Collective oscillations of a stored deuteron beam close to the quantum limit
J. Slim, N.N. Nikolaev, F. Rathmann, A. Wirzba, A. Nass, V. Hejny, J. Pretz, H. Soltner, F. Abusaif, A. Aggarwal, A. Aksentev, A. Andres, L. Barion, G. Ciullo, S. Dymov, R. Gebel, M. Gaisser, K. Grigoryev, D. Grzonka, O. Javakhishvili, A. Kacharava, V. Kamerdzhiev, S. Karanth, I. Keshelashvili, P. Lenisa, N. Lomidze, B. Lorentz, A. Magiera, D. Mchedlishvili, F. Müller, A. Pesce, V. Poncza, D. Prasuhn, A. Saleev, V. Shmakova, H.Ströher, M. Tabidze, G. Tagliente, Y. Valdau, T. Wagner, C. Weidemann, A. Wro?ska, M. Żurek
CCollective oscillations of a stored deuteronbeam close to the quantum limit
J. Slim , N.N. Nikolaev , F. Rathmann , A. Wirzba , A. Nass , V. Hejny , J. Pretz , H. Soltner ,F. Abusaif , A. Aggarwal , A. Aksentev , A. Andres , L. Barion , G. Ciullo , S. Dymov , R. Gebel ,M. Gaisser , K. Grigoryev , D. Grzonka , O. Javakhishvili , A. Kacharava , V. Kamerdzhiev ,S. Karanth , I. Keshelashvili , P. Lenisa , N. Lomidze , B. Lorentz , A. Magiera , D. Mchedlishvili ,F. Müller , A. Pesce , V. Poncza , D. Prasuhn , A. Saleev , V. Shmakova , H. Ströher , M. Tabidze ,G. Tagliente , Y. Valdau , T. Wagner , C. Weidemann , A. Wro´nska , and M. ˙Zurek III. Physikalisches Institut B, RWTH Aachen University, 52056 Aachen, Germany L.D. Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia Institut für Kernphysik, Forschungszentrum Jülich, 52425 Jülich, Germany Institute for Advanced Simulation, Forschungszentrum Jülich, 52425 Jülich, Germany Zentralinstitut für Engineering, Elektronik und Analytik, Forschungszentrum Jülich, 52425 Jülich, Germany Marian Smoluchowski Institute of Physics, Jagiellonian University, 30348 Cracow, Poland Institute for Nuclear Research, Russian Academy of Sciences, 117312 Moscow, Russia University of Ferrara and Istituto Nazionale di Fisica Nucleare, 44100 Ferrara, Italy Laboratory of Nuclear Problems, Joint Institute for Nuclear Research, 141980 Dubna, Russia Department of Electrical and Computer Engineering, Agricultural University of Georgia, 0159 Tbilisi, Georgia High Energy Physics Institute, Tbilisi State University, 0186 Tbilisi, Georgia GSI Helmholtzzentrum für Schwerionenforschung, 64291 Darmstadt, Germany Istituto Nazionale di Fisica Nucleare sez. Bari, 70125 Bari, Italy Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
January 20, 2021
We investigated coherent betatron oscillations of a deuteron beam in the stor-age ring COSY, excited by a detuned radio-frequency Wien filter. These beamoscillations were detected by conventional beam position monitors, read outwith lock-in amplifiers. The response of the stored beam to the detuned Wienfilter was modelled using the ring lattice and time-dependent 3D field maps ofthe radio-frequency Wien filter. The influence of uncertain system parametersrelated to manufacturing tolerances and electronics was investigated using the a r X i v : . [ nu c l - e x ] J a n olynomial chaos expansion. With the currently available apparatus, we showthat oscillation amplitudes down to µ m can be detected. Future measure-ments of the electric dipole moment of protons will, however, require control ofthe relative position of counter-propagating beams in the sub-picometer range.Since the stored beam can be considered as a rarefied gas of uncorrelated par-ticles, we moreover demonstrate that the amplitudes of the zero-point betatronoscillations of individual particles are within a factor of 10 of the Heisenberguncertainty limit. As a consequence of this, we conclude that quantum me-chanics does not preclude the control of the beam centroids to sub-picometeraccuracy. The smallest Lorentz force exerted on a single particle that we havebeen able to determine is
10 aN . Experiments searching for electric dipole moments (EDMs) of charged particles using storagerings are at the forefront of the incessant quest to find new physics beyond the Standard Modelof particle physics. These investigations bear the potential to shed light on the origin of theanomalously large matter-antimatter asymmetry in the Universe ( ), for which the combinedpredictions of the Standard Models of particle physics and of cosmology fall short of the exper-imentally observed asymmetry by about seven to eight orders of magnitude ( ).In order to cancel out systematic effects in a fully-electric frozen-spin storage ring aimingat a proton EDM sensitivity of − e cm , simultaneous measurements of the EDM-driven spinrotations of the counter-propagating beams are necessary. For this purpose, it is imperativeto control the relative vertical displacement of two centers of gravity of two beams with anaccuracy of about ( ). One may wonder whether such an enormously small accuracyvalue is not prohibited by Heisenberg’s uncertainty relation.2ere we report on the measurement of the amplitude of collectively excited vertical os-cillations of a deuteron beam orbiting in the magnetic storage ring COSY at a momentum ofabout
970 MeV / c ( ). The data were taken in 2018 in the course of a dedicated experimentin the framework of systematic beam and spin dynamics studies for the deuteron EDM ex-periment (so-called precursor experiment), presently carried out by the JEDI collaboration atCOSY ( ). One of the central devices in the precursor experiment is the radio-frequency(RF) Wien filter (WF), shown in Fig. 1, which was designed to provide a cancellation of theelectric and magnetic forces acting on the particle. In this operation mode, the WF affects onlythe particle spins, but does not perturb the beam orbit ( ). A slightly detuned WF, however,exerts a non-vanishing Lorentz force on the orbiting beam particles. It is shown that collectivebeam oscillation amplitudes down to µ m can be detected with the currently available equip-ment. Our approach to measuring ultra-small displacements complements other measurementsof ultra-small forces using different techniques ( ).In a storage ring, the beam can be viewed as a rarefied gas of uncorrelated particles. In-dividual particles are confined vertically in the harmonic oscillator potential. Apart from theirconventional individual betatron motions, all the particles participate in one and the same col-lective and coherent oscillation that is driven by the WF. Therefore, the upper bound of theamplitude of the collective oscillation of the beam corresponds to the upper bound of the oscil-lation amplitude of a single particle.From a comparison of the energy of driven transverse beam oscillations to the zero-pointenergy of a particle in the oscillator potential, it is concluded that the amplitude of the WF-driven single-particle oscillation is only about a factor of ten larger than the quantum limit ofHeisenberg’s uncertainty relation. A further improvement of the sensitivity to coherent beamoscillations will become possible when an even more precise adjustment of the driving circuitof the WF ( ) will be applied. 3 (a) CAD drawing of the design of the RF Wien filter. 1:RF feed, 2: beam pipe, 3: inner mounting cylinder, 4:inner support structure, 5: lower electrode, 6: insulator, 7:RF connector, and 8: vacuum vessel. (b) Photograph with a view along the beam axisshowing the gold-plated copper electrodes, whichhave a length of . . Figure 1: The waveguide RF Wien filter is mounted inside a cylindrical vessel. The effectivelength of the device amounts to (cid:96) = 1 .
