# Hybrid multi-bend achromat lattice with sextupole cancellation across
straight section

Jiajie Tan  Jianhao Xu  Penghui Yang  Zhenghe Bai<EMAIL_ADDRESS>Lin Wang
<EMAIL_ADDRESS>National Synchrotron Radiation Laboratory, University of
Science and Technology of China, Hefei 230029, China

###### Abstract

The hybrid multi-bend achromat (HMBA) lattice concept is adopted in some
diffraction-limited storage ring designs, which can permit relatively large
on-momentum dynamic aperture and relatively weak sextupoles. In a typical HMBA
lattice, the main arc section is constrained by the transverse phase advances
making –$\it{I}$ transformation for sextupole cancellation. In this paper, a
new HMBA lattice concept with sextupole cancellation across straight section
is proposed, where –$\it{I}$ is made between adjacent dispersion bumps of two
lattice cells. This makes the main arc section free of the phase advance
constraint, and as a result, the number of bending magnets (bends) in the
lattice cell and the cell tunes can be easily changed, thus providing more
choices for lattice design. To achieve the large phase advances required for
–$\it{I}$ in this new concept, split bend is used as the matching bend, which
is a bend split into two pieces with a quadrupole in between. The split bend
also serves to reduce the emittance, and the large phase advances also give
low beta functions in the straight section enhancing the insertion device
brightness. Besides, for a given emittance goal, this new HMBA lattice can
have less bends than the typical HMBA lattice due to stronger focusing in bend
unit cells, which is beneficial for saving space and suppressing intra-beam
scattering effect. Two lattices are given as examples to demonstrate this new
concept and show its linear and nonlinear properties, and further extension is
also discussed.

##  I Introduction

We can relate the natural emittance of an electron storage ring to several
parameters as

$\varepsilon_{x}={F}\frac{E^{2}}{J_{x}N_{b}^{3}},$ (1)

where $F$ is a ring lattice dependent factor, $E$ is the electron energy,
$J_{x}$ is the horizontal damping partition number, and $N_{b}$ is the number
of bending magnets (bends) in the ring. Due to the inverse third-power
dependence of $\varepsilon_{x}$ on $N_{b}$, multi-bend achromat (MBA) lattices
[1] are used for designing diffraction-limited storage rings (DLSRs). Among
them, the hybrid MBA (HMBA) lattice concept [2], developed by ESRF-EBS, has
been adopted in some DLSR designs for its relatively large on-momentum dynamic
aperture (DA) and relatively weak sextupoles. This is attributed to the
creation of a pair of dispersion bumps on both sides of the lattice arc
section, separated by a –$\it{I}$ transformation with horizontal and vertical
phase advances of (3$\pi$, $\pi$), which can make a high efficiency of
chromaticity correction and very effective sextupole cancellation. However,
the constraint of phase advances also limits the flexibility of HMBA lattice.
The number of bends in the lattice cell can not be easily changed, and it is
also difficult to reduce the lattice factor $F$.

Some variants of HMBA lattice have been proposed to pursue better performances
and different requirements for DLSRs. The H7BA lattice of APS-U [3] introduced
reverse bends (RBs) [4] to reduce the emittance; the H6BA lattices of Diamond-
II [5] and HALF [6] have an additional straight section in the middle part
replacing the central bend to accommodate more insertion devices (IDs). In
these lattices, the phase advances between two pumps for –$\it{I}$ remain
unchanged. Choosing other phase advances for –$\it{I}$ can give different
number of bends, such as (5$\pi$, $\pi$) for an H10BA lattice [7] and ($\pi$,
$\pi$) for the PETRA IV lattice [8]. However, such a choice is discontinuous.
The number of bends in the lattice cell is restricted to a large extent, and
so are the cell tunes. Variable number of bends and adjustable cell tunes are
beneficial for lattice design, which can for example reduce the emittance and
better satisfy other needs.

Compared to some conventional MBA lattices [9, 10, 11], HMBA lattices need
more bends to reach the same emittance goal [12] due to relatively weak
focusing in their bend unit cells (corresponding to relatively large lattice
factor $F$), which will make weaker dipole fields and thus longer damping
times. For example, the horizontal tune of the H7BA lattice cell is obviously
smaller than that of the SLS-2 7BA lattice cell. Longer damping times can
cause more serious intra-beam scattering (IBS) effect. This has been reflected
in recently-designed medium-energy HMBA lattices with emittances of only tens
of pm$\cdot$rad [13, 14], where higher energies of 3.5$\sim$4.0 GeV are chosen
and also damping wigglers (DWs) are employed to significantly enhance
radiation damping for suppressing IBS-induced emittance increase and reducing
emittance. Therefore, if the HMBA lattice factor $F$ is reduced by increasing
the focusing of bend unit cells, less bends can be needed for a given
emittance goal, which is good for suppressing IBS and saving space. However,
this is also limited by the phase advance constraint in the arc section.

Moving the phase advance constraint from the main arc section to the both
sides of the lattice cell would solve the problems above. In this paper we
propose a new HMBA lattice concept with sextupole cancellation across straight
section, in which –$\it{I}$ is made between adjacent bumps of two cells. Due
to that the main arc section is no longer constrained by the phase advances,
it is easy to change the number of bends and the cell tunes. However, to
realize this concept, we need to increase the phase advances of both sides of
the cell to about (3$\pi$, $\pi$) for making –$\it{I}$, which are too large in
the normal magnet layout. Inspired by the studies in Refs. [15, 16] we use
split bend as the matching bend to increase the phase advances, and also the
beta functions in the straight section are lowered. The former can also reduce
the emittance, and the latter enhances the ID brightness. Besides, the
relatively strong focusing in the split bends can also make the focusing in
other bends stronger due to the need for optics matching, thus reducing the
lattice factor $F$.

The content of this paper is organized as follows. In Sec. II, we describe the
new HMBA lattice concept, and then some related properties of split bend cell
employed in the concept are studied using simplified lattice model. In Sec.
III, two HMBA lattices are presented as examples to illustrate the lattice
concept and also to show the lattice properties. In Sec. IV, possible lattice
extension based on the sextupole cancellation of the concept is discussed.
Finally, we summarize the conclusion in Sec. IV.

##  II Lattice concept

Figure 1:  Schematic of –$\it{I}$ transformation between two dispersion bumps
for the typical HMBA lattice (upper) and new HMBA lattice (lower).

The upper part of Fig. 1 shows the schematic of –$\it{I}$ transformation made
between two dispersion bumps of the typical HMBA lattice cell. The number of
bends in the lattice cell and the cell tunes are largely dependent on the
inner part of the arc section, which is however constrained by the phase
advances for –$\it{I}$. To have better flexibility, here we propose a new HMBA
lattice concept with sextupole cancellation across straight section. As shown
in the lower part of Fig. 1, –$\it{I}$ is made between adjacent dispersion
bumps of two lattice cells, which will make the inner part of the arc section
free of the phase advance constraint. To do so, we need to get –$\it{I}$ with
the matching sections and straight section, which is however difficult to
realize with a normal magnet layout. In most HMBA lattice designs [2, 3, 5,
6], the horizontal and vertical phase advances of the matching sections and
straight section, denoted as ($\Delta\phi_{x}$, $\Delta\phi_{y}$), are about
(0.9×2$\pi$, 0.35×2$\pi$), which are far from the required (1.5×2$\pi$,
0.5×2$\pi$), especially in the horizontal plane. SOLEIL designed an HMBA
lattice [17] with very small horizontal beta function in the straight section,
which has a larger $\Delta\phi_{x}$ of about 1.2×2$\pi$. But it is still not
enough to satisfy the required value for –$\it{I}$.

