Experimental test of the new analytic matrix formalism for spin dynamics

We recently started testing Chao's proposed new matrix formalism for describing the spin dynamics due to a single spin resonance. The Chao formalism is probably the first fundamental improvement of the Froissart-Stora equation in that it allows analytic calculations of the beam polarization's behavior inside a resonance. We tested the Chao formalism using a $1.85\text{ }\text{ }\mathrm{GeV}/c$ polarized deuteron beam stored in COSY, by sweeping an rf dipole's frequency through 200 Hz, while varying the distance from the sweep's end frequency to an rf-induced spin resonance's central frequency. Since the Froissart-Stora equation itself can make no prediction inside a resonance, we compared our experimental data with the predictions of the Chao formalism and those of an empirical two-fluid model based on the Froissart-Stora equation. The data strongly favor the Chao formalism.


I. INTRODUCTION
There has been considerable interest in polarized scattering experiments at storage rings such as the MIT-Bates Storage Ring [1], COSY [2], RHIC at Brookhaven [3], and HERA at DESY [4,5]. Many such polarized beam experiments benefit from the ability to precisely control the beam's polarization. The polarization of a stored beam can be manipulated in a well-controlled way by ramping an rf magnet's frequency through an rf-induced spin resonance. The Froissart-Stora formula [6] has been widely used to relate the beam polarizations before and after crossing a resonance. However, it is only valid for a constant-rate linear crossing from far below to far above the spin resonance. A matrix formalism was recently proposed by Chao [7] to treat many experimental conditions that the Froissart-Stora formula cannot treat. The Chao formalism can be used to calculate the polarization at any point inside an arbitrary piecewise linear crossing pattern. Thus, it allows one to calculate the spin dynamics when spin manipulating stored polarized beams outside the Froissart-Stora validity region. Our experiment tested the Chao formalism, using a 1:85 GeV=c vertically polarized deuteron beam stored in COSY, by sweeping an rf dipole's frequency near or through an rf-induced spin resonance.
In an ideal flat circular storage ring or accelerator, with no horizontal bending magnetic fields, each particle's spin precesses around the vertical magnetic fields of the ring's bending dipoles. The spin tune s , which is the number of spin precessions during one turn around the ring, is proportional to the particle's energy where G g ÿ 2=2 is its gyromagnetic anomaly (for the deuteron G d ÿ0:142 987) and is its Lorentz energy factor. The vertical polarization can be perturbed by an rf magnet's horizontal rf magnetic field. This perturbation can induce an rf depolarizing resonance [6,8,9], which can be used to spin manipulate the stored polarized particles [10 -24], such as deuterons. The rf-induced spin resonance's frequency is where f c is the deuteron's circulation frequency and k is an integer.
Ramping an rf magnet's frequency through a spin resonance with strength can flip the stored beam's polarization. When the rf frequency is ramped at a constant rate by a range f, from far below to far above a resonance, during a ramp time t, the Froissart-Stora equation [6] can relate the beam's initial polarization P i to its final polarization P after crossing the resonance, PHYSICAL REVIEW SPECIAL TOPICS -ACCELERATORS AND BEAMS 10, 041001 (2007)

II. CHAO FORMALISM PREDICTION
The paper [7] by Chao developed a matrix formalism for describing the spin dynamics during the crossing of an isolated spin resonance in a synchrotron. The formalism was developed by analytically solving the spinor equation of motion near an isolated spin resonance for two cases: (i) a constant distance between the spin tune s G and the rf-induced resonance tune rf k f rf =f c ; (ii) a linearly changing distance between s and the instantaneous rf . For each case a time-dependent matrix describing the spinor evolution was obtained. If a spin resonance is crossed with a piecewise linear crossing pattern, matrices corresponding to the individual linear segments can be multiplied sequentially to find the final spinor state, which determines the polarization.
To test the Chao formalism, we devised the experiment illustrated in Fig. 1. The frequency of an rf dipole was ramped over a frequency range f, which started at a frequency f start away from a spin resonance and ended at a frequency f end near or sometimes inside the resonance, which was centered at f r . The frequency range f and ramp time t were both held fixed, while f start and therefore f end were varied. The rf dipole was turned off abruptly at f end to preserve the vertical polarization component at the instant of the turn off. The beam's vertical polarization was then measured.
For the experiment shown in Fig. 1, the final spinor state is given by [7] h g end U ÿ; c ; end ; start h g start ; where h and g are the spinor components, start and end are the particle's ''times'' 2f c t at the ramp's start and end, respectively, ÿ is the crossing rate, c is the particle's time at the resonance crossing, is the resonance strength, and U ÿ; c ; end ; start is a 2 2 matrix given explicitly in Eq. (52) of Ref. [7]. The final vertical polarization P is obtained from Eq. (11) of Ref. [7]: Since the ramp starts far from the spin resonance, the initial spinor is simply that of a pure vertical polarization. We related the Chao parameters start , end , ÿ, and c to our experimental parameters f c , f r , f end , f, and t using start ÿf c t; (6) end f c t; We then used Eqs. (4) and (5)  The red solid line in Fig. 2 is a single-particle prediction (f r spread 0) that ignores the f r spread caused by the beam's momentum spread, which may be significant in a real beam. Thus, we included the f r spread by folding the single-particle prediction curve together with Gaussians representing different f r spreads. These predictions are shown in Fig. 2 by the different color dotted lines. Note that the f r spread smoothes the polarization oscillations; their amplitude is reduced as the f r spread increases.

III. TWO-FLUID MODEL PREDICTION
The Froissart-Stora formula equation (3) is not valid for the experiment shown in Fig. 1 because the spin resonance is not crossed completely. However, a two-fluid model based on the Froissart-Stora formula equation (3) may be useful in cases when the beam's f r spread is significantly greater than the single-particle resonance width w 2f c [10].
The model assumes that the frequency ramp only affects those beam particles with resonance frequencies within the ramp's frequency range f start < f r < f end . The final polarization of these particles is obtained using Eq. (3), the Froissart-Stora formula. The model assumes that the remaining fraction of particles retains their initial polarization P i . The two-fluid/FS polarization is the average of these two beam fractions' polarizations. Thus, for a beam with a density function f r , the beam's final polarization is Assuming a Gaussian distribution of the f r spread, we used Eq. (10) to obtain predictions for a few different widths of the f r spread. These predictions are plotted in Fig. 3.

IV. APPARATUS
The apparatus used for this experiment, including the COSY storage ring [25][26][27][28], the EDDA detector [29,30], the rf dipole, the electron cooler [31], the low energy polarimeter [32], the injector cyclotron, and the polarized ion source [33][34][35] are shown in Fig. 4. The beam emerging from the polarized D ÿ ion source was accelerated by the cyclotron to COSY's deuteron injection energy of about 75.7 MeV. Then the low energy polarimeter measured the beam's polarization before injection into COSY to monitor the stable operation of the cyclotron and ion source. The D ÿ beam was next strip injected into COSY.
For the Chao formalism test, we used the electron cooler at injection energy to reduce the beam's size and momentum spread. A 20.6 keV electron beam cooled the deuteron beam to its equilibrium emittances in both the longitudinal and transverse dimensions. The beam was then accelerated to the experimental momentum of 1:85 GeV=c. The rf acceleration cavity was turned off and shorted during COSY's flattop; thus, there were no synchrotron oscillations.
We manipulated the deuteron's polarization using a ferrite-core rf dipole, with an 8-turn copper coil, which produced a uniform radial magnetic field. The rf dipole was part of an RLC resonant circuit, which operated near 917 kHz, typically at an rf voltage of 3.1 kV rms producing an rf R B rms dl of 0:60 0:03 T mm. The EDDA detector [29,30] was used to measure the beam's polarization in COSY. We reduced its systematic The measured 1; 1 vector polarization, before spin manipulation, was about 63%.

