Four-dimensional Phase Space Research Articles

Four-Dimensional Painlevé-Type Difference Equations

We focus on Fuchsian equations with four accessory parameters and three singular points. We see that the Fuchsian equations admit a “degeneration scheme” in some sense, which is expected to give rise to a degeneration scheme of discrete isomonodromic deformation equations with four-dimensional phase space. We compute an example of discrete isomonodromic deformation equations of a certain Fuchsian equation.

Four-dimensional phase-space reconstruction of flat and magnetized beams using neural networks and differentiable simulations

Beams with cross-plane coupling or extreme asymmetries between the two transverse phase spaces are often encountered in particle accelerators. Flat beams with large transverse-emittance ratios are critical for future linear colliders. Similarly, magnetized beams with significant cross-plane coupling are expected to enhance the performance of electron cooling in hadron beams. Preparing these beams requires precise control and characterization of the four-dimensional transverse phase space. In this study, we employ generative phase-space reconstruction techniques to rapidly characterize magnetized and flat-beam phase-space distributions using a conventional quadrupole-scan method. The reconstruction technique is experimentally demonstrated on an electron beam produced at the Argonne Wakefield Accelerator and successfully benchmarked against conventional diagnostics techniques. Specifically, we show that predicted beam parameters from the reconstructed phase-space distributions (e.g., as magnetization and flat-beam emittances) are in excellent agreement with those measured from the conventional diagnostic methods. Published by the American Physical Society 2024

Accelerator beam phase space tomography using machine learning to account for variations in beamline components

We describe a technique for reconstruction of the four-dimensional transverse phase space of a beam in an accelerator beamline, taking into account the presence of unknown errors on the strengths of magnets used in the data collection. Use of machine learning allows rapid reconstruction of the phase-space distribution while at the same time providing estimates of the magnet errors. The technique is demonstrated using experimental data from CLARA, an accelerator test facility at Daresbury Laboratory.

Technical Project of the Ion Beam Diagnostics System in Phase Space

Increasing requirements for modern acceleration complexes require precise data on ion beams injected into the accelerator. This paper presents a system of ion beam diagnostics that implements the four-slit method and makes it possible to measure the beam current density distribution in the four-dimensional phase space (PS) with high precision. Technical solutions and design features aimed at obtaining the design parameters of the system are described: the range of investigated beam currents from 1 pA to 20 mA with energy up to 50 keV, spatial resolution from 6 mm to 50 μm.

Observation of fixed lines induced by a nonlinear resonance in the CERN Super Proton Synchrotron

The motion of systems with linear restoring forces and recurring nonlinear perturbations is of central importance in physics. When a system’s natural oscillation frequencies and the frequency of the nonlinear restoring forces satisfy certain algebraic relations, the dynamics become resonant. In accelerator physics, an understanding of resonances and nonlinear dynamics is crucial for avoiding the loss of beam particles. Here we confirm the theoretical prediction of the dynamics for a single two-dimensional coupled resonance by observing so-called fixed lines. Specifically, we use the CERN Super Proton Synchrotron to measure the position of a particle beam at discrete locations around the accelerator. These measurements allow us to construct the Poincaré surface of section, which captures the main features of the dynamics in a periodic system. In our setting, any resonant particle passing through the Poincaré surface of section lies on a curve embedded in a four-dimensional phase space, the fixed line. These findings are relevant for mitigating beam degradation and thus for achieving high-intensity and high-brightness beams, as required for both current and future accelerator projects.

Four-dimensional emittance measurements of ultrafast electron diffraction optics corrected up to sextupole order

Ultrafast electron diffraction (UED) is a technique in which short-pulse electron beams can probe the femtosecond-scale evolution of atomic structure in matter driven far from equilibrium. As an accelerator physics challenge, UED imposes stringent constraints on the brightness of the probe electron beam. The low normalized emittance employed in UED, often at the 10-nm scale and below, is particularly sensitive to both applied field aberrations and space-charge effects. The role of aberrations is increasingly important in small probe systems that often feature multiple orders of magnitude variations in beam size during transport. In this work, we report the correction of normal quadrupole, skew quadrupole, and sextupole aberrations via dedicated corrector elements in an ultrafast electron microdiffraction beamline. To do this, we generate precise four-dimensional phase space maps of rf-compressed electron beams and find excellent agreement with aberration-free space-charge simulations. Finally, we discuss the role a probe-forming aperture can play in improving the brightness of bunches with appreciable space-charge effects.

