Centrifugal quantum states of neutrons

Centrifugal quantum states of neutrons V.V. Nesvizhevsky and A.K. Petukhov Institut Laue-Langevin (ILL), 6 rue Jules Horowitz, F-38042, Grenoble, France, e-mail: [email protected] K.V. Protasov Laboratoire de Physique Subatomique et de Cosmologie (LPSC), IN2P3-CNRS, UJFG, 53, Avenue des Martyrs, F-38026, Grenoble, France A.Yu. Voronin arXiv:0806.3871v1 [quant-ph] 24 Jun 2008 P.N. Lebedev Physical Institute, 53 Leninsky prospekt,119991, Moscow, Russia We propose a method for observation of the quasi-stationary states of neutrons, localized near the curved mirror surface. The bounding effective well is formed by the centrifugal potential and the mirror Fermi-potential. This phenomenon is an example of an exactly solvable ”quantum bouncer” problem that could be studied experimentally. It could provide a promising tool for studying fundamental neutron-matter interactions, as well as quantum neutron optics and surface physics effects. We develop formalism, which describes quantitatively the neutron motion near the mirror surface. The effects of mirror roughness are taken into account. I. INTRODUCTION The ”centrifugal states” of neutrons is a quantum analog of the so called whispering gallery wave, the phenomenon which in brief consists in the wave localization near the curved surface of a scatterer. It is known in acoustics since ancient times and was explained by Baron Rayleigh in his ”Theory of Sound” [1, 2]. The whispering gallery waves in optics is an object of growing interest during the last decade [3, 4]. In the following we will be interested in the matter-wave aspect of the whispering gallery wave phenomenon, namely the large-angle neutron scattering on a curved mirror. Such a scattering can be explained in terms of states of a quantum particle above a mirror in a linear potential - the so called ”quantum bouncer”[5, 6, 7, 8, 9, 10, 11]. The neutron quantum motion in the Earth’s gravitational field above a flat mirror is another example of such a ”quantum bouncer”, which was observed recently [12]. We will show that the centrifugal quantum bouncer and the gravitational quantum bouncer have many common features. Therefore we compare these two phenomena and discuss motivation for their studies. Experimental observation and study of the gravitational states is a challenging problem which brings rich physical information for searches for extensions of the Standard model or for studying interaction of a quantum system with gravitational field [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], for constraining spin-independent extra shortrange forces [27, 28, 29], hypothetical axion-mediated spin-matter interactions [30] and in surface physics. Indeed any additional interaction between mirror bulk and neutron with the characteristic range of the gravitational states of a few micrometers would modify the quantum states and thus could be detected. A natural extension of the mentioned experimental activity consists in approaching ultimate sensitivity for extra interactions at shorter characteristic ranges. Evidently, the quantum states characteristic size has to be decreased. To achieve this goal one needs to study novel approaches [31]. We will show that the promising method consists in localization of cold neutrons near a curved mirror surface due to the superposition of the centrifugal potential and the Fermi potential of the mirror. In such a case the quasi-stationary ”centrifugal” quantum states play essential role in the neutron flux dynamics. In the limit of the centrifugal quantum states spatial size being much smaller than the curved mirror radius this problem is reduced to the simple case of a quantum particle in a linear potential above a mirror. Measurement of the gravitationally bound and centrifugal quantum states of neutrons could be considered as a kind of confirmation of the equivalence principle for a quantum particle [32, 33, 34, 35, 36]. Both problems (the gravitational and the centrifugal ones) provide perfect experimental laboratory for studying neutron quantum optics phenomena, quantum revivals and localization [37, 38, 39, 40, 