Potential of the neutron Lloyd's mirror interferometer for the search for new interactions
PPotential of the neutron Lloyd’s mirror interferometerfor the search for new interactions
Yu. N. Pokotilovski Joint Institute for Nuclear Research141980 Dubna, Moscow region, Russia
Abstract
We discuss the potential of the neutron Lloyd’s mirror interferometer in asearch for new interactions at small scales. We consider three hypotheti-cal interactions that may be tested using the interferometer. The chameleonscalar field proposed to solve the enigma of accelerating expansion of the Uni-verse produces interaction between particles and matter. The axion-like spin-dependent coupling between neutron and nuclei or/and electrons may causeP- and T-non-invariant interaction with matter. Hypothetical non-Newtoniangravitational interactions mediates additional short-range potential betweenneutrons and bulk matter. These interactions between the neutron and themirror of the Lloyd’s type neutron interferometer cause phase shift of neutronwaves. We estimate the sensitivity and systematic effects of possible experi-ments.PACS: 14.80.Mz; 12.20.Fv; 29.90.+r; 33.25.+kKeywords: Chameleon scalar fields; Axion; Non-Newtonian gravity; Neutron interferometers
1. Introduction
It is believed that the Standard model is a low energy approximation of some more fundamental theory.Most popular extensions of the Standard Model: supersymmetry and superstring theories, predict the exis-tence of new particles and hence new interactions. These new particles were not detected up to now becauseof their too large mass, or because of too weak interaction with ordinary matter. This last case is of interestin our discussion of a search for new hypothetical weak interactions of different nature.The possible existence of new interactions with macroscopic ranges and weak coupling to matter cur-rently attracts increasing attention. Significant number of experiments has been performed to search fornew forces in a wide range of distance scales. Here we consider possibilities of the neutron Lloyd’s mirrorinterferometer in searching for some of these new interactions.The Lloyd’s mirror interferometer (see Fig. 1) well known in the light optics has not yet been discussedin the experimental neutron optics.The geometric phase shift is determined by the difference of path lengths of the reflected and non-reflected beams: ϕ geom = ϕ II,geom − ϕ I,geom = k (cid:16)(cid:113) L + ( b + a ) − (cid:113) L + ( b − a ) (cid:17) ≈ kab/L, (1) e-mail: [email protected] a r X i v : . [ nu c l - e x ] N ov here k is the neutron wave vector, L , b and a are given in the Fig. 1 caption. The last equation is validwith relative precision better than ab/L . The geometric phase shift linearly depends on the interferencecoordinate b . It means that the interference pattern I ∼ sin ( πab/λ n L ) in the absence of any potentialsis sinusoidal with high precision: ab/L ∼ − at a ∼ b ∼ − cm and L = 1 m. The period ofoscillations in the interference pattern is Λ osc = λ n L/ (2 a ) , where λ n is the neutron wavelength, and is ∼ µ m for the thermal neutrons energy range and reasonable parameters of the interferometer. But forvery cold neutrons in the µ eV-energy range the period of the interference oscillations approaches dozens of µ m, and an interference picture may be registered with a narrow (about ∼ µ m) slit at a detector windowor with modern high resolution position sensitive neutron detectors.The idea of possible application of the Lloyd’s mirror interferometer for the search for new hypotheticalinteraction between matter and particles consists in measuring the neutron wave phase shift produced by ahypothetical mirror-neutron potential. We here consider three actively discussed hypothetical interactions:the cosmological scalar fields proposed to explain the accelerated expansion of the Universe, the axion-like spin-dependent pseudoscalar nucleon-nucleon and/or nucleon-electron interaction, and hypotheticaldeviation of the gravitation law from the Newtonian one at small distances (non-Newtonian gravity).