Figure 2:  Layout of the simplified unit cell model with split bend. In the
model, $q1$ and $q3$ are horizontally focusing quadrupoles and $q2$ is
defocusing quadrupole. Figure 3:  Cell horizontal phase advance $\mu_{x}$ vs.
quadrupole strength $k1$ for scaned solutions, with colour denoting the
emittance ratio $\varepsilon_{x}/\varepsilon_{TME}$.

Figure 4:  Left: Pareto front of split-bend quadrupole strength $k1$ and
emittance ratio $\varepsilon_{x}/\varepsilon_{DBA}$ for a DBA cell model,
where the colour denotes the cell horizontal phase advance. Right: $k1$ vs.
minimum $\varepsilon_{x}/\varepsilon_{DBA}$ for the DBA cell model with phase
advances of about (3$\pi$, $\pi$), where the colour denotes the horizontal
beta function at the middle of straight section.

Inspired by the studies in [15, 16], we introduce split bends as the matching
bends so as to increase the phase advances ($\Delta\phi_{x}$,
$\Delta\phi_{y}$). The split bend is such a bend which is split into two
pieces with a quadrupole in between. It increases phase advances by
introducing additional focusing. We first use a simplified unit cell model
shown in Fig. 2 to study the focusing property of split bend. Similar cell
model has been mentioned in Ref. [15], where the split bend cell was studied
with fixed phase advances. Thin lens approximation is used for quadrupoles in
this model. In the study, for simplicity, the ratio of the bend length to the
total drift length is set to 1. Here we set the length of drift $d1$ to 1 m;
the half of the split bend, $b1$, has a length of 1 m and a bending angle
$\theta$/2 of 1°. The whole ring consists of 180 identical cells. The
strengths $k1$, $k2$, $k3$ of three quadrupole families $q1$, $q2$, $q3$ are
allowed to vary in the ranges of [0, 5], [-5, 0] and [0, 5] $m^{-1}$,
respectively. We scanned the quadrupole strengths with the step of 0.01
$m^{-1}$.

Fig. 3 presents the results of scanned solutions. The emittances of solutions
are compared to the theoretical minimum emittance (TME) [18],
$\varepsilon_{TME}$, with the same bending angle $\theta$, which are shown as
ratios of $\varepsilon_{x}/\varepsilon_{TME}$ in the figure. The solutions
with emittances larger than 5×$\varepsilon_{TME}$ are not shown in the figure.
The split bend cell becomes a TME-like cell when $k1=0$. When $k1$ has a small
value, the horizontal phase advance of the unit cell, $\mu_{x}$, is less than
0.5×2$\pi$. As $k1$ increases, $\mu_{x}$ shifts its maximum towards 2$\pi$,
and lower emittance can be obtained. This means that we can get a horizontal
phase advance of 1.5×2$\pi$ with two split bend unit cells. The red area
indicates that the emittance can be reduced to lower than $\varepsilon_{TME}$
with appropriate values of $k1$ and $\mu_{x}$.

Our goal is to make an HMBA lattice with –$\it{I}$ across straight section by
employing split bends. The –$\it{I}$ part of this HMBA lattice can be seen as
a double-bend achromat (DBA) lattice cell with two split bends, which has
horizontal and vertical phase advances of about (3$\pi$, $\pi$). Then we study
the DBA cell with split bends using a simplified model similar to the unit
cell model above. As shown at the top of Fig. 5, the layout from the first
quadrupole to the center of the DBA cell is like the model in Fig. 2. The
difference is that the parameters and positions of magnets are not symmetric
about the quadrupole inserted in the split bend (hereafter called split-bend
quadrupole). In the DBA cell model, the sum of bending angles of two pieces of
the split bend is also set to $\theta$=2° and the sum of their lengths is 2 m,
and the sum of two different drifts between quadrupoles is 2 m (i.e. here the
ratio of the bend length to the total length of two drifts is also 1).
Besides, the length of the straight section is set to 5 m. With these
settings, the NSGA-III algorithm [19] was used to study the DBA cell, with
magnet parameters and positions as variables.

In the study, the optimization objectives are to minimize the split-bend
quadrupole strength $k1$ and the emittance ratio
$\varepsilon_{x}/\varepsilon_{DBA}$, where $\varepsilon_{DBA}$ is the minimum
emittance of DBA lattice, with $\varepsilon_{DBA}=3\times\varepsilon_{TME}$
[20]. For the constraints, first we only considered the dispersion-free
condition: the absolute dispersion in the straight section is less than 0.001
m. The optimized Pareto front of the two objectives is shown in the left plot
of Fig. 4, and the horizontal phase advance of the DBA cell is also calculated
and shown in the figure for these Pareto solutions. We see that as $k1$
increases, the minimum emittance obtained decreases roughly linearly, and the
cell horizontal phase advance increases. When $k1=0$, i.e. the usual bend is
used, the emittance is about $1.5\times\varepsilon_{DBA}$.

Then, the cell phase advance condition was further considered as the second
constraint in the study. For this constraint, we set the horizontal phase
advance in the range of 1.48$\sim$1.52×2$\pi$ and the vertical in the range of
0.48$\sim$0.52×2$\pi$. Note that the horizontal and vertical phase advances
can not be exactly equal to 3$\pi$ and $\pi$, respectively, for having a
stable solution. Due to that the dispersion-free and phase advance constraints
we set here are very stringent, it is very difficult to obtain the true Pareto
front for the two objectives [21]. So we scanned $k1$ so as to obtain better
results, and at each value of $k1$, $\varepsilon_{x}/\varepsilon_{DBA}$ was
minimized. The optimization results are shown in the right plot of Fig. 4, and
for these solutions, their horizontal beta functions at the middle of straight
section, $\beta_{x,0}$, are also shown as additional information. We see that
with the phase advance constraint further considered, the emittance also
decreases roughly linearly with increasing $k1$ and the minimum emittance is
slightly less than $0.5\times\varepsilon_{DBA}$. We also see that as $k1$
increases, $\beta_{x,0}$ increases but is less than 3 m. The optimum beta
function for matching the electron and photon beam phase spaces to enhance the
ID brightness is $\beta_{x,0}=L/\pi$ [22] with $\it{L}$ being the ID length.
It is interesting to see that at smaller $k1$, the emittance is larger and
$\beta_{x,0}$ is too small (less than 1 m), both of which are not good for
brightness enhancement. While at larger $k1$, both lower emittance and
reasonably low $\beta_{x,0}$ can promise higher brightness.

Such a DBA lattice cell with phase advances of about (3$\pi$, $\pi$) is shown
in Fig. 5. The emittance of this DBA lattice is 0.64$\times\varepsilon_{DBA}$.
The split bend of this DBA lattice is not divided into two identical pieces as
that of the unit cell shown in Fig. 2. The outer piece at lower dispersion has
a shorter length and higher dipole field compared to the inner piece. As the
same principle of longitudinal gradient bend (LGB) [23, 24] for reducing the
emittance, distributing high dipole field at the position with low dispersion
can help to suppress the quantum excitation, thus giving small dispersion
action $\mathcal{H}_{x}$. Besides, the horizontal and vertical beta functions
in the straight section have small values so that the ID brightness can be
further enhanced.

Figure 5:  Simplified model layout and optical functions of a DBA lattice cell
with split bends, which has horizontal and vertical phase advances of about
(3$\pi$, $\pi$). In the layout, split bends are shown in blue and thin-lens
quadrupoles are shown as arrows.

