Precision Measurement of the Position-space Wave Functions of Gravitationally Bound Ultracold Neutrons
aa r X i v : . [ h e p - e x ] A p r Precision Measurement of the Position-spaceWave Functions of Gravitationally BoundUltracold Neutrons
Y. Kamiya ∗ , G. Ichikawa † , and S. Komamiya Department of Physics, Graduate School of Science,and International Center for Elementary Particle Physics,The University of Tokyo, Tokyo 113-0033, Japan
Abstract
Gravity is the most familiar force at our natural length scale. However, itis still exotic from the view point of particle physics. The first experimentalstudy of quantum effects under gravity was performed using a cold neu-tron beam in 1975. Following this, an investigation of gravitationally boundquantum states using ultracold neutrons was started in 2002. This quantumbound system is now well understood, and one can use it as a tunable toolto probe gravity. In this paper, we review a recent measurement of position-space wave functions of such gravitationally bound states, and discuss issuesrelated to this analysis, such as neutron loss models in a thin neutron guide,the formulation of phase space quantum mechanics, and UCN position sensi-tive detectors. The quantum modulation of neutron bound states measuredin this experiment shows good agreement with the prediction from quantummechanics.
Introduction
Phenomena due to the gravitational field have been well understood at themacroscopic scales. However, there are only a few cases of experiments at ∗ email: [email protected] † currently at Department of Physics, Nagoya University {− ¯ h m d d z + V ( z ) } ψ n ( z ) = E n ψ n ( z ) (1)with the linear potential V ( z ) = ( mgz, z ≥ , ∞ , z ≤ , (2)2here ψ n and E n are the eigenfunctions and eigenenergies, ¯ h is the reducedPlanck constant, m is the neutron mass, and g is the standard gravitationalacceleration. Equation (1) can be rewritten in dimensionless form as( d d ξ n − ξ n ) ψ n ( ξ n ) = 0 , (3)where ξ n ≡ z/z − E n /E and the characteristic length and energy are givenby z = ( ¯ h m g ) / ∼ µ m , (4) E = ( mg ¯ h / ∼ . . (5)This is the Airy equation and solutions are described by Airy special func-tions, Ai( ξ n ) and Bi( ξ n ). Calculated probability distributions for the first fivestates are illustrated in Fig. 1. Note that ξ n = 0, where z n ≡ z E n /E , arethe classical turning points of a classical bouncing ball, which are denoted inthe cross points of each eigenenergy and the potential line of mgz in Fig. 1. E n e r g y ( p e V ) E E E E E (microns) z height V(z) = mgz z z z z z Figure 1: Probability distributions of gravitationally bound states of neu-trons for the first five states [10]. Horizontal lines indicate eigenenegies E n and vertical lines show the classical turning points z n for each state.This system is a suitable device with which to test the inverse squarelaw of standard Newtonian gravity and to search for new gravity-like short-range interactions around these scales. The first limit to a non-Newtonian3orce with a Yukawa-type interaction potential using this microscopic systemwas reported in [11]. The limits for a CP-violating Yukawa-type potentialwere shown in [12, 13]. Several experimental schemes have been proposedto improve the sensitivity to such hypothetical new physics. One idea toachieve a better resolution on the measurement of the characteristic length isto utilize a convex reflection mirror to magnify the neutron distributions [14,15]. In this review, we discuss an experiment using this scheme to preciselymeasure the UCN position-space wave functions, performed by the group ofS. Komamiya of the University of Tokyo [16]. Experiments exploiting otherideas for measuring the energy scale, i.e. the energy differences betweenquantum states, are proposed by observing resonance transitions induced bya magnetic field [17–20] and mechanical vibrations [21]. These projects arecalled GRANIT and qBounce, respectively. The first measurement of theresonance transition from the ground to the third state was reported by thegroup of H. Abele in [22] . For details of those projects, please see reviewpapers [23] and [24] in this special issue. Precision measurement and issues related to this anal-ysis
The precision measurement with a convex magnification mirror [16] was per-formed using a UCN source provided at ILL(Institut Laue-Langevin) [25].The velocity distribution was measured by a TOF method and is well char-acterized by a Gaussian distribution with mean of 9.4 m/s and standard de-viation of 2.8 m/s. Figure 2(a) shows a schematic drawing of the experiment.The entire system was mounted on an anti-vibration table and magneticallyshielded by a Permalloy sheet. The main components can be separated intothree parts: a vertically thin collimating guide, a magnification mirror(rod),and a pixelated position sensitive detector. The layout of these componentsis shown in Fig. 2(b).