2. Chameleon scalar field
There is evidence of the accelerated expansion of the Universe. The nature of this effect is one of themost exciting problems in physics and cosmology. It is not clear yet whether the explanation of the factthat gravity becomes repulsive at large distances should be found within General Relativity or within a newtheory of gravitation. One possibility to explain this fact is to modify the General Relativity Theory, andthere was a number of proposals of this kind. Among various ideas proposed to explain this astronomicalobservation in a different way, one of popular variants is a new matter component of the Universe – acosmological scalar field of the quintessence type [1] dominating the present day density of the Universe(for the recent reviews see [2, 3]).Acting on cosmological distances the mass of this field should be of the order of the Hubble constant: ¯ hH /c = 10 − eV /c .The massless scalar fields appearing in string and supergravity theories couple to matter with gravita-tional strength. Because of direct coupling to matter with a strength of gravity, the existence of light scalarfields leads to a violation of the equivalence principle. In the absence of self-interaction of the scalar field,the experimental constraints on such a field are very strict, requiring its coupling to matter to be unnaturallysmall.The solution proposed in [4, 5, 6, 7, 8, 9] consists in the introduction of the coupling of the scalar fieldwith matter of such a form that as a result of self-interaction and the interaction of the scalar field withmatter, the mass of the scalar field depends on the local matter environment.In the proposed variant, the coupling to matter is of the order as expected from string theory, but is verysmall on cosmological scales. In the environment of the high matter density, the mass of the field increases,the interaction range strongly decreases, and the equivalence principle is not violated in laboratory exper-iments for the search for the long-range fifth force. The field is confined inside the matter screening itsexistence to the external world. In this way the chameleon fields evade tests of the equivalence principleand the fifth force experiments even if these fields are strongly coupled to matter. As a result of the screen-ing effect the laboratory gravitational experiments are unable to set an upper limit on the strength of thechameleon-matter coupling.The deviations of results of measurements of gravity forces at macroscopic distances from calculationsbased on Newtonian physics can be seen in the experiments of Galileo-, E¨otv¨os-or Cavendish-type [10]performed with macro-bodies. At smaller distances ( − − − ) cm the effect of these forces can be2bserved in measurements of the Casimir force between closely placed macro-bodies (for review see [11])or in the atomic force microscopy experiments. Casimir force measurements may to some degree evade thescreening and probe the interactions of the chameleon field at the micrometer range despite the presence ofthe screening effect [9, 12, 13].At even smaller distances such experiments are not sensitive enough, and high precision particle scat-tering experiments may play their role. In view of absence of electric charge the experiments with neutronsare more sensitive than with charged particles, electromagnetic effects in scattering of neutrons by nucleiare generally known and can be accounted for with high precision [14, 15].As regards the chameleon interaction of elementary particles with bulk matter, it was mentioned in [16]that neutron should not show a screening effect - the chameleon-induced interaction potential of bulk matterwith neutron can be observed. It was also proposed in [16] to search for the chameleon field through energyshift of ultracold neutrons in the vicinity of reflecting horizontal mirror. From the already performed exper-iments on the observation of gravitational levels of neutrons, the constraints on parameters, characterizingthe