16 m . The technical details are described in Refs. (
8, 9 ). The Cooler Synchrotron (COSY) (
4, 17 ) at Forschungszentrum Jülich is a storage ring witha circumference of approximately
184 m . Its principal elements used for the experiments areindicated in Fig. 2. For the investigations presented here, the two key devices are the RF WF,based on a parallel-plates waveguide ( ), and a conventional electrostatic beam position mon-itor (BPM) that is used to monitor the beam oscillations ( ). The WF generates orthogonaland highly-homogeneous electric and magnetic fields. In the present experiment, the WF wasoperated in the mode with the electric field pointing vertically upward (in y -direction), whereasthe magnetic field points radially outward (in x -direction), and the beam moves in z -direction(see coordinate system in Fig. 2). The effective length of the WF amounts to (cid:96) = 1 .
16 m , (1)4igure 2: Schematic diagram of the cooler synchrotron and storage ring (COSY) with the maincomponents, especially the focusing/defocussing magnets (quadrupoles) and the bending mag-nets (dipoles). Indicated are the position of the RF WF and the location of the beam posi-tion monitor (BPM 17), used to observe the beam oscillations. Further components such asthe electron cooler ( ), the WASA ( ) and the JEPO (
21, 22 ) polarimeters, and theSiberian snake ( ) are also shown. The coordinate system used is indicated.which corresponds to the length over which the RF fields in the WF can be considered relevant(see Refs. (
8, 9 ) for further technical details).As a spin rotator for the forthcoming deuteron EDM (precursor) experiment ( ), the WFis designed to operate in resonance with the spin precession of the orbiting deuterons ( ),and at a vanishing Lorentz force, given by F y = q ( E y + v z · B x ) , (2)where q denotes the elementary charge and v z represents the velocity of the beam particles.Unlike in conventional DC Wien filters, the electric and magnetic fields of the RF WF are5enerated simultaneously by exciting the transverse electromagnetic (TEM) mode.When the electric and magnetic fields in the WF are mismatched, i.e. , when the electric andmagnetic forces no longer cancel each other, the RF fields excite vertical collective beam oscil-lations at the frequency at which the WF is operated. As already mentioned in the introduction,the collective motion is shared by all particles in the bunch, and the amplitude of the collectivebeam oscillation is identical to the oscillation amplitude of each individual particle.With a vanishing Lorentz force, the beam performs idle vertical (and horizontal) betatronoscillations y ( t ) = y (0) (cid:115) β y ( t ) β y (0) cos [ ψ y ( t )] , (3)where β y ( t ) is the betatron amplitude function. With the beam revolution period of T =2 π/ω rev , the betatron phase advance ψ y ( t ) satisfies ψ y ( t + T ) − ψ y ( t ) = ω y T = 2 πν y , andthe vertical betatron tune is given by ν y = ω y /ω rev .On the other hand, a mismatched WF generates stroboscopically, i.e. , once per turn, a kickof the vertical velocity of the stored particle, given by ∆ v y ( nT ) = − ζω y cos( n ω WF T ) . (4)Here n denotes the turn number, ω WF denotes the angular velocity of the RF in the WF, andthe change ∆ y of the vertical position y in the WF is neglected. The amplitude of the velocitychange is given by the momentum change accumulated during the time interval ∆ t = (cid:96)/v z theparticle spends per turn inside the WF, ζω y = F y ∆ tγm , (5)where F y denotes the amplitude of the vertical component of the Lorentz force, and γ and m are the Lorentz-factor and the mass of the particle, respectively.In the approximation of a constant amplitude function, the betatron motion can be describedin terms of the complex variable z = y − iv y /ω y , common for the oscillatory motion. With the6nitial condition z (0) = 0 , after n turns, the solution for z ( n ) behind the WF reads z ( n ) = iζ · exp( inω y T ) − exp( in ω WF T )exp( i ( ω y − ω WF ) T ) − { ω WF → − ω WF } . (6)This expression serves as the initial condition for the idle betatron motion during the subsequent ( n + 1) turn. Driven by the mismatched WF, all beam particles participate in one and the samecollective and coherent oscillation, and according to Eq. (6), the beam oscillates at the WFfrequency ω WF . A lock-in amplifier may be used to selectively measure the correspondingFourier component of the beam oscillation y = (cid:15) y cos( n ω WF T ) from the output of a BPM. Itsamplitude is given by (cid:15) y = ζ · sin(2 πν y )cos(2 πν WF ) − cos(2 πν y ) , (7)where the vertical betatron tune is given by ν y = ω y /ω rev and the WF tune by ν WF = ω WF /ω rev .When the WF tune is close to the vertical betatron tune, a resonant enhancement of the beamoscillation amplitude (cid:15) y occurs. Equation (7) describes Hooke’s law, F y = k H (cid:15) y , and Hooke’sconstant is given by k H = (cid:12)(cid:12)(cid:12)(cid:12) γmω y ∆ t · cos(2 πν WF ) − cos(2 πν y )sin(2 πν y ) (cid:12)(cid:12)(cid:12)(cid:12) . (8)We compare the measured amplitude of the WF driven oscillations to the amplitude Q ofthe zero-point betatron oscillations with energy (cid:126) ω y / , which entails Q = (cid:126) / ( mγω y ) . For thepresent experiment, we obtain Q = 82 √ γν y nm . (9)With the actual COSY values for the betatron tune ν y and the Lorentz-factor γ of the beam (seeTable 1, Appendix A), the quantum limit of the vertical oscillations, driven by the WF, amountsto Q ≈
41 nm . (10)The interpretation of the measured oscillation amplitudes in terms of the WF parametersrequires numerical simulations of the performance of the WF as an element of the storage7ing ( ). The details relevant to the present study are described below; the corresponding beamsimulations carried out are consistent with the available experimental results on the propertiesof COSY ( ). The control of the Lorentz force of the waveguide RF Wien filter is based on the wave-mismatchprinciple ( ). An impedance mismatch is introduced at the load part of the device to deliber-ately create reflections that generate a standing wave pattern inside the WF ( ). These standingwaves can be represented by the complex-valued field quotient Z q , defined as the ratio of thetotal electric to the total magnetic field strength, Z q = E total H total = E + + E − H + − H − = E + + Γ · E + H + − Γ · H + = Z w − Γ = Z dW − Γ , (11)where the superscripts ’ + ’ and ’ − ’ refer to the forward and backward direction of propagation, Z w is the wave impedance, Z ≈