Effective dimensions of infinite-dimensional Hilbert spaces: A phase-space approach.

By employing Husimi quasiprobability distributions, we show that a bounded portion of an unbounded phase space induces a finite effective dimension in an infinite-dimensional Hilbert space. We compare our general expressions with numerical results for the spin-boson Dicke model in the chaotic energy regime, restricting its unbounded four-dimensional phase space to a classically chaotic energy shell. This effective dimension can be employed to characterize quantum phenomena in infinite-dimensional systems, such as localization and scarring.

Orbits of charged particles trapped in a dipole magnetic field.

Motion of a test charged particle in a dipole magnetic field can be reduced to a two degree-of-freedom Hamiltonian system due to the axisymmetry of the dipole field. We carried out a systematic study of orbits of low-energy trapped charged particles in the dipole field via calculation of their Lyapunov characteristic exponents (LCEs) with random initial conditions in the four-dimensional phase space. Since there is at most one positive LCE, these orbits are classified as chaotic ones with one positive LCE and quasi-periodic ones with vanishing LCEs. The dependence of the fraction of quasi-periodic orbits in the phase space on the particle energy is given, which reveals a discrete spectrum, confirming the results of earlier studies. It is also found that most quasi-periodic orbits are confined near the equatorial plane and away from the dipole except for some at very low energies, while chaotic ones are ergodic. The distribution of the maximum LCE (mLCE) appears to vary gradually in the phase space and chaotic orbits with very low values of the mLCE wander near quasi-periodic orbits for a significant amount of time before merging into the sea of chaos.

Simulation of space-charge effects using a quantum Schrödinger approach

Space-charge effects play an important role in high intensity and high brightness particle accelerators. These effects were generally studied self-consistently by solving the Vlasov-Poisson equations using a particle-in-cell method in the accelerator community. In this paper, we propose an alternative method to simulate the space-charge effects in the accelerator. Instead of solving the Vlasov-Poisson equations, the proposed approach solves the Schr\"odinger-Poisson equations to study the space-charge effects in accelerators. Using a quantum Schr\"odinger approach reduces the original problem from six- or four-dimensional phase space down to three or two spatial dimensions. It also provides a possibility to simulate accelerator beam physics on quantum computers by evolving the wave function through quantum gates. Benchmarks of a coasting proton beam through a focusing drift defocusing drift lattice show excellent agreement between the quantum Schr\"odinger method and the particle-in-cell method.

Identification of quantum scars via phase-space localization measures

There is no unique way to quantify the degree of delocalization of quantum states in unbounded continuous spaces. In this work, we explore a recently introduced localization measure that quantifies the portion of the classical phase space occupied by a quantum state. The measure is based on the α-moments of the Husimi function and is known as the Rényi occupation of order α. With this quantity and random pure states, we find a general expression to identify states that are maximally delocalized in phase space. Using this expression and the Dicke model, which is an interacting spin-boson model with an unbounded four-dimensional phase space, we show that the Rényi occupations with α>1 are highly effective at revealing quantum scars. Furthermore, by analyzing the high moments (α>1) of the Husimi function, we are able to identify qualitatively and quantitatively the unstable periodic orbits that scar some of the eigenstates of the model.

