Observation of gravitationally induced vertical striation of polarized ultracold neutrons by spinecho spectroscopy
Article (Published Version)
Afach, S, Ayres, N J, Ban, G, Bison, G, Bodek, K, Chowdhuri, Z, Daum, M, Fertl, M, Franke, B, Griffith, W C, Grujić, Z D, Harris, P G, Heil, W, Hélaine, V, Kasprzak, M et al. (2015) Observation of gravitationally induced vertical striation of polarized ultracold neutrons by spin-echo spectroscopy. Physical Review Letters, 115 (16). a162502 1-6. ISSN 0031-9007
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http://sro.sussex.ac.uk week ending PRL 115, 162502 (2015) PHYSICAL REVIEW LETTERS 16 OCTOBER 2015
Observation of Gravitationally Induced Vertical Striation of Polarized Ultracold Neutrons by Spin-Echo Spectroscopy
† ‡ S. Afach,1,2,3 N. J. Ayres,4 G. Ban,5 G. Bison,1 K. Bodek,6 Z. Chowdhuri,1 M. Daum,1 M. Fertl,1,2, B. Franke,1,2, W. C. Griffith,4 Z. D. Grujić,7 P. G. Harris,4 W. Heil,8 V. Hélaine,1,5,§ M. Kasprzak,7 Y. Kermaidic,9 K. Kirch,1,2 ∥ P. Knowles,7, H.-C. Koch,7,8 S. Komposch,1,2 A. Kozela,10 J. Krempel,2 B. Lauss,1 T. Lefort,5 Y. Lemière,5 A. Mtchedlishvili,1 M. Musgrave,4 O. Naviliat-Cuncic,5,¶ J. M. Pendlebury,4,* F. M. Piegsa,2 G. Pignol,9 C. Plonka-Spehr,11 P. N. Prashanth,12 G. Quéméner,5 M. Rawlik,2 D. Rebreyend,9 D. Ries,1,2 S. Roccia,13 D. Rozpedzik,6 P. Schmidt-Wellenburg,1,** N. Severijns,12 J. A. Thorne,4 A. Weis,7 E. Wursten,12 G. Wyszynski,6 J. Zejma,6 J. Zenner,1,2,11 and G. Zsigmond1 1Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland 2ETH Zürich, Institute for Particle Physics, CH-8093 Zürich, Switzerland 3Hans Berger Department of Neurology, Jena University Hospital, D-07747 Jena, Germany 4Department of Physics and Astronomy, University of Sussex, Falmer, Brighton BN1 9QH, United Kingdom 5LPC Caen, ENSICAEN, Université de Caen, CNRS/IN2P3, 14050 Caen Cedex, France 6Marian Smoluchowski Institute of Physics, Jagiellonian University, 30-059 Cracow, Poland 7Physics Department, University of Fribourg, CH-1700 Fribourg, Switzerland 8Institut für Physik, Johannes-Gutenberg-Universität, D-55128 Mainz, Germany 9LPSC, Université Grenoble Alpes, CNRS/IN2P3, 38026 Grenoble Cedex, France 10Henryk Niedwodniczanski Institute for Nuclear Physics, 31-342 Cracow, Poland 11Institut für Kernchemie, Johannes-Gutenberg-Universität, D-55128 Mainz, Germany 12Instituut voor Kern- en Stralingsfysica, University of Leuven, B-3001 Leuven, Belgium 13CSNSM, Université Paris Sud, CNRS/IN2P3, 91405 Orsay Campus, France (Received 1 June 2015; published 16 October 2015) We describe a spin-echo method for ultracold neutrons (UCNs) confined in a precession chamber and exposed to a jB0j¼1 μT magnetic field. We have demonstrated that the analysis of UCN spin-echo resonance signals in combination with knowledge of the ambient magnetic field provides an excellent method by which to reconstruct the energy spectrum of a confined ensemble of neutrons. The method takes advantage of the relative dephasing of spins arising from a gravitationally induced striation of stored UCNs of different energies, and also permits an improved determination of the vertical magnetic-field gradient with an exceptional accuracy of 1.1 pT=cm. This novel combination of a well-known nuclear resonance method and gravitationally induced vertical striation is unique in the realm of nuclear and particle physics and should prove to be invaluable for the assessment of systematic effects in precision experiments such as searches for an electric dipole moment of the neutron or the measurement of the neutron lifetime.
