††thanks: Email<EMAIL_ADDRESS>
# Numerical construction of quasi-invariants in double- and multi-bend
achromat lattices

Yongjun Li  Brookhaven National Laboratory, Upton, New York 11973, USA  Kilean
Hwang  Lawrence Berkeley National Laboratory, Berkeley 94720, California, USA
Chad Mitchell  Lawrence Berkeley National Laboratory, Berkeley 94720,
California, USA  Robert Rainer  Brookhaven National Laboratory, Upton, New
York 11973, USA  Robert Ryne  Lawrence Berkeley National Laboratory, Berkeley
94720, California, USA  Victor Smaluk  Brookhaven National Laboratory, Upton,
New York 11973, USA

###### Abstract

A numerical method to construct quasi-invariants (QI) has been explored in
double- and multi-bend achromat (DBA and MBA) lattices. The Fermilab’s
Integrable Optics Test Accelerator (IOTA) and the University of Maryland
Electron Ring (UMER) built with exact invariants, pave the way for discovery
of a new type of nonlinear lattice. But it might be impossible to construct
exact invariants in various achromatic lattices for dedicated synchrotron
light source storage rings. Quasi-invariants, however, were found to be
constructible by tuning the sextupoles and octupoles in the DBA and MBA
lattices. In our exploration, QIs were constructed by confining the simulated
trajectories to tori in the phase space at a certain Poincaré section, while
simultaneously maintaining monochromatic betatron tunes. The lattices with QIs
share some important features as integrable ones, such as a large dynamic
aperture, confined trajectory tori, robustness to resonances and errors, and a
large amplitude-dependent tune-spread.

##  I introduction

The Integrable Optics Test Accelerator (IOTA) Antipov  _et al._ (2017) and
University of Maryland Electron (UMER) Ruisard  _et al._ (2019) constructed
with one or more exact invariants Danilov (2008); Danilov and Nagaitsev
(2010), paves the way for discovery of a new type of nonlinear lattice. To be
integrable, however, imposes strong constraints on the linear optics pattern
in order to accomodate special nonlinear magnetic inserts. The storage rings
used as dedicated synchrotron light sources, are designed in a different way:
a linear achromat lattice with a desired beam emittance is designed first;
then nonlinear dynamics is optimized with sextupoles and/or octupoles. These
nonlinear magnets usually control the low order resonance driving terms of the
single-turn Hamiltonian Dragt (2011) to obtain sufficient dynamic apertures
(DA). Under these conditions it is generally difficult, if not completely
impossible, to construct exact invariants on those rings. Quasi-invariants
(QI), however, were found to be constructible in the achromat lattices. This
paper introduces our exploration into numerically constructing and applying
such QI in both double-bend achromat (DBA) and multi-bend achromat (MBA)
lattices. The motivation for constructing such lattices is that, although they
are not completely integrable, particle motion within their DA are much less
chaotic and more robust to the resonances. Lattices such as the one seen at
the IOTA, usually have a large amplitude-dependent betatron tune-spread which
can increase instability and space charge thresholds due to improved Landau
damping. This research was also motivated by related studies such as the
square matrix method Yu (2017) and constant Courant-Synder variants method
Borland ; Sun and Borland (2017).

To further explain our approach, the remaining sections are outlined as
follows: Sect. II explains the conception of Poisson commuting invariants in
integrable Hamiltonian systems, and the numerical approach of constructing
linear [action–angle]-like QIs through the symplectic tracking simulations.
Sect. III and IV introduce the nonlinear properties of two constructed
nonlinear lattices respectively: the existing NSLS-II DBA lattice ring, and an
under-designed diffraction-limited MBA lattice ring. Some detailed studies
focusing on the DBA lattice are then further explained: a technique to
modulate the action-like variant to reshape the tori is illustrated in Sect.
V; a simulation of the kicked beam decoherence to observe the large nonlinear
tune-spread is shown in Sect. VII; The periodicity of QI is introduced in
Sect. VI. Some discussion and a brief summary are given in Sect. VIII.