In addition, the focusing in the split bend of this DBA lattice is relatively
strong. When other bends are inserted in the middle part to make the new HMBA
lattice, the focusing in these bends can also become strong due to the need
for better optics matching, which will be shown in the practical lattice
design of the next section.

##  III Examples

As examples, an H8BA lattice and an H6BA lattice with sextupole cancellation
across straight section were preliminarily designed. The two designs have the
same energy of 3 GeV and the same storage ring circumference of 460.8 m. The
H8BA design has 20 lattice cells, and the H6BA design has 24 cells. The NSGA-
III algorithm was also used in the lattice design. Following Ref. [7], in the
linear optics optimization, not only the natural emittance but also the
natural chromaticities and integrated strengths of sextupoles were optimized,
so that the lattice solutions with potentially good nonlinear dynamics can be
easily got. The main storage ring parameters of the two designs are listed in
Table 1.

Table 1:  Main storage ring parameters of two lattices Parameters | H8BA | H6BA
---|---|---
Energy [GeV] | 3
Circumference [m] | 460.8
Number of cells | 20 | 24
Length of the straight |  |
section [m] | 5
Natural emittance [pm$\cdot$rad] | 74.5 | 136.2
Betatron tunes (H/V) | 56.14/15.18 | 52.21/17.13
Natural chromaticities |  |
(H/V) | -106.6/-100.0 | -94.3/-88.9
Momentum compaction | 1.29×$10^{-4}$ | 1.47×$10^{-4}$
Damping partition |  |
numbers (H/V/L) | 1.91/1.0/1.09 | 1.58/1.0/1.41
Natural damping times |  |
(H/V/L) [ms] | 13.8/26.3/24.0 | 14.3/22.6/16.0
Energy lost per turn [keV] | 351.7 | 407.8
Total absolute bending |  |
angle [°] | 410.4 | 406.7
$\beta_{x}$/$\beta_{y}$ at the middle of |  |
straight section [m] | 2.88/1.83 | 2.48/2.39

Fig. 6 shows the magnet layout and linear optics of the designed H8BA lattice
cell. Due to that the matching bend, i.e. the split bend, has two pieces of
bends, here this lattice is regarded as an H8BA lattice. In this lattice cell,
the horizontal and vertical phase advances between the starting position and
the peak of the first dispersion bump are about half of (3$\pi$, $\pi$).
Except that the two central bends are combined-function bends, the other main
bends are LGBs, and two families of RBs are also employed to help reduce the
natural emittance down to 74.5 pm·rad. The beta functions at the middle of the
straight section have values of 1.5$\sim$3 m. Such small beta functions and
the ultra-low emittance can significantly increase the ID brightness.

Figure 6:  Magnet layout and optical functions of the H8BA lattice cell. Main
bends are in blue, RBs in cyan, quadrupoles in red and sextupoles in green.

This H8BA lattice can be compared with the recent H9BA lattice of SSRL-X [13],
which also uses split bends and has low beta functions in the straight
section. But the main arc section of the H9BA lattice is constrained by
–$\it{I}$. The horizontal tune of the H9BA lattice cell is about 2.76, while
the H8BA lattice cell has a larger horizontal tune of about 2.81, though it
has smaller number of bends. This means that the bend unit cells of the H8BA
lattice have stronger focusing. For an emittance goal, smaller number of bends
together with stronger focusing will generally give higher dipole fields and
thus shorter damping times, which is beneficial for suppressing the IBS
induced emittance increase.

In the H8BA lattice, three families of sextupoles were used to correct the
chromaticities and optimize the nonlinear dynamics. Fig. 7 shows the optimized
on-momentum DA, tracked with ELEGANT [25]. We see that the horizontal 4D DA is
about -8$\sim$6 mm, which is relatively large considering the ultra-low
emittance and the small horizontal beta function. With RF cavity included, the
DA shrinks due to the path lengthening effect as in other HMBA lattices.

Figure 7:  On-momentum DA of the H8BA lattice, tracked at the middle of the
straight section.

Due to the –$\it{I}$ made across the straight section, the sextupole resonance
driving terms [26] are not cancelled when observed in the straight section,
and they are cancelled when observed in the middle part of the arc section.
Fig. 8 shows the transverse phase space tori tracked at two locations. We see
that the horizontal phase space tori are much distorted at the middle of the
straight section. Besides, IDs installed in straight sections may disturb the
nonlinear cancellation, while the low beta functions in straight sections are
beneficial for reducing the tune shifts and beta-beat produced by IDs. Here we
will not further study the ID effect and compensation. Anyway, for this
lattice, we can adopt the on-axis swap-out injection [27], which can allow a
DA of only $\sim$1 mm.

Figure 8:  Horizontal (top) and vertical (bottom) phase space tori of the H8BA
lattice, tracked at the middle of the straight section (left) and the middle
of the arc section (right).

To reflect the flexibility of the new lattice concept, an H6BA lattice is also
designed, shown in Fig. 9, which removes the two central combined-function
bends from the H8BA lattice. The natural emittance is 136.2 pm·rad. Recently,
PETRA IV also adopted an H6BA lattice [8] with split bends used and low beta
in the straight, but the two central bend cells are constrained by –$\it{I}$
with transverse phase advances of ($\pi$, $\pi$). The horizontal tune of the
H6BA lattice cell here is near 2.2, obviously larger than the value of less
than 1.8 for the PETRA IV H6BA lattice cell, which is beneficial for reducing
the emittance. As said before, this can also reduce the damping times for an
emittance goal so as to better suppress the IBS effect. It is interesting to
note that the central two bends can also be seen as a split bend. If we regard
the split bend as a single bend, then this H6BA lattice can be seen as a
triple-bend achromat lattice.

Figure 9:  Magnet layout and optical functions of the H6BA lattice cell.
Octupoles are in pink, and the other magnets are as in Fig. 6.

Fig. 10 shows the horizontal DAs for a range of momentum deviations, optimized
with three sextupole families and one octupole family. We see that the on-
momentum horizontal DA is larger than 6 mm and the dynamic momentum acceptance
is large. The local momentum aperture (LMA) is shown in Fig. 11. Based on this
error-free LMA, the Touschek lifetime was calculated, which is about 9.7 hours
for 10%-coupling beam with bunch charge of 1 nC. In the calculation, the
frequency of RF cavity is 500 MHz, and the bunch is assumed to be lengthened
by a factor of 4 with harmonic cavity.

Figure 10:  Horizontal DAs of the H6BA lattice for relative momentum
deviations of -6%$\sim$6%, tracked at the middle of the straight section.
Figure 11:  Local momentum aperture along one H6BA lattice cell. The RF
momentum acceptance is set to about 5%.

The H6BA design has more straight sections than the H8BA, and we can place
some DWs in straight sections to further reduce the emittance and also to
reduce the damping times for suppressing IBS induced emittance increase. With
4 DWs employed, each with length of 4.2 m, peak field of 2.2 T and period of
105 mm, the natural emittance is reduced to 88.4 pm·rad. And the damping times
(H/V/L) are reduced to 8.3/10.6/6.1 ms with $J_{x}$ of 1.27, smaller than that
without DW. When including IBS, the horizontal emittance increases to 99.2
pm·rad for the beam as in the Touschek lifetime calculation. If full-coupling
beam is used, the horizontal emittance can be further reduced and the
horizontal emittance increase due to IBS can also be better suppressed.