DOI: 10.1103/PhysRevLett.115.162502 PACS numbers: 29.30.Hs, 07.55.Ge, 76.60.Lz
Spin-echo resonances, first observed by Hahn in 1950 [1], neutrons (UCNs) [5–11] where coherence times T2 of several have proven to be a very powerful tool in nuclear magnetic hundred seconds have been achieved. The defining charac- resonance (NMR) experiments for identifying different teristic of UCNs is that they are reflected from appropriate sources of depolarization in spin-polarized samples. In material surfaces at all angles of incidence, even at room particular, the π=2 − π pulse sequence, proposed by Carr temperature, conserving kinetic energy as they do so. They and Purcell [2], is widely applied in distinguishing the can, therefore, be confined for observation times that are intrinsic spin-lattice or spin-spin coherence time T2 from commensurate with neutron β decay: inelastic scattering the global coherence time T2 dominated by dephasing of with phonons from the storage-vessel walls contributes precessing spins due to spatially varying magnetic fields. Γ ≈ 10−3 s−1 to their loss rate. This peculiarity also makes Another established application of the spin-echo technique is them ideal for neutron lifetime measurements. A measure- in neutron scattering [3] where it is used to resolve correlation ment of the evolution of the energy spectrum in these UCN times below 1 μs. Essentially, the spin-echo technique can be storage experiments can give insight into energy dependent applied in solid and soft matter samples where the intrinsic systematic effects and could help to understand the disagree- component is long compared to dephasing, which can even ment between beam and storage experiments [12]. ≪ 1 be the case for T2 s. Two similar components of Ultracold neutrons have kinetic energies Ekin of the same depolarization [4] exist in experiments using ultracold order of magnitude as their gravitational potential energy
0031-9007=15=115(16)=162502(6) 162502-1 © 2015 American Physical Society week ending PRL 115, 162502 (2015) PHYSICAL REVIEW LETTERS 16 OCTOBER 2015 ∂ Epot above the lower confining surface. For any given ω ϵ γ − Bz 2 − ϵ ð Þ¼ n hB0i ðH= hhð ÞiÞ ; ð2Þ UCN, E ¼ Ekin þ Epot is constant during storage, although ∂z for the ensemble, the average energy decreases over time because faster neutrons have a higher loss rate. These where hB0i is the volume-averaged magnetic field mag- energy-dependent losses result in a time dependence of nitude in the precession chamber for B0 up ðþÞ or down T − γ −2π 29 1646943 69 both depolarization terms, making 2 a function of the free- ð Þ, and n ¼ × . ð Þ MHz=T [25] is the precession time. For neutrons, mg ≈ 1.03 neV=cm, so, e.g., gyromagnetic ratio of the neutron. This induces an energy- UCNs with kinetic energy of 200 neV can rise about dependent relative phase ΦðϵÞ¼ωðϵÞT, where T is the free 200 cm in the gravitational field. Therefore, it is convenient spin-precession (FSP) time. It leads to a relative dephasing (and common) to refer to the energy ϵ of UCNs in terms of which, when averaged over the energy spectrum, can be the maximum height (in cm) attainable within Earth’s interpreted as a gravitationally enhanced depolarization. gravitational field, and we do so here during our analysis, This relative dephasing can be studied by applying a π=2 − although we revert to E (in neV) for our final results. π − π=2 spin-echo sequence, varying the time t1 at which Two channels of field-induced depolarization of UCNs the π pulse of duration tπ is applied while keeping constant exist: the known intrinsic depolarization due to inhomo- the total duration T þ tπ between the end of the first and geneous magnetic fields [13–18], and an energy-dependent start of the second π=2 pulse. This second π=2 pulse, at the relative dephasing for different energies in a vertical end of the sequence, is required to transfer information magnetic-field gradient. The latter shows an interplay about the spin-precession phase onto the measurable between gravitationally defined spatial distributions of longitudinal spin component. For