##  II numerical construction of quasi-invariants

The Liouville integrability of a Hamiltonian system means there exists a
maximal set of Poisson commuting invariants, i.e., functions on the phase
space whose Poisson brackets with the Hamiltonian of the system and with each
other, vanish Liouville (1855); Arnold . Systems with these conditions are
completely integrable, distinguishing them from partial integrable systems by
having less invariants. By ignoring radiation and longitudinal acceleration, a
charged particle’s transverse motion in a storage ring is a 4-dimensional
Hamiltonian system. If we let the canonical coordinates of the system be
$\boldsymbol{z}=(x,p_{x};y,p_{y})$, and its single-turn Hamiltonian be $H$,the
free variable is the path length of the reference particle $s=n\times C$,
where $n=1,2,\cdots$, and $C$ is the circumference of a ring. For any function
$f(s,\boldsymbol{z})$, its value changes with $s$ either because of its
explicit $s$ dependence, or because it depends on $\boldsymbol{z}$ and changes
because $\boldsymbol{z}$ changes. The total derivative of $f$ therefore can be
written as

$f^{\prime}=\frac{\partial f}{\partial s}+\left[f,H\right],$

where $[,]$ is the Poisson bracket. A quantity $f(\boldsymbol{z})$ is a
constant of the motion if it is not explicitly $s$ dependent, and satisfies
the condition

$\left[f,H\right]=0.$

If four such independent invariants $f_{i}$ with $i=1,2,3,4$ exist and they
commute with each other,

$\left[f_{i},f_{j}\right]=0,$

i.e., any two of them form a pair of Poisson commuting set, the lattice is
completely Liouville integrable. In the linear beam dynamics, uncoupled
action-angle variables $J_{x}-\phi_{x};\;J_{y}-\phi_{y}$ are the most common
used Poisson commuting pairs.

In the existing, conventional storage rings (i.e., non-IOTA type), various
linear magnets are used to achieve desirable linear optics properties, e.g., a
low beam emittance. Once the linear lattice and the nonlinear magnets
locations are fixed, the single-turn Hamiltonian $H$ is the function of the
coordinates $\boldsymbol{z}$ and the nonlinear magnet’s excitations
$K_{i},\;\text{with}\;i\geq 2$. It is difficult, if not impossible, to
construct even one exact invariant in the conventional storage rings. However,
the topology of trajectories observed at a Poincaré section for a completely
integrable system is well-known, i.e., all its trajectories are confined to
tori with well-defined and stable tunes. The Poincaré section is the
intersection of a periodic orbit in the state space of a continuous dynamical
system with a certain lower-dimensional subspace, which is transverse to the
flow of the system. When only linear magnets are present, the linear
action–angle variables $J-\phi$ are a pair of commuting invariants, and $\phi$
is constant regardless of the betatron amplitude. Based on this picture, we
can imitate the linear motion to construct a pair of QIs as illustrated in
Fig. 1. Unlike the linear case, however, the angle variant can be dependent on
the amplitude of actions.

Figure 1:  Schematic illustration of a rotating trajectory observed at a
Pioncaré section with normalized coordinates ($\bar{x},\;\bar{p}_{x}$). The
fluctuations of action $J$ and phase advances $\phi$ in multiple-turn tracking
simulations are the objectives to be minimized.

The uncoupled action-like variant in the quadratic form is constructed as

$J^{2}_{x}(x,p_{x})=\bar{x}^{2}+\bar{p}_{x}^{2}=\gamma_{x}x^{2}+2\alpha_{x}xp_{x}+\beta_{x}p_{x}^{2},$
(1)

in the horizontal plane. Here the normalized coordinates $\bar{x}$ and
$\bar{p}_{x}$ are read as

$\displaystyle\begin{bmatrix}\bar{x}\\\
\bar{p}_{x}\end{bmatrix}=\begin{bmatrix}\frac{1}{\sqrt{\beta_{x}}}&0\\\
\frac{\alpha_{x}}{\sqrt{\beta_{x}}}&\sqrt{\beta_{x}}\end{bmatrix}\begin{bmatrix}x\\\
p_{x}\end{bmatrix}.$ (2)

Here $\alpha_{x},\;\beta_{x},\text{and}\;\gamma_{x}$ are the horizontal Twiss
parameters Courant and Snyder (1958) at a given longitudinal location, where
the Poincaré section is observed.