Collimating guide
The guide settles gravitationally bound quantum states. To clearly distin-guish each quantum state, which have energy differences of order of ∆ E ∼ t ∼ ¯ h/ ∆ E ∼ Recently, a new limit for the CP-Violating Yukawa-type potential using the resonancemethod was reported in [26]. It also shows a limit for the chameleon field [27–29], a darkenergy candidates. agnetic shieldGranite tableAnti−vibration tableVacuum chamberAl window(100 µm) UCN
HeliumNeutron shutter Inclinometer20 ° ~ 2.5 mm200 mmCeilingBottom mirror Magnification rodPixelated detector Z (a)(b) Main components100 µm xzy
I II III IV
Collimating guide z = 100 µmz = 0 µm 3 mm7.9 mm (3 mm radius)
Figure 2: Schematic drawing of the precision measurement experiment usinga convex magnification mirror [16]. (a) is an overview of the system and (b)shows the geometry of the main components.5etup) to form each quantum states. The bottom mirror is made of polishedglass with a roughness of R a (arithmetic average) = 0 . µ m. The ceilingscatterer is a Gd-Ti-Zr alloy (54/35/11) deposited on glass, with a Fermi po-tential tuned to be nearly zero, and a roughness of R a = 0 . µ m [30], whichscatters out neutrons in the higher states. The guide selects lower stateswith appropriate populations to improve the contrast of the quantum spa-tial modulation . The neutron loss models for the scatterer or rough surfacehave been discussed in detail in [31–33], and they are still interesting issuesnot only for UCN guiding applications but also for storage experiments suchas neutron EDM measurements. In the experiment reviewed in this paper,empirical models of the loss rates are adopted [16]. The loss rate by the scat-terer, Γ n , is assumed to be proportional to the probability of finding neutronin the roughness region, and is given byΓ n = γ Z hh − δ | ˜ ψ n | dz , (6)where γ is a scaling constant, estimated from data to be 9 . +0 . − . × s − , h is the height of the guide (100 µ m), δ is the roughness of the scatterer(0 . µ m), and ˜ ψ n are deformed wave functions in the guide. Neutron lossesat the bottom mirror due to absorption, non-specular reflection, up scatteringand other processes are modeled empirically as B n = β g √ s m ˜ E n , (7)in which the loss rate is assumed to be proportional to the classical bouncingnumber per unit time, g/ v z,n,max , where β is a scaling constant (estimatedto be 0 . +0 . − . ), ˜ v z,n,max ≡ q E n /m is the maximum vertical velocity of aneutron in the n th state, and ˜ E n is the eigenenergy of the deformed wavefunctions. The transmissivity of the guide for each state can be written as˜ p n ∝ * exp[ − lv x (Γ n + B n )] + v x , (8)where < ... > v x indicate the average over the neutron horizontal velocities.By applying a diabatic transition from region II to III, the population distri-bution (a probability in the paper [16]) of neutrons for each state at the endof region III is estimated as in Fig. 3(a). The eigenfunctions modulate somewhat coherently, especially around the lower region,except for the ground state, as seen in Fig. 1. To improve the contrast of the quantummodulation, a method using a negative step of several tens of µ m, which transfers neutronsfrom the ground state to higher states, was considered in [9]. Results were reportedin [34–36].
00 403020100 (a)
Principal quantum number 0.060.04 P r obab ili t y p n n ( / ) V e l o c i t y (µm) 100 120 mm s (b) Height v z z Figure 3: (a) is an estimated population distribution (probability distribu-tion of the guide transmission) at the end of region III, and (b) shows thecorresponding Wigner phase space distribution [16].
