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We studied the feasibility of a hard x-ray FEL oscillator (XFELO) based on a 3 to 4 GeV storage ring considered for the low-emittance upgrade of NSLS-II. We present a more detailed derivation of a formula for the small-gain gain calculation for 3 GeV XFELO published in the proceedings of IPAC'21 [1]. We modified the small-signal low-gain formula developed by K.J. Kim, et.al. [4{6] so that the gain can be derived without taking the \no focusing approximation" and a strong focusing can be applied. In this formula, the gain is cast in the form of a product of two factors with one of them depending only on the harmonic number, undulator period, and gap. Using this factor, we show that it is favorable to use harmonic lasing to achieve hard x-ray FEL working in the small-signal low-gain regime with the medium-energy electron beam (3-4 GeV). Our formula also allows FEL optimization by varying the vertical gradient of the undulator, the vertical dispersion, and the horizontal and vertical focusing, independently. Since a quite high peak current is required for the FEL, the collective effects of beam dynamics in medium-energy synchrotrons significantly affect the electron beam parameters. We carried out a multiple-parameter optimization taking collective effects into account and the result indicates the XFELO is feasible for storage ring energy as low as 3 GeV, with local correction of betatron coupling.

We present a set of exact system solutions to a model we developed to study wave function collapse in the quantum spin measurement process. Specifically, we calculated the wave function evolution for a simple harmonic oscillator of spin \frac{1}{2}, with its magnetic moment in interaction with a magnetic field, coupled to an environment that is a bath of harmonic oscillators. The system's time evolution is described by the direct product of two independent Hilbert spaces: one that is defined by an effective Hamiltonian, which represents a damped simple harmonic oscillator with its potential well divided into two, based on the spin and the other that represents the effect of the bath, i.e., the Brownian motion. The initial states of this set of wave functions form an orthonormal basis, defined as the eigenstates of the system. If the system is initially in one of these states, the final result is predetermined, i.e., the measurement is deterministic. If the bath is initially in the ground state,and the wave function is initially a wave packet at the origin, it collapses into one of the two potential wells depending on the initial spin. If the initial spin is a vector in the Bloch sphere not parallel to the magnetic field, the final distribution among the two potential wells is given by the Born rule applied to the initial spin state with the well-known ground state width. Hence, the result is also predetermined. We discuss its implications to the Bell theorem[1]. We end with a summary of the implications for the understanding of the statistical interpretation of quantum mechanics., Comment: 20 pages 6 gigures

To analyze nonlinear dynamic systems, we developed a new technique based on the square matrix method. We propose this technique called the \convergence map" for generating particle stability diagrams similar to the frequency maps widely used in accelerator physics to estimate dynamic aperture. The convergence map provides similar information as the frequency map but in a much shorter computing time. The dynamic equation can be rewritten in terms of action-angle variables provided by the square matrix derived from the accelerator lattice. The convergence map is obtained by solving the exact nonlinear equation iteratively by the perturbation method using Fourier transform and studying convergence. When the iteration is convergent, the solution is expressed as a quasi-periodic analytical function as a highly accurate approximation, and hence the motion is stable. The border of stable motion determines the dynamical aperture. As an example, we applied the new method to the nonlinear optimization of the NSLS-II storage ring and demonstrated a dynamic aperture comparable to or larger than the nominal one obtained by particle tracking. The computation speed of the convergence map is 30 to 300 times faster than the speed of the particle tracking, depending on the size of the ring lattice (number of superperiods). The computation speed ratio is larger for complex lattices with low symmetry, such as particle colliders.