Chaotic switching in driven-dissipative Bose-Hubbard dimers: When a flip bifurcation meets a T-point in $ \mathbb{R}^4 $

<p style='text-indent:20px;'>The Bose-Hubbard dimer model is a celebrated fundamental quantum mechanical model that accounts for the dynamics of bosons at two interacting sites. It has been realized experimentally by two coupled, driven and lossy photonic crystal nanocavities, which are optical devices that operate with only a few hundred photons due to their extremely small size. Our work focuses on characterizing the different dynamics that arise in the semiclassical approximation of such driven-dissipative photonic Bose-Hubbard dimers. Mathematically, this system is a four-dimensional autonomous vector field that describes two specific coupled oscillators, where both the amplitude and the phase are important. We perform a bifurcation analysis of this system to identify regions of different behavior as the pump power <inline-formula><tex-math id="M2">\begin{document}$ f $\end{document}</tex-math></inline-formula> and the detuning <inline-formula><tex-math id="M3">\begin{document}$ \delta $\end{document}</tex-math></inline-formula> of the driving signal are varied, for the case of fixed positive coupling. The bifurcation diagram in the <inline-formula><tex-math id="M4">\begin{document}$ (f, \delta) $\end{document}</tex-math></inline-formula>-plane is organized by two points of codimension-two bifurcations——a <inline-formula><tex-math id="M5">\begin{document}$ \mathbb{Z}_2 $\end{document}</tex-math></inline-formula>-equivariant homoclinic flip bifurcation and a Bykov T-point——and provides a roadmap for the observable dynamics, including different types of chaotic behavior. To illustrate the overall structure and different accumulation processes of bifurcation curves and associated regions, our bifurcation analysis is complemented by the computation of kneading invariants and of maximum Lyapunov exponents in the <inline-formula><tex-math id="M6">\begin{document}$ (f, \delta) $\end{document}</tex-math></inline-formula>-plane. The bifurcation diagram displays a menagerie of dynamical behavior and offers insights into the theory of global bifurcations in a four-dimensional phase space, including novel bifurcation phenomena such as degenerate singular heteroclinic cycles.</p>

Current distribution in magnetically confined 2DEG: semiclassical and quantum mechanical treatment

In the ballistic regime we study both semiclassically and quantum mechanically the electron’s dynamics in two-dimensional electron gas in the presence of an inhomogeneous magnetic field applied perpendicular to the plane. The magnetic field is constant inside four separate circular regions which are located at the four corners of a square of side length larger than the diameter of the circles, while outside the circles the magnetic field is zero. We carry out the stability analysis of the periodic orbits and for given initial conditions numerically calculate the two-dimensional invariant torus embedded in the four-dimensional phase space. Applying the Bohr–Sommerfeld and the Einstein–Brillouin–Keller semiclassical quantization methods we obtain the energy levels for different magnetic field strengths. We also perform exact quantum calculations solving numerically the discretized version of the Schrödinger equation. In our calculations, we consider only those bound states that are localized to the neighborhood of the four magnetic disks. We show that the semiclassical results are in good agreement with those found from our quantum calculations. Moreover, the current distribution and the phase of the different wave functions enable us to deduce the two quantum numbers n 1 and n 2 characterizing the energy levels in the semiclassical methods. Finally, we present two examples in which the quantum state shows a similar structure to the previous states, but these are special in the following sense. One of them is a scar state localized to the neighborhood of the periodic orbit while this orbit is already unstable. In the case of the other state, the current density is circulating in two rings in opposite directions. Thus, it is not consistent with the classical motion in the neighborhood of the periodic orbit.

Connes distance of 2D harmonic oscillators in quantum phase space* *Project supported by the Key Research and Development Project of Guangdong Province, China (Grant No. 2020B0303300001), the National Natural Science Foundation of China (Grant No. 11911530750), the Guangdong Basic and Applied Basic Research Foundation, China (Grant No. 2019A1515011703), the Fundamental Research Funds for the Central Universities, China (Grant No. 2019MS109),

We study the Connes distance of quantum states of two-dimensional (2D) harmonic oscillators in phase space. Using the Hilbert–Schmidt operatorial formulation, we construct a boson Fock space and a quantum Hilbert space, and obtain the Dirac operator and a spectral triple corresponding to a four-dimensional (4D) quantum phase space. Based on the ball condition, we obtain some constraint relations about the optimal elements. We construct the corresponding optimal elements and then derive the Connes distance between two arbitrary Fock states of 2D quantum harmonic oscillators. We prove that these two-dimensional distances satisfy the Pythagoras theorem. These results are significant for the study of geometric structures of noncommutative spaces, and it can also help us to study the physical properties of quantum systems in some kinds of noncommutative spaces.