377 Ω is the vacuum wave impedance, d = 100 mm is thedistance between the electrodes, W = 182 mm is their width ( ), and Γ is the reflection coef-ficient that controls the amplitude and phase of the reflected wave. During the measurementsdescribed here, the WF was typically operated at a net input RF power of
600 W .The field quotient Z q is controlled via a specially designed RF circuit ( ). By altering Γ via two variable vacuum capacitors, called C L and C T , a wide range of Z q values can be cov-ered, and the matching point corresponding to the minimum induced vertical beam oscillationamplitude may be determined. In this experiment, the electric field of the WF is oriented vertically and the magnetic field hor-izontally. This implies that the oscillations mainly take place along the y -axis [see Eq. (2)]. For8
50 100 1500510152025
Figure 3: Vertical and horizontal beta-functions along the circumference of COSY ( ). Thevertical dashed lines mark the location of the WF and of the BPM used during the measurementof the beam oscillations.the detection of the vertical beam oscillations, a conventional beam position monitor (BPM) hasbeen employed. In order to be most sensitive, BPM 17 located in the straight section opposite tothe WF (see Fig. 2) with a large vertical β function was used, β BPM y ≈ . , while at theWF location, β WF y ≈ . , as shown in Fig. 3. The arguments to pick BPM 17 are furtherdiscussed below in Sec. 4.1.In order to measure small beam oscillations, a technique based on lock-in amplifiers wasdeveloped ( ). These devices operate in the frequency domain and lock onto a signal whosefrequency is set as a reference, which is particularly useful in an electromagnetically noisyenvironment. Each measurement consisted of two subsequent machine cycles of duration,as depicted in Fig. 14 in Appendix B.A stored beam bunch circulating at a revolution frequency of f rev that passes through a BPMinduces a voltage signal on all its four electrodes, as indicated in the BPM readout scheme, HF2LI 50 MHz Lock-in Amplifier, Zurich Instruments AG, 8005 Zurich, Switzerland, . Experimental Physics and Industrial Control System, https://epics.anl.gov/index.php . efsignalrefsignalrefsignalrefsignallock−in amplifiersCOSY RF referenceWien filter RF referencetopleftrightbottomCOSYbeam D a t a ac qu i s iti on E p i c s Figure 4: Readout scheme of the COSY BPM 17. The signals of the four electrodes are fedinto lock-in amplifiers. The differential signal of each electrode is analyzed at the two refer-ence frequencies given by the COSY RF and the Wien filter frequency. The resulting Fourieramplitudes of the signals are recorded in the EPICS archiving system of COSY.shown in Fig. 4. For the detection of vertical beam oscillations, only the voltage signals U t ,b from the top (t) and bottom (b) electrodes are considered. These signals are trains of shortpulses with the repetition frequency f rev . In view of Eq. (3) and as far as the Fourier spectrumof the beam oscillations is concerned, without loss of generality the BPM can be considered tobe located right behind the WF, and the induced voltages can be represented by U t, b = [ U ± ∆ U (∆ y )] cos( ω rev t ) , (12)where the index "t" refers to the + sign and the index "b" to the − sign, respectively. Here, U denotes the voltage induced when the beam passes exactly through the center of the BPM,10nd ∆ U (∆ y ) represents the voltage variation induced by a beam that is vertically displaced by ∆ y . The harmonic factor cos( ω rev t ) describes the pulse repetition frequency, and cos( ω rev t ) = 1 applies to multiples of the orbital period t = nT .For small beam displacements, the BPM operates in its linear regime, which implies thatthe induced voltages take the form ∆ U (∆ y ) = κ · ∆ y · U , (13)where κ is a calibration factor that needs to be determined. At a momentum of
970 MeV / c , therevolution frequency of deuterons orbiting in COSY is f rev ≈
750 kHz . The WF is operated atthe k th sideband of the spin-precession frequency, given by f WF = ω WF π = ( k + ν s ) f rev = k · f rev + f s . (14)Here ν s = Gγ denotes the spin tune, i.e. , the number of spin precessions per revolution, G ≈− . is the magnetic anomaly of the deuteron, and the spin precession frequency f s = ν s f rev ( ). It should be noted that the beam oscillations do not depend on the actual choice of thesideband. In the present experiment the WF was operated at k = − , which corresponds to f WF ≈
871 kHz .The induced oscillations of amplitude (cid:15) y contribute to Eq. (12) the harmonic voltage vari-ation ∆ U ( y ( t )) . The BPM in conjunction with the lock-in amplifiers is used to measure attimes t = nT the beam positions at the reference frequencies, i.e. , at f WF and at f rev . Given that y ( t ) can be evaluated at the spin precession frequency, the BPM signals of the upper and lowerelectrodes can be written as follows U t, b ( t ) = [ U ± ∆ U (∆ y ) ± ∆ U ( y ( t ))] cos ( ω rev t ) , = [ U ± ∆ U (∆ y ) ± κ(cid:15) y U cos ( ω s t )] cos ( ω rev t ) , = [ U ± κ ∆ yU ] cos ( ω rev t ) ± κ(cid:15) y U cos ( ω ∆ t ) ± κ(cid:15) y U cos ( ω Σ t ) . (15) For the considerations presented in this paper, negative and positive frequencies are considered equivalent. ω ∆ and ω Σ represent sidebands of the WF frequency at ω ∆ = ω rev − ω s = ω WF | k =1 , and ω Σ = ω rev + ω s = ω WF | k = − . (16)In order to measure the beam oscillations, four lock-in amplifiers ( ) were used, two for thehorizontal and two for the vertical direction. For each direction, one lock-in amplifier detectsthe Fourier amplitudes at f rev ≈
750 kHz and a second one at f Σ = f rev + f s ≈
871 kHz . Thelock-in amplifiers receive reference frequencies from the signal generator of the WF and fromthe master oscillator of COSY. The four Fourier amplitudes of the top and bottom electrodesare determined practically in real-time, yielding A revt, b = U ± κ ∆ yU , and A Σ t, b = ∓ κ(cid:15) y U . (17)The amplitude of the vertical oscillation (cid:15) y can then be determined from A Σ t − A Σ b A revt + A revb = ˆ (cid:15) y = κ U U (cid:15) y = 12 κ(cid:15) y . (18)The uncalibrated raw asymmetry of the four Fourier amplitudes is denoted by ˆ (cid:15) y . The determi-nation of the calibration constant κ , required to calibrate the vertical oscillation amplitude, isdescribed in detail in Appendix B. It amounts to κ = (5 . ± . · − µ m − . (19)During the experiments, the vertical betatron tune of the machine amounted to about ν y ≈ . . The frequency f Σ ≈