convenience, we analyze the neutrons and a vertical magnetic-field gradient the system in a rotating frame precessing with ω0 ¼ γ hB0i ∂ ∂ n Bz= z and has only recently been studied theoretically and B0 > 0 inside the storage volume, and we only [4,19,20]. It is the subject of this investigation which uses a consider a linear gradient ∂Bz=∂z along the primary B0 new form of spin-echo spectroscopy that also circumvents field axis z. Therefore, in the rotating frame, the spins of effects provoked by the time-dependent softening of the UCNs with energy E precess with average frequency (to UCN spectrum. Prior to this, the spin-echo resonance first order in ∂Bz=∂z) method was used in a study of Berry’s phase with UCNs to cancel any dynamic phase shift [21] with a ∂B ω ðϵÞ¼−γ z ðH=2 − hhðϵÞiÞ: ð3Þ coherence time of approximately 850 ms. r n ∂z We have measured UCN spin-echo signals using an apparatus dedicated to the neutron electric dipole moment Figure 1 illustrates the evolution of the spins in the rotating (nEDM) experiment [22] with a vertical magnetic field of frame. The resonance condition under which the spins of all ≈ 0 0 −1 μ B0 ð ; ; Þ T and depolarization times of more than different energies refocus is established when T ¼ 2t1.At 450 s. These measurements, together with first-order this time we measure the highest polarization, which is knowledge of the magnetic-field gradients, were used to essentially the polarization hαðT;ϵÞiϵ without the gravita- determine the energy spectrum of the UCN after storage as tionally induced dephasing. If we change the start time t1 of well as a precise value for a common offset to the vertical the π pulse to values shorter than T=2, the resonance magnetic-field gradients. The technique has the potential to condition will have been met before we apply the second analyze and correct for the so-called geometric-phase effect π=2 pulse. As the spins of the different energy classes then [15,23], which is currently the most important systematic proceed to fan out again, the final polarization decreases. effect in experiments searching for an nEDM. Similarly, if the π pulse is applied at times larger than T=2 The equilibrium density distribution of UCNs as a the refocusing is not yet complete by the time the second function of height h above the lower confinement surface π=2 pulse is applied. ρ in ap bottleffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with diffusely scattering walls is ðhÞ¼ For a particular UCN energy ϵ and fixed T, the observed ρð0Þ 1 − h=ϵ [24]. This leads to a center-of-mass offset polarization after the second π=2 pulse can be written as of H=2 − hhðϵÞi relative to the center plane of a cylindrical ϵ α ϵ ω ϵ − 2 precession chamber of radius r and height H, where Pð ;t1Þ¼ ðT; Þ cos ½ rð ÞðT t1Þ : ð4Þ ( 2 ϵ ϵ 5 for 162502-2 week ending PRL 115, 162502 (2015) PHYSICAL REVIEW LETTERS 16 OCTOBER 2015 (a) (b) (c)(d) (e) (f) (g) We have used the MCUCN package [26] with a detailed model of the nEDM experiment at Paul Scherrer Institute rf (PSI) [22] (H ¼ 12 cm, r ¼ 23.5 cm) and simulated the measurement with a π=2 − π − π=2 spin-flip sequence (tip–flip–tip) for a fully polarized initial population of t /2 t1 t t t t /2 1 f UCNs. This simulation was carried out for two different z z z z UCN energy spectra pðϵÞ, parametrized as in Eq. (7) below, y y y y each with three different constant vertical gradients of x x x x ∂ ∂ 100 400 z z z Bz= z ¼ , 200, and pT=cm for a free precession time of T ¼ 92 s. The resulting simulated spin-echo y y x x x resonances are shown in the insets of Fig. 2 which can be described by Eq. (5). In particular, we used FIG. 1 (color online). Illustration of a UCN spin-echo meas- αðT;ϵÞ¼α0 exp ½−TΓ2ðϵÞ , with a transverse relaxation urement of duration T ¼ 2t1 þ tf in the frame rotating at rate as derived in [20] of frequency ω ¼ ω0 for a negative gradient ∂B =∂z: (a) a rf z polarized ensemble of UCNs is loaded into the cell, (a)–(b) an γ2 8 3 ∂ 2 ∂ 2 H3 ϵ ∂ 2 π 2 n r Bz Bz ð Þ Bz initial = pulse tips all spins (black arrow) into the equatorial Γ2ðϵÞ¼a þ þ ; v ϵ 