force of chameleon-matter interaction were obtained in [16].Chameleons can also couple to photons. It was proposed in [17, 18] to search for in a closed vacuumcavity for the afterglow effect resulting from the chameleon-photon interaction in a magnetic field. TheGammeV-CHASE [19, 20] and ADMX [21] experiments based on this approach are intended to measure(constrain) the coupling of chameleon scalar field to matter and photons.In the approach proposed here only the chameleon-matter interaction is measured irrespective of the ex-istence of the chameleon-photon interaction. The approach is based on the standard method of measurementthe phase shift of a neutron wave in the interaction potential.Testing the interaction of particles with matter at small distances may be interesting irrespective of anyparticular variant of the theory.In one of popular variants of the chameleon scalar field theory [4, 5, 6, 7, 8, 9], the chameleon effectivepotential is V eff ( φ ) = V ( φ ) + e βφ/M Pl ρ, (2)where V ( φ ) = Λ + Λ n φ n . (3)is the scalar field potential, M P l is the Planck mass, ρ is the local energy density of the environment, Λ = (¯ h c ρ d.e. ) / =2.4 meV is the dark energy scale, ρ d.e. ≈ . × − erg/cm is the dark energy density,and β is the interaction parameter not predicted by the theory.The chameleon interaction potential of a neutron with bulk matter (mirror) was calculated in [16]: V ( z ) = β mM P l λ (cid:16) n √ (cid:17) / (2+ n ) (cid:16) zλ (cid:17) / (2+ n ) = β · . · − eV (cid:16) n √ (cid:17) / (2+ n ) (cid:16) zλ (cid:17) / (2+ n ) = V (cid:16) zλ (cid:17) / (2+ n ) , (4) V = β · . · − eV (cid:16) n √ (cid:17) / (2+ n ) , (5)where m is the neutron mass and λ = ¯ hc/ Λ = 82 µm .To reduce the strong effect of Earth’s gravity the mirror of the interferometer is vertical.The neutron wave vector k (cid:48) in the potential V is k (cid:48) = k − mV ¯ h , k (cid:48) = k − mVk ¯ h . (6)3he phase shift due to the chameleon-mediated interaction potential of a neutron with the mirror, de-pending on the distance from the mirror, is obtained by integration along trajectories ϕ = (cid:72) k (cid:48) ds = ϕ II − ϕ I ,where ϕ I and ϕ II are the phases obtained along trajectories I and II: ϕ I = k (cid:113) L + ( b − a ) − mV (cid:113) b − a ) /L ) k ¯ h λ α n − (cid:90) L (cid:16) a + b − aL x (cid:17) α n − dx == ϕ I,geom − γ (cid:113) b − a ) /L ) λ α n − α n ( b − a ) (cid:16) b α n − a α n (cid:17) (7)and ϕ II = k (cid:113) L + ( b + a ) − mV (cid:113) b + a ) /L ) k ¯ h λ α n − (cid:34)(cid:90) l (cid:16) a − b + aL x (cid:17) α n − dx + (cid:90) Ll (cid:16) b + aL x − a (cid:17) α n − dx (cid:35) == ϕ II,geom + γ (cid:113) b + a ) /L ) λ α n − α n ( b + a ) (cid:16) b α n + a α n (cid:17) . (8)Here l = ( aL ) / ( a + b ) is the x-coordinate of the beam II reflection point from the mirror, γ =( mV L ) / ( k ¯ h ) , and α n = (4 + n ) / (2 + n ) .The phase shift from the chameleon neutron-mirror potential ϕ cham = ϕ II,cham − ϕ I,cham == γλ α n − α n (cid:104) b α n − a α n b − a (cid:113) b − a ) /L ) − b α n + a α n b + a (cid:113) b + a ) /L ) (cid:105) ≈≈ γλ α n − α n ab b α n − − a α n − b − a . (9)For a non-strictly vertical mirror, the component of the Earth’s gravity normal to the surface of themirror produces the potential V gr = κmgz , where g is the acceleration of gravity, and the coefficient κ depends on the angle θ between gravity vector and the mirror plane. At θ = 10 ” , we have κ ≈ × − .This linear potential leads to additional phase shift ϕ gr = ϕ II,gr − ϕ I,gr = κgm k ¯ h ( a + b ) (cid:104) ( b + a ) (cid:113) L + ( b − a ) − ( b + a ) (cid:113) L + ( b + a ) (cid:105) ≈ κgm k ¯ h abLa + b , (10)calculated in analogy with Eqs.