Matrix formalism for spin dynamics near a single depolarization resonance

A matrix formalism is developed to describe the spin dynamics in a synchrotron near a single depolarization resonance as the particle energy (and therefore its spin precession frequency) is varied in a prescribed pattern as a function of time such as during acceleration. This formalism is ﬁrst applied to the case of crossing the resonance with a constant crossing speed and a ﬁnite total step size, and then applied also to other more involved cases when the single resonance is crossed repeatedly in a prescribed manner consisting of linear ramping segments or sudden jumps. How repeated crossings produce an interference behavior is discussed using the results obtained. For a polarized beam with ﬁnite energy spread, a spin echo experiment is suggested to explore this interference effect.


I. INTRODUCTION
Dynamics of spin motion in a circular accelerator, such as a synchrotron or a storage ring, is conveniently described using a spinor algebra technique [1][2][3][4][5][6]. If the accelerator has a planar geometry with its nominal bending magnetic field in the vertical y direction, the particle spin will precess rapidly around theŷ axis. This nominal spin precession frequency is equal to G times the particle's orbital revolution frequency, where G is a fundamental property of the particle given by G g ÿ 2=2 with g the gyromaetgnic ratio of the particle, and is the particle's energy in units of its rest mass. The quantity G is called the spin tune of the particle. In the case of a synchrotron, as the particle is being accelerated by a constant rate, its spin tune increases linearly with time.
Spin motion is strongly affected if the orbital motion of the particle in the circular accelerator experiences field deviations from the nominal bending fields, and if its spin tune is near a depolarization resonance G (1) where specifies the resonance location (for example, integer for imperfection resonances, integer vertical betatron tune for intrinsic resonances, and higher betatron tune side bands for higher order depolarization resonances). In this situation, the perturbation on spin motion can be characterized by a single complex quantity, the resonance strength , which can be expressed as an appropriate Fourier harmonic of the field deviations around the accelerator [1][2][3][4]6,7]. Given the depolarization resonance strength , as will be assumed in this paper, a question arises as to how the spin motion depends on the pattern in which the spin tune is varied with time in the neighborhood of condition (1). For example, one would be interested in knowing what happens to the spin as the particle is accelerated through a depolarization resonance.
One such analysis has been performed by Froissart and Stora [7], yielding the well-known Froissart-Stora formula that relates the spin polarizations after and before crossing the resonance. The pattern assumed in the Froissart-Stora formula is that of a linear rise of spin tune from ÿ1 to 1 with a given constant rate of change.
In practice and in experimental studies of spin dynamics [8], sometimes it would be useful to calculate other patterns of resonance crossing. One example is to start with a spin tune a finite distance (instead of ÿ1) below the resonance, cross the resonance linearly with time, and then park the spin tune a finite distance (instead of 1) above the resonance after crossing. Another example could be crossing the resonance back and forth multiple times linearly in time, and park the spin tune at some location after the process. Still another possibility is to cross the resonance by a sudden jump of spin tune. To study these patterns will require an extension of the Froissart-Stora analysis. In this paper, we offer a matrix formalism based on spinor algebra that allows calculation for spin tune patterns that consist of an arbitrary combination of piecewise constant, piecewise linear, and sudden jumps. It is suggested that this matrix formalism can be used to treat practical experimental conditions beyond the Froissart-Stora case. Cases thus made calculable offer a range of potentially interesting, and sometimes unexpected, behavior to be studied experimentally.
Note that spin dynamics of a sudden spin tune jump and that of a linear ramping of spin tune over a finite step have both been available in the literature [1][2][3][4]6,7]. The matrix formalism proposed here allows combining these results for various practical applications.
Several example applications will be described in subsequent sections. In applications that include multiple resonance crossings, we will explore the constructive and destructive interference effects of these crossings. A condition for destructive interference between two crossings, for example, might be useful if one wishes to compensate a particularly strong depolarization resonance by another artificially induced resonance.
Experimentally, these interference effects are expected to be most readily observable when the polarized beam has a small energy spread. Ways to produce a beam with small energy spread should help greatly the exploration of these effects. For a polarized beam of larger energy spreads, these effects are not readily observable but an echo experiment is suggested. In this echo experiment, two resonance crossings normally considered to be far separated can still interference with each other to produce an echo at an unexpected long time after the crossings.