9π ∂x ∂y 16 ∂z plane, (b)–(c) the UCN spins precess with ωrðEÞ for t1, and fan ð Þ out. Low-energy UCNs (blue) see a larger field for a negative ð6Þ gradient ∂Bz=∂z than higher-energy UCNs (magenta), whereas spins that precess at ω0 (i.e., with no center-of-mass offset) are stationary and are oriented along y.Aπ pulse (c)–(d) then flips where vðϵÞ, and HðϵÞ¼ϵ ∀ ϵ ≤ H and HðϵÞ¼H ∀ ϵ >H the spins around the x axis, after which (d)–(e) the UCNs are the energy-dependent mean velocity, and the effective continue to precess in the same direction, eventually refocusing height of the UCN as introduced in [24] for a cylindrical at 2t1 (black arrow). As they continue to precess beyond this time cell; a is a free fit parameter. We found it useful to model they fan out again (e)–(f) until finally (f)–(g) a second π=2 pulse the UCN spectrum for a fixed precession time T as is applied. All pulses used for spin manipulation have the same field strength B applied along the same axis (x). A single x ϵ1=2 oscillator is used, with the output being gated on only during the A · pðϵÞ¼ 2 blue time windows; hence, all of the pulses are phase coherent 1 ðE0−ϵÞ=b þ e with each other. All energy-dependent effects during the pulses 1 1 2π cancel, since a full rotation about is made and only dephasing × 2 − 2 ; ð7Þ ϵ− E0 E1 =c E2−E0−E1 =c during FSP has to be taken into account. 1 þ e½ ð þ Þ 1 þ eð Þ 0.02 0.06 Case A Case B 0.018 0.8 0.8 0.05 0.016 0.4 ) 0.4 1 t ( ) 0.014 1 P t 0 0.04 ( P 0 0.012 −0.4 0 23 46 69 92 0.01 0.03 −0.4 t1(s) 0 23 46 69 92 Intensity Intensity 0.008 t1(s) 0.02 0.006 0.004 0.01 2 χ /dof = 169/143 χ2 0.002 Center of mass offset: 2.3+0.3 mm /dof = 159/143 −0.2 Center of mass offset: 6.3(3) mm 0 0 20 40 60 80 100 120 140 160 180 200 220 10 20 30 40 50 60 70 80 90 100 Energy (neV) Energy (neV) (a) (b) FIG. 2 (color online). Simulated spin-echo signals and extracted spectra. The inserts show the simulated spin-echo signals using MCUCN [26]. On each plot, three resonances are shown, using the gray input spectra with three different vertical field gradients: blue (100 pT=cm), green (200 pT=cm), and red (400 pT=cm). The points with their uncertainties come from the simulation. The red lines on the main plots are the best simultaneous fits to the simulated spectra. The colors (from bright yellow to dark green) represent the χ2 value for each line; dark green corresponds to a 68.3% C.L while bright yellow lines have χ2 values close to the minimum. The original input spectra are represented by the gray areas, which give center-of-mass offsets of 1.93 mm [case (A)] and 6.46 mm [case (B)]. 162502-3 week ending PRL 115, 162502 (2015) PHYSICAL REVIEW LETTERS 16 OCTOBER 2015 where E0, b, and E0 þ E1, c are the points of inflection and simultaneous spin-analyzing system [30]. The magnetic- slopes of the rising and falling energy edges, respectively, field measurement with the CsM array permitted us to while E0 Rþ E1 þ E2 is the high-energy cutoff. A is chosen calculate the magnetic-field gradients using a polynomial E2 such that 0 pðT;ϵÞ dϵ ¼ 1. This energy spectrum is based field decomposition up to second order, as described in on a very general distribution nðϵÞdϵ ∝ ϵ1=2dϵ from the Ref. [31]. Data were taken for a constant FSP duration of 216 low-energy tail of a Maxwell-Boltzmann distribution, T ¼ s, with three different trim-coil current settings for 0 allowing for low- and high-energy cutoffs. A simultaneous B0 < (down), thereby applying vertical gradients of ∂ ∂ −18 38 nonlinear least-squares fit (algorithm: “trust-region- Bz= z ¼ , 10, and pT=cm. The uncertainties of reflective”) to the signals of all three gradients yielded the gradient measurements are dominated by systematics a, b, c, E0, E1, and E2 for each of the two spectra. The due to individual sensor offsets, and amount to 12–14 upper and lower confidence intervals were estimated by pT=cm (see [31,32] for a determination of these carrying out random scans of the parameter space around gradient errors). However, under our conditions, such χ2 < χ2 ∼5 105 individual, but constant, offsets lead