spontaneously broken U (1) A symmetry [9][10][11], also called PQ symmetry. The energy scale at which the symmetry breaking occurs f A is inversely proportional both to the axion mass m a and its couplings with the standard model fields (G aγγ , G af f , G aqq ). The experimental constraints on these couplings suggest that f A > 10 9 GeV, so that QCD axions generally interact weakly with ordinary matter and have a mass well below the eV (m a ∼ 10 −6 − 10 −2 eV). Axions are then interesting in at least two respects: because they signal a new Physics beyond the Standard Model at a relatively low energy scale f A , and because such light and weakly interacting particles represent a natural candidate for Dark Matter. This has motivated the introduction of many theoretical models, some of which can deviate significantly from the original Peccei-Quinn (PQ) axion. Indeed, since the PQ proposal, a wide range of axion-like particles (ALPs) has been studied, from ultra-light axions (ULAs) [12] with masses m a ∼ 10 −22 eV to heavy GUT axions, with masses up to 1 TeV [13]. In particular, the weakly interacting ULAs have attracted interest as a possible cold dark matter (CDM) component [14].
Nonetheless, the experimental search for ALPs has proven to be very challenging. Many experiments have been devised to detect ALPs by exploiting their coupling with photons: searches for polarization anomalies in the light propagating through a magnetic field (PVLAS) [15], "light shining through a wall" experiments (OSQAR) [16], detection via the Primakoff effect (CAST) [17] and more recently by means of geometric phases [18] and QFT effects in the axion-photon mixing [19]. Astrophysical observations and terrestrial experiments over the decades have restricted the allowed regions in parameter space, and further constraints might come from the analysis of the axion-nucleon and axion-lepton interactions, as suggested for instance in [20]. Despite the efforts, no evidence for ALPs has been found up to now, and this motivates the search for alternative approaches in their detection.
On the other hand quantum information theory and the study of the entanglement have developed consistently over the last decades. The characterization and analysis of entanglement has found application in the most disparate fields, from quantum biology [21] to the study of fundamental interactions [22][23][24][25]. These applications stem from the fact that the emergence of entanglement between two (or more) physical objects is strictly connected to the presence of a quantum interaction between them. If the interaction Hamiltonian does not have a completely degenerate spectrum and cannot be expressed as the sum of local terms acting separately on every single object, its action rises a non-zero entanglement even if the initial state is separable [25][26][27].
In particular, in this letter we study the axion-mediated fermion-fermion interaction and propose a new approach to the detection of axions and ALPs, joining the two seemingly unrelated topics of entanglement and fundamental pseudoscalar particles. The approach is based on the fact that the Yukawa-like interaction between two fermionic fields and a pseudoscalar field in the non-relativistic limit gives rise to a two-body potential that acts as a source of entanglement for the two fermions. A measurement of such an entanglement may provide an evidence for the pseudoscalar interaction and thus for the existence of axions. Obviously, the two fermions interact with each other in many other ways, i.e. gravitationally, magnetically, etc. Hence, one of the main goals of the arXiv:1910.01533v2 [hep-ph] 14 Oct 2019 letter is to show how it is possible to design an experimental setup able to extract the axions-induced entanglement contribution from the others, and whose parameters fall within the range of today's technologies.
The letter is organized as follows. We first recap the axionfermion pseudoscalar interaction and the corresponding twofermions potential in the non-relativistic limit. From the knowledge of the potential we compute the time-dependent entanglement, quantified through the 2-Renyi entropy. Under suitable conditions we show that it is possible to extract the axions-induced entanglement from all the other terms. In addition, to overcome the problem of a direct observation of an entanglement entropy we also individuate a two-body spin correlation function that plays the role of an entanglement witness, which vanishes if and only if the entanglement vanishes. Finally, we discuss the possibility to realize an experiment based on this kind of approach.
Fermion-Fermion interaction induced by Axions -Let us start by recalling the main features of the axion-mediated fermion-fermion interaction. Below the PQ symmetry breaking scale the coupling of axions with fermions is described by a Yukawa pseudoscalar vertex [28]. If φ is the axion field and ψ 1 , ψ 2 are the fermion fields, the interaction term reads where γ 5 is the product of Dirac matrices iγ 0 γ 1 γ 2 γ 3 and g pj are the effective axion-fermion coupling constants, which depend critically on the fermions considered and the underlying axion (or ALP) model. For QCD axions the effective couplings are proportional to the ratio m f f A , with m f the fermion mass [29]. Due to the constraints on the axion decay constant (f A > 10 9 GeV), the couplings are small g pj 1. Accordingly, the scattering amplitudes are well approximated by the leading order in the perturbative expansion. For the scattering where the pseudoscalar free propagator with momentum q = p 1 − p 1 appears, and m is the axion mass. Substituting the plane wave solutions and taking the non-relativistic limit, one obtains an expression for the Fourier transform of the two-fermion potential V (q). It can be shown that this twobody potential, in real space and for two identical fermions, has the form [30] where we have assumed g p1 = g p2 = g p , M is the mass of the fermions, r (r r r) is the modulus (the unit vector) of the relative distance between the fermions and δ 3 (r) the Dirac delta.
