On the Evolution of Ion Bunch Profile in the Presence of Longitudinal Coherent Electron Cooling

In the presence of longitudinal coherent electron cooling, the evolution of the line-density profile of a circulating ion bunch can be described by the 1-D Fokker-Planck equation. We show that, in the absence of diffusion, the 1-D equation can be solved analytically for certain dependence of cooling force on the synchrotron amplitude. For more general cases with arbitrary diffusion, we solved the 1-D Fokker-Planck equation numerically and the numerical solutions have been compared with results from macro-particle tracking.


I. INTRODUCTION
The future electron-ion collider demands a strong hadron cooling technique to reach the luminosity level where all the relevant physics can be fully covered. As a potential candidate to provide such a cooling technique, the concept of Coherent electron Cooling (CeC) and its variants have been extensively investigated [1][2][3][4][5]. Both analytical and simulation tools have been developed to predict the ion bunch evolution in the presence of CeC, which is essential for diagnosing as well as optimizing the cooling system [6,7]. While simulation through macro-particle tracking is the most straightforward approach in predicting the ion bunch evolution under cooling, analytical tools are needed in both benchmarking the simulation code and providing a fast estimate for the ion bunch profile.
In case that the system has only longitudinal cooling, the evolution of the ion bunch under cooling can be described by the 1-D Fokker-Planck equation. The analytical tools are being developed to solve the 1-D Fokker-Planck equation with given cooling rate and diffusion coefficient. We review the 1-D Fokker-Planck equation and its equilibrium solutions in section II. In section III, we derived an analytical solution of the 1-D Fokker-Planck equation for a specific dependence of the cooling rate on synchrotron oscillation amplitude, in the limit of vanishing diffusion coefficient. Section IV contains our approach of numerically solving the 1-D Fokker-Planck equation for finite diffusion coefficient and arbitrary dependence of the cooling rate on synchrotron oscillation amplitude. It is shown that the numerical solution agrees well with the analytical solution at the proper limits taken by the latter. In section V, we present the numerical solution for parameters of the proof of CeC principle experiment and compare it with that obtained from macroparticle tracking in section VI. We summarize in section VII.

II. 1-D FOKKER-PLANCK EQUATION
In the cooling section of a CeC system, a circulating ion sees a coherent energy kick induced by itself to correct its energy error as well as a random energy kick induced by its neighbours (electrons and ions). In addition, the circulating ions also get random kicks from the Intra-beam * gawang@bnl.gov scattering (IBS). In the presence of the cooling force and the random diffusive kicks, the evolution of the longitudinal phase space density is described by the 1-D Fokker-Planck equation [8,9] where   , F I t is the longitudinal phase space density averaged over one synchrotron oscillation, I is the amplitude of synchrotron oscillation,   are the cooling rate and the diffusion coefficient averaged over one synchrotron oscillation. The diffusion coefficients are to be calculated from the summation of all random kicks. Throughout the context, we assume that the ion bunch length is much smaller than the fundamental wavelength of the RF cavities and consequently we can apply small amplitude approximation for the synchrotron oscillation. The action-angle variables under this approximation are given by [10] 2 cos P I w  (2) and 2 sin I w where  is the RF phase and is the normalized energy deviation of the ion. The unperturbed motion of the ions can be derived from the Hamiltonian and the distribution function,   , F I t , satisfies the following relation with N being the total number of ions in the bunch. It follows from eq. (2), eq. (3) and eq. (6) that and hence the distribution function in canonical variable,   , P  , is given by At equilibrium, the first term at the L.H.S. of eq. (1) vanishes and we obtain the following solution for the equilibrium distribution of the ion beam:

III. ANALYTICAL SOLUTIONS IN THE LIMIT OF ZERO DIFFUSION
and     Before proceeding any further, we consider a case when the coefficient   Comparing eq. (17) with eq. (10) yields and Solving eq. (19) generates and the solution of eq. (18) is with C given by eq. (22). At 0 t t  , the solution must satisfy a given initial condition and imposing the initial condition to eq. (23) leads to For any value of eq. (25) requires Inserting eq. (23) and (28) into eq. (11) yields the solution of eq. (10) which satisfy the initial condition of eq. (24) The solution of eq. (32) reads where the product logarithm function, i.e.
Inserting eq. (33) into eq. (29) generates We take the initial ion distribution as 2 e I is determined by the bunch length of the electron beam which can be significantly shorter than that of the ion beam (see eq. (42)). For the proof of CeC principle experiment at RHIC, the electron bunch is more than two orders of magnitude shorter than the ion bunch.
where ion I is a parameter determined by the longitudinal emittance of the ion bunch. Inserting eq. (26) and eq. (31) into eq. (22), and taking 0 0 t  lead to Making use of eq. (37) and (38), eq. (36) becomes Using eq. (8), the line number density profile of the ion beam is then given by the following expression where z is the longitudinal location along the ion bunch, e l is one half of electron full bunch with ,max e  being the maximal RF phase for an ion with synchrotron oscillation action of e I . From eq. (37), we also obtained the initial RMS bunch length of ion beam as and the initial RMS energy spread of the ion beam as As an example of applying eq. (41), Fig. 1 shows line density of an ion bunch after being cooled by a short electron bunch sitting in its center with cooling rate profile given by eq. 10  , a dense core appears around the ion bunch center. Since the local blip appears much faster than the overall reduction of the longitudinal emittance, it could potentially be used as a diagnostic tool for optimizing cooling in commission a CeC system. However, the formation of the core is both due to the localized cooling as well as the absence of a mechanism to drive the particles out of the dense region. As we will see in the next section, the core is smoothed out for non-zero diffusion coefficient since the diffusive kicks tend to move particle out of the dense region.

IV. NUMERICAL SOLUTIONS FOR FINITE DIFFUSION
In the presence of non-zero diffusion coefficient, finding analytical solution of eq. (10) is usually difficult and numerical approach is pursued. Using the following definitions of normalized variables: eq. (1) can be re-written as where and The difference equation derived from eq. (46) reads for 2 j N   with N being the index of the last bin in the grid of r , for j N  . In deriving eq. (51) and (52), the following boundary conditions are applied: Numerical solution of eq. (50)-(52) is obtained by applying the subroutine, TRIDAG, from Numerical Recipes [11]. After obtaining the phase space density,   , R r t , the line number density of the ion bunch is given by and to compare with analytical result in eq. (41), we take The numerical method is applicable to arbitrary dependence of diffusion coefficient on synchrotron oscillation amplitude. In fig. 2 (magenta dots), to illustrate the effects of diffusion, we take with 0 100 D  as an example. As shown in Fig.2, the numerical solution (blue dashes) reproduces the analytical result (red solid) in the absence of diffusion. For non-vanishing diffusion coefficient (magenta dots), however, the central blip is smoothed out. As we will see in the next section, the smoothing effects due to diffusion does not rely on the specific form of cooling rate and diffusion coefficient, i.e. eq. (58), and similar influences of diffusion to the ion bunch profile are observed for a more realistic cooling profile derived for the FEL-based CeC simulation.