(6)-(9).The Coriolis phase shift due to Earth’s rotation [22] is ϕ Cor = 2 m ¯ h (Ω · A ) , (11)where Ω is the vector of angular rotation of Earth and A is the vector of the area enclosed by the interfer-ometer beams.As A = ( abL ) / ( a + b ) , for the location of the Institute Laue-Langevin (where a good very cold neutronsource has been constructed [23]) we have ϕ Cor = 0 . abL ) / (( a + b )) rad (a,b,L in cm). As expected it issimilar to the gravitational phase shift in its dependence both of the slit and the interference coordinates.We must also calculate the phase shift of the neutron wave along beam II at the point of reflection.Neglecting the imaginary part of the potential of the mirror, the amplitude of the reflected wave is r = e − iϕ refl , with the phase ϕ refl = 2 arccos( k ⊥ /k b ) ≈ π − δϕ refl , (12)4here δϕ refl = 2 k ⊥ /k b ≈ π − kk b a + bL , (13) k ⊥ is the neutron wave vector component normal to the mirror’s surface, and k b is the boundary wave vectorof the mirror. This phase shift linearly depends on b similarly to the geometric phase shift ϕ geom .The reflected and non-reflected beams follow slightly different paths in the interferometer. Thereforein the vertical arrangement of the reflecting mirror they spent different times in Earth’s gravitational fieldwith ∆ t = 2 ab/ ( Lv ) , where v is the neutron velocity. The difference in vertical shifts of the reflected andnon-reflected beams is ∆ h = 2 gab/v , and the phase shift due to this difference ∆ ϕ vert = kg abL/v . (14)With our parameters of the interferometer this value is of the order ∼ − .The total measured phase shift is ϕ = ϕ geom + ϕ cham + ϕ gr + ϕ Cor + ϕ refl . (15)The gravitational phase shift can be suppressed by installing the mirror vertically with the highest pos-sible precision. On the other hand the gravitational phase shift may be used for calibration of the theinterferometer by rotation around horizontal axis. The phase shifts due to Earth rotation ϕ Cor and reflection ϕ refl may be calculated and taken into account in analysis of the interference curve.Figure 2 shows the calculated phase shift ϕ cham for an idealized Lloyd’s mirror interferometer (strictlymonochromatic neutrons, the width of the slit is zero, the detector resolution is perfect) with parameters:the neutron wave length λ n =100 ˚A, (the neutron velocity 40 m/s), L =1 m, a = 100 µ m, β = 10 , at n =1and n =6. Shown also are the gravitational phase shift ϕ gr at c = 5 × − (deviation of the mirror fromverticality is 10”), and δϕ refl = π − phase shift of the ray II at reflection ( k b = 10 cm − ).It is essential that the sought phase shift due to hypothetical chameleon potential depends on the inter-ference coordinate nonlinearly. Effect of the hypothetical interaction has to be inferred from analysis of theinterference pattern after subtracting off the effects of Earth gravity, Coriolis and reflection.Figure 3 demonstrates the calculated interference pattern for the same parameters of the interferometeras in Fig. 2 for two cases: (1) with the geometrical phase shift ϕ geom , the gravitational phase shift ϕ gr at κ = 5 × − , and the phase shift of the ray II at reflection δϕ refl = π − ϕ refl ( k b = 10 cm − ) taken intoaccount; (2) the same plus the phase shift due to the chameleon field with matter interaction parameters β = 10 , n = 1 .After subtracting all the phase shifts except purely geometrical the interference pattern should be stronglysinusoidal with the period of oscillations determined by the geometric phase shift: Λ osc = λ n L/ (2 a ) . Thenumber of oscillations in an interference pattern with the coordinate less than b is n osc = 2 ab/ ( λ n L ) .It follows from these calculations that the effect of the chameleon interaction of a neutron with mattermay be tested in the range of strong coupling with the parameter of interaction down to β ∼ or lower.Existing constraints on the parameters β and n may be found in Fig. 1 of Ref. [16]. For example theallowed range of parameters for the strong coupling regime β (cid:29) are: < β < × for n=1, < β < × for n=2, and β < for n > . It is seen that the Lloyd’s mirror interferometer maybe able to constrain the chameleon field in the large coupling area of the theory parameters.