##  IV Extension

The nonlinear cancellation scheme proposed above can be extended to produce
other MBA lattices. If the emittances of DLSRs are further reduced, such as
the $\sim$10 pm·rad emittance design for MAX IV future upgrade [28], the
number of bends in an MBA lattice will increase. If conventional MBA lattices
are used, like a 19BA lattice for MAX IV upgrade [28], the magnet strengths
will be extremely strong and the magnet layout will be unusually compact. A
possible solution to mitigate the magnet and space issue is to use an HMBA-
like lattice with few dispersion bumps. If there are only one pair of bumps in
an MBA lattice with many bends, like the ALS-U 9BA lattice [29] and the H10BA
lattice [7], it is very hard to have good off-momentum nonlinear dynamics for
reasonable beam lifetime. Introducing another pair of bumps may improve the
off-momentum nonlinear dynamics [30]. Fig. 12 shows the schematic of an MBA
lattice with two pairs of –$\it{I}$ separated bumps. The two pairs of bumps
are non-interleaved: the inner pair is inside the cell and the outer pair is
the same as the scheme proposed in this paper. This non-interleaved layout is
good for improving nonlinear dynamics performance.

Besides, due to that the inner part of the arc section in the new HMBA lattice
is free, it can be easily extended to a combined lattice, H$m$BA-H$n$BA
lattice ($m$$\neq$$n$), without affecting the nonlinear cancellation.

Figure 12:  Schematic of an MBA lattice with two pairs of –$\it{I}$ separated
dispersion bumps.

##  V Conclusion

In this paper, we studied a new HMBA lattice concept to make the main arc
section free of the phase advance constraint so as to increase the flexibility
of lattice. In the concept, –$\it{I}$ transformation for sextupole
cancellation is moved to both sides of the lattice cell. To increase the phase
advances for satisfying the –$\it{I}$, split bends are used as matching bends,
which also reduce the emittance, and beta functions in the straight section
are lowered, which is also good for enhancing ID brightness. The bend unit
cells of the new HMBA lattice can have stronger focusing than those of the
typical HMBA lattice, which is beneficial for reducing the emittance and also
for suppressing the IBS effect for a given emittance goal. The DA of the new
lattice is relatively large, but the phase space tori are distorted in the
straight section due to that the nonlinear cancellation is made across the
straight section. This cancellation can also be extended to produce other MBA
lattices for DLSRs.

###### Acknowledgements.

This work was supported by the National Key Research and Development Program
of China under Grant No. 2016YFA0402000 and the National Natural Science
Foundation of China under Grant Nos. 11875259 and 12205299.

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# Decoupling of static and dynamic criticality in a driven Mott insulator

A. de la Torre  Department of Physics, California Institute of Technology,
Pasadena, CA 91125, USA  Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena, CA 91125, USA  K. L. Seyler
Department of Physics, California Institute of Technology, Pasadena, CA 91125,
USA  Institute for Quantum Information and Matter, California Institute of
Technology, Pasadena, CA 91125, USA  M. Buchhold  Department of Physics,
California Institute of Technology, Pasadena, CA 91125, USA  Institute for
Quantum Information and Matter, California Institute of Technology, Pasadena,
CA 91125, USA  Y. Baum  Department of Physics, California Institute of
Technology, Pasadena, CA 91125, USA  Institute for Quantum Information and
Matter, California Institute of Technology, Pasadena, CA 91125, USA  G. Zhang
Department of Physics, University of California, San Diego, La Jolla,
California 92093, USA  N.J. Laurita  Department of Physics, California
Institute of Technology, Pasadena, CA 91125, USA  J.W. Harter  Department of
Physics, California Institute of Technology, Pasadena, CA 91125, USA
Institute for Quantum Information and Matter, California Institute of
Technology, Pasadena, CA 91125, USA  L. Zhao  Department of Physics,
California Institute of Technology, Pasadena, CA 91125, USA  Institute for
Quantum Information and Matter, California Institute of Technology, Pasadena,
CA 91125, USA  I. Phinney  Department of Physics, California Institute of
Technology, Pasadena, CA 91125, USA  Institute for Quantum Information and
Matter, California Institute of Technology, Pasadena, CA 91125, USA  X. Chen
Department of Physics, Boston College, Chestnut Hill, MA, 02467, USA
Materials Department, University of California, Santa Barbara, CA, 93106, USA
S. D. Wilson  Materials Department, University of California, Santa Barbara,
CA, 93106, USA  G. Cao  Department of Physics, University of Colorado,
Boulder, CO 80309, USA  R. D. Averitt  Department of Physics, University of
California, San Diego, La Jolla, California 92093, USA  G. Refael  Department
of Physics, California Institute of Technology, Pasadena, CA 91125, USA
Institute for Quantum Information and Matter, California Institute of
Technology, Pasadena, CA 91125, USA  D. Hsieh<EMAIL_ADDRESS>Department
of Physics, California Institute of Technology, Pasadena, CA 91125, USA
Institute for Quantum Information and Matter, California Institute of
Technology, Pasadena, CA 91125, USA

Dynamically driven interacting quantum many-body systems have the potential to
exhibit properties that defy the laws of equilibrium statistical mechanics. A
widely studied model is the impulsively driven antiferromagnetic (AFM) Mott
insulator, which is predicted to realize exotic transient phenomena including
dynamical phase transitions into thermally forbidden states Werner  _et al._
(2012); Sandri and Fabrizio (2013); Tsuji  _et al._ (2013) and highly non-
thermal magnon distributions Walldorf  _et al._ (2019). However such far-from-
equilibrium regimes, where conventional time-dependent Ginzburg-Landau
descriptions fail, are experimentally challenging to prepare and to probe
especially in solid state systems. Here we use a combination of time-resolved
second harmonic optical polarimetry and coherent magnon spectroscopy to
interrogate $n$-type photo-doping induced ultrafast magnetic order parameter
(MOP) dynamics in the Mott insulator Sr2IrO4. We find signatures of an unusual
far-from-equilibrium critical regime in which the divergences of the magnetic
correlation length and relaxation time are decoupled. This violation of
conventional thermal critical behavior arises from the interplay of photo-
doping and non-thermal magnon population induced demagnetization effects. Our
findings, embodied in a non-equilibrium “phase diagram”, provide a blueprint
for engineering the out-of-equilibrium properties of quantum matter, with
potential applications to terahertz spintronics technologies.

The low energy electronic structure of the strongly spin-orbit coupled Mott
insulator Sr2IrO4 consists of a completely filled band of spin-orbital
entangled pseudospin $J_{\rm eff}=3/2$ states and a narrow half-filled band of
$J_{\rm eff}=1/2$ states, which splits into lower and upper Hubbard bands due
to on-site Coulomb repulsion Kim  _et al._ (2008). Short-range AFM
correlations between $J_{\rm eff}=1/2$ moments within a layer are established
well above room temperature Fujiyama  _et al._ (2012); Vale  _et al._ (2015).
But owing to weak interlayer exchange coupling ($J_{c}<10^{-3}J_{ab}$) Vale
_et al._ (2015); Porras  _et al._ (2019), three-dimensional (3D) long-range
AFM ordering only occurs below a Néel temperature $T_{\rm N}=230$ K. The
ordered structure exhibits a weak in-plane ferromagnetic moment in each layer
due to pseudospin canting, which is staggered along the $c$-axis (Fig. 1a).
Resonant x-ray and neutron scattering measurements of La-doped Sr2IrO4 have
shown that a small concentration of $n$-type carriers destroys $c$-axis
ordering, leaving behind a paramagnetic state with short-range 2D AFM
correlations Chen  _et al._ (2015); Gretarsson  _et al._ (2016); Pincini  _et
al._ (2017). As such, $n$-type photo-doping, which may be simulated by
optically pumping electrons from the electronically inert $J_{\rm eff}=3/2$
band into the upper Hubbard band, provides a potential pathway to impulsively
quench long-range AFM order.