V. EVALUATION FOR FEL-BASED COHERENT ELECTRON COOLING
The assumed dependence of cooling rate on synchrotron oscillation amplitude in eq. (30) is convenient for obtaining analytical solution, eq. (39), as well as validating the numerical method of solving eq. (50)-(52). In practice, however, the dependences of both the cooling rate and diffusion coefficient on synchrotron amplitude, after averaged over one synchrotron oscillation period, should be determined by the specific cooling scheme such as the longitudinal profile of the electron bunch, whether the electron bunch is painted around the ion bunch, and how cooling force and diffusive kick depend on local electron properties. The single pass energy kick received by an ion as it travels through the storage ring is given by sin j n j n j n j n ion j n e j n IBS j n The local cooling time, 0 T , and its inversion, the local cooling rate, 0  , are related to the coherent energy kick, g  , by For each circulation around the ring, the reduction of the ion's longitudinal oscillation action due to cooling is given by where into eq. (65) and using eq. (61) give where rev T is the revolution period of the ion and we use eq. (2) in deriving the second equation of (68 and for 1 r  , we used the following relation to derive eq. (70) 2 72) and (73), we assume that the center of the electron bunch is located at e z , the ion beam line density does not vary significantly over the range of the cooling electron beam, and the electron beam has uniform line density. The last term of the R.H.S. of eq. (60) is responsible for the accumulated energy kicks due to Intra-beam scattering (IBS) while ions traveling through the ring. , j n Z is a random number uniformly distributed from -1 to 1 and the kick strength, IBS d , is to be determined by the IBS growth rate as obtained from Piwinski's formulas [12] (APPENDIX B).
For an ion with synchrotron oscillation action, I , a random energy kick in its th i circulation, Using eq. (71), the diffusion coefficient in the Fokker-Planck equation, eq. (1), is given by [13]       The normalized diffusion coefficient, where in deriving eq. (78), we used eqs. (43) With the help from eq. (76) and the following relation  I  I  IBS  ion  ion   I  I  D  I  e  I  I  I  I Fig. 4 (right) shows the ion bunch current profile around the bunch center after ~1 minute of cooling with the nominal (Green) and artificially reduced diffusion coefficients (Blue, Magenta and Black). As shown in fig. 5 (right), the local blip is fully suppressed for the nominal parameters and starts to be visible if the normalized diffusion coefficient is reduced by two orders of magnitudes or more. According to eqs. (72), (73) and (81), the normalized diffusion coefficients due to CeC, 0,cec D , decreases linearly with g  and one way to reduce diffusion in the CeC process is to reduce selfinduced energy kick, g  . However, the local cooling time, 0 T , increases proportionally with 1 g   and hence the normalized maximal diffusion coefficient due to IBS, 0,IBS D , increases linearly with 1 g   . In addition, increasing 0 T will also make the overall process slower, defeating the purpose of using the blip as a fast-diagnostic signal.  Tables 1  and 2. The cooling rate and diffusion coefficients are calculated from eqs. (69) and (92). The red curve shows the initial ion bunch current profile and the green curve shows the ion bunch profile after 1 minute of cooling. The blue, magenta and black curve shows the ion bunch profiles after 1 minute of cooling with diffusion coefficient reduced by a factor of 10, 100 and 1000 from the nominal values (green). The left plot shows the overall bunch profile and the right plot shows the zoomed-in region around the bunch center.