3. Axion-like spin-dependent interaction.
There are general theoretical indications of the existence of interactions coupling mass to particle spin[26, 27, 28, 29, 30]. Experimental search for these forces is promising way to discover new physics.5n the other hand, a number of concrete proposals were published of new light, scalar or pseudoscalar,vector or pseudovector weakly interacting bosons. The masses and the coupling of these new hypotheticalparticles to nucleons, leptons, and photons are not predicted by the proposed models.The popular solution of the strong CP problem is the existence of a light pseudoscalar boson - the axion[31]. The axion coupling to fermions has general view g aff = C f m f /f a , where C f is the model dependentfactor. Here f a is the scale of Peccei-Quinn symmetry breaking which is not predicted so that the axion mayhave a priori mass in a very large range: ( − < m a < ) eV. The main part of this mass range fromboth – low and high mass boundaries – was excluded in result of numerous experiments and constraintsfrom astrophysical considerations [32]. Astrophysical bounds are based on some assumptions concerningthe axion and photon fluxes produced in stellar plasma.More recent constraints limit the axion mass to ( − < m a < − ) eV with respectively very smallcoupling constants to quarks and photon [32].The axion is one of the best candidates for the cold dark matter of the Universe [33, 34].Axions can mediate a P- and T-reversal violating monopole-dipole interaction potential between spinand matter (polarized and unpolarized nucleons or electrons) [35]: V mon − dip ( r ) = g s g p ¯ h σ · n πm n (cid:16) λr + 1 r (cid:17) e − r/λ , (16)where g s and g p are dimensionless coupling constants of the scalar and pseudoscalar vertices (unpolarizedand polarized particles), m n the nucleon mass at the polarized vertex, the nucleon spin s = ¯ h σ / , r is thedistance between the nucleons, λ = ¯ h/ ( m a c ) is the range of the force, m a - the axion mass, and n = r /r isthe unitary vector.Several laboratory searches (mostly by the torsion pendulum method) provided constraints on theproduct of the scalar and pseudoscalar couplings at macroscopic distances λ > − cm (see reviews[36, 37, 38, 39]).There are also experiments on the search for the monopole-dipole interactions in which the polarizedprobe is an elementary particle: neutron [40, 41, 42, 43], or atoms and nuclei [44, 45], correction in [46].For the monopole-monopole interaction due to exchange of the pseudo-scalar boson [35] V mon − mon ( r ) = g s π ¯ hcr e − r/λ (17)the limit on the scalar coupling constant g s can be inferred from the experimental search for the ”fifth force”in the form of the Yukawa-type gravity potential U ( r ) = α GM me − r/λ /r : g s = 4 πGm n α ¯ hc ≈ − α , (18)where α is the ”fifth force” Yukawa-type interaction constant.It follows from the experimental tests of gravitation at small distances (see reviews in [36, 37]) that g s is limited by the value − − − in the interaction range 1 cm > λ > − cm. The sensitivity of theseexperiments falls with decreasing the interaction range below ∼ g p < − from astrophysical considerations [47,39].It is seen that the constraints obtained and expected from further laboratory searches are weak comparedto the limit on the product g s g p < − inferred from the above mentioned separate constraints on g s and g p . Although laboratory experiments may not lead to bounds that are strongest numerically, measurements6ade in terrestrial laboratories produce the most reliable results. The direct experimental constraints on themonopole-dipole interaction may be useful for limiting more general class of low-mass bosons irrespectiveof any particular theoretical model. In what follows the constraint on product g s g p may be used for thelimits on the coupling