Recently, both ultrafast optical and free-electron laser-based resonant x-ray
scattering (RXS) methods have been used to probe photo-doping induced
pseudospin dynamics in iridates Dean  _et al._ (2016); Krupin  _et al._
(2016); Afanasiev  _et al._ (2019); Mazzone  _et al._ (2021); Pastor  _et al._
(2021). In the case of Sr2IrO4, one study showed that it could be made
magneto-optically active by aligning the canted moments in an external
magnetic field Afanasiev  _et al._ (2019). Transient magnetization dynamics
were then probed in the near-equilibrium weak photo-doping limit using time-
resolved magneto-optical Kerr effect (MOKE) spectroscopy. The recovery time of
the MOP was observed to diverge at $T_{\rm N}$, which was explained within a
time-dependent Ginzburg-Landau framework. In another pair of studies, time-
resolved RXS was used to probe MOP dynamics without a field in a far-from-
equilibrium strong photo-doping regime Dean  _et al._ (2016); Krupin  _et al._
(2016). A non-thermal melting of the MOP was achieved, but no critical
dynamics were resolved across this phase transition.

The behavior of a system near a dynamical phase boundary should be sensitive
to small changes in excitation density. In order to resolve out-of-equilibrium
critical dynamics, it is therefore imperative that the excitation density be
uniform throughout the probed volume and finely sampled near the phase
boundary. Since the pump excitation density invariably decays with depth into
the sample, one should in principle probe exclusively within a small interval
of depth to minimize the effects of averaging over different excitation
densities, which might otherwise obscure features of criticality (see
Supplementary sections S1 and S2).

This condition is well met using an optical second harmonic generation
rotational anisotropy (SHG-RA) technique (Fig. 1b), which was recently shown
to be directly sensitive to the MOP in the topmost layer of Sr2IrO4 in zero
magnetic field Di Matteo and Norman (2016); Seyler  _et al._ (2020). We note
that while an additional hidden order has been reported in Sr2IrO4 Jeong  _et
al._ (2017); Murayama  _et al._ (2021), there is no indication that SHG
couples to any hidden order Seyler  _et al._ (2020), contrary to a previous
interpretation Zhao  _et al._ (2016). In an SHG-RA experiment, light is
focused obliquely onto the (001) face of Sr2IrO4 and the intensity of light
reflected at twice the incident frequency $I(2\omega)$ is measured as the
scattering plane is rotated about the surface normal. In equilibrium, SHG-RA
patterns acquired above $T_{\rm N}$ exhibit four-fold ($C_{4}$) rotational
symmetry, arising from a bulk electric-quadrupole (EQ) radiation from its
centrosymmetric crystallographic point group Seyler  _et al._ (2020). Below
$T_{\rm N}$, an additional surface magnetization-induced electric-dipole (ED)
contribution turns on and interferes with the EQ contribution, lowering the
symmetry of the SHG-RA pattern from $C_{4}$ to $C_{1}$ (Fig. 1c).

To study the effects of $n$-type photo-doping, we developed time-resolved
pump-probe SHG-RA using the apparatus depicted in Figure 1b. The pump beam,
which was tuned on resonance with the $J_{\rm eff}=3/2$ band to upper Hubbard
band transition, and probe beam were focused within a single magnetic domain
for our experiments. Figure 1d shows transient SHG-RA data acquired at the
instant of pump excitation ($t$ = 0) as a function of pump fluence ($F$). In
the un-pumped case ($F$ = 0), the $C_{1}$ symmetry of the SHG-RA pattern is
manifested through the presence of a dominant intensity lobe, indicating a
finite ED contribution and thus a finite MOP. As $F$ increases, the intensity
of the dominant lobe decreases linearly until it plateaus above a critical
fluence $F_{\rm c}\approx 0.9$ mJ/cm2. Beyond $F_{\rm c}$, the non-magnetic
$C_{4}$ symmetry is fully restored to the SHG-RA pattern, signaling a
collapsed MOP. The slightly higher $C_{4}$-EQ yield from the pump-induced
compared to heating-induced non-magnetic state likely arises from un-relaxed
bulk magneto-elastic deformations that will be discussed in more detail below.
The prompt change in the SHG-RA pattern within the time resolution of our
instrument ($<200$ fs) for all fluences is consistent with a photo-doping
scenario. Specifically, AFM order is suppressed by magnetic defects that are
left in the wake of propagating doublons (Fig. 1b), which are generated on the
time scale of the nearest-neighbor hopping ($<20$ fs) Takahashi  _et al._
(2002); Piovera  _et al._ (2016); Afanasiev  _et al._ (2019). Moreover, the
observed critical fluence corresponds to an excitation density ($n_{\rm ex}$)
of approximately 0.05 per iridium site, which is close to the reported
critical La-doping level to suppress AFM order Chen  _et al._ (2015);
Gretarsson  _et al._ (2016), suggestive of an optical $n$-type doping induced
quench mechanism.

More detailed comparisons between the properties of Sr2IrO4 in and out-of-
equilibrium can be made by mapping the fluence versus temperature magnetic
“phase diagram”. This is accomplished by collecting an array of SHG-RA
patterns immediately after optical excitation across different starting
temperatures and pump fluences and then tracking the onset of the ED
contribution. For $F$ = 0, the ED contribution emerges below $T_{\rm N}$ as
expected (Fig. 2a). With increasing $F$, this transition shifts monotonically
to lower temperatures, indicating that $F_{\rm c}$ increases upon cooling. By
plotting $T_{\rm N}$ as a function of $n_{\rm ex}$, or equivalently $F_{\rm
c}$ as a function of $T$, a sharp out-of-equilibrium magnetic phase boundary
is identified. Despite an instantaneous electronic temperature that already
far exceeds $T_{\rm N}$ at $F\approx 0.2$ mJ/cm2 based on the electronic
specific heat of Sr2IrO4 (see Supplementary section S3), the photo-doping and
La-doping phase boundaries nearly coincide over the small La-doping range
where AFM order exists in equilibrium Chen  _et al._ (2015); Gretarsson  _et
al._ (2016) (Fig. 2b). This indicates that heat transfer from the charge to
pseudospin subsystem is negligible near $t$ = 0. Since chemical doping differs
from photo-doping in non-trivial ways, including the introduction of disorder
that is known to nucleate non-magnetic metallic puddles in La-doped Sr2IrO4
Chen  _et al._ (2015); Battisti  _et al._ (2017), we refrain from drawing more
detailed comparisons.

At later time delays, the charge, pseudospin and lattice subsystems are
typically expected to thermalize and thus the out-of-equilibrium phase
diagrams mapped at $t\approx 0$ and $t\gg 0$ should be different. Figure 2c
shows the intensity of the dominant SHG-RA lobe over a range of fluence
values, converted into effective temperatures using the total specific heat of
Sr2IrO4 (see Methods), acquired at $t=10$ ps, which far exceeds the reported
charge and lattice relaxation timescales but is much shorter than the
timescale for heat to escape the probed region Hsieh  _et al._ (2012); Piovera
_et al._ (2016). At low fluences we find excellent agreement with the un-
pumped temperature dependence data, confirming a pure optical heating effect.
Surprisingly however, there is increasing bifurcation of the curves above
$F\approx 0.8$ mJ/cm2, leading to a growing mismatch between the out-of-
equilibrium phase boundary mapped at $t=10$ ps and that calculated assuming
pure optical heating as a function of fluence (Fig. 2d). This indicates that
thermalization of the pseudospin subsystem is impeded at high fluences.

