Twiss parameters and evolution of quantum harmonic-oscillator states

Dattoli, G.; Gallerano, G. P.; Mari, Carlo; Torre, A.; Richetta, M.

doi:10.1007/bf02727200

Cited by 4 publications

(7 citation statements)

References 12 publications

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“…The first set of parametrization schemes for MðtÞ was developed by Teng and Edwards [9][10][11] and Ripken [12][13][14], some of which have been adopted in lattice design and particle tracking codes, such as the MAD code [15,16]. A class of different parametrizations by directly generalizing the Twiss parameters to higher dimensions has also been developed by Dattoli et al [17][18][19]. However, in contrast to the original CS theory, these parametrization schemes are designed from mathematical considerations, and fail to connect with physical parameters of the beam.…”

Section: Introductionmentioning

confidence: 99%

Analytical methods for describing charged particle dynamics in general focusing lattices using generalized Courant-Snyder theory

Qin

Davidson

Burby

et al. 2014

Phys. Rev. ST Accel. Beams

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The dynamics of charged particles in general linear focusing lattices with quadrupole, skew-quadrupole, dipole, and solenoidal components, as well as torsion of the fiducial orbit and variation of beam energy is parametrized using a generalized Courant-Snyder (CS) theory, which extends the original CS theory for one degree of freedom to higher dimensions. The envelope function is generalized into an envelope matrix, and the phase advance is generalized into a 4D symplectic rotation, or a Uð2Þ element. The 1D envelope equation, also known as the Ermakov-Milne-Pinney equation in quantum mechanics, is generalized to an envelope matrix equation in higher dimensions. Other components of the original CS theory, such as the transfer matrix, Twiss functions, and CS invariant (also known as the Lewis invariant) all have their counterparts, with remarkably similar expressions, in the generalized theory. The gauge group structure of the generalized theory is analyzed. By fixing the gauge freedom with a desired symmetry, the generalized CS parametrization assumes the form of the modified Iwasawa decomposition, whose importance in phase space optics and phase space quantum mechanics has been recently realized. This gauge fixing also symmetrizes the generalized envelope equation and expresses the theory using only the generalized Twiss function β. The generalized phase advance completely determines the spectral and structural stability properties of a general focusing lattice. For structural stability, the generalized CS theory enables application of the Krein-Moser theory to greatly simplify the stability analysis. The generalized CS theory provides an effective tool to study coupled dynamics and to discover more optimized lattice designs in the larger parameter space of general focusing lattices.

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Section: Introductionmentioning

confidence: 99%

Analytical methods for describing charged particle dynamics in general focusing lattices using generalized Courant-Snyder theory

Qin

Davidson

Burby

et al. 2014

Phys. Rev. ST Accel. Beams

View full text Add to dashboard Cite

show abstract

“…An important generalization was constructed in 1992 by Dattoli et al [25][26][27], who posed the question: ''Can we adapt the coupled motion formalism to get a picture closer to the Courant-Snyder formulation?'' [27].…”

Section: Introduction and Theoretical Modelmentioning

confidence: 99%

Generalized Courant-Snyder theory for coupled transverse dynamics of charged particles in electromagnetic focusing lattices

Qin

Davidson

2009

Phys. Rev. ST Accel. Beams

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The Courant-Snyder theory gives a complete description of the uncoupled transverse dynamics of charged particles in electromagnetic focusing lattices. In this paper, the Courant-Snyder theory is generalized to the case of coupled transverse dynamics with two degrees of freedom. The generalized theory has the same structure as the original Courant-Snyder theory for one degree of freedom. The four basic components of the original Courant-Snyder theory, i.e., the envelope equation, phase advance, transfer matrix, and the Courant-Snyder invariant, all have their counterparts, with remarkably similar expressions, in the generalized theory presented here. In the generalized theory, the envelope function is generalized into an envelope matrix, and the envelope equation becomes a matrix envelope equation with matrix operations that are noncommutative. The generalized theory gives a new parametrization of the 4D symplectic transfer matrix that has the same structure as the parametrization of the 2D symplectic transfer matrix in the original Courant-Snyder theory. All of the parameters used in the generalized Courant-Snyder theory correspond to physical quantities of importance, and this parametrization can provide a valuable framework for accelerator design and particle simulation studies. A time-dependent canonical transformation is used to develop the generalized Courant-Snyder theory. Applications of the new theory to strongly and weakly coupled dynamics are given. It is shown that the stability of coupled dynamics can be determined by the generalized phase advance developed. Two stability criteria are given, which recover the known results about sum and difference resonances in the weakly coupled limit.

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“…The new method suggested by the above constructive proof of the existence of matched solution only requires solving Eq. (25) once with an arbitrary initial condition. We can construct the one-period map M(T) using any matched or unmatched solution of Eq.…”

Section: Stability Analysis and Matched Lattice Functionsmentioning

confidence: 99%

“…We can construct the one-period map M(T) using any matched or unmatched solution of Eq. (25), then the eigenvectors of M(T) can be calculated. When the set of bases satisfying Eqs.…”

Section: Stability Analysis and Matched Lattice Functionsmentioning

confidence: 99%

“…For coupled lattices, several parameterization schemes have been developed. [18][19][20][21][22][23][24][25][26] But none of these schemes is as effective for coupled lattices as the CS theory is for uncoupled lattices. Recently, we have developed a generalized CourantSnyder theory for coupled lattices, 17,[27][28][29][30][31][32][33][34] which generalizes every important aspect of the original CS theory to higher dimensions.…”

Section: Introductionmentioning

confidence: 99%

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Spectral and structural stability properties of charged particle dynamics in coupled latticesa)

et al. 2015

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It has been realized in recent years that coupled focusing lattices in accelerators and storage rings have significant advantages over conventional uncoupled focusing lattices, especially for high-intensity charged particle beams. A theoretical framework and associated tools for analyzing the spectral and structural stability properties of coupled lattices are formulated in this paper, based on the recently developed generalized Courant-Snyder theory for coupled lattices. It is shown that for periodic coupled lattices that are spectrally and structurally stable, the matrix envelope equation must admit matched solutions. Using the technique of normal form and pre-Iwasawa decomposition, a new method is developed to replace the (inefficient) shooting method for finding matched solutions for the matrix envelope equation. Stability properties of a continuously rotating quadrupole lattice are investigated. The Krein collision process for destabilization of the lattice is demonstrated. V C 2015 AIP Publishing LLC. [http://dx

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Twiss parameters and evolution of quantum harmonic-oscillator states

Cited by 4 publications

References 12 publications

Analytical methods for describing charged particle dynamics in general focusing lattices using generalized Courant-Snyder theory

Analytical methods for describing charged particle dynamics in general focusing lattices using generalized Courant-Snyder theory

Generalized Courant-Snyder theory for coupled transverse dynamics of charged particles in electromagnetic focusing lattices

Spectral and structural stability properties of charged particle dynamics in coupled latticesa)

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