constant of this more general interaction.It follows from Eq. (16) that the potential between the layer of substance and the nucleon separated bythe distance x from the surface is: V mon − dip ( x ) = ± g s g p ¯ h N λ m n ( e − x/λ − e − ( x + d ) /λ ) = ± V e − x/λ ( d (cid:29) λ ) , (19)where V = g s g p ¯ h N λ/ (4 m n ) , N is the nucleon density in the layer, d is the layer’s thickness. The ”+” and”-” depends on the nucleon spin projection on x-axis (the surface normal).Phase shifts of beams I and II due to interaction of Eq. (20) are calculated similarly to Eqs. (7) and (8): ϕ I = k (cid:113) L + ( b − a ) + mV k ¯ h L (cid:113) L + ( b − a ) L (cid:90) exp (cid:104) − (cid:16) a + b − aL x (cid:17) /λ (cid:105) dx == ϕ geom.I + mV k ¯ h (cid:113) L + ( b − a ) λb − a ( e − a/λ − e − b/λ ) = ϕ geom.I + ϕ pot.I (20)and ϕ II = k (cid:113) L + ( b + a ) + mV k ¯ h L (cid:113) L + ( b + a ) (cid:34) l (cid:90) exp (cid:104) − (cid:16) a − b + aL x (cid:17) /λ (cid:105) dx + L (cid:90) l exp (cid:104) − (cid:16) b + aL x − a (cid:17) /λ (cid:105) dx (cid:35) = ϕ geom.II + mV k ¯ h (cid:113) L + ( b + a ) λb + a (2 − e − a/λ − e − b/λ ) == ϕ geom.II + ϕ pot.II . (21)For the spin-dependent potential of Eq.(19) the signs of potential V and, respectively, the phase shifts ϕ pot are opposite for two spin orientations in respect to the mirror surface normal.The difference in these phase shifts, measured in the experiment ( ϕ + I − ϕ + II ) − ( ϕ − I − ϕ − II ) = 2( ϕ I − ϕ II ) = δϕ. (22)The geometric, gravitational phase shifts and phase shift of the beam II at reflection calculated earlier donot depend on spin.The phase shift due to axion interaction is ϕ ax = 2 γλ (cid:104) a (1 − e − b/λ ) − b (1 − e − a/λ ) (cid:105) / ( b − a ) , (23)where γ = g s g p N λL/ (4 k ) . At b = a , and λ/a (cid:28) , ϕ ax → γλ/a .Figure 4 shows the neutron wave phase shift ϕ ax for different interaction range λ . Lloyd’s mirrorinterferometer has the following parameters: neutron wave length 100 ˚A (neutron velocity 40 m/s), L =1 m, a = 100 µ m, and the interaction strength g s g p = 10 − .The possible sensitivity seen from this figure shows that constraints on the monopole-dipole interactionwhich can be obtained with the method of the neutron Lloyd’s mirror interferometry is competing with bestconstraints obtained by other methods (see Ref. [39]).7igure 5 shows the calculated interference pattern due to an axion-like spin-dependent interaction. Thegradient of the external magnetic field ∇ ( µB ) normal to the mirror plane ( µ n is the neutron magneticmoment) may produce the phase hift effect on polarized neutrons, similar to the effect of gravitational force F gr = mg (see Eq. (10)). Simple calculation gives that magnetic field gradient 0.01 Oe/cm is equivalent to ∼ × − of the Earth gravitation.A significant increase in sensitivity may be achieved in the range of small λ ( λ/a (cid:28) if the geometryshown in Fig. 1(2) is used, where the slit is located in close vicinity to the surface of additional (upper),non-reflecting mirror. The axion-like potential is produced in this case by both mirrors, but with oppositesigns in accordance to Eq. (20).To avoid multiple reflections, the boundary wave vector of the reflecting mirror must satisfy the con-dition k b ≤ ka/L , or the neutron beam incident on the slit should be collimated so the the first half ofthe reflecting mirror is not illuminated by the neutrons. In this geometry the phase shift due to axion-likemonopole-dipole interaction of the neutron with both mirrors is ϕ ax = 2 γλa ( e ( b − a ) /λ − e − a/λ + e − b/λ − / ( b − a ) . (24)In this case ϕ ax → γ (1 − e − a/λ ) → γ as b → a , for λ/a (cid:28) compared to γλ/a for the case of onemirror (Eq. (23).The gain in sensitivity at λ/a (cid:28) compared to the case of Fig. 4 is illustrated at Fig. 6.