871 kHz , at which the WF is operated, is well separated from thelowest intrinsic spin resonances at
297 kHz ,
453 kHz , , and . The numerical values used for the simulation calculations are listed in Table 1 of Appendix A. An intrinsic depolarizing resonance is encountered, when the betatron motion of the particles is in sync withthe spin motion, hence, when the condition f s = ν s f rev = f y = ( nP ± ν (cid:48) y ) f rev is fulfilled ( ), where n ∈ N , P denotes the superperiodicity of the lattice, and ν (cid:48) y the fractional tune. During the experiments described here, P = 1 (see also Fig. 3). C L and C T , are driven by stepper motors.They constitute the main dynamical elements of the driving circuit. Each pair of capacitor val-ues yields a well-defined field quotient | Z q | , as shown in Fig. 5 (a). Away from the matchingpoint, a phase shift ∠ Z q occurs between electric and magnetic fields, as shown in Fig. 5 (b).The corresponding Lorentz force leads to the measured beam oscillations, i.e. , the function (cid:15) y = f ( C L , C T ) , which can be vizualized in the form of a 2D map, as shown in Fig. 6. Theexperimental data were taken on a grid of (7 × points of C L and C T , with correspondinggrid spacings of (94 . ± .
0) pF for C L and (95 . ± .
0) pF for C T . Each grid spacing corre-sponds to 1000 steps of the corresponding stepper motors. The calibration of the capacitances C L and C T as a function of step number is discussed in detail in ( ). The grid spans over C L ∈ [318 . , .
58] pF and C T ∈ [428 . , .
79] pF . The uncertainties of the grid spac-ings are systematic ones. It should be emphasized that the technique of inducing collectivebeam oscillations can be used later in the actual EDM experiments as a tool to investigate thesystematic effects of beam oscillations.The map of the measured and calibrated vertical beam oscillations (cid:15) y is shown in Fig. 6. Theparameters of the matching point are given by C L = (697 . ± .
0) pF , and C T = (503 . ± .
0) pF , (20)and the corresponding minimal beam oscillation amplitude at the location of BPM 17 amountsto (cid:15) min y = (1 . ± . µ m . (21)The maximum measured amplitude of driven beam oscillations at a strongly mismatched point The individually measured uncertainties of the capacitors are actually much smaller than the stated uncertaintyof . . However, other factors, such as the capacitances and inductances of the connectors and cables and theirpower dependencies, also contribute to the aforementioned uncertainties. a) Magnitude of the field quotient | Z q | , evaluated inte-grally, where | Z q | int = (cid:82) | Z q | d (cid:96) . Ideally, with | Z q | closeto
176 Ω , the electric and magnetic forces are equal. (b) Phase of the field quotient ∠ Z q evaluated integrally,where ∠ Z int q = (cid:82) ∠ Z q d (cid:96) . A non-vanishing ∠ Z q impliesa phase shift between the electric and magnetic fields. Figure 5: Simulated integral magnitude (a) and phase of the field quotient Z q (b) at each pointof the C L and C T grid, indicated by the blue points, (cid:96) denotes the effective length of the WF[see Eq. (1)]. Besides the matching point [see Eq. (20)], (7 × grid points were investigated.with
600 W of input RF power amounts to (cid:15) max y = (66 . ± . µ m . (22)In Fig. 7 (a), the data measured at the matching point [Eq. (20)] are shown. Each samplewas recorded by the lock-in amplifiers with an integration time set to . , corresponding to anaverage of measurements. A Monte Carlo error propagation model was applied to treat theuncertainties of the still uncalibrated raw position asymmetries ˆ (cid:15) y and the calibration coefficient κ ( ). The results are fitted with a normal distribution, as shown in Fig. 7 (b), from which themean value µ (cid:15) y and the error of the measured beam oscillations σ (cid:15) y are estimated. The latterrepresents the systematic error of the measurement. It should be noted that the map shown inFig. 6 is actually a function of all the circuit elements. The uncertainties of (cid:15) y are influenced bythe uncertainties of all circuit elements and also by the ones of the BPM itself, which includeits readout electronics, i.e. , the lock-in amplifiers.To appreciate the result given in Eq. (21), one can compare the oscillation amplitude to the14igure 6: Measured amplitudes of beam oscillations (cid:15) exp y at BMP 17, plotted on a grid as afunction of the variable capacitor values C L and C T . To avoid crowding up the map, the errorbars of the data points were omitted, these are shown in Fig. 13 instead. The parameters of thematching point are given in Eq. (20). σ vertical beam size. The latter has been deduced from the σ beam emittance (cid:15) b y and theamplitude of the β function at the BPM position, yielding σ BPM y = (cid:113) β BPM y (cid:15) b y ≈ . . (23)Although in the present experiment the beam emittance was not monitored, as a reference valuefor a well-cooled beam in the above numerical estimate of σ BPM y , the experimental result for the σ beam emittance of . protons in COSY, (cid:15) b y = (0 . ± . µ m ( ), has been rescaledto the conditions of the present experiment. It is noteworthy that with the present equipment itis possible to access coherent beam oscillations with amplitudes that are more than three ordersof magnitude smaller than the beam size. 15
00 1000 1500 2000 2500 30000.511.522.53 (a) Measured oscillation amplitudes (cid:15) y using datasamples of . duration, each sample reflects theaverage of measurements of the lock-in ampli-fiers. (b) Probability density distribution f (cid:15) y of the mea-sured data, fitted with a Gaussian to determine meanand standard deviation. Figure 7: Measured beam oscillations at the matching point [Eq. (20)] of the map shown inFig. 6. The samples shown in panel (a) were acquired during a data taking period of
108 min ,using 36 machine fills (cycles).