41, 42, 43, 44]. Evident advantages of using cold neutrons consist in much higher statistics attainable, broad accessibility of cold neutron beams as well as in crucial reduction of many false effects compared to the experiments with the gravitationally bound quantum states of neutrons due to approximately ∼ 105 times higher energies of the quantum states involved. The phenomenon of the centrifugal quantum states of neutrons and the method of their experimental observation are described in Chapter 2. In Chapter 3 we develop formalism, which describes neutron motion near the curved mirror surface; the properties of the centrifugal quantum states are discussed in Chapter 4. We will show that cold neutrons with the velocity of ∼ 103 m/s are well suited for such a kind of experiments. Time-dependent approach is considered in Chapter 5. The effects of mirror roughness are taken into account in Chapter 6. 2 II. PRINCIPLE OF OBSERVATION If the neutron energy is much larger than the scatterer Fermi potential most neutrons are scattered to small angles. However some neutrons could be captured into long-living centrifugal quasi-stationary states localized near the curved scatterer surface and thus could be detected at large deflection angles. The curved mirror surface plays a role of a wave-guide and the centrifugal states play a role of radial modes in such a wave-guide. The spectral dependence of transmission probability is determined by the existence of the centrifugal states in such a system. Similarly, in the gravitational state experiment one measures the slit-size dependence of the transmission probability of the wave-guide between a mirror and above-placed absorber [45, 46, 47, 48, 49, 50, 51, 52, 53, 54]. The characteristic energy scale of the gravitational state problem is ε0 = 0.6 peV , and the characteristic length scale is l0 = 5.87 µm. The mirror Fermi-potential could be considered as infinitely high and sharp. This approximation is justified as far as l0 is much larger than the characteristic range of the Fermi-potential increase (typically < 1 nm), and ε0 is much smaller than the characteristic value of the mirror Fermi-potential (typically ∼ 10−7 eV). The methods for experimental observation of the gravitationally bound quantum states of neutrons are based on relatively large value of the characteristic length l0 , which allowed direct measurement of the shapes of neutron density distribution in the quantum states using two following complementary methods. First approach consisted in scanning the neutron density above mirror using a flat horizontal scatterer/absorber at variable height. The second method is based on use of position-sensitive detectors of UCN with high spatial resolution of ∼ 1 µm. In analogy with the gravitational well the centrifugal quantum well is formed by effective centrifugal potential and repulsive Fermi-potential of a curved mirror as shown in Fig.1 and Fig.2. The effective acceleration near the curved mirror surface could be approximated as a = v 2 /R , where v is the neutron velocity and R is the mirror radius. We have significant freedom to choose values of v and R. In particular it would be advantageous to increase the neutron velocity and to decrease the mirror radius in order to get higher centrifugal acceleration a. In such a case the quantum well that confines the radial motion of neutrons near the curved mirror surface becomes narrower, while the energy of radial motion in the corresponding quantum states increases. This enables us to eliminate many possible systematic effects. The radial motion energy could be as high as the mirror Fermi-potential. In this case, we could use the Fermi-potential of a mirror as a ”filter” for the quantum states. For an ideal cylindrical mirror with perfect shape and zero roughness made of low-absorbing material neutron losses in such quasi-stationary quantum states occur via tunneling of neutrons through triangle potential barrier shown in Fig.2. The lifetime of deeply bound states is long. The lifetime of the quasi-stationary quantum states with energy close to the barrier edge is short; such neutrons tunnel rapidly