Magnification mirror
A Ni coated cylindrical rod is used as a magnifying convex mirror. Its radiusis 3 mm and it is placed to have a grazing angle of 20 deg. for horizontallymoving neutrons at the bottom floor level ( z = 0). Figure 4 shows themagnification power as a function of the height [10]. It gives about 20 timesmagnification around the lower region of z ∼ µ m. Before depositing the Nicoating, the cylinder was polished at the Research Center for Ultra-PrecisionScience and Technology, Osaka University. The roughness of the rod afterthe depositing was measured to be R a = 1 . W ( z, p z ) ≡ π ¯ h Z ∞−∞ δη e − i ¯ h p z η < z + 12 η | ˆ ρ | z − η > , (9)where p z is the momentum and ˆ ρ is a density operator. The Wigner distri-bution is known as a phase space formulation of quantum mechanics, and iswidely used for quantum optics, for example in the study of decoherence [39].As an application to massive particles, one can find a paper which shows aphase-space tomography of the Wigner distribution for a coherent atomicbeam in a double-slit experiment [40]. Figure 3(b) shows the Wigner distri-bution constructed from the estimated populations in our experiment [16].The time evolution of the Wigner distribution is calculated by the evolution7 ( m) µ Figure 4: Magnification power M of the cylindrical rod as a function of theheight z [10]. It is about 20 times magnification around the lower region.of the density operator described by the Liouville - von Neumann equation ∂ ˆ ρ∂t = − i ¯ h [ ˆ H, ˆ ρ ] , (10)where ˆ H = ˆ p z m + V (ˆ z ) . (11)By evaluating the kinetic part as T = − p z m ∂∂z W ( z, p z ) , (12)and the potential part as ∞ X l =0 U l = ∞ X l =0 ( − l (¯ h/ l (2 l + 1)! d l +1 V ( z ) dz l +1 ∂ l +1 ∂p l +1 z W ( z, p z ) , (13)one can obtain the Quantum Liouville Equation for the Wigner distribution[41] ∂∂t + p z m ∂∂z − dV ( z ) dz ∂∂p z ! W ( z, p z ) = ∞ X l =1 U l . (14)In the case of V ( z ) = mgz , the right hand side of the equation vanishesand it becomes the classical Liouville equation. Therefore, in region IV, onecan treat the evolution of each phase point of the Wigner distribution as a8lassical path under gravity. Figure 5 shows the measured data and the besttheoretical estimation using this model. The corresponding p-value is 0.715,and the experimental data support the quantum features described by thephase space formulation using the Wigner distribution [16]. m ) µ C oun t s ( / n 2n 5n 10 n 15n 3n 50n 4 n=1 Position at the detector surface (mm)
Figure 5: Black crosses show measured neutron position distribution [16].Modulated distribution was clearly measured. The result shows good agree-ment with the quantum expectation calculated using phase space formaliza-tion by Wigner distribution (p-value is 0.715). The best estimated line isshown by a red solid curve and the other lines indicate contributions of eachstate to the distribution. For definitions of fitting parameters, best fit values,systematic uncertainties, and the other details, see [16].
Position sensitive detector for UCNs
A back-thinned CCD (HAMAMATSU S7030-1008) with thin Ti- B-Ti layersis used for the position sensitive detector in the experiment [16]. Its pixelsize is 24 µ m × µ m and the thickness of the active volume is about 20 µ m.The B layer converts neutrons into charged particles by the nuclear reaction B(n, α ) Li. The secondary particles are emitted in a nearly back-to-backconfiguration. One of them deposits its kinetic energy in the active areaand creates a charge cluster, which typically spreads into nine pixels. Theweighted center of the charge cluster is a good estimation of the incidentneutron position. The thickness of the layers are 20 nm and 200 nm for Ti9nd B, respectively, and they are formed by evaporating directly on theCCD surface. The spatial resolution is measured to be 3 . ± . µ m byevaluating the line spread function (LSF) (see Fig. 6) using very cold neutronbeams at ILL [10]. For the detail of the evaluation scheme, see [15, 42]. Aneutron converter of Li using Li(n, α ) H reaction was also investigated. Itis concluded that the use of B gives better spatial resolution [42].
Position (mm)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 m ) µ C oun t( / Gd Figure 6: Fitting results of a Gd shadow pattern [10]. Spatial resolution ismeasured to be 3 . ± . µ m by evaluating these line spread functions. Fordetails of evaluation method, see [15, 42].Other detectors using a silicon pixel device, Timepix [43], with LiF and B converter were studied in [44]. Its pixel pitch is 55 µ m × µ m and thethickness of its silicon layer is about 300 µ m. The spatial resolution for the LiF converter was evaluated by LSF to be 2 . µ m, corresponding to about5 . µ m in FWHM of the point spread function (PSF). The performance ofthe B converter is estimated by Monte-Carlo simulations to be better than3 µ m in FWHM of the PSF.Another detector concept of Uranium coated plastic nuclear tracker (CR39)[9] was used in several experiments [8, 34–36, 45]. Usually two fission frag-ments are emitted from a thin U coating, and one of the daughter nucleimakes a track of defects in the CR39. By chemical etching, the diameterof track points is increased up to 1 µ m, allowing us to scan these vertexesusing an optical microscope. Position resolution is around 1 µ m [9]. Bycarefully analyzing the vertex shape, the spatial resolution can be improvedto 0 . µ m [45]. 10 ummary The quantum system of a gravitationally bound neutron is one of the mostsuitable tools to investigate gravity or gravity-like hypothetical interactionsaround the scale of 10 µ m in length or 1 peV in energy. After establishingthis research field by the pioneering work in observation of the quantum state[6], experimental schemes for precision measurements of these characteristicscales have developed rapidly, and nowadays, one can establish limits forparameter spaces of new physics experimentally [11–13,26]. Furthermore onecan start discussing a phase space formulation of quantum physics for thegravitationally bound quantum state [16]. An experiment for a possible testof phase space formalization using spatial interference is under preparation. Acknowledgments
The authors would like to thank Valery V. Nesvizhevsky (Institut Laue-Langevin), Hartmut Abele (Vienna University of Technology), William Snow(Indiana University), Peter Geltenbolt (Institut Laue-Langevin), and all par-ticipants in the GRANIT-2014 Workshop for interesting discussions and help-ful suggestions. This material is based upon work supported by JSPS KAK-ENHI Grants No. 20340050 and No. 24340045, and Grant-in-Aid for JSPSFellows No. 22.1661.