To improve our understanding of the measured results, a computer code was developed to modelthe beam dynamics in the COSY storage ring. The modeled storage ring consists of a sequenceof drift regions, quadrupole and dipole magnets, the WF, and BPMs. These elements are rep-resented by transfer matrices, which are well understood and documented in the literature ( ).In the model of the ring, the actual settings of the beam optics elements of COSY were thoseused at the time when the experiment took place. Simulations are based on the Hamiltonianformulation as presented in Ref. ( ). The WF is modeled by a time-dependent matrix that alsotakes into account the arrival time of the particles.In order to be able to perform reliable beam simulations, we have placed great emphasis ongood spatial resolution and the accuracy of the 3D field maps inside the WF , computed using Each field map consists of · points, 200 points along the x axis ( x ∈ [ − , , 200 points alongthe y axis ( y ∈ [ − , , and 50 points along the z (WF) axis ( z ∈ [ − (cid:96)/ , + (cid:96)/ , where (cid:96) is specified in a) 3D electric field distribution of the component E y . (b) 3D magnetic field distribution of the flux densitycomponent B x . Figure 8: Examples of the main electric and magnetic field components inside the waveguideRF Wien filter at the matching point [see Eq. (20)] with an input RF power of
600 W . Theelectric field component in (a) points vertically upward ( y -direction), while the component ofthe magnetic flux density in (b) points radially outward ( x -direction).a 3D electromagnetic simulation tool . The fringe fields of the WF are included, because theyare of particular importance for the beam oscillations, as will be discussed later. An exampleof the computed 3D fields of the WF at the experimentally determined matching point is shownin Fig. 8. The beam-tracking simulations use the three vector components of the electric andmagnetic fields. The Wien filter is implemented as an RF kicker, as described by Eq. (4).Inside COSY there are 32 BPMs available to control the horizontal and vertical beam po-sition during operation. In order to select one of them with a good sensitivity to determine thebeam oscillations induced by the WF, a number of particles were tracked, as described above,and the orbit response induced by a field change at the location of the WF was calculated at eachBPM location . As a result, BPM 17, located about
70 m downstream of the WF (see Fig. 2),
Eq. (1). Electromagnetic and circuit simulations were performed using CST, from Dassault Systèmes, Vélizy-Villacoublay, France, . In the preparatory stage, simulations were carried out using the Software Toolkit for Charged-Particle andX-Ray Simulations BMAD ( ). E y and B x . When mismatched, the WF generatesperiodic transverse perturbations of the trajectory as a result of the velocity kicks in the form ofbeam oscillations. Switching off the WF eliminates such oscillations. The maximum amplitudeof the observed oscillations in the simulation is then used to represent (cid:15) y . The two simulatedvertical beam oscillation amplitudes of BPM 17 and WF read (cid:15) BPM y = (1 . ± . µ m , and (cid:15) WF y = (0 . ± . µ m . (24)A detailed description of the determination of the uncertainties of the beam simulations is dis-cussed in the next section.For each and every measured point on the C L versus C T grid, a beam dynamics simulationwas carried out. For each of these points, a 3D field map of the WF was generated and thenused for the beam tracking simulations. The results of these simulations are shown in Fig. 9,and are later compared with the results of the measurements. The accuracy with which the Lorentz force and the resulting amplitudes of the beam oscillationscan be tuned depends on the accuracy with which the field quotient Z q can be integrally setto the desired value. Z q depends on the hardware elements in the driving circuit. In order toevaluate the effects of uncertainties of these elements, extensive coupled circuit electromagneticsimulations have been conducted, as discussed in Ref. ( ). The uncertainties involved are listedin Table 6 and shown in Fig. 16 of Ref. ( ). As far as the Lorentz force is concerned, mostimportant are the uncertainties of the fixed inductance L f and the fixed resistance R f . Once theseuncertainties are known, one can compute the electric and magnetic fields and the correspondingLorentz force, including their corresponding errors. Figure 10 shows a few examples of the main18igure 9: Simulated amplitudes of beam oscillations (cid:15) sim y as a function of the variable capacitorvalues C L and C T . To avoid overcrowding the map, the error bars of the data points were omittedhere and are shown in Fig. 13.components of the electric and magnetic fields, computed with the above mentioned circuituncertainties. As will be explained below, these 3D fields, together with their uncertainties, aresubsequently used as input to the beam simulations.The algorithm used to compute the uncertainties of the beam simulations is the polynomialchaos expansion (PCE), as explained in Refs. (
9, 16 ) and in Appendix C. The PCE has beenproven in many applications in science and engineering to be just as accurate as the computa-tionally much more expensive Monte-Carlo counterpart (
9, 30–32 ).To compute the uncertainties σ (cid:15) y , the PCE algorithm requires a random set of the simulated (cid:15) y , alongside a set of randomized input parameters according to their uncertainties to generatethe output. The set of (cid:15) y is produced using a number of beam-tracking simulations, wherefor each instance, a 3D field map of the WF is generated, according to the randomized inputparameters. An example of the electric and magnetic fields evaluated at the center of the WF19 a) Electric field component E y ( z ) under circuit uncer-tainties. (b) Magnetic field component B x ( z ) under circuit un-certainties. Figure 10: examples of the electric and magnetic fields as a function of z along the beamaxis under the circuit uncertainties, specified in the list of uncertainties in Table 6 of ( ).for the matching case [see Eq. (20)] is shown in Fig. 10. The magnitudes of the fields vary as afunction of the uncertainties of the driving circuit ( ). The numerical tracking of the particlesthrough these fields generates a collection of different (cid:15) y values that the PCE algorithm canuse to project the output onto orthogonal polynomial functions. These functions serve as basisfunctions, from which the expansion coefficients are determined that are used to generate a largesample of outputs to compute the uncertainties of the beam simulations.In Fig. 11 (a), the simulated values of (cid:15) y are shown for the matching case. The detailed stepsto achieve this result are discussed in Appendix C. As shown in Fig. 11(b), fitting these datato a Gaussian yields a standard deviation of σ (cid:15) y = 0 . µ m . This number is of considerableimportance, because, given the uncertainties of the driving circuit, it sets the lower limit that canbe achieved by minimizing the amplitude of the vertical beam oscillations when more sharplytuning the driving circuit of the WF. The same procedure is performed on each point of the mapshown in Fig. 9. 20
000 4000 6000 8000 100000.80.911.11.21.31.4 (a) Simulated oscillation amplitudes under uncertain-ties at the matching point (Eq. (20)). Of the simu-lations that were carried out, only are shown here. (b) Probability density distribution f (cid:15) y of the simulations from panel (a), fitted by a Gaus-sian to determine mean and standard deviation. Figure 11: Results of the sparce PCE algorithm to compute the uncertainties of the simulatedvertical beam oscillations at BPM 17.