into the mirror bulk. If we vary continuously the centrifugal acceleration (by means of changing the neutron velocity), we will vary the height and the width of the triangle barrier correspondingly and so far the lifetime of the quasi-stationary quantum states. Due to the very fast (exponential) increase of the mentioned lifetime as a function of the barrier width we will get a step-wise dependence of the neutron flux parallel to the mirror surface as a function of the neutron velocity. Analogous step-wise dependence of the total neutron flux as a function of the slit size was observed in case of the gravitationally bound quantum states. An alternative method for observation and studying the centrifugal quantum states consists in measuring velocity distribution in the quantum states using a position-sensitive neutron detector, placed at some distance from the curved mirror. Such a method was used as well in the experimental studies of the gravitationally bound quantum states of neutrons. It is natural to chose the neutron velocity within the range of maximum intensity of standard neutron sources (neutron reactors or spallation sources) around ∼ 103 m/s. The cylindrical mirror radius has to be equal to a few centimeters in this case that is just optimal for its production. Neutron beams with high intensity are available in many neutron centers around the world; they could be angularly and spatially collimated; time-of-flight and polarization analysis are available at standard neutron-scattering instruments. Evidently, the characteristic size of the centrifugal quantum states is much smaller than the mirror radius and the effective centrifugal acceleration could be approximated as constant with high accuracy. On the other hand, Fermi-potential of a mirror can not be considered as infinitely high. Just the opposite: quantum states energy is close to the value of Fermi-potential. Therefore in contrast to the gravitationally bound quantum states neutrons tunnel deeply into the mirror (compared to the characteristic size of the wave-functions). This phenomenon has to be taken into account. Another essential difference is related to the effects of surface roughness: as far as the characteristic scale of the centrifugal quantum states is much smaller, the roughness effects are much larger; therefore constraints for the cylindrical mirror surface are even more severe than those for flat mirrors in the gravitational experiments. We will show rigorously in the following chapters that the centrifugal quantum states could be described in very similar way as the gravitational quantum states although they are formed by completely different physical potentials. Large difference in characteristic scales of the quantum states in two cases requires different approaches for their experimental observation and study. 3 FIG. 1: A scheme of the neutron centrifugal experiment. 1 - the classical trajectories of incoming and outcoming neutrons, 2 the collimators, 3 - the cylindrical mirror, 4 - the detector. Cylindrical coordinates ρ − ϕ used throughout the paper are shown. III. FORMAL SOLUTION Neutrons with given energy E, scattered by the curved mirror obey the following Schrödinger equation in the cylindrical coordinates:   2   ~2 ~2 ∂ 2 ∂ 1 ∂ − − + + U (ρ, ϕ) − E Ψ(ρ, ϕ) = 0 (1) 2M ∂ρ2 ρ ∂ρ 2M ρ2 ∂ϕ2 Here M is the neutron mass, ρ is the radial distance, measured from the center of mirror curvature (see Fig.1), ϕ is the angle and U (ρ, ϕ) is the mirror Fermi potential. We will use the following step-like dependence for the mirror Fermi-potential: U (ρ, ϕ) = U0 Θ(ρ − R) (Θ(ϕ) − Θ(ϕ − ϕ0 )) where R is the mirror curvature radius, and the angle ϕ0 is determined by the mirror length Lmirr and the mirror curvature radius R via: ϕ0 = Lmirr 2πR In the equation (1) we omitted trivial dependence on z coordinate along the curved mirror axis. By standard √ substitution Ψ(ρ, ϕ) = Φ(ρ, ϕ)/ ρ the equation (1) is transformed to the following form:  2     2  ~2 ∂ ∂ 1 ~2 + U (ρ, ϕ) − E Φ(ρ, ϕ) = 0 (2) − + − 2M ∂ρ2 2M ρ2 ∂ϕ2 4 Now the problem is formulated as follows. The incoming neutron flux is known at the curved mirror entrance. We