The variant $\phi_{x}$ corresponds to the phase advance completed in a single
revolution, and is constructed as

$\displaystyle\phi_{x}$ $\displaystyle=\Phi_{x,i+1}-\Phi_{x,i}$
$\displaystyle=\arctan\left({\frac{\bar{p}_{x,i+1}}{\bar{x}_{i+1}}}\right)-\arctan\left({\frac{\bar{p}_{x,i}}{\bar{x}_{i}}}\right)+k\cdot
2\pi,$

where $k$ is the integer part of betatron tune.

With a given one-turn Hamiltonian $H(K_{i},x,p_{x},y,p_{y})$, by tuning its
nonlinear knobs $K_{i}$, one can minimize their absolute derivative expressed
by the Poisson bracket

$\begin{cases}\min\limits_{K_{i}}\bigr{\lvert}[J_{x}(x,p_{x}),H(K_{i},x,p_{x},y,p_{y})]\bigr{\rvert}\rightarrow
0\\\
\min\limits_{K_{i}}\bigr{\lvert}[\phi_{x}(x,p_{x}),H(K_{i},x,p_{x},y,p_{y})]\bigr{\rvert}\rightarrow
0\end{cases},$ (3)

at different coordinates $(x,\;p_{x};y\;p_{y})$.

In order to compute the Poisson brackets in Eq. (3), we need to extract the
single-turn Hamiltonian $H$ explicitly. For a given ring lattice, each magnet
has its own Lie generator, and the whole ring’s single-turn Hamiltonian can be
obtained by concatenating these Lie generators sequentially Dragt (2011). An
alternative is to reconstruct the Hamiltonian from the single-turn Taylor map
Dragt and Finn (1976) obtained with the truncated power series algorithm
(TPSA) Berz (1988). The Hamiltonians obtained with the above two techniques
are still approximate because one must truncate the maps at a certain order.
In principle, one can use an approximate Hamiltonian to calculate the
derivative of a function $f$ with the Poisson bracket, but it is needed to
estimate and control the accuracy carefully. Therefore, instead of using an
approximate Hamiltonian to compute the Poisson brackets, we implemented
element-by-element tracking from given initial conditions to compute the turn-
by-turn evolution of $J_{x}$. The tracking was implemented with a kick-drift
symplectic integrator Yoshida (1990). The reason for using symplectic
integrator is to preserve the geometry of the Hamiltonian system. To ensure
the actions $J_{x}$ are QIs from different initial conditions within the
needed DA, multiple particles with different $J_{x,0}$ were launched. The
available nonlinear knobs were simultaneously tuned to minimize turn-by-turn
fluctuations as illustrated in Fig. 2.

Figure 2:  Schematic illustration of the fluctuation of actions $\Delta J_{x}$
starting from different initial amplitudes. Usually the fluctuations increase
gradually with the initial amplitude.

Confining the fluctuation of the angle variant $\phi_{x}$ is equivalent to
maintaining a monochromatic betatron tune. Instead of directly calculating the
fluctuation of phase advances, turn-by-turn complex eigenmodes $\bar{x}\pm
i\bar{p}_{x}$ Chao (2002) was analyzed in the frequency domain. The reason for
using the eigenmodes here is to determine if the fractional tune is below or
above the half integer. The amplitudes of two leading harmonics were computed
utilizing the Numerical Analysis of Fundamental Frequencies (NAFF) technique
Laskar (2003). By tuning the nonlinear knobs, the ratio between the two
leading harmonics $r=\frac{A_{2}}{A_{1}}$ is suppressed, as well as with
other, smaller amplitude harmonics as illustrated in Fig. 3. Note that here we
only require that the tune for each initial condition be monochromatic, but
allow it to be amplitude dependent as well. In the same way, another pair of
QIs ($J_{y}-\phi_{y}$) can be constructed in the vertical plane.