to a constant offset the best value with limit ( × samples; yellow to ∂ ∂ dark green curves in Fig. 2) Gz of the true Bz= z that can be determined to within 1–2 pT=cm using the treatment described below. The measured initial polarization α0 ¼ 0.861ð2Þ, 2 2 p χ ¼ χ 1 þ Fαðn; n − pÞ ; ð8Þ 0.865(2), 0.861(2) was prepared by passing UCNs from limit min n − p the UCN source at the PSI [33] through a 5 T solenoid magnet. After a filling time of 24 s, the UCN shutter in the where Fα is the inverse cumulative function of the α 68 3% ground electrode was closed. For each setting, one fixed statistical F distribution for an ¼ . confidence level ω 30 2075 30 2031 6 − 143 radio frequency rf ¼ . ; . , and 30.2013 Hz with p ¼ free parameters and ðn pÞ¼ degrees of 199 freedom [27]. For the numerical integration of Eq. (5) we was used. The Hg comagnetometer measured the mean ω γ used energy bins of equal height: ΔhðϵÞ¼0.02 mm, magnetic field B0 via Hg ¼ Hg · hB0i [34]. π 2 − π − π 2 which is sufficiently small that any systematic error from A full phase coherent = = spin-flip sequence the finite energy bins would be less than the statistical was applied to the neutrons in the cell before the UCN resolution of each data point. Figure 2 compares spectra shutter was then opened again, and the UCNs were detected extracted from the fit to the input spectra of the simulation. in a pair of spin-state-specific detector channels. A meas- We note that the results become less reliable at higher urement consisted of several cycles with t1 increasing in energies; this is to be expected, since once the neutrons steps of 9 s. Figure 3 shows the three measured spin-echo populate the bottle more or less uniformly, it becomes resonances. As for the simulated cases, Eq. (5) was fitted to increasingly difficult to discriminate between them. the data, using the same fitting procedure and, in addition, ω We measured three spin-echo profiles (see Fig. 3) using randomly varying the initial start values. The frequency r Δω ω − the apparatus described in Ref. [28], improved with an from Eq. (3) was corrected for the difference ¼ rf γ γ ω array of 16 cesium-vapor magnetometers (CsMs) [29] and a j nj= Hg Hg and was also extended by a term for the second-order gradient in z, which was available from the field measurement 0.6 ∂B ω ðϵÞ¼γ − z þ G ðH=2 − hhðϵÞiÞ 0.4 r n ∂z z 2 ) 0.2 ∂ 1 Bz 2 − 2 − ϵ 2 − Δω ( t þ 2 fhh i h½H= hð Þ ig ; ð9Þ P 0 ∂z R −0.2 2 þ6 2 2 where hh i¼ −6 z =H dz ¼ 12 cm is the spectrum-in- −0.4 18 36 54 72 90 108 126 144 162 180 198 216 dependent quadratic expectation value. The fit parameter 0.04 Gz was included to accommodate the above-mentioned r 0 −0.04 common gradient offset. As the observed polarization at 2 0 18 36 54 72 90 108 126 144 162 180 198 216 t1 ¼ T= was the same for all three measurements, we used t (s) Γ ϵ 1 an averaged common 2ð Þ for the fit. Figure 4 shows the extracted energy spectrum. The best fit parameters were FIG. 3 (color online). Measured UCN spin-echo signal for three 5 8 1 1 10 0 4 1þ0.9 1=2 Gz ¼ . ð . Þ pT=cm, a ¼ . ð Þ, b ¼ −0.0 neV , c ¼ nominal magnetic field gradients. Red squares: ∂B =∂z ¼ þ1.2 1=2 þ4.3 þ4.5 z 2.7 neV , E0 ¼ 7.7 neV, and E1 ¼ 28:7 neV, −18 pT=cm; black diamonds: 10 pT=cm; blue circles: −0.2 −1.4 −4.5 200 1 1=2 38 pT=cm. The lines represent the best fit to the data of with E2 fixed at neV. Lower bounds for b ¼ neV 2 5 1=2 Eq. (5), using Eqs. (7) and (9). The residuals are r ¼ Pðt1Þ − and c ¼ . neV were chosen to give smooth physical fit χ2 Pðt1Þexp. edges while remained essentially constant for even lower 162502-4 week ending PRL 115, 162502 (2015) PHYSICAL REVIEW LETTERS 16 OCTOBER 2015 0.035 Best fit to extract the energy spectrum of stored UCNs, but also to CMoff C.L.68.3% 0.03 max determine a common gradient offset with a resolution of CMoff C.L.68.3% min 1.1 pT=cm, and, thus, to determine the magnetic-field Fraction 0.015 H ≈ 40 cm. Using an array of vector magnetometers, as χ2/dof = 85/66 Center of mass offset: 5.4+0.6 mm −0.4 described in, e.g., Ref. 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