have to find the neutron flux at the exit of the mirror with the angle coordinate ϕ0 . The measured neutron current component, parallel to the mirror surface is:   i~ ∂Ψ∗ (ρ, ϕ) ∂Ψ(ρ, ϕ) ∗ J(ρ, ϕ) = Ψ(ρ, ϕ) (3) − Ψ (ρ, ϕ) 2M ρ ∂ϕ ∂ϕ We start with the formal solution of the equation (1) in the domain 0 ≤ ϕ ≤ ϕ0 . We express a solution of the equation (1) as a series expansion in the complete set of basis functions χµ (ρ) [55, 56, 57, 58] : X Φ(ρ, ϕ) = χµ (ρ) (cµ exp(iµϕ) + dµ exp(−iµϕ)) (4) µ 4 where cµ and dµ are the expansion coefficients. The basis functions χµ (ρ) are solutions of the following eigenvalue problem :  2    ~2 (µ2 − 1/4) ∂ ~2 + U Θ(ρ − R) − E χµ (ρ) = − χµ (ρ) (5) − 0 2 2M ∂ρ 2M ρ2 χµ (ρ → 0) = 0 (6) √ (7) χµ (ρ → ∞) = sin( 2M Eρ + δµ ) Here ~2 (1/4 − µ2 )/(2M ) ≡ −~2 η 2 /(2M ) is the eigenvalue and δµ is the scattering phase. µ plays a role of the angular momentum. Let us note that the energy E is a fixed parameter in the equation (5), while µ is the angular momentum eigenvalue to be found. For the above mentioned eigenvalue problem (5,6,7) self-adjointness of the corresponding Hamiltonian of radial motion is required for the completeness of the basis set χµ [59]. One can prove that this requirement and the boundary conditions (6,7) are equivalent to the following condition for the eigenstates phases: δµ′ − δµ = πk (8) where k is integer. In this case, the functions χµ are orthogonal to each other with the weight of 1/ρ2 on the interval [0, ∞). Note, that there is no uniqueness condition for the wave function as long as ϕ0 < 2π. So far µ is no longer an integer value in our problem. For given positive energy E > 0, the values µ form discrete spectrum of real values if η 2 ≥ 0 and continuum spectrum of complex values if η 2 < 0. The flux (3) through a band with the radial coordinates (ρ1 , ρ2 ) orthogonal to the mirror surface in the mentioned basis can be expressed as: Z ρ2 X Z ρ2 χ∗µ′ (ρ)χµ (ρ) 
~ J(ρ, ϕ)dρ = F (ϕ) = dρ µ′ cµ c∗µ′ exp(i(µ − µ′ )ϕ) − dµ d∗µ′ exp(−i(µ − µ′ )ϕ) (9) Re 2 M ρ ρ1 ρ1 ′ µ,µ In the following we will be interested in the flux F (ϕ) evolution as a function of the angle ϕ, which indicates the neutron density along the curved mirror. The neutron density is ”initially” (i.e. for ϕ = 0) localized by the collimator near the surface of the mirror in the band (ρ1, ρ2). Due to ”dephasing” of ϕ-dependent exponents in the expression (9) the neutron density within the band (ρ1, ρ2) decays rapidly when ϕ increases. We will find the rate of such a decay in the following sections. In particular we will show that such a rate is determined by the lifetime of the quasi-stationary states formed by the superposition of the centrifugal potential and the Fermi potential of the mirror. IV. CENTRIFUGAL QUASI-STATIONARY STATES To study the neutron states, localized near the mirror surface, we will expand the expression for the centrifugal energy in the equation (5) in the vicinity of ρ = R. We introduce the deviation from the mirror surface z = ρ − R and get the following equation in the first order of small ratio z/R:   2 ~2 ∂ 2 2 µn − 1/4 − + U0 Θ(z) + ~ (1 − 2z/R) − E χn (z) = 0 (10) 2M ∂z 2 2M R2 We will be interested in those solutions with different angular momenta µn , which correspond to the states of neutron, moving parallel to the mirror surface. Such neutrons with given energy E = M v 2 /2 possess angular momentum µn close to the classical value µ0 = M vR/~. Let us mention that the value of µ0 is extremely high µ0 ∼ 5 108 if v = 1000 m/s and R = 2.5 cm (parameters which can be realized in experimental setup) . Introducing new variables ∆n = µ0 − µn , where ∆n ≪ µ0 and εn = ~2 µ0 ∆n /(M R2 ) and keeping leading terms in µ0 we get the following equation:   ~2 ∂ 2 M v2 − (11) + U Θ(z) − z − ε 0 n χn (z) = 0 2M ∂z 2 R Let us mention that the eigenvalue εn plays a role of energy in the above equation only formally. In fact it defines the angular momentum eigenvalue µn = µ0 − εn M R 2 µ0 ~2 (12) 5 5,5 5,0 4,5 4,0 n=2 3,5 V 3,0 2,5 n 2,0 n=1 1,5 1,0 0,5 0,0 -5 -4 -3 -2 -1 0 1 2 3 4 5 z FIG. 2: A sketch of the effective potential in the mirror surface vicinity. The