Figure 3:  Schematic illustration of the spectrum analysis with the turn-by-
turn trajectory data $\bar{x}\pm i\bar{p}_{x}$. The ratio between two leading
harmonics amplitudes $\frac{A_{2}}{A_{1}}$ is the minimization objective in
order to obtain monochromatic tune.

Since we want to minimize the fluctuations of four different variants with
different initial conditions simultaneously, the construction of such a
nonlinear lattice becomes a typical multi-objective (MO) optimization problem:

  * •

given a set of nonlinear knobs $K_{i}$ within their allowed ranges;

  * •

subject to some constraints, such as certain desired chromaticities;

  * •

simultaneously minimize the objective functions, i.e., $\frac{\Delta
J_{x,y}}{J_{x,y}}$ and $r_{x,y}$ of multi-particles launched from different
initial conditions.

The MO optimization techniques are widely used in the accelerator community
now, and more details can be found in Deb (2001).

Thus far, we have only discussed uncoupled linear lattices. When linear
coupling is present, a parameterization, such as the one depicted in ref.
Edwards and Teng (1973), is needed.

##  III Applied to double-bend achromat

In this Section, we introduce a constructed nonlinear DBA lattice for the
NSLS-II main storage ring. NSLS-II BNL  is a dedicated $3^{\text{rd}}$
generation medium energy (3 GeV) light source operated by Brookhaven National
Laboratory. The storage ring’s lattice is a typical DBA structure with main
parameters listed in Table 1. Its linear optics for one cell is illustrated in
Fig. 4. The whole ring is composed of 30 such cells. In this configuration,
three families of chromatic sextupoles are used to correct its chromaticity to
$+7$. Then six families of harmonic sextupoles in dispersion-free sections are
the tuning knobs for the multi-objective optimizer to construct four QIs.

Table 1: Main parameters of NSLS-II storage ring Parameters |  Values
---|---
Hor. emit. ($nm$) |  2.1
Natural chrom. (x/y) |  -101/-40
Tune (x/y) |  33.22/16.26
Energy spread |  $5.1\times 10^{-4}$
Damp. partition (x/y/s) |  1.0/1.0/2.0
Figure 4:  (Colored) The linear optics and magnet layout for one cell of the
NSLS-II storage ring. The red blocks represent sextupoles. Six harmonic
sextupoles were used in searching QIs.

Below we present the nonlinear lattice performance of an optimized solution
using the tracking simulation code elegant Borland (2000). All the tracking
simulations in this paper were implemented with this code unless stated
otherwise. First a sufficient on-momentum DA (through 1,024 turns particle
tracking) has been confirmed as shown in Fig. 5. Each stable initial condition
is also colored with the tune diffusion rate obtained with the NAFF technique.

Figure 5:  (Colored) DA of the DBA lattice observed at the center of long
straight section, i.e., $s=0$ in Fig. 4. It was colored with the tune
diffusion rate.

The fluctuations of action $J_{x,y}$ at different amplitudes are illustrated
in Fig. 6. The fluctuation increases gradually with the amplitude of initial
condition. The spectrum analysis using the NAFF technique indicates that some
harmonics gradually become stronger as well (Fig. 7). The amplitude ratios
between two leading harmonics increase as shown in Fig. 8.

Figure 6:  (Colored) Fluctuation of $J_{x,y}$ of the DBA lattice starting from
different initial conditions in the the horizontal (left) and vertical (right)
planes. Figure 7:  Some leading harmonics extracted from a set of turn-by-turn
data $\bar{x}\pm i\bar{p}_{x}$ using the NAFF technique for the DBA lattice.
Only the first two leading harmonics ratio is used for the minimization
objective .