Here σ σ σ i is the three-dimensional vector whose components are the three Pauli operators defined on the i-th fermion. Assumingr r r coinciding with the z-direction and neglecting the term proportional to δ 3 (r r r), we have

Entanglement and axion induced fermion-fermion interaction -
The Hamiltonian H p in eq. (3) describes the interaction between two identical non-relativistic fermions at distance r mediated by axions. The spectrum of H p is not completely degenerate spectrum and H p cannot be written as the sum of two local terms each acting on a single fermion. As a consequence, the presence of this interaction can be seen as a quantum channel that, differently from a classical one, can create (or destroy) entanglement [26,27]. We start by considering a system of two identical spin-1 2 fermions (for instance electrons or neutrons) which at the initial time t = 0 are in a separable, i.e. non-entangled state. The state at t = 0 is then of the form |ψ(0) = |ψ 1 |ψ 2 , with the two local states |ψ 1 and |ψ 2 given by |ψ i = cos(θ) |↑ i + e ıφ sin(θ) |↓ i . Here |↑ i and |↓ i denote the eigenstates of the magnetic moment along the direction joining the two fermions (z-direction).
Assuming that the relative distance between the fermions is large enough (r > 10 −10 m), we can neglect the weak and the strong nuclear interactions. Then, along with the axionmediated interaction H p , we need to consider the gravitational and the electromagnetic interactions, since fermions, in general, have a non-vanishing magnetic moment, mass and electric charge (apart from neutrons and neutrinos). However, a fundamental requirement for an interaction to induce entanglement in a system is that the initial state can be seen as a linear combination of states for which the interaction assumes different values [26]. Particles such as electrons, protons, or neutrons are characterized by precise values of charge and mass that do not depend on their spin states. Therefore any state depending solely on the spin of the particles will react to gravitational and electrostatic interactions in the same way. Hence their action on the evolution of the state amounts to a global phase factor that does not affect the entanglement of the system. This result is not in contrast with several recent works, see for example Ref. [22,23,31] that suggest the entanglement as a probe for the quantum nature of the gravity. In fact, in these papers, position-dependent states are considered, and the evolution induced by the gravitational and the electrostatic interactions cannot be reduced to a global phase factor, generating a non-vanishing entanglement. In our case the local states differ only for the spin state and not for the position. Then the entanglement owing to the Coulomb and the gravitational interaction vanishes.
Thus only the dipole-dipole magnetic interaction provides a further contribution to the entanglement of the two fermions. This interaction is described, for spin-1/2 fermions, by the Hamiltonian where g is the g-factor of the fermions, and q e is the charge of the electron.
Once that the initial state |ψ(0) is fixed, the state at t > 0 can be obtained as |ψ(t) = U (t) |ψ(0) , where the time evolution operator U (t) is unitary since we consider our system to be closed (we neglect any other interaction with the surrounding world). Thus, U (t) can be written as U (t) = exp[−ıt(H T )], where the total Hamiltonian H T is the sum of the magnenetic (H µ ) and the axion term (H p ), i.e. H T = H p + H µ , and reads In eq. (5) the parameter A = g 2 qe 2 64πM 2 is the strength of the magnetic interaction, B = 4g 2 p g 2 q 2 e = 4g 2 p αg 2 is the relative weight of the axion interaction and α denotes the fine structure constant.