A hallmark of the La-doping induced paramagnetic state is the persistence of
short-range intralayer AFM correlations, manifested through remnant dispersive
2D magnons detected by resonant inelastic x-ray scattering Gretarsson  _et
al._ (2016); Pincini  _et al._ (2017). To search for similar intralayer
correlations in the photo-induced paramagnetic phase of Sr2IrO4, we leverage
the fact that the zone center 2D magnon mode with $B_{2g}$ symmetry is weakly
gapped ($\approx 2$ meV) and Raman active Gim  _et al._ (2016); Gretarsson
_et al._ (2017). This allows the magnon to be coherently excited by our pump
pulse via impulsive stimulated Raman scattering and optically tracked in the
time domain using ultrafast MOKE spectroscopy Seifert and Balents (2019).
Figures 2e,f show typical SHG and MOKE transients acquired in the high pump
fluence ($F\gg F_{\rm c}$) regime. Despite the complete collapse of the MOP
over the displayed 10 ps time window (Fig. 2e), coherent oscillations of the
infinite wavelength $B_{2g}$ magnon continue to be supported (Fig. 2f). This
provides clear evidence of a significant 2D magnetic correlation length in the
transient paramagnetic state, suggesting a non-thermal analogue of La-doped
Sr2IrO4 is realized. These results are consistent with and complementary to a
time-resolved RXS study of Sr2IrO4 Dean  _et al._ (2016), which showed that
photo-doping strongly suppresses an AFM Bragg peak but does not alter the main
high energy and short wavelength features of the intralayer magnon spectrum
(see Supplementary section S4).

Having comprehensively mapped the transient magnetic phase diagram of Sr2IrO4,
we now finely examine the out-of-equilibrium critical dynamics. Figures 3a,b
show the time-dependent change in intensity $\Delta I(2\omega)$ of the
dominant SHG-RA lobe acquired at $T=80$ K over a range of pump fluences. For
$F\ll F_{\rm c}$ (Fig. 3a), we observe a rapid drop in $\Delta I(2\omega)$ at
$t$ = 0 signifying a reduction of the MOP. This is followed by an exponential
recovery with rise time $\tau\approx 1–2$ ps towards a value negatively offset
from the $t<0$ intensity, which we showed (Fig. 2c,d) represents a thermalized
state with a slightly elevated temperature. As $F$ increases, $\tau$ rises
slightly but exhibits no discontinuity at $F_{\rm c}$ despite the MOP
vanishing. However as $F$ exceeds $F_{\rm c}$, the point where $C_{4}$
symmetry is restored, $\tau$ continues to grow and eventually diverges around
$F^{*}\approx 1.6$ mJ/cm2 (Fig. 3b,c). Similar dynamics are observed if
instead $F$ is kept fixed and $T$ is varied (inset Fig. 3a). These data reveal
that the divergence of the magnetic correlation length and relaxation time
occur along separated critical lines in the out-of-equilibrium phase diagram
marked by $F_{\rm c}$ and $F^{*}$ respectively (inset Fig. 3c). Such a
decoupling is forbidden in equilibrium and is also not observed across
dynamical phase transitions in weakly correlated ferromagnets Tengdin  _et
al._ (2018) and charge density wave systems Zong  _et al._ (2019).

We note the presence of a subtle exponential drop just after $t$ = 0 that
becomes more pronounced at large $F$. This causes $\Delta I(2\omega)$ to drop
below the instantaneous EQ value (gray bar in Fig. 1 and Figs. 3a,b and
ultimately plateau at the high temperature EQ value upon reaching $F^{*}$.
Given that magneto-elastic deformations are known to occur below $T_{\rm N}$
in Sr2IrO4 Porras  _et al._ (2019) and can cause small changes in the EQ
response, a possible origin of this exponential component is lattice
relaxation following impulsive suppression of the MOP. Therefore, only at
large $F$ when $\tau$ sufficiently exceeds the relevant phonon timescales do
the magneto-elastic deformations have time to fully relax (see Supplementary
section S5). Although a direct confirmation awaits time-resolved
crystallography measurements, the main conclusions of our work are independent
of this interpretation.

To check whether the divergence of $\tau$ simply results from a prolonged
photo-dopant lifetime, we performed simultaneous transient linear reflectivity
($\Delta R/R$) and SHG-RA measurements below $T_{N}$ to directly compare the
charge and pseudospin dynamics. As shown in Figures 4a-c, the generation of
photo-dopants and their subsequent suppression of the MOP both occur within
the time resolution of our experiment. However, unlike the SHG response, we
observe no saturation in the amplitude of $\Delta R/R$ and no significant
change in its characteristic recovery time ($\tau_{0}\approx 1$ ps) as a
function of fluence (inset Fig. 4a), demonstrating that the slow dynamics
observed at $F^{*}$ occur exclusively in the pseudospin sector. This suggests
that $\tau$ must be governed by the relaxation of excess magnons that are
emitted upon doublon decay. One possibility is that the magnons rapidly
thermalize to a higher temperature and that $\tau$ represents the timescale
for the hot pseudospin subsystem to cool via heat exchange with the lattice
and charge subsystems. However, a three-temperature model describing this
process predicts that magnons should cool faster at higher excitation
densities. Moreover, it fails to produce any diverging timescale (see
Supplementary sections S3 and S6). This points to the alternative possibility
that $\tau$ represents the timescale for the pseudospin subsystem to
internally thermalize.

The rate-limiting step for recovering long-range magnetic order is the
establishment of $c$-axis correlations (Fig. 2e,f). Therefore $\tau$ must be
set by the thermalization time of interlayer $c$-axis magnons. However the
large mismatch between $J_{c}$ ($\approx$ 10 $\mu$eV)Vale  _et al._ (2015);
Porras  _et al._ (2019) and both the charge gap ($\approx 250$ meV)de la Torre
_et al._ (2015) and lowest optical phonon energy ($\approx 10$ meV)Gretarsson
_et al._ (2017) of Sr2IrO4 should impede $c$-axis magnon thermalization via
charge or phonon excitation processes, leaving magnon-magnon scattering as the
dominant thermalization channel. Building on the seminal work of Hohenberg and
Halperin Hohenberg and Halperin (1977), we study the critical dynamics after
doublon decay of this effectively closed $c$-axis magnon subsystem via a
general stochastic Langevin equation for a real order parameter in one spatial
dimension (see Supplementary section S7):