Figure 8:  Ratio of two leading harmonics amplitudes gradually increases with
the initial amplitude in the horizontal (left) and vertical (right) planes for
the DBA lattice.

One of the features of an integrable system is that the trajectories are
confined to tori in the phase space, which are shown in Fig. 9. Although
trajectories begin to deviate from the desired ellipse gradually when the
amplitude increases, they are still confined to their own tori. It indicates
some high order terms might be needed in order to construct a better QI in Eq.
(1).

Figure 9:  Simulated trajectories of the DBA lattice starting from different
initial conditions in the horizontal (left) and vertical (right) phase space.
Within the DA, although the trajectories deviate from desired circles
gradually, they are still confined to thin tori.

Like the IOTA rings, this lattice also provides a large amplitude-dependent
tune-spread as illustrated in Fig. 10. This property can increase instability
and space charge thresholds through an improved Landau damping effect.

Figure 10:  (Colored) Tune footprint of the DBA lattice in the tune space. A
large amplitude-dependent tune-spread is observed, and various resonance lines
can be crossed.

The robustness of this lattice has been confirmed in both the simulation and
experiment. After adding various realistic errors to the magnets in the
simulation, a sufficient DA still remains. Experimentally this lattice had
been successfully commissioned at NSLS-II. Comparing with the current
operation sextupole configuration, under a same condition of $400\;mA$ stored
beam current, the needed bunch-by-bunch feedback amplitude gain could be
reduced by 50% and 75% in the horizontal and vertical planes respectively
thanks to the larger chromaticity and nonlinear tune-spread.

##  IV Applied to multi-bend achromat

Now the low emittance light source ring design is entering a new era. Various
MBA-type lattices already reach diffraction-limited horizontal emittances to
deliver much brighter X-ray beams. It is interesting to explore if it is
possible to construct some QIs in the MBA lattice as well. The ESRF-EBS type
hybrid MBA lattice Farvacque  _et al._ (2013) has been widely adopted by other
facilities. It is also being considered as one of options for the future NSLS-
II upgrade. An underdesigned 7-BA lattice’s linear optics for one cell is
illustrated in Fig. 11, in which several reverse bends are incorporated
Riemann and Streun (2019); Zhang and Huang (2019) as well. The main parameters
are listed in Table 2.

Table 2: Main parameters of the test hybrid MBA ring Parameters |  Values
---|---
Hor. emit. ($pm$) |  31
Natural chrom. (x/y) |  -125/-108
Tune (x/y) |  73.19/28.62
Energy spread |  $7.1\times 10^{-4}$
Damp. partition (x/y/s) |  2.0/1.0/1.0
Figure 11:  (Colored) The linear optics and magnet layout for one cell of an
ESRF-EBS type hybrid 7BA lattice. The red blocks represent sextupoles. Six
chromatic sextupoles grouped into five families inside two dispersive bumps
are used to correct the chromaticity with three extra degrees of freedom. Four
harmonic sextupoles and four dispersive octupoles (green blocks) are also
available for nonlinear dynamics optimization.

A two-stage optimization has been implemented. First, both chromatic and
harmonic sextupoles were used to correct chromaticities and minimize the
fluctuations of the QIs. Then four octupoles inside the dispersive bumps were
used to further minimize their residual fluctuations. A sufficient DA is
obtained as shown in Fig. 12.

Figure 12:  (Colored) DA of the MBA lattice in the transverse $x-y$ plane
colored with the tune diffusion rate.

The MBA lattice is found more challenging than the DBA lattice in constructing
QIs. The constructed QIs have larger fluctuations, especially in the vertical
plane as illustrated in Fig. 13 and 14. Nevertheless, their trajectories are
still confined to tori in the horizontal plane as seen in Fig. 15. More
importantly, a large amplitude spread is also observed in its tune footprint
as shown in Fig. 16.