II. EQUATION OF MOTION
We assume the spin dynamics is determined by one and only one depolarization resonance of strength at G . We assume G is near the resonance and its distance to the resonance is a prescribed function of time. In particular, it may occur that the spin tune crosses the resonance with some prescribed manner.
In spinor notation, the spin state of a particle is described by a two-component complex spinor , and the spin dynamics is described by where is the time variable (advancing by 2 per revolution of the particle around the circular accelerator), G depends on as where is prescribed by the way the resonance is approached or crossed. This spinor equation has implicitly assumed that and as functions of time vary slowly compared to spin precession. This is a good assumption because spin precession is fast. Spin diffusion effects are ignored.
We define Z where 0 refers to the initial time when the spin dynamics is launched, and make the transformation with Pauli matrices It can be shown that We then let to obtain where a prime designates d=d.
It follows from Eq. (8) that jfj 2 jgj 2 constant in time (9) because jfj 2 jgj 2 0 can be easily shown to vanish. The initial conditions f 0 and g 0 are normalized so that jf 0 j 2 jg 0 j 2 1. As a result, This condition holds even when and are functions of .
We will primarily be interested in P y , the y component of polarization, in this planar accelerator. This y component at any time is given by P y y y y 1 y 1 jfj 2 ÿ jgj 2 2jfj 2 ÿ 1: It is sometime useful to eliminate g from Eq. (8), which yields Note that, for the case when is constant in time, it is jj 2 that plays the role in determining the polarization; the phase of does not matter.
We still need to specify the initial conditions of the spin at time 0 (one could mentally set 0 to be a large negative value). Let us designate 0 0 and 0 0 . We assume that these parameters have been held at these values, and that the spin has been in an eigenstate, from ÿ1 until 0 . The initial spin eigenstate is given by f g The initial polarization of this eigenstate is which approaches 100% when j 0 j j 0 j as one would expect because the spin tune is far from the resonance.
Note that being in a pure eigenstate, the spin is initially 100% polarized although its y component jP y 0 j < 1. Indeed, one may compute the x and z components of the polarization to obtain and P 2 x 0 P 2 z 0 P 2 y 0 1.

III. CONSTANT AND
We are now ready to treat some special cases. Consider the case when and stay at constant values from time 1 to 2 , i.e., when The spin motion in this period can be written in a matrix form, where the map that brings the spin state from 1 to is designated by a matrix T 0 ; 0 ; 1 . In this simple case, it is given by solving Eq. (8), One can check that the normalization is preserved with this map, i.e., jfj 2 jgj 2 jf 1 j 2 jg 1 j 2 .
The special case when 0 0 gives Another special case occurs when m where m is an integer, i.e., when the accumulated spin precession angle ÿ 1 is an integral multiple of 2. Then Other than an overall phase factor, the spin motion in this period is a unit transformation. This period from 1 to is therefore spin transparent, and has no net effect on the spin motion. Still another special case is when m 1 2 , in which case we have

A. Application 1
As a first application of this case of constant and constant , consider a spin launched at time 0 with initial conditions (13), and let and be kept at their initial values 0 and 0 . The spin will stay in the eigenstate, with In this case, polarization in this time period is preserved, P y jfj 2 ÿ jgj 2 jf 0 j 2 ÿ jg 0 j 2 j 0 j : (24)

B. Application 2
As a second application, consider the case when the resonance strength stays constant at the value 0 , while the spin tune crosses the resonance at time 0 according to a tune-jump pattern, where the sign is such that the resonance is crossed from below when A > 0 and from above if A < 0. In order for the single-resonance approximation, i.e., Eq. (2), to hold, the sudden change of is assumed to take place over at least several spin precession periods. At the same time, it should be fast enough to qualify for a sudden jump. These conditions together require that the time during which the jump takes place, measured in number of turns n jump , should satisfy 1=G n jump 1=, where is to be defined in Eq. (27).
The spin state after crossing is given by When the initial conditions are specified by (13), then f g >0 e ÿi=2A 0 e ÿi=2sgnAÿ 0 jAj 2 where Again one can verify that the normalization jfj 2 jgj 2 1 holds. The polarization after the spin tune jump is found to be [1,9] This polarization P y oscillates in time around the mean value jAjA 2 ÿ j 0 j 2 =A 2 j 0 j 2 3=2 with frequency , and it oscillates between the maximum value jAj=A 2 j 0 j 2 1=2 and the minimum value jAjA 2 ÿ 3j 0 j 2 =A 2 j 0 j 2 3=2 , where the maximum value is actually equal to the initial polarization at time 0 . Note that although the final polarization depends on the observation time , it does not depend on the initial time 0 because the spin was launched in its eigenstate.