The simulations yield the net Lorentz force exerted by the WF on beam particles and the cor-responding oscillation amplitudes for each measured point of the C L versus C T map, shown inFig. 6. The only variables in this case are the field maps of the WF itself. After turns,the beam position is computed at the same location in the ring, where the measurement usingBPM 17 took place (see Fig. 2). The net Lorentz force is a result of local cancellations betweenthe electric and magnetic field components, as illustrated in Fig. 12 for the matching point givenin Eq. (20) with the minimal measured oscillation amplitude.In Fig. 12(a), the local Lorentz force is shown along the trajectory for 5 randomly chosenpasses though the WF. The trajectory of the same particle changes from pass to pass, therebydifferent WF fields and consequently different values of the Lorentz force F y will be pickedup. As shown in Fig. 12, even at the matching point, the matching is still imperfect, and thelargest local F y contributions are caused by the fringe fields at the entrance and exit of theWF. Despite the different location of the particle in the vertical and horizontal phase space21
00 400 600 800 1000-60-40-200204060 turn-15 turn-23 turn-68 turn-69 turn-83 (a) Local Lorentz force F y ( z ) exerted on a singledeuteron for different passes though the WF. The turnnumbers used here were randomly selected between 1to 100. The fields were evaluated at the crosses and theinterconnecting lines are to guide the eye. datafit (b) Integral Lorentz force F y ( n ) evaluated along thetrajectory. Each point represents an overall kick ex-erted per turn n . The points marked in blue correspondto the integrated local Lorentz force of the individualturns shown in panel (a). Figure 12: Simulation of the local and integrated Lorentz force in the WF at the matching pointof Eq. (20). Depending on the initial coordinates in the vertical and horizontal phase space, theparticle travels along different trajectories, and therefore picks up different field components F y .at the entrance of the WF upon subsequent passes, the integration of these local forces alongthe particle trajectories exhibits nevertheless a perfectly harmonic time dependence with thefrequency f s , as shown in Fig. 12(b). The points encircled in blue correspond to the randomlyselected passes through the WF, shown in Fig. 12(a).In the left panel of Fig. 13, the amplitude of the simulated Lorentz force F sim y is plottedversus the simulated oscillation amplitude (cid:15) sim y at the WF position. As expected, it exactlyfollows Hooke’s law with a spring constant of k H = (151 . ± .
2) MeV / m . In Fig. 13(b), themeasured amplitudes are compared with the ones simulated for the location of BPM 17. Thetwo sets (cid:15) exp y and (cid:15) sim y are in very good agreement with each other. The horizontal and verticalerror bars are derived from the uncertainties of the measurements and simulations, representedby the width of the distributions, as shown in Figs. 10(b) and 7(b). It is important to note thatthe error bars refer to systematic uncertainties and should not be confused with statistical ones.This implies that repetitions of either the measurements or the simulations will neither reduce22 a) Simulated Lorentz force F sim y at the WF locationas function of the oscillation amplitude (cid:15) sim y , fittedwith the function F sim y = a · (cid:15) sim y + b . (b) Simulated beam oscillation amplitude (cid:15) sim y versusthe measured oscillation amplitude (cid:15) exp y at the BPM,fitted with the function (cid:15) sim y = c · (cid:15) exp y + d . Figure 13: (a): Simulated amplitude of the Lorentz force at the WF location as function ofthe simulated beam oscillation amplitudes (cid:15) sim y . (b): Simulated versus measured vertical beamoscillation amplitudes at the location of BPM 17. The horizontal error bars of the measuredamplitudes (cid:15) exp y originate from the readout electronics of BPM 17 and the calibration factor κ (see Appendix B), whereas the vertical ones are determined by the circuit uncertainties usingthe PCE method, as described in Appendix C.the systematic error of the readout electronics of BPM 17, nor will it affect the uncertainties ofthe elements of the driving circuit.The fit shown in Fig. 13 yields χ / ndf = 45 . / , very close to unity ( ). The linear fityields a slope of . ± . , which is perfectly consistent with unity. The intercept parameterof the fit yields ( − . ± . µ m , and within three standard deviations, it agrees with zero.The very good agreement between measurements and simulations reflects our good under-standing of both the electromagnetic fields generated in the WF and of the underlying beamdynamics in the machine. This point is further substantiated by comparing the simulated am-plitudes at the WF and BPM positions with the estimated amplitudes expected from rescaling23ased on the β functions , taking into account the numerical values, listed in Table 1 of Ap-pendix A, (cid:15) WF y (cid:12)(cid:12) sim = (0 . ± . µ m , (cid:115) β WF y β BPM y (cid:15) BPM y | sim = (cid:15) WF y (cid:12)(cid:12) est = (0 . ± . µ m . (25)The good agreement between these two numerical values in Eq. (25) indicates that the observa-tion of the oscillation amplitude at one location in the ring can be reliably transfered to someother place in the ring by use of Eq. (3). The above quoted value of (cid:15) WF y (cid:12)(cid:12) sim = 0 . µ m is abouta factor of 10 larger than the quantum limit of the vertical oscillation amplitude Q , given inEq. (10).In searches for EDMs in dedicated all-electric storage rings, a continuous monitoring ofthe orbits of the two counter-rotating beams is mandatory during data acquisition within thehorizontal spin-coherence time ( ). When intrabeam scattering can be neglected ( ), whichis arguably justified within the horizontal spin-coherence time, the beam can be described as ararefied gas of particles, i.e. , the zero-point oscillations of individual particles are uncorrelated.Therefore, the quantum limit of the centroid of a bunch containing N particles can be estimatedvia Q bunch = Q/ √ N . For a bunch of N = 10 stored particles, one obtains Q bunch (cid:39) . ,indicating that the systematic uncertainty of the beam oscillations will be rather limited by thesensitivity of the BPMs.Finally, a satisfactory agreement has been achieved between Hooke’s constant, simulatedusing the electromagnetic fields in the WF and the β functions of the COSY lattice, and thetheoretical approximation of the no-lattice model assuming constant β functions of Eq. (8),yielding k simH = (151 . ± .