Figure 13:  (Colored) Fluctuation of $J_{x,y}$ of the MBA lattice in the the
horizontal (left) and vertical (right) planes.

Figure 14:  Ratio of two leading harmonics amplitude of the MBA lattice in the
the horizontal (left) and vertical (right) planes.

Figure 15:  Simulated trajectories of the MBA lattice in the horizontal (left)
and vertical (right) phase space. The vertical trajectories begin to smear out
from thin tori gradually, but some patterns are visible. Figure 16:  (Colored)
A large amplitude-dependent tune-spread is observed in the MBA lattice
constructed with QIs.

While observing the trajectories of the vertical phase space in Fig. 15, at a
large amplitude, they significantly deviate from our desired shape, but
regular patterns can be recognized. That means the motion is still regular
rather than chaotic, but a complicated QI expression including high order
polynomial terms are needed. Thus far it is clear for us how to construct
better QIs analytically.

##  V Modulation on quasi-invariant

An exact invariant of nonlinear integrable Hamiltonian system couldn’t be in a
simple quadratic form as we constructed. We can add extra high order
polynomial terms to construct more complicated quasi-variant. In this section,
we demonstrate an alternative method by periodically modulating the quasi-
invariant $J_{x}$ to obtain triangle-shaped tori,

$\displaystyle{}J(\phi_{x})$ $\displaystyle=J_{0}+\Delta J_{x}(\phi_{x})$
$\displaystyle=J_{0}\left\\{1+\delta\sin\left[n\left(\phi_{x}-\frac{\pi}{2n}\right)\right]\right\\},$
(4)

where $\delta=\frac{\Delta J_{x}}{J_{x}}$ and $n=3$. The modulation (top) on
$J_{x}$ and the expected torus are illustrated in Fig. 17,

Figure 17:  (Colored) A azimuthal periodical modulation (top) on the action QI
can reshape a circle torus into a triangle-shaped one (bottom).

Applying the above modulation into the optimization objectives in the DBA
lattice, a new set of sextupole settings were obtained. A tracking simulation
confirmed that the tori have been reshaped as expected in Fig. 18. Such phase
space manipulation technique might be useful when a non-symmetric DA is
desired. For example, a slightly larger inboard DA is preferred to capture
off-axis injected beam from there.

Figure 18:  Particle trajectories are confined to triangle-shaped tori as
confirmed by the tracking simulation. Usually a non-symmetric DA can be
obtained with this modulation technique.

This modulation could introduce some weak harmonics into the betatron
spectrum. For example, a weak harmonic sitting near $1/3$ is visible through
the NAFF analysis in Fig. 19. Because the modulation amplitude is usually
weak, it usually doesn’t spoil the overall performance of the nonlinear
lattice. Note that the triangle-shaped tori are not caused by the betatron
tune’s approaching to a third-order resonance. As seen in Fig. 19, the main
betatron tune is around 0.22, and still far away from $1/3$.

Figure 19:  A weak $1/3$ harmonic is visible when a triangle-shaped modulation
is added on a torus. The dominated tune is still around 0.22. The tori are
modulated as tri-angle shapes but not because the main tune is approaching the
third order resonance.

##  VI Periodicity of quasi-invariants

In our examples, the QIs were constructed at a specific Poincaré section,
e.g., at the center of straight section in our examples. Therefore it is not
surprised to observe distorted tori at other locations. An example of
distorted tori observed at a location inside the achromat of the NSLS-II DBA
lattice is illustrated in Fig. 20. The one-cell map behaves like an “optical
anastigmat”, which has regular tori only at the periodical locations, i.e.,
its entrance and exit. But distorted tori, even the chaos can be observed
somewhere in-between.

Figure 20:  Tori are regular only when observed at the long straight section,
but could be distorted at other locations.