$\partial_{t}\varphi_{k}\left(t\right)=-\left({k^{2}+\tau}_{0}^{-1}\right)\varphi_{k}\left(t\right)-\lambda\int\frac{dk_{1}}{2\pi}\int{\frac{dk_{2}}{2\pi}{\
\varphi}_{k_{1}}\left(t\right)\varphi_{k_{2}}\left(t\right)\varphi_{{k-k}_{1}-k_{2}}\left(t\right)+\xi_{k}\left(t\right)}$
(1)

where $\varphi_{k}\left(t\right)$ is the $k$-th Fourier component of the MOP
field $\varphi\left(z,t\right)$ that parameterizes the ordering of the
intralayer Néel vector along the $c$-axis (Fig. 4d), $\tau_{0}^{-1}$ is a
temperature-dependent mass term that is equivalent to the thermalization rate
in a linearized Boltzmann equation Sieberer  _et al._ (2016), and $\lambda$
parameterizes the interaction between magnons. Going beyond conventional time-
dependent Ginzburg-Landau descriptions, we introduce a Gaussian noise term
$\xi_{k}\left(t\right)$ with
$\langle\xi_{k}(t)\xi_{k^{\prime}}(t^{\prime})\rangle=\delta(t-t^{\prime})\delta(k+k^{\prime})(k^{2}+\tau_{0}^{-1})[2n_{k}(t)+1]$
that imprints a non-equilibrium fluctuation-dissipation relation for some
general magnon distribution function $n_{k}\left(t\right)$, in which the modes
transverse to the $c$-axis are assumed to relax much faster and are thus
integrated out Dolgirev  _et al._ (2020). To leading order, the equation of
motion for the MOP $\Phi\left(t\right)=\langle\varphi_{k=0}(t)\rangle$ is
given by
$\partial_{t}\Phi\left(t\right)=-\tau^{-1}\Phi\left(t\right)-\lambda{\Phi\left(t\right)}^{3}$,
and the Boltzmann equation for the magnon distribution is given by
$\partial_{t}n_{k}\left(t\right)=-\left|\tau^{-1}\right|\left[n_{k}\left(t\right)-n_{k}^{T}\right]$,
where $\tau^{-1}=\tau_{0}^{-1}-\lambda N_{\rm mag}$ and $n_{k}^{T}$ is the
Bose distribution function corresponding to the final equilibrated magnon
temperature $T$. These equations show that, in a closed system, the rate at
which the MOP recovers to its equilibrium value at $T$ via magnon
thermalization is a decreasing function of the total number of excited magnons
($N_{\rm mag}$). This provides an explanation for all key features of the data
in Fig. 3c. In the low pump fluence limit where $N_{\rm mag}\ll
1/\tau_{0}\lambda$, $\tau$ approaches a lower bound that is naturally set by
the photo-dopant decay time $\tau_{0}$. As $N_{\rm mag}$ grows with fluence,
$\tau$ exhibits a power law increase following the reduced curvature of the
Mexican hat potential. Finally once $N_{\rm mag}$ reaches $1/\tau_{0}\lambda$
at the critical fluence $F^{*}$, $\tau$ diverges even though the total energy
of the magnetic subsystem remains sub-critical (see Supplementary section S7).
For $F\geq F^{*}$, thermalization can still eventually occur through the weak
coupling between $c$-axis magnons and phonons, which is likely responsible for
the slow ($>100$ ps) MOP recovery observed by time-resolved RXS measurements
at high fluences Dean  _et al._ (2016); Krupin  _et al._ (2016). Ultimately,
on even longer timescales, the system cools back to the initial temperature
via heat diffusion out of the pumped region (Fig. 4d).

Altogether, our results provide experimental signatures that static and
dynamic critical behavior can be decoupled in a Mott antiferromagnet upon
driving due to a subtle interplay of photo-doping and magnon thermalization
processes (Fig. 4d). The hierarchy of magnetic, phononic and charge gap energy
scales that enables this behavior in Sr2IrO4 is typical for layered transition
metal oxide based AFM Mott insulators including the high-$T_{c}$
superconducting cuprates, suggesting that our reported phenomena may be
operational in numerous layered quantum materials. This raises the intriguing
prospect of fine-tuning parameters such as $\tau_{0}$, $F_{\rm c}$ and $F^{*}$
to optimize desired out-of-equilibrium properties. For example, extending
$\tau_{0}$ may be a pathway for realizing metastable electronic instabilities
in a photo-doped Mott insulator, while the combination of a small $\tau_{0}$
and $F_{\rm c}$ with a large $F^{*}$ would be favorable for low-power high-
speed AFM switching applications.

##  I Acknowledgements

We thank L. Balents, A. Cavalleri, S. K. Cushing, E. Demler, S. Di. Matteo, M.
Endres, B. Fine, N. Gedik, D. Kennes, O. Mehio, H. Ning, M. Norman and M.
Sentef for useful discussions. This work is supported by ARO MURI Grant No.
W911NF-16-1-0361. D.H. also acknowledges support for instrumentation from the
David and Lucile Packard Foundation and from the Institute for Quantum
Information and Matter (IQIM), an NSF Physics Frontiers Center (PHY-1733907).
A.d.l.T. acknowledges support from the Swiss National Science Foundation
through an Early Postdoc Mobility Fellowship (P2GEP2${}_{1}65044$). M.B.
acknowledges support from the Alexander von Humboldt foundation. N.J.L.
acknowledges support from an IQIM Fellowship. G.C. acknowledges NSF support
via a grant DMR-1712101.

##  II Competing Interests

The authors declare no competing interests.

##  III Methods

###  III.1 Sample growth

Single crystals of Sr2IrO4 were grown using a self-flux technique from off-
stoichiometric quantities of IrO2, SrCO3 and SrCl2. The ground mixtures of
powders were melted at 1470∘C in partially capped platinum crucibles. The
soaking phase of the synthesis lasted for $>20$ hours and was followed by a
slow cooling at 2∘C/hr to reach 1400∘C. From this point the crucible is
brought to room temperature through a rapid cooling at a rate of 100∘C/hr.
Crystals were cleaved along the (001) face just prior to measurements and
immediately pumped down to a pressure $<5\times 10^{-6}$ torr in an optical
cryostat.

###  III.2 Time-resolved spectroscopic probes

Time-resolved SHG-RA: Experiments were performed by splitting light from a
regeneratively amplified Ti:sapphire laser, which produces 100 fs pulses with
central wavelength $\lambda$ = 800 nm at a repetition rate of 100 kHz, into
pump and probe arms. The pump beam feeds an optical parametric amplifier,
whose 1400 nm output beam is linearly polarized and sent through a delay line
before being focused at normal incidence onto a 90 $\mu$m spot (FWHM) located
within a single magnetic domain of a Sr2IrO4 crystal. As shown in Fig. 1b, the
circularly polarized probe laser pulse ($\lambda$ = 800 nm) traverses a linear
input polarizer, phase mask and collimating lens and is focused at oblique
incidence ($\approx 10^{\circ}$) onto a 30 $\mu$m spot within the pumped
sample area using an objective lens. The reflected SHG pulse ($\lambda$ = 400
nm) is re-collimated by the objective lens, linearly polarized by an output
polarizer and then deflected onto a CCD camera by a triad of dichroic mirrors,
which measures the SHG intensity at a fixed time delay with respect to the
pump. The probe fluence was fixed to $1.5$ mJ/cm2. A complete instantaneous
SHG-RA pattern is acquired by mechanically co-rotating the input polarizer,
phase mask and output polarizer, which implements a rotation of the probe
scattering plane about the sample surface normal, and then projecting the SHG
light reflected at each scattering plane angle onto different positions along
a circular locus of points on the CCD camera [see Ref. Harter  _et al._ (2015)
for further details]. The procedure for converting raw CCD data to the polar
SHG-RA plots shown in the main text is described in the Supplementary section
S8. The SHG dynamics shown in Fig. 3 were reproduced under different
polarization geometries and different scattering plane angles as shown in
Supplementary section S9.

Time-resolved linear reflectivity: Transient reflectivity measurements were
performed using exactly the same setup as that used for time-resolved SHG-RA
as described above. Identical experimental conditions were preserved for the
reflectivity and SHG-RA measurements, with the only change between
experimental runs being the spectral filtering optics before the CCD camera to
either isolate the 800 nm or 400 nm output.