2) MeV / m , and k thH = 207 MeV / m . (26) The uncertainty of the β functions amounts to about 10%, as discussed in Ref. ( ). k thH , calculated using the numerical values listed in Table 1 of Ap-pendix A, is about a factor of 1.4 larger than the simulated one. The given uncertainty of k simH does not include the systematic scale uncertainty of the BPM calibration factor κ [see Eq. (29)in Appendix A]. At the matching point [see Eq. (20)], the Lorentz force amounts to F WF y = k simH · (cid:15) WF y (cid:12)(cid:12) sim ≈
66 eV / m = 10 . , (27)where the intercept parameter has been ignored because of its smallness. As part of several studies to investigate the performance of the waveguide RF Wien filter, ex-ploratory data were taken to provide a benchmark on the sensitivity to very weak collectivevertical beam oscillations of deuterons stored in the COSY ring. To a good approximation,the beam can be viewed as a rarefied gas of uncorrelated particles, and the sensitivity limit isapplicable to the classical motion of individual particles, propagating along the ring circum-ference in the confining oscillatory potential. Simulations of the beam dynamics in the COSYring equipped with an RF Wien filter suggest that with the present apparatus, the sensitivityto collective beam oscillations on the sub-micron level is within a factor of 10 of the ampli-tude of zero-point quantum oscillations of the stored deuterons. From the perspective of futureEDM experiments, our finding confirms that the separation of the centroids of two counter-propagating beams may be determined to sub-picometer accuracy – the limitation is given bythe sensitivity of the presently employed BPMs.The reported excellent agreement between simulated and experimentally observed verti-cal beam oscillations at COSY suggests that a further increase in sensitivity to collective beamoscillations is possible. Specifically, the simulation on finer capacitor grids indicates that by fur-ther optimization of the WF settings to C L = (692 . ± .
00) pF and C T = (495 . ± .
00) pF ,25n oscillation amplitude at the WF location of (cid:15) y = (0 . ± . µ m may be achieved. Thusin that case, the vertical oscillation amplitude would only be about a factor of 2 away from thequantum limit, with a corresponding Lorentz force of F y ∼ . Acknowledgments
This work has been performed in the framework of the JEDI collaboration and is supported byan ERC Advanced Grant of the European Union (proposal number 694340). In addition, it wassupported by the Russian Fund for Basic Research (Grant No. 18-02-40092 MEGA) and by theShota Rustaveli National Science Foundation of the Republic of Georgia (SRNSFG Grant No.DI-18-298:
High precision polarimetry for charged-particle EDM searches in storage rings ). A Quantities used in the beam simulations
In order to provide a consistent calculation of all effects in the storage ring, the beam simulationswere carried out using the set of quantities given in Table 1 as an input. The vertical machinetune ν y is a result of simulations with the known COSY lattice, reflecting the actual currents ofthe magnetic elements in the machine at the time when the experiment was conducted. The sim-ulations provide the uncalibrated parameters of the vertical beam oscillations to about per millaccuracy, and giving the kinematic, ring, and WF parameters to four digits appears thereforesufficient. It should be noted that within the simulation calculations carried out in the contextof the present work, all quantities have been computed to double precision (machine epsilon of . · − ). Of the physical quantities, the highest sensitivity to the vertical betatron tune isexhibited by the theoretical estimate for Hooke’s constant, d k thH / d ν y ≈ · MeV / m . Thelargest uncertainty contributing to the error of the detected oscillation amplitudes arises fromthe BPM calibration factor κ , given in Eq. (29). It amounts to about . and is considered asystematic scale-factor uncertainty (see Appendix B).26able 1: Numerical values used for the beam simulations. The genuinely independent inputparameters are listed in bold face. The derived quantities are displayed in normal font and aretruncated to four decimal places. Quantity Symbol Valuedeuteron beam momentum p . / c deuteron mass m . / c deuteron G factor G − . Lorentz factor β . Lorentz factor γ . COSY circumference L COSY . revolution frequency f rev
750 603 . vertical machine tune ν y . vertical β function at BPM 17 β BPM y . vertical β function at WF β WF y . effective length WF (cid:96) . frequency WF f WF
871 000 . tune WF ν WF = f WF f rev . B Calibration of the BPM
The complex amplitudes measured by the lock-in amplifiers describe the magnitude and phaseof each signal, and are here expressed by the corresponding real and imaginary components,denoted by X and Y , respectively, i.e. , A = X + iY . Examples of the data recorded at the sumfrequency f Σ and at the revolution frequency f rev are shown in Fig. 14. The effect of switchingon the power amplifiers of the WF at t = 60 s is clearly visible. In both panels, one observes aseparation of the quantities recorded by the top and bottom electrodes in the µ V range for bothfrequencies after the WF is switched on. This separation is much more pronounced at the WFfrequency than at the revolution frequency.The quantities A revt and A revb , given in Eqs. (17), are related to a vertical beam offset ∆ y in27 a) Real ( X Σ ) and imaginary part ( Y Σ ) of the complexFourier amplitudes A Σ at the WF frequency. (b) Real ( X rev ) and imaginary part ( Y rev ) of the complexFourier amplitudes A rev at the revolution frequency. Figure 14: Fourier amplitudes A = X + iY for the top and bottom electrodes of BPM 17recorded by the lock-in amplifier as a function of time in the cycle at a strongly mismatchedpoint ( C L = 907 .
79 pF and C T = 885 .