Quantum localization measures in phase space.

Measuring the degree of localization of quantum states in phase space is essential for the description of the dynamics and equilibration of quantum systems, but this topic is far from being understood. There is no unique way to measure localization, and individual measures can reflect different aspects of the same quantum state. Here we present a general scheme to define localization in measure spaces, which is based on what we call Rényi occupations, from which any measure of localization can be derived. We apply this scheme to the four-dimensional unbounded phase space of the interacting spin-boson Dicke model. In particular, we make a detailed comparison of two localization measures based on the Husimi function in the regime where the model is chaotic, namely, one that projects the Husimi function over the finite phase space of the spin and another that uses the Husimi function defined over classical energy shells. We elucidate the origin of their differences, showing that in unbounded spaces the definition of maximal delocalization requires a bounded reference subspace, with different selections leading to contextual answers.

Second-order modes of transverse oscillations in intense beams: Chernin’s matrix formalism versus Vlasov–Poisson equations

We study an intense space charge dominated beam in the realm of beam optics of particle accelerators. We assume an anisotropic beam with different external focussing constants and with different phase space emittances in the two transverse directions. We consider the root mean square moment equations of the beam with linear transverse coupling of particle motion ( $$x$$ and $$y$$ ). Magnetic focussing fields and the space charge Coulomb’s self-field forces are expressed in a matrix format and describe the governing equations for the beam outer envelope. By linearizing the equations, we investigate the collective response of ensemble of particles. This response is stated as coherent mode frequencies for the second-order even (envelope) and odd (tilting) oscillation modes of the beam cross section. We discuss the main features of the eigenfrequencies in detail. The approach known as Chernin’s matrix formalism is conceptually equivalent to the second-order moments of the full Vlasov–Poisson kinetic equation in four-dimensional phase space. We show that the relations obtained via matrix approach are in full agreement with the eigenfrequencies of transverse oscillations obtained by the linearized kinetic Vlasov–Poisson equations.

Smale-Williams solenoids in autonomous system with saddle equilibrium.

We construct an autonomous low-dimensional system of differential equations by replacement of real-valued variables with complex-valued variables in a self-oscillating system with homoclinic loops of a saddle. We provide analytical and numerical indications and argue that the emerging chaotic attractor is a uniformly hyperbolic chaotic attractor of Smale-Williams type. The four-dimensional phase space of the flow consists of two parts: a vicinity of a saddle equilibrium with two pairs of equal eigenvalues where the angular variable undergoes a Bernoulli map, and a region which ensures that the trajectories return to the origin without angular variable changing. The trajectories of the flow approach and leave the vicinity of the saddle equilibrium with the arguments of complex variables undergoing a Bernoulli map on each return. This makes possible the formation of the attractor of a Smale-Williams type in Poincaré cross section. In essence, our model resembles complex amplitude equations governing the dynamics of wave envelops or spatial Fourier modes. We discuss the roughness and generality of our scheme.

Transverse phase space characterization in an accelerator test facility

We compare three techniques for characterising the transverse phase space distribution of the beam in CLARA FE (the Compact Linear Accelerator for Research and Applications Front End, at Daresbury Laboratory, UK): emittance and optics measurements using screens at three separate beamline locations; quadrupole scans; and phase space tomography. We find that where the beam distribution has significant structure (as in the case of CLARA FE at the time the measurements presented here were made) tomography analysis is the most reliable way to obtain a meaningful characterisation of the transverse beam properties. We present the first experimental results from four-dimensional phase space tomography: our results show that this technique can provide an insight into beam properties that are of importance for optimising machine performance.