Time-resolved MOKE: Experiments were performed by splitting light from a
regeneratively amplified Ti:sapphire laser (repetition rate 1 kHz; pulse
energy 6 mJ; pulse duration 100 fs; central wavelength $\lambda$ = 800 nm)
into pump and probe arms. The pump arm feeds a collinear dual-stage optical
parametric amplifier, whose 1300 nm output beam is circularly polarized and
then focused onto the sample at normal incidence. The probe arm ($\lambda$ =
800 nm) is linearly S-polarized and focused near normal incidence onto the
sample. To measure the polarization rotation (Kerr) angle of the reflected
probe light, the reflected probe beam is sent through a half wave plate
followed by a Wollaston prism. These optics are oriented such that in the
absence of a pump beam ($\theta_{k}=0$, $F=4.6$ mJ/cm2), two equally intense
beams of orthogonal linear polarization are produced, which are fed into the
two input channels of a balanced photo-detector.

###  III.3 Conversion from $F$ to $n_{\rm ex}$

The number of photo-excitations $n_{\rm ex}$ created by the pump beam was
calculated using the following equation:

$n_{\rm ex}=\frac{F\left(1-R\right)V_{\rm u.c.}}{N_{\rm Ir}E_{\rm ph}\delta}$
(2)

where $F$ is the pump fluence, $R$ = 0.2 and $\delta$ = 100 nm are the sample
reflectivity and penetration depth respectively at the pump wavelength
$\lambda$ = 1400 nm Lee  _et al._ (2012); Nichols  _et al._ (2013), $V_{\rm
u.c.}$ is the unit cell volume of Sr2IrO4, $N_{\rm Ir}$ is the number of Ir
atoms per unit cell, and $E_{\rm ph}$ the pump photon energy (0.89 eV).

###  III.4 Exponential fitting procedure for linear reflectivity

Transient linear reflectivity curves were fit to a double exponential function
of the form $\Delta R/R=Ae^{-t/\tau_{0}}+Be^{-t/\tau_{1}}+C$. A high quality
of fit is achieved as shown in Fig. 4a. The timescales extracted for
$\tau_{0}$ ($\approx$ 1 ps) and $\tau_{1}$ ( $\approx$ 200 fs) agree with the
two exponential timescales associated with the decay of photo-induced midgap
states observed by ultrafast angle-resolved photoemission spectroscopy
measurements Piovera  _et al._ (2016), attributed to acoustic phonon and
optical phonon mediated photo-carrier relaxation processes respectively. Since
the rate limiting step for photo-carrier relaxation is $\tau_{0}$, that is
what we plot in the inset of Fig. 4a. We note that Ref. Piovera  _et al._
(2016), also reported an absence of any fluence dependence for $\tau_{0}$,
which is consistent with our observations (inset Fig. 4a).

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Figure 1: Instantaneous photo-doping response of magnetic order in Sr2IrO4. a,
Intralayer magnetic order of Sr2IrO4. Each layer possesses a non-zero net
moment due to pseudospin canting, which are staggered along the $c$-axis. b,
Schematic of the time-resolved SHG-RA setup. A circularly polarized probe
laser pulse ($\lambda$ = 800 nm) traverses a linear polarizer, phase mask,
collimating lens and is focused onto the sample at oblique incidence using an
objective lens. The reflected SHG pulse ($\lambda$ = 400 nm) is re-collimated
by the objective, linearly polarized and deflected onto a CCD camera by a
dichroic mirror. Different scattering plane angles are accessed by
mechanically co-rotating select optics. Data shown throughout the manuscript
are acquired in the Pin-Sout polarization channel (see Methods). A time-
delayed linearly polarized pump pulse resonant with the $J_{\rm eff}=3/2$ to
upper Hubbard band energy ($\lambda$ = 1400 nm) is focused normally onto the
sample. Regions of different color on the sample represent distinct magnetic
domains identified by SHG imaging. Inset: schematic of doublon generation and
propagation. c, Temperature dependence of the change in SHG intensity $\Delta
I(2\omega)$ relative to $T$ = 300 K for an un-pumped sample acquired at the
angle of maximum intensity (pink). Dashed line is a guide to the eye. d, Pump
fluence dependence of $\Delta I(2\omega)$ at $t$ = 0 and $T$ = 80 K with
corresponding SHG-RA patterns. Fits to the data below (above) $F_{\rm c}$
including both EQ and ED terms (solely EQ term) are overlaid. Height of the
gray bar represents the change in EQ SHG intensity with respect to the
equilibrium value as discussed in the main text. Horizontal error bars
represent the systematic error in calculating $n_{\rm ex}$. Figure 2: Out-of-
equilibrium magnetic phase diagrams. a, Temperature dependence of $\Delta
I(2\omega)$ at $t$ = 0.3 ps for select pump excitation densities $n_{\rm ex}$
and, b, the out-of-equilibrium phase diagram mapped using such datasets (at
$t$ = 0.3 ps). The reported La-doping dependence of $T_{\rm N}$ is overlaid.
Horizontal error bars represent the systematic error in calculating $n_{\rm
ex}$. c, Comparison of the temperature dependence of $\Delta I(2\omega)$ for
an un-pumped sample with the fluence dependence of $\Delta I(2\omega)$
collected at $t$ = 10 ps and $T$ = 80 K, where $F$ is converted into an
effective temperature by assuming all of the pump energy goes into quasi-
equilibrium heating. d, The out-of-equilibrium phase diagram mapped at $t$ =
10 ps (black dashed line). Overlaid are the phase boundary reproduced from
panel b (white line) and that calculated assuming all of the pump energy goes
into quasi-equilibrium heating (red line). e, High fluence $\Delta I(2\omega)$
and f, MOKE transients acquired below and above $T_{\rm N}$. Figure 3: Out-of-
equilibrium critical dynamics of the magnetic order parameter. $\Delta
I(2\omega)$ transients acquired at $T$ = 80 K for select fluences a, below and
b, above $F_{\rm c}$. The height of the gray bar denotes the excess EQ SHG
intensity at $t=0$ as discussed in the main text. Inset shows analogous data
acquired at fixed $F=0.55$ mJ/cm2 for different temperatures. Solid lines are
fits of the recovery to a single exponential function plus a constant
determined by the long time offset of the SHG intensity. c, Plot of fitted
exponential relaxation times versus fluence superposed with a fit to the
function $\tau=\tau_{0}(1–F/F^{*})^{-1}$, with $\tau_{0}$ fixed to 1.2 ps by
the transient reflectivity data (Fig. 4a). The values of $F_{\rm c}$ and
$F^{*}$ at $T$ = 80 K based on the data in panels a, b and the fit (black
line) respectively are explicitly marked. The decoupling of the divergence in
magnetic correlation length (defined by $F_{\rm c}$) from the divergence in
relaxation time (defined by $F^{*}$) is illustrated in the inset based on the
data in panels a and b. The arrows are color coded in correspondence to the
curves in panels a and b and point to the location in the out-of-equilibrium
phase diagram being accessed. Schematics show the the out-of-equilibrium free
energy landscape after doublon decay based on the Langevin theory described in
the text. Figure 4: Magnon population induced critical slowing down.
Simultaneous measurements of the a, transient reflectivity and b-c, transient
SHG intensity acquired under identical experimental conditions (except for
fluence) at $T$ = 80 K. Inset of panel a: fluence dependence of the intensity
minimum and characteristic recovery time $\tau_{0}$ extracted through double
exponential fits (red curve) to the $\Delta R/R$ data (see Methods). d,
Schematic of the complete temporal evolution of (top) the real space $c$-axis
magnetic correlations and (bottom) the magnon distribution function
$n\left(\omega\right)$ versus energy $\omega$ following pump excitation. Over
the time window $0<t\lesssim\tau$ (shaded orange), $n\left(\omega\right)$
departs from Bose-Einstein form.