##  VII decoherence due to nonlinearity

The amplitude-dependent tune-spread due to the nonlinearity of betatron
oscillation can be observed by kicking a bunched beam transversely Meller  _et
al._ (1987). A 4-dimensional multi-particle simulation was used to confirm it.
After being kicked, the beam motion will decohere as the individual betatron
phases of the particles disperse. The phase space distribution of the beam
spreads from a localized bunch to an annulus which occupies all betatron
phases (see Fig. 21), and the observed centroid of the beam will show a
decaying oscillation (see Fig. 22). From the decoherence factor curve (the red
line in Fig. 22), the local tune-shift-with-amplitude coefficient can be
determined, and confirmed with the spectrum analysis of the single particle
tracking data in Fig. 23.

Figure 21:  After being kicked, the phase space distribution of the beam
spreads from a localized bunch to an annulus which occupies all betatron
phases eventually. Figure 22:  (Colored) After being kicked, the centroid of
the beam will show a decaying oscillation. The red line is known as the
decoherence factor in ref. Meller  _et al._ (1987), and can be used to
determine the local tune-shift-with-amplitude. Figure 23:  The tune-shift-
with-amplitude curve obtained by the single particle tracking simulation. The
horizontal axis is measured in the unit of the square of beam size
$\sigma_{x}^{2}$. The local slope at the kicked beam amplitude (at the dashed
line) agrees with the decoherence factor computed from the centroid motion in
Fig. 22.

##  VIII Discussion and summary

Thus far, we only studied the on-momentum particle motion. Practically a
sufficient off-momentum acceptance is required as well. In the two examples
shown previously, only the on-momentum lattice were constructed with four
quasi-variants, then the off-momentum acceptances were checked with tracking
simulation. They were found already having sufficient momentum acceptance. In
case the momentum acceptance needs to be optimized simultaneously, we expanded
the method to include fixed off-momentum particles. For a fixed off-momentum
particle, its dispersive reference orbit needs to be computed first, and then
its action-like variants are constructed with the momentum-dependent Twiss
parameters
$\alpha(\delta,K_{i}),\;\beta(\delta,K_{i})\;\text{and}\;\gamma(\delta,K_{i})$
in Eq. (2). The momentum-dependent Twiss parameters also depend on the
nonlinear magnets excitations. For a given nonlinear magnet setting, the
reference closed orbit and corresponding momentum-dependent Twiss parameters
need to be updated prior to the normalization.

We demonstrated that a quadratic form of the action– and angle–like QIs are
constructible in the conventional DBA and MBA lattices by tuning conventional
sextupoles and octupoles. The obtained QIs are only valid within a limited DA.
However, most particles trajectories are regular and confined to tori, their
betatron tunes are well-defined and stable. As real IOTA-type lattice, a large
nonlinear tune-spread also exists, which can provide extra Landau damping.

###### Acknowledgements.

We would like to thank L-H. Yu (BNL), M. Borland, R. Lindberg, Y. Sun (ANL)
and Y. Hao (MSU), E. Stern, A. Valishev, A. Romanov (FNAL), N. Kuklev
(UChicago), G. Xu (IHEP) for the stimulating and collaborative discussions.
This research is supported by the U.S. Department of Energy under Contract No.
DE-SC0012704 (BNL) and DE-AC02-05CH11231 (LBNL). Co-authors from LBNL would
like to acknowledge the support from the U.S. DOE Early Career Research
Program under the Office of High Energy Physics.