Since the operator U (t) is a unitary operator for any time t ≥ 0, the state |ψ(t) remains a pure state for any t, although in general, it is entangled. The amount of entanglement between the two fermions in |ψ(t) can be quantified using different measures of the entanglement. We consider the 2-Renyi entropy [32][33][34][35], that can be associated to experimental quantities [36][37][38]. The 2-Renyi entropy is defined as S 2 = − ln(P(ρ i (t))), where ρ i (t) = Tr j =i (|ψ(t) ψ(t)|) is the reduced density matrix obtained projecting |ψ(t) on the Hilbert space defined on one of the two femions and P(ρ i (t)) = Trρ 2 i (t) is the purity of |ψ(t) . For the time dependent state |ψ(t) , the 2-Renyi entropy is where Γ(B) = 6A r 3 1 − B 3 e −rm 3 + 3rm + r 2 m 2 . As we can see from the expression of Γ(B), the entanglement derives from both the dipole-dipole magnetic interaction and the presence of the axions. However the second contribution can be isolated by properly setting the time interval of the entanglement measurement. Indeed, from eq. (6), by setting t = nt * where n is a positive integer and t * = πr 3 6A , we have that the entanglement induced by the magnetic interaction vanishes, and we are left with the axion contribution alone: In the case in which g p 1, the last line of eq. (7) shows that the entropy is proportional to B 2 and then to g  with the mass. However, when the Yukawa damping becomes relevant, the entanglement is lowered and exponentially goes to zero. It is a matter of fact, however, that direct entropy measurements are not easy to accomplish. Although various approaches to measure the 2-Renyi Entropy have been proposed [36][37][38], they are tied to specific experimental devices and no general method has been developed. To overcome such a difficulty it is possible to make use of an entanglement witness, i.e. a quantity strictly related to the state and the dynamic of the system under analysis, whose value signals the presence or the absence of entanglement. In our system, setting to zero the phase φ of the initial state in eq. (5), such a witness can be identified with the two-spins correlation functions C yz = σ y 1 σ z 2 ≡ σ z 1 σ y 2 , where the angular parentheses denote the average calculated on the state of the system. Indeed, with this choice, it is straightforward to show that for any time greater than zero the above correlation functions are equal to C yz (t) = − sin(2θ) sin (Γ(B)t), which, for any θ = kπ 2 with k integer, vanishes only when the entanglement (namely the 2-Renyi entropy of eq. (6)) is zero. Setting t = nt * = nπr 3 6A the correlation function becomes that vanishes only in absence of interaction between axions and fermions, i.e. only if axions do not exist. In the case nπB 1, from eq. (8), we can see that the entanglement witness shows a dependence on the interaction strength proportional to g 2 p rather than to g 4 p making the detection easier to be achieved. Plots of the entanglement witness at t = t * are shown in the lower panel of fig. ((1)).
The experimental setup -As it is well known, one of the main problems when we are interested in the experimental analysis of the entanglement of a system is the finiteness of the coherence time [40]. In realistic systems, the coherent superposition characterizing the pure quantum states (that can show non-vanishing entanglement) is destroyed by noisy interactions that every quantum systems shares with the surrounding world. Up to now, in our analysis, we have completely neglected such kind of interaction assuming that our system is closed. However this assumption is realistic only for a finite time interval, which is known as coherence time. Beyond that time the effects of decoherence can no longer be neglected. It is therefore essential, for the experiment to be faithful, that the observation time be lesser than the coherence time. In modern experimental setups, the coherence time is becoming longer and longer and a duration of more than 1 second has been achieved [41]. This has to be compared with the minimum time interval needed to isolate the axion contribution to the entanglement (t * ). Notice that t * is proportional to M 2 and then, it is strongly dependent on the fermions considered in the experiment. For a system of two electrons, we have A = 2.6 × 10 −23 cm 2 . By fixing the relative distance r = 0.1 µm, we have t * = 0.7 ms, which is shorter than the coherence times accessible to the present experimental setups. By contrast, considering neutrons, the mass increase of 3 orders of magnitude determines a much larger time, even at very short relative distances. For instance, at r = 0.01 µm, with the parameter A = 2.8 × 10 −29 cm 2 we have t * 0.7 s. This might still fall below the coherence time currently attainable in experiments. Another limitation comes from the Yukawa damping factor e −mr , which strongly suppresses the axion-mediated interaction, and then the entanglement, outside a limited spatial region r < 1 m . Of course, the smaller the axion mass, the larger the spatial region where the experiment is efficient. To give an idea, for an axion mass m 10 −2 eV, a Yukawa damping of the order of 0.9 can be obtained for a distance r max ∼ 10 µm, which can be obtained in the experimental apparatus.
Conclusions -In a system made of two spin-1 2 fermions we have analyzed the dynamics induced by the axion-mediated fermion-fermion interaction in the non-relativistic regime and have shown that it is characterized by the rising of an entanglement between the two fermions. This source of entanglement must be added to the dipole-dipole interaction of magnetic origin. However, by suitably tuning the observation time t = nt * and the distance between the two fermions, one can get rid of the magnetic source of entanglement. In this way any residual entanglement-entropy can be seen as a direct consequence of the presence of axions.
This entanglement can be increased choosing n integer and greater than one in such a way that nt * is less than the decoherence time of the experimental device. Moreover, to overcome possible difflculties in direct entropy measurement, we have introduced a spin-spin correlation function which we have proved to be a suitable entanglement witness that vanishes if and only if the entanglement goes to zero. Such witness has also the advantage to show, at t = t * , a dependence proportional to g 2 p rather than to g 4 p as the 2-Renyi entropy. This fact allows to extend the range of applicability of our experiment of several orders of magnitude. The method we propose can likely probe a coupling constant range g p = 10 −5 − 10 −1 for any axion mass up to m 0.3 eV , being particularly efficient for ALPs with low masses and large coupling constants [42]. For coupling constants below 10 −5 and for masses beyond m 0.3 eV measurements are limited by the current experimental precision. Improvements in this respect may render wider regions of parameter space accessible in the near future.