The goodness of global fitting in the reconstruction scheme

In the reconstruction scheme, the results for predictions from chaotic time series are accurate. This work introduces the chi-square test for goodness-of-fit and the statistic R-squared in this scenario. These new features facilitate the choice between different predictors and improve the predictive capacity of the LinMapTS package. New version program summaryProgram Title: LinMapTSProgram Files doi:http://dx.doi.org/10.17632/pnhy9zymrp.2Licensing provisions: GPLv3Programming language: Maple 17Journal reference of previous version: P R.L. Alves, L.G.S. Duarte, L.A.C.P. da Mota, Comput. Phys. Commun. 215 (2017) 265–268Does the new version supersede the previous version?: YesReasons for the new version: The global fitting captures the underlying dynamics from a time series. It is quite convenient to test if a map is proper to describe the time evolutions of an observable or not. The introduction of the chi-square test for goodness-of-fit in the reconstruction scheme attends this demand [1].On the other hand, a satisfactory global map may be more accurate than another. The statistic R-squared is a powerful aid to one’s choosing the best predictor. [2].Summary of revisions: If the command LinGfiTS has the optional argument Analysis=1, then the output now presents both results of the test and the statistic R-squared for the global fitting.To illustrate the computational implementation of these new features, we revisit some predictions of chaotic time series with the LinMapTS package [3,4]. The commands below are sufficient to reconstruct the state vectors and to forecast of the dynamical variable of order 723 (XP) if one analyzes the time series – for a Lorenz System – stored in the file ts37.txt. In the next step, this communication presents the commands for the analysis of a time series of the real-world – a chaotic circuit [5,3,6] – and outputs of the new features and results for different polynomial predictors. [Display omitted] [Display omitted] Fig. 1 presents well-succeed global fittings for the Lorenz System (see Fig. 1(a)) and the chaotic voltage in a four-dimensional reconstruct phase space (see Fig. 1(b)). In both study cases, the most accurate forecasts correspond to the largest R-squared between all polynomial predictors (see Tables 1 and 2). So these applications suggest that the new features presented in this communication may improve the forecast capability significantly.Nature of problem: Time series analysis and improving forecast capability.Solution method: The method of solution is published in [3].Additional comments including restrictions and unusual features: Depending on the data inputted, the chi-square test may not run properly. In this case, one must introduce the optional argument ChiSquare=0 to perform the graphical analysis and compute the statistic R-squared.Declaration of competing interestThe authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.AcknowledgmentsL.G.S. Duarte and L.A.C.P. da Mota wish to thank Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ) – Registro n.o8182/UERJ/2013 and Deliberacão n.o25/2013 – for the Research Grant.

Four-dimensional phase space measurement using multiple two-dimensional profiles

Abstract Knowledge of 4D beam phase space is essential to characterize beam motion along a beamline which includes elements that couple the transverse planes. A 4D phase space measurement method using multiple 2D profiles is proposed in this paper. The effectiveness of the method is validated by numerical simulations. Finally, we present a successful application of the method at the Xi’an Proton Application Facility to measure the coupled 4D phase space after a skew quadrupole.

Pair-cat codes: autonomous error-correction with low-order nonlinearity

We introduce a driven-dissipative two-mode bosonic system whose reservoir causes simultaneous loss of two photons in each mode and whose steady states are superpositions of pair-coherent/Barut-Girardello coherent states. We show how quantum information encoded in a steady-state subspace of this system is exponentially immune to phase drifts (cavity dephasing) in both modes. Additionally, it is possible to protect information from arbitrary photon loss in either (but not simultaneously both) of the modes by continuously monitoring the difference between the expected photon numbers of the logical states. Despite employing more resources, the two-mode scheme enjoys two advantages over its one-mode cat-qubit counterpart with regards to implementation using current circuit QED technology. First, monitoring the photon number difference can be done without turning off the currently implementable dissipative stabilizing process. Second, a lower average photon number per mode is required to enjoy a level of protection at least as good as that of the cat-codes. We discuss circuit QED proposals to stabilize the code states, perform gates, and protect against photon loss via either active syndrome measurement or an autonomous procedure. We introduce quasiprobability distributions allowing us to represent two-mode states of fixed photon number difference in a two-dimensional complex plane, instead of the full four-dimensional two-mode phase space. The two-mode codes are generalized to multiple modes in an extension of the stabilizer formalism to non-diagonalizable stabilizers. The M-mode codes can protect against either arbitrary photon losses in up to M − 1 modes or arbitrary losses and gains in any one mode.