potential step at z = 0 is equal to the mirror Fermipotential in units ε0 = (~2 M v 4 /(2R2 ))1/3 . The potential slope at z 6= 0 is governed by the centrifugal effective acceleration a = v 2 /R. while the neutron energy E is a fixed parameter in our problem. The value εn can be interpreted as the radial motion energy within the above used linear expansion of the centrifugal potential in the vicinity of the curved mirror radius R. The equation (11) describes the neutron motion in constant effective field a = −v 2 /R superposed with the mirror Fermi potential U0 Θ(z). The sketch of corresponding potential is shown in Fig.2. The regular solution of the equation (11) is given by the well-known Airy function [60]:  Ai(z0 − z/l0 − εn /ε0 ) if z > 0 χn (z) ∼ (13) Ai(−z/l0 − εn /ε0 ) if z ≤ 0 Here l0 = (~2 R/(2M 2 v 2 ))1/3 (14) is the characteristic distance scale of the problem, and ε0 = (~2 M v 4 /(2R2 ))1/3 (15) is the characteristic energy scale, z0 = U0 /ε0 . For the typical experimental setup parameters U0 = 150 neV, v = 1000 m/s and R = 2.5 cm the above mentioned scales are l0 = 0.04 µm and ε0 = 15.3 neV and z0 ≃ 10. The above mentioned effective potential supports existence of the quasi-stationary states. They correspond to the solution of the equation (11) with the outgoing wave boundary condition:  Bi(z0 − z/l0 − εn /ε0 ) + i Ai(z0 − z/l0 − εn /ε0 ) if z > 0 (16) χ en (z) ∼ Ai(−z/l0 − εn /ε0 ) if z ≤ 0 The complex energies of such quasi-stationary states can be found from the matching of logarithmic derivative at z = 0: εn ≡ ε0 λn Ai (−λn ) (Bi(z0 − λn ) + i Ai(z0 − λn )) = Ai(−λn ) (Bi′ (z0 − λn ) + i Ai′ (z0 − λn )) ′ (17) (18) The real and imaginary parts of eigen-value λ, obtained by numerical solution of the equation (18) for two lowest states are shown as a function of dimensionless variable z0 in Fig.3 and Fig.4. 6 n=1 5,0 n=2 4,5 4,0 Re 3,5 3,0 2,5 2,0 1,5 1,0 1 2 3 4 5 6 Z 0 FIG. 3: The real part of two lowest eigen-values λ as a function of z0 obtained by numerical integration of the equation (18). n=1 n=2 0,0 -0,1 -0,2 Im -0,3 -0,4 -0,5 -0,6 -0,7 -0,8 1 2 3 4 5 6 Z 0 FIG. 4: The imaginary part of two lowest eigen-values λ as a function of z0 obtained by numerical integration of the equation (18). One can get semiclassical approximation for the widths of the centrifugal quasi-stationary states if |λn | ≫ 1, z0 ≫ |λn | [49]. In this case one can use the asymptotic expressions for the Airy functions of large argument to get the following equation: r λn 2 tan( λ3/2 − π/4) = 1 − 2i exp(−4/3(z0 − λn )3/2 ) (19) z0 − λn 3 n Also one can get semiclassical approximation for the width Γn and λn from the above expression, valid for large n: s ( 34 π(2n − 1/2))2/3 3 (20) λn ≃ ( π(2n − 1/2))2/3 − 4 z0 − ( 34 π(2n − 1/2))2/3 7 n=1 n=2 time of flight 0,1 T, s 0,01 1E-3 1E-4 1E-5 1200 1300 1400 1500 1600 1700 V (m/s) FIG. 5: The life-time of the two lowest neutron centrifugal quasi-stationary states is shown as a function of the neutron velocity. The mirror curvature radius equals R=2.5 cm, the mirror length is 5 cm, and the mirror Fermi potential is U0 = 150 neV. Nearly horizontal solid line indicates the time of flight along the curved mirror. Γn √ z0 − λn ≃ 4ε0 exp(−4/3(z0 − λn )3/2 ) z0 (21) In the above expressions n = 1, 2, ... is an integer number. The angular momentum eigenvalue, corresponding to the complex energy εn of the quasi-stationary states, obtains positive imaginary part, according to (12): Im µn = Γn R 2~v (22) The energy and the width of the quasi-stationary states depends strongly on the centrifugal acceleration |a| = v 2 /R. Small acceleration a results in broad barrier, which separates the states in the effective well from continuum. Indeed, z0 = U0 /ε0 = U0 [(2R2 )/(~2 M v 4 )]1/3 increases if v decreases. The widths of the quasi-stationary states decrease exponentially as it is seen from the expression (21). Besides that, the effective well becomes broader and new quasistationary states appear with decreasing of a (in analogy with appearance of new bound states with increasing the size of the well). The equation (19) enables us to estimate the critical values of the neutron