VI. MACRO-PARTICLE TRACKING
The numerical method of solving Fokker-Planck equation as developed in section IV has the advantages of requiring minimal computational time (a few minutes in a pc) and resources. However, its applicability is limited to the scheme where the cooling rate and diffusion coefficient do not change significantly throughout the process, the ion bunch length is small so that the Hamiltonian of eq. (5) is accurate and the cooling force is proportional to the energy deviation of the particle. To cross-check the results obtained in section V as well as to provide a more versatile tool for general cooling scheme, we have developed a macro-particle tracking code. As shown in fig. 6, typically 0.2~6 millions of macro-ions are generated when the simulation starts. The longitudinal coordinates of each macro-ion are then updated according to the rf voltage it sees and the phase slip factor of the lattice: where q is the charge of the ion, m is the mass of the ion, c is speed of light,  is the energy deviation of the macro-ion in unit of where x  and y  are the one turn phase advances of horizontal and vertical betatron motion. A random 3-D Langevin kick is applied to each macro-ion every turn to account for effects due to IBS. The R.M.S. amplitude of the kick is determined by the growth rate as calculated from the Piwinski's formula (APPENDIX B). Local ion line density is used in the IBS growth rate calculations.
To implement the one-turn update due to CeC, we first estimate how the ions are mixed from turn to turn by synchrotron oscillation. The synchrotron period for the CeC experiment is about 4000 revolutions. With the RMS ion bunch length of 3 ns, the average longitudinal slippage of a typical ion in one revolution is 1 4 3 0.9 4000 ns c mm which is ~30 times larger than the optical wavelength of the FEL amplifier (30.5 μm). Consequently, no phase information is preserved after one revolution and the incoherent kicks due to neighbor ions (and cooling electrons) can be implemented as a random Langevin kick as shown in eq. (60).
As shown in Fig. 7, the prediction from solving Fokker-Planck equation (red-dash) agrees very well with that obtained from macro-particle tracking if linear cooling force is adopted in the tracking (magenta). It is worth noting that the peak current from solving Fokker-Planck equation (red-dash) is about 1~2% higher than that obtained from tracking with linear cooling force (magenta), which is likely due to the static diffusion coefficients assumed in the Fokker-Planck equation, while the tracking algorithm calculates diffusive kick from the updated bunch profile. Fig. 7 also shows that the cooling effect is less pronounced if more realistic Sinusoidal cooling force is applied in the tracking, which is to be expected as ions with large synchrotron amplitude get reduced cooling or even anti-cooling during the process. (50)-(52) (red) and through macro-particle tracking (green and magenta). The green curve shows the tracking results with sinusoidal cooling force as described by eq. (60), while the magenta curve shows the tracking results with linear cooling force given by eq. (63).
The technique described above are also used to study the tolerance of a CeC system to the noise level in the electron beam. Comparing the results from macro-particle tracking with that from numerically solving Fokker-Planck equation is illustrative for the limitation of the latter. Fig. 8 shows the current profiles of an ion bunch after 40 minutes of cooling by electrons with various noise levels. Parameters in tables 1 and 2 are used in generating fig. 8. Fig. 8 (a) is obtained by numerically solving eq. (50)-(52) and fig. 8 (b) is the results of macro-particle tracking. While results from both approaches predict that heating due to diffusion dominates the CeC process once the noise in the electron beam is more than a factor of 3 higher than its natural noise level, the peak currents of cooled bunch in fig. 8 (b) are all lower than that shown in fig. 8 (a) due to the approximation of linear cooling force applied in the Fokker-Planck solver. On the other hand, the witness bunch (the black curve) in fig. 8 (b) has slightly higher peak current than that in fig. 8 (a) since the IBS rate decreases with the peak current but in Fokker-Planck equation, the diffusion coefficient does not vary with time. Moreover, the number of particles is conserved in fig. 8 (a) as a result of using small amplitude Hamiltonian, i.e. eq. (5), in the Fokker-Planck equation. In tracking, particles are lost once they move out of the RF bucket and consequently bunch intensity reduces once the bunch length becomes comparable with the RF wavelength as shown in fig. 8(b).

VII. DISCUSSION AND SUMMARY
In summary, we have developed a set of tools to predict the ion bunch profile in the presence of longitudinal coherent electron cooling, which include an analytical expression for vanishing diffusion coefficient, a numerical algorithm for solving Fokker-Planck equation with linear cooling force and arbitrary dependence of cooling rate and diffusion coefficient on synchrotron oscillation action, and a macro-particle tracking code for arbitrary cooling force. In their applicable regime, we have found good agreement among these tools and used them to predict the performance of the CeC system in the proof of CeC principle experiment.
One of the insights achieved from these studies is that, for the proof of the CeC principle experiment, the ion bunch profile will not show any local blip during the cooling process due to IBS and diffusion induced by neighbor particles in the cooling section. Since any stochastic cooling mechanism, including CeC, inevitably introduces diffusive kicks due to neighbor ions, it limits the feasibility to use the high-frequency components in the beam current profile as a diagnostic tool for optimizing cooling.