The nonlinear dynamics of a system with periodic structure can be analyzed using a square matrix. We show that because the special property of the square matrix constructed for nonlinear dynamics, we can reduce the dimension of the matrix from the original large number for high order calculation to low dimension in the first step of the analysis. Then a stable Jordan decomposition is obtained with much lower dimension. The Jordan decomposition leads to a transformation to a new variable, which is an accurate action-angle variable, in good agreement with trajectories and tune obtained from tracking. And more importantly, the deviation from constancy of the new action-angle variable provides a measure of the stability of the phase space trajectories and tune fluctuation. Thus the square matrix theory shows a good potential in theoretical understanding of a complicated dynamical system to guide the optimization of dynamical apertures. The method is illustrated by many examples of comparison between theory and numerical simulation. In particular, we show that the square matrix method can be used for fast optimization to reduce the nonlinearity of a system.

The nonlinear dynamics of a system can be analyzed using a square matrix. If off resonance, the lead vector of a Jordan chain in a left eigenspace of the square matrix is an accurate action- angle variable for sufficiently high power order. The deviation from constancy of the action-angle variable provides a measure of the stability of a trajectory. However, near resonance or the stability boundary, the fluctuation increases rapidly and the lead vector no longer represents an accurate action-angle variable. In this paper we show that near resonance or stability boundary, it is possible to find a set of linear combinations of the vectors in the degenerate Jordan chains as the action-angle variables by an iteration procedure so that the fluctuation is minimized. Using the Henon-Heiles problem as an example on resonance, we show that when compared with conventional canonical perturbation theory, the iteration leads to result in much more close agreement with the forward integration, and the iteration is convergent even very close to the stability boundary. It is further shown that the action-angle variables found in the iteration can be used to find another action-angle variables even much more closer to rigid rotations (KAM invariants). The fast convergent result is not in the form of polynomials, it is an exponential function with a rational function in the exponent more similar to a Laurent series than a Taylor series. Hence the method provides a new way to probe the boundary of the region where KAM tori exist., Comment: 13 pages, 19 figures

Nonlinear dynamics has an important role when designing modern synchrotron lattices. In this letter, we introduce a new method of using a square matrix to analyze periodic nonlinear dynamical systems [1, 2]. Applying the method to the National Synchrotron Light Source II storage ring lattice has helped to mitigate the chaotic motion within its dynamic aperture. For a given dynamical system, the vector space of a square matrix can be separated into different low dimension invariant subspaces according to their eigenvalues. When Jordan decomposition is applied to one of the eigenspaces, it yields a set of accurate action-angle variables. The distortion of the new action-angle variables provides a measure of the nonlinearity. Our studies show that the common convention of confining the tune-shift with amplitude to avoid the crossing of resonance lines may not be absolutely necessary. We demonstrate that the third order resonance can be almost perfectly compensated with this technique. The method itself is general, and could be applied to other nonlinear systems., Comment: 5 pages, 5 figures

Applying a technique developed in a recent work[1] to calculate wavefunction evolution in a dissipative system with Ohmic friction, we show that the wavelength of the wavefunction decays exponentially, while the Brownian motion width gradually increases. In an interference experiment, when these two parameters become equal, the Brownian motion erases the fringes, the system thus approaches classical limit. We show that the wavelength decay is an observable phenomenon., Comment: 12 pages, 3 Postscript figures, uses standard latex

Applying a technique developed recently [1,2] for an harmonic oscillator coupled to a bath of harmonic oscillators, we present an exact solution for the tunneling problem in an Ohmic dissipative system with inverted harmonic potential. The result shows that while the dissipation tends to suppress the tunneling, the Brownian motion tends to enhance the tunneling. Whether the tunneling rate increases or not would then depend on the initial conditions. We give a specific formula to calculate the tunneling probability determined by various parameters and the initial conditions., Comment: The reference 2 is changed to Phys. Lett. from Phys. Rev. Lett

For a dissipative system with Ohmic friction, we obtain a simple and exact solution for the wave function of the system plus the bath. It is described by the direct product in two independent Hilbert space. One of them is described by an effective Hamiltonian, the other represents the effect of the bath, i.e., the Brownian motion, thus clarifying the structure of the wave function of the system whose energy is dissipated by its interaction with the bath. No path integral technology is needed in this treatment. The derivation of the Weisskopf-Wigner line width theory follows easily., Comment: 10 pages, BNL-49275