velocity vc , which correspond to the appearance of new states in the effective well: z0 = λ0n = (3/2π(n − 3/4))2/3 Taking into account that z0 = U0 /ε0 we conclude:  1/4 U03 2R2 vcn = (3/2π(n − 3/4))2 ~2 M (23) The accuracy of the above approximation increases with n. The lifetime of the two lowest quasi-stationary quantum states as a function of the neutron velocity, obtained from solving the equation (18) is shown in Fig.5 for the mirror with the Fermi potential U0 = 150 neV (sapphire) and in Fig.6 for the mirror with the Fermi potential U0 = 54 neV (silicium). The critical velocity values scales with the 3/4 Fermi potential as vc ∼ U0 . The above mentioned quasi-stationary states play essential role in neutron density evolution near the mirror surface as a function of ϕ. We will show that under certain conditions the expansion (9) can be substituted by a few effective terms, corresponding to the contribution of quasi-stationary states. To clarify the role of quasi-stationary states it would be more convenient to use time-dependent formalism. 8 n=1 n=2 time of flight 0,01 T, s 1E-3 1E-4 560 580 600 620 640 660 680 700 720 740 760 780 800 820 840 V (m/s) FIG. 6: The life-time of the two lowest neutron centrifugal quasi-stationary states is shown as a function of the neutron velocity. The mirror curvature radius equals R=2.5 cm, mirror length is 5 cm, and mirror Fermi potential is U0 = 54 neV. Nearly horizontal solid line indicates the time of flight along the curved mirror. V. TIME-DEPENDENT APPROACH Let us return to the expansion (9) for the neutron current through the band of dimension h = ρ2 − ρ1 ≪ R, orthogonal to the mirror surface. Taking into account very large values of the angular momenta of the neutron ”near-surface” states (µ ∼ µ0 ≃ 5 108 ), we can use the semi-classical character of motion along the ϕ variable. It is characterized by contribution of fast oscillating exponents exp(iµϕ). This enables us to treat ϕ as a classical variable. Namely we will assume that neutrons follow classical ”trajectory” along ϕ: ϕ = ωt = vt R The evolution along ϕ is then substituted by the evolution of the time-dependent wave-function. Taking into account the relation between the angular momentum and the energy eigenvalues (12) we come to the following expression:  X Z ρ2 v Re χ∗n′ (ρ)χn (ρ)dρ (cn c∗n′ exp(i(ε′n − εn )t/~)) (24) F (t) = R ρ1 ′ n,n with functions χn (ρ) being solutions of (11). In the above expression we neglected back-scattering and put coefficients dn = 0. Also we took into account that the size of the band ρ2 − ρ1 = h ≪ R, where the neutron density is measured. Thus we arrive to the problem of the time evolution (instead of the angular variable ϕ evolution) of initially localized wave-packet in the band with radial dimension h = ρ2 − ρ1, which moves in the effective homogeneous field a = v 2 /R. As it was shown in [61] the integration over energies in (24) results in two terms. One term reflects the existence of S-matrix poles in the complex energy plain, which are situated close to the real axis and correspond to the complex energies of quasi-stationary states. So far this term describes the decay of the quasi-stationary states and the characteristic time scale is given by the corresponding widths τn = ~/Γn . The second term reflects the non-resonant contribution of pall other energies (which do not match with energies of the quasi-stationary states). The characteristic time τcl = 2hR/v 2 for such neutrons is equal to the classical time of passage of distance h with constant acceleration v 2 /R. This time of passage is much smaller than the time, which the neutron spends in the quasi-stationary states τcl ≪ τn . For the times τcl ≪ t ≤ τn the quasi-stationary states contribution is dominant. This enables us to neglect non- 9 1,0 0,6 F/F 0 0,8 0,4 0,2 0,0 1200 1300 1400 1500 1600 1700 1800 V (m/s) FIG. 7: The relative flux of neutrons, deflected by the curved mirror as a function of the neutron velocity. F0 is the flux calculated at v = 1200 m/s. The mirror curvature radius is R = 2.5 cm, the mirror length is Lmirr = 5 cm and the mirror Fermi potential U0 = 150 neV. resonant