4. Non-Newtonian gravity
New short-distance spin-independent forces are frequently predicted in theories expanding the StandardModel. These interactions can violate the Equivalence Principle if they depend on the composition ofbodies, or the sort of particles.Precision experiments to search for deviations from Newton’s inverse square law and of violation theWeak Equivalence Principle have been performed in a number laboratories (reviews may be found in [10,36, 37, 38, 39].The pioneering ideas of the multi-dimensional models first formulated in the first half of the XX-thcentury (G. Nordstr¨om, T. Kaluza and O. Klein) received renewed interest in [48, 49, 50]. The developmentof supergravity and superstring theories required for their consistency extra-dimensions. A more recentpromising development contained in [51, 52, 53, 54, 55] proposed mechanisms in which the StandardModel fields are located on the 4-dimensional brane while gravity propagates to the (4+n)-bulk with alarger number of dimensions. As a result the gravitational law may be different from the Newtonian one.The frequently used parametrization of new spin-independent hypothetical short-range interaction po-tential has the Yukawa-type form U ( r ) = αGM mr e − r/λ , (25)where G is the Newtonian gravitational constant, M and m are the masses of gravitating bodies, α is thedimensionless parameter characterizing the strength of the new force relative to gravity, and λ = ¯ h/ ( m c ) is the Compton wave length of the particle with the mass m . The mass m can be the mass of the newscalar field responsible for the short-range interaction. In this case α ∼ g s - the product of the scalarcoupling constants. Or the mass m can be the mass of the lightest Kaluza-Klein state (which is the leadingorder mode) when the short-range interaction comes from the extra-dimensional expansion of the StandardModel.The strength α is constrained to be below unity for λ ≥ µ m, [39, 56], but for shorter distancesthe measurements are not as sensitive being complicated by the Casimir and electrostatic forces [11]. Thesensitivity reached in the experiments aiming to test spin-independent interactions between elementary8articles and matter are orders of magnitude less sensitive: at λ = 100 µ m it is at the level α > [57]with loss of sensitivity at lower distances.The potential following from the interaction of Eq. (25) between the layer of substance and a neutronseparated by the distance x from the surface is: V Y uk ( x ) = 2 παm n N Gλ e − x/λ = V e − x/λ , (26)where N ≈ ρ/m n is the nucleon density in the layer, ρ is density of the mirror, and V = 2 παm n N Gλ The potential of Eq. (26) has the same coordinate dependence as the axion-like interaction potential ofEq. (19), therefore the expressions for the phase shifts are similar to Eq. (23) with γ = 2 παρλ m n L/ ( k ¯ h ) .Fig. 7 shows the phase shifts due to the non-Newtonian interaction of Eq. (26) at the same parametersof the interferometer of Fig. 2, and ρ = 10 g/cm .For the ”inverted” Lloyd’s mirror geometry when the reflecting mirror has much lower density so thatits gravitational effect is insignificant compared to the effect of the upper mirror the phase shift is ϕ Y uk = 2 γλb − a (cid:104) a ( e − a/λ − e ( b − a ) /λ ) + b (1 − e − a/λ ) (cid:105) . (27)At b → a , and λ/a → , ϕ Y uk → γ with significant gain in sensitivity for λ/a (cid:28) compared to thegeometry of Fig. 1(1) and Eq. (23).To avoid multiple reflections in this case, the geometry of Fig. 1(3) may be applied, in which thereflecting mirror has only half length compared to Fig. 1(1). The gain ib sensitivity is illustrated in Fig. 8.