## References

  * Antipov  _et al._ (2017) Sergey Antipov, Daniel Broemmelsiek, David Bruhwiler, Dean Edstrom, Elvin Harms, Valery Lebedev, Jerry Leibfritz, Sergei Nagaitsev, Chong-Shik Park, Henryk Piekarz, _et al._ , “IOTA (integrable optics test accelerator): facility and experimental beam physics program,” Journal of Instrumentation 12, T03002 (2017).
  * Ruisard  _et al._ (2019) K Ruisard, HB Komkov, B Beaudoin, I Haber, D Matthew, and T Koeth, “Single-invariant nonlinear optics for a small electron recirculator,” Physical Review Accelerators and Beams 22, 041601 (2019).
  * Danilov (2008) V Danilov, “Practical solutions for nonlinear accelerator lattice with stable nearly regular motion,” Physical Review Special Topics-Accelerators and Beams 11, 114001 (2008).
  * Danilov and Nagaitsev (2010) V Danilov and S Nagaitsev, “Nonlinear accelerator lattices with one and two analytic invariants,” Physical Review Special Topics-Accelerators and Beams 13, 084002 (2010).
  * Dragt (2011) Alex J Dragt, “Lie methods for nonlinear dynamics with applications to accelerator physics,” unpublished (2011).
  * Yu (2017) Li Hua Yu, “Analysis of nonlinear dynamics by square matrix method,” Physical Review Accelerators and Beams 20, 034001 (2017).
  * (7) M. Borland, personal communication.
  * Sun and Borland (2017) Yipeng Sun and Michael Borland, “Comparison of Nonlinear Dynamics Optimization Methods for APS-U,” in  _2nd North American Particle Accelerator Conference_ (2017) p. WEPOB15.
  * Liouville (1855) Joseph Liouville, “Note sur l’intégration des équations différentielles de la dynamique, présentée au bureau des longitudes le 29 juin 1853.” Journal de Mathématiques pures et appliquées , 137–138 (1855).
  * (10) VI Arnold, “Mathematical Methods of Classical Mechanics, Springer–Verlag, Berlin and New York, 1989,” .
  * Courant and Snyder (1958) E.D Courant and H.S Snyder, “Theory of the alternating-gradient synchrotron,” Annals of Physics 3, 1 – 48 (1958).
  * Dragt and Finn (1976) Alex J Dragt and John M Finn, “Lie series and invariant functions for analytic symplectic maps,” Journal of Mathematical Physics 17, 2215–2227 (1976).
  * Berz (1988) A Berz, “Differential algebraic description of beam dynamics to very high orders,” Part. Accel. 24, 109–124 (1988).
  * Yoshida (1990) H. Yoshida, “Construction of higher order symplectic integrators,” Phys. Lett. A150, 262–268 (1990).
  * Chao (2002) A W Chao,  _Lecture Notes on Topics in Accelerator Physics_ (SLAC, Stanford, CA, 2002).
  * Laskar (2003) J. Laskar, “Frequency Map Analysis and Particle Accelerators,” in  _20th Particle Accelerator Conference (PAC 03)_ (2003) p. 378.
  * Deb (2001) Kalyanmoy Deb,  _Multi-Objective Optimization Using Evolutionary Algorithms_ (Wiley, 2001).
  * Edwards and Teng (1973) DA Edwards and LC Teng, “Parametrization of linear coupled motion in periodic systems,” IEEE Transactions on nuclear science 20, 885–888 (1973).
  * (19) BNL, https://www.bnl.gov/nsls2/project/PDR/.
  * Borland (2000) M. Borland, “elegant: A Flexible SDDS-Compliant Code for Accelerator Simulation,” in  _Advanced Photon Source LS-287_ (2000).
  * Farvacque  _et al._ (2013) Laurent Farvacque, N Carmignani, J Chavanne, A Franchi, G Le Bec, S Liuzzo, B Nash, T Perron, P Raimondi, _et al._ , “A low-emittance lattice for the ESRF,” Proc. IPAC’13 , 79–81 (2013).
  * Riemann and Streun (2019) B Riemann and A Streun, “Low emittance lattice design from first principles: Reverse bending and longitudinal gradient bends,” Physical Review Accelerators and Beams 22, 021601 (2019).
  * Zhang and Huang (2019) Tong Zhang and Xiaobiao Huang, “Numerical optimization of a low emittance lattice cell,” Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 923, 55–63 (2019).
  * Meller  _et al._ (1987) R. E. Meller, A. W. Chao, J. M. Peterson, Stephen G. Peggs, and M. Furman, “Decoherence of Kicked Beams,” SSC-N-360 (1987).