contribution in the expansion (24) and to take into account only the quasi-stationary states contribution. v X F (t) ≈ |Cn′ |2 exp(−Γn′ t) (25) R ′ n Here n′ indicates the quasi-stationary state number, |Cn′ |2 is the initial population of a given quasi-stationary state. The sharp increase in the quasi-stationary states lifetime (21) with decreasing the velocity below vn (23) can be used for experimental observation of such states. Indeed, when the neutron velocity decreases the contribution of new quasi-stationary states increases rapidly. This results to the step-like dependence of the deflected neutron flux as a function of v. There are no quasi-stationary states for v ≫ vc1 and therefor all neutrons traverse the mirror without being deflected. There are many quasi-stationary states in the opposite limit v ≪ vc1 and therefor we deal with the classical reflection from the curved mirror. In Fig.7 the flux of deflected neutrons is shown as a function of the neutron velocity. Under the assumptions made above the problem of deflection of cold neutrons (v ∼ 103 m/s) by the curved mirror is analogous to the problem of the passage of ultra-cold neutrons through the slit between a horizontal mirror and an absorber in the presence of the Earth’s gravitational field, studied in details in [45, 46, 47, 48, 49, 50, 51, 52]. In the cited experiments the spatial density of neutrons in the gravitational states was scanned by changing the position of the absorber above the mirror. In case of neutron motion along the curved mirror surface the initial velocity variation results in changing the spatial dimension of the effective well, which bounds the neutron near the surface, ensuring the ”scanning” of the quasi-stationary states. The experimental observation of the centrifugal states of neutrons could be however complicated by the diffuse scattering of neutrons from the the mirror surface roughness. Below we will estimate the additional broadening of the centrifugal states due to the scattering on rough surface. VI. EFFECT OF ROUGHNESS The effect of roughness consists in transferring the high neutron velocity parallel to the mirror surface into velocity component normal to the surface. As a result the quasi-stationary centrifugal states acquire additional ”ionization” width. The detailed theory of neutron rough-surface interaction can be found in [50, 51]. To obtain a simple estimation of such a width we will follow the method, developed in [49]. Namely, in the frame related to the neutron the mirror roughness appears as a time-dependent variation of the mirror position. We will start by treating a simple case of 10 harmonic dependence: U (z) = U0 Θ(z + br sin(ωr t)) Here br is the roughness amplitude and ωr is the roughness frequency, which can be related to the angular velocity of neutrons ω, the mirror curvature radius R and the characteristic length of roughness lr via ωr = ωR/lr . Then the equation describing the evolution of initially localized wave-packet gets the time-dependent right-hand side:   M v2 ~2 ∂ 2 dΨ(z, t) + U Θ(z − b sin(ω t)) − = − z Ψ(z, t) (26) i~ 0 r r dt 2M ∂z 2 R A solution of such an equation could be expanded in the set of eigen-functions of right-hand side Hamiltonian, taken at instant t:   ~2 ∂ 2 M v2 − + U Θ(z − b sin(ω t)) − z − ε (t) un (z, t) = 0 (27) 0 r r n 2M ∂z 2 R The corresponding expansion is: Ψ(z, t) = X Cn (t)un (z, t) exp(−i Z t ε(τ )/~dτ ) (28) 0 n Substitution of (28) into (26) yields in the coupled equation system for time-dependent amplitudes Cn (t): X d dCn (t) =− hun | |uk iCk (t) exp(−iωnk (t)) dt dt (29) k Here ωnk (t) = Rt 0 (εk (τ ) − εn (τ ))/~dτ . It follows directly from (27), that hun | hun | dU d dt |uk i |uk i = dt εn − εk In the following we will consider roughness small enough, so that U (z, t) ≈ U0 Θ(z)+U0br ωr cos(ωr t)δ(z). The coupling matrix elements are: hun | dU (z, t) |uk i = br U0 ωr cos(ωr t)un (0, t)uk (0, t) ≈ br U0 ωr cos(ωr t)un (0, t = 0)uk (0, t = 0) dt Using an analog of the Fermi ”golden rule” we get the following expression for ionization probability of centrifugal state n per unit of time: Pion = 2πb2r U02 |un (0, t = 0)uf (0, t = 0)|2 δ(εn + ~ωr − Ef )dkf ~ (30) Here index f labels the eigen-state of the final state of continuum spectrum with energy Ef and wave-number kf . We assume that Ef ≫ εn for the neutron√velocity v ∼ 103 m/s and for realistic roughness parameters. This enables us to use the free-wave expression uf = 1/ 2π exp(ikf z) for the final state wave-function. Taking into account the explicit form of un given by (16) and its semiclassical asymptotic we get simple estimation for Pion of the n-th quasi-stationary state: n Pion ≈ b2r U02 p ~2 Rl0 (z0 − λn ) 2M Ef (31) Taking into account the explicit expressions for the characteristic length (14) and energy (15) scales of the problem, we get for the case z0 ≫ λn : Pion ≈ b2r U0 v 2 M 2 p ~2 R 2M Ef (32) To get the ionization width of the centrifugal state one should integrate the obtained probability with the spectral function of roughness f (ω), which provides the square of roughness amplitude as a function frequency: Z ∞ b 2 U0 v 2 M 2 b2r f (ω)U0 v 2 M 2 p dω = r q (33) Γi = ~ ~2 R 2M (εn + ~ω) 0 ~R 2M Ef 11 2nm 1nm 0,6 0,5 F/F 0 0,4 0,3 0,2 0,1 0,0 1200 1300 1400 1500 1600 1700 1800 V (m/s) FIG. 8: The relative flux of neutrons, deflected by the curved sapphire mirror as a function of the neutron velocity. F0 is the flux calculated at v = 1200 m/s and zero roughness. The mirror curvature radius equals R = 2.5 cm, the mirror length is Lmirr = 5 cm, the mirror Fermi potential is U0 = 150 neV . Solid line corresponds to the roughness amplitude br = 1 nm, dashed line corresponds to the roughness amplitude br = 2 nm and the roughness length lr = 1 µm. Here b2r is the mean square roughness, and Ef is the mean ionization energy in the sense defined. An important feature of the obtained result is the square dependence of ionization width on the neutron velocity and the roughness amplitude (for the case of small amplitudes, studied above). It constraints severely the roughness amplitudes acceptable for observation the centrifugal states. In Fig.8 we plot the neutron flux, deflected by the curved sapphire mirror in the presence of roughness with the amplitudes br = 1 nm and br = 2 nm. Fig.9 demonstrates the neutron flux, deflected by the curved silicon mirror in the presence of roughness with the amplitudes br = 1 nm and br = 3 nm. Thus the effect of roughness is reduced if the Fermi potential is low. Indeed, according to (23) and (33) we expect the following scaling law: 17/8 Pion ∼ U0 So the roughness amplitude of a sapphire mirror surface should be smaller than 1 nm (and 4 nm for silicon mirror) to allow observation of the centrifugal states. VII. CONCLUSIONS We proposed a method for observation of the quasi-stationary states of neutrons, localized near a curved mirror surface. The effective bounding well is formed by a superposition of the centrifugal potential and the mirror Fermi potential. Reduction of the initial neutron velocity results in the spatial size increase of such a centrifugal trap which, in its turn, results in appearance of the quasi-stationary states in the spectrum of the system. This could be observed via step-like dependence of the deflected neutron flux. We show that several centrifugal states can be observed for instance with a sapphire mirror (Fermi potential U0 = 150 neV), with the curvature radius R = 2.5 cm, the length Lmirr ≈ 5 cm and the surface roughness amplitude < 1 nm. The critical velocities corresponding to the steps in the deflected flux are v1 = 1700 m/s and v2 = 1350 m/s. The characteristic spatial dimension of the mentioned centrifugal states is l0 ≈ 0.04 µm. In case of a silicon mirror with the same shape (Fermi potential U0 = 54 neV) the corresponding critical velocities values are v1 = 810 m/s and v2 = 650 m/s. Such neutron states could provide a promising tool for studding different types of neutron-matter interactions with the characteristic range of a few tens nm. 12 b=3nm b=1 nm 1,0 0,8 F/F 0 0,6 0,4 0,2 0,0 500 600 700 800 900 1000 V (m/s) FIG. 9: The relative flux of neutrons, deflected by the curved silicon mirror as a function of the neutron velocity. 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