5. Feasibility
As mentioned above, the interference may be measured step by step shifting a narrow slit with thewidth δb (cid:28) Λ osc ∼ − µ m. A better option is to use the coordinate detector measuring in this way allthe interference picture simultaneously. The current spatial resolution of position-sensitive slow neutrondetectors is at the level of 5 µ m with electronic registration [58] and about 1 µ m with the plastic nucleartrack detection technique [59]. With a µ m thick B neutron converter the efficiency of registration of the100 ˚A–wave-length neutrons may approach 100 %.From Ref. [23] where the neutron phase density at the PF-2 very cold (VCN) channel at the InstituteLaue-Langevin was measured to be 0.25 cm − (m/s) − at v=50 m/s it is possible to estimate the VCN fluxdensity as φ V CN = 1 . × cm − s − (m/s) − ≈ × cm − s − ˚A − (at the boundary velocity of theneutron guide 6.5 m/s). On the other hand Ref. [60] gives the larger value φ V CN = 4 × cm − s − (m/s) − for the same channel.Using the Zernike theorem it is possible to calculate the width d sl of the slit necessary to satisfy goodcoherence within the coherence aperture ω , i.e. the maximum angle between diverging interfering beams: x = πωd sl /λ n ≤ , if the slit is irradiated with an incoherent neutron flux. As ω = 2 b max /L , the slitwidth d sl ≤ Lλ n / (2 πb max ) ≈ µ m at b max = 1 mm (20 orders of interference at the period of interference Λ osc = λ n L/ (2 a ) = 50 µ m at a = 100 µ m).At the monochromaticity 5 ˚A (the coherence length l coh = 20 λ n ), the slit width d sl = 2 µ m, the lengthof the slit 3 cm, the divergence of the incident beam determined by the the VCN guide boundary velocityof 6.5 m/s: Ω = 6 . /
100 = 0 . , the interference aperture ω = 2 b/L = 0 . /
100 = 2 × − , andhence ω/ Ω = 0 . , we have the count rate to all interference curve with a width of 1 mm (20 orders ofinterference) is given by × × · − × × . ∼ s − . In one day measurement the numberof events in one period of interference is ∼ × . It is enough to observe the phase shift of ∼ . corresponding to the effect at β = 10 . 9istinctive feature of the Lloyd’s mirror interferometer is the possibility to register all the interferencepattern simultaneously along z-coordinate starting from z=0 (Fig. 1). The measured interference patternis then analyzed as regards the presence of the sought for effects, after the corrections taking the knowngravity, Coriolis, and reflection phase shifts into account.The LLL-type interferometers [24, 25] may be used in principle to search for new hypothetical interac-tions placing a piece of matter in the vicinity of interfering beams. But the geometry of these interferometersdoes not permit probing hypothetical short-range interactions: the axion-like or non-Newtonian gravity inthis way.We may estimate sensitivity of the LLL-type interferometer to the chameleon potential, which is actuallynot short-range. The phase shift in this case is ϕ LLL = 2 γ (cid:113) a/L ) λ α n − (2 a ) α n − , (28)where a is the half distance between the beams of the LLL-interferometer. The sensitivity of the Lloyd’smirror and the LLL-interferometers is determined by the factor La α n − /k , which is much in favor of theLloyd’s mirror interferometer.In the interferometers of the LLL-type [24, 25] an interference pattern is obtained point by point byrotation of a phase flag introduced into the beams. In the case of the VCN three grating interferometers(for example [61, 62, 63]) the phase shift between the beams is realized by the same method or by shiftingposition of the grating.The neutron Lloyd’s mirror experiments may be performed with monochromatic very cold neutrons, orin the time-of-flight mode using large wave length range, for example 80-120 ˚A. The pseudo-random mod-ulation [64, 65] is used in the correlation time-of-flight spectrometry. It was realized in the very low neutronenergy range [66]. In this case a two-dimensional interference coordinate – time-of-flight registration givessignificant statistical gain.As in the VCN interferometers based on three gratings, in the Lloyd’s mirror neutron interferometer thespace between the beams is small (parts of mm). Therefore it hardly can be used in experiments where somedevices are introduced in the beams, or between the beams (for example to investigate non-local quantum-mechanical effects). But it may be applicable to search for short-range interactions when they are producedby a reflecting mirror.
6. Acknowledgement
The author is indepted to an anonymous referee for the comments on the first version of the manuscriptand suggestions.
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