References [1] R. Colella, A. W. Overhauser, and S. A Werner, Phys. Rev. Lett. ,1472 (1975)[2] J. J. Sakurai, Phys. Rev. D , 2993 (1980)[3] J. L. Staudenmann et al. , Phys. Rev. A , 1419 (1980)[4] S. A. Werner et al. , Physica B , 22 (1988)[5] K. C. Littrell, B. E. Allman, and S. A. Werner, Phys. Rev. A , 1767(1997)[6] V. V. Nesvizhevsky et al. , Nature , 297 (2002)[7] V. V. Nesvizhevsky et al. , Phys. Rev. D , 102002 (2003)[8] V. V. Nesvizhevsky et al. , Eur. Phys. J. C , 479 (2005)119] V. V. Nesvizhevsky et al. , Nucl. Instrum. Meth. A , 754 (2000)[10] G. Ichikawa, PhD. Dissertation, The University of Tokyo (2013)[11] V. V. Nesvizhevsky and K. V. Protasov, Class. Quantum Grav. , 4557(2004)[12] S. Baeßler it et al., Phys. Rev. D , 075006 (2007)[13] S. Baeßler it et al., Nucl. Instrum. Meth. A , 149 (2009)[14] V. V. Nesvizhevsky, Preprint ILL 96NE14T (1994)[15] T. Sanuki et al. , Nucl. Instrum. Meth. A , 657 (2009)[16] G. Ichikawa et al. , Phys. Rev. Lett. , 071101 (2014)[17] P. Schmidt-Wellenburg et al. , Nucl. Instrum. Meth. A , 267 (2009)[18] M. Kreuz et al. , Nucl. Instrum. Meth. A , 326 (2009)[19] S. Baeßler et al. (GRANIT Collaboration) , C. R. Physique 12, 707 (2011)[20] V. V. Nesvizhevsky (GRANIT Collaboration) , Mod. Phys. Lett. A 27,1230006 (2012)[21] H. Abele et al. , Phys. Rev. D , 065019 (2010)[22] T. Jenke et al. , Nature Phys. , 468 (2011)[23] D. Roulier et al. , Adv. High En. Phys. this issue (2014)[24] G. Cronenberg et al. , Adv. High En. Phys. this issue (2014)[25] A. Steyeri et al. , Phys. Lett. A , 347 (1986)[26] T. Jenke et al. , Phys. Rev. Lett. , 151105 (2014)[27] J. Khoury and A. Weltman, Phys. Rev. Lett. , 171104 (2004)[28] D. F. Mota and D. J. A. Shaw, Phys. Rev. Lett. , 151102 (2006)[29] J. Khoury, Class. Quantum Grav. , 214004 (2013)[30] Y. Kamiya et al. , KURRI Progress Report CO1-1 (2008)[31] A. E. Meyerovich and V. V. Nesvizhevsky, Phys. Rev. A , 063616(2006) 1232] A. Yu. Voronin et al. , Phys. Rev. D 73, 044029 (2006)[33] A. Westphal et al. , Eur. Phys. J. C , 367 (2007)[34] V. V. Nesvizhevsky et al. , ILL Annual Report (2004)[35] H. Abele et al. , Nucl. Phys. A , 593c (2009)[36] T. Jenke et al. , Nucl. Instrum. Meth. A , 318 (2009)[37] E. Wigner, Phys. Rev , 749 (1932)[38] M. Hillery et al. , Phys. Rep. , 121 (1984)[39] L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cam-bridge University Press 1995)[40] Ch. Kurtsiefer, T Pfau, and J. Mlynek, Nature , 150 (1997)[41] W. P. Schleich,
Quantum Optics in Phase Space (Wiley-VCH 2001)[42] S. Kawasaki et al. , Nucl. Instrum. Meth. A , 42 (2010)[43] X. Llopart et al. , Nucl. Instrum. Meth. A , 485 (2007)[44] J. Jakubek et al. , Nucl. Instrum. Meth. A , 651 (2009)[45] T. Jenke et al. , Nucl. Instrum. Meth. A732