58 pF ), at the WF frequency (a), and at the revolutionfrequency (b). In both panels, the stored beam current is shown in black. The cycle starts rightafter injection is completed at t = 0 s , beam preparation continues until t = 55 s , and the WFis switched on and data acquisition starts at t = 60 s . At t = 156 s the WF is switched off anddata acquisition stops.Table 2: Current variation ∆ I (in % of the maximum admissible current) in the vertical steerersto generate bumps and the corresponding position change of the vertical orbit ∆ y at the locationof BPM 17. ∆ I (steerer) [%] ∆ y [ mm ] − − . ± . − − . ± . − − . ± . − − . ± . − − . ± . − . ± . − . ± . − . ± . . ± . . ± . . ± . the following way, R = A revt − A revb A revt + A revb = κ U ∆ y U = κ ∆ y . (28)28igure 15: Calibration curve of BPM 17. The ratio R , defined in Eq. (28), depends on theintroduced vertical beam offset ∆ y at the BPM.The calibration constant κ is experimentally determined by introducing local vertical beambumps in the ring at the location of BPM 17. These displacements ∆ y are invoked by alteringthe current of a set of vertical steerers, listed in Table 2. The calibration factor κ is obtained byfitting. Figure 15 shows the ratio R as a function of the vertical orbit variation ∆ y , exhibiting anearly linear relation. The slope corresponds to κ = (5 . ± . · − µ m − . (29) C Uncertainty of the simulations
The uncertainties of the simulated amplitudes of the beam oscillations are computed using thePolynomial Chaos Expansion (PCE) algorithm. The functionality of the algorithm is explainedbelow for one of the simulated data points of the map shown in Fig. 9.The PCE algorithm offers an alternative to the well-known Monte-Carlo (MC) method with-out compromising the intended accuracy. It uses orthogonal polynomials to represent randomlychanging variables to describe observables by means of a finite (truncated) series [for moredetails, see, e.g., Ref. ( )]. When the defined criteria of convergence are met, the expansion29 lgorithm 1: Sparse Polynomial Chaos Expansion algorithm
Data:
Generates Gaussian-distributed ensemble of uncertain circuit parameters usingthe Latin Hypercube Sampling (LHS) scheme X i Result:
Compute uncertainty of (cid:15) y Standardize input data X i −→ ξ i ;Guess hyperbolic truncation norm, q -norm;Start with lowest possible expansion order p ;Generate basis functions H p ( ξ i ) ( p th order Hermite polynomials);Generate hyperbolically truncated set of basis functions H qp ( ξ i ) ;Apply Least-Angle Regression (LAR) algorithm;Estimate optimum sparse set of basis functions H q ∗ p ( ξ i ) ;Compute expansion coefficients C j given Γ = (cid:80) C j H q ∗ p ( ξ i ) ;Compute estimated values of ˆ (cid:15) y = Γ · (cid:15) y ;Compute leave-one-out error LOO err ;Check convergence condition( LOO err < − ); while not convergent do Enhance model (vary p and q ); if convergent then Generate large sample of ˆ (cid:15) y ;Estimate statistical parameters;Terminate algorithm; else Enrich input samples X i ;Repeat algorithm;coefficients can be used to generate an arbitrarily large sample of observables, from which theuncertainties can be computed to the desired statistical accuracy.The PCE algorithm has been compared with the MC method in many applications and hasbeen shown to provide very reliable results. ( ). The PCE requires much fewer simulations toconverge compared to the MC method. For instance, for the present case, 200 beam trackingsimulations per point in the 2D the map of beam oscillations, shown in Fig. 9 were sufficient toreach convergence.In cases where the number of random input variables m is larger than , the PCE methodoffers clear advantages over the MC method. The reason is that the number of basis functions in30he PCE method increases enormously as a consequence of the tensor product of the involvedpolynomials. Therefore, the algorithm has been improved further to allow for a reduction ofthe number of simulations required. Such an approach is also adopted here, as described in Al-gorithm 1. The hyperbolic truncation scheme together with the Least-Angle Regression (LAR)method form a sparse version of the original algorithm.An m -dimensional set is first created, representing N combinations of simultaneous randomvariables. Many methods can be used to generate such sets, and here the Latin-hypercubesample scheme is adopted ( ). Subsequently, the set is standardized for convergence reasons.Depending on the distribution of the data, the basis functions, here Hermite polynomials, aredetermined. The number of basis functions restricts the lower limit of the number of simulations(full-wave and tracking) which are usually computationally expensive. As a rule of thumb,with N basis functions, the PCE algorithm requires at least . × N (in this case, full-wave)simulations to converge. The number of basis functions itself can, however, be reduced by thehyperbolic truncation scheme that eliminates higher-order terms that do not have a significantimpact on the observation objects (
34, 35 ). Furthermore, by applying the LAR algorithm, thenumber of remaining basis functions can be further reduced substantially, whereby the problembecomes computationally solvable in a very efficient fashion.The matching point, specified in Eq. (20), yields the minimum measurable beam oscilla-tions, as given by Eq. (21). This experimental result can be estimated using the beam-trackingcalculations. Subsequently, the concrete steps of the application of the PCE algorithm are dis-cussed.All the reasonable sources of uncertainties of the circuit are represented by 15 random pa-rameters that are allowed to vary simultaneously. At first, a sample of ( × ) entries isgenerated using the Latin-hypercube sampling scheme. As an example of this sample, the vari-ation of the three circuit elements C L , C T , and the load resistor R f is shown in Fig. 16(a).31 a) Sample of C L , C T , and R f used inthe PCE calculations showing a subsetof the 15-dimensional input of randomcircuit uncertainties. standard truncationhyperbolic truncationLAR (b) Truncation schemes of the PCEalgorithm. -6 -4 -2
20 40 60 80 (c) Expansion coefficients on a semi-log scale. 91 coefficients have beenselected after applying the LAR algo-rithm to the matching point.
PCE Tracking (d) Comparison between the trackingresults and the PCE with respect to theoscillation amplitude, determined usingthe expansion coefficients of (c).
Figure 16: Intermediate results of the PCE algorithm applied at the matching point [seeEq. (20)]. Quantitative results of the PCE algorithm are summarized in Table 3.All the uncertain parameters in the electromagnetic circuit simulations are used to gen-erate the electric and magnetic fields shown in Fig. 10. These are subsequently used in thebeam-tracking calculations. For the matching point of the map [Eq. (20)], N = 200 full-wavesimulations were conducted. The import of these field maps into the beam-tracking calculationsresulted in a set of N = 200 values of (cid:15) y . This set is not directly used to conduct the statisticalanalysis. Instead, in conjunction with the input samples, these data are used as input to thesparse PCE algorithm. 32able 3: PCE simulation parameters of the matching point in Eq. (20).parameter valueorder of expansion p dimension m hyperbolic truncation q . leave-one-out error LOO err . · − number of used basis functions P LAR N p = 6 and a truncation norm q = 0 . , executing thePCE algorithm required basis functions to converge, reflected by the low value of the leave-one-out error LOO err = 1 . · − . Subsequently, the expansion coefficients are computed,qualitatively depicted in Fig. 16(c). It is shown in Fig. 16(d) that the PCE algorithm perfectlyreproduces the tracking results using these expansion coefficients. Finally, these coefficientsare used to reconstruct a larger sample of (cid:15) y to estimate the error σ (cid:15) y . Figure 11 shows reconstructed samples. The PCE parameters used are summarized in Table 3.The fitting of these results with a Gaussian, as depicted in panel (b) of Fig. 11, yields astandard deviation of σ (cid:15) y = 0 . µ m . The same technique is repeated for each point in themap. References
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