V. EXPERIMENTAL TESTS
The deuteron circulation frequency in the COSY ring was f c 1:147 43 MHz at 1:850 GeV=c, where its Lorentz energy factor was 1:4046. With these parameters, Eq.
We experimentally determined f r and the resonance's full width at half maximum (FWHM) w by measuring the polarization, after running the rf dipole at different fixed frequencies near 917.0 kHz. In this study [36] was set at about 1:4 10 ÿ6 ; we obtained f r 916 992 10 Hz and w 23 2 Hz. This measured width w was dominated by the f r spread due to the beam's p=p since the resonance's natural width 2f c was only 3 Hz. We calibrated the strength of an rf-dipole-induced spin resonance against the rf dipole's voltage. This involved ramping the rf dipole's frequency through the resonance with various ramp times t while keeping the rf dipole's frequency range f and its voltage fixed; then we measured the final polarization after each frequency ramp. We found the strength by fitting these data to Eq. (3) with as a fit parameter [36].
To experimentally test the Chao formalism, we ramped the rf dipole's frequency over a range f, which started at f start and ended at f end near the rf resonance frequency f r , as shown in Fig. 1. After reaching f end , the rf dipole was turned off abruptly in a few s. We then measured the beam's final polarization. This procedure was repeated at different values of f end while holding fixed: the frequency ramp range at f 200 Hz, the ramp time at t 4 s, and the resonance strength at 9:56 10 ÿ6 . The measured polarization ratio, averaged for all spin states, is plotted against f end ÿ f r in Fig. 5. Predictions of the Chao formalism for 0 and 23 Hz FWHM Gaussian f r spreads and of the two-fluid model are shown in Fig. 5 by the dotted red, solid green, and dashed blue lines, respectively. To compare how well the different predictions agree with the data, we calculated 2 =N for each prediction. The 2 analysis included only the data's statistical errors and ignored systematic errors; thus, the 2 =N values were rather large. Note that the fits of all three predictions are extremely sensitive to the value of f r , as shown in Fig. 8(a). Thus, we chose f r 916 994 Hz, which simul-taneously minimizes the 2 =N for all three predictions and is certainly consistent with the 916 992 10 Hz measured earlier [36]. The 2 analysis in Fig. 5 strongly favors the Chao formalism prediction for the measured 23 Hz FWHM f r spread.
We next did a similar study with the rf dipole's ramp time t set at 0.2 s. This faster ramp resulted in an onlypartial spin flip when the resonance was fully crossed. These data are shown in Fig. 6, which also shows the Chao formalism and two-fluid model predictions.
A blowup of the region in Fig. 6, where the oscillations were expected, is shown in Fig. 7. Most data points fall almost exactly on top of the green solid line, supporting the validity of the 23 Hz prediction of the Chao formalism. With no f r spread, the Chao formalism predicts largeamplitude oscillations of the polarization. However, the f r spread smoothes these oscillations leaving only a small wiggle in the predicted polarization. Note that we chose f r 916:987 kHz because, as shown in Fig. 8, it was the only f r value, which gave a 2 =N minimum below 100 for any of the three predictions for either Fig. 6 [see Fig. 8 To further improve the Chao test, we made a prediction for the study shown in Fig. 1 using a 4 times faster crossing rate than in Figs. 6 and 7. This t 100 ms prediction is shown in Fig. 9; note that folding in the 23 Hz FWHM Gaussian now only partly smoothes the oscillations in Fig. 9 because their period is now longer due to the faster crossing rate. For the 23 Hz FWHM f r spread, the predicted maximum peak-to-peak amplitude of the oscilla-      tions is about 15%; this should allow a more convincing test of the Chao formalism. We plan to soon test this prediction at COSY.

VI. SUMMARY
We used 1:85 GeV=c vertically polarized deuterons stored in COSY to experimentally test the recently proposed Chao matrix formalism for describing the spin dynamics during crossing of an isolated spin resonance in a synchrotron. The Chao formalism allows predictions for experiments where the Froissart-Stora formula is not valid. We conducted such an experiment at COSY by ramping an rf dipole's frequency through a range ending near a spin resonance; both the frequency range and ramp time were fixed while we varied the ramp's start and therefore end frequencies. We compared our experimental data with the predictions of the Chao formalism and of the two-fluid model. Our data strongly favor the validity of the Chao formalism for the measured [36] resonance width of 23 Hz.