C. Application 3
As a third application, consider an experiment that turns off the resonance after a tune jump, The spin state after switching off the resonance is f g One can then calculate the polarization P y > 1 and find that it does not depend on after resonance is switched off, and is given by Eq. (28) with set to 1 , as one might expect.

D. Application 4
In this application, we keep the resonance strength 0 fixed, but make a spin tune jump across the resonance at time 0, and jump across the resonance back again at time 1 > 0, i.e.,

<
: and as seen in Fig. 1.
This example is of interest to study the interference effect of two resonance crossings. In general, one may consider crossing two nearby resonances in one tune ramping process. In this case, we try to gain some insight by considering crossing a single resonance twice. In this example, the crossings are performed by sudden tune jumps forward and backward, and the two crossings are separated in time by 1 .
The spin state after the second crossing, > 1 , is given by f g Polarization is then found to be, after some algebra, The second and third terms of Eq. (32) combine to give an oscillatory contribution P osc of the polarization after the double spin tune jumps. The oscillation frequency is . The average polarization P av is given by the first term of Eq. (32). We see that P osc and P av depend on 1 , the time separation between the two jumps.

E. Interference and spin echo
Here we continue the previous application and explore the interference between the two jump crossings and suggest an echo phenomenon to be explored experimentally.
According to Eq. (32), a destructive interference between the two crossings will yield a maximum average polarization, which occurs with 1 even multiple of ; (33) P av jAj ; P osc 0: As mentioned earlier, Eq. (33) is the condition when the spin motion between the two spin tune jumps is transparent. When that happens, the two equal and opposite jumps have a complete destructive interference. The final polarization keeps its initial value, and there is no loss of polarization.
If the spin tune jumps not to A but to some other value A 0 for the time period between 0 and 1 , the condition for destructive interference still occurs at A 02 jj 2 p 1 2m. This is valid even if A 0 is negative, i.e., the resonance is not actually crossed.
A constructive interference yields a minimum average polarization, which occurs with 1 odd multiple of ; (34) P av jAj 5 4 ÿ 8A 2 2 8A 4 ; P osc 4jAj 5 2 ÿ A 2 2 ÿ 2A 2 cos: It should be pointed out that this interference behavior is independent of whether jAj is small or large compared with jj. It also does not matter whether 1 is small or large as long as 1 is an even or odd multiple of . In other words, at least in principle, after crossing a resonance, the potential of interfering strongly with a next resonance crossing lasts indefinitely in time, and occurs even when the resonance is crossed ''cleanly'' in each crossing (meaning jAj jj). In an acceleration process, for example, it is not how long ago a resonance has been crossed or how far the spin tune has passed the resonance that determines whether this resonance crossing can be considered an isolated event. The effect of crossing a resonance carries an infinitely long memory.
The above result applies to the case of a single particle. For a beam of particles with, for example, a finite energy spread among the particles, an averaging on our results over the beam's energy distribution will have to be carried out. It is then expected that some of the interference features we observe here will be diluted [10].
More specifically for the double-jump application here, the polarization for an off-energy particle after the first jump will be determined by f g 1 >>0 where = is the relative energy deviation of the particle, while after the second jump, we have f g The beam polarization will then be obtained by integrating the single-particle polarization over the distribution of the beam. Figure 2 is an illustration of the transition from a single particle to a beam. Parameters used are j 0 j 0:08, G 4, A 0:1, and 1 16. The red curve shows the polarization P y as a function of for a single on-momentum particle. It exhibits the case of a complete destructive interference. The green and blue curves are what happens to a beam with Gaussian energy distribution around the on-momentum particle with rms 0:1% and 0.3%, respectively.
For the blue curve, one sees that the polarization after each jump settles down to some equilibrium value, and each jump reduces the polarization level by a factor A 2 ÿ j 0 j 2 =A 2 j 0 j 2 0:22, which is what is expected if each jump is to be regarded as an isolated event and if the oscillatory term of polarization after a jump is ignored [keeping only the first term in Eq. (28)]. After each jump, the memory of the jump is kept for a time which for the parameters of the blue curve gives memory 300. After a time of memory , the jump can be considered an isolated event.
One should, however, keep in mind that individual particles will remember the jump much longer. This effect provides the possibility of echo phenomena to be explored in a circular accelerator. In the above example, one manifestation of echo shows up as a secondary excitation around 2 1 785. Note that there is an echo even when 1 is much larger than memory .
To show the echo more clearly, Fig. 3 gives another case when the energy spread is increased to 0:8%, 1 is extended to 1 41 (it does not matter whether the interference is nominally constructive or destructive), and A increased to 0.15. The three vertical black lines indicate the locations of 0; 1 ; 2 1 . One sees that, although the large energy spread of the beam has apparently completely suppressed any interference effect, there remains an echo of polarization near time 2 1 .

IV. CONSTANT , LINEAR
In this case, we consider a resonance strength that is constant in time, while the spin tune has a linear dependence, with ÿ specifying the rate of change of the spin tune (ÿ can have either sign) and c specifying the time when spin tune crosses the resonance (by linear extrapolation if resonance is not actually crossed). Solution of spin motion for this linear ramping of spin tune over a finite time period, leading to Eqs. (51) and (52), has been obtained in [3,4,6,7], and Ref. [4] has expressed the result in a matrix form. This result is nevertheless reproduced below for completeness, and will be used as part of the matrix formalism to be developed.
We consider the time range over which Eq. (38) applies to be from 1 to 2 . Between 1 and 2 , we have from Eq. (12) Make a transformation f Fe ÿiÿ=4ÿ c 2 and change variable from to then we obtain where a i 2 sgnÿ ÿ jj 2 4jÿj : Note that in this case only the absolute value jj enters. (The phase of matters only as an overall phase in determining g, which does not have physical consequences.) Given F, polarization is given by P y 2jFj 2 ÿ 1.
Two independent solutions for F can be expressed in terms of the degenerate hypergeometric function 1 F 1 [3,4,7,11], It is useful here to give explicit expressions also for their derivatives, It is easy to show that the Wronskian is a constant in x, and is equal to 1.

The general solution for F is
where C 1;2 are determined by the initial conditions. Given f 1 and g 1 at initial time 1 , we have the initial conditions, in terms of the x variable, We then have, now expressed in matrix form, For time 2 > > 1 , we have The outer two matrices of the above product bring the f; g representation to F, dF=dx, while the middle two matrices describe the dynamics of F, dF=dx. When 1 , we have U 1.

A. Application 5
As an application of the case with linear ramping in spin tune, let us start with initial condition (13) at time 0 , and cross the resonance in a sloped-step pattern, (53) as shown in Fig. 4.
The crossing speed is specified by the slope ÿ. The spin tune swings from ÿA to A linearly. We will consider ÿ and A to have the same sign. When A > 0, ÿ > 0, the resonance is crossed from below as shown in Fig. 4. When A < 0, ÿ < 0, the resonance is crossed from above. The following analysis applies to both cases. The resonance strength has a constant value throughout the process. During the crossing, 2 > > 1 , the spin state is given by f g We now define four quantities Since y 1 x and dy 2 =dxx are even functions of x, y 2 x and dy 1 =dxx are odd functions of x, and x 1 ÿx 2 , we have After some algebra, we find by using Eq. (56) that, for 2 > > 1 , the polarization is given by where y 1;2 are evaluated at x jÿj p and A 2 jj 2 p .
We then proceed to calculate the spin state after crossing, > 2 , f g After some algebra, we find jÿj p ; where A is the normalized step size, is the normalized resonance width.
The red curves in Fig. 6 are the mean, the upper bound, and the lower bound values of the final polarization as functions of for four cases A 0:5 (upper left), 1 (upper right), 2 (lower left), and 10 (lower right). The green curves are the Froissart-Stora formula [7] P final;FS 2e ÿ2 2 ÿ 1: The Froissart-Stora formula of course does not depend on A. The green curves in all four subfigures are therefore identical. Naturally, the Froissart-Stora prediction is accurate when A such as the lower right subfigure shows, and it becomes rather inaccurate when A becomes comparable or less than as seen, e.g., in the upper left subfigure.
In an actual spin resonance crossing, it is therefore necessary to confirm first whether the experimental conditions are such that the Froissart-Stora formula is applicable. With the inclusion of the finite step size, our result offers the following observations: (i) Polarization after crossing differs from Froissart-Stora formula when jAj & jj.
(ii) In general, the final polarization will depend on the time of observation. The oscillation frequency after crossing , if can be measured, offers another way to measure jj.
(iii) A complete spin flip for large , as predicted by Froissart-Stora formula is modified in such a way that the spin flip efficiency will start to decrease if approaches A [12].
It might be useful to experimentally test some of these observations beyond the Froissart-Stora regime. One curiosity occurs when the final polarization does not oscillate under some special conditions of parameters. This occurs when D 1 iD 2 and therefore P osc 0. From  Fig. 6, one sees an example when A 1 and 0:915 where a node point appears. Indeed, a numerical calculation reconfirms this observation.

C. Application 6: More on interference
As the next application, let us cross the resonance twice in a back and forth pattern shown in Fig. 7, We shall try to investigate the interference behavior between the two crossings. We consider the case A > 0, ÿ > 0.
Spin state after crossing ( > 2A=ÿ) is given by f g As before, polarization after the two resonance crossings, given by 2jfj 2 ÿ 1, oscillates in time with frequency A 2 jj 2 p around a mean value. It is found to be given by Eq.    crossings. One notices the quasiperiodic behavior of the polarization in both and A. To explore the condition for the interference pattern, consider the spin precession angle accumulated between the two resonance crossings. The exact value of has to be found by the hypergeometric functions as our results indicate, but as an approximation, we write Z A=ÿ We then make the following observations from Figs. 8 and 9: (i) Fig. 8 shows the occurrence of nodes when the final polarization ceases to oscillate in time. Following the discussion in Eq. (33), one might anticipate a destructive interference to occur when is equal to an even multiple of .
This conclusion agrees with the figure. It is further noted that when a destructive interference occurs, not only the final polarization ceases to oscillate, but also it is equal to the initial eigenstate polarization P y A= . There is therefore no net depolarization in the double resonance crossings.

(ii) When
A , it seems from Fig. 9 that destructive nodes occur when is equal to an even multiple of although with an apparent small shift that depends on , (iii) When a constructive interference occurs, one might expect a maximum loss of polarization that to some degree resembles crossing a single resonance of strength 2. This indeed seems to be the case when 1 < A, as seen in Fig. 9 with 0:1. The minimum level of polarization is approximately equal to the Froissart-Stora prediction of 2e ÿ22 2 ÿ 1 0:55. However, this expectation breaks down when is increased. In fact, when is increased to become comparable or larger than 1, the polarization increases contrary to expectation and becomes close to 100% as seen, for example, in the case 1, indicating the two crossings tend to cancel each other completely regardless of the value of A, i.e., regardless of how far apart the two crossings might be intended.
It is clear from these observations that the interference behavior of resonance crossings contains some surprises, and exhibits a potentially rich area to be explored.

D. Application 7
As the next application, we cross the resonance three times, twice in the same direction, and once by a jump, as shown in Fig. 10, We again assume A > 0, ÿ > 0. Spin state after crossing ( > 2A=ÿ) is given by f g The final polarization again oscillates according to Eq. (61), where D 1;2 are given by Eq. (67) but the matrix M of Eq. (68) is replaced by where Y 0 s are defined as before. The first and the third matrices on the right-hand side describe the spin motion during the two linear ramping periods, while the middle matrix is a result of the spin tune jump. Figures 11 and 12 are the same as Figs. 8 and 9 earlier for the present crossing pattern. Figure 11 is polarization as a function of for A 0:5; 1; 2; 10. Figure 12 shows the polarization as a function of A for 0:1; 0:5; 1. Note that the interference pattern is not as pronounced in this case because, unlike the previous case, there are three resonance crossings interplaying. It is conceivable that by replacing the middle tune-jump matrix in Eq. (74) by another matrix reflecting a redefinition of the spin state, one could extend the applicability of our result approximately to the case of crossing two separate resonances whose distance between them is much larger than their resonance strengths. This direction is not explored in this report.

V. SUMMARY
We have developed a matrix formalism to treat spin dynamics near a single depolarization resonance in a synchrotron. It is applicable to the following operations of the synchrotron: (i) constant spin tune, constant resonance strength; (ii) sudden jumps in either the spin tune or the resonance strength; (iii) constant resonance strength, linearly varying spin tune crossing or not crossing the resonance; (iv) any piecewise combination of the above, including repeated crossing of the resonance, each crossing having its own crossing speed. Once a ramping procedure is given in terms of the above piecewise segments, a sequential multiplication of matrices, either in the form of T ; 1 ; 2 of Eq. (18) or in the form of U ÿ; c ; 1 ; 2 in Eq. (52), will give the spin state at any time .
This matrix formalism can be applied to various experimental conditions such as to extend the Froissart-Stora analysis to that with a finite spin tune change and to study interference effects of repeated resonance crossings. For a polarized beam with large energy spread, an echo phenomenon is suggested as a way to study these interference effects.
In the appendix, the matrix formalism is